# Patent

## Title
Interferometer and method for controlling the coalescence of a pair of photons

## 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)}(|21,02−|01,22)). 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)}·(|21,02+|01,22)), 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)}·(|21,02+|01,22)+eiτ·1/√{square root over (2)}·(|21,02−|01,22)), the weights of the sum being a function of the delay.

## Background
CROSS-REFERENCE TO RELATED APPLICATIONS
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
The present invention relates to an interferometer and method for controlling the coalescence of a pair of photons.
BACKGROUND
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.
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.
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.
In greater detail, the optical source 2 is a coherent type of source, such as a laser source for example.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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).
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.
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:
a) both the first and second converted photons pass through the optical beam splitter 14;
b) both the first and the second converted photon are reflected by the optical beam splitter 14;
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
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.
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.
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).
In general, the described phenomenon is usually referred to as coalescence, or that the first and second converted photons are coalescent.
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.
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.
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.
By using the so-called Dirac notation, the input state to the optical beam splitter 14 is normally indicated |11|12, 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)}(|21,02−|01,22), from which it can be inferred that there is only one antisymmetric state with two coalescent photons in output from the HOM interferometer 1.
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.
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.
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.
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.
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.
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 |e1|o2 and state |o1|e2. 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 |e1|o2) 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 |o1|e2). 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.
In practice, the input state to the optical beam splitter 14 is still coherent and can be expressed as:




  
    
      
        
          
            
              
                
                e
                〉
              
              1
            
            ⁢
            
              
                
                o
                〉
              
              2
            
          
          +
          
            
              e
              
                i
                ⁢
                
                  
                
                ⁢
                φ
              
            
            ⁢
            
              
                
                o
                〉
              
              1
            
            ⁢
            
              
                
                e
                〉
              
              2
            
          
        
        
          2
        
      
    
    
      
        (
        1
        )
      
    
  




where φ is a function of the phase shift introduced by the polarization phase-shifter 8.

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.
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.
Conversely, in the case where φ=π, the following is obtained:




  
    
      
        
          
            
              
                
                e
                〉
              
              1
            
            ⁢
            
              
                
                o
                〉
              
              2
            
          
          -
          
            
              
                
                o
                〉
              
              1
            
            ⁢
            
              
                
                e
                〉
              
              2
            
          
        
        
          2
        
      
    
    
      
        (
        2
        )
      
    
  




namely an antisymmetric state is obtained, also known as a singlet state. In particular, relation (2) can also be expressed as:





  
    
      
        
          
            
              
              
                1
                
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  1
                
              
              〉
            
            ⁢
            
              
              
                1
                
                  2
                  ⁢
                  o
                
              
              〉
            
          
          -
          
            
              
              
                1
                
                  1
                  ⁢
                  o
                
              
              〉
            
            ⁢
            
              
              
                1
                
                  2
                  ⁢
                  e
                
              
              〉
            
          
        
        
          2
        
      
    
    
      
        (
        3
        )
      
    
  




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.

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).
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.
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.
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.
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.
    
    


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:




  
    
      
        
          
          ψ
          〉
        
        =
        
          
            C
            
              2
            
          
          ⁢
          
            
              ∫
              
                -
                L
              
              0
            
            ⁢
            
              
                d
                z
              
              ⁢
              
                
                  ∫
                  0
                  
                    +
                    ∞
                  
                
                ⁢
                
                  
                    d
                    
                      v
                      p
                    
                  
                  ⁢
                  
                    
                      E
                      p
                      
                        (
                        +
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        v
                        p
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        v
                        p
                      
                      ⁢
                      Λ
                      ⁢
                      
                        
                      
                      ⁢
                      z
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      
                        +
                        ∞
                      
                    
                    ⁢
                    
                      
                        d
                        v
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          Dvz
                        
                      
                      ×
                      ×
                      
                        [
                        
                          
                            
                              
                                
                                  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
                                  
                                
                                )
                              
                            
                          
                        
                        ]
                      
                      ⁢
                      
                        
                        0
                        〉
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




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, Ep(+)(υp) is the spectral distribution of the pump, namely of the electromagnetic radiation emitted by the optical source 2, Ωp is the central pump frequency and L is the length of the crystal 4, measured along the propagation direction of the pump. Moreover, {circumflex over (α)}1e† and {circumflex over (α)}2e† are the creation operators related to the extraordinary photon and to the first and second optical paths 20, 22, respectively; {circumflex over (α)}1o† and {circumflex over (α)}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:





  
    
      
        
          Λ
          =
          
            
              1
              
                u
                p
              
            
            -
            
              
                1
                2
              
              ⁢
              
                (
                
                  
                    1
                    
                      u
                      e
                    
                  
                  +
                  
                    1
                    
                      u
                      o
                    
                  
                
                )
              
            
          
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        (
        5
        )
      
    
  
  
    
      
        
          D
          =
          
            (
            
              
                1
                
                  u
                  
                    e
                    ⁢
                    
                      
                    
                  
                
              
              -
              
                1
                
                  u
                  o
                
              
            
            )
          
        
        ,
      
    
    
      
        (
        6
        )
      
    
  




where up, ue and uo are the group velocities in the crystal 4 of the pump, the extraordinary photon and the ordinary photon, respectively.

In practice, equation (4) permits revealing observance of the energy conservation condition in the generation process of the extraordinary photon and the ordinary photon.
The output state from the optical beam splitter 14 can thus be expressed as:




  
    
      
        
          
          ψ
          〉
        
        =
        
          
            
              C
              ⁢
              
                
              
              ⁢
              
                e
                
                  
                    -
                    
                      i
                      ⁡
                      
                        (
                        
                          
                            Ω
                            p
                          
                          2
                        
                        )
                      
                    
                  
                  ⁢
                  ζ
                
              
            
            
              2
              ⁢
              
                2
              
            
          
          ⁢
          
            
              ∫
              
                -
                L
              
              0
            
            ⁢
            
              
                d
                z
              
              ⁢
              
                
                  ∫
                  0
                  
                    +
                    ∞
                  
                
                ⁢
                
                  
                    d
                    
                      v
                      p
                    
                  
                  ⁢
                  
                    
                      E
                      p
                      
                        (
                        +
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        v
                        p
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          v
                          p
                        
                        ⁡
                        
                          (
                          
                            
                              Λ
                              ⁢
                              
                                
                              
                              ⁢
                              z
                            
                            -
                            
                              ζ
                              2
                            
                          
                          )
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      
                        +
                        ∞
                      
                    
                    ⁢
                    
                      
                        d
                        v
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          D
                          ⁢
                          
                            
                          
                          ⁢
                          vz
                        
                      
                      ×
                      
                        {
                        
                          
                            
                              
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      α
                                                      ^
                                                    
                                                    
                                                      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
                                                      
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                    ]
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        e
                                        
                                          i
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          v
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          ζ
                                        
                                      
                                      ±
                                      
                                        e
                                        
                                          
                                            -
                                            i
                                          
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          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
                                                    
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  ]
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      e
                                      
                                        i
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        ζ
                                      
                                    
                                    ∓
                                    
                                      e
                                      
                                        
                                          -
                                          i
                                        
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        ζ
                                      
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                      ⁢
                      
                        
                        0
                        〉
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




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.

The above-mentioned (normalized) probabilities 1e1o, 1e2o, 2e1o and 2e2o, hereinafter respectively indicated as HOM1e1o, HOM1e2o, HOM2e1o and HOM2e2o, can thus be expressed as:




  
    
      
        
          
            
              
                HOM
                
                  1
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  o
                
              
              ⁡
              
                (
                ς
                )
              
            
            =
            
              
                1
                4
              
              [
              
                1
                ±
                
                  
                    Tri
                    ⁡
                    
                      (
                      
                        ς
                        DL
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        σ
                        p
                        2
                      
                      ⁢
                      
                        
                          Λ
                          2
                        
                        
                          D
                          
                            2
                            ⁢
                            
                              
                            
                          
                        
                      
                      ⁢
                      
                        ς
                        2
                      
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              
                HOM
                
                  1
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  o
                
              
              ⁡
              
                (
                ς
                )
              
            
            =
            
              
                1
                4
              
              [
              
                1
                ∓
                
                  
                    Tri
                    ⁡
                    
                      (
                      
                        ς
                        DL
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        σ
                        p
                        2
                      
                      ⁢
                      
                        
                          Λ
                          2
                        
                        
                          D
                          
                            2
                            ⁢
                            
                              
                            
                          
                        
                      
                      ⁢
                      
                        ς
                        2
                      
                    
                  
                
              
              ]
            
          
          ,
          
            

          
          ⁢
          
            
              
                HOM
                
                  2
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  o
                
              
              ⁡
              
                (
                ς
                )
              
            
            =
            
              
                1
                4
              
              [
              
                1
                ∓
                
                  
                    Tri
                    ⁡
                    
                      (
                      
                        ς
                        DL
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        σ
                        p
                        2
                      
                      ⁢
                      
                        
                          Λ
                          2
                        
                        
                          D
                          2
                        
                      
                      ⁢
                      
                        ς
                        2
                      
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                HOM
                
                  2
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  o
                
              
              ⁡
              
                (
                ς
                )
              
            
            =
            
              
                1
                4
              
              [
              
                1
                ±
                
                  
                    Tri
                    ⁡
                    
                      (
                      
                        ς
                        DL
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        σ
                        p
                        2
                      
                      ⁢
                      
                        
                          Λ
                          2
                        
                        
                          D
                          2
                        
                      
                      ⁢
                      
                        ς
                        2
                      
                    
                  
                
              
              ]
            
          
          ,
        
      
    
    
      
        (
        8
        )
      
    
  




where the following holds:





  
    
      
        
          Tri
          ⁡
          
            (
            x
            )
          
        
        =
        
          {
          
            
              
                0
              
              
                
                  
                    for
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                      x
                      
                    
                  
                  >
                  
                    1
                    2
                  
                
              
            
            
              
                
                  (
                  
                    1
                    -
                    
                      2
                      ⁢
                      
                        
                        x
                        
                      
                    
                  
                  )
                
              
              
                
                  
                    for
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                      x
                      
                    
                  
                  ≤
                  
                    1
                    2
                  
                
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




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) HOM1e1o and HOM2e2o 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.
As shown in FIG. 6, in the case where the input state is antisymmetric, it is found that the probabilities HOM1e1o and HOM2e2o 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.
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.
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
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.
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)}(|21,02−|01,22)), 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)}·(|21,02+|01,22)). 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)}·(|21,02+|01,22)+eiτ·1/√{square root over (2)}·(|21,02−|01,22)), the weights of the sum being a function of the delay.



BRIEF DESCRIPTION OF THE DRAWINGS
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:
FIGS. 1 and 4 show block diagrams of a known type of interferometer;
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;
FIG. 3 shows a cross-section of the two emission cones shown in FIG. 2, taken along a section line shown in FIG. 2;
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;
FIG. 7 shows a block diagram of a first embodiment of the interferometer according to the present invention;
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;
FIG. 9 shows probability density functions related to the outputs of the interferometric subsystem;
FIG. 10 shows the envelope profile of a probability density function related to the outputs of the interferometer shown in FIG. 7;
FIG. 11 shows probability density functions related to the outputs of the interferometer shown in FIG. 7;
FIG. 12 shows a block diagram of a second embodiment of the interferometer according to the present invention;
FIG. 13 shows the envelope profile of a probability density function related to the outputs of the interferometer shown in FIG. 12; and
FIG. 14 shows probability density functions related to the outputs of the interferometer shown in FIG. 12.



DETAILED DESCRIPTION
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Independently of the constructional details, the pump delay line 89 conveys the second crystal pulses to the second crystal 84.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
From a quantum viewpoint, the output state from the first interferometer beam splitter 64 can be expressed as:




  
    
      
        
          
          ψ
          〉
        
        =
        
          
            
              C
              ⁢
              
                
              
              ⁢
              
                e
                
                  
                    -
                    
                      i
                      ⁡
                      
                        (
                        
                          
                            Ω
                            p
                          
                          2
                        
                        )
                      
                    
                  
                  ⁢
                  θ
                
              
            
            
              2
              ⁢
              
                2
              
            
          
          ⁢
          
            
              ∫
              
                -
                L
              
              0
            
            ⁢
            
              
                d
                z
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  
                    +
                    ∞
                  
                
                ⁢
                
                  
                    d
                    
                      v
                      p
                    
                  
                  ⁢
                  
                    
                      E
                      p
                      
                        (
                        +
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        v
                        p
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          v
                          p
                        
                        ⁡
                        
                          (
                          
                            
                              Λ
                              ⁢
                              
                                
                              
                              ⁢
                              z
                            
                            ⁢
                            
                              
                            
                            -
                            
                              θ
                              2
                            
                          
                          )
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      
                        +
                        ∞
                      
                    
                    ⁢
                    
                      
                        d
                        v
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          Dvz
                        
                      
                      ×
                      
                        {
                        
                          
                            
                              
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    
                                                      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
                                                      
                                                    
                                                    )
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                    ]
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        e
                                        
                                          i
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          v
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          θ
                                        
                                      
                                      +
                                      
                                        e
                                        
                                          
                                            -
                                            i
                                          
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          v
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          θ
                                        
                                      
                                    
                                    )
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  [
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    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
                                                    
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  ]
                                
                                ⁢
                                
                                  (
                                  
                                    
                                      e
                                      
                                        i
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                    
                                    -
                                    
                                      e
                                      
                                        
                                          -
                                          i
                                        
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                        }
                      
                      ⁢
                      
                        
                        0
                        〉
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




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.

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:




  
    
      
        
          
          ψ
          〉
        
        =
        
          
            
              CL
              ⁢
              
                
              
              ⁢
              
                e
                
                  
                    -
                    i
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      Ω
                      p
                    
                    2
                  
                  ⁢
                  
                    (
                    
                      θ
                      +
                      ς
                    
                    )
                  
                
              
            
            
              4
              ⁢
              
                2
              
            
          
          ⁢
          
            
              ∫
              
                -
                L
              
              0
            
            ⁢
            
              
                d
                z
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  
                    +
                    ∞
                  
                
                ⁢
                
                  
                    d
                    
                      v
                      p
                    
                  
                  ⁢
                  
                    
                      E
                      p
                      
                        (
                        +
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        v
                        p
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          v
                          p
                        
                        ⁡
                        
                          [
                          
                            Λ
                            ⁢
                            
                              
                            
                            ⁢
                            z
                            ⁢
                            
                              
                            
                            ⁢
                            
                              
                                θ
                                +
                                ς
                              
                              2
                            
                          
                          ]
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      
                        +
                        ∞
                      
                    
                    ⁢
                    
                      
                        d
                        v
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          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
                                                    
                                                  
                                                  )
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                  ]
                                
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    
                                      e
                                      
                                        i
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                    
                                    +
                                    
                                      e
                                      
                                        
                                          -
                                          i
                                        
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                    
                                  
                                  )
                                
                                ⁢
                                
                                   
                                  
                                    
                                      [
                                      
                                        
                                          
                                            
                                              
                                                e
                                                
                                                  
                                                    i
                                                    ⁡
                                                    
                                                      (
                                                      
                                                        
                                                          
                                                            v
                                                            p
                                                          
                                                          +
                                                          
                                                            Ω
                                                            p
                                                          
                                                        
                                                        2
                                                      
                                                      )
                                                    
                                                  
                                                  ⁢
                                                  ς
                                                
                                              
                                              +
                                            
                                          
                                        
                                        
                                          
                                            
                                              e
                                              
                                                
                                                  -
                                                  i
                                                
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    
                                                      
                                                        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
                                                  
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                            
                          
                          
                            
                              
                                (
                                
                                  
                                    e
                                    
                                      i
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      v
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  -
                                  
                                    e
                                    
                                      
                                        -
                                        i
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      v
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                
                                )
                              
                            
                          
                          
                            
                              
                                 
                                
                                  
                                    [
                                    
                                      
                                        e
                                        
                                          i
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          v
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          ς
                                        
                                      
                                      +
                                      
                                        e
                                        
                                          
                                            -
                                            i
                                          
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          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
                                                  
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                            
                          
                          
                            
                              
                                (
                                
                                  
                                    e
                                    
                                      i
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      v
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                  +
                                  
                                    e
                                    
                                      
                                        -
                                        i
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      v
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      θ
                                    
                                  
                                
                                )
                              
                            
                          
                          
                            
                              
                                 
                                
                                   
                                  
                                    
                                      [
                                      
                                        
                                          e
                                          
                                            i
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              (
                                              
                                                
                                                  
                                                    v
                                                    p
                                                  
                                                  +
                                                  
                                                    Ω
                                                    p
                                                  
                                                
                                                2
                                              
                                              )
                                            
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            ς
                                          
                                        
                                        -
                                        
                                          e
                                          
                                            
                                              -
                                              
                                                i
                                                ⁡
                                                
                                                  (
                                                  
                                                    
                                                      
                                                        
                                                          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
                                            
                                          
                                          )
                                        
                                      
                                    
                                  
                                
                                ]
                              
                            
                          
                          
                            
                              
                                
                                  (
                                  
                                    
                                      e
                                      
                                        i
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                    
                                    -
                                    
                                      e
                                      
                                        
                                          -
                                          i
                                        
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        θ
                                      
                                    
                                  
                                  )
                                
                                ⁡
                                
                                  [
                                  
                                    
                                      e
                                      
                                        i
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        ς
                                      
                                    
                                    -
                                    
                                      e
                                      
                                        
                                          -
                                          i
                                        
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        v
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        ς
                                      
                                    
                                  
                                  ]
                                
                              
                            
                          
                        
                        ]
                      
                      ⁢
                      
                        
                        0
                        〉
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




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.

In the case where θ=0 and ζ=0, equation (11) becomes:




  
    
      
        
          
          ψ
          〉
        
        =
        
          
            -
            
              
                CL
                ⁢
                
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      i
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        Ω
                        p
                      
                      2
                    
                    ⁢
                    
                      (
                      
                        θ
                        +
                        ς
                      
                      )
                    
                  
                
              
              
                4
                ⁢
                
                  2
                
              
            
          
          ⁢
          
            
              ∫
              
                -
                L
              
              0
            
            ⁢
            
              
                d
                z
              
              ⁢
              
                
              
              ⁢
              
                
                  ∫
                  
                    -
                    ∞
                  
                  
                    +
                    ∞
                  
                
                ⁢
                
                  
                    d
                    
                      v
                      p
                    
                  
                  ⁢
                  
                    
                      E
                      p
                      
                        (
                        +
                        )
                      
                    
                    ⁡
                    
                      (
                      
                        v
                        p
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      i
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          v
                          p
                        
                        ⁡
                        
                          [
                          
                            
                              Λ
                              ⁢
                              
                                
                              
                              ⁢
                              z
                            
                            -
                            
                              
                                θ
                                +
                                ς
                              
                              2
                            
                          
                          ]
                        
                      
                    
                  
                  ⁢
                  
                    
                      ∫
                      
                        -
                        ∞
                      
                      
                        +
                        ∞
                      
                    
                    ⁢
                    
                      
                        d
                        v
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            i
                          
                          ⁢
                          
                            
                          
                          ⁢
                          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
        )
      
    
  




In a manner in itself known, it is also possible to determine the (normalized) probabilities 1e1o, 1e2o, 2e1o and 2e2o, henceforth respectively indicated as MZ1e1o, MZ1e2o, MZ2e1o and MZ2e2o, which can be expressed as:




  
    
      
        
          
            MZ
            
              1
              ⁢
              e
              ⁢
              
                
              
              ⁢
              1
              ⁢
              o
            
          
          ⁡
          
            (
            
              θ
              ,
              ς
            
            )
          
        
        =
        
          
            1
            4
          
          ⁢
          
            {
            
              
                
                  
                    1
                    +
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        Cos
                        ⁡
                        
                          (
                          
                            
                              Ω
                              p
                            
                            ⁢
                            ζ
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              
                                σ
                                p
                                2
                              
                              2
                            
                          
                          ⁢
                          
                            
                              (
                              ζ
                              )
                            
                            2
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        Cos
                        ⁡
                        
                          (
                          
                            
                              Ω
                              p
                            
                            ⁢
                            ζ
                          
                          )
                        
                      
                      ⁢
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              θ
                              DL
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            σ
                                            p
                                            2
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        
                                          (
                                          
                                            
                                              2
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              
                                                
                                                  Λ
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  θ
                                                
                                                D
                                              
                                            
                                            -
                                            ζ
                                          
                                          )
                                        
                                        2
                                      
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                      
                                      ⁢
                                      
                                        
                                          σ
                                          p
                                          2
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        (
                                        
                                          
                                            2
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              
                                                Λ
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                θ
                                              
                                              D
                                            
                                          
                                          +
                                          ζ
                                        
                                        )
                                      
                                      2
                                    
                                  
                                
                              
                            
                          
                          ]
                        
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        Tri
                        ⁡
                        
                          (
                          
                            
                              ζ
                              +
                              θ
                            
                            DL
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              
                                σ
                                p
                                2
                              
                              2
                            
                          
                          ⁢
                          
                            
                              (
                              
                                2
                                ⁢
                                
                                  Λ
                                  D
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      ζ
                                      +
                                      θ
                                    
                                    )
                                  
                                  2
                                
                              
                              )
                            
                            2
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        Tri
                        ⁡
                        
                          (
                          
                            ζ
                            DL
                          
                          )
                        
                      
                      ⁢
                      
                        
                          e
                          
                            
                              -
                              
                                
                                  σ
                                  p
                                  2
                                
                                2
                              
                            
                            ⁢
                            
                              (
                              
                                2
                                ⁢
                                
                                  
                                
                                ⁢
                                
                                  Λ
                                  D
                                
                                ⁢
                                ζ
                              
                              )
                            
                          
                        
                        2
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      Tri
                      ⁡
                      
                        (
                        
                          
                            ζ
                            -
                            θ
                          
                          DL
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          
                            
                          
                          ⁢
                          
                            
                              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
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          e
                                          
                                            
                                              -
                                              
                                                
                                                  σ
                                                  p
                                                  2
                                                
                                                2
                                              
                                            
                                            ⁢
                                            
                                              
                                                (
                                                ζ
                                                )
                                              
                                              2
                                            
                                          
                                        
                                      
                                      +
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          1
                                          4
                                        
                                        ⁢
                                        
                                          Cos
                                          ⁡
                                          
                                            (
                                            
                                              
                                                Ω
                                                p
                                              
                                              ⁢
                                              ζ
                                            
                                            )
                                          
                                        
                                        ⁢
                                        
                                          
                                            Tri
                                            ⁡
                                            
                                              (
                                              
                                                θ
                                                DL
                                              
                                              )
                                            
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  
                                                    
                                                      e
                                                      
                                                        
                                                          -
                                                          
                                                            
                                                          
                                                          ⁢
                                                          
                                                            
                                                              σ
                                                              p
                                                              2
                                                            
                                                            2
                                                          
                                                        
                                                        ⁢
                                                        
                                                          
                                                            (
                                                            
                                                              
                                                                2
                                                                ⁢
                                                                
                                                                  
                                                                
                                                                ⁢
                                                                
                                                                  Λθ
                                                                  D
                                                                
                                                              
                                                              -
                                                              ζ
                                                            
                                                            )
                                                          
                                                          2
                                                        
                                                      
                                                    
                                                    +
                                                  
                                                
                                              
                                              
                                                
                                                  
                                                    e
                                                    
                                                      
                                                        -
                                                        
                                                          
                                                            σ
                                                            p
                                                            2
                                                          
                                                          2
                                                        
                                                      
                                                      ⁢
                                                      
                                                        
                                                          (
                                                          
                                                            
                                                              2
                                                              ⁢
                                                              
                                                                
                                                              
                                                              ⁢
                                                              
                                                                
                                                                  Λ
                                                                  ⁢
                                                                  
                                                                    
                                                                  
                                                                  ⁢
                                                                  θ
                                                                
                                                                D
                                                              
                                                            
                                                            +
                                                            ζ
                                                          
                                                          )
                                                        
                                                        2
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                            ]
                                          
                                        
                                      
                                      -
                                    
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    1
                                    4
                                  
                                  ⁢
                                  
                                    Tri
                                    ⁡
                                    
                                      (
                                      
                                        
                                          ζ
                                          +
                                          θ
                                        
                                        DL
                                      
                                      )
                                    
                                  
                                  ⁢
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            σ
                                            p
                                            2
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        
                                          (
                                          
                                            2
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              Λ
                                              D
                                            
                                            ⁢
                                            
                                              (
                                              
                                                ζ
                                                +
                                                θ
                                              
                                              )
                                            
                                          
                                          )
                                        
                                        2
                                      
                                    
                                  
                                
                                +
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              1
                              2
                            
                            ⁢
                            
                              Tri
                              ⁡
                              
                                (
                                
                                  ζ
                                  DL
                                
                                )
                              
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      σ
                                      p
                                      2
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  
                                    (
                                    
                                      2
                                      ⁢
                                      
                                        Λ
                                        D
                                      
                                      ⁢
                                      ζ
                                    
                                    )
                                  
                                  2
                                
                              
                            
                          
                          -
                        
                      
                    
                  
                
              
              
                
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      Tri
                      ⁡
                      
                        (
                        
                          
                            ζ
                            -
                            θ
                          
                          DL
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          
                            
                              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
                                
                                )
                              
                            
                            ⁢
                            
                              e
                              
                                
                                  -
                                  
                                    
                                      σ
                                      p
                                      2
                                    
                                    2
                                  
                                
                                ⁢
                                
                                  
                                    (
                                    ζ
                                    )
                                  
                                  2
                                
                              
                            
                          
                          -
                        
                      
                    
                    
                      
                        
                          
                            
                              1
                              4
                            
                            ⁢
                            
                              Cos
                              ⁡
                              
                                (
                                
                                  
                                    Ω
                                    p
                                  
                                  ⁢
                                  ζ
                                
                                )
                              
                            
                            ⁢
                            
                              
                                Tri
                                ⁡
                                
                                  (
                                  
                                    θ
                                    DL
                                  
                                  )
                                
                              
                              ⁡
                              
                                [
                                
                                  
                                    
                                      
                                        
                                          e
                                          
                                            
                                              -
                                              
                                                
                                                  σ
                                                  p
                                                  2
                                                
                                                2
                                              
                                            
                                            ⁢
                                            
                                              
                                                (
                                                
                                                  
                                                    2
                                                    ⁢
                                                    
                                                      Λθ
                                                      D
                                                    
                                                  
                                                  -
                                                  ζ
                                                
                                                )
                                              
                                              2
                                            
                                          
                                        
                                        +
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          e
                                          
                                            
                                              -
                                              
                                                
                                                  σ
                                                  p
                                                  2
                                                
                                                2
                                              
                                            
                                            ⁢
                                            
                                              
                                                (
                                                
                                                  
                                                    2
                                                    ⁢
                                                    
                                                      
                                                    
                                                    ⁢
                                                    
                                                      Λθ
                                                      D
                                                    
                                                  
                                                  +
                                                  ζ
                                                
                                                )
                                              
                                              2
                                            
                                          
                                        
                                        ⁢
                                        
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                            
                          
                          +
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        Tri
                        ⁡
                        
                          (
                          
                            
                              ζ
                              +
                              θ
                            
                            DL
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              
                                σ
                                p
                                2
                              
                              2
                            
                          
                          ⁢
                          
                            
                              (
                              
                                2
                                ⁢
                                
                                  Λ
                                  D
                                
                                ⁢
                                
                                  (
                                  
                                    ζ
                                    +
                                    θ
                                  
                                  )
                                
                              
                              )
                            
                            2
                          
                        
                      
                    
                    -
                  
                
              
              
                
                  
                    
                      
                        1
                        2
                      
                      ⁢
                      
                        Tri
                        ⁡
                        
                          (
                          
                            ζ
                            DL
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              
                                σ
                                p
                                2
                              
                              2
                            
                          
                          ⁢
                          
                            
                              (
                              
                                2
                                ⁢
                                
                                  Λ
                                  D
                                
                                ⁢
                                ζ
                              
                              )
                            
                            2
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      1
                      4
                    
                    ⁢
                    
                      Tri
                      ⁡
                      
                        (
                        
                          
                            ζ
                            -
                            θ
                          
                          DL
                        
                        )
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          
                            
                              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
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                
                                  σ
                                  p
                                  2
                                
                                2
                              
                            
                            ⁢
                            
                              
                                (
                                ζ
                                )
                              
                              2
                            
                          
                        
                      
                      -
                    
                  
                
                
                  
                    
                      
                        
                          1
                          4
                        
                        ⁢
                        
                          Cos
                          ⁡
                          
                            (
                            
                              
                                Ω
                                p
                              
                              ⁢
                              ζ
                            
                            )
                          
                        
                        ⁢
                        
                          
                            Tri
                            ⁡
                            
                              (
                              
                                θ
                                DL
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      e
                                      
                                        
                                          -
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                        
                                        ⁢
                                        
                                          
                                            (
                                            
                                              
                                                2
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                
                                                  
                                                    Λ
                                                    ⁢
                                                    
                                                      
                                                    
                                                    ⁢
                                                    θ
                                                  
                                                  D
                                                
                                              
                                              -
                                              ζ
                                            
                                            )
                                          
                                          2
                                        
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    e
                                    
                                      
                                        -
                                        
                                          
                                            σ
                                            p
                                            2
                                          
                                          2
                                        
                                      
                                      ⁢
                                      
                                        
                                          (
                                          
                                            
                                              2
                                              ⁢
                                              
                                                
                                              
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                                                D
                                              
                                            
                                            +
                                            ζ
                                          
                                          )
                                        
                                        2
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        
                          1
                          4
                        
                        ⁢
                        
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                          ⁡
                          
                            (
                            
                              
                                ζ
                                +
                                θ
                              
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                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                
                                  σ
                                  p
                                  2
                                
                                2
                              
                            
                            ⁢
                            
                              
                                (
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Λ
                                    D
                                  
                                  ⁢
                                  
                                    (
                                    
                                      ζ
                                      +
                                      θ
                                    
                                    )
                                  
                                
                                )
                              
                              2
                            
                          
                        
                      
                      -
                    
                  
                
                
                  
                    
                      
                        
                          1
                          2
                        
                        ⁢
                        
                          Tri
                          ⁡
                          
                            (
                            
                              ζ
                              DL
                            
                            )
                          
                        
                        ⁢
                        
                          e
                          
                            
                              -
                              
                                
                                  σ
                                  p
                                  2
                                
                                2
                              
                            
                            ⁢
                            
                              
                                (
                                
                                  2
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  
                                    Λθ
                                    D
                                  
                                  ⁢
                                  ζ
                                
                                )
                              
                              2
                            
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        1
                        4
                      
                      ⁢
                      
                        Tri
                        ⁡
                        
                          (
                          
                            
                              ζ
                              -
                              θ
                            
                            DL
                          
                          )
                        
                      
                      ⁢
                      
                        e
                        
                          
                            -
                            
                              
                                σ
                                p
                                2
                              
                              2
                            
                          
                          ⁢
                          
                            
                              (
                              
                                2
                                ⁢
                                
                                  Λ
                                  D
                                
                                ⁢
                                
                                  (
                                  
                                    ζ
                                    -
                                    θ
                                  
                                  )
                                
                              
                              )
                            
                            2
                          
                        
                      
                    
                  
                
              
              }
            
            .
          
        
      
    
    
      
        
      
    
  




With reference to equations (13), FIG. 8 shows an envelope of probability MZ1e1o as a function of ζ, obtained for ζ=0. Moreover, FIG. 9, also obtained for θ=0, shows the probability MZ1e1o (equal to probability MZ2e2o) and the probability MZ1e2o (equal to probability MZ2e1o).

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 MZ1e1o 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 Is, 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)}(|21,02+|01,22).
Upon removing the assumption that the first and second compensation elements 78, 80 and the second crystal 84 are absent, the following occurs.
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 Ia, in a certain sense “cascaded to” the symmetric state interferential device Is. More precisely, the antisymmetric state interferential device Ia shares outputs with the symmetric state interferential device Is, 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 Ia 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)}(|21,02−|01,22).
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:




  
    
      
        
          
          ψ
          〉
        
        =
        
          
            
              
                e
                
                  
                    -
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                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  
                    (
                    
                      
                        
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                          L
                          c
                        
                      
                    
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              ⁢
              CL
              ⁢
              
                
              
              ⁢
              
                e
                
                  i
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
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                      p
                    
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                  ⁢
                  
                    (
                    
                      
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            8
          
          ⁢
          
            
              ∫
              
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            ⁢
            
              
                d
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                  ⁢
                  
                    
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                        (
                        +
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                      (
                      
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                      )
                    
                  
                  ⁢
                  
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                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          v
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                        ⁡
                        
                          [
                          
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                            ⁢
                            
                              
                            
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                            ⁢
                            
                              
                            
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                                    )
                                  
                                
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                                -
                                
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                                +
                                
                                  
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                                      c
                                    
                                  
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                              )
                            
                          
                        
                      
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        (
        14
        )
      
    
  
  
    
      
        
          
        
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          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              C
              ⁢
              
                
              
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                e
                
                  
                    -
                    
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                      ⁡
                      
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            4
          
          ⁢
          
            
              ∫
              
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              0
            
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                  ⁢
                  
                    
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                      p
                      
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                        +
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                      )
                    
                  
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                          v
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                              ζ
                              2
                            
                          
                          )
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              -
              ∞
            
            
              +
              ∞
            
          
          ⁢
          
            
              d
              v
            
            ⁢
            
              
            
            ⁢
            
              e
              
                
                  -
                  i
                
                ⁢
                
                  
                
                ⁢
                Dvz
              
            
            ×
            
              {
              
                
                  
                    
                      
                        
                          
                            [
                            
                              
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                
                                                  
                                                    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
                                        
                                      
                                      )
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                      
                      
                        
                          
                            
                              (
                              
                                
                                  e
                                  
                                    i
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    v
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    ζ
                                  
                                
                                +
                                
                                  e
                                  
                                    
                                      -
                                      i
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    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
                                  
                                
                                )
                              
                            
                          
                        
                      
                      ]
                    
                  
                
                
                  
                    
                      (
                      
                        
                          e
                          
                            i
                            ⁢
                            
                              
                            
                            ⁢
                            v
                            ⁢
                            
                              
                            
                            ⁢
                            ζ
                          
                        
                        -
                        
                          e
                          
                            
                              -
                              i
                            
                            ⁢
                            
                              
                            
                            ⁢
                            v
                            ⁢
                            
                              
                            
                            ⁢
                            ζ
                          
                        
                      
                      )
                    
                  
                
              
              }
            
            ⁢
            
              
              0
              〉
            
          
        
      
    
    
      
        
      
    
  




where, indicating the extraordinary and ordinary refractive indices of the first crystal 54 as ne and no, gives





  
    
      K
      e
    
    =
    
      
        
          n
          e
        
        ⁢
        
          Ω
          p
        
      
      
        2
        ⁢
        c
      
    
  
  ,
  
    
      K
      o
    
    =
    
      
        
          n
          o
        
        ⁢
        
          Ω
          p
        
      
      
        2
        ⁢
        c
      
    
  
  ,
  
    G
    =
    
      (
      
        
          1
          
            u
            e
          
        
        +
        
          1
          
            u
            o
          
        
      
      )
    
  
  ,




and Lc is the length of the first and second compensation elements 78, 80.

It is also possible to determine the (normalized) probabilities 1e1o, 1e2o, 2e1o and 2e2o, henceforth respectively indicated as P1e1o, P1e2o, P2e1o and P2e2o, which can be expressed as:




  
    
      
        
          
            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
        )
      
    
  




where the indices i and j can assume values of “1” or “2”, and where the following hold:





  
    
      
        
          
            
              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
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    
                                                      D
                                                      ⁢
                                                      
                                                        
                                                      
                                                    
                                                  
                                                  ⁢
                                                  
                                                    
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                -
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    +
                                                    ς
                                                    +
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  +
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                -
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            σ
                                            p
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                -
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    +
                                                    ς
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                +
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    -
                                                    ς
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                    ⁢
                                    
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                -
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    -
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                        
                                      
                                      )
                                    
                                  
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          
                                            
                                              e
                                              
                                                -
                                                
                                                  
                                                    
                                                      
                                                        σ
                                                        p
                                                        2
                                                      
                                                      2
                                                    
                                                    ⁡
                                                    
                                                      [
                                                      
                                                        
                                                          
                                                            Λ
                                                            D
                                                          
                                                          ⁢
                                                          
                                                            (
                                                            θ
                                                            )
                                                          
                                                        
                                                        +
                                                        
                                                          
                                                            
                                                              2
                                                              ⁢
                                                              GL
                                                            
                                                            +
                                                            θ
                                                            -
                                                            
                                                              2
                                                              ⁢
                                                              τ
                                                            
                                                          
                                                          2
                                                        
                                                      
                                                      ]
                                                    
                                                  
                                                  2
                                                
                                              
                                            
                                            -
                                          
                                        
                                      
                                      
                                        
                                          
                                            e
                                            
                                              -
                                              
                                                
                                                  
                                                    
                                                      σ
                                                      p
                                                      2
                                                    
                                                    2
                                                  
                                                  ⁡
                                                  
                                                    [
                                                    
                                                      
                                                        
                                                          Λ
                                                          D
                                                        
                                                        ⁢
                                                        
                                                          (
                                                          θ
                                                          )
                                                        
                                                      
                                                      -
                                                      
                                                        
                                                          
                                                            2
                                                            ⁢
                                                            GL
                                                          
                                                          +
                                                          θ
                                                          -
                                                          
                                                            2
                                                            ⁢
                                                            τ
                                                          
                                                        
                                                        2
                                                      
                                                    
                                                    ]
                                                  
                                                
                                                2
                                              
                                            
                                          
                                        
                                      
                                    
                                    ]
                                  
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              
                                
                                  Tri
                                  ⁡
                                  
                                    (
                                    
                                      
                                        θ
                                        -
                                        ς
                                      
                                      
                                        2
                                        ⁢
                                        DL
                                      
                                    
                                    )
                                  
                                
                                ⁡
                                
                                  [
                                  
                                    
                                      
                                        
                                          e
                                          
                                            
                                              -
                                              
                                                
                                                  
                                                    
                                                      σ
                                                      p
                                                      2
                                                    
                                                    2
                                                  
                                                  ⁡
                                                  
                                                    [
                                                    
                                                      
                                                        
                                                          Λ
                                                          D
                                                        
                                                        ⁢
                                                        
                                                          (
                                                          
                                                            θ
                                                            -
                                                            ς
                                                          
                                                          )
                                                        
                                                      
                                                      +
                                                      
                                                        
                                                          
                                                            2
                                                            ⁢
                                                            GL
                                                          
                                                          +
                                                          θ
                                                          -
                                                          
                                                            2
                                                            ⁢
                                                            τ
                                                          
                                                        
                                                        2
                                                      
                                                    
                                                    ]
                                                  
                                                
                                                2
                                              
                                            
                                            -
                                          
                                        
                                      
                                    
                                    
                                      
                                        
                                          e
                                          
                                            -
                                            
                                              
                                                
                                                  
                                                    σ
                                                    p
                                                    2
                                                  
                                                  2
                                                
                                                ⁡
                                                
                                                  [
                                                  
                                                    
                                                      
                                                        Λ
                                                        D
                                                      
                                                      ⁢
                                                      
                                                        (
                                                        
                                                          θ
                                                          -
                                                          ς
                                                        
                                                        )
                                                      
                                                    
                                                    -
                                                    
                                                      
                                                        
                                                          2
                                                          ⁢
                                                          GL
                                                        
                                                        +
                                                        θ
                                                        -
                                                        
                                                          2
                                                          ⁢
                                                          τ
                                                        
                                                      
                                                      2
                                                    
                                                  
                                                  ]
                                                
                                              
                                              2
                                            
                                          
                                        
                                      
                                    
                                  
                                  ]
                                
                              
                              +
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                +
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                -
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    +
                                                    ς
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  +
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                      
                                    
                                    2
                                  
                                
                              
                            
                            ]
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                -
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    -
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  +
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                -
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    -
                                                    ς
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                    +
                                  
                                
                              
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                -
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    -
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  +
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  +
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                  
                                
                                )
                              
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      
                                        e
                                        
                                          -
                                          
                                            
                                              
                                                
                                                  σ
                                                  p
                                                  2
                                                
                                                2
                                              
                                              ⁡
                                              
                                                [
                                                
                                                  
                                                    
                                                      Λ
                                                      D
                                                    
                                                    ⁢
                                                    
                                                      (
                                                      θ
                                                      )
                                                    
                                                  
                                                  +
                                                  
                                                    
                                                      
                                                        2
                                                        ⁢
                                                        GL
                                                      
                                                      +
                                                      θ
                                                      -
                                                      
                                                        2
                                                        ⁢
                                                        τ
                                                      
                                                    
                                                    2
                                                  
                                                
                                                ]
                                              
                                            
                                            2
                                          
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    θ
                                                    )
                                                  
                                                
                                                -
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                  
                                
                              
                              ]
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            Tri
                            ⁡
                            
                              (
                              
                                
                                  θ
                                  -
                                  ς
                                
                                
                                  2
                                  ⁢
                                  DL
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      -
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                +
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    -
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                +
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                -
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    +
                                                    ς
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  +
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                -
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  +
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                
                              
                              )
                            
                          
                          ⁡
                          
                            [
                            
                              
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                +
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    -
                                                    ς
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                -
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    -
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                        
                                      
                                      )
                                    
                                  
                                  ⁡
                                  
                                    [
                                    
                                      
                                        
                                          
                                            
                                              e
                                              
                                                -
                                                
                                                  
                                                    
                                                      
                                                        σ
                                                        p
                                                        2
                                                      
                                                      2
                                                    
                                                    ⁡
                                                    
                                                      [
                                                      
                                                        
                                                          
                                                            Λ
                                                            D
                                                          
                                                          ⁢
                                                          
                                                            (
                                                            θ
                                                            )
                                                          
                                                        
                                                        +
                                                        
                                                          
                                                            
                                                              2
                                                              ⁢
                                                              GL
                                                            
                                                            -
                                                            θ
                                                            -
                                                            
                                                              2
                                                              ⁢
                                                              τ
                                                            
                                                          
                                                          2
                                                        
                                                      
                                                      ]
                                                    
                                                  
                                                  2
                                                
                                              
                                            
                                            -
                                          
                                        
                                      
                                      
                                        
                                          
                                            e
                                            
                                              -
                                              
                                                
                                                  
                                                    
                                                      σ
                                                      p
                                                      2
                                                    
                                                    2
                                                  
                                                  ⁡
                                                  
                                                    [
                                                    
                                                      
                                                        
                                                          Λ
                                                          D
                                                        
                                                        ⁢
                                                        
                                                          (
                                                          θ
                                                          )
                                                        
                                                      
                                                      +
                                                      
                                                        
                                                          
                                                            2
                                                            ⁢
                                                            GL
                                                          
                                                          -
                                                          θ
                                                          -
                                                          
                                                            2
                                                            ⁢
                                                            τ
                                                          
                                                        
                                                        2
                                                      
                                                    
                                                    ]
                                                  
                                                
                                                2
                                              
                                            
                                          
                                        
                                      
                                    
                                    ]
                                  
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                
                                  Tri
                                  ⁡
                                  
                                    (
                                    
                                      
                                        θ
                                        -
                                        ς
                                      
                                      
                                        2
                                        ⁢
                                        DL
                                      
                                    
                                    )
                                  
                                
                                ⁡
                                
                                  [
                                  
                                    
                                      
                                        
                                          
                                            e
                                            
                                              -
                                              
                                                
                                                  
                                                    
                                                      σ
                                                      p
                                                      2
                                                    
                                                    2
                                                  
                                                  ⁡
                                                  
                                                    [
                                                    
                                                      
                                                        
                                                          Λ
                                                          D
                                                        
                                                        ⁢
                                                        
                                                          (
                                                          
                                                            θ
                                                            -
                                                            ς
                                                          
                                                          )
                                                        
                                                      
                                                      -
                                                      
                                                        
                                                          
                                                            2
                                                            ⁢
                                                            GL
                                                          
                                                          +
                                                          θ
                                                          -
                                                          
                                                            2
                                                            ⁢
                                                            τ
                                                          
                                                        
                                                        2
                                                      
                                                    
                                                    ]
                                                  
                                                
                                                2
                                              
                                            
                                          
                                          -
                                        
                                      
                                    
                                    
                                      
                                        
                                          e
                                          
                                            -
                                            
                                              
                                                
                                                  
                                                    σ
                                                    p
                                                    2
                                                  
                                                  2
                                                
                                                ⁡
                                                
                                                  [
                                                  
                                                    
                                                      
                                                        Λ
                                                        D
                                                      
                                                      ⁢
                                                      
                                                        (
                                                        
                                                          θ
                                                          -
                                                          ς
                                                        
                                                        )
                                                      
                                                    
                                                    -
                                                    
                                                      
                                                        
                                                          2
                                                          ⁢
                                                          GL
                                                        
                                                        -
                                                        θ
                                                        -
                                                        
                                                          2
                                                          ⁢
                                                          τ
                                                        
                                                      
                                                      2
                                                    
                                                  
                                                  ]
                                                
                                              
                                              2
                                            
                                          
                                        
                                      
                                    
                                  
                                  ]
                                
                              
                              -
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                +
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    +
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              +
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                  
                                
                                )
                              
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      
                                        e
                                        
                                          -
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              ⁡
                                              
                                                [
                                                
                                                  
                                                    
                                                      Λ
                                                      D
                                                    
                                                    ⁢
                                                    
                                                      (
                                                      
                                                        θ
                                                        +
                                                        ς
                                                      
                                                      )
                                                    
                                                  
                                                  +
                                                  
                                                    
                                                      
                                                        2
                                                        ⁢
                                                        GL
                                                      
                                                      +
                                                      θ
                                                      +
                                                      ς
                                                      -
                                                      
                                                        2
                                                        ⁢
                                                        τ
                                                      
                                                    
                                                    2
                                                  
                                                
                                                ]
                                              
                                            
                                            2
                                          
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                -
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    +
                                                    ς
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                  
                                
                              
                              ]
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                -
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            σ
                                            p
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    -
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  +
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                  
                                
                                )
                              
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      
                                        e
                                        
                                          -
                                          
                                            
                                              
                                                
                                                  σ
                                                  p
                                                  2
                                                
                                                2
                                              
                                              ⁡
                                              
                                                [
                                                
                                                  
                                                    
                                                      Λ
                                                      D
                                                    
                                                    ⁢
                                                    
                                                      (
                                                      
                                                        θ
                                                        +
                                                        ς
                                                      
                                                      )
                                                    
                                                  
                                                  +
                                                  
                                                    
                                                      
                                                        2
                                                        ⁢
                                                        GL
                                                      
                                                      +
                                                      θ
                                                      -
                                                      ς
                                                      -
                                                      
                                                        2
                                                        ⁢
                                                        τ
                                                      
                                                    
                                                    2
                                                  
                                                
                                                ]
                                              
                                            
                                            2
                                          
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
                                      e
                                      
                                        -
                                        
                                          
                                            
                                              
                                                σ
                                                p
                                                2
                                              
                                              2
                                            
                                            ⁡
                                            
                                              [
                                              
                                                
                                                  
                                                    Λ
                                                    D
                                                  
                                                  ⁢
                                                  
                                                    (
                                                    
                                                      θ
                                                      +
                                                      ς
                                                    
                                                    )
                                                  
                                                
                                                -
                                                
                                                  
                                                    
                                                      2
                                                      ⁢
                                                      GL
                                                    
                                                    +
                                                    θ
                                                    -
                                                    ς
                                                    -
                                                    
                                                      2
                                                      ⁢
                                                      τ
                                                    
                                                  
                                                  2
                                                
                                              
                                              ]
                                            
                                          
                                          2
                                        
                                      
                                    
                                  
                                
                              
                              ]
                            
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          Tri
                          ⁡
                          
                            (
                            
                              
                                θ
                                -
                                ς
                              
                              
                                2
                                ⁢
                                DL
                              
                            
                            )
                          
                        
                        ⁡
                        
                          [
                          
                            
                              
                                
                                  
                                    e
                                    
                                      -
                                      
                                        
                                          
                                            
                                              σ
                                              p
                                              2
                                            
                                            2
                                          
                                          ⁡
                                          
                                            [
                                            
                                              
                                                
                                                  Λ
                                                  D
                                                
                                                ⁢
                                                
                                                  (
                                                  
                                                    θ
                                                    -
                                                    ς
                                                  
                                                  )
                                                
                                              
                                              -
                                              
                                                
                                                  
                                                    2
                                                    ⁢
                                                    GL
                                                  
                                                  +
                                                  θ
                                                  -
                                                  ς
                                                  -
                                                  
                                                    2
                                                    ⁢
                                                    τ
                                                  
                                                
                                                2
                                              
                                            
                                            ]
                                          
                                        
                                        2
                                      
                                    
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            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
                                  
                                
                                )
                              
                            
                            ⁡
                            
                              [
                              
                                
                                  
                                    
                                      
                                        e
                                        
                                          -
                                          
                                            
                                              
                                                
                                                  σ
                                                  p
                                                  2
                                                
                                                2
                                              
                                              ⁡
                                              
                                                [
                                                
                                                  
                                                    
                                                      Λ
                                                      D
                                                    
                                                    ⁢
                                                    
                                                      (
                                                      θ
                                                      )
                                                    
                                                  
                                                  +
                                                  
                                                    
                                                      
                                                        2
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                                                      +
                                                      θ
                                                      -
                                                      
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                                                        ⁢
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                                                    2
                                                  
                                                
                                                ]
                                              
                                            
                                            2
                                          
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
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                                          2
                                        
                                      
                                    
                                  
                                
                              
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                        -
                      
                    
                  
                  
                    
                      
                        
                          
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                                  -
                                
                              
                            
                            
                              
                                
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                                                D
                                              
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                                                  +
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                }
              
            
          
        
      
    
    
      
        (
        16
        )
      
    
  




With reference to equation (15), FIG. 10 shows an envelope of probability P1e1o as a function of τ, obtained for θ=0 and ζ=0.

Moreover, FIG. 11, also obtained for θ=0 and ζ=0, shows, as a function of τ, the probability P1e1o, 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 P2e2o, namely the probability of having an extraordinary photon—ordinary photon pair on the second output of the second interferometer beam splitter 98.
In detail, both probability P1e1o and probability P2e2o 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:

1/√{square root over (2)}(|21,02+|01,22)+eiτ·1/√{square root over (2)}(|21,02−|01,22)  (17)

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.

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.
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.
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.
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.
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 64b 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.
In this case, the output state of the second interferometer beam splitter 98 becomes:




  
    
      
        
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                                                            +
                                                            
                                                              e
                                                              
                                                                
                                                                  -
                                                                  i
                                                                
                                                                ⁢
                                                                
                                                                  
                                                                
                                                                ⁢
                                                                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
                                                                            
                                                                          
                                                                          )
                                                                        
                                                                      
                                                                    
                                                                  
                                                                
                                                              
                                                            
                                                          
                                                          ]
                                                        
                                                        ⁢
                                                        
                                                           
                                                          
                                                            (
                                                            
                                                              
                                                                e
                                                                
                                                                  i
                                                                  ⁢
                                                                  
                                                                    
                                                                  
                                                                  ⁢
                                                                  v
                                                                  ⁢
                                                                  
                                                                    
                                                                  
                                                                  ⁢
                                                                  ζ
                                                                
                                                              
                                                              +
                                                              
                                                                e
                                                                
                                                                  
                                                                    -
                                                                    i
                                                                  
                                                                  ⁢
                                                                  
                                                                    
                                                                  
                                                                  ⁢
                                                                  v
                                                                  ⁢
                                                                  
                                                                    
                                                                  
                                                                  ⁢
                                                                  ζ
                                                                
                                                              
                                                            
                                                            )
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                              ⁢
                                              
                                                
                                              
                                              }
                                            
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            
                                              
                                              
                                                
                                              
                                              ⁢
                                              0
                                              ⁢
                                              
                                                
                                              
                                              〉
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        18
        )
      
    
  




The probabilities P1e1o, P1e2o, P2e1o and P2e2o can therefore be expressed as:




  
    
      
        
          
            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
        )
      
    
  




where the indices i and j can assume values of “1” or “2”, and where the following hold:





  
    
      
        
          
        
        ⁢
        
          
            
              
                HOM
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  o
                
              
              ⁡
              
                (
                ς
                )
              
            
            =
            
              [
              
                1
                +
                
                  
                    Tri
                    ⁡
                    
                      (
                      
                        ς
                        DL
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        σ
                        p
                        2
                      
                      ⁢
                      
                        
                          Λ
                          2
                        
                        
                          D
                          2
                        
                      
                      ⁢
                      
                        ς
                        2
                      
                    
                  
                
              
              ]
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                HOM
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  o
                
              
              ⁡
              
                (
                ς
                )
              
            
            =
            
              [
              
                1
                +
                
                  
                    Tri
                    ⁡
                    
                      (
                      
                        ς
                        DL
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        σ
                        p
                        2
                      
                      ⁢
                      
                        
                          Λ
                          2
                        
                        
                          D
                          2
                        
                      
                      ⁢
                      
                        ς
                        2
                      
                    
                  
                
              
              ]
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                HOM
                
                  1
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  o
                
              
              ⁡
              
                (
                ς
                )
              
            
            =
            
              [
              
                1
                -
                
                  
                    Tri
                    ⁡
                    
                      (
                      
                        ς
                        DL
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        σ
                        p
                        2
                      
                      ⁢
                      
                        
                          Λ
                          2
                        
                        
                          D
                          2
                        
                      
                      ⁢
                      
                        ς
                        2
                      
                    
                  
                
              
              ]
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                HOM
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  o
                
              
              ⁡
              
                (
                ς
                )
              
            
            =
            
              [
              
                1
                -
                
                  
                    Tri
                    ⁡
                    
                      (
                      
                        ς
                        DL
                      
                      )
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        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
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      
                        
                          σ
                          p
                          2
                        
                        2
                      
                    
                    ⁢
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Λθ
                              D
                            
                          
                          -
                          ζ
                        
                        )
                      
                      2
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                PBMZ
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  o
                
              
              ⁡
              
                (
                
                  θ
                  ,
                  ς
                
                )
              
            
            =
            
              1
              +
              
                
                  Cos
                  ⁡
                  
                    (
                    
                      
                        Ω
                        p
                      
                      ⁢
                      ζ
                    
                    )
                  
                
                ⁢
                
                  Tri
                  ⁡
                  
                    (
                    
                      θ
                      DL
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      
                        
                          σ
                          p
                          2
                        
                        2
                      
                    
                    ⁢
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                              
                            
                            ⁢
                            
                              Λθ
                              D
                            
                          
                          -
                          ζ
                        
                        )
                      
                      2
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                PBMZ
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  o
                
              
              ⁡
              
                (
                
                  θ
                  ,
                  ς
                
                )
              
            
            =
            
              1
              -
              
                
                  Cos
                  ⁡
                  
                    (
                    
                      
                        Ω
                        p
                      
                      ⁢
                      ζ
                    
                    )
                  
                
                ⁢
                
                  Tri
                  ⁡
                  
                    (
                    
                      θ
                      DL
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      
                        
                          σ
                          p
                          2
                        
                        2
                      
                    
                    ⁢
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                              Λθ
                              D
                            
                          
                          -
                          ζ
                        
                        )
                      
                      2
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                PBMZ
                
                  2
                  ⁢
                  
                    
                  
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  o
                
              
              ⁡
              
                (
                
                  θ
                  ,
                  ς
                
                )
              
            
            =
            
              1
              -
              
                
                  Cos
                  ⁡
                  
                    (
                    
                      
                        Ω
                        p
                      
                      ⁢
                      ζ
                    
                    )
                  
                
                ⁢
                
                  Tri
                  ⁡
                  
                    (
                    
                      θ
                      DL
                    
                    )
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      
                        
                          σ
                          p
                          2
                        
                        2
                      
                    
                    ⁢
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            
                              Λθ
                              D
                            
                          
                          -
                          ζ
                        
                        )
                      
                      2
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                I
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  1
                  ⁢
                  o
                
              
              ⁡
              
                (
                
                  τ
                  ,
                  θ
                  ,
                  ς
                
                )
              
            
            =
            
              
                -
                
                  
                    I
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      e
                      ⁢
                      
                        
                      
                      ⁢
                      2
                      ⁢
                      o
                    
                  
                  ⁡
                  
                    (
                    
                      τ
                      ,
                      θ
                      ,
                      ς
                    
                    )
                  
                
              
              =
              
                
                  
                    -
                    
                      Sin
                      ⁡
                      
                        [
                        
                          
                            
                              Ω
                              p
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    2
                                    ⁢
                                    τ
                                  
                                  -
                                  θ
                                  -
                                  ς
                                
                                2
                              
                              )
                            
                          
                          -
                          
                            2
                            ⁢
                            
                              L
                              ⁡
                              
                                (
                                
                                  
                                    K
                                    e
                                  
                                  +
                                  
                                    K
                                    o
                                  
                                
                                )
                              
                            
                          
                        
                        ]
                      
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            
                              +
                              
                                
                                  Tri
                                  ⁡
                                  
                                    (
                                    
                                      
                                        θ
                                        +
                                        ς
                                      
                                      
                                        2
                                        ⁢
                                        DL
                                      
                                    
                                    )
                                  
                                
                                [
                                
                                  e
                                  
                                    
                                      -
                                      
                                        
                                          σ
                                          p
                                          2
                                        
                                        2
                                      
                                    
                                    ⁢
                                    
                                      
                                        ⌈
                                        
                                          
                                            
                                              Λ
                                              D
                                            
                                            ⁢
                                            
                                              (
                                              
                                                θ
                                                +
                                                ς
                                              
                                              )
                                            
                                          
                                          -
                                          
                                            
                                              
                                                2
                                                ⁢
                                                GL
                                              
                                              +
                                              θ
                                              +
                                              ς
                                              -
                                              
                                                2
                                                ⁢
                                                τ
                                              
                                            
                                            2
                                          
                                        
                                        ⌉
                                      
                                      2
                                    
                                  
                                
                                ]
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              Tri
                              ⁡
                              
                                (
                                
                                  
                                    θ
                                    -
                                    ς
                                  
                                  
                                    2
                                    ⁢
                                    DL
                                  
                                
                                )
                              
                            
                            [
                            
                              e
                              
                                -
                                
                                  
                                    
                                      
                                        σ
                                        p
                                        2
                                      
                                      2
                                    
                                    ⁡
                                    
                                      [
                                      
                                        
                                          
                                            Λ
                                            D
                                          
                                          ⁢
                                          
                                            (
                                            
                                              θ
                                              +
                                              ς
                                            
                                            )
                                          
                                        
                                        +
                                        
                                          
                                            
                                              2
                                              ⁢
                                              GL
                                            
                                            +
                                            θ
                                            -
                                            ς
                                            -
                                            
                                              2
                                              ⁢
                                              τ
                                            
                                          
                                          2
                                        
                                      
                                      ]
                                    
                                  
                                  2
                                
                              
                            
                            ]
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Sin
                    ⁡
                    
                      [
                      
                        
                          
                            Ω
                            p
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  2
                                  ⁢
                                  τ
                                
                                -
                                θ
                                +
                                ς
                              
                              2
                            
                            )
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            L
                            ⁡
                            
                              (
                              
                                
                                  K
                                  e
                                
                                +
                                
                                  K
                                  o
                                
                              
                              )
                            
                          
                        
                      
                      ]
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            
                              +
                              
                                
                                  Tri
                                  ⁡
                                  
                                    (
                                    
                                      
                                        θ
                                        +
                                        ς
                                      
                                      
                                        2
                                        ⁢
                                        DL
                                      
                                    
                                    )
                                  
                                
                                [
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            σ
                                            p
                                            2
                                          
                                          2
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                Λ
                                                D
                                              
                                              ⁢
                                              
                                                (
                                                
                                                  θ
                                                  +
                                                  ς
                                                
                                                )
                                              
                                            
                                            +
                                            
                                              
                                                
                                                  2
                                                  ⁢
                                                  GL
                                                
                                                +
                                                θ
                                                -
                                                ς
                                                -
                                                
                                                  2
                                                  ⁢
                                                  τ
                                                
                                              
                                              2
                                            
                                          
                                          ]
                                        
                                      
                                      2
                                    
                                  
                                
                                ]
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              Tri
                              ⁡
                              
                                (
                                
                                  
                                    θ
                                    -
                                    ς
                                  
                                  
                                    2
                                    ⁢
                                    DL
                                  
                                
                                )
                              
                            
                            [
                            
                              e
                              
                                -
                                
                                  
                                    
                                      
                                        σ
                                        p
                                        2
                                      
                                      2
                                    
                                    ⁡
                                    
                                      [
                                      
                                        
                                          
                                            Λ
                                            D
                                          
                                          ⁢
                                          
                                            (
                                            
                                              θ
                                              +
                                              ς
                                            
                                            )
                                          
                                        
                                        +
                                        
                                          
                                            
                                              2
                                              ⁢
                                              GL
                                            
                                            +
                                            θ
                                            -
                                            ς
                                            -
                                            
                                              2
                                              ⁢
                                              τ
                                            
                                          
                                          2
                                        
                                      
                                      ]
                                    
                                  
                                  2
                                
                              
                            
                            ]
                          
                        
                      
                    
                    }
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              
                I
                
                  1
                  ⁢
                  
                    
                  
                  ⁢
                  e
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  o
                
              
              ⁡
              
                (
                
                  τ
                  ,
                  θ
                  ,
                  ς
                
                )
              
            
            =
            
              
                -
                
                  
                    I
                    
                      2
                      ⁢
                      
                        
                      
                      ⁢
                      e
                      ⁢
                      
                        
                      
                      ⁢
                      1
                      ⁢
                      o
                    
                  
                  ⁡
                  
                    (
                    
                      τ
                      ,
                      θ
                      ,
                      ς
                    
                    )
                  
                
              
              =
              
                
                  
                    -
                    
                      Sin
                      ⁡
                      
                        [
                        
                          
                            
                              Ω
                              p
                            
                            ⁡
                            
                              (
                              
                                
                                  
                                    2
                                    ⁢
                                    τ
                                  
                                  -
                                  θ
                                  -
                                  ς
                                
                                2
                              
                              )
                            
                          
                          -
                          
                            2
                            ⁢
                            
                              L
                              ⁡
                              
                                (
                                
                                  
                                    K
                                    e
                                  
                                  +
                                  
                                    K
                                    o
                                  
                                
                                )
                              
                            
                          
                        
                        ]
                      
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            
                              -
                              
                                
                                  Tri
                                  ⁡
                                  
                                    (
                                    
                                      
                                        θ
                                        +
                                        ς
                                      
                                      
                                        2
                                        ⁢
                                        DL
                                      
                                    
                                    )
                                  
                                
                                [
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            σ
                                            p
                                            2
                                          
                                          2
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                Λ
                                                D
                                              
                                              ⁢
                                              
                                                (
                                                
                                                  θ
                                                  +
                                                  ς
                                                
                                                )
                                              
                                            
                                            -
                                            
                                              
                                                
                                                  2
                                                  ⁢
                                                  GL
                                                
                                                +
                                                θ
                                                +
                                                ς
                                                -
                                                
                                                  2
                                                  ⁢
                                                  τ
                                                
                                              
                                              2
                                            
                                          
                                          ]
                                        
                                      
                                      2
                                    
                                  
                                
                                ]
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              Tri
                              ⁡
                              
                                (
                                
                                  
                                    θ
                                    -
                                    ς
                                  
                                  
                                    2
                                    ⁢
                                    DL
                                  
                                
                                )
                              
                            
                            [
                            
                              e
                              
                                -
                                
                                  
                                    
                                      
                                        σ
                                        p
                                        2
                                      
                                      2
                                    
                                    ⁡
                                    
                                      [
                                      
                                        
                                          
                                            Λ
                                            D
                                          
                                          ⁢
                                          
                                            (
                                            
                                              θ
                                              -
                                              ς
                                            
                                            )
                                          
                                        
                                        -
                                        
                                          
                                            
                                              2
                                              ⁢
                                              GL
                                            
                                            +
                                            θ
                                            +
                                            ς
                                            -
                                            
                                              2
                                              ⁢
                                              τ
                                            
                                          
                                          2
                                        
                                      
                                      ]
                                    
                                  
                                  2
                                
                              
                            
                            ]
                          
                        
                      
                    
                    }
                  
                
                -
                
                  
                    Sin
                    ⁡
                    
                      [
                      
                        
                          
                            Ω
                            p
                          
                          ⁡
                          
                            (
                            
                              
                                
                                  2
                                  ⁢
                                  τ
                                
                                -
                                θ
                                +
                                ς
                              
                              2
                            
                            )
                          
                        
                        -
                        
                          2
                          ⁢
                          
                            L
                            ⁡
                            
                              (
                              
                                
                                  K
                                  e
                                
                                +
                                
                                  K
                                  o
                                
                              
                              )
                            
                          
                        
                      
                      ]
                    
                  
                  ⁢
                  
                    {
                    
                      
                        
                          
                            
                              -
                              
                                
                                  Tri
                                  ⁡
                                  
                                    (
                                    
                                      
                                        θ
                                        +
                                        ς
                                      
                                      
                                        2
                                        ⁢
                                        DL
                                      
                                    
                                    )
                                  
                                
                                [
                                
                                  e
                                  
                                    -
                                    
                                      
                                        
                                          
                                            σ
                                            p
                                            2
                                          
                                          2
                                        
                                        ⁡
                                        
                                          [
                                          
                                            
                                              
                                                Λ
                                                D
                                              
                                              ⁢
                                              
                                                (
                                                
                                                  θ
                                                  +
                                                  ς
                                                
                                                )
                                              
                                            
                                            -
                                            
                                              
                                                
                                                  2
                                                  ⁢
                                                  GL
                                                
                                                +
                                                θ
                                                -
                                                ς
                                                -
                                                
                                                  2
                                                  ⁢
                                                  τ
                                                
                                              
                                              2
                                            
                                          
                                          ]
                                        
                                      
                                      2
                                    
                                  
                                
                                ]
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              Tri
                              ⁡
                              
                                (
                                
                                  
                                    θ
                                    -
                                    ς
                                  
                                  
                                    2
                                    ⁢
                                    DL
                                  
                                
                                )
                              
                            
                            [
                            
                              e
                              
                                -
                                
                                  
                                    
                                      
                                        σ
                                        p
                                        2
                                      
                                      2
                                    
                                    ⁡
                                    
                                      [
                                      
                                        
                                          
                                            Λ
                                            D
                                          
                                          ⁢
                                          
                                            (
                                            
                                              θ
                                              +
                                              ς
                                            
                                            )
                                          
                                        
                                        +
                                        
                                          
                                            
                                              2
                                              ⁢
                                              GL
                                            
                                            +
                                            θ
                                            -
                                            ς
                                            -
                                            
                                              2
                                              ⁢
                                              τ
                                            
                                          
                                          2
                                        
                                      
                                      ]
                                    
                                  
                                  2
                                
                              
                            
                            ]
                          
                        
                      
                    
                    }
                  
                
              
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




With reference to equation (19), FIG. 13 shows an envelope of probability P1e1o 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 P1e1o and P2e2o. In practice, the output state from the second interferometer beam splitter 98 can be expressed by formula (17) for the second embodiment as well.
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.
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.

## Claims

1. An interferometer system comprising:
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;
a first interferometer configured to receive the first pump pulse and generate an antisymmetric state with two coalescent photons (1/ √{square root over (2)}(|21,02−|01,22)); and
a second interferometer configured to receive the second pump pulse and to generate a symmetric state with two coalescent photons(1/ √{square root over (2)}·(|21,02+|01,22)) ;
said interferometer system being configured to control the probabilities with which two coalescent photons are emitted from a first output or a second output of the second interferometer;
wherein the first and second interferometers are connected so that the interferometer system 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)}·(|21,02+|01,22)+eiτ·1/√{square root over (2)}, ·(|21,02−|01,22)), the weights of the sum being a function of the delay,
wherein the first and second interferometers define a first optical path and a second optical path; and wherein the second interferometer comprises a first crystal of the optically non-linear type and configured to receive the second pump pulse, and first and second optical beam splitters, the first and second optical paths extending between the first crystal and the second optical beam splitter, the first optical beam splitter being interposed between the first crystal and the second optical beam splitter, and being crossed by the first and second optical paths.

2. The interferometer system according to claim 1, wherein the first interferometer comprises the second optical beam splitter and a second crystal of the optically non-linear type and configured to receive the first pump pulse, the second crystal being crossed by the first and second optical paths and being interposed between the first and second optical beam splitters.

3. The interferometer system according to claim 2, wherein the first and second crystals are adapted to emit a first and a second pair of photons, respectively, upon the reception of the second and first pump pulses, respectively; and wherein the first and second optical paths are such that, considered any one of the first and second pairs of photons, a photon of the considered pair is emitted along the first optical path, and the other photon is emitted along the second optical path.

4. The interferometer system according to claim 3, wherein the second interferometer comprises first optical delay means arranged along the first optical path, interposed between the first crystal and the first optical beam splitter, and adapted to delay in an electronically controllable manner the photon of the first pair of photons that propagates along the first optical path.

5. The interferometer system according to claim 4, wherein the first interferometer comprises second optical delay means arranged along the first optical path, interposed between the second crystal and the second optical beam splitter, and adapted to delay in an electronically controllable manner the photons of the first and second pairs of photons that propagate along the first optical path.

6. The interferometer system according to claim 2, further comprising first and second birefringent optical elements, the first and second crystals and first and second birefringent optical elements having respective optical axes; the first and second birefringent optical elements are arranged along the first and second optical paths, respectively, are interposed between the first and second crystals and have the same length, equal to the length of the second crystal, each of the first and second birefringent optical elements further having a position that corresponds to the position of the second crystal, rotated by approximately 90° about the optical axis.

7. The interferometer system according to claim 1, wherein the first optical beam splitter is of the polarizing or 50/50 type.

8. The interferometer system according to claim 1, wherein the first interferometer forms a Hong-Ou-Mandel interferometer.