PATENT ABSTRACT
An interferometer for controlling the coalescence of a pair of photons, including: an optical source, which generates a first and a second pump pulse coherent with each other and shifted in time by a delay; and a first interferometric stage, which receives the first pump pulse and generates an antisymmetric state with two coalescent photons (1/√{square root over (2)}(|2 1 ,0 2           −|0 1 ,2 2           )). The interferometer also includes a second interferometric stage, which receives the second pump pulse and generates a symmetric state with two coalescent photons (1/√{square root over (2)}·(|2 1 ,0 2           +|0 1 ,2 2           )), the first and the second interferometric stages being connected in a manner such that the interferometer outputs a final state equal to a weighted sum of the antisymmetric state and of the symmetric state (1/√{square root over (2)}·(|2 1 ,0 2           +|0 1 ,2 2           )+e iτ ·1/√{square root over (2)}·(|2 1 ,0 2           −|0 1 ,2 2

PATENT DESCRIPTION
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims priority to Italian Patent Application No. TO2011A000602 filed on 8 Jul. 2011, the contents of which are incorporated herein, in their entirety, by this reference. 
       TECHNICAL FIELD 
       [0002]    The present invention relates to an interferometer and method for controlling the coalescence of a pair of photons. 
       BACKGROUND 
       [0003]    As is known, devices have been proposed that are able to reveal the quantum behaviour of Nature. In particular, devices are known that enable detection of the quantum behaviour of particles such as, for example, photons. The operation of these devices is based on single or multiple photon state processing and finds application in particularly advanced sectors such as, for example, quantum computing, quantum cryptography, quantum communications and even random number generation. 
         [0004]    In particular, the so-called Hong-Ou-Mandel interferometer, also known as the HOM interferometer, is also known, an example of which is shown in  FIG. 1 , where it is indicated with reference numeral  1 . 
         [0005]    In detail, the HOM interferometer  1  comprises an optical source  2 , a crystal  4 , a delay line  6 , a polarization phase-shifter  8 , first and second mirrors  10 ,  12  and an optical beam splitter  14 . The HOM interferometer  1  also comprises a beam stopper  16 . 
         [0006]    In greater detail, the optical source  2  is a coherent type of source, such as a laser source for example. 
         [0007]    The crystal  4  is an optically non-linear crystal such as, for example, a non-centrosymmetric crystal (for example, a crystal of barium borate, BBO), and is aligned with the optical source  2 . 
         [0008]    The delay line  6  is of the optical type and is formed, for example, by a so-called optical trombone. In use, when it is crossed by a photon, the delay line  6  delays it for a predetermined time. 
         [0009]    Whereas with regard to the polarization phase-shifter  8 , this is formed by a birefringent crystal, which delays photons having different polarizations differently, namely by introducing controlled phase shifting, typically not greater than the wavelength of the photons. For example, the polarization phase-shifter  8  can be voltage controlled. 
         [0010]    In greater detail, the crystal  4  defines, together with the delay line  6 , the polarization phase-shifter  8  and the first mirror  10 , a first optical path  20 . The crystal  4  also defines, together with the second mirror  12 , a second optical path  22 . The delay line  6  is able to alter the optical length of the first optical path  20  with respect to the second optical path  22 . 
         [0011]    The optical beam splitter  14  is of the so-called 50/50 type and has a first and a second input and a first and a second output. From the classical point of view, given an incident signal on any input of the first and second inputs, two signals will be generated on the first and second outputs of the optical beam splitter  14  having power half that of the incident signal. 
         [0012]    The first and second optical paths  20 ,  22  are optically and respectively connected to the first and second inputs of the optical beam splitter  14 . 
         [0013]    Operationally, the optical source  2  is able to emit electromagnetic pulses formed of photons at the same frequency, which are commonly referred to as pump photons; these electromagnetic pulses, and therefore the pump photons, impinge on the crystal  4 . 
         [0014]    In particular, assuming that a pump photon impinge on the crystal  4 , due to the phenomenon of spontaneous parametric down-conversion (SPDC), the crystal  4  can generate a pair of converted photons, one of which propagates along the first optical path  20 , while the other propagates along the second optical path  22 . Instead, in the case where there is no spontaneous parametric down-conversion, the pump photon passes through the crystal  4  and is absorbed by the beam stopper  16 , which, for this purpose, is arranged in front of the crystal  4 , with which it is aligned. 
         [0015]    In the case where two converted photons are generated, both of them propagate until reaching a respective input of the optical beam splitter  14 . In this regard, generally, but not necessarily, the optical beam splitter  14  is formed by a pair of prisms suitable for receiving electromagnetic signals that propagate in free space, such that the term “input” implies a corresponding direction of propagation of an electromagnetic signal or a photon that impinges on the optical beam splitter  14 , while the term “output” implies a corresponding direction of propagation of an electromagnetic signal or a photon that moves away from the optical beam splitter  14 . 
         [0016]    The term “first converted photon” is used to indicate the photon of the pair of converted photons that propagates along the first optical path  20 ; this photon arrives at the first input of the optical beam splitter  14  after having passed through the delay line  6  and the polarization phase-shifter  8 , and also after being reflected by the first mirror  10 . Furthermore, when the first converted photon impinges on the optical beam splitter  14 , it can, alternatively and with the same probability, pass through the optical beam splitter  14 , leaving from the first output of the optical beam splitter  14 , or be reflected by the optical beam splitter  14 , leaving from the second output of the optical beam splitter  14 . 
         [0017]    Similarly, the term “second converted photon” is used to indicate the photon of the pair of converted photons that propagates along the second optical path  22 ; this photon arrives at the second input of the optical beam splitter  14  after being reflected by the second mirror  12 . Furthermore, when the second converted photon impinges on the optical beam splitter  14 , it can, alternatively and with the same probability, pass through the optical beam splitter  14 , leaving from the second output of the optical beam splitter  14 , or can be reflected by the optical beam splitter  14 , leaving from the first output of the optical beam splitter  14 . 
         [0018]    In other words, in the case where the first converted photon passes through the optical beam splitter  14 , it does not change its direction of propagation. Conversely, when the first converted photon is reflected by the optical beam splitter  14 , the direction of propagation is changed. Furthermore, the crystal  4 , the first and second mirrors  10 ,  12  and the optical beam splitter  14  are arranged such that, in case of reflection, the first converted photon propagates with a direction of propagation equal to the direction of propagation with which the second converted photon impinges on the optical beam splitter  14 . 
         [0019]    Completely symmetrical considerations can be made regarding the second converted photon. In fact, in the case where second converted photon passes through the optical beam splitter  14 , it does not change its direction of propagation. Conversely, when the second converted photon is reflected by the optical beam splitter  14 , the direction of propagation is changed. Furthermore, the crystal  4 , the first and second mirrors  10 ,  12  and the optical beam splitter  14  are arranged such that, in case of reflection, the second converted photon propagates with a direction of propagation equal to the direction of propagation with which the first converted photon impinges on the optical beam splitter  14 . 
         [0020]    Thus, in the case where first converted photon passes through the optical beam splitter  14  and the second converted photon is reflected by the optical beam splitter  14 , they subsequently propagate along a same direction of propagation (first output of the optical beam splitter). Similarly, in the case where the first converted photon is reflected by the optical beam splitter  14  and the second photon passes through the optical beam splitter  14 , they subsequently propagates along a same direction of propagation (second output of the optical beam splitter). 
         [0021]    In order to detect the quantum behaviour of the photons, it is possible to prepare a first and a second photodetector  30 ,  32 , both of the single photon type, i.e. able to detect individual photons. For example, the first and second photodetectors  30 ,  32  could be Geiger-mode avalanche photodiodes, also known as single-photon avalanche photodiodes (SPAD). 
         [0022]    The first photodetector  30  is placed to detect the first converted photon if it passes through the optical beam splitter  14  and the second converted photon if it is reflected by the optical beam splitter  14 . Similarly, the second photodetector  32  is placed to detect the first converted photon if it is reflected by the optical beam splitter  14  and the second converted photon if it passes through the optical beam splitter  14 . 
         [0023]    There are therefore four different possible scenarios for output from the optical beam splitter  14 , detectable by using the first and second photodetectors  30 ,  32  and known as Feynman paths: 
         [0024]    a) both the first and second converted photons pass through the optical beam splitter  14 ; 
         [0025]    b) both the first and the second converted photon are reflected by the optical beam splitter  14 ; 
         [0026]    c) the first converted photon passes through the optical beam splitter  14 , while the second converted photon is reflected by the optical beam splitter  14 ; and 
         [0027]    d) the second converted photon passes through the optical beam splitter  14 , while the first converted photon is reflected by the optical beam splitter  14 . 
         [0028]    From the quantum viewpoint, assuming (temporarily) for simplicity that the first and second converted photons have the same polarization, and assuming that the delay line  6  and the polarization phase-shifter  8  are such as to compensate possible differences in length between the first and second optical paths  20 ,  22 , in a manner such that the first and second converted photons impinge on the optical beam splitter  14  at the same time, contrary to that foreseen by classical optics, it is found that scenarios a) a b) can never occur. In other words, the first and second converted photons always find themselves on the same output of the optical beam splitter  14 . In particular, the probability that the first and second converted photons are on the first output of the optical beam splitter  14  is 0.5, and the probability that they are on the second output of the optical beam splitter  14  is 0.5. 
         [0029]    Thus, the first and second converted photons cannot both be reflected or both be transmitted, and therefore both be detected by the same photodetector, whether this be the first photodetector  30  or the second photodetector  32 . By monitoring the first and second photodetectors  30 ,  32 , a total lack of coincidence between their measurements is therefore verified. This is due to destructive interference between different alternatives; in the case in point, between scenario a) and scenario b). 
         [0030]    In general, the described phenomenon is usually referred to as coalescence, or that the first and second converted photons are coalescent. 
         [0031]    More precisely, the phenomenon of coalescence of a pair of photons occurs when, as in the example described, it is not possible to distinguish between scenario a) and scenario b), for example, by measuring different arrival times of the first and second converted photons at the first and second photodetectors  30 ,  32 . 
         [0032]    According to one interpretation of the phenomenon of the coalescence of two photons, the optical beam splitter  14  is a linear device capable of discriminating between symmetric states, namely between invariant states with respect to particle exchange, and asymmetric states, also known as antisymmetric states. 
         [0033]    In detail, a state of a system with two photons having the same polarization is symmetric, precisely because it is invariant with respect to the exchange of the two photons. In fact, when two photons impinge on two different inputs of an optical beam splitter of the 50/50 type, they are propagated together in output from the optical beam splitter, along the same direction. 
         [0034]    By using the so-called Dirac notation, the input state to the optical beam splitter  14  is normally indicated |1 1           |1 2           , where the subscripts “1” and “2” refer to the first and the second optical path  20 ,  22  respectively. Regarding this, it is normal to still refer to the first output of the optical beam splitter  14  as the first optical path  20 , and still refer to the second output of the optical beam splitter  14  as the second optical path  22 , so that it is possible to express the output state of the optical beam splitter  14  as 1/√{square root over (2)}(|2 1 ,0 2           −|0 1 ,2 2           ), from which it can be inferred that there is only one antisymmetric state with two coalescent photons in output from the HOM interferometer  1 . 
         [0035]    More in detail, it is possible to remove the simplifying assumption in which the first and second converted photons have the same polarization, something that, in effect, does not occur in the case where, as shown in  FIG. 1 , they are generated by the crystal  4  through type-II spontaneous parametric down-conversion. In fact, in this case, the pump photon is annihilated and the first and second converted photons are generated with orthogonal polarizations; in other words, if one of the first and second converted photons propagates along the first optical path  20  with horizontal polarization, the other photon propagates on the second optical path  22  with vertical polarization, or vice versa. 
         [0036]    In particular, in the case of so-called type-II spontaneous parametric down-conversion, it is found that the first and second converted photons are orthogonally polarized and satisfy the so-called phase-matching conditions, i.e. the conditions of conservation of energy and linear moment. 
         [0037]    It thus occurs that one of the first and second converted photons is polarized in a direction parallel to the optical axis of the crystal  4 , also known as the extraordinary direction, while the other is polarized in a direction perpendicular to the optical axis of the crystal  4 , also known as the ordinary direction. Furthermore, due to conservation of the linear moment, as shown in  FIG. 2 , the first and second converted photons are respectively emitted along a first and a second emission cone  34 ,  36 , respectively corresponding to the extraordinary direction and the ordinary direction. 
         [0038]    More in detail, assuming the degenerate case, namely the case where the first and second converted photons both have frequencies equal to half of the frequency of the pump photon, the first and second converted photons emerge from the crystal  4  forming a precise angle, for example, respectively equal to ±3°, with respect to the direction of propagation of the pump photon, as shown in  FIGS. 2 and 3  for example. 
         [0039]    The first and second emission cones  34 ,  36  intersect along a first and a second line  38 ,  40 , along which it is therefore possible to detect both the first and the second converted photons. In other words, along the first and second lines  38 ,  40  it is possible to detect polarized photons both along the extraordinary direction and along the ordinary direction. Therefore, the HOM interferometer  1  is such that the first and second optical paths  20 ,  22  respectively lie along the first and second lines  38 ,  40 , which undergo obvious changes following the interaction of the first and second emission cones  34 ,  36  with the components of the HOM interferometer  1 , such as, for example, the first and second mirrors  10 ,  12 . 
         [0040]    That having been said, using the term “extraordinary photon” to indicate the photon, of the first and second converted photons, polarized parallel to the optical axis of the crystal  4 , and using the term “ordinary photon” to indicate the photon, of the first and second converted photons, polarized perpendicularly to the optical axis of the crystal  4 , it is possible to discriminate between state |e 1           |o 2            and state |o 1           |e 2           . In other words, it is possible to distinguish a scenario in which the extraordinary photon (also known as the “signal”) and the ordinary photon (also known as the “idler”) respectively propagate along the first and second optical path  20 ,  22  (state |e 1           |o 2           ) from a scenario in which the extraordinary photon and the ordinary photon respectively propagate along the second and the first optical path  22 ,  20  (state |o 1           |e 2           ). This is due to the fact that the extraordinary photon and the ordinary photon propagate in the crystal  4  with different group velocities, and so the corresponding emissions are temporally distinguishable. 
         [0041]    In practice, the input state to the optical beam splitter  14  is still coherent and can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                            
                           e 
                           〉 
                         
                         1 
                       
                        
                       
                         
                            
                           o 
                           〉 
                         
                         2 
                       
                     
                     + 
                     
                       
                          
                         
                            
                            
                           
                               
                           
                            
                           ϕ 
                         
                       
                        
                       
                         
                            
                           o 
                           〉 
                         
                         1 
                       
                        
                       
                         
                            
                           e 
                           〉 
                         
                         2 
                       
                     
                   
                   
                     2 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where φ is a function of the phase shift introduced by the polarization phase-shifter  8 . 
         [0042]    In practice, by means of the delay line  6  and the polarization phase-shifter  8 , it is possible to control the input state to the optical beam splitter  14 , as well as the superposition of the wave function of the two-photon system on the optical beam splitter  14 . 
         [0043]    For example, when φ=0, the input state is symmetric and so coalescence of the photons occurs. Here, φ is such that the condition φ=0 is obtained when the delay introduced by the delay line  6  is sufficient to compensate the difference in optical length between the first and second optical paths  20 ,  22 , and when the polarization phase-shifter  8  is inactive. 
         [0044]    Conversely, in the case where φ=π, the following is obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                            
                           e 
                           〉 
                         
                         1 
                       
                        
                       
                         
                            
                           o 
                           〉 
                         
                         2 
                       
                     
                     - 
                     
                       
                         
                            
                           o 
                           〉 
                         
                         1 
                       
                        
                       
                         
                            
                           e 
                           〉 
                         
                         2 
                       
                     
                   
                   
                     2 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    namely an antisymmetric state is obtained, also known as a singlet state. In particular, relation (2) can also be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         
                           1 
                           
                             e 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         〉 
                       
                        
                       
                          
                         
                           1 
                           
                             2 
                              
                             o 
                           
                         
                         〉 
                       
                     
                     - 
                     
                       
                          
                         
                           1 
                           
                             1 
                              
                             o 
                           
                         
                         〉 
                       
                        
                       
                          
                         
                           1 
                           
                             2 
                              
                             e 
                           
                         
                         〉 
                       
                     
                   
                   
                     2 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where subscripts “1” and “2” still refer to the first and second optical paths  20 ,  22  and subscripts “e” and “o” refer to the extraordinary photon and the ordinary photon. 
         [0045]    It can then be checked, both mathematically and experimentally, that the singlet state does not change, i.e. that the state on output from the optical beam splitter  14  still takes form (2), equivalent to form (3). 
         [0046]    Therefore, the coalescence of the extraordinary photon and the ordinary photon does not occur in output from the optical beam splitter  14 . Conversely, anti-coalescence occurs, since the extraordinary photon and the ordinary photon are always present on different outputs of the optical beam splitter  14 . In other words, the optical beam splitter  14  implements a projection of the input state in the symmetric and antisymmetric subspaces, this projection also being known as Bell measurement. 
         [0047]    By way of example, anti-coalescence can be detected, as shown in  FIG. 4 , by using a third and a fourth photodetectors  42 ,  44 , and two further optical beam splitters, which are referred to as the first and second measurement splitters  46 ,  48 . In particular, the first and second measurement splitters  46 ,  48  are polarizing optical beam splitters, each of which is able to let one of either the extraordinary photon or the ordinary photon pass through and to reflect the other one, so as to spatially separate the extraordinary photon and the ordinary photon. For example, the first measurement splitter  46  can be arranged on the first output of the optical beam splitter  14  in a manner such that, in the case where the extraordinary photon or the ordinary photon emerge from the first output of the optical beam splitter  14 , they are respectively directed towards the first photodetector  30  and the third photodetector  42 . 
         [0048]    Similarly, the second measurement splitter  48  can be arranged on the second output of the optical beam splitter  14  in a manner such that, in the case where extraordinary photon or the ordinary photon emerge from the second output of the optical beam splitter  14 , they are respectively directed towards the second photodetector  32  and the fourth photodetector  44 . 
         [0049]    In practice, by counting the readings of the first, second, third and fourth photodetectors  30 ,  32 ,  42 ,  44 , it is possible to determine measurements related to the so-called probabilities 1e1o, 1e2o, 2e1o and 2e2o, i.e. the probabilities that:
       both the extraordinary photon and the ordinary photon are on the first output of the optical beam splitter  14 ;   the extraordinary photon and the ordinary photon are, respectively, on the first and the second output of the optical beam splitter  14 ;   the extraordinary photon and the ordinary photon are, respectively, on the second and the first output of the optical beam splitter  14 ; and   both the extraordinary photon and the ordinary photon are on the second output of the optical beam splitter  14 .       
 
         [0054]    More in detail, the description concerning the input state to the optical beam splitter  14  can be rendered mathematically more accurate in relation to the physical phenomenon. In fact, by considering the longitudinal components of the electromagnetic fields associated with the photons, the emission state from the crystal  4  can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     ψ 
                     〉 
                   
                   = 
                   
                     
                       C 
                       
                         2 
                       
                     
                      
                     
                       
                         ∫ 
                         
                           - 
                           L 
                         
                         0 
                       
                        
                       
                         
                            
                           z 
                         
                          
                         
                           
                             ∫ 
                             0 
                             
                               + 
                               ∞ 
                             
                           
                            
                           
                             
                                
                               
                                 v 
                                 p 
                               
                             
                              
                             
                               
                                 E 
                                 p 
                                 
                                   ( 
                                   + 
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   v 
                                   p 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   v 
                                   p 
                                 
                                  
                                 Λ 
                                  
                                 
                                     
                                 
                                  
                                 z 
                               
                             
                              
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 
                                   + 
                                   ∞ 
                                 
                               
                                
                               
                                 
                                    
                                   v 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
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                                      
                                     
                                         
                                     
                                      
                                     Dvz 
                                   
                                 
                                 × 
                                 × 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             a 
                                             ^ 
                                           
                                           
                                             1 
                                              
                                             e 
                                           
                                           † 
                                         
                                          
                                         
                                           ( 
                                           
                                             v 
                                             + 
                                             
                                               
                                                 
                                                   v 
                                                   p 
                                                 
                                                 + 
                                                 
                                                   Ω 
                                                   p 
                                                 
                                               
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           
                                             a 
                                             ^ 
                                           
                                           
                                             2 
                                              
                                             o 
                                           
                                           † 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               - 
                                               v 
                                             
                                             + 
                                             
                                               
                                                 
                                                   v 
                                                   p 
                                                 
                                                 + 
                                                 
                                                   Ω 
                                                   p 
                                                 
                                               
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           
                                             a 
                                             ^ 
                                           
                                           
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                                              
                                             e 
                                           
                                           † 
                                         
                                          
                                         
                                           ( 
                                           
                                             v 
                                             + 
                                             
                                               
                                                 
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                                                   p 
                                                 
                                                 + 
                                                 
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                                                   p 
                                                 
                                               
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           
                                             a 
                                             ^ 
                                           
                                           
                                             1 
                                              
                                             o 
                                           
                                           † 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               - 
                                               v 
                                             
                                             + 
                                             
                                               
                                                 
                                                   v 
                                                   p 
                                                 
                                                 + 
                                                 
                                                   Ω 
                                                   p 
                                                 
                                               
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                  
                                 
                                    
                                   0 
                                   〉 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where C is a constant that depends on the power of the optical source  2 , the interaction volume of the pump photons with the crystal  4  and the second-order, non-linear, effective tensor of the crystal  4 . In addition, E p   (+) (υ p ) is the spectral distribution of the pump, namely of the electromagnetic radiation emitted by the optical source  2 , Ω p  is the central pump frequency and L is the length of the crystal  4 , measured along the propagation direction of the pump. Moreover, â 1e   †  and â 2e   †  are the creation operators related to the extraordinary photon and to the first and second optical paths  20 ,  22 , respectively; â 1o   †  and â 2o   †  are the creation operators related to the ordinary photon and to the first and second optical paths  20 ,  22 , respectively. The following also hold: 
         [0000]    
       
         
           
             
               
                 
                   
                     Λ 
                     = 
                     
                       
                         1 
                         
                           u 
                           p 
                         
                       
                       - 
                       
                         
                           1 
                           2 
                         
                          
                         
                           ( 
                           
                             
                               1 
                               
                                 u 
                                 e 
                               
                             
                             + 
                             
                               1 
                               
                                 u 
                                 o 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     D 
                     = 
                     
                       ( 
                       
                         
                           1 
                           
                             u 
                             
                               e 
                                
                               
                                   
                               
                             
                           
                         
                         - 
                         
                           1 
                           
                             u 
                             o 
                           
                         
                       
                       ) 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where u p , u e  and u o  are the group velocities in the crystal  4  of the pump, the extraordinary photon and the ordinary photon, respectively. 
         [0055]    In practice, equation (4) permits revealing observance of the energy conservation condition in the generation process of the extraordinary photon and the ordinary photon. 
         [0056]    The output state from the optical beam splitter  14  can thus be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     ψ 
                     〉 
                   
                   = 
                   
                     
                       
                         C 
                          
                         
                             
                         
                          
                         
                            
                           
                             
                               - 
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       Ω 
                                       p 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                             
                              
                             ζ 
                           
                         
                       
                       
                         2 
                          
                         
                           2 
                         
                       
                     
                      
                     
                       
                         ∫ 
                         
                           - 
                           L 
                         
                         0 
                       
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                   ( 
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         [0000]    where symbol “±” is intended to be “+” in the case when φ=o (symmetric input state) and “−” when φ=π (antisymmetric input state); furthermore, in equation (7), ζ is a time that is a function of the delay introduced by the delay line  6 . 
         [0057]    The above-mentioned (normalized) probabilities 1e1o, 1e2o, 2e1o and 2e2o, hereinafter respectively indicated as HOM 1e1o , HOM 1e2o , HOM 2e1o  and HOM 2e2o , can thus be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           HOM 
                           
                             1 
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                             e 
                              
                             
                                 
                             
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                           ( 
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                       = 
                       
                         
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                                   ς 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         
                           HOM 
                           
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                       = 
                       
                         
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                                 ( 
                                 
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                                   ς 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         
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                             2 
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                           ( 
                           ς 
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                       = 
                       
                         
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                                   ς 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         
                           HOM 
                           
                             2 
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                             e 
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                             2 
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                             o 
                           
                         
                          
                         
                           ( 
                           ς 
                           ) 
                         
                       
                       = 
                       
                         
                           1 
                           4 
                         
                         [ 
                         
                           1 
                           ± 
                           
                             
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                                
                               
                                 ( 
                                 
                                   ς 
                                   DL 
                                 
                                 ) 
                               
                             
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                                
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                 
                                   σ 
                                   p 
                                   2 
                                 
                                  
                                 
                                   
                                     Λ 
                                     2 
                                   
                                   
                                     D 
                                     2 
                                   
                                 
                                  
                                 
                                   ς 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the following holds: 
         [0000]    
       
         
           
             
               
                 
                   
                     Tri 
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               
                                  
                                 x 
                                  
                               
                             
                             &gt; 
                             
                               1 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               1 
                               - 
                               
                                 2 
                                  
                                 
                                    
                                   x 
                                    
                                 
                               
                             
                             ) 
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               
                                  
                                 x 
                                  
                               
                             
                             ≤ 
                             
                               1 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0058]    As shown in  FIG. 5 , in the case where the input state is symmetric, it is found that the probabilities (more precisely, the corresponding probability density functions) HOM 1e1o  and HOM 2e2o  have a maximum of 0.5 when ζ=0. In other words, coalescence occurs, since the first and second converted photons are on the same output of the optical beam splitter  14 , which, with a probability of 0.5, is the first output or, with the same probability, is the second output of the optical beam splitter  14 . 
         [0059]    As shown in  FIG. 6 , in the case where the input state is antisymmetric, it is found that the probabilities HOM 1e1o  and HOM 2e2o  are null when ζ=0. In other words, anti-coalescence occurs, since the extraordinary photon and the ordinary photon are never present on the same output of the optical beam splitter  14 . 
         [0060]    From a practical standpoint, the HOM interferometer  1  has been used, among other things, in measuring the temporal uncertainty associated with the emission of a pair of photons by spontaneous parametric down-conversion, which is typically less than a picosecond. Still by way of example, the HOM interferometer  1  has also been used to test the Bell inequalities, to measure the tunneling time of the photons, to perform quantum teleportation, and even to build quantum logic gates, to clone quantum states and to implement quantum key distribution schemes. 
         [0061]    Even though the HOM interferometer  1  enables generating the coalescence of two photons and consequently lends itself to various applications based on the quantum characteristics of matter, it does not allow its outputs to be controlled, i.e. the probabilities with which the two coalescent photons appear on the first or on the second output of the optical beam splitter  14 . 
       SUMMARY 
       [0062]    An object of one or more embodiments of the present invention is to provide an interferometer that at least partially resolves the drawbacks of the known art. According to one or more embodiments of the present invention, an interferometer and method for controlling the coalescence of a pair of photons are provided. 
         [0063]    In an embodiment, an interferometer for controlling the coalescence of a pair of photons includes an optical source configured to generate a first pump pulse and a second pump pulse coherent with each other and shifted in time by a delay. The interferometer includes first interferometric means configured to receive the first pump pulse and generate an antisymmetric state with two coalescent photons (1√{square root over (2)}(|2 1 ,0 2           −|0 1 ,2 2           )), and second interferometric means configured to receive the second pump pulse and to generate a symmetric state with two coalescent photons (1/√{square root over (2)}·(|2 1 ,0 2           +|0 1 ,2 2           )). The first and second interferometric means are connected so that the interferometer is configured to output a final state equal to a weighted sum of the antisymmetric state and the symmetric state (1/√{square root over (2)}·(|2 1 ,0 2           +|0 1 ,2 2           )+e iτ ·1/√{square root over (2)}·(|2 1 ,0 2           −|0 1 ,2 2           )), the weights of the sum being a function of the delay. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0064]    For a better understanding of the invention, some embodiments will now be described, purely by way of non-limitative example and with reference to the attached drawings, where: 
           [0065]      FIGS. 1 and 4  show block diagrams of a known type of interferometer; 
           [0066]      FIG. 2  schematically shows a perspective view of a crystal of the interferometer shown in  FIGS. 1 and 4 , and of two emission cones, related to an extraordinary photon and an ordinary photon; 
           [0067]      FIG. 3  shows a cross-section of the two emission cones shown in  FIG. 2 , taken along a section line shown in  FIG. 2 ; 
           [0068]      FIGS. 5 and 6  show probability density functions related to the outputs of the interferometer shown in  FIGS. 1 and 4 , in the cases of symmetric and antisymmetric input states, respectively; 
           [0069]      FIG. 7  shows a block diagram of a first embodiment of the interferometer according to the present invention; 
           [0070]      FIG. 8  shows the envelope profile of a probability density function related to the outputs of an interferometric subsystem defined by a portion of the interferometer shown in  FIG. 7 ; 
           [0071]      FIG. 9  shows probability density functions related to the outputs of the interferometric subsystem; 
           [0072]      FIG. 10  shows the envelope profile of a probability density function related to the outputs of the interferometer shown in  FIG. 7 ; 
           [0073]      FIG. 11  shows probability density functions related to the outputs of the interferometer shown in  FIG. 7 ; 
           [0074]      FIG. 12  shows a block diagram of a second embodiment of the interferometer according to the present invention; 
           [0075]      FIG. 13  shows the envelope profile of a probability density function related to the outputs of the interferometer shown in  FIG. 12 ; and 
           [0076]      FIG. 14  shows probability density functions related to the outputs of the interferometer shown in  FIG. 12 . 
       
    
    
     DETAILED DESCRIPTION 
       [0077]      FIG. 7  shows an interferometer for controlling the coalescence of a pair of photons, indicated as a whole by reference numeral  50  and which is henceforth referred to as the interferometer  50 . 
         [0078]    In detail, the interferometer  50  comprises an optical source  51  of the coherent type, such as a laser source for example, and an optical pump beam splitter  52 , which together form a pump photon generator  53 . 
         [0079]    In particular, the optical source  51  is able to generate electromagnetic pulses, for example, at a wavelength of 415 nm, which are received by the optical pump beam splitter  52 . In detail, the optical pump beam splitter  52  has an input, on which it receives electromagnetic pulses generated by the optical source  51 , and first and second outputs. 
         [0080]    In use, for each electromagnetic pulse that impinges on its input, the optical pump beam splitter  52  generates a corresponding pump pulse, described further on, on each of the first and second outputs. 
         [0081]    The interferometer  50  also comprises a first crystal  54 , a first delay line  56 , a first polarization phase-shifter  58 , first and second mirrors  60 ,  62  and an optical beam splitter that is referred to as the first interferometer beam splitter  64 . 
         [0082]    The first delay line  56 , in itself known and which will henceforth be referred to as the first interferometer delay line  56 , is of the optical type and is formed, for example, by an optical trombone. In use, when it is passed through by a photon, the first interferometer delay line  56  delays it for a predetermined and electronically controllable time. 
         [0083]    The first polarization phase-shifter  58 , also of a type in itself known, is formed, for example, by a birefringent crystal, which, when passed through by photons, phase shifts them differently according to the polarizations of the photons. Furthermore, the first polarization phase-shifter  58  can be controlled, by voltage for example, so as to vary the phase shift that it introduces. 
         [0084]    The first crystal  54  is an optically non-linear crystal, such as, for example, a non-centrosymmetric crystal (for example, a crystal of barium borate, BBO), and is arranged so as to receive the pump pulses originating from the first output of the optical pump beam splitter  52 , which will henceforth be referred to as first crystal pulses. Although not shown, the first crystal pulses can be directed to the first crystal  54  in a manner which is in itself known, for example, by opportune mirrors and/or lenses and/or waveguides. 
         [0085]    A first beam stopper  66  (optional) is arranged in front of and aligned with the first crystal  54 , having the function of absorbing the photons of the first crystal pulses that pass through the first crystal  54  without giving rise to phenomena of spontaneous parametric down-conversion, as described further on. 
         [0086]    The first crystal  54  defines, together with the first interferometer delay line  56 , the first polarization phase-shifter  58  and the first mirror  60 , a first optical path  70 , which connects the first crystal  54  to the first interferometer beam splitter  64  in a way such that the first interferometer delay line  56  and the first polarization phase-shifter  58  are interposed between the first crystal  54  and the first mirror  60 . The first crystal  54  also defines, together with the second mirror  62 , a second optical path  72 , which also connects the first crystal  54  to the first interferometer beam splitter  64 . 
         [0087]    Although not shown in  FIG. 7 , the first and second optical paths  70 ,  72  are arranged, in a manner which is in itself known, along the intersections of the two emission cones along which the first crystal  54  emits, following the phenomenon of spontaneous parametric down-conversion of a photon of a first crystal pulse, the corresponding extraordinary photon and the corresponding ordinary photon. In these terms, it can be assumed that the first crystal  54  has a first and a second output, each one defined by a corresponding line between the two lines of intersection along which the above-mentioned two emission cones intersect. The first and second optical paths  70  and  72  originate respectively from the first and second outputs of the first crystal  54 ; for the purposes of the present invention, it is indifferent which of the first and second optical path  70 ,  72  derives its origin from the first output of the first crystal  54  and which one derives it from the second output. It should also be noted that, in  FIG. 7 , the angles formed by the first and second optical paths  70 ,  72  are purely qualitative. 
         [0088]    The first interferometer beam splitter  64  is of the so-called 50/50 type and has a first and second input, and a first and second output. For example, the first interferometer beam splitter  64  can be formed by a pair of prisms. 
         [0089]    As already mentioned, the first and second optical paths  70 ,  72  optically connect to the first interferometer beam splitter  64 , defining precisely the first and second input, since, as already established with regard to the HOM interferometer, the term “input” means a corresponding direction of propagation of an electromagnetic signal or photon that impinges on the first interferometer beam splitter  64 . The definition of the first and second inputs of the first interferometer beam splitter  64  also entails the definition of the first and second outputs of the first interferometer beam splitter  64 , which imply corresponding directions of propagation of electromagnetic signals or photons that move away from the optical beam splitter. 
         [0090]    For completeness, hereinafter reference will be made to the first output of the first interferometer beam splitter  64  to indicate the direction of propagation taken by the photons that impinge the first optical beam splitter  64  after having followed the first optical path  70  and that pass through the first interferometer beam splitter  64  without being reflected, this direction of propagation being coincident with the direction of propagation of the photons that impinge on the first optical beam splitter  64  after having followed the second optical path  72  and are reflected by it. Similarly, hereinafter reference will be made to the second output of the first interferometer beam splitter  64  to indicate the direction of propagation taken by the photons that impinge on the first interferometer beam splitter  64  after having followed the second optical path  72  and that pass through the first interferometer beam splitter  64  without being reflected, this direction of propagation being coincident with the direction of propagation of the photons that impinge on the first optical beam splitter  64  after having followed the first optical path  70  and are reflected by it. 
         [0091]    The interferometer  50  also comprises a third and fourth mirror  74 ,  76 , a first and second compensation element  78 ,  80  (described in detail further on) and a second crystal  84 , the latter being an optically non-linear crystal, identical to the first crystal  54  and oriented in the same manner. For example, each one of the first and second crystals  54 ,  84  is parallelepipedal and is cut so that the optical axis forms an angle with the longitudinal axis of the parallelepiped, for example, equal to 42°; in addition, these parallelepipeds obviously have the same length. 
         [0092]    In greater detail, the first and second compensation elements  78 ,  80  are each formed by a crystal identical to the second crystal  84 , with regard to both the material and the geometrical shape; they therefore have the same length as the second crystal  84 . Furthermore, each of the first and second compensation elements  78 ,  80  is rotated 90° with respect to the second crystal  84 ; in particular, each of the first and second compensation elements  78 ,  80  has a position (orientation) that corresponds to the position of the second crystal  84 , but rotated 90° with respect to the optical axis (whether clockwise or anticlockwise is indifferent). In this way, downstream of the second crystal  84 , the ordinary photon and the extraordinary photon generated by the first crystal  54  maintain the same time delay they have in output from the first crystal  54 , since each one passes through two birefringent optical elements (the second crystal and, in alternative, the first or the second compensation element) having ordinary and extraordinary refractive indices exchanged between them. In practice, the first and second compensation elements  78 ,  80  perform the function of making the output state from the first interferometer beam splitter  64  temporally equal to the output state from second crystal  84 . 
         [0093]    The third and fourth mirrors  74 ,  76  are arranged to receive, respectively, the photons originating from the first and second outputs of the first interferometer beam splitter  64 , and to reflect them in the direction of the second crystal  84 . 
         [0094]    In even greater detail, the first interferometer beam splitter  64 , the third mirror  74 , the first compensation element  78  and the second crystal  84  form a third optical path  86 , which connects the first output of the first interferometer beam splitter  64  to the second crystal  84 , and is such that the first compensation element  78  is interposed between the third mirror  74  and the second crystal  84 . In practice, the third optical path  86  is the continuation of the first optical path  70 . 
         [0095]    Similarly, the first interferometer beam splitter  64 , the fourth mirror  76 , the second compensation element  80  and the second crystal  84  form a fourth optical path  88 , which connects the second output of the first interferometer beam splitter  64  to the second crystal  84 , and is such that the second compensation element  80  is interposed between the fourth mirror  76  and the second crystal  84 . In practice, the fourth optical path  88  is the continuation of the second optical path  72 . 
         [0096]    Similarly to what happens for the first and second optical paths  70 ,  72 , the third and fourth optical paths  86 ,  88  also develop along two lines of intersection of the two emission cones of the first crystal  54 , which change, in a manner which is in itself known, following interaction with the components of the interferometer  50 , and in particular following interaction with the first and second mirrors  60 ,  62 , the first interferometer beam splitter  64 , the third and the fourth mirrors  76 ,  78 , and the first and second compensation elements  78 ,  80 . 
         [0097]    The interferometer  50  also comprises a further delay line  89 , which is henceforth referred to as the pump delay line  89 . This pump delay line  89  is arranged to receive the pump pulses originating from the second output of the optical pump beam splitter  52 , which are henceforth referred to as second crystal pulses, each of which corresponds to a respective first crystal pulse. Moreover, the pump delay line  89  can be formed in free space, for example, by using an optical trombone, or can be of the guided type; in addition, in a manner which is in itself known, the entity of the delay introduced by the pump delay line  89 , namely the time employed by the photons to pass through the pump delay line  89 , is variable in an electrically controllable manner. 
         [0098]    Independently of the constructional details, the pump delay line  89  conveys the second crystal pulses to the second crystal  84 . 
         [0099]    The interferometer  50  also comprises a further delay line  90 , which is referred to as the second interferometer delay line  90 , and a second polarization phase-shifter  92 , as well as a fifth and sixth mirror  94 ,  96 , and a further optical beam splitter  98 , which is referred to as the second interferometer beam splitter  98 . The second interferometer beam splitter  98  can be the same as the first interferometer beam splitter  64 . 
         [0100]    The second crystal  84 , the second interferometer delay line  90 , the second polarization phase-shifter  92 , the fifth mirror  94  and the second interferometer beam splitter  98  form a fifth optical path  100 , which connects the second crystal  84  to the second interferometer beam splitter  98 , and is such that the second interferometer delay line  90  and the second polarization phase-shifter  92  are interposed between the second crystal  84  and the mirror  94 . In practice, the fifth optical path  100  is a continuation of the third optical path  86 . 
         [0101]    Furthermore, the second crystal  84 , the sixth mirror  96  and the second interferometer beam splitter  98  form a sixth optical path  102 , which connects the second crystal  84  to the second interferometer beam splitter  98 . In practice, the sixth optical path  102  is a continuation of the fourth optical path  88 . 
         [0102]    In even greater detail, the fifth and sixth optical paths  100 ,  102  optically connect to the second interferometer beam splitter  98 , defining the first and second inputs, in a similar manner to what happens for the first and second optical paths  70 ,  72  and the first interferometer beam splitter  64 . 
         [0103]    Although not shown, the fifth and sixth optical paths  100 ,  102  are arranged, in a manner which is in itself known, along the intersections of the two cones along which the second crystal  84  emits, following the phenomenon of spontaneous parametric down-conversion of a photon of a second crystal pulse, the corresponding extraordinary photon and the corresponding ordinary photon, in a manner similar to what happens on the first crystal  54  and the first and second optical paths  70 ,  72 . In addition, these intersections overlap the intersections of the two cones along which the first crystal  54  emits, following the phenomenon of spontaneous parametric down-conversion of a photon of a first crystal pulse, the corresponding extraordinary photon and the corresponding ordinary photon. 
         [0104]    The interferometer  50  is also such that any photon originating from the third optical path  86  that impinges on the second crystal  84  propagates, after having passed through the second crystal  84 , along the fifth optical path  100 . In addition, the interferometer  50  is such that any photon originating from the fourth optical path  88  that impinges on the second crystal  84  propagates, after having passed through the second crystal  84 , along the sixth optical path  102 . 
         [0105]    In practice, a photon originating from the third optical path  86  propagates, after having passed through the second crystal  84 , along the same path followed by a photon generated by the second crystal  84  through spontaneous parametric down-conversion and propagated along the fifth optical path  100 . Similarly, a photon originating from the fourth optical path  88  propagates, after having passed through the second crystal  84 , along the same path followed by a photon generated by the second crystal  84  through spontaneous parametric down-conversion and propagated along the sixth optical path  102 . Thus, considering photons generated by the first crystal  54  and photons generated by the second crystal  84 , if the corresponding first and second crystal pulses that generated them are temporally delayed in a manner such as to compensate the different optical lengths of the optical paths covered by the photons generated by the first crystal  54  and by the photons generated by the second crystal  84 , it is not possible to determine, on the basis of the optical paths followed, which crystal generated them. 
         [0106]    For practical purposes, the first, third and fifth optical paths  70 ,  86 ,  100  are consecutive to each other and form a first extended path, while the second, fourth and sixth optical paths  72 ,  88 ,  102 , also consecutive to each other, form a second extended path. 
         [0107]    The interferometer also comprises a second beam stopper  104 , arranged in front of the second crystal  84  and having the function of absorbing the photons of the second crystal pulses that pass through the second crystal  84  without giving rise to the phenomenon of spontaneous parametric down-conversion. 
         [0108]    As described in detail in the following, coalescence control is obtained on the first and second outputs of the second interferometer beam splitter  98 . To this end, in a manner which is in itself known, it is possible to monitor the first and second outputs of the second interferometer beam splitter  98 , for example through the first, second, third and fourth photodetectors  30 ,  32 ,  42 ,  44 , as well as through the first and second measurement splitters  46 ,  48  described in connection with the HOM interferometer  1 . In particular, in the example in  FIG. 7 , the first and third photodetectors  30 ,  42  monitor the first output of the second interferometer beam splitter  98 , and therefore the first extended path, while the second and fourth photodetectors  32 ,  44  monitor the second output of the second interferometer beam splitter  98 , and therefore the second extended path. It is assumed that the fifth optical path  100  passes through the second interferometer beam splitter  98  and arrives to the first measurement splitter  46 , and that the sixth optical path  102  passes through the second interferometer beam splitter  98  and arrives to the second measurement splitter  48 . 
         [0109]    Operationally, assuming a pair of first and second crystal pulses that are mutually corresponding, namely generated by the optical pump beam splitter  52  starting from the same electromagnetic pulse, and referring to them respectively as the symmetric pump pulse and the antisymmetric pump pulse, the interferometer  50  behaves in the following manner. 
         [0110]    In detail, one of the photons of the symmetric pump pulse can generate, through spontaneous parametric down-conversion inside the first crystal  54 , a first extraordinary photon and a first ordinary photon, each of which can propagate along the first or the second optical path  70 ,  72 . 
         [0111]    From a quantum viewpoint, the output state from the first interferometer beam splitter  64  can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0000]    where subscripts “1” and “2” refer to the first and second optical paths  70 ,  72 , respectively (equivalently, to the first and second extended path), while subscripts “e” and “o” refer to the first extraordinary photon and the first ordinary photon; in addition, θ is a time that is a function of the delay introduced by the first interferometer delay line  56 . In particular, θ is such that the condition θ=0 is obtained when the first and second optical paths  70 ,  72  have the same optical length. 
         [0112]    Assuming that the first and second compensation elements  78 ,  80  and the second crystal  84  are absent, or rather assuming that the interferometer  50  is a so-called Mach-Zender interferometer, it is possible to express the output state from the second interferometer beam splitter  98  as: 
         [0000]    
       
         
           
             
               
                 
                   
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                                                    
                                                   
                                                       
                                                   
                                                    
                                                   v 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   ϑ 
                                                 
                                               
                                               + 
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                      
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   v 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   ϑ 
                                                 
                                               
                                             
                                             ) 
                                           
                                            
                                           
                                               
                                             
                                               
                                                 [ 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                            
                                                           
                                                             
                                                              
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             ) 
                                                             
                                                             
                                                              
                                                             ς 
                                                           
                                                         
                                                         + 
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                          
                                                         
                                                           
                                                             - 
                                                              
                                                           
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             ) 
                                                           
                                                            
                                                           ς 
                                                         
                                                       
                                                     
                                                   
                                                 
                                                 ] 
                                               
                                               + 
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           [ 
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             a 
                                                             ^ 
                                                           
                                                           
                                                             1 
                                                              
                                                             e 
                                                           
                                                           † 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             v 
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             1 
                                                              
                                                             o 
                                                             
                                                             † 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             a 
                                                             ^ 
                                                           
                                                           
                                                             2 
                                                              
                                                             e 
                                                           
                                                           † 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             v 
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             a 
                                                             ^ 
                                                           
                                                           
                                                             1 
                                                              
                                                             o 
                                                           
                                                           † 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                                
                                               
                                                  
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 v 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 ϑ 
                                               
                                             
                                             - 
                                             
                                                
                                               
                                                 
                                                   - 
                                                    
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 v 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 ϑ 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       
                                         
                                             
                                           
                                             
                                               [ 
                                               
                                                 
                                                    
                                                   
                                                      
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     v 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     ς 
                                                   
                                                 
                                                 + 
                                                 
                                                    
                                                   
                                                     
                                                       - 
                                                        
                                                     
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     v 
                                                      
                                                     
                                                         
                                                     
                                                      
                                                     ς 
                                                   
                                                 
                                               
                                               ] 
                                             
                                             + 
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           [ 
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             a 
                                                             ^ 
                                                           
                                                           
                                                             1 
                                                              
                                                             e 
                                                           
                                                           † 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             v 
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             2 
                                                              
                                                             o 
                                                             
                                                             † 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             a 
                                                             ^ 
                                                           
                                                           
                                                             2 
                                                              
                                                             e 
                                                           
                                                           † 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             v 
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             a 
                                                             ^ 
                                                           
                                                           
                                                             1 
                                                              
                                                             o 
                                                           
                                                           † 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                                
                                               
                                                  
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 v 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 ϑ 
                                               
                                             
                                             + 
                                             
                                                
                                               
                                                 
                                                   - 
                                                    
                                                 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 v 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 ϑ 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     
                                       
                                         
                                             
                                           
                                               
                                             
                                               
                                                 [ 
                                                 
                                                   
                                                      
                                                     
                                                        
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                           
                                                           2 
                                                         
                                                         ) 
                                                       
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       ς 
                                                     
                                                   
                                                   - 
                                                   
                                                      
                                                     
                                                       
                                                         - 
                                                         
                                                            
                                                            
                                                           
                                                             ( 
                                                             
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                              
                                                             
                                                               
                                                             
                                                             
                                                             2 
                                                             
                                                             ) 
                                                           
                                                         
                                                       
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       ς 
                                                     
                                                   
                                                 
                                                 ] 
                                               
                                               + 
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           [ 
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       a 
                                                       ^ 
                                                     
                                                     
                                                       1 
                                                        
                                                       e 
                                                     
                                                     † 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       v 
                                                       + 
                                                       
                                                         
                                                           
                                                             v 
                                                             p 
                                                           
                                                           + 
                                                           
                                                             Ω 
                                                             p 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         a 
                                                         ^ 
                                                       
                                                       
                                                         2 
                                                          
                                                         o 
                                                       
                                                       † 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           - 
                                                           v 
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                   - 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       a 
                                                       ^ 
                                                     
                                                     
                                                       2 
                                                        
                                                       e 
                                                     
                                                     † 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       v 
                                                       + 
                                                       
                                                         
                                                           
                                                             v 
                                                             p 
                                                           
                                                           + 
                                                           
                                                             Ω 
                                                             p 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                     
                                                       a 
                                                       ^ 
                                                     
                                                     
                                                       1 
                                                        
                                                       o 
                                                     
                                                     † 
                                                   
                                                    
                                                   
                                                     ( 
                                                     
                                                       
                                                         - 
                                                         v 
                                                       
                                                       + 
                                                       
                                                         
                                                           
                                                             v 
                                                             p 
                                                           
                                                           + 
                                                           
                                                             Ω 
                                                             p 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             ( 
                                             
                                               
                                                  
                                                 
                                                    
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   v 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   ϑ 
                                                 
                                               
                                               - 
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                      
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   v 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   ϑ 
                                                 
                                               
                                             
                                             ) 
                                           
                                            
                                           
                                             [ 
                                             
                                               
                                                  
                                                 
                                                    
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   v 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   ς 
                                                 
                                               
                                               - 
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                      
                                                   
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   v 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   ς 
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                  
                                 
                                    
                                   0 
                                   〉 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where ζ is a time that is a function of the delay introduced by the second interferometer delay line  90 . In particular, the condition ζ=0 is obtained when the fifth and sixth optical paths  100 ,  102  have the same optical length. It should also be noted that when reference is made to the output states of the second interferometer beam splitter  98 , subscripts “1” and “2” refer, strictly speaking, to the first and second extended paths. 
         [0113]    In the case where θ=0 and ζ=0, equation (11) becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     ψ 
                     〉 
                   
                   = 
                   
                     
                       - 
                       
                         
                           CL 
                            
                           
                               
                           
                            
                           
                              
                             
                               
                                 - 
                                  
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   Ω 
                                   p 
                                 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   ϑ 
                                   + 
                                   ς 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           4 
                            
                           
                             2 
                           
                         
                       
                     
                      
                     
                       
                         ∫ 
                         
                           - 
                           L 
                         
                         0 
                       
                        
                       
                         
                            
                           z 
                         
                          
                         
                             
                         
                          
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             
                               + 
                               ∞ 
                             
                           
                            
                           
                             
                                
                               
                                 v 
                                 p 
                               
                             
                              
                             
                               
                                 E 
                                 p 
                                 
                                   ( 
                                   + 
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   v 
                                   p 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                  
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     v 
                                     p 
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         Λ 
                                          
                                         
                                             
                                         
                                          
                                         z 
                                       
                                       - 
                                       
                                         
                                           ϑ 
                                           + 
                                           ς 
                                         
                                         2 
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                              
                             
                               
                                 ∫ 
                                 
                                   - 
                                   ∞ 
                                 
                                 
                                   + 
                                   ∞ 
                                 
                               
                                
                               
                                 
                                    
                                   v 
                                 
                                  
                                 
                                     
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                        
                                     
                                      
                                     
                                         
                                     
                                      
                                     DLvz 
                                   
                                 
                                 × 
                                 × 
                                 
                                   [ 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           a 
                                                           ^ 
                                                         
                                                         
                                                           1 
                                                            
                                                           e 
                                                         
                                                         † 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           v 
                                                           + 
                                                           
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                       
                                                         
                                                           
                                                             a 
                                                             ^ 
                                                           
                                                           
                                                             1 
                                                              
                                                             o 
                                                           
                                                           † 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   
                                                     a 
                                                     ^ 
                                                   
                                                   
                                                     2 
                                                      
                                                     e 
                                                   
                                                   † 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     v 
                                                     + 
                                                     
                                                       
                                                         
                                                           v 
                                                           p 
                                                         
                                                         + 
                                                         
                                                           Ω 
                                                           p 
                                                         
                                                       
                                                       2 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             
                                               a 
                                               ^ 
                                             
                                             
                                               2 
                                                
                                               o 
                                             
                                             † 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 - 
                                                 v 
                                               
                                               + 
                                               
                                                 
                                                   
                                                     v 
                                                     p 
                                                   
                                                   + 
                                                   
                                                     Ω 
                                                     p 
                                                   
                                                 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                  
                                 
                                    
                                   0 
                                   〉 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0114]    In a manner in itself known, it is also possible to determine the (normalized) probabilities 1e1o, 1e2o, 2e1o and 2e2o, henceforth respectively indicated as MZ 1e1o , MZ 1e2o , MZ 2e1o  and MZ 2e2o , which can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       MZ 
                       
                         1 
                          
                         e 
                          
                         
                             
                         
                          
                         1 
                          
                         o 
                       
                     
                      
                     
                       ( 
                       
                         ϑ 
                         , 
                         ς 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       4 
                     
                      
                     
                       { 
                       
                         
                           
                             
                               1 
                               + 
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   Cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         Ω 
                                         p 
                                       
                                        
                                       ζ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       
                                         
                                           σ 
                                           p 
                                           2 
                                         
                                         2 
                                       
                                     
                                      
                                     
                                       
                                         ( 
                                         ζ 
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   4 
                                 
                                  
                                 
                                   Cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         Ω 
                                         p 
                                       
                                        
                                       ζ 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         ϑ 
                                         DL 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 
                                                   - 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         2 
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         
                                                           
                                                             Λ 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             ϑ 
                                                           
                                                           D 
                                                         
                                                       
                                                       - 
                                                       ζ 
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               
                                                 - 
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   
                                                     σ 
                                                     p 
                                                     2 
                                                   
                                                   2 
                                                 
                                               
                                                
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         
                                                           Λ 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           ϑ 
                                                         
                                                         D 
                                                       
                                                     
                                                     + 
                                                     ζ 
                                                   
                                                   ) 
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   4 
                                 
                                  
                                 
                                   Tri 
                                    
                                   
                                     ( 
                                     
                                       
                                         ζ 
                                         + 
                                         ϑ 
                                       
                                       DL 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       
                                         
                                           σ 
                                           p 
                                           2 
                                         
                                         2 
                                       
                                     
                                      
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             Λ 
                                             D 
                                           
                                            
                                           
                                             
                                               ( 
                                               
                                                 ζ 
                                                 + 
                                                 ϑ 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   Tri 
                                    
                                   
                                     ( 
                                     
                                       ζ 
                                       DL 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                      
                                     
                                       
                                         - 
                                         
                                           
                                             σ 
                                             p 
                                             2 
                                           
                                           2 
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           
                                             Λ 
                                             D 
                                           
                                            
                                           ζ 
                                         
                                         ) 
                                       
                                     
                                   
                                   2 
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 4 
                               
                                
                               
                                 Tri 
                                  
                                 
                                   ( 
                                   
                                     
                                       ζ 
                                       - 
                                       ϑ 
                                     
                                     DL 
                                   
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     
                                         
                                     
                                      
                                     
                                       
                                         σ 
                                         p 
                                         2 
                                       
                                       2 
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         2 
                                          
                                         
                                           Λ 
                                           D 
                                         
                                          
                                         
                                           ( 
                                           
                                             ζ 
                                             - 
                                             ϑ 
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       MZ 
                       
                         2 
                          
                         e 
                          
                         
                             
                         
                          
                         2 
                          
                         o 
                       
                     
                      
                     
                       ( 
                       
                         ϑ 
                         , 
                         ς 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       4 
                     
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           
                                             
                                               
                                                 1 
                                                 + 
                                                 
                                                   
                                                     1 
                                                     2 
                                                   
                                                    
                                                   
                                                     Cos 
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           Ω 
                                                           p 
                                                         
                                                          
                                                         ζ 
                                                       
                                                       ) 
                                                     
                                                   
                                                    
                                                   
                                                      
                                                     
                                                       
                                                         - 
                                                         
                                                           
                                                             σ 
                                                             p 
                                                             2 
                                                           
                                                           2 
                                                         
                                                       
                                                        
                                                       
                                                         
                                                           ( 
                                                           ζ 
                                                           ) 
                                                         
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                                 + 
                                               
                                             
                                           
                                           
                                             
                                               
                                                 
                                                   
                                                     1 
                                                     4 
                                                   
                                                    
                                                   
                                                     Cos 
                                                      
                                                     
                                                       ( 
                                                       
                                                         
                                                           Ω 
                                                           p 
                                                         
                                                          
                                                         ζ 
                                                       
                                                       ) 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       Tri 
                                                        
                                                       
                                                         ( 
                                                         
                                                           ϑ 
                                                           DL 
                                                         
                                                         ) 
                                                       
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             - 
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             2 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             Λϑ 
                                                             D 
                                                             
                                                             
                                                             - 
                                                             ζ 
                                                             
                                                             ) 
                                                             
                                                             2 
                                                             
                                                             
                                                             
                                                             + 
                                                             
                                                           
                                                         
                                                         
                                                           
                                                             
                                                              
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                             
                                                             
                                                              
                                                             
                                                             
                                                             ( 
                                                             
                                                             
                                                             2 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             
                                                             Λ 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             ϑ 
                                                             
                                                             D 
                                                             
                                                             
                                                             + 
                                                             ζ 
                                                             
                                                             ) 
                                                             
                                                             2 
                                                             
                                                             
                                                             
                                                           
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                         
                                       
                                     
                                     
                                       
                                         
                                           
                                             
                                               1 
                                               4 
                                             
                                              
                                             
                                               Tri 
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     ζ 
                                                     + 
                                                     ϑ 
                                                   
                                                   DL 
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                                
                                               
                                                 
                                                   - 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     ( 
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         Λ 
                                                         D 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           ζ 
                                                           + 
                                                           ϑ 
                                                         
                                                         ) 
                                                       
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                           + 
                                         
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         Tri 
                                          
                                         
                                           ( 
                                           
                                             ζ 
                                             DL 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         
                                           
                                             - 
                                             
                                               
                                                 σ 
                                                 p 
                                                 2 
                                               
                                               2 
                                             
                                           
                                            
                                           
                                             
                                               ( 
                                               
                                                 2 
                                                  
                                                 
                                                   Λ 
                                                   D 
                                                 
                                                  
                                                 ζ 
                                               
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                     
                                     - 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 4 
                               
                                
                               
                                 Tri 
                                  
                                 
                                   ( 
                                   
                                     
                                       ζ 
                                       - 
                                       ϑ 
                                     
                                     DL 
                                   
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     
                                       
                                         σ 
                                         p 
                                         2 
                                       
                                       2 
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         2 
                                          
                                         
                                           Λ 
                                           D 
                                         
                                          
                                         
                                           ( 
                                           
                                             ζ 
                                             - 
                                             ϑ 
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       MZ 
                       
                         1 
                          
                         e 
                          
                         
                             
                         
                          
                         2 
                          
                         o 
                       
                     
                      
                     
                       ( 
                       
                         ϑ 
                         , 
                         ς 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       4 
                     
                      
                     
                       { 
                       
                         
                           
                             
                               
                                 
                                   
                                     1 
                                     - 
                                     
                                       
                                         1 
                                         2 
                                       
                                        
                                       
                                         Cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               Ω 
                                               p 
                                             
                                              
                                             ζ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                          
                                         
                                           
                                             - 
                                             
                                               
                                                 σ 
                                                 p 
                                                 2 
                                               
                                               2 
                                             
                                           
                                            
                                           
                                             
                                               ( 
                                               ζ 
                                               ) 
                                             
                                             2 
                                           
                                         
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         1 
                                         4 
                                       
                                        
                                       
                                         Cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               Ω 
                                               p 
                                             
                                              
                                             ζ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         
                                           Tri 
                                            
                                           
                                             ( 
                                             
                                               ϑ 
                                               DL 
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           [ 
                                           
                                             
                                               
                                                 
                                                   
                                                      
                                                     
                                                       
                                                         - 
                                                         
                                                           
                                                             σ 
                                                             p 
                                                             2 
                                                           
                                                           2 
                                                         
                                                       
                                                        
                                                       
                                                         
                                                           ( 
                                                           
                                                             
                                                             2 
                                                              
                                                             
                                                             Λϑ 
                                                             D 
                                                             
                                                             
                                                             - 
                                                             ζ 
                                                           
                                                           ) 
                                                         
                                                         2 
                                                       
                                                     
                                                   
                                                   + 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   
                                                      
                                                     
                                                       
                                                         - 
                                                         
                                                           
                                                             σ 
                                                             p 
                                                             2 
                                                           
                                                           2 
                                                         
                                                       
                                                        
                                                       
                                                         
                                                           ( 
                                                           
                                                             
                                                             2 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             Λϑ 
                                                             D 
                                                             
                                                             
                                                             + 
                                                             ζ 
                                                           
                                                           ) 
                                                         
                                                         2 
                                                       
                                                     
                                                   
                                                    
                                                   
                                                       
                                                   
                                                 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   4 
                                 
                                  
                                 
                                   Tri 
                                    
                                   
                                     ( 
                                     
                                       
                                         ζ 
                                         + 
                                         ϑ 
                                       
                                       DL 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       
                                         
                                           σ 
                                           p 
                                           2 
                                         
                                         2 
                                       
                                     
                                      
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             Λ 
                                             D 
                                           
                                            
                                           
                                             ( 
                                             
                                               ζ 
                                               + 
                                               ϑ 
                                             
                                             ) 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   1 
                                   2 
                                 
                                  
                                 
                                   Tri 
                                    
                                   
                                     ( 
                                     
                                       ζ 
                                       DL 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       
                                         
                                           σ 
                                           p 
                                           2 
                                         
                                         2 
                                       
                                     
                                      
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             Λ 
                                             D 
                                           
                                            
                                           ζ 
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 1 
                                 4 
                               
                                
                               
                                 Tri 
                                  
                                 
                                   ( 
                                   
                                     
                                       ζ 
                                       - 
                                       ϑ 
                                     
                                     DL 
                                   
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     
                                       
                                         σ 
                                         p 
                                         2 
                                       
                                       2 
                                     
                                   
                                    
                                   
                                     
                                       ( 
                                       
                                         2 
                                          
                                         
                                           Λ 
                                           D 
                                         
                                          
                                         
                                           ( 
                                           
                                             ζ 
                                             - 
                                             ϑ 
                                           
                                           ) 
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       MZ 
                       
                         2 
                          
                         e 
                          
                         
                             
                         
                          
                         1 
                          
                         o 
                       
                     
                      
                     
                       ( 
                       
                         ϑ 
                         , 
                         ς 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       4 
                     
                      
                     
                       
                         { 
                         
                           
                             
                               
                                 1 
                                 - 
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   
                                     Cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           Ω 
                                           p 
                                         
                                          
                                         ς 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     
                                       
                                         - 
                                         
                                           
                                             σ 
                                             p 
                                             2 
                                           
                                           2 
                                         
                                       
                                        
                                       
                                         
                                           ( 
                                           ζ 
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     1 
                                     4 
                                   
                                    
                                   
                                     Cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           Ω 
                                           p 
                                         
                                          
                                         ζ 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       Tri 
                                        
                                       
                                         ( 
                                         
                                           ϑ 
                                           DL 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       ( 
                                                       
                                                         
                                                           2 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           
                                                             
                                                             Λ 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             ϑ 
                                                             
                                                             D 
                                                           
                                                         
                                                         - 
                                                         ζ 
                                                       
                                                       ) 
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                                
                                               
                                                 
                                                   - 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                     ( 
                                                     
                                                       
                                                         2 
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         
                                                           Λϑ 
                                                           D 
                                                         
                                                       
                                                       + 
                                                       ζ 
                                                     
                                                     ) 
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     1 
                                     4 
                                   
                                    
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ζ 
                                           + 
                                           ϑ 
                                         
                                         DL 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     
                                       
                                         - 
                                         
                                           
                                             σ 
                                             p 
                                             2 
                                           
                                           2 
                                         
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               Λ 
                                               D 
                                             
                                              
                                             
                                               ( 
                                               
                                                 ζ 
                                                 + 
                                                 ϑ 
                                               
                                               ) 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                 
                                 - 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     1 
                                     2 
                                   
                                    
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         ζ 
                                         DL 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                      
                                     
                                       
                                         - 
                                         
                                           
                                             σ 
                                             p 
                                             2 
                                           
                                           2 
                                         
                                       
                                        
                                       
                                         
                                           ( 
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               Λϑ 
                                               D 
                                             
                                              
                                             ζ 
                                           
                                           ) 
                                         
                                         2 
                                       
                                     
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   1 
                                   4 
                                 
                                  
                                 
                                   Tri 
                                    
                                   
                                     ( 
                                     
                                       
                                         ζ 
                                         - 
                                         ϑ 
                                       
                                       DL 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       
                                         
                                           σ 
                                           p 
                                           2 
                                         
                                         2 
                                       
                                     
                                      
                                     
                                       
                                         ( 
                                         
                                           2 
                                            
                                           
                                             Λ 
                                             D 
                                           
                                            
                                           
                                             ( 
                                             
                                               ζ 
                                               - 
                                               ϑ 
                                             
                                             ) 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                       . 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    With reference to equations (13),  FIG. 8  shows an envelope of probability MZ 1e1o  as a function of ζ, obtained for ζ= 0 . Moreover,  FIG. 9 , also obtained for θ=0, shows the probability MZ 1e1o  (equal to probability MZ 2e2o ) and the probability MZ 1e2o  (equal to probability MZ 2e1o ). 
         [0115]    In particular,  FIG. 9  shows how it is possible to change the symmetry of the output state from the second interferometer beam splitter  98  by altering ζ (within the wavelength), i.e. by altering the delay introduced by the second interferometer delay line  90 . In addition, it can be noted how probability MZ 1e1o  is at its maximum and equal to 0.5 for ζ=0, i.e. coalescence occurs. In other words, the first crystal  54 , the first interferometer delay line  56 , the first polarization phase-shifter  58 , the first and second mirrors  60 ,  62 , the first interferometer beam splitter  64 , the third and fourth mirror  74 ,  76 , the second interferometer delay line  90 , the second polarization phase-shifter  92 , the fifth and sixth mirrors  94 ,  96  and the second interferometer beam splitter  98  form a symmetric state interferential device I s , the output state of which is a symmetric state with two coalescent photons, which can be expressed in a simplified manner (without considering the difference in polarization) as 1/√{square root over (2)}(|2 1 ,0 2           +|0 1 ,2 2           ). 
         [0116]    Upon removing the assumption that the first and second compensation elements  78 ,  80  and the second crystal  84  are absent, the following occurs. 
         [0117]    Operationally, one of the photons of the antisymmetric pump pulse can generate, through spontaneous parametric down-conversion inside the second crystal  84 , a second extraordinary photon and a second ordinary photon, each of which can propagate along the fifth or the sixth optical path  100 ,  102 , namely along the first or the second extended path. In practice, the second crystal  84 , the second interferometer delay line  90 , the second polarization phase-shifter  92 , the fifth and sixth mirrors  94 ,  96  and the second interferometer beam splitter  98  form an antisymmetric state interferential device I a , in a certain sense “cascaded to” the symmetric state interferential device I s . More precisely, the antisymmetric state interferential device I a  shares outputs with the symmetric state interferential device I s , namely the second interferometer beam splitter  98 , as well as the second interferometer delay line  90  and the second polarization phase-shifter  92 . Furthermore, the antisymmetric state interferential device I a  is a Hong-Ou-Mandel interferometer and as such provides, on its own, an antisymmetric state with two coalescent photons, which can be expressed in a simplified manner as 1/√{square root over ( 2 )}(|2 1 ,0 2           −|0 1 ,2 2           ). 
         [0118]    In greater detail, indicating with τ a time that is a function of the delay introduced by delay line  89 , namely the delay with which the antisymmetric pump pulse impinges on the second crystal  84  with respect to the time at which the symmetric pump pulse impinges on the first crystal  54 , the output state from the second interferometer beam splitter  98  becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                      
                     ψ 
                     〉 
                   
                   = 
                   
                     
                       
                         
                            
                           
                             
                               - 
                                
                             
                              
                             
                                 
                             
                              
                             2 
                              
                             
                               ( 
                               
                                 
                                   
                                     K 
                                     e 
                                   
                                    
                                   L 
                                 
                                 + 
                                 
                                   
                                     K 
                                     o 
                                   
                                    
                                   
                                     L 
                                     c 
                                   
                                 
                               
                               ) 
                             
                           
                         
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                                                   Ω 
                                                   p 
                                                 
                                               
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             
                                               a 
                                               ^ 
                                             
                                             
                                               2 
                                                
                                               o 
                                             
                                             † 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 - 
                                                 v 
                                               
                                               + 
                                               
                                                 
                                                   
                                                     v 
                                                     p 
                                                   
                                                   + 
                                                   
                                                     Ω 
                                                     p 
                                                   
                                                 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                         - 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             a 
                                             ^ 
                                           
                                           
                                             2 
                                              
                                             e 
                                           
                                           † 
                                         
                                          
                                         
                                           ( 
                                           
                                             v 
                                             + 
                                             
                                               
                                                 
                                                   v 
                                                   p 
                                                 
                                                 + 
                                                 
                                                   Ω 
                                                   p 
                                                 
                                               
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             a 
                                             ^ 
                                           
                                           
                                             1 
                                              
                                             o 
                                           
                                           † 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               - 
                                               v 
                                             
                                             + 
                                             
                                               
                                                 
                                                   v 
                                                   p 
                                                 
                                                 + 
                                                 
                                                   Ω 
                                                   p 
                                                 
                                               
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                      
                                     
                                        
                                        
                                       
                                           
                                       
                                        
                                       v 
                                        
                                       
                                           
                                       
                                        
                                       ζ 
                                     
                                   
                                   - 
                                   
                                      
                                     
                                       
                                         - 
                                          
                                       
                                        
                                       
                                           
                                       
                                        
                                       v 
                                        
                                       
                                           
                                       
                                        
                                       ζ 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                        
                       
                          
                         0 
                         〉 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
         [0000]    where, indicating the extraordinary and ordinary refractive indices of the first crystal  54  as n e  and n o , gives 
         [0000]    
       
         
           
             
               
                 K 
                 e 
               
               = 
               
                 
                   
                     n 
                     e 
                   
                    
                   
                     Ω 
                     p 
                   
                 
                 
                   2 
                    
                   c 
                 
               
             
             , 
             
               
                 K 
                 o 
               
               = 
               
                 
                   
                     n 
                     o 
                   
                    
                   
                     Ω 
                     p 
                   
                 
                 
                   2 
                    
                   c 
                 
               
             
             , 
             
               G 
               = 
               
                 ( 
                 
                   
                     1 
                     
                       u 
                       e 
                     
                   
                   + 
                   
                     1 
                     
                       u 
                       o 
                     
                   
                 
                 ) 
               
             
             , 
           
         
       
     
         [0000]    and L c  is the length of the first and second compensation elements  78 ,  80 . 
         [0119]    It is also possible to determine the (normalized) probabilities 1e1o, 1e2o, 2e1o and 2e2o, henceforth respectively indicated as P 1e1o , P 1e2o , P 2e1o  and P 2e2o , which can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       iejo 
                     
                      
                     
                       ( 
                       ς 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                           
                             HOM 
                             iejo 
                           
                            
                           
                             ( 
                             ς 
                             ) 
                           
                         
                       
                       + 
                       
                         2 
                          
                         
                           
                             MZ 
                             iejo 
                           
                            
                           
                             ( 
                             
                               ϑ 
                               , 
                               ς 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           I 
                           iejo 
                         
                          
                         
                           ( 
                           
                             τ 
                             , 
                             ϑ 
                             , 
                             ς 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           ∑ 
                           
                             i 
                             , 
                             j 
                           
                         
                         2 
                       
                        
                       
                         [ 
                         
                           
                             2 
                              
                             
                               
                                 HOM 
                                 iejo 
                               
                                
                               
                                 ( 
                                 ς 
                                 ) 
                               
                             
                           
                           + 
                           
                             2 
                              
                             
                               
                                 MZ 
                                 iejo 
                               
                                
                               
                                 ( 
                                 
                                   ϑ 
                                   , 
                                   ς 
                                 
                                 ) 
                               
                             
                           
                           - 
                           
                             
                               I 
                               iejo 
                             
                              
                             
                               ( 
                               
                                 τ 
                                 , 
                                 ϑ 
                                 , 
                                 ς 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the indices i and j can assume values of “1” or “2”, and where the following hold: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         I 
                         
                           1 
                            
                           e 
                            
                           
                               
                           
                            
                           1 
                            
                           o 
                         
                       
                        
                       
                         ( 
                         
                           τ 
                           , 
                           ϑ 
                           , 
                           ς 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                       - 
                                       ς 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       Tri 
                                        
                                       
                                         ( 
                                         
                                           
                                             ϑ 
                                             + 
                                             ς 
                                           
                                           
                                             2 
                                              
                                             DL 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             
                                                             D 
                                                              
                                                             
                                                               
                                                             
                                                             
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             + 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           - 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             - 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           + 
                                                           ς 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                       + 
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                       + 
                                       ς 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       Tri 
                                        
                                       
                                         ( 
                                         
                                           
                                             ϑ 
                                             + 
                                             ς 
                                           
                                           
                                             2 
                                              
                                             DL 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           + 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                                
                                               
                                                   
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           - 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             - 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           - 
                                                           ς 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                       + 
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           2 
                                            
                                           
                                             
                                               Tri 
                                                
                                               
                                                 ( 
                                                 
                                                   ϑ 
                                                   
                                                     2 
                                                      
                                                     DL 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     
                                                       
                                                          
                                                         
                                                           - 
                                                           
                                                             
                                                             
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                             
                                                              
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             ϑ 
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ] 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                       
                                                       - 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                        
                                                       
                                                         - 
                                                         
                                                           
                                                             
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                             
                                                              
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             ϑ 
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ] 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                         + 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             Tri 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   ϑ 
                                                   - 
                                                   ς 
                                                 
                                                 
                                                   2 
                                                    
                                                   DL 
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             [ 
                                             
                                               
                                                 
                                                   
                                                      
                                                     
                                                       
                                                         - 
                                                         
                                                           
                                                             
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                             
                                                              
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ] 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       - 
                                                     
                                                   
                                                 
                                               
                                               
                                                 
                                                   
                                                      
                                                     
                                                       - 
                                                       
                                                         
                                                           
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                           
                                                            
                                                           
                                                             [ 
                                                             
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ] 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         + 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           + 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             + 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       - 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         I 
                         
                           2 
                            
                           e 
                            
                           
                               
                           
                            
                           1 
                            
                           o 
                         
                       
                        
                       
                         ( 
                         
                           τ 
                           , 
                           ϑ 
                           , 
                           ς 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           - 
                           
                             Cos 
                              
                             
                               [ 
                               
                                 
                                   
                                     Ω 
                                     p 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           2 
                                            
                                           τ 
                                         
                                         - 
                                         ϑ 
                                         - 
                                         ς 
                                       
                                       2 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   2 
                                    
                                   
                                     L 
                                      
                                     
                                       ( 
                                       
                                         
                                           K 
                                           e 
                                         
                                         + 
                                         
                                           K 
                                           o 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       Tri 
                                        
                                       
                                         ( 
                                         
                                           
                                             ϑ 
                                             + 
                                             ς 
                                           
                                           
                                             2 
                                              
                                             DL 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                 
                                               
                                               2 
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           - 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             - 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           + 
                                                           ς 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                       - 
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                       + 
                                       ς 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       Tri 
                                        
                                       
                                         ( 
                                         
                                           
                                             ϑ 
                                             + 
                                             ς 
                                           
                                           
                                             2 
                                              
                                             DL 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               + 
                                             
                                           
                                         
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           - 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             + 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             - 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           - 
                                                           ς 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                       + 
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     2 
                                      
                                     
                                       
                                         Tri 
                                          
                                         
                                           ( 
                                           
                                             ϑ 
                                             
                                               2 
                                                
                                               DL 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 
                                                    
                                                   
                                                     - 
                                                     
                                                       
                                                         
                                                           
                                                             σ 
                                                             p 
                                                             2 
                                                           
                                                           2 
                                                         
                                                          
                                                         
                                                           [ 
                                                           
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             ϑ 
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ] 
                                                         
                                                       
                                                       2 
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             ϑ 
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       Tri 
                                        
                                       
                                         ( 
                                         
                                           
                                             ϑ 
                                             - 
                                             ς 
                                           
                                           
                                             2 
                                              
                                             DL 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           + 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           + 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             + 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       - 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         I 
                         
                           1 
                            
                           e 
                            
                           
                               
                           
                            
                           2 
                            
                           o 
                         
                       
                        
                       
                         ( 
                         
                           τ 
                           , 
                           ϑ 
                           , 
                           ς 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                       - 
                                       ς 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       Tri 
                                        
                                       
                                         ( 
                                         
                                           
                                             ϑ 
                                             + 
                                             ς 
                                           
                                           
                                             2 
                                              
                                             DL 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           - 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             + 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           + 
                                                           ς 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                       - 
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                       + 
                                       ς 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       Tri 
                                        
                                       
                                         ( 
                                         
                                           
                                             ϑ 
                                             + 
                                             ς 
                                           
                                           
                                             2 
                                              
                                             DL 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       [ 
                                       
                                         
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           + 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                               - 
                                             
                                           
                                         
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           - 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             - 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           - 
                                                           ς 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                       - 
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           2 
                                            
                                           
                                             
                                               Tri 
                                                
                                               
                                                 ( 
                                                 
                                                   ϑ 
                                                   
                                                     2 
                                                      
                                                     DL 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                              
                                             
                                               [ 
                                               
                                                 
                                                   
                                                     
                                                       
                                                          
                                                         
                                                           - 
                                                           
                                                             
                                                             
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                             
                                                              
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             ϑ 
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             - 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ] 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                       
                                                       - 
                                                     
                                                   
                                                 
                                                 
                                                   
                                                     
                                                        
                                                       
                                                         - 
                                                         
                                                           
                                                             
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                             
                                                              
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             ϑ 
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             - 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ] 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                               ] 
                                             
                                           
                                         
                                         - 
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             Tri 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   ϑ 
                                                   - 
                                                   ς 
                                                 
                                                 
                                                   2 
                                                    
                                                   DL 
                                                 
                                               
                                               ) 
                                             
                                           
                                            
                                           
                                             [ 
                                             
                                               
                                                 
                                                   
                                                     
                                                        
                                                       
                                                         - 
                                                         
                                                           
                                                             
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                             
                                                              
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ] 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                     
                                                     - 
                                                   
                                                 
                                               
                                               
                                                 
                                                   
                                                      
                                                     
                                                       - 
                                                       
                                                         
                                                           
                                                             
                                                             σ 
                                                             p 
                                                             2 
                                                             
                                                             2 
                                                           
                                                            
                                                           
                                                             [ 
                                                             
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             - 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ] 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                             ] 
                                           
                                         
                                         - 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           + 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       
                                                         σ 
                                                         p 
                                                         2 
                                                       
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         + 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             + 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       - 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         I 
                         
                           2 
                            
                           e 
                            
                           
                               
                           
                            
                           2 
                            
                           o 
                         
                       
                        
                       
                         ( 
                         
                           τ 
                           , 
                           ϑ 
                           , 
                           ς 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           - 
                           
                             Cos 
                              
                             
                               [ 
                               
                                 
                                   
                                     Ω 
                                     p 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         
                                           2 
                                            
                                           τ 
                                         
                                         - 
                                         ϑ 
                                         - 
                                         ς 
                                       
                                       2 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   2 
                                    
                                   
                                     L 
                                      
                                     
                                       ( 
                                       
                                         
                                           K 
                                           e 
                                         
                                         + 
                                         
                                           K 
                                           o 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     + 
                                     
                                       
                                         Tri 
                                          
                                         
                                           ( 
                                           
                                             
                                               ϑ 
                                               + 
                                               ς 
                                             
                                             
                                               2 
                                                
                                               DL 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 
                                                    
                                                   
                                                     - 
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                          
                                                         
                                                           [ 
                                                           
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                           
                                                           ] 
                                                         
                                                       
                                                       2 
                                                     
                                                   
                                                 
                                                 - 
                                               
                                             
                                           
                                           
                                             
                                               
                                                  
                                                 
                                                   - 
                                                   
                                                     
                                                       
                                                         
                                                           σ 
                                                           p 
                                                           2 
                                                         
                                                         2 
                                                       
                                                        
                                                       
                                                         [ 
                                                         
                                                           
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                           
                                                           - 
                                                           
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ] 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     Tri 
                                      
                                     
                                       ( 
                                       
                                         
                                           ϑ 
                                           - 
                                           ς 
                                         
                                         
                                           2 
                                            
                                           DL 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     [ 
                                     
                                       
                                         
                                           
                                             
                                                
                                               
                                                 - 
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                      
                                                     
                                                       [ 
                                                       
                                                         
                                                           
                                                             Λ 
                                                             D 
                                                           
                                                            
                                                           
                                                             ( 
                                                             
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             
                                                             ) 
                                                           
                                                         
                                                         - 
                                                         
                                                           
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                           
                                                           2 
                                                         
                                                       
                                                       ] 
                                                     
                                                   
                                                   2 
                                                 
                                               
                                             
                                             - 
                                           
                                         
                                       
                                       
                                         
                                           
                                              
                                             
                                               - 
                                               
                                                 
                                                   
                                                     
                                                       σ 
                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
                                                    
                                                   
                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             ϑ 
                                                             - 
                                                             ς 
                                                           
                                                           ) 
                                                         
                                                       
                                                       + 
                                                       
                                                         
                                                           
                                                             2 
                                                              
                                                             GL 
                                                           
                                                           + 
                                                           ϑ 
                                                           + 
                                                           ς 
                                                           - 
                                                           
                                                             2 
                                                              
                                                             τ 
                                                           
                                                         
                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                         
                                       
                                     
                                     ] 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                       + 
                       
                         
                           Cos 
                            
                           
                             [ 
                             
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         2 
                                          
                                         τ 
                                       
                                       - 
                                       ϑ 
                                       + 
                                       ς 
                                     
                                     2 
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 2 
                                  
                                 
                                   L 
                                    
                                   
                                     ( 
                                     
                                       
                                         K 
                                         e 
                                       
                                       + 
                                       
                                         K 
                                         o 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   
                                     + 
                                     
                                       
                                         Tri 
                                          
                                         
                                           ( 
                                           
                                             
                                               ϑ 
                                               + 
                                               ς 
                                             
                                             
                                               2 
                                                
                                               DL 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         [ 
                                         
                                           
                                             
                                               
                                                 
                                                    
                                                   
                                                     - 
                                                     
                                                       
                                                         
                                                           
                                                             σ 
                                                             p 
                                                             2 
                                                           
                                                           2 
                                                         
                                                          
                                                         
                                                           [ 
                                                           
                                                             
                                                             
                                                             Λ 
                                                             D 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             ϑ 
                                                             + 
                                                             ς 
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             2 
                                                              
                                                             GL 
                                                             
                                                             + 
                                                             ϑ 
                                                             - 
                                                             ς 
                                                             - 
                                                             
                                                             2 
                                                              
                                                             τ 
                                                             
                                                             
                                                             2 
                                                             
                                                           
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                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With reference to equation (15),  FIG. 10  shows an envelope of probability P 1e1o  as a function of τ, obtained for θ=0 and ζ=0. 
         [0120]    Moreover,  FIG. 11 , also obtained for θ=0 and ζ=0, shows, as a function of τ, the probability P 1e1o , namely the probability of having an extraordinary photon—ordinary photon pair on the first output of the second interferometer beam splitter  98 , and the probability P 2e2o , namely the probability of having an extraordinary photon—ordinary photon pair on the second output of the second interferometer beam splitter  98 . 
         [0121]    In detail, both probability P 1e1o  and probability P 2e2o  have an oscillating trend and are in the range between 0 and 1. In particular, for ζ=0 and θ=0, the output state from the second interferometer beam splitter  98  can be expressed as: 
         [0000]      1/√{square root over (2)}(|2 1 ,0 2           +|0 1 ,2 2           )+ e   iτ ·1/√{square root over (2)}(|2 1 ,0 2           −|0 1 ,2 2           )  (17)
 
         [0000]    i.e. the weighted sum of a symmetric state with two coalescent photons and an antisymmetric state with two coalescent photons. The weight of the symmetric state with two coalescent photons is unitary, while the weight of the antisymmetric state with two coalescent photons is a function of τ, i.e. it depends on the delay introduced by the pump delay line  89 . 
         [0122]    It should be noted that in order to pump the first and second crystals  54 ,  84  in a coherent manner, the pump delay line  89  is sized such that τ does not exceed the characteristic coherence time of the electromagnetic pulses emitted by the optical source  52 ; in this way, the pulses of each pair of first and second crystal pulses are mutually coherent. In addition, τ is chosen so that the value τ=0 corresponds to the case where the first extraordinary photon and the first ordinary photon possibly generated by the first crystal  54  impinge on the second crystal  84  simultaneously with the antisymmetric pump pulse. 
         [0123]    Since the delay introduced by the pump delay line  89  is electronically controllable, in a manner which is in itself known, it is therefore possible to control the output state from the interferometer  50 . 
         [0124]    In other words, by altering the temporal phase shift between the symmetric pump pulse and the antisymmetric pump pulse, it is possible to control the probabilities with which the pair of coalescent photons appear on the first or on the second output of the second interferometer beam splitter  98 . 
         [0125]    It may also be noted that the description provided for the embodiment shown in  FIG. 7  can be extended in the case where phenomena of type-I spontaneous parametric down-conversion occur inside the first and second crystals  54 ,  84 , i.e. phenomena where the generated photons have the same polarization. In this case, the first and second measurement splitters  46 ,  48  are replaced by corresponding substitutive optical beam splitters, of the 50/50 type, while the first and second compensation elements  78 ,  80  are absent; in addition, two narrow-band interference filters, centred on the wavelength of the photons emitted through spontaneous parametric down-conversion, are interposed between the second interferometer beam splitter  98  and the substitutive optical beam splitters. 
         [0126]    According to a different embodiment, still regarding the case of type-II spontaneous parametric down-conversion, it is also possible to replace the first interferometer beam splitter  64  with a substitutive optical beam splitter  64   b  is of the polarizing type, as shown in  FIG. 12 . Instead, the second interferometer beam splitter  98  continues to be of the 50/50 type. 
         [0127]    In this case, the output state of the second interferometer beam splitter  98  becomes: 
         [0000]    
       
         
           
             
               
                 
                   
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                                                         † 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             - 
                                                             v 
                                                           
                                                           + 
                                                           
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                           
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                         ] 
                                       
                                     
                                      
                                     
                                         
                                       
                                         
                                           
                                             [ 
                                             
                                               
                                                 
                                                   
                                                     
                                                       
                                                          
                                                         
                                                            
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           
                                                             v 
                                                              
                                                             
                                                             ( 
                                                             ϑ 
                                                             ) 
                                                             
                                                           
                                                         
                                                       
                                                        
                                                       
                                                          
                                                         
                                                           
                                                             - 
                                                              
                                                           
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                           
                                                            
                                                           
                                                             ( 
                                                             ς 
                                                             ) 
                                                           
                                                         
                                                       
                                                     
                                                     - 
                                                   
                                                 
                                               
                                               
                                                 
                                                   
                                                     
                                                        
                                                       
                                                         
                                                           - 
                                                            
                                                         
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         
                                                           v 
                                                            
                                                           
                                                             ( 
                                                             ϑ 
                                                             ) 
                                                           
                                                         
                                                       
                                                     
                                                      
                                                     
                                                        
                                                       
                                                          
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         
                                                           
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                           
                                                           2 
                                                         
                                                          
                                                         
                                                           ( 
                                                           ς 
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                             ] 
                                           
                                            
                                           
                                              
                                             0 
                                             〉 
                                           
                                         
                                         + 
                                         
                                           
                                             
                                               C 
                                                
                                               
                                                   
                                               
                                                
                                               
                                                  
                                                 
                                                   
                                                     - 
                                                     
                                                        
                                                        
                                                       
                                                         ( 
                                                         
                                                           
                                                             Ω 
                                                             p 
                                                           
                                                           2 
                                                         
                                                         ) 
                                                       
                                                     
                                                   
                                                    
                                                   ζ 
                                                 
                                               
                                             
                                             4 
                                           
                                            
                                           
                                             
                                               ∫ 
                                               
                                                 - 
                                                 L 
                                               
                                               0 
                                             
                                              
                                             
                                               
                                                  
                                                 z 
                                               
                                                
                                               
                                                 
                                                   ∫ 
                                                   0 
                                                   
                                                     + 
                                                     ∞ 
                                                   
                                                 
                                                  
                                                 
                                                   
                                                      
                                                     
                                                       v 
                                                       p 
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       E 
                                                       p 
                                                       
                                                         ( 
                                                         + 
                                                         ) 
                                                       
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         v 
                                                         p 
                                                       
                                                       ) 
                                                     
                                                   
                                                    
                                                   
                                                      
                                                     
                                                        
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                         
                                                           v 
                                                           p 
                                                         
                                                          
                                                         
                                                           ( 
                                                           
                                                             
                                                             Λ 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             z 
                                                             
                                                             - 
                                                             
                                                             ζ 
                                                             2 
                                                             
                                                           
                                                           ) 
                                                         
                                                       
                                                     
                                                   
                                                    
                                                   
                                                     
                                                       ∫ 
                                                       
                                                         - 
                                                         ∞ 
                                                       
                                                       
                                                         + 
                                                         ∞ 
                                                       
                                                     
                                                      
                                                     
                                                       
                                                          
                                                         v 
                                                       
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                          
                                                         
                                                           
                                                             - 
                                                              
                                                           
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           D 
                                                            
                                                           
                                                               
                                                           
                                                            
                                                           vz 
                                                         
                                                       
                                                       × 
                                                       
                                                         { 
                                                         
                                                           
                                                             
                                                             
                                                             
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             1 
                                                              
                                                             e 
                                                             
                                                             † 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             v 
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             1 
                                                              
                                                             o 
                                                             
                                                             † 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             2 
                                                              
                                                             e 
                                                             
                                                             † 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             2 
                                                              
                                                             o 
                                                             
                                                             † 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             ] 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                              
                                                             
                                                              
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             v 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             ζ 
                                                             
                                                             
                                                             + 
                                                             
                                                              
                                                             
                                                             
                                                             - 
                                                              
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             v 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             ζ 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             + 
                                                             
                                                             
                                                           
                                                           
                                                             
                                                             
                                                               
                                                             
                                                             
                                                             [ 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             1 
                                                              
                                                             e 
                                                             
                                                             † 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             v 
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             2 
                                                              
                                                             o 
                                                             
                                                             † 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             - 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             2 
                                                              
                                                             e 
                                                             
                                                             † 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             v 
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             a 
                                                             ^ 
                                                             
                                                             
                                                             1 
                                                              
                                                             o 
                                                             
                                                             † 
                                                             
                                                              
                                                             
                                                             ( 
                                                             
                                                             
                                                             - 
                                                             v 
                                                             
                                                             + 
                                                             
                                                             
                                                             
                                                             v 
                                                             p 
                                                             
                                                             + 
                                                             
                                                             Ω 
                                                             p 
                                                             
                                                             
                                                             2 
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             
                                                             ] 
                                                             
                                                              
                                                             
                                                               
                                                             
                                                             ( 
                                                             
                                                             
                                                              
                                                             
                                                              
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             v 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             ζ 
                                                             
                                                             
                                                             + 
                                                             
                                                              
                                                             
                                                             
                                                             - 
                                                              
                                                             
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             v 
                                                              
                                                             
                                                               
                                                             
                                                              
                                                             ζ 
                                                             
                                                             
                                                             
                                                             ) 
                                                             
                                                             
                                                             
                                                             
                                                             
                                                           
                                                         
                                                          
                                                         
                                                             
                                                         
                                                         } 
                                                       
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       
                                                          
                                                         
                                                             
                                                         
                                                          
                                                         0 
                                                          
                                                         
                                                             
                                                         
                                                         〉 
                                                       
                                                     
                                                   
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0128]    The probabilities P 1e1o , P 1e2o , P 2e1o  and P 2e2o  can therefore be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       P 
                       iejo 
                     
                      
                     
                       ( 
                       ς 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         2 
                          
                         
                           
                             HOM 
                             iejo 
                           
                            
                           
                             ( 
                             ς 
                             ) 
                           
                         
                       
                       + 
                       
                         2 
                          
                         
                           
                             MZ 
                             iejo 
                           
                            
                           
                             ( 
                             
                               ϑ 
                               , 
                               ς 
                             
                             ) 
                           
                         
                       
                       - 
                       
                         
                           I 
                           iejo 
                         
                          
                         
                           ( 
                           
                             τ 
                             , 
                             ϑ 
                             , 
                             ς 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           ∑ 
                           
                             i 
                             , 
                             j 
                           
                         
                         2 
                       
                        
                       
                         [ 
                         
                           
                             
                               HOM 
                               iejo 
                             
                              
                             
                               ( 
                               ς 
                               ) 
                             
                           
                           + 
                           
                             
                               PBMZ 
                               iejo 
                             
                              
                             
                               ( 
                               
                                 ϑ 
                                 , 
                                 ς 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               I 
                               iejo 
                             
                              
                             
                               ( 
                               
                                 τ 
                                 , 
                                 ϑ 
                                 , 
                                 ς 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the indices i and j can assume values of “1” or “2”, and where the following hold: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           HOM 
                           
                             1 
                              
                             
                                 
                             
                              
                             e 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                          
                         
                           ( 
                           ς 
                           ) 
                         
                       
                       = 
                       
                         [ 
                         
                           1 
                           + 
                           
                             
                               Tri 
                                
                               
                                 ( 
                                 
                                   ς 
                                   DL 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                 
                                   σ 
                                   p 
                                   2 
                                 
                                  
                                 
                                   
                                     Λ 
                                     2 
                                   
                                   
                                     D 
                                     2 
                                   
                                 
                                  
                                 
                                   ς 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           HOM 
                           
                             2 
                              
                             
                                 
                             
                              
                             e 
                              
                             
                                 
                             
                              
                             2 
                              
                             o 
                           
                         
                          
                         
                           ( 
                           ς 
                           ) 
                         
                       
                       = 
                       
                         [ 
                         
                           1 
                           + 
                           
                             
                               Tri 
                                
                               
                                 ( 
                                 
                                   ς 
                                   DL 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                 
                                   σ 
                                   p 
                                   2 
                                 
                                  
                                 
                                   
                                     Λ 
                                     2 
                                   
                                   
                                     D 
                                     2 
                                   
                                 
                                  
                                 
                                   ς 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           HOM 
                           
                             1 
                              
                             e 
                              
                             
                                 
                             
                              
                             2 
                              
                             o 
                           
                         
                          
                         
                           ( 
                           ς 
                           ) 
                         
                       
                       = 
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               Tri 
                                
                               
                                 ( 
                                 
                                   ς 
                                   DL 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                 
                                   σ 
                                   p 
                                   2 
                                 
                                  
                                 
                                   
                                     Λ 
                                     2 
                                   
                                   
                                     D 
                                     2 
                                   
                                 
                                  
                                 
                                   ς 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           HOM 
                           
                             2 
                              
                             
                                 
                             
                              
                             e 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                          
                         
                           ( 
                           ς 
                           ) 
                         
                       
                       = 
                       
                         [ 
                         
                           1 
                           - 
                           
                             
                               Tri 
                                
                               
                                 ( 
                                 
                                   ς 
                                   DL 
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                   2 
                                 
                                  
                                 
                                   σ 
                                   p 
                                   2 
                                 
                                  
                                 
                                   
                                     Λ 
                                     2 
                                   
                                   
                                     D 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                     
                                   
                                 
                                  
                                 
                                   ς 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           PBMZ 
                           
                             1 
                              
                             
                                 
                             
                              
                             e 
                              
                             
                                 
                             
                              
                             1 
                              
                             o 
                           
                         
                          
                         
                           ( 
                           
                             ϑ 
                             , 
                             ς 
                           
                           ) 
                         
                       
                       = 
                       
                         1 
                         + 
                         
                           
                             Cos 
                              
                             
                               ( 
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 ζ 
                               
                               ) 
                             
                           
                            
                           
                             Tri 
                              
                             
                               ( 
                               
                                 ϑ 
                                 DL 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                   
                                     σ 
                                     p 
                                     2 
                                   
                                   2 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Λϑ 
                                         D 
                                       
                                     
                                     - 
                                     ζ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           PBMZ 
                           
                             2 
                              
                             
                                 
                             
                              
                             e 
                              
                             
                                 
                             
                              
                             2 
                              
                             o 
                           
                         
                          
                         
                           ( 
                           
                             ϑ 
                             , 
                             ς 
                           
                           ) 
                         
                       
                       = 
                       
                         1 
                         + 
                         
                           
                             Cos 
                              
                             
                               ( 
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 ζ 
                               
                               ) 
                             
                           
                            
                           
                             Tri 
                              
                             
                               ( 
                               
                                 ϑ 
                                 DL 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                   
                                     σ 
                                     p 
                                     2 
                                   
                                   2 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       
                                         Λϑ 
                                         D 
                                       
                                     
                                     - 
                                     ζ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           PBMZ 
                           
                             1 
                              
                             
                                 
                             
                              
                             e 
                              
                             
                                 
                             
                              
                             2 
                              
                             o 
                           
                         
                          
                         
                           ( 
                           
                             ϑ 
                             , 
                             ς 
                           
                           ) 
                         
                       
                       = 
                       
                         1 
                         - 
                         
                           
                             Cos 
                              
                             
                               ( 
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 ζ 
                               
                               ) 
                             
                           
                            
                           
                             Tri 
                              
                             
                               ( 
                               
                                 ϑ 
                                 DL 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                   
                                     σ 
                                     p 
                                     2 
                                   
                                   2 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         Λϑ 
                                         D 
                                       
                                     
                                     - 
                                     ζ 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
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                           PBMZ 
                           
                             2 
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                             1 
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                             o 
                           
                         
                          
                         
                           ( 
                           
                             ϑ 
                             , 
                             ς 
                           
                           ) 
                         
                       
                       = 
                       
                         1 
                         - 
                         
                           
                             Cos 
                              
                             
                               ( 
                               
                                 
                                   Ω 
                                   p 
                                 
                                  
                                 ζ 
                               
                               ) 
                             
                           
                            
                           
                             Tri 
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                               ( 
                               
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                               ) 
                             
                           
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                                 - 
                                 
                                   
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                                     2 
                                   
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                                   ( 
                                   
                                     
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                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
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                          
                         
                           ( 
                           
                             τ 
                             , 
                             ϑ 
                             , 
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                           ) 
                         
                       
                       = 
                       
                         
                           - 
                           
                             
                               I 
                               
                                 2 
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                                  
                                 e 
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                                 2 
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                                 o 
                               
                             
                              
                             
                               ( 
                               
                                 τ 
                                 , 
                                 ϑ 
                                 , 
                                 ς 
                               
                               ) 
                             
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                 Sin 
                                  
                                 
                                   [ 
                                   
                                     
                                       
                                         Ω 
                                         p 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               2 
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                                               τ 
                                             
                                             - 
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                                             - 
                                             ς 
                                           
                                           2 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     
                                       2 
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                                         L 
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                                           ( 
                                           
                                             
                                               K 
                                               e 
                                             
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                                   ] 
                                 
                               
                             
                              
                             
                               { 
                               
                                 
                                   
                                     
                                       
                                         + 
                                         
                                           
                                             Tri 
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                                               ( 
                                               
                                                 
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                                                   + 
                                                   ς 
                                                 
                                                 
                                                   2 
                                                    
                                                   DL 
                                                 
                                               
                                               ) 
                                             
                                           
                                           [ 
                                           
                                              
                                             
                                               
                                                 - 
                                                 
                                                   
                                                     σ 
                                                     p 
                                                     2 
                                                   
                                                   2 
                                                 
                                               
                                                
                                               
                                                 
                                                   ⌈ 
                                                   
                                                     
                                                       
                                                         Λ 
                                                         D 
                                                       
                                                        
                                                       
                                                         ( 
                                                         
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                                                           + 
                                                           ς 
                                                         
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                                                           2 
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                                                           2 
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                                                       2 
                                                     
                                                   
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                                                 2 
                                               
                                             
                                           
                                           ] 
                                         
                                       
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                                           ( 
                                           
                                             
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                                               2 
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                                               DL 
                                             
                                           
                                           ) 
                                         
                                       
                                       [ 
                                       
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                                                   p 
                                                   2 
                                                 
                                                 2 
                                               
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                                                 [ 
                                                 
                                                   
                                                     
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                                                       ( 
                                                       
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                                                 ] 
                                               
                                             
                                             2 
                                           
                                         
                                       
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                               } 
                             
                           
                           - 
                           
                             
                               Sin 
                                
                               
                                 [ 
                                 
                                   
                                     
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                                      
                                     
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                                             2 
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                                         2 
                                       
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                                 ] 
                               
                             
                              
                             
                               { 
                               
                                 
                                   
                                     
                                       
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                                                   ς 
                                                 
                                                 
                                                   2 
                                                    
                                                   DL 
                                                 
                                               
                                               ) 
                                             
                                           
                                           [ 
                                           
                                              
                                             
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                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
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                                                     [ 
                                                     
                                                       
                                                         
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                                                 2 
                                               
                                             
                                           
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                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
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                                           ( 
                                           
                                             
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                                               2 
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                                           ) 
                                         
                                       
                                       [ 
                                       
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                                                   p 
                                                   2 
                                                 
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                                                 [ 
                                                 
                                                   
                                                     
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                                                     2 
                                                   
                                                 
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                                             2 
                                           
                                         
                                       
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                               } 
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
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                           I 
                           
                             1 
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                             2 
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                             o 
                           
                         
                          
                         
                           ( 
                           
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                             , 
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                           ) 
                         
                       
                       = 
                       
                         
                           - 
                           
                             
                               I 
                               
                                 2 
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                                 e 
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                                 o 
                               
                             
                              
                             
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                               ) 
                             
                           
                         
                         = 
                         
                           
                             
                               - 
                               
                                 Sin 
                                  
                                 
                                   [ 
                                   
                                     
                                       
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                                         p 
                                       
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                                           2 
                                         
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                                   ] 
                                 
                               
                             
                              
                             
                               { 
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           
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                                                   ς 
                                                 
                                                 
                                                   2 
                                                    
                                                   DL 
                                                 
                                               
                                               ) 
                                             
                                           
                                           [ 
                                           
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                                                       2 
                                                     
                                                     2 
                                                   
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                                                     [ 
                                                     
                                                       
                                                         
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                                                           ( 
                                                           
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                                         Tri 
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                                               2 
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                                                   p 
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                                                 ] 
                                               
                                             
                                             2 
                                           
                                         
                                       
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                           - 
                           
                             
                               Sin 
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                                 [ 
                                 
                                   
                                     
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                                 ] 
                               
                             
                              
                             
                               { 
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           
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                                               ( 
                                               
                                                 
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                                                   2 
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                                           [ 
                                           
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                                                       p 
                                                       2 
                                                     
                                                     2 
                                                   
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                                                     [ 
                                                     
                                                       
                                                         
                                                           Λ 
                                                           D 
                                                         
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                                                           ( 
                                                           
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                                                             2 
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                                                             2 
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                                                         2 
                                                       
                                                     
                                                     ] 
                                                   
                                                 
                                                 2 
                                               
                                             
                                           
                                           ] 
                                         
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         Tri 
                                          
                                         
                                           ( 
                                           
                                             
                                               ϑ 
                                               - 
                                               ς 
                                             
                                             
                                               2 
                                                
                                               DL 
                                             
                                           
                                           ) 
                                         
                                       
                                       [ 
                                       
                                          
                                         
                                           - 
                                           
                                             
                                               
                                                 
                                                   σ 
                                                   p 
                                                   2 
                                                 
                                                 2 
                                               
                                                
                                               
                                                 [ 
                                                 
                                                   
                                                     
                                                       Λ 
                                                       D 
                                                     
                                                      
                                                     
                                                       ( 
                                                       
                                                         ϑ 
                                                         + 
                                                         ς 
                                                       
                                                       ) 
                                                     
                                                   
                                                   + 
                                                   
                                                     
                                                       
                                                         2 
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                                                       + 
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                                                       - 
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                                                       - 
                                                       
                                                         2 
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                                                         τ 
                                                       
                                                     
                                                     2 
                                                   
                                                 
                                                 ] 
                                               
                                             
                                             2 
                                           
                                         
                                       
                                       ] 
                                     
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0129]    With reference to equation (19),  FIG. 13  shows an envelope of probability P 1e1o  as a function of τ, obtained for θ=0 and ζ=0. Moreover,  FIG. 14 , also obtained for θ=0 and ζ=0, shows, as a function of τ, the probabilities P 1e1o  and P 2e2o . In practice, the output state from the second interferometer beam splitter  98  can be expressed by formula (17) for the second embodiment as well. 
         [0130]    The advantages that can be obtained with the present interferometer clearly emerge from the foregoing description. In particular, the present interferometer enables controlling the coalescence of a pair of photons, and in particular the probabilities with which a pair of coalescent photons is emitted from the first or from the second output of the second interferometer beam splitter  98 . In other words, the present interferometer functions as an electronically controllable two-photon switch. 
         [0131]    Finally, it is clear that changes and modifications can be made to the described interferometer without leaving the scope of protection of the present invention.