Patent Publication Number: US-2023142878-A1

Title: Superconducting quantum circuit

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is based upon and claims the benefit of the priority of Japanese patent application No. 2021-184241, filed on Nov. 11, 2021, the disclosure of which is incorporated herein in its entirety by reference thereto. 
     FIELD 
     The present invention relates to a superconducting quantum circuit, qubit circuit, qubit coupler, and quantum computer. 
     BACKGROUND 
     There has been widespread development of quantum computers using superconducting quantum circuits. Such a quantum computer generally includes a microwave LC resonance circuit which is made up of a superconductor and includes a nonlinear element including Josephson junctions (e.g., Superconducting Quantum Interference Device, SQUID). 
     The microwave LC resonance circuit is formed as a planer circuit with a superconducting material deposited on a semiconductor substrate.
     PTL 1: Japanese Unexamined Patent Application Publication (Translation of PCT Application) No. 2021-500737A   PTL 2: Japanese Unexamined Patent Application Publication No. 2021-108308A   

     SUMMARY 
     As described later in detail, the techniques disclosed the literatures in Citation List have a problem that it is difficult to adjust a resonance operation point. 
     Accordingly, it is an object of the present disclosure to provide a superconducting quantum circuit solving the above problem. 
     According to an aspect of the present disclosure, there is provided a superconducting quantum circuit including a plurality of SQUIDs (Superconducting Quantum Interference Devices) connected in parallel, each of the plurality of SQUIDs including a first superconducting line, a first Josephson junction, a second superconducting line, and a second Josephson junction connected in a loop, wherein a junction area of the first Josephson junction and a junction area of the second Josephson junction are different from each other, the plurality of SQUIDs configured to be mutually different in either one or both of: a sum of the junction area of the first Josephson junction and the junction area of the second Josephson junction; and a ratio of the junction area of the first Josephson junction to the junction area of the second Josephson junction. 
     According to the present disclosure, it is possible to facilitate the adjustment of a resonance operation point. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1 A  is a diagram illustrating a related technology. 
         FIG.  1 B  is a diagram illustrating a related technology. 
         FIG.  1 C  is a diagram illustrating a related technology. 
         FIG.  2 A  is a diagram illustrating an example embodiment. 
         FIG.  2 B  is a diagram illustrating an example embodiment. 
         FIG.  3    is a diagram schematically illustrating a configuration example of an example embodiment. 
         FIG.  4    is a diagram schematically illustrating a configuration example of an example embodiment. 
         FIG.  5    is a diagram schematically illustrating a variation of an example embodiment. 
         FIG.  6 A  is a diagram illustrating a configuration example of another example embodiment. 
         FIG.  6 B  is a diagram schematically illustrating a configuration example of another example embodiment. 
         FIG.  7    is a diagram illustrating a variation of another example embodiment. 
         FIG.  8    is a diagram illustrating a variation of another example embodiment. 
         FIG.  9    is a diagram illustrating still another example embodiment. 
     
    
    
     EXAMPLE EMBODIMENTS 
     In the following, the above-mentioned problem is described and then some example embodiments will be described. 
     A SQUID behaves as a variable inductance which depends on a magnitude of a magnetic flux Φ penetrating a loop surface of the SQUID. Therefore, it is possible to adjust circuit characteristics such as a resonance frequency by applying a DC current to a control line coupled via a mutual inductance to the SQUID. 
     An effective critical current value I c  of a SQUID depends on the magnetic flux Φ. An inductance (self-inductance) L is inversely proportional to the critical current value I c . A self-inductance L of the SQUID can be given as follows: 
         L=Φ   0 /(2 I   c )∝1/ I   c   (1)
 
     where Φ 0  is a magnetic flux quantum (Φ 0 =h/2e, where h is the Planck constant and e is an elementary charge). That L is inversely proportional to I c  is derived as below. When taking in account of a shielding current to counteract an external magnetic field flowing through the SQUID, a parameter β in the following equation (2) is introduced. β may be approximated to 1 for simplicity. 
       β=2 L·I   c /Φ 0   (2)
 
     When two Josephson junctions of the SQUID have the same critical current value I 0 , a total current I flowing through the SQUID is given by the following equation (3): 
         I=I   0  sin(γ A )+ I   0  sin(γ B )  (3)
 
     where γA and γB are respective phase shifts (phase differences) in the two Josephson junctions, and have the relationship given by the following equation (4): 
       γ B−γA= 2πΦ/Φ 0   (4)
 
     where Φ is a magnetic flux (external magnetic flux) penetrating through a loop of the SQUID. 
     From the equations (3) and (4), a maximum value I max  of the current I flowing through the SQUID is given as follows: 
         I   max =2 I   0 |cos(πΦ/Φ 0 )|  (5)
 
     I max  is 2I 0  when the magnetic flux t is an integral multiple (including zero) of the magnetic flux quantum to (i.e., Φ/Φ 0 =n), and zero when it is a half integer multiple (Φ/Φ 0 =½+n). 
     In a case where two Josephson junctions of the SQUID have the same critical current value I 0 , i.e., the SQUID being symmetric, there is only one operation point (resonance operation point) (the magnetic flux phase=πΦ/Φ 0 =nπ, the maximum resonance frequency), where a gradient of a resonance frequency with respect to the magnetic flux Φ becomes zero and coherence is improved, as described later. Note that the resonant operation point indicates a resonance frequency set by a DC magnetic field Φ dc  applied to the SQUID. A resonator using a SQUID generally has an inductance component Lc other than that of the SQUID. Thus, the inductance of the resonator is given as Lc+L, where L is an inductance of the SQUID. However, letting Lc=0 for the sake of simplicity, based the above equation (1) (where β in the above equation (2) is set to 1), a resonance angular frequency at the resonance operation point is given by the following equation (6): 
     
       
         
           
             
               
                 
                   ω 
                   = 
                   
                     
                       1 
                       
                         LC 
                       
                     
                     = 
                     
                       
                         1 
                         
                           
                             
                               ( 
                               
                                 
                                   Φ 
                                   0 
                                 
                                 
                                   2 
                                   ⁢ 
                                   
                                     I 
                                     0 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             C 
                           
                         
                       
                       = 
                       
                         
                           
                             2 
                             ⁢ 
                             
                               I 
                               0 
                             
                           
                         
                         
                           
                             
                               Φ 
                               0 
                             
                             ⁢ 
                             C 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     When a resonance frequency (angular frequency) has a gradient with respect to a magnetic flux Φ, the resonance frequency is varied due to, for example, a magnetic field noise present in an environment where the SQUID is arranged. Therefore, when a high degree of coherence is required, it is desirable for the resonator (SQUID) to have an operation point at which a gradient of the resonance frequency with respect to the magnetic flux is small. It is, however, known that there is a trade-off between sensitivity to a magnetic flux and ability to adjust parameters and it is difficult to achieve both. 
     In contrast to a resonator using a SQUID with two Josephson junctions inserted in a superconducting loop, an LC resonator using a single Josephson junction, as illustrated in  FIG.  1 A , has an exceptionally low sensitivity to a magnetic flux Φ, but it becomes almost impossible (extremely difficult) to adjust a parameter(s) of the resonator. It is noted that, in  FIG.  1 A , a resonance mode of a superconducting LC resonance circuit is nonlinear due to nonlinearity of the Josephson junction and operates as a qubit, which is a quantum two-level system having two states, with energy levels unequally spaced. 
     In order to cause a resonator using a SQUID to have an operation point with a low sensitivity to a magnetic flux Φ, an asymmetric SQUID such as one illustrated in  FIG.  1 B  is commonly used (e.g., refer to PTLs (Patent Literatures) 1 and 2).  FIG.  1 B  illustrates a lumped element resonator using an asymmetric SQUID. Referring to  FIG.  1 B , the SQUID  10  has a loop structure in which a first superconducting line  103 , a first Josephson junction  101 , a second superconducting line  104 , and a second Josephson junction  102  are connected in a loop. In the first and the second Josephson junctions, in each of which an insulator (not shown) with a thickness on an order of nanometer is sandwiched by the first and the second superconducting lines  103  and  104 , a superconducting current flows due to tunneling effect of Cooper pairs in a superconductor/insulator/superconductor structure, where a Cooper pair is a pair of free electrons within a solid that act together as one quasiparticle and in large numbers give rise to superconductivity. In  FIG.  1 B , reference numerals  12  and  13  designate an input/output capacitor (coupling capacitor) and an input/output line, respectively. A signal (input signal or output signal) on the input/output line  13  is AC coupled to a SQUID  10 . A signal source (e.g., a current source not shown) supplies a direct current to a flux line  14  with on end grounded, which functions as a magnetic field generator to generate a magnetic flux Φ through the SQUID  10 . That is, the magnetic flux generated by the flux line  14  penetrates through a loop of the SQUID  10  from front to back of the drawing, or vice versa. 
     In the SQUID  10 , a critical current value I 0 (1+x) of the first Josephson junction  101  and a critical current value I 0 (1−x) of the second Josephson junction  102  are different (where 0&lt;x&lt;1). Note that a critical current value of a Josephson junction is proportional to a junction area thereof. Therefore, by adjusting a ratio of a junction area of the first Josephson junction  101  to that of the second Josephson junction  102 , a ratio of a critical current value of the first Josephson junction  101  to that of the second Josephson junction  102  can be adjusted. 
     An inductance of the SQUID  10  and a capacitor  11  form a parallel resonance circuit. In the SQUID  10 , a first node  105  on the first superconducting line  103  and a second node  106  on the second superconducting line  104  are connected to opposite electrodes of the capacitor  11  and shunted by the capacitor  11 . As shown in  FIG.  1 B , the SQUID  10  may be configured to have one end grounded. 
     The resonance frequency f of the resonator using the asymmetric SQUID  10  illustrated in  FIG.  1 B  is maximized when Φ/Φ 0  (a value obtained by dividing the magnetic flux Φ penetrating through the SQUID  10  by the magnetic flux quantum ( 130   0 ) is zero, while minimized when Φ/Φ 0  is one half, bringing a gradient with respect to the magnetic flux Φ zero. 
     In the SQUID  10 , when the critical currents of the first and the second Josephson junctions  101  and  102  are I 0 (1+x) and I 0 (1−x), the maximum value of a current that can flow through the SQUID  10  may be evaluated using the following equation (7): 
     
       
         
           
             
               
                 
                   2 
                   ⁢ 
                   
                     I 
                     0 
                   
                   ⁢ 
                   
                     
                       
                         
                           cos 
                           2 
                         
                         ( 
                         
                           π 
                           ⁢ 
                           
                             Φ 
                             
                               Φ 
                               0 
                             
                           
                         
                         ) 
                       
                       + 
                       
                         
                           x 
                           2 
                         
                         ⁢ 
                         
                           
                             sin 
                             2 
                           
                           ( 
                           
                             π 
                             ⁢ 
                             
                               Φ 
                               
                                 Φ 
                                 0 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     In equation (7), since 0&lt;x&lt;1, the maximum value of the current flowing through the SQUID  10  is 2I 0  when the magnetic flux Φ is an integer multiple of the magnetic flux quantum Φ 0 , and the minimum value thereof is 2I 0 x when the magnetic flux Φ is a half integer multiple of the magnetic flux quantum Φ 0 . The minimum value 2I 0 x is x times the maximum value and is equal to a difference of the critical currents I 0 (1+x)−I 0 (1−x) between the first and the second Josephson junctions  101  and  102 . Further, from equation (7), when x=0, the minimum value of the current flowing through the SQUID  10  is zero. 
       FIG.  1 C  is a diagram illustrating the relationship between the resonance frequency f of the resonator using the asymmetric SQUID  10  shown in  FIG.  1 B  and the magnetic flux Φ penetrating through the loop of the SQUID  10 . A horizontal axis (X) is the value (ranging from 0 to 1) obtained by dividing the magnetic flux Φ penetrating through the loop of the SQUID  10  by the magnetic flux quantum Φ 0 . A vertical axis (Y) is a resonance frequency f (in GHz (gigahertz)). 
     The above equation (2) indicates that the inductance L of the SQUID  10  is inversely proportional to the critical current value. Therefore, from the equation (7), the resonance frequency of the resonator using the SQUID  10  is maximized when Φ/Φ 0  (termed as a magnetic flux phase, where Φ is a magnetic flux penetrating through the loop of the SQUID  10  and Φ 0  is the magnetic flux quantum) is zero (integer), while minimized when Φ/Φ 0  is one half (half-integer), with a zero gradient with respect to the magnetic flux, as shown in  FIG.  1 C . Further, from the above equation (6) and (7), the resonance frequency f in  FIG.  1 C  is given as follows: 
     
       
         
           
             
               
                 
                   f 
                   = 
                   
                     
                       ω 
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                     = 
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             LC 
                           
                         
                       
                       = 
                       
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 I 
                                 0 
                               
                               ⁢ 
                               
                                 
                                   
                                     { 
                                     
                                       
                                         
                                           cos 
                                           2 
                                         
                                         ( 
                                         
                                           π 
                                           ⁢ 
                                           
                                             Φ 
                                             
                                               Φ 
                                               0 
                                             
                                           
                                         
                                         ) 
                                       
                                       + 
                                       
                                         
                                           x 
                                           2 
                                         
                                         ⁢ 
                                         
                                           
                                             sin 
                                             2 
                                           
                                           ( 
                                           
                                             π 
                                             ⁢ 
                                             
                                               Φ 
                                               
                                                 Φ 
                                                 0 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     } 
                                   
                                   0 
                                 
                               
                             
                           
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 β 
                                 ⁢ 
                                 
                                   Φ 
                                   0 
                                 
                                 ⁢ 
                                 C 
                               
                             
                           
                         
                         = 
                         
                           
                             
                               
                                 I 
                                 0 
                               
                             
                             
                               π 
                               ⁢ 
                               
                                 
                                   2 
                                   ⁢ 
                                   β 
                                   ⁢ 
                                   
                                     Φ 
                                     0 
                                   
                                   ⁢ 
                                   C 
                                 
                               
                             
                           
                           ⁢ 
                           
                             
                               { 
                               
                                 
                                   
                                     cos 
                                     2 
                                   
                                   ( 
                                   
                                     π 
                                     ⁢ 
                                     
                                       Φ 
                                       
                                         Φ 
                                         0 
                                       
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   
                                     x 
                                     2 
                                   
                                   ⁢ 
                                   
                                     
                                       sin 
                                       2 
                                     
                                     ( 
                                     
                                       π 
                                       ⁢ 
                                       
                                         Φ 
                                         
                                           Φ 
                                           0 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               } 
                             
                             
                               1 
                               4 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Letting 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           g 
                           ⁡ 
                           ( 
                           θ 
                           ) 
                         
                         = 
                         
                           
                             
                               cos 
                               2 
                             
                             ⁢ 
                             θ 
                           
                           + 
                           
                             
                               x 
                               2 
                             
                             ⁢ 
                             
                               sin 
                               2 
                             
                             ⁢ 
                             θ 
                           
                         
                       
                     
                     
                       
                         ( 
                         
                           
                             where 
                             ⁢ 
                                 
                             θ 
                           
                           = 
                           
                             π 
                             ⁢ 
                             
                               Φ 
                               
                                 Φ 
                                 0 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     a first-order differential of g(θ) with respect to θ is: 
         g ′(θ)=2( x   2 −1)cos θ sin θ   (10)
 
     A second-order differential is: 
         g ″(θ)=2( x   2 −1)cos 2θ   (11)
 
     x 2 −1&lt;0 since 0&lt;x&lt;1, and in the range of 0≤θ≤π, maximal (maximum) are at θ=0 and π (the horizontal axis X=0, 1 in  FIG.  1 C ) and minimal (minimum) at θ=π/2 (the horizontal axis X=½ in  FIG.  1 C ), at each of which a gradient with respect to a magnetic flux phase θ (Φ/Φ 0 ) is zero. Note that the minimum is √{square root over (x)} times the maximum. Since g″(θ)=0, inflection points are πΦ/Φ 0 =π/4, 3π/4 between 0≤θπ≤π (X=¼, ¾ in  FIG.  1 C ). 
     An asymmetric SQUID can increase the number of the resonance operation points to two (the magnetic flux phase at 0 or 0.5, the maximum or minimum resonance frequency) whereas a resonator using a symmetric SQUID can have only one (the magnetic flux phase at 0, the maximum resonance frequency). 
     Resonators according to the following example embodiments are realized by lines (wirings) formed by a superconducting material on a substrate. The substrate is silicon, but other electronic materials such as sapphire or compound semiconductor materials (group IV, III-V and II-VI) may be used. The substrate is preferably a single crystal but may be polycrystalline or amorphous. As the line material, Nb (niobium) or Al (aluminum) may be used, though not limited thereto. Any metal that becomes superconductive at a cryogenic temperature may be used, such as niobium nitride, indium (In), lead (Pb), tin (Sn), rhenium (Re), palladium (Pd), titanium (Ti), molybdenum (Mo), tantalum (Ta), tantalum nitride, and an alloy containing at least one of the above. In order to achieve superconductivity, the resonator circuit is used in a temperature environment of about 10 mK (millikelvin) achieved by a refrigerator. 
       FIG.  2 A  is a diagram illustrating a first example embodiment. In  FIG.  2 A , two SQUIDs  10 A and  10 B are shown as a plurality of SQUIDs disposed in parallel, for the sake of simplicity, but the number of SQUIDs is not limited to two. The SQUIDs  10 A and  10 B connected in parallel are configured as an asymmetric SQUID  10 . The SQUID  10 A is configured to have a critical current value of a first Josephson junction  101 A different from a critical current value of a second Josephson junction  102 A. The SQUID  10 B is configured to have a critical current value of a first Josephson junction  101 B different from a critical current value of a second Josephson junction  102 B. The SQUIDs  10 A and  10 B are configured in such a way that a sum (or one half) of the critical current values of the first and the second Josephson junctions or/and a ratio of the critical current value of the first Josephson junction to that of the second Josephson junction is/are different from each other between the SQUIDs  10 A and  10 B. 
     In  FIG.  2 A , the critical current value I 0 (1+x) of the first Josephson junction  101 A and the critical current value I 0 (1−x) of the second Josephson junction  102 A of the SQUID  10 A are different (0&lt;x&lt;1). I 0  is one half (an average value) of the sum I 0 (1+x)+I 0 (1−x)=2I 0  of the critical current values of the first and the second Josephson junctions  101 A and  102 A of the SQUID  10 A. As described above, in the SQUID  10 A, the critical current value I 0 (1+x) of the first Josephson junction  101 A corresponds (is proportional) to a junction area (size) of the first Josephson junction  101 A, and the critical current value I 0 (1−x) of the second Josephson junction  102 A corresponds (is proportional) to a junction area of the second Josephson junction  102 A. The first and the second Josephson junctions  101 A and  102 A are made of the same insulating material. One half of the sum of the critical current values of the first and the second Josephson junctions  101 A and  102 A can be made to correspond to one half of the sum of the junction areas of the first and the second Josephson junctions  101 A and  102 A, assuming linearity holds. In  FIG.  2 A , reference numerals  12  and  13  designate an input/output (IO) capacitor and an input/output (IO) line, respectively. A power supply (current source) not shown supplies a direct current to a flux line  14 A, which functions as a magnetic field generator to generate a magnetic flux ( 130 A penetrating through a loop surface of the SQUID  10 A. 
     The critical current value I 0 ′(1+x′) of the first Josephson junction  101 B and the critical current value I 0 ′(1−x′) of the second Josephson junction  102 B of the SQUID  10 B are different (0&lt;x′&lt;1). I 0 ′ is one half (an average value) of the sum I 0 ′(1+x′)+I 0 ′(1−x′)=2I 0 ′ of the critical current values of the first and the second Josephson junctions  101 B and  102 B of the SQUID  10 B. In the SQUID  10 B, the critical current value I 0 ′(1+x′) of the first Josephson junction  101 B corresponds (is proportional) to a junction area of the first Josephson junction  101 B, and the critical current value I 0 ′(1−x′) of the second Josephson junction  102 B corresponds (is proportional) to a junction area of the second Josephson junction  102 B. The first and the second Josephson junctions  101 B and  102 B are made of the same insulating material. One half of a sum of the critical current values of the first and the second Josephson junctions  101 B and  102 B can be made to correspond to one half of the sum of the junction areas of the first and the second Josephson junctions  101 B and  102 B. A power supply (current source) not shown supplies a direct current to a flux line  14 B, which functions as a magnetic field generator to generate a magnetic flux ΦB penetrating through the loop surface of the SQUID  10 B. 
     A first node  105 A of the SQUID  10 A, a first node  105 B of the SQUID  10 B, and one end of the capacitor  11  (Cavity Capacitor; a capacitance which the resonator  20  has) are commonly connected to a node  107  (common connection node), which is connected to the input/output (IO) line  13  via the input/output (IO) capacitor  12 . A second node  106 A of the SQUID  10 A, a second node  106 B of the SQUID  10 B, and the other end of the capacitor  11  are connected in common to a node  108 , which is connected to ground. 
     An inductance of each of the SQUIDs  10 A and  10 B forms a parallel resonator together with the capacitor  11 . The first node  105 A on a first superconducting line  103 A and the second node  106 A on a second superconducting line  104 A of the SQUID  10 A are connected to opposite electrodes of the capacitor  11  and shunted by the capacitor  11 . The first node  105 B on a first superconducting line  103 B and the second node  106 B on a second superconducting line  104 B of the SQUID  10 B are connected to opposite electrodes of the capacitor  11  and shunted by the capacitor  11 . As shown in  FIG.  2 A , the SQUIDs  10 A and  10 B may be configured to have one end grounded. 
     The resonator  20  is constituted as an LC resonator in which the SQUIDs  10 A and  10 B, and the capacitor  11  which the resonator  20  has are connected in parallel. 
     In this case, an effective inductance of the resonator  20  is inversely proportional to a sum of the effective critical current values of the SQUIDs  10 A and  10 B. That is, letting the inductances of the SQUIDs  10 A and  10 B, are L A  and L B , respectively, the parallel inductance L is as follows. 
         L=L   A   ×L   B /( L   A   +L   B )  (12)
 
     From the equation (1) where β in the equation (2) is set to 1, when a current flowing through the SQUIDs  10 A and  10 B are I A  and I B , respectively, then: 
         L   A =Φ 0 /(2 I   A )  (13)
 
         L   B =Φ 0 /(2 I   B )  (14)
 
     By substituting equation (13) and (14) into equation (12), the following equation (15) is obtained: 
     
       
         
           
             
               
                 
                   L 
                   = 
                   
                     
                       
                         Φ 
                         0 
                       
                       / 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                             I 
                             A 
                           
                         
                         ) 
                       
                       * 
                       
                         Φ 
                         0 
                       
                       / 
                       
                         ( 
                         
                           2 
                           ⁢ 
                           
                             I 
                             B 
                           
                         
                         ) 
                       
                       / 
                       
                         { 
                         
                           
                             
                               Φ 
                               0 
                             
                             / 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   I 
                                   A 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               Φ 
                               0 
                             
                             / 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   I 
                                   B 
                                 
                               
                               ) 
                             
                           
                         
                         } 
                       
                     
                     = 
                     
                       
                         Φ 
                         0 
                       
                       / 
                       
                         { 
                         
                           2 
                           ⁢ 
                           
                             ( 
                             
                               
                                 I 
                                 A 
                               
                               + 
                               
                                 I 
                                 B 
                               
                             
                             ) 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     In each of the SQUIDs  10 A and  10 B, when Φ/Φ 0  (magnetic flux phase: a ratio of the magnetic flux Φ to the magnetic flux quantum Φ 0 ) is an integer (n) or a half integer (½+n), a gradient of the critical current with respect to the magnetic flux Φ is zero. Therefore, in each of the SQUIDs  10 A and  10 B, when the magnetic flux phase Φ/Φ 0  is an integer and half-integer, each of the SQUIDs  10 A and  10 B has sensitivity to the magnetic flux Φ suppressed. 
     By setting the maximum and minimum values of the critical currents of N SQUIDs connected in parallel to mutually different values, it is possible to achieve at maximum 2 N  resonance operation points with mutually different resonance frequencies. 
     The critical current values I A1  and I A2  of the two Josephson junctions  101 A and  102 A of the SQUID  10 A are different from each other as follows: 
         I   A1   =I   0 (1+ x )  (16)
 
         I   A2   =I   0 (1− x )  (17)
 
     where I 0  is one half (an average critical current value) of a sum of the critical current values of the first and the second Josephson junctions  101 A and  102 A of the SQUID  10 A, and x is a parameter representing a degree of asymmetry of the SQUID  10 A (0&lt;x&lt;1). 
     The critical current values I B1 ′ and I B2 ′ of the two Josephson junctions  101 B and  102 B of the SQUID  10 B are different from each other as follows: 
         I   B1   ′=I   0 ′(1+ x ′)  (18)
 
         I   B2   ′=I   0 ′(1− x ′)  (19)
 
     where I 0 ′ is one half (an average critical current value) of a sum of the critical current values of the first and the second Josephson junctions  101 B and  102 B of the SQUID  10 B, and x′ is a parameter representing a degree of asymmetry of the SQUID  10 B (0&lt;x′&lt;1). 
     Letting r be a ratio of the critical current value I A1  to I A2  of the SQUID  10 A, 
         r =(1− x )/(1+ x )  (20)
 
       then,  x  is given as 
         x =(1− r )/(1+ r )  (21)
 
     The degree of asymmetry x corresponds one-to-one to the ratio r of the critical current value I A1  to I A2  of the SQUID  10 A. Likewise, letting r′ be a ratio of the critical current value I B1 ′ to I B2 ′ of the SQUID  10 B, 
         r ′=(1− x ′)/(1+ x ′)  (22)
 
     then, x′ is given as 
         x ′=(1− r ′)/(1+ r ′)  (23)
 
     The degree of asymmetry x′ corresponds one-to-one to the ratio r′ of the critical current value I B1 ′ to I B2 ′ of the SQUID  10 B. 
     In the SQUID  10 A, currents flowing through the first and the second Josephson junctions  101 A and  102 A are I 0 (1+x) and I 0 (1−x), respectively. From the above equation (7), a critical current value of the SQUID  10 A (a maximum value of the current that can flow through the SQUID 10 A) can be given by the following equation (24): 
     
       
         
           
             
               
                 
                   2 
                   ⁢ 
                   
                     I 
                     0 
                   
                   ⁢ 
                   
                     
                       
                         
                           cos 
                           2 
                         
                         ( 
                         
                           π 
                           ⁢ 
                           
                             
                               Φ 
                               A 
                             
                             
                               Φ 
                               0 
                             
                           
                         
                         ) 
                       
                       + 
                       
                         
                           x 
                           2 
                         
                         ⁢ 
                         
                           
                             sin 
                             2 
                           
                           ( 
                           
                             π 
                             ⁢ 
                             
                               
                                 Φ 
                                 A 
                               
                               
                                 Φ 
                                 0 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     where Φ A  is a magnetic flux penetrating through the loop of the SQUID  10 A. 
     In SQUID  10 B, currents flowing through the first and the second Josephson junctions  101 B and  102 B are I 0 ′(1+x′) and I 0 ′(1−x′), respectively. A critical current value of the SQUID  10 B (a maximum value of the current that can flow through the SQUID) can be given by the following equation (25): 
     
       
         
           
             
               
                 
                   2 
                   ⁢ 
                   
                     I 
                     0 
                     ′ 
                   
                   ⁢ 
                   
                     
                       
                         
                           cos 
                           2 
                         
                         ( 
                         
                           π 
                           ⁢ 
                           
                             
                               Φ 
                               B 
                             
                             
                               Φ 
                               0 
                             
                           
                         
                         ) 
                       
                       + 
                       
                         
                           x 
                           ′2 
                         
                         ⁢ 
                         
                           
                             sin 
                             2 
                           
                           ( 
                           
                             π 
                             ⁢ 
                             
                               
                                 Φ 
                                 B 
                               
                               
                                 Φ 
                                 0 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     where Φ B  is a magnetic flux penetrating through the loop of the SQUID  10 B. 
     From the above equation (24), the critical current value I A  of the SQUID  10 A takes: 
     a maximum value: 2I 0  when the magnetic flux Φ A  is an integral multiple of the magnetic flux quantum Φ 0 ; and 
     a minimum value: 2I 0 x when the magnetic flux Φ A  is a half-integer multiple of the magnetic flux quantum Φ 0 . 
     From the above equation (25), the critical current value I B  of the SQUID  10 B takes: 
     a maximum value: 2I 0 ′ when the magnetic flux Φ B  is an integral multiple of the magnetic flux quantum Φ 0 ; and 
     a minimum value: 2I 0 ′x′ when the magnetic flux Φ B  is a half-integer multiple of the magnetic flux quantum Φ 0 . 
     With respect to the magnetic fluxes Φ A  and Φ B  penetrating through the loops of the SQUIDs  10 A and  10 B, respectively, there are four combinations of a sum (I A +I B ) of the current values flowing through the SQUIDs  10 A and  10 B, respectively: 
         a ) 2 I   0 +2 I   0 ′(Φ A/Φ   0   =n, ΦB/Φ   0   =n ′)  (26)
 
         b ) 2 xI   0 +2 I   0 ′(Φ A/Φ   0 =½+ n, ΦB/Φ   0   =n ′)  (27)
 
         c ) 2 I   0 +2 x′I   0 ′(Φ A/Φ   0   =n, ΦB/Φ   0   =n′+ ½)  (28)
 
         d ) 2 xI   0 +2 x′I   0 ′(Φ A/Φ   0 =½+ n, ΦB/Φ   0 =½+ n ′)  (29)
 
     These correspond to the resonance operation points. That is, there are four resonance operation points in a range where Φ A /Φ 0  and Φ B /Φ 0  are from 0 to ½. 
     In the following, the resonance frequency of the resonator  20  is assumed to be given by the following equation (30): 
     
       
         
           
             
               
                 
                   f 
                   = 
                   
                     
                       ω 
                       
                         2 
                         ⁢ 
                         π 
                       
                     
                     = 
                     
                       
                         1 
                         
                           2 
                           ⁢ 
                           π 
                           ⁢ 
                           
                             LC 
                           
                         
                       
                       = 
                       
                         
                           1 
                           
                             2 
                             ⁢ 
                             π 
                             ⁢ 
                             
                               
                                 
                                   { 
                                   
                                     
                                       Φ 
                                       0 
                                     
                                     / 
                                     2 
                                     ⁢ 
                                     
                                       ( 
                                       
                                         
                                           I 
                                           A 
                                         
                                         + 
                                         
                                           I 
                                           B 
                                         
                                       
                                     
                                   
                                   } 
                                 
                                 ⁢ 
                                 C 
                               
                             
                           
                         
                         = 
                         
                           
                             
                               
                                 I 
                                 A 
                               
                               + 
                               
                                 I 
                                 B 
                               
                             
                           
                           
                             π 
                             ⁢ 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   Φ 
                                   0 
                                 
                                 ⁢ 
                                 C 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     The resonance frequencies f a , f b , f c , and f d  at the four resonance operation points of the above equation (26) to (29) are given by equations (31) to (34): 
     
       
         
           
             
               
                 
                   
                     f 
                     a 
                   
                   = 
                   
                     
                       
                         
                           I 
                           0 
                         
                         + 
                         
                           I 
                           0 
                           ′ 
                         
                       
                     
                     
                       π 
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           
                             Φ 
                             0 
                           
                           ⁢ 
                           C 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     f 
                     b 
                   
                   = 
                   
                     
                       
                         
                           xI 
                           0 
                         
                         + 
                         
                           I 
                           0 
                           ′ 
                         
                       
                     
                     
                       π 
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           
                             Φ 
                             0 
                           
                           ⁢ 
                           C 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     f 
                     c 
                   
                   = 
                   
                     
                       
                         
                           I 
                           0 
                         
                         + 
                         
                           
                             x 
                             ′ 
                           
                           ⁢ 
                           
                             I 
                             0 
                             ′ 
                           
                         
                       
                     
                     
                       π 
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           
                             Φ 
                             0 
                           
                           ⁢ 
                           C 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     f 
                     d 
                   
                   = 
                   
                     
                       
                         
                           xI 
                           0 
                         
                         + 
                         
                           
                             x 
                             ′ 
                           
                           ⁢ 
                           
                             I 
                             0 
                             ′ 
                           
                         
                       
                     
                     
                       π 
                       ⁢ 
                       
                         
                           2 
                           ⁢ 
                           
                             Φ 
                             0 
                           
                           ⁢ 
                           C 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     (A) In a case where the SQUIDs  10 A and  10 B have average critical current values equal but asymmetries different to each other, i.e., I 0 =I 0 ′ and x≠x′, 
         f   a   &gt;f   b   , f   c   &gt;f   d   (35)
 
     A magnitude relationship between f b  and f c  is swapped depending on a magnitude relationship between x and x′.
 
When x&gt;x′,
 
         f   a   &gt;f   b   &gt;f   c   &gt;f   d   (36)
 
       When  x&lt;x′,    
         f   a   &gt;f   c   &gt;f   b   &gt;f   d   (37)
 
     Therefore, there are four different resonance operation points.
 
(B) In a case where the SQUIDs  10 A and  10 B have asymmetries equal but average critical current values different to each other, i.e., I 0 ≠I 0 ′, and x=x′,
 
         f   a   &gt;f   b   ,f   c   &gt;f   d   (38)
 
     A magnitude relationship between f b  and f c  is swapped depending on a magnitude relationship between I 0  and I 0 ′.
 
When I 0 &lt;I 0 ′,
 
         f   a   &gt;f   b   &gt;f   c   &gt;f   d   (39)
 
     When I 0 &gt;I 0 ′, 
         f   a   &gt;f   c   &gt;f   b   &gt;f   d   (40)
 
     Therefore, there are four different resonance operation points.
 
(C) In a case where the SQUIDs  10 A and  10 B have average critical current values and asymmetries, both different to each other, i.e., I 0 ≠I 0 ′, and x≠x′,
 
         f   a   &gt;f   b   , f   c   &gt;f   d   (41)
 
     A magnitude relationship between f b  and f c  is swapped depending on a magnitude relationship between I 0  and I 0 ′ and that between x and x′. 
     That is, 
       when  I   0   ′/I   0 &gt;(1− x )/(1− x ′),
 
         f   a   &gt;f   b   &gt;f   c   &gt;f d (42)
 
       When  I   0   ′/I   0 &lt;(1− x )/(1− x ′),
 
         f   a   &gt;f   c   &gt;f   b   &gt;f   d   (43)
 
     Therefore, there are four different resonance operation points. 
       However, when  I   0   ′/I   0 =(1− x )/(1− x ′),
 
         f   a   &gt;f   b   =f   c   &gt;f   d   (44)
 
     In this case, the number of resonance operation points is degenerated to three. Therefore, in the case (C) where the SQUIDs  10 A and  10 B have the average critical current values (I 0 , I 0 ′) and the asymmetries (x, x′) both different from each other, the SQUIDs  10 A and  10 B may have the average critical current values and the asymmetries set so as to have four different resonance operation points.
 
(D) In a case where the SQUIDs  10 A and  10 B have average critical current values and asymmetries, both equal to each other, i.e., I 0 =I 0 ′, and x=x′,
 
         f   a   &gt;f   b   =f   c   &gt;f   d   (45)
 
     There are three resonance operation points. 
     In the two asymmetric SQUIDs  10 A and  10 B connected in parallel, when the current values (I 0 , I 0 ′), which are one half of the sum (2I 0 , 2I 0 ′) of critical current values of two Josephson junctions of each SQUID, and/or the parameters x and x′ representing a degree of asymmetry (corresponding to a ratio between the critical current values of the two Josephson junctions) are different to each other, 2 2 =4 different resonance operation points can be achieved. Likewise, in a case of N asymmetric SQUIDs connected in parallel, 2N different resonance operation points can be achieved by varying the current value I 0  and/or I 0 ′ which is one half of the sum: 2I 0  and/or 2I 0 ′ of the critical current values of the two Josephson junctions of each SQUID and the value of the parameter x and/or x′ among N asymmetric SQUIDs so as to avoid the situation (degeneration) described in (C) above. 
     In the present example embodiment described above, in order to adjust a resonance operation point (an operation point that has a magnetic field gradient of a value zero and is resistant to magnetic field noise) of the resonator  20  that includes the SQUIDs  10 A and  10 B, a direct current is applied from the flux lines  14 A and  14 B to apply a static magnetic field to the SQUIDs  10 A and  10 B, respectively. It is noted that in  FIG.  2 A , with a signal of frequency ω 0  being supplied from an input/output line  13  and the resonance frequency when a static magnetic field is applied to the SQUIDs  10 A and  10 B being ω 0 , by applying from the flux lines  14 A and  14 B a sufficiently strong pump beam (microwave current+direct current) of frequency ω p  close to twice the resonance frequency ω 0 , parametric oscillation may be invoked outside an operation point resistant to magnetic field noise under conditions where there is a magnetic field gradient. 
       FIG.  2 B  is a diagram showing a calculation result of resonance frequencies of the resonator  20  of  FIG.  2 A  using a contour line diagram. X-axis corresponds to Φ A /Φ 0  and Y-axis corresponds to Φ B /Φ 0  (Φ A  and Φ B  are magnetic fluxes penetrating through loops of the SQUIDs  10 A and  10 B of  FIG.  2 A , respectively). In  FIG.  2 B , the higher the value of the resonance frequency, the darker the grayscale. A valley (gradient=0) at (X, Y)=(0.5, 0.5), a top (gradient=0) at (X, Y)=(0, 0), (0, 1), (1, 0), (1, 1), and a medium level at (X, Y)=(0.5, 0), (0.5, 1), (0, 0.5), (1, 0.5). 
     In general, when a circuit pattern forms a large loop, a magnetic field is generated from the loop and interferes with other circuits. When an area of the loop increases, an unwanted signal is induced in the loop due to influence of an external magnetic field. Therefore, it is desirable to increase a distance between the SQUIDs  10 A and  10 B to reduce contribution of a closed loop current due to the loops between the adjacent SQUIDs  10 A and  10 B. For instance, for the SQUIDs  10 A and  10 B processed to a micrometer size, a distance therebetween may be on the order of millimeters. 
       FIG.  3    is a diagram illustrating a lumped element resonator  20 .  FIG.  3    schematically illustrates a part of a wiring pattern (plane circuit) of the resonator  20  with SQUIDs  10 A and  10 B and a single electrode  15  formed on the circuit surface (main surface) of a silicon substrate. Areas (gray colored) of the electrode  15  and a ground pattern  16 , indicate areas where a superconducting thin film is vapor-deposited on a silicon substrate, and a white portion  18  indicates an exposed area of the silicon substrate (a gap of a coplanar waveguide). The electrode  15  is of a cruciform shape with four arms extending to top, bottom, left and right. The resonator  20  is formed of a coplanar plane circuit in which a signal line and the ground pattern  16  surrounding the signal line (signal electrode) are placed on the same plane on the silicon substrate. In  FIG.  3   , a capacitor  11  in  FIG.  2 A  is formed in a gap between the electrode  15  and the ground pattern  16  facing each other. In  FIG.  3   , one end of each of the two SQUIDs  10 A and  10 B is connected to one end of the electrode  15 , and the other end of each of the two SQUIDs  10 A and  10 B is connected to the ground pattern  16 . The electrode  15  has a cruciform shape in which a first pattern (first and second arms) having both ends along a length connected to one ends of the SQUIDs  10 A and  10 B intersects a second pattern (third and four arms) having one end along a length capacitively coupled to an input/output line  13 . It is noted that the planar shape of the electrode  15  is not limited to the example illustrated in  FIG.  3   . 
     The electrode  15  and the ground pattern  16  may be made of superconducting materials such as Nb and Al. The SQUIDs  10 A and  10 B may also be constituted by wiring patterns of an Nb—Al based superconducting conductor formed on the silicon substrate. The Josephson junctions may be formed by using known techniques (e.g., a thin Al film may be formed on a Nb wiring and an AlOx film with a predetermined thickness may be formed by thermally oxidizing the Al surface, and then an upper Nb film may be deposited). 
     A power supply (current source) not shown in the drawing supplies a direct current signal to each of the flux lines  14 A and  14 B. The ground pattern  16  is provided on both longitudinal sides of each of the flux lines  14 A and  14 B. The ground pattern  16  are arranged facing via a gap with each longitudinal side of each of the flux lines  14 A and  14 B. The flux lines  14 A and  14 B have longitudinal one ends made in contact with one longitudinal sides of line-shaped ground patterns (ground lines)  16 A and  16 B, respectively. The ground lines  16 A and  16 B face the SQUIDs  10 A and  10 B, respectively, on other longitudinal sides. On the ground pattern  16  (the ground pattern provided facing a side of each of the flux lines  14 A and  14 B in the longitudinal direction with a gap therebetween), notches  17 A and  17 B are provided running along ground lines  16 A- 1  and  16 B- 1  that are made in contact with the longitudinal one ends of the flux lines  14 A and  14 B, respectively, and extend in directions orthogonal to the longitudinal directions of the flux lines  14 A and  14 B. 
     A current flowing through the flux line  14 A (or  14 B) is divided at the one longitudinal end thereof to the ground line  16 A- 1  and a ground line  16 A- 2  (or the ground line  16 B- 1  and a ground line  16 B- 2 ). A current flowing through the ground line  16 A- 2  (or  16 B- 2 ) and a current flowing through the ground line  16 A- 1  (or  16 B- 1 ) in an opposite direction do not cancel out a magnetic field applied to the loop of the SQUID  10 A (or the SQUID  10 B). That is, a line length of the ground line  16 A- 1  extending along the notch  17 A is longer than the ground line  16 A- 2  by approximately a length of the notch  17 A, and a magnetic field generated by the current flowing through the ground line  16 A- 1  (a first magnetic field penetrating through a loop of the SQUID  10 A) is larger than a magnetic field generated by the current flowing through the ground line  16 A- 2  (a second magnetic field penetrating through the loop of the SQUID  10 A in the opposite direction to the first magnetic field). As a result, the configuration of the flux line  14 A and the ground lines  16 A- 1  and  16 A- 2  illustrated in  FIG.  3    enables efficient generation of the magnetic field applied to the loop of the SQUID  10 A. Likewise, regarding the flux line  14 B, since a magnetic field generated by a current flowing through the ground line  16 B- 1  (a first magnetic field penetrating through a loop of the SQUID  10 B) is larger than a magnetic field generated by a current flowing through the ground line  16 B- 2  (a second magnetic field penetrating through the loop of the SQUID  10 B in an opposite direction to the first magnetic field), the magnetic field applied to the loop of the SQUID  10 B can be efficiently generated. Line widths of the ground lines  16 A- 1  and  16 A- 2  (or  16 B- 1  and  16 B- 2 ) do not have to be the same and may differ from each other, such as the ground line  16 A- 1  (or  16 B- 1 ) being wider than the ground line  16 A- 2  (or  16 B- 2 ). It is noted that the flux lines  14 A and  14 B illustrated in  FIG.  3    are merely examples, and any configuration other than that in  FIG.  3    may, as a matter of course, be used as long as it satisfies a condition for efficiently generating a magnetic field applied to the loop of the SQUID. 
     In  FIG.  3   , the resonator  20  has two SQUIDs  10 A and  10 B connected in parallel, but the number of SQUIDs is not limited to two. 
     In an example illustrated in  FIG.  4   , four SQUIDs  10 A,  10 B,  10 C, and  10 D are connected between the electrode  15  and the ground pattern  16 , each shown in  FIG.  3   . Flux lines  14 A,  14 B,  14 C, and  14 D are provided for the four SQUIDs  10 A,  10 B,  10 C, and  10 D, respectively, supplying a magnetic flux Φ to a loop of each SQUID. A power supplies (current source) not shown supply a direct current signal to each of the flux lines  14 A,  14 B,  14 C, and  14 D. For the four SQUIDs  10 A,  10 B,  10 C, and  10 D connected in parallel, by setting the average critical current values and the asymmetries (a ratio between the critical current values of two Josephson junctions of a SQUID) of two Josephson junctions to values different from each other, it becomes possible to achieve 2 4 =16 resonance operation points having mutually different resonance frequencies. 
     In  FIGS.  2 A,  3 , and  4   , lumped element resonators are illustrated. However, the resonator of the example embodiment is not limited to the lumped element type, and a distributed element resonator such as a λ/4 resonator illustrated in  FIG.  5    may be used. Referring to  FIG.  5   , a waveguide (λ/4 waveguide)  19  having a length close to a quarter of a resonance wavelength A. (a wavelength of a standing wave) is provided between an input/output capacitor  12  and a node  107  which is a common node between first nodes  105 A and  105 B of the SQUIDs  10 A and  10 B. The λ/4 waveguide  19  is terminated at the ground via the SQUIDs  10 A and  10 B. In  FIG.  5   , a capacitor  11  illustrated in  FIG.  2 A  is not shown which is connected in parallel to the SQUIDs  10 A and  10 B of the lumped element resonator  20 . In the distributed element resonator  20  in  FIG.  5   , a distributed capacitance includes capacitance components between the SQUIDs  10 A and  10 B and the ground pattern and a capacitance component between the λ/4 waveguide  19  and the ground pattern. Since each of the Josephson junctions  101 A,  102 A,  101 B, and  102 B of the SQUIDs  10 A and  10 B also has a minute capacitance component, the distributed capacitance may include the capacitance components of these Josephson junctions. 
       FIGS.  6 A and  6 B  are diagrams illustrating a second example embodiment of the present invention. Referring to  FIG.  6 B , a resonator  20  includes two electrodes  15 A and  15 B bridged by two SQUIDs  10 A and  10 B. As illustrated in  FIGS.  6 A and  6 B , first nodes  105 A and  105 B of the two SQUIDs  10 A and  10 B are connected to the first electrode  15 A (i.e., a common connection node  107  of the first nodes  105 A and  105 B is the first electrode  15 A). The first electrode  15 A is connected to a first input/output line  13 A via a first input/output capacitor  12 A. Second nodes  106 A and  106 B of the two SQUIDs  10 A and  10 B are connected to the second electrode  15 B (i.e., a common connection node  108  of the second nodes  106 A and  106 B is the second electrode  15 B). The second electrode  15 B is connected to a second input/output line  13 B via a second input/output capacitor  12 B. 
     In  FIG.  6 B , the flux lines  14 A and  14 B supplied with currents supplied thereto generate magnetic fluxes penetrating through the loops of the SQUIDs  10 A and  10 B, respectively. The flux lines  14 A and  14 B, the ground lines  16 A- 1 ,  16 A- 2 ,  16 B- 1 , and  16 B- 2 , and the notches  17 A and  17 B have the same patterns and functions as those in  FIG.  3    described above. Since the example illustrated in  FIG.  6 B  is a plane circuit, the magnetic fluxes penetrating through the loops of three or more SQUIDs cannot be individually manipulated from the flux lines, as illustrated in  FIG.  4   . 
     In the second example embodiment, the resonator  20  may be constituted as a distributed element resonator such as a λ/2 resonator as shown in  FIG.  7   . The first nodes  105 A and  105 B of the SQUIDs  10 A and  10 B are connected to one end of a waveguide (λ/4 waveguide)  19 A having a length close to a quarter of a resonance wavelength A. (the wavelength of a standing wave), and the other end of the waveguide  19 A is connected to the first input/output line  13 A via the first input/output capacitor  12 A. The second nodes  106 A and  106 B of the SQUIDs  10 A and  10 B are connected to one end of a waveguide (λ/4 waveguide)  19 B having a length close to a quarter of the resonance wavelength (the wavelength of the standing wave), and the other end of the waveguide  19 B is connected to the second input/output line  13 B via the second input/output capacitor  12 B. In  FIG.  7   , the capacitor  11  is not shown which is connected in parallel to the SQUIDs  10 A and  10 B of the lumped element resonator  20  as illustrated in  FIG.  2   . In the distributed element resonator  20  in  FIG.  7   , a distributed capacitance includes, for instance, capacitance components between the SQUIDs  10 A and  10 B and the ground pattern and capacitance components between the λ/4 waveguides  19 A and  19 B and the ground pattern. Since each of the Josephson junctions  101 A,  102 A,  101 B, and  102 B of the SQUIDs  10 A and  10 B also has a minute capacitance component, the distributed capacitance may include the capacitance components of these Josephson junctions. 
     In  FIG.  6 A , a connection node of the first nodes  105 A and  105 B and a connection node of the second nodes  106 A and  106 B of the SQUIDs  10 A and  10 B are connected to the first and the second input/output lines  13 A and  13 B, via the first and the second input/output capacitors  12 A and  12 B, respectively, but the second example embodiment is not limited to such a configuration. As a variation of the second example embodiment, the connection node of the first nodes  105 A and  105 B and the connection node of the second nodes  106 A and  106 B of the SQUIDs  10 A and  10 B may be connected to another qubit (not shown) and to ground, respectively. In a case where one connection node is connected to ground and the other is connected to an input/output line, the circuit operates as a qubit. As illustrated in  FIG.  8   , in a case where a node  107  which is a common connection node of the first nodes  105 A and  105 B of the SQUIDs  10 A and  10 B, and a node  108  which is a common connection node of the second nodes  106 A and  106 B, are connected to first and second qubits (quantum bits)  22 A and  22 B, respectively, the resonator  20  operates as a qubit coupler which causes a plurality of qubits to be mutually couple. 
     As stated above, the resonator  20  of the example embodiment may be used as a qubit or as a qubit coupler. The following describes an example of using the resonator of the example embodiment described above, as a qubit circuit used in a quantum computer. The quantum computer is assumed to be a quantum annealing computer to compute a solution to a combinatorial optimization problem that can be mapped onto an Ising model. In the quantum computer illustrated in  FIG.  9   , four resonators  20 A to  20 D interconnects with each other via a coupling circuit (qubit coupler)  21 . The coupling circuit  21 , which couples the four resonators  20 A to  20 D, may include a Josephson junction  213 . The resonators  20 A and  20 B are connected via capacitors  211 A and  211 B to one end of a superconducting conductor (electrode)  212 - 1  of the coupling circuit  21 , where the other end of the superconducting conductor  212 - 1  is connected to one end of the Josephson junction  213 . The resonators  20 C and  20 D are connected via capacitors  211 C and  211 D to one end of a superconducting conductor (electrode)  212 - 2  of the coupling circuit  21 , where the other end of the superconducting conductor  212 - 2  is connected to the other end of the Josephson junction  213 . In  FIG.  9   , the quantum computer with four resonators  20 A to  20 D (qubits) is illustrated, but a quantum computer on which any number of the resonators are integrated may be realized by using the configuration illustrated in  FIG.  9   , as a unit structure and arranging and connecting a plurality of the unit structures. 
     Each disclosure of Patent Literatures 1 and 2 cited above is incorporated herein in its entirety by reference thereto. It is to be noted that it is possible to modify or adjust the example embodiments or examples within the whole disclosure of the present invention (including the Claims) and based on the basic technical concept thereof. Further, it is possible to variously combine or select a wide variety of the disclosed elements (including the individual elements of the individual claims, the individual elements of the individual examples and the individual elements of the individual figures) within the scope of the Claims of the present invention. That is, it is self-explanatory that the present invention includes any types of variations and modifications to be done by a skilled person according to the whole disclosure including the Claims, and the technical concept of the present invention. 
     APPENDIX 
     The following describes the derivation of the equation (7). Currents I 1  and I 2  flowing through the two Josephson junctions of an asymmetric SQUID are as follows: 
         I   1   =I   0 (1+ x )  (A.1)
 
         I   2   =I   0 (1− x )  (A.2)
 
         I   1   +I   2 =2 I   0   (A.3)
 
     The current I flowing through the asymmetric SQUID is given as follows: 
         I=I   0 (1+ x )*sin(γ A )+ I   0 (1− x )*sin(γ B )  (A.4)
 
     The equation (A.4) can be rewritten by using 
       γ B−γA= 2πΦ/Φ 0   (A.5)
 
     as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   
                                     
                                       I 
                                       = 
                                       
                                         
                                           
                                             
                                               I 
                                               0 
                                             
                                             ( 
                                             
                                               1 
                                               + 
                                               x 
                                             
                                             ) 
                                           
                                           * 
                                           
                                             sin 
                                             ⁡ 
                                             ( 
                                             
                                               γ 
                                               A 
                                             
                                             ) 
                                           
                                         
                                         + 
                                         
                                           
                                             
                                               I 
                                               0 
                                             
                                             ( 
                                             
                                               1 
                                               - 
                                               x 
                                             
                                             ) 
                                           
                                           * 
                                           
                                             sin 
                                             ⁡ 
                                             ( 
                                             
                                               
                                                 γ 
                                                 A 
                                               
                                               - 
                                               
                                                 2 
                                                 ⁢ 
                                                 π 
                                                 ⁢ 
                                                 Φ 
                                                 / 
                                                 
                                                   Φ 
                                                   0 
                                                 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                     
                                     } 
                                   
                                   = 
                                   
                                     
                                       
                                         
                                           I 
                                           0 
                                         
                                         ⁢ 
                                         
                                           { 
                                           
                                             
                                               sin 
                                               ⁡ 
                                               ( 
                                               
                                                 γ 
                                                 A 
                                               
                                               ) 
                                             
                                             + 
                                             
                                               sin 
                                               ⁡ 
                                               ( 
                                               
                                                 
                                                   γ 
                                                   A 
                                                 
                                                 - 
                                                 
                                                   2 
                                                   ⁢ 
                                                   π 
                                                   ⁢ 
                                                   Φ 
                                                   / 
                                                   
                                                     Φ 
                                                     0 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                           } 
                                         
                                       
                                       + 
                                       
                                         
                                           xI 
                                           0 
                                         
                                         ⁢ 
                                         
                                           { 
                                           
                                             
                                               sin 
                                               ⁡ 
                                               ( 
                                               
                                                 γ 
                                                 A 
                                               
                                               ) 
                                             
                                             - 
                                             
                                               sin 
                                               ⁡ 
                                               ( 
                                               
                                                 
                                                   γ 
                                                   ⁢ 
                                                   A 
                                                 
                                                 - 
                                                 
                                                   2 
                                                   ⁢ 
                                                   π 
                                                   ⁢ 
                                                   Φ 
                                                   / 
                                                   
                                                     Φ 
                                                     0 
                                                   
                                                 
                                               
                                               ) 
                                             
                                           
                                           } 
                                         
                                       
                                     
                                     = 
                                     
                                       2 
                                       ⁢ 
                                       
                                         
                                           I 
                                           0 
                                         
                                         [ 
                                         
                                           
                                             cos 
                                             ⁡ 
                                             ( 
                                             
                                               π 
                                               ⁢ 
                                               Φ 
                                               / 
                                               
                                                 Φ 
                                                 0 
                                               
                                             
                                             ) 
                                           
                                           ⁢ 
                                           sin 
                                           ⁢ 
                                           
                                             { 
                                             
                                               
                                                 γ 
                                                 A 
                                               
                                               - 
                                               
                                                 π 
                                                 ⁢ 
                                                 Φ 
                                                 / 
                                                 
                                                   Φ 
                                                   0 
                                                 
                                               
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               } 
                             
                             ] 
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               
                                 xI 
                                 0 
                               
                               [ 
                               
                                 
                                   sin 
                                   ⁡ 
                                   ( 
                                   
                                     π 
                                     ⁢ 
                                     Φ 
                                     / 
                                     
                                       Φ 
                                       0 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 cos 
                                 ⁢ 
                                 
                                   { 
                                   
                                     
                                       γ 
                                       A 
                                     
                                     - 
                                     
                                       π 
                                       ⁢ 
                                       Φ 
                                       / 
                                       
                                         Φ 
                                         0 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                       } 
                     
                     ] 
                   
                   = 
                   
                     
                       
                         2 
                         ⁢ 
                         
                           
                             I 
                             0 
                           
                           [ 
                           
                             
                               cos 
                               ⁡ 
                               ( 
                               
                                 π 
                                 ⁢ 
                                 Φ 
                                 / 
                                 
                                   Φ 
                                   0 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     sin 
                                     ⁡ 
                                     ( 
                                     
                                       γ 
                                       A 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     cos 
                                     ⁡ 
                                     ( 
                                     
                                       π 
                                       ⁢ 
                                       Φ 
                                       / 
                                       
                                         Φ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 - 
                                 
                                   
                                     sin 
                                     ⁡ 
                                     ( 
                                     
                                       π 
                                       ⁢ 
                                       Φ 
                                       / 
                                       
                                         Φ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     cos 
                                     ⁡ 
                                     ( 
                                     
                                       γ 
                                       A 
                                     
                                     ) 
                                   
                                 
                               
                               } 
                             
                           
                           ] 
                         
                       
                       + 
                       
                         2 
                         ⁢ 
                         
                           
                             xI 
                             0 
                           
                           [ 
                           
                             
                               sin 
                               ⁡ 
                               ( 
                               
                                 π 
                                 ⁢ 
                                 Φ 
                                 / 
                                 
                                   Φ 
                                   0 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               { 
                               
                                 
                                   
                                     cos 
                                     ⁡ 
                                     ( 
                                     
                                       γ 
                                       A 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     cos 
                                     ⁡ 
                                     ( 
                                     
                                       π 
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       Φ 
                                       / 
                                       
                                         Φ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     sin 
                                     ⁡ 
                                     ( 
                                     
                                       γ 
                                       A 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     sin 
                                     ⁡ 
                                     ( 
                                     
                                       π 
                                       ⁢ 
                                       Φ 
                                       / 
                                       
                                         Φ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               } 
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           2 
                           ⁢ 
                           
                             I 
                             0 
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   cos 
                                   2 
                                 
                                 ( 
                                 
                                   π 
                                   ⁢ 
                                   Φ 
                                   / 
                                   
                                     Φ 
                                     0 
                                   
                                 
                                 ) 
                               
                               + 
                               
                                 x 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ( 
                                   
                                     π 
                                     ⁢ 
                                     Φ 
                                     / 
                                     
                                       Φ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                             } 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             ( 
                             
                               γ 
                               A 
                             
                             ) 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             I 
                             0 
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   - 
                                   
                                     sin 
                                     ⁡ 
                                     ( 
                                     
                                       π 
                                       ⁢ 
                                       Φ 
                                       / 
                                       
                                         Φ 
                                         0 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     π 
                                     ⁢ 
                                     Φ 
                                     / 
                                     
                                       Φ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 x 
                                 ⁢ 
                                 
                                   cos 
                                   ⁡ 
                                   ( 
                                   
                                     π 
                                     ⁢ 
                                     Φ 
                                     / 
                                     
                                       Φ 
                                       x 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   sin 
                                   ⁡ 
                                   ( 
                                   
                                     π 
                                     ⁢ 
                                     Φ 
                                     / 
                                     
                                       Φ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                             } 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               γ 
                               A 
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           2 
                           ⁢ 
                           
                             I 
                             0 
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   cos 
                                   2 
                                 
                                 ( 
                                 
                                   π 
                                   ⁢ 
                                   Φ 
                                   / 
                                   
                                     Φ 
                                     0 
                                   
                                 
                                 ) 
                               
                               + 
                               
                                 x 
                                 ⁢ 
                                 
                                   
                                     sin 
                                     2 
                                   
                                   ( 
                                   
                                     π 
                                     ⁢ 
                                     Φ 
                                     / 
                                     
                                       Φ 
                                       0 
                                     
                                   
                                   ) 
                                 
                               
                             
                             } 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             ( 
                             
                               γ 
                               A 
                             
                             ) 
                           
                         
                         - 
                         
                           2 
                           ⁢ 
                           
                             
                               I 
                               0 
                             
                             ( 
                             
                               1 
                               - 
                               x 
                             
                             ) 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             ( 
                             
                               π 
                               ⁢ 
                               Φ 
                               / 
                               
                                 Φ 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               π 
                               ⁢ 
                               Φ 
                               / 
                               
                                 Φ 
                                 0 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             ( 
                             
                               γ 
                               A 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     . 
                         
                     6 
                   
                   ) 
                 
               
             
           
         
       
     
     Letting 
       α={cos 2 (πΦ/Φ 0 )+ x  sin 2 (πΦ/Φ 0 )}  (A.7)
 
       and 
       β=(1− x )sin(πΦ/Φ 0 )cos(πΦ/Φ 0 )  (A.8)
 
     the equation (A.6) can be rewritten as follows: 
         I= 2 I   0 [α sin(γ A )+β cos(γ A )]=2 I   0 √(α 2 +β 2 )sin(γ A   +C )   (A.9)
 
       where 
       cos( C )=α/√(α 2 +β 2 ), sin( C )=β/√(α 2 +β 2 )  (A.10)
 
     Calculating inside the square root of equation (A.9) gives the following equation (A.11): 
     
       
         
           
             
               
                 
                   
                     √ 
                     
                       ( 
                       
                         
                           α 
                           2 
                         
                         + 
                         
                           β 
                           2 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           [ 
                           
                             
                               
                                 cos 
                                 2 
                               
                               ( 
                               
                                 π 
                                 ⁢ 
                                 Φ 
                                 / 
                                 
                                   Φ 
                                   0 
                                 
                               
                               ) 
                             
                             + 
                             
                               x 
                               ⁢ 
                               
                                 
                                   sin 
                                   2 
                                 
                                 ( 
                                 
                                   π 
                                   ⁢ 
                                   Φ 
                                   / 
                                   
                                     Φ 
                                     0 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       + 
                     
                   
                 
               
               
                 
                   ( 
                   
                     A 
                     . 
                         
                     11 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         ( 
                         
                           1 
                           - 
                           x 
                         
                         ) 
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         ( 
                         
                           π 
                           ⁢ 
                           Φ 
                           / 
                           
                             Φ 
                             0 
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         ( 
                         
                           π 
                           ⁢ 
                           Φ 
                           / 
                           
                             Φ 
                             0 
                           
                         
                         ) 
                       
                     
                     ] 
                   
                   2 
                 
                 ) 
               
               
                 1 
                 / 
                 2 
               
             
             = 
             
               ( 
               
                 [ 
                 
                   
                     
                       cos 
                       4 
                     
                     ( 
                     
                       π 
                       ⁢ 
                       Φ 
                       / 
                       
                         Φ 
                         0 
                       
                     
                     ) 
                   
                   + 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     cos 
                     2 
                   
                   ( 
                   
                     π 
                     ⁢ 
                     Φ 
                     / 
                     
                       Φ 
                       0 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   
                     sin 
                     2 
                   
                   ( 
                   
                     π 
                     ⁢ 
                     Φ 
                     / 
                     
                       Φ 
                       0 
                     
                   
                   ) 
                 
               
               ] 
             
             + 
             
               
                 x 
                 2 
               
               [ 
               
                 
                   
                     sin 
                     4 
                   
                   ( 
                   
                     π 
                     ⁢ 
                     Φ 
                     / 
                     
                       Φ 
                       0 
                     
                   
                   ) 
                 
                 + 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         cos 
                         2 
                       
                       ( 
                       
                         π 
                         ⁢ 
                         Φ 
                         / 
                         
                           Φ 
                           0 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       
                         sin 
                         2 
                       
                       ( 
                       
                         π 
                         ⁢ 
                         Φ 
                         / 
                         
                           Φ 
                           0 
                         
                       
                       ) 
                     
                   
                   ] 
                 
                 ) 
               
               
                 1 
                 / 
                 2 
               
             
             = 
           
         
       
       
         
           
             ( 
             
               
                 [ 
                 
                   
                     
                       cos 
                       2 
                     
                     ( 
                     
                       π 
                       ⁢ 
                       
                         Φ 
                         0 
                       
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           cos 
                           2 
                         
                         ( 
                         
                           π 
                           ⁢ 
                           Φ 
                           / 
                           
                             Φ 
                             0 
                           
                         
                         ) 
                       
                       + 
                       
                         
                           sin 
                           2 
                         
                         ( 
                         
                           π 
                           ⁢ 
                           Φ 
                           / 
                           
                             Φ 
                             0 
                           
                         
                         ) 
                       
                     
                   
                 
                 ] 
               
               + 
             
           
         
       
       
         
           
             
               
                 
                   
                     x 
                     2 
                   
                   [ 
                   
                     
                       
                         sin 
                         2 
                       
                       ( 
                       
                         π 
                         ⁢ 
                         Φ 
                         / 
                         
                           Φ 
                           0 
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             sin 
                             2 
                           
                           ( 
                           
                             π 
                             ⁢ 
                             Φ 
                             / 
                             
                               Φ 
                               0 
                             
                           
                           ) 
                         
                         + 
                         
                           
                             cos 
                             2 
                           
                           ( 
                           
                             π 
                             ⁢ 
                             Φ 
                             / 
                             
                               Φ 
                               0 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                   
                   ] 
                 
                 ) 
               
               
                 1 
                 / 
                 2 
               
             
             = 
           
         
       
       
         
           
             √ 
             
               { 
               
                 
                   
                     cos 
                     2 
                   
                   ( 
                   
                     π 
                     ⁢ 
                     Φ 
                     / 
                     
                       Φ 
                       0 
                     
                   
                   ) 
                 
                 + 
                 
                   
                     x 
                     2 
                   
                   ⁢ 
                   
                     
                       sin 
                       2 
                     
                     ( 
                     
                       π 
                       ⁢ 
                       Φ 
                       / 
                       
                         Φ 
                         0 
                       
                     
                     ) 
                   
                 
               
               } 
             
           
         
       
     
     From above, the equation (A.9) can be rewritten as follows: 
         I= 2 I   0 {cos 2 (πΦ/Φ 0 )+ x   2  sin 2 (πΦ/Φ 0 )} 1/2  sin(γ A+C )=
 
         A  sin(γ A+C )  (A.12)
 
       where 
         A= 2 I   0 {cos 2 (θ)+ x   2  sin 2 (θ)} 1/2 (θ=πΦ/Φ 0 )  (A.13)
 
       Since 
       | I|≤A   (A.14)
 
     the amplitude A in equation (A.13) can be regarded as the maximum value (the critical current value) of the current flowing through the asymmetric SQUID.