Patent ID: 12188999

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 Icof a SQUID depends on the magnetic flux Φ. An inductance (self-inductance) L is inversely proportional to the critical current value Ic. A self-inductance L of the SQUID can be given as follows:
L=Φ0/(2Ic)∝1/Ic(1)
where Φ0is a magnetic flux quantum (Φ0=h/2e, where h is the Planck constant and e is an elementary charge). That L is inversely proportional to Icis 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=2L·Ic/Φ0(2)

When two Josephson junctions of the SQUID have the same critical current value I0, a total current I flowing through the SQUID is given by the following equation (3):
I=I0sin(γA)+I0sin(γB)  (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):
γB−γA=2πΦ/Φ0(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 Imaxof the current I flowing through the SQUID is given as follows:
Imax=2I0|cos(πΦ/Φ0)|  (5)

Imaxis 2I0when the magnetic flux t is an integral multiple (including zero) of the magnetic flux quantum to (i.e., Φ/Φ0=n), and zero when it is a half integer multiple (Φ/Φ0=½+n).

In a case where two Josephson junctions of the SQUID have the same critical current value I0, i.e., the SQUID being symmetric, there is only one operation point (resonance operation point) (the magnetic flux phase=πΦ/Φ0=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 Φdcapplied 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=1LC=1(Φ02⁢I0)⁢C=2⁢I0Φ0⁢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 inFIG.1A, 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, inFIG.1A, 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 inFIG.1Bis commonly used (e.g., refer to PTLs (Patent Literatures) 1 and 2).FIG.1Billustrates a lumped element resonator using an asymmetric SQUID. Referring toFIG.1B, the SQUID10has a loop structure in which a first superconducting line103, a first Josephson junction101, a second superconducting line104, and a second Josephson junction102are 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 lines103and104, 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. InFIG.1B, reference numerals12and13designate an input/output capacitor (coupling capacitor) and an input/output line, respectively. A signal (input signal or output signal) on the input/output line13is AC coupled to a SQUID10. A signal source (e.g., a current source not shown) supplies a direct current to a flux line14with on end grounded, which functions as a magnetic field generator to generate a magnetic flux Φ through the SQUID10. That is, the magnetic flux generated by the flux line14penetrates through a loop of the SQUID10from front to back of the drawing, or vice versa.

In the SQUID10, a critical current value I0(1+x) of the first Josephson junction101and a critical current value I0(1−x) of the second Josephson junction102are different (where 0<x<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 junction101to that of the second Josephson junction102, a ratio of a critical current value of the first Josephson junction101to that of the second Josephson junction102can be adjusted.

An inductance of the SQUID10and a capacitor11form a parallel resonance circuit. In the SQUID10, a first node105on the first superconducting line103and a second node106on the second superconducting line104are connected to opposite electrodes of the capacitor11and shunted by the capacitor11. As shown inFIG.1B, the SQUID10may be configured to have one end grounded.

The resonance frequency f of the resonator using the asymmetric SQUID10illustrated inFIG.1Bis maximized when Φ/Φ0(a value obtained by dividing the magnetic flux Φ penetrating through the SQUID10by the magnetic flux quantum (1300) is zero, while minimized when Φ/Φ0is one half, bringing a gradient with respect to the magnetic flux Φ zero.

In the SQUID10, when the critical currents of the first and the second Josephson junctions101and102are I0(1+x) and I0(1−x), the maximum value of a current that can flow through the SQUID10may be evaluated using the following equation (7):

2⁢I0⁢cos2(π⁢ΦΦ0)+x2⁢sin2(π⁢ΦΦ0)(7)

In equation (7), since 0<x<1, the maximum value of the current flowing through the SQUID10is 2I0when the magnetic flux Φ is an integer multiple of the magnetic flux quantum Φ0, and the minimum value thereof is 2I0x when the magnetic flux Φ is a half integer multiple of the magnetic flux quantum Φ0. The minimum value 2I0x is x times the maximum value and is equal to a difference of the critical currents I0(1+x)−I0(1−x) between the first and the second Josephson junctions101and102. Further, from equation (7), when x=0, the minimum value of the current flowing through the SQUID10is zero.

FIG.1Cis a diagram illustrating the relationship between the resonance frequency f of the resonator using the asymmetric SQUID10shown inFIG.1Band the magnetic flux Φ penetrating through the loop of the SQUID10. 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 SQUID10by the magnetic flux quantum Φ0. A vertical axis (Y) is a resonance frequency f (in GHz (gigahertz)).

The above equation (2) indicates that the inductance L of the SQUID10is inversely proportional to the critical current value. Therefore, from the equation (7), the resonance frequency of the resonator using the SQUID10is maximized when Φ/Φ0(termed as a magnetic flux phase, where Φ is a magnetic flux penetrating through the loop of the SQUID10and Φ0is the magnetic flux quantum) is zero (integer), while minimized when Φ/Φ0is one half (half-integer), with a zero gradient with respect to the magnetic flux, as shown inFIG.1C. Further, from the above equation (6) and (7), the resonance frequency f inFIG.1Cis given as follows:

f=ω2⁢π=12⁢π⁢LC=2⁢I0⁢{cos2(π⁢ΦΦ0)+x2⁢sin2(π⁢ΦΦ0)}02⁢π⁢β⁢Φ0⁢C=I0π⁢2⁢β⁢Φ0⁢C⁢{cos2(π⁢ΦΦ0)+x2⁢sin2(π⁢ΦΦ0)}14(8)
Letting

g⁡(θ)=cos2⁢θ+x2⁢sin2⁢θ(where⁢θ=π⁢ΦΦ0)(9)
a first-order differential of g(θ) with respect to θ is:
g′(θ)=2(x2−1)cos θ sin θ   (10)

A second-order differential is:
g″(θ)=2(x2−1)cos 2θ   (11)

x2−1<0 since 0<x<1, and in the range of 0≤θ≤π, maximal (maximum) are at θ=0 and π (the horizontal axis X=0, 1 inFIG.1C) and minimal (minimum) at θ=π/2 (the horizontal axis X=½ inFIG.1C), at each of which a gradient with respect to a magnetic flux phase θ (Φ/Φ0) is zero. Note that the minimum is √{square root over (x)} times the maximum. Since g″(θ)=0, inflection points are πΦ/Φ0=π/4, 3π/4 between 0≤θπ≤π (X=¼, ¾ inFIG.1C).

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.2Ais a diagram illustrating a first example embodiment. InFIG.2A, two SQUIDs10A and10B 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 SQUIDs10A and10B connected in parallel are configured as an asymmetric SQUID10. The SQUID10A is configured to have a critical current value of a first Josephson junction101A different from a critical current value of a second Josephson junction102A. The SQUID10B is configured to have a critical current value of a first Josephson junction101B different from a critical current value of a second Josephson junction102B. The SQUIDs10A and10B 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 SQUIDs10A and10B.

InFIG.2A, the critical current value I0(1+x) of the first Josephson junction101A and the critical current value I0(1−x) of the second Josephson junction102A of the SQUID10A are different (0<x<1). I0is one half (an average value) of the sum I0(1+x)+I0(1−x)=2I0of the critical current values of the first and the second Josephson junctions101A and102A of the SQUID10A. As described above, in the SQUID10A, the critical current value I0(1+x) of the first Josephson junction101A corresponds (is proportional) to a junction area (size) of the first Josephson junction101A, and the critical current value I0(1−x) of the second Josephson junction102A corresponds (is proportional) to a junction area of the second Josephson junction102A. The first and the second Josephson junctions101A and102A are made of the same insulating material. One half of the sum of the critical current values of the first and the second Josephson junctions101A and102A can be made to correspond to one half of the sum of the junction areas of the first and the second Josephson junctions101A and102A, assuming linearity holds. InFIG.2A, reference numerals12and13designate 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 line14A, which functions as a magnetic field generator to generate a magnetic flux (130A penetrating through a loop surface of the SQUID10A.

The critical current value I0′(1+x′) of the first Josephson junction101B and the critical current value I0′(1−x′) of the second Josephson junction102B of the SQUID10B are different (0<x′<1). I0′ is one half (an average value) of the sum I0′(1+x′)+I0′(1−x′)=2I0′ of the critical current values of the first and the second Josephson junctions101B and102B of the SQUID10B. In the SQUID10B, the critical current value I0′(1+x′) of the first Josephson junction101B corresponds (is proportional) to a junction area of the first Josephson junction101B, and the critical current value I0′(1−x′) of the second Josephson junction102B corresponds (is proportional) to a junction area of the second Josephson junction102B. The first and the second Josephson junctions101B and102B are made of the same insulating material. One half of a sum of the critical current values of the first and the second Josephson junctions101B and102B can be made to correspond to one half of the sum of the junction areas of the first and the second Josephson junctions101B and102B. A power supply (current source) not shown supplies a direct current to a flux line14B, which functions as a magnetic field generator to generate a magnetic flux ΦB penetrating through the loop surface of the SQUID10B.

A first node105A of the SQUID10A, a first node105B of the SQUID10B, and one end of the capacitor11(Cavity Capacitor; a capacitance which the resonator20has) are commonly connected to a node107(common connection node), which is connected to the input/output (IO) line13via the input/output (IO) capacitor12. A second node106A of the SQUID10A, a second node106B of the SQUID10B, and the other end of the capacitor11are connected in common to a node108, which is connected to ground.

An inductance of each of the SQUIDs10A and10B forms a parallel resonator together with the capacitor11. The first node105A on a first superconducting line103A and the second node106A on a second superconducting line104A of the SQUID10A are connected to opposite electrodes of the capacitor11and shunted by the capacitor11. The first node105B on a first superconducting line103B and the second node106B on a second superconducting line104B of the SQUID10B are connected to opposite electrodes of the capacitor11and shunted by the capacitor11. As shown inFIG.2A, the SQUIDs10A and10B may be configured to have one end grounded.

The resonator20is constituted as an LC resonator in which the SQUIDs10A and10B, and the capacitor11which the resonator20has are connected in parallel.

In this case, an effective inductance of the resonator20is inversely proportional to a sum of the effective critical current values of the SQUIDs10A and10B. That is, letting the inductances of the SQUIDs10A and10B, are LAand LB, respectively, the parallel inductance L is as follows.
L=LA×LB/(LA+LB)  (12)

From the equation (1) where β in the equation (2) is set to 1, when a current flowing through the SQUIDs10A and10B are IAand IB, respectively, then:
LA=Φ0/(2IA)  (13)
LB=Φ0/(2IB)  (14)

By substituting equation (13) and (14) into equation (12), the following equation (15) is obtained:

L=Φ0/(2⁢IA)*Φ0/(2⁢IB)/{Φ0/(2⁢IA)+Φ0/(2⁢IB)}=Φ0/{2⁢(IA+IB)}(15)

In each of the SQUIDs10A and10B, when Φ/Φ0(magnetic flux phase: a ratio of the magnetic flux Φ to the magnetic flux quantum Φ0) 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 SQUIDs10A and10B, when the magnetic flux phase Φ/Φ0is an integer and half-integer, each of the SQUIDs10A and10B 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 2Nresonance operation points with mutually different resonance frequencies.

The critical current values IA1and IA2of the two Josephson junctions101A and102A of the SQUID10A are different from each other as follows:
IA1=I0(1+x)  (16)
IA2=I0(1−x)  (17)

where I0is one half (an average critical current value) of a sum of the critical current values of the first and the second Josephson junctions101A and102A of the SQUID10A, and x is a parameter representing a degree of asymmetry of the SQUID10A (0<x<1).

The critical current values IB1′ and IB2′ of the two Josephson junctions101B and102B of the SQUID10B are different from each other as follows:
IB1′=I0′(1+x′)  (18)
IB2′=I0′(1−x′)  (19)
where I0′ is one half (an average critical current value) of a sum of the critical current values of the first and the second Josephson junctions101B and102B of the SQUID10B, and x′ is a parameter representing a degree of asymmetry of the SQUID10B (0<x′<1).

Letting r be a ratio of the critical current value IA1to IA2of the SQUID10A,
r=(1−x)/(1+x)  (20)
then,xis 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 IA1to IA2of the SQUID10A. Likewise, letting r′ be a ratio of the critical current value IB1′ to IB2′ of the SQUID10B,
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 IB1′ to IB2′ of the SQUID10B.

In the SQUID10A, currents flowing through the first and the second Josephson junctions101A and102A are I0(1+x) and I0(1−x), respectively. From the above equation (7), a critical current value of the SQUID10A (a maximum value of the current that can flow through the SQUID10A) can be given by the following equation (24):

2⁢I0⁢cos2(π⁢ΦAΦ0)+x2⁢sin2(π⁢ΦAΦ0)(24)
where ΦAis a magnetic flux penetrating through the loop of the SQUID10A.

In SQUID10B, currents flowing through the first and the second Josephson junctions101B and102B are I0′(1+x′) and I0′(1−x′), respectively. A critical current value of the SQUID10B (a maximum value of the current that can flow through the SQUID) can be given by the following equation (25):

2⁢I0′⁢cos2(π⁢ΦBΦ0)+x′2⁢sin2(π⁢ΦBΦ0)(25)
where ΦBis a magnetic flux penetrating through the loop of the SQUID10B.

From the above equation (24), the critical current value IAof the SQUID10A takes:

a maximum value: 2I0when the magnetic flux ΦAis an integral multiple of the magnetic flux quantum Φ0; and

a minimum value: 2I0x when the magnetic flux ΦAis a half-integer multiple of the magnetic flux quantum Φ0.

From the above equation (25), the critical current value IBof the SQUID10B takes:

a maximum value: 2I0′ when the magnetic flux ΦBis an integral multiple of the magnetic flux quantum Φ0; and

a minimum value: 2I0′x′ when the magnetic flux ΦBis a half-integer multiple of the magnetic flux quantum Φ0.

With respect to the magnetic fluxes ΦAand ΦBpenetrating through the loops of the SQUIDs10A and10B, respectively, there are four combinations of a sum (IA+IB) of the current values flowing through the SQUIDs10A and10B, respectively:
a) 2I0+2I0′(ΦA/Φ0=n, ΦB/Φ0=n′)  (26)
b) 2xI0+2I0′(ΦA/Φ0=½+n, ΦB/Φ0=n′)  (27)
c) 2I0+2x′I0′(ΦA/Φ0=n, ΦB/Φ0=n′+½)  (28)
d) 2xI0+2x′I0′(ΦA/Φ0=½+n, ΦB/Φ0=½+n′)  (29)
These correspond to the resonance operation points. That is, there are four resonance operation points in a range where ΦA/Φ0and ΦB/Φ0are from 0 to ½.

In the following, the resonance frequency of the resonator20is assumed to be given by the following equation (30):

f=ω2⁢π=12⁢π⁢LC=12⁢π⁢{Φ0/2⁢(IA+IB}⁢C=IA+IBπ⁢2⁢Φ0⁢C(30)

The resonance frequencies fa, fb, fc, and fdat the four resonance operation points of the above equation (26) to (29) are given by equations (31) to (34):

fa=I0+I0′π⁢2⁢Φ0⁢C(31)fb=xI0+I0′π⁢2⁢Φ0⁢C(32)fc=I0+x′⁢I0′π⁢2⁢Φ0⁢C(33)fd=xI0+x′⁢I0′π⁢2⁢Φ0⁢C(34)
(A) In a case where the SQUIDs10A and10B have average critical current values equal but asymmetries different to each other, i.e., I0=I0′ and x≠x′,
fa>fb, fc>fd(35)
A magnitude relationship between fband fcis swapped depending on a magnitude relationship between x and x′.
When x>x′,
fa>fb>fc>fd(36)
Whenx<x′,
fa>fc>fb>fd(37)
Therefore, there are four different resonance operation points.
(B) In a case where the SQUIDs10A and10B have asymmetries equal but average critical current values different to each other, i.e., I0≠I0′, and x=x′,
fa>fb,fc>fd(38)
A magnitude relationship between fband fcis swapped depending on a magnitude relationship between I0and I0′.
When I0<I0′,
fa>fb>fc>fd(39)
When I0>I0′,
fa>fc>fb>fd(40)
Therefore, there are four different resonance operation points.
(C) In a case where the SQUIDs10A and10B have average critical current values and asymmetries, both different to each other, i.e., I0≠I0′, and x≠x′,
fa>fb, fc>fd(41)
A magnitude relationship between fband fcis swapped depending on a magnitude relationship between I0and I0′ and that between x and x′.
That is,
whenI0′/I0>(1−x)/(1−x′),
fa>fb>fc>fd(42)
WhenI0′/I0<(1−x)/(1−x′),
fa>fc>fb>fd(43)
Therefore, there are four different resonance operation points.
However, whenI0′/I0=(1−x)/(1−x′),
fa>fb=fc>fd(44)
In this case, the number of resonance operation points is degenerated to three. Therefore, in the case (C) where the SQUIDs10A and10B have the average critical current values (I0, I0′) and the asymmetries (x, x′) both different from each other, the SQUIDs10A and10B 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 SQUIDs10A and10B have average critical current values and asymmetries, both equal to each other, i.e., I0=I0′, and x=x′,
fa>fb=fc>fd(45)
There are three resonance operation points.

In the two asymmetric SQUIDs10A and10B connected in parallel, when the current values (I0, I0′), which are one half of the sum (2I0, 2I0′) 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, 22=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 I0and/or I0′ which is one half of the sum: 2I0and/or 2I0′ 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 resonator20that includes the SQUIDs10A and10B, a direct current is applied from the flux lines14A and14B to apply a static magnetic field to the SQUIDs10A and10B, respectively. It is noted that inFIG.2A, with a signal of frequency ω0being supplied from an input/output line13and the resonance frequency when a static magnetic field is applied to the SQUIDs10A and10B being ω0, by applying from the flux lines14A and14B a sufficiently strong pump beam (microwave current+direct current) of frequency ωpclose to twice the resonance frequency ω0, parametric oscillation may be invoked outside an operation point resistant to magnetic field noise under conditions where there is a magnetic field gradient.

FIG.2Bis a diagram showing a calculation result of resonance frequencies of the resonator20ofFIG.2Ausing a contour line diagram. X-axis corresponds to ΦA/Φ0and Y-axis corresponds to ΦB/Φ0(ΦAand ΦBare magnetic fluxes penetrating through loops of the SQUIDs10A and10B ofFIG.2A, respectively). InFIG.2B, 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 SQUIDs10A and10B to reduce contribution of a closed loop current due to the loops between the adjacent SQUIDs10A and10B. For instance, for the SQUIDs10A and10B processed to a micrometer size, a distance therebetween may be on the order of millimeters.

FIG.3is a diagram illustrating a lumped element resonator20.FIG.3schematically illustrates a part of a wiring pattern (plane circuit) of the resonator20with SQUIDs10A and10B and a single electrode15formed on the circuit surface (main surface) of a silicon substrate. Areas (gray colored) of the electrode15and a ground pattern16, indicate areas where a superconducting thin film is vapor-deposited on a silicon substrate, and a white portion18indicates an exposed area of the silicon substrate (a gap of a coplanar waveguide). The electrode15is of a cruciform shape with four arms extending to top, bottom, left and right. The resonator20is formed of a coplanar plane circuit in which a signal line and the ground pattern16surrounding the signal line (signal electrode) are placed on the same plane on the silicon substrate. InFIG.3, a capacitor11inFIG.2Ais formed in a gap between the electrode15and the ground pattern16facing each other. InFIG.3, one end of each of the two SQUIDs10A and10B is connected to one end of the electrode15, and the other end of each of the two SQUIDs10A and10B is connected to the ground pattern16. The electrode15has a cruciform shape in which a first pattern (first and second arms) having both ends along a length connected to one ends of the SQUIDs10A and10B intersects a second pattern (third and four arms) having one end along a length capacitively coupled to an input/output line13. It is noted that the planar shape of the electrode15is not limited to the example illustrated inFIG.3.

The electrode15and the ground pattern16may be made of superconducting materials such as Nb and Al. The SQUIDs10A and10B 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 lines14A and14B. The ground pattern16is provided on both longitudinal sides of each of the flux lines14A and14B. The ground pattern16are arranged facing via a gap with each longitudinal side of each of the flux lines14A and14B. The flux lines14A and14B have longitudinal one ends made in contact with one longitudinal sides of line-shaped ground patterns (ground lines)16A and16B, respectively. The ground lines16A and16B face the SQUIDs10A and10B, respectively, on other longitudinal sides. On the ground pattern16(the ground pattern provided facing a side of each of the flux lines14A and14B in the longitudinal direction with a gap therebetween), notches17A and17B are provided running along ground lines16A-1and16B-1that are made in contact with the longitudinal one ends of the flux lines14A and14B, respectively, and extend in directions orthogonal to the longitudinal directions of the flux lines14A and14B.

A current flowing through the flux line14A (or14B) is divided at the one longitudinal end thereof to the ground line16A-1and a ground line16A-2(or the ground line16B-1and a ground line16B-2). A current flowing through the ground line16A-2(or16B-2) and a current flowing through the ground line16A-1(or16B-1) in an opposite direction do not cancel out a magnetic field applied to the loop of the SQUID10A (or the SQUID10B). That is, a line length of the ground line16A-1extending along the notch17A is longer than the ground line16A-2by approximately a length of the notch17A, and a magnetic field generated by the current flowing through the ground line16A-1(a first magnetic field penetrating through a loop of the SQUID10A) is larger than a magnetic field generated by the current flowing through the ground line16A-2(a second magnetic field penetrating through the loop of the SQUID10A in the opposite direction to the first magnetic field). As a result, the configuration of the flux line14A and the ground lines16A-1and16A-2illustrated inFIG.3enables efficient generation of the magnetic field applied to the loop of the SQUID10A. Likewise, regarding the flux line14B, since a magnetic field generated by a current flowing through the ground line16B-1(a first magnetic field penetrating through a loop of the SQUID10B) is larger than a magnetic field generated by a current flowing through the ground line16B-2(a second magnetic field penetrating through the loop of the SQUID10B in an opposite direction to the first magnetic field), the magnetic field applied to the loop of the SQUID10B can be efficiently generated. Line widths of the ground lines16A-1and16A-2(or16B-1and16B-2) do not have to be the same and may differ from each other, such as the ground line16A-1(or16B-1) being wider than the ground line16A-2(or16B-2). It is noted that the flux lines14A and14B illustrated inFIG.3are merely examples, and any configuration other than that inFIG.3may, 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.

InFIG.3, the resonator20has two SQUIDs10A and10B connected in parallel, but the number of SQUIDs is not limited to two.

In an example illustrated inFIG.4, four SQUIDs10A,10B,10C, and10D are connected between the electrode15and the ground pattern16, each shown inFIG.3. Flux lines14A,14B,14C, and14D are provided for the four SQUIDs10A,10B,10C, and10D, 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 lines14A,14B,14C, and14D. For the four SQUIDs10A,10B,10C, and10D 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 24=16 resonance operation points having mutually different resonance frequencies.

InFIGS.2A,3, and4, 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 inFIG.5may be used. Referring toFIG.5, a waveguide (λ/4 waveguide)19having a length close to a quarter of a resonance wavelength A. (a wavelength of a standing wave) is provided between an input/output capacitor12and a node107which is a common node between first nodes105A and105B of the SQUIDs10A and10B. The λ/4 waveguide19is terminated at the ground via the SQUIDs10A and10B. InFIG.5, a capacitor11illustrated inFIG.2Ais not shown which is connected in parallel to the SQUIDs10A and10B of the lumped element resonator20. In the distributed element resonator20inFIG.5, a distributed capacitance includes capacitance components between the SQUIDs10A and10B and the ground pattern and a capacitance component between the λ/4 waveguide19and the ground pattern. Since each of the Josephson junctions101A,102A,101B, and102B of the SQUIDs10A and10B also has a minute capacitance component, the distributed capacitance may include the capacitance components of these Josephson junctions.

FIGS.6A and6Bare diagrams illustrating a second example embodiment of the present invention. Referring toFIG.6B, a resonator20includes two electrodes15A and15B bridged by two SQUIDs10A and10B. As illustrated inFIGS.6A and6B, first nodes105A and105B of the two SQUIDs10A and10B are connected to the first electrode15A (i.e., a common connection node107of the first nodes105A and105B is the first electrode15A). The first electrode15A is connected to a first input/output line13A via a first input/output capacitor12A. Second nodes106A and106B of the two SQUIDs10A and10B are connected to the second electrode15B (i.e., a common connection node108of the second nodes106A and106B is the second electrode15B). The second electrode15B is connected to a second input/output line13B via a second input/output capacitor12B.

InFIG.6B, the flux lines14A and14B supplied with currents supplied thereto generate magnetic fluxes penetrating through the loops of the SQUIDs10A and10B, respectively. The flux lines14A and14B, the ground lines16A-1,16A-2,16B-1, and16B-2, and the notches17A and17B have the same patterns and functions as those inFIG.3described above. Since the example illustrated inFIG.6Bis 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 inFIG.4.

In the second example embodiment, the resonator20may be constituted as a distributed element resonator such as a λ/2 resonator as shown inFIG.7. The first nodes105A and105B of the SQUIDs10A and10B are connected to one end of a waveguide (λ/4 waveguide)19A having a length close to a quarter of a resonance wavelength A. (the wavelength of a standing wave), and the other end of the waveguide19A is connected to the first input/output line13A via the first input/output capacitor12A. The second nodes106A and106B of the SQUIDs10A and10B are connected to one end of a waveguide (λ/4 waveguide)19B having a length close to a quarter of the resonance wavelength (the wavelength of the standing wave), and the other end of the waveguide19B is connected to the second input/output line13B via the second input/output capacitor12B. InFIG.7, the capacitor11is not shown which is connected in parallel to the SQUIDs10A and10B of the lumped element resonator20as illustrated inFIG.2. In the distributed element resonator20inFIG.7, a distributed capacitance includes, for instance, capacitance components between the SQUIDs10A and10B and the ground pattern and capacitance components between the λ/4 waveguides19A and19B and the ground pattern. Since each of the Josephson junctions101A,102A,101B, and102B of the SQUIDs10A and10B also has a minute capacitance component, the distributed capacitance may include the capacitance components of these Josephson junctions.

InFIG.6A, a connection node of the first nodes105A and105B and a connection node of the second nodes106A and106B of the SQUIDs10A and10B are connected to the first and the second input/output lines13A and13B, via the first and the second input/output capacitors12A and12B, 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 nodes105A and105B and the connection node of the second nodes106A and106B of the SQUIDs10A and10B 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 inFIG.8, in a case where a node107which is a common connection node of the first nodes105A and105B of the SQUIDs10A and10B, and a node108which is a common connection node of the second nodes106A and106B, are connected to first and second qubits (quantum bits)22A and22B, respectively, the resonator20operates as a qubit coupler which causes a plurality of qubits to be mutually couple.

As stated above, the resonator20of 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 inFIG.9, four resonators20A to20D interconnects with each other via a coupling circuit (qubit coupler)21. The coupling circuit21, which couples the four resonators20A to20D, may include a Josephson junction213. The resonators20A and20B are connected via capacitors211A and211B to one end of a superconducting conductor (electrode)212-1of the coupling circuit21, where the other end of the superconducting conductor212-1is connected to one end of the Josephson junction213. The resonators20C and20D are connected via capacitors211C and211D to one end of a superconducting conductor (electrode)212-2of the coupling circuit21, where the other end of the superconducting conductor212-2is connected to the other end of the Josephson junction213. InFIG.9, the quantum computer with four resonators20A to20D (qubits) is illustrated, but a quantum computer on which any number of the resonators are integrated may be realized by using the configuration illustrated inFIG.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 I1and I2flowing through the two Josephson junctions of an asymmetric SQUID are as follows:
I1=I0(1+x)  (A.1)
I2=I0(1−x)  (A.2)
I1+I2=2I0(A.3)

The current I flowing through the asymmetric SQUID is given as follows:
I=I0(1+x)*sin(γA)+I0(1−x)*sin(γB)  (A.4)

The equation (A.4) can be rewritten by using
γB−γA=2πΦ/Φ0(A.5)
as follows:

I=I0(1+x)*sin⁡(γA)+I0(1-x)*sin⁡(γA-2⁢π⁢Φ/Φ0)}=I0⁢{sin⁡(γA)+sin⁡(γA-2⁢π⁢Φ/Φ0)}+xI0⁢{sin⁡(γA)-sin⁡(γ⁢A-2⁢π⁢Φ/Φ0)}=2⁢I0[cos⁡(π⁢Φ/Φ0)⁢sin⁢{γA-π⁢Φ/Φ0)}]+2⁢xI0[sin⁡(π⁢Φ/Φ0)⁢cos⁢{γA-π⁢Φ/Φ0)}]=2⁢I0[cos⁡(π⁢Φ/Φ0)⁢{sin⁡(γA)⁢cos⁡(π⁢Φ/Φ0)-sin⁡(π⁢Φ/Φ0)⁢cos⁡(γA)}]+2⁢xI0[sin⁡(π⁢Φ/Φ0)⁢{cos⁡(γA)⁢cos⁡(π⁢π⁢Φ/Φ0)+sin⁡(γA)⁢sin⁡(π⁢Φ/Φ0)}]=2⁢I0⁢{cos2(π⁢Φ/Φ0)+x⁢sin2(π⁢Φ/Φ0)}⁢sin⁡(γA)+2⁢I0⁢{-sin⁡(π⁢Φ/Φ0)⁢cos⁡(π⁢Φ/Φ0)+x⁢cos⁡(π⁢Φ/Φx)⁢sin⁡(π⁢Φ/Φ0)}⁢cos⁡(γA)=2⁢I0⁢{cos2(π⁢Φ/Φ0)+x⁢sin2(π⁢Φ/Φ0)}⁢sin⁡(γA)-2⁢I0(1-x)⁢sin⁡(π⁢Φ/Φ0)⁢cos⁡(π⁢Φ/Φ0)⁢cos⁡(γA)(A.6)
Letting
α={cos2(πΦ/Φ0)+xsin2(πΦ/Φ0)}  (A.7)
and
β=(1−x)sin(πΦ/Φ0)cos(πΦ/Φ0)  (A.8)
the equation (A.6) can be rewritten as follows:
I=2I0[α sin(γA)+β cos(γA)]=2I0√(α2+β2)sin(γA+C)   (A.9)
where
cos(C)=α/√(α2+β2), sin(C)=β/√(α2+β2)  (A.10)

Calculating inside the square root of equation (A.9) gives the following equation (A.11):

√(α2+β2)=([cos2(π⁢Φ/Φ0)+x⁢sin2(π⁢Φ/Φ0)]2+(A.11)[(1-x)⁢sin⁡(π⁢Φ/Φ0)⁢cos⁡(π⁢Φ/Φ0)]2)1/2=([cos4(π⁢Φ/Φ0)+cos2(π⁢Φ/Φ0)⁢sin2(π⁢Φ/Φ0)]+x2[sin4(π⁢Φ/Φ0)+cos2(π⁢Φ/Φ0)⁢sin2(π⁢Φ/Φ0)])1/2=([cos2(π⁢Φ0)⁢(cos2(π⁢Φ/Φ0)+sin2(π⁢Φ/Φ0)]+x2[sin2(π⁢Φ/Φ0)⁢(sin2(π⁢Φ/Φ0)+cos2(π⁢Φ/Φ0))])1/2=√{cos2(π⁢Φ/Φ0)+x2⁢sin2(π⁢Φ/Φ0)}

From above, the equation (A.9) can be rewritten as follows:
I=2I0{cos2(πΦ/Φ0)+x2sin2(πΦ/Φ0)}1/2sin(γA+C)=
Asin(γA+C)  (A.12)
where
A=2I0{cos2(θ)+x2sin2(θ)}1/2(θ=πΦ/Φ0)  (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.