阅读说明：本技术 Josephson行波参量放大器 (Josephson traveling wave parametric amplifier ) 是由 维萨·维斯特林 朱哈·哈塞尔 于 2020-01-08 设计创作，主要内容包括：根据本发明的一个示例方面,提供了一种行波参量放大器,其包括：波导传输线,其中包括至少十个Josephson元件,其中至少十个Josephson元件中的每一个都包括回路,具有在回路的一半上的恰好一个第一尺寸的Josephson结以及在回路的另一半上的至少两个第二尺寸的Josephson结,第二尺寸大于第一尺寸；磁通偏置线,被配置为生成穿过至少一个回路中的每一个的磁通量；以及一组电阻器,与磁通偏置线耦接。(According to an example aspect of the present invention, there is provided a traveling wave parametric amplifier comprising: a waveguide transmission line comprising at least ten Josephson elements, wherein each of the at least ten Josephson elements comprises a loop having exactly one Josephson junction of a first size on one half of the loop and at least two Josephson junctions of a second size on the other half of the loop, the second size being larger than the first size; a flux bias line configured to generate a magnetic flux through each of the at least one loop; and a set of resistors coupled to the flux bias lines.)

1. A traveling wave parametric amplifier, comprising:

-a waveguide transmission line comprising at least ten Josephson elements, wherein each of the at least ten Josephson elements comprises a loop with exactly one Josephson junction of a first size on one half of the loop and at least two Josephson junctions of a second size on a second half of the loop, the second size being larger than the first size;

-a flux bias line configured to generate a magnetic flux through each of at least one of the loops; and

-a set of resistors coupled with the flux bias line.

2. The traveling wave parametric amplifier of claim 1, wherein each of the at least ten Josephson elements exhibits no or negligible kerr nonlinearity contributions, and wherein each of the at least ten Josephson elements exhibits a three-wave mixing.

3. The traveling wave parametric amplifier of any one of claims 1 to 2, wherein a ratio of the Josephson energy of the first sized junction to the Josephson energy of the second sized junction is configured to partially or completely cancel the Kerr nonlinearity.

4. A travelling wave parametric amplifier according to claim 3, wherein the ratio of the Josephson energies is configured by the area of the junction.

5. The traveling wave parametric amplifier of claim 3, wherein the ratio of the Josephson energies is configured by a superconducting critical current density of the junction.

6. The traveling wave parametric amplifier of any one of claims 1 to 5, wherein the traveling wave parametric amplifier is configured to enable a current to be generated in the flux bias line such that a magnetic flux passing through the loop corresponds to an operating point that minimizes the Kerr nonlinearity.

7. The traveling wave parametric amplifier of any one of claims 1 to 6, wherein the magnetic flux passing through each of the at least one loops is 0.40 times the magnetic flux quantum, and wherein each of the at least ten Josephson elements comprises exactly two second-sized junctions on the second half of the loop.

8. The traveling wave parametric amplifier of claim 7, wherein the Josephson energy of the first size junction is 0.27 times the Josephson energy of the second size junction.

9. A travelling wave parametric amplifier according to any one of claims 3 to 5, wherein the ratio of the Josephson energies and the magnetic flux through the loop are configured such that operation is first order insensitive to variations in the smaller Josephson energies.

10. The traveling wave parametric amplifier of any one of claims 1 to 9, wherein the waveguide transmission line comprises more than fifteen Josephson elements.

11. The traveling wave parametric amplifier of any one of claims 1 to 10, wherein the flux bias line forms an upper or lower electrode of a parallel plate forming a branch capacitance of the waveguide transmission line.

12. The traveling wave parametric amplifier of claim 11, wherein the flux bias line is connected to a ground plane of the waveguide transmission line through a resistor having a smaller value than a reactive impedance of the branch capacitor at a frequency at which the traveling wave parametric amplifier is configured to amplify.

13. The traveling wave parametric amplifier of any one of claims 11 to 12, wherein the traveling wave parametric amplifier is configured to apply a direct current in the flux bias line to generate a magnetic field gradient.

14. The traveling wave parametric amplifier of claim 13, wherein each of the loops is configured in a gradient configuration to be insensitive to a uniform magnetic field from the environment.

15. The traveling wave parametric amplifier of any one of claims 11 to 14, wherein the waveguide transmission line comprises two groups of Josephson elements, the groups being separated from each other by a branch capacitor arranged on the waveguide transmission line.

16. The traveling wave parametric amplifier of claims 11 to 15, wherein the values of the branch capacitances are not constant along the length of the transmission line to compensate for microwave attenuation along the transmission line.

17. The traveling wave parametric amplifier of any one of claims 11 to 15, wherein the value of the critical current is not constant along the length of the transmission line to compensate for microwave attenuation along the transmission line.

18. The travelling wave parametric amplifier of claim 16 or 17, wherein the branch capacitance value variation or critical current value variation is designed configured such that: the corresponding current distribution along the transmission line is constant or nearly constant when exposed to a pumping microwave signal.

19. The traveling wave parametric amplifier of any one of claims 16, 17, or 18, wherein the microwave attenuation is achieved by dielectric losses of the branch capacitance.

20. The traveling wave parametric amplifier of any one of claims 16, 17, 18, or 19, wherein the microwave attenuation is achieved by losses in the resistors of the waveguide transmission line.

21. The traveling wave parametric amplifier of any one of claims 12 or 20, wherein both the resistor and branch capacitance values are configured to be optimized for a uniform microwave current distribution along the transmission line.

22. The traveling wave parametric amplifier of any one of claims 1 to 21, further comprising an impedance matching device at least one end of the waveguide transmission line.

23. The traveling wave parametric amplifier of claim 22, wherein the impedance matching device comprises a tapered transmission line matching element.

24. The traveling wave parametric amplifier of claim 23, wherein the tapered transmission line matching element comprises a Klopfenstein taper.

25. The traveling wave parametric amplifier of claim 23, wherein the tapered transmission line matching element comprises an exponential taper.

26. A method of fabricating a traveling wave parametric amplifier, comprising:

-providing a waveguide transmission line comprising at least ten Josephson elements, wherein each of the at least ten Josephson elements comprises a loop with exactly one Josephson junction of a first size on one half of the loop and at least two Josephson junctions of a second size on a second half of the loop, the second size being larger than the first size;

-providing a magnetic flux bias line configured to generate a magnetic field across each of at least one of the loops; and

-providing a set of resistors coupled to the flux bias line.

Technical Field

The present invention relates to superconducting Traveling Wave Parametric Amplifiers (TWPAs).

Background

A parametric amplifier is actually a mixer in which a weaker input signal can be amplified by mixing it with a stronger pumping signal, resulting in a stronger output signal. Parametric amplifiers rely on the nonlinear response of the physical system to produce amplification. Such amplifiers may include standing wave parametric amplifiers or traveling wave parametric amplifiers, wherein the traveling wave parametric amplifiers use a series of nonlinear elements distributed along a transmission line, such as coplanar waveguides, for example. Where the nonlinear element comprises a Josephson (Josephson) junction, the amplifier may be referred to as a Josephson Travelling Wave Parametric Amplifier (JTWPA). In JTWPA, the Josephson junction remains in a superconducting state and carries the supercurrent.

In use, a signal is added to the strong oscillator signal, a sum signal is generated in which an amplitude envelope (amplitude envelope) exhibits a variation with frequency as the difference between the signal frequency and the oscillator frequency. Since the phase velocity depends on the amplitude in a waveguide transmission line, the phase of the sum signal at the end of the line will vary according to the difference between the two frequencies. In effect, the nonlinear waveguide transmission line converts amplitude modulation to phase modulation. If the non-linearity is strong enough, this will result in a gain at the signal frequency.

Disclosure of Invention

According to some aspects, the subject matter of the independent claims is provided. Some embodiments are defined in the dependent claims.

According to a first aspect of the present invention, there is provided a travelling wave parametric amplifier comprising: a waveguide transmission line comprising at least ten Josephson elements, wherein each of the at least ten Josephson elements comprises a loop having exactly one Josephson junction of a first size on one half of the loop and at least two Josephson junctions of a second size on the other half of the loop, the second size being larger than the first size; a flux bias line configured to generate a magnetic flux through each of the at least one loop; and a set of resistors coupled to the flux bias lines.

According to a second aspect of the present invention, there is provided a method of manufacturing a travelling wave parametric amplifier, comprising: providing a waveguide transmission line comprising at least ten Josephson elements, wherein each of the at least ten Josephson elements comprises a loop having exactly one Josephson junction of a first size on one half of the loop and at least two Josephson junctions of a second size on the other half of the loop, the second size being larger than the first size; providing a magnetic flux bias line configured to generate a magnetic field through each of the at least one loop; and providing a set of resistors coupled to the flux bias lines.

Drawings

FIG. 1 illustrates an example amplifier according to at least some embodiments of the inventions;

FIG. 2 shows an example Josephson element according to document [2 ];

FIG. 3 illustrates an example Josephson element in accordance with at least some embodiments of the invention; and

FIG. 4 is a flow chart of a method according to at least some embodiments of the inventions.

Detailed Description

According to the solution disclosed herein, by appropriately selecting the parameters, the traveling wave parametric amplifier can be made less sensitive to dimensional errors of the smaller Josephson junction in a Josephson element comprising larger and smaller Josephson junctions. Furthermore, a gradient layout of a Josephson element is disclosed which makes the element sensitive only to magnetic field gradients, and flux bias lines in a travelling wave parametric amplifier are provided to generate such field gradients. Furthermore, the transmission line may be tapered. In general, using one or more of these enhancements, a traveling wave parametric amplifier may be more suitable for use in practical applications because its operation is less sensitive to disturbances and manufacturing defects.

FIG. 1 illustrates an example amplifier according to at least some embodiments of the inventions. In general, in quantum computing, for example, a signal may decay to even a single photon or near single photon state (region) due to transmission. Detecting such signals presents challenges due to their low amplitude. Thus, a suitable amplifier may be used to increase the amplitude of the received signals, into which the information encoded may be recovered, before they are supplied to the detector elements. As another example, single photon state communication may be used to communicate encryption keys in a secure manner using quantum communication, making imperceptible eavesdropping very difficult.

The present disclosure focuses on superconducting implementations of TWPAs in which the central trace of the transmission line is an array of elements based on Josephson junctions (referred to as Josephson elements), which constitute nonlinear inductances. The non-linearity allows a mixing process that provides power gain for weak signals that propagate in the same direction as the strong radio frequency, rf, pump tone (pump tone). The intensity of the pump tone is measured by the ratio between the pump current amplitude Ip and the critical current Ic of the Josephson element. The nature of the non-linearity depends on the arrangement of the Josephson junctions within the element. The simplest implementation is to use a single Josephson junction as the nonlinear element: the relative Taylor spread of the inductance is a constant plus a term proportional to (Ip/Ic) ^2, i.e., Kerr nonlinearity. Although the kerr term results in the required mixing process, it also changes the wave vector of the pump tone, which effect must be compensated for by dispersion engineering. The balance of the wavevectors, also known as phase matching, allows the TWPA gain to grow exponentially as a function of device length. Since the transmission line has embedded therein the typical narrow-band dispersion characteristic, the center frequency of the gain is a fixed amount in this example of a TWPA.

There is a goal to achieve a new solution without kerr non-linearity by introducing magnetic flux degrees of freedom into the Josephson element. In the Taylor expansion of the inductance, this alternative nonlinearity is a term proportional to Ip/Ic. Kerr-free operation is beneficial because no dispersion engineering is required to achieve phase matching. The pumping frequency at which the center frequency of the gain is set can be freely selected. The main features of a typical kerr-free element include (i) a superconducting magnetic pick-up loop, which can be described as two half-loops connected together, (ii) interference of unequal numbers of halves of a Josephson junction, and (iii) a finite magnetic flux bias that causes a shield current to flow in the loop according to the principle of fluxization. In current implementations, the particular weaknesses of the kerr-free elements are (i) sensitivity to magnetic interference and (ii) sensitivity to inhomogeneities in the magnetic bias field, especially in arrays consisting of a plurality of elements.

Furthermore, a common problem in TWPAs is the depletion of the pump current. This is due to dissipation in the transmission line or power transfer from the pump to the amplified signal in case the TWPA is operating near saturation. Pump depletion limits TWPA gain because the mixing process depends on a suitable ratio between Ip and Ic. Another common problem in TWPAs is the manufacturing spread (fabrication spread) of ics, resulting in non-uniformity of transmission line electrical parameters.

The JTWPA of fig. 1 includes a waveguide that includes Josephson elements 110 and parallel plate capacitors 120. The Josephson elements 110 are connected to each other by waveguides capable of transmitting electromagnetic waves, as is known in the art. A part of the waveguide is shown in fig. 1 with an input port on the left side arranged to receive the signal to be amplified and the strong oscillator signal, which are mixed in the waveguide in a non-linear Josephson element 110. At the output port on the right side, a phase-modulated amplified signal is obtained as an output. For example, both the wiring layer elements 101 may include a superconductor covered with an insulator.

In general, Josephson elements, such as single junction, superconducting quantum interference devices (SQUIDs), asymmetric SQUIDs, or more complex Josephson elements, such as magnetic flux qubit circuits, can be described using effective potential energy:

here E_jIs Josephson energy, andis a superconducting phase. c. C₂The term is related to the critical current and linear part of the Josephson inductance, c₃The term is related to 3-wave mixing, and c₄The term is related to 4-wave mixing, which is also known as kerr nonlinearity.

Typically single junctions and SQUIDs, including asymmetric SQUIDs, having c₃0, so no 3-wave mixing occurs, and the non-linearity is provided by the kerr term. 3-wave mixing implies the ability to pump the input frequency at twice, which is ideal. The 3-wave mixing can be activated by injecting a direct current, but the kerr term will remain non-zero.

The non-linearity provided by the kerr term is associated with the need for resonant phase matching, and in fact the pump signal is given small phase increments at regular intervals along the transmission line. This is because the pump has a different phase speed than the signal (frequency fP) and idle (idler) (frequency fI). This phase mismatch increases with pump power. Conservation of energy means that there is an idle frequency at the output, whose frequency is at the "mirror" of the signal frequency with respect to the pump, fI-2 fP-fS. In detail, in kerr mode, the phase mismatch and gain depend on the same parameters, kerr non-linearity. In the case of 3-wave mixing, these three frequencies are correlated by fS + fI ═ fP. To minimize the amount of reflection, both ends of the TWPA also need to have good impedance matching at each frequency fI, fS, and fP.

Therefore, it is preferable to use 3-wave mixing (i.e., using c with Kerr nonlinearity suppressed)₃Term) without using 4-wave mixing to operate the TWPA. Thus, the amplifier can be constructed without the equipment to provide the periodic phase increment required for the kerr mode. In the 3-wave mode, the phase mismatch and gain depend on different nonlinear terms.

In particular, the present invention sets out to solve or at least mitigate the following problems: first, susceptibility to magnetic interference in kerr-free TWPAs. The interference may impair the ultra low noise performance of the TWPA. Second, sensitivity to inhomogeneities in the magnetic bias field in a kerr-free TWPA. Third, the fabrication profile of the Josephson junction affects the critical current in the TWPA. These effects cause changes in the transmission line impedance, which is a potential source of reflection. The reflections may cause standing waves that will periodically introduce into the frequency response of the TWPA or even prevent the mixing process from providing gain. Fourth, depletion of the pumping current in the TWPA. This limits the maximum gain of the TWPA.

Zorin in [1]Therein describes a solutionSolution (b) wherein c₃And c₄The balance between the mixing can be controlled by applying a suitable external magnetic field to the rf-SQUID. Thus, dominant 3-wave mixing can be achieved in the system of Zorin.

Frattini et al in [2] describe a magnetic flux qubit circuit that simultaneously nulls the Kerr mixing term and maximizes the 3-wave mixing term. This circuit, named "Superconducting Nonlinear Inductive eLement" by the author of document [2] (i.e., SNAIL), is modified as described herein to implement the Josephson eLement 110 in this embodiment. In detail, in [2], the Josephson element has three large Josephson junctions on one half of the loop and one small Josephson junction on the other half of the loop. In the present solution, a Josephson element is used with at least two large Josephson junctions on one half of the loop and one small Josephson junction on the other half of the loop. This will be shown later in fig. 2 and 3.

The JTWPA of fig. 1 has parallel plate capacitors 120 interspersed between Josephson elements 110 in the waveguide. Two Josephson elements 110 between each two parallel plate capacitors 120 is an example and the invention is not limited in this regard and, in fact, in embodiments, there may be three or more Josephson elements 110 between each two parallel plate capacitors 120. The parallel plate capacitor 120 forms the majority of the shunt capacitance (shunt capacitance) of the transmission line. The JTWPA of fig. 1 is a coplanar waveguide.

The JTWPA of fig. 1 is also equipped with a flux bias line FBL 130. The flux bias line 130 is a two-port circuit that takes a serpentine path, ranging from one side of the coplanar waveguide to the other. As shown in fig. 1, the flux bias line 130 forms the upper electrode of the parallel plate capacitor 120 at the location where it crosses the other side of the waveguide. The flux bias line 130 is connected to the ground plane of the transmission line through a resistor 140, the value of the resistor 140 being much smaller than the reactive impedance (reactive impedance) of the capacitor 120 at the relevant frequencies f1, fS and fP. The purpose of the resistor 140 is to provide an rf path from the parallel plate capacitor 120 to ground. Meanwhile, resistor 140 and flux bias line 130 implement similar potentials for the ground plane at frequencies fI, fS, and fP.

As shown, the flux bias line 130 extends parallel to the waveguide on one side of the waveguide and then extends to the other side of the waveguide at a location corresponding to one of the parallel plate capacitors 120 to extend parallel to the waveguide again on the other side of the waveguide. In the case where the flux bias line 130 extends parallel to the waveguide, it may be connected with resistors 140, as shown, each of which may form a loop around a contact hole 150. Resistor 140 includes metal layers in the present multi-layer JTWPA. The resistor 140 partially covers the superconducting material to form the contact, where the resistive aspect of the resistor 140 is created where the resistor 140 covers an insulator rather than a superconductor.

The operating parameters of the Josephson elements 110 include that these elements have at least two large Josephson junctions on one half of the loop and one small Josephson junction on the other half of the loop. In particular, it is possible to have two and only two large Josephson junctions on one half of the loop and one and only one small Josephson junction on the other half of the loop. Furthermore, the critical current of a small junction is smaller by an alpha factor compared to the critical current of a larger junction. In the present Josephson element 110, alpha may be 0.27. Furthermore, in the present solution, the magnetic flux of the circuit through the element 110 may be 0.40 times the quantum of the magnetic flux. Thus, one combination of parameters may be two large Josephson junctions and one small Josephson junction, which are related by 0.27, with a magnetic field of 0.40 times the quantum of magnetic flux.

The dissipation of the resistor increases the dielectric loss of the parallel plate capacitor 120. The dc current in the flux bias line 130 generates a magnetic field gradient for the Josephson element 110. Resistors 140 prevent this current from leaking to the ground plane, and they also prevent superconducting loops from forming from the ground plane and the intersection. Such superconducting loops may result in flux quantification. A current source floating with respect to the waveguide may be provided for generating a direct current in the flux bias line 130.

The amount of dissipation in the transmission line can be expressed in terms of the effective loss tangent (loss tangent) of the parallel plate capacitor 120. When the characteristic impedance along the transmission line is constant, both the pumping current and the pumping voltage experience an exponential decay due to dissipation. It would be desirable if a fixed ratio were maintained between Ip and Ic to ensure that the mixing process remained strong despite dissipation. For this purpose, a position-dependent capacitance or a position-dependent critical current can be applied. An expression for the position dependent branch capacitance consisting essentially of the parallel plate capacitor 120 is derived below. The capacitance change maintains a fixed pumping current amplitude along the transmission line at the expense of a faster decay of the pumping voltage amplitude. From input to output to TWPA, the branch capacitance 120 will increase. The characteristic impedance will be reduced accordingly and an impedance matching device may be employed at the output of the device. Examples of impedance matching devices are Klopfenstein tapering and exponential tapering

In the following, the following notation is used:

a: physical length per unit cell

G: branch conductance of unit cell

V: voltage of

C: unit cell capacitor

C₀: line capacitance at input, i.e. at x ═ 0

tan δ: loss tangent of C

ω: angular frequency

L: unit cell inductor

x: physical coordinates

Z: characteristic impedance

The power dissipated in a unit cell is read as "Re { VG^*V^*H 2 ", and the total dissipation from the TWPA input to location x is the integral:

write V ═ ZI where the current magnitude | I | is assumed to be constant, andfurther, G ═ omega. C ═ tan. delta. was inserted,

importantly, this dissipation does not change with possible changes in C. On the other hand, considering the power delivered to location x, i.e.,

such a self-consistent solution of the constant | I | presents itself as

The technical effects achieved by the present embodiment include the elimination of magnetic shielding for superconducting circuits, which typically include a combination of a high-permeability (high-permeability) layer and a superconducting layer. The gradient design of the Josephson element relaxes the magnetic shielding requirements of the kerr-free TWPA, allowing savings in system cost and size. The gradient layout of the kerr-free Josephson element makes the element sensitive only to the magnetic field gradient, and not also to the amplitude of the magnetic field. Furthermore, the ability to maintain the ratio between the pump current and the critical current at a fixed value enables higher TWPA gains to be achieved. The choice of parameters for a kerr-free Josephson element makes the cell first order insensitive to errors in the minimum Josephson junction dimensions. In addition, flux bias line 130 generates the necessary magnetic field gradients and is connected to the transmission line ground through a low value resistor. The gradual change in transmission line impedance along the line maintains a constant ratio between Ip and Ic.

Fig. 2 shows an example Josephson element according to document [2 ]. In the upper half of the image, the Josephson element is shown with three large junctions on one half of the loop and one small Josephson junction on the other half of the loop. As shown, the Josephson energies of the junctions correlate with each other by a ratio α.

The lower half of the figure shows the parameter set α being 0.29, Φ_ext/Φ₀An example potential of 0.41. In other wordsHere, the external magnetic field is 0.41 times the magnetic flux quantum. This achieves third order non-linearity without fourth order non-linearity, in other words, c₃Not equal to 0 and c₄＝0。

In the case of a Josephson element with one small junction and n large junctions, the parameter set can be determined as follows. The inductive energy of the Josephson element can be expressed as

WhereinIs the superconducting phase at the small junction, alpha is the ratio of the junction dimensions, E_JIs the Josephson energy of the large junction,is a reduced external magnetic flux:

Φ_extis an external magnetic flux and phi₀Is the magnetic flux quantum, natural constant h/(2 e). Where h is the Planck constant and e is the electron charge.

Stage 1: seek to be asMinimum value of the inductive energy of the function of (a). At said minimumIs shown asThe search may be limited to the parameter space α<1/n to avoid having multiple minima. In a parameter space alpha>In 1/n there is a risk thatThere is more than one minimum value for some of the values of (a). The multiple minimum case results in undesirable hysteresis of the Josephson element.

And (2) stage: for describing inductance energy around a minimumDependent effective potential U_effTaylor expansion is performed.

And (3) stage: c. C₂As a andis investigated to establish a function of where dc₂/dα＝0。

And (4) stage: c. C₄As a andis investigated to establish, where c₄＝0。

And (5) stage: establishing an optimal parameter pair (alpha, phi)_ext) Wherein dc₂Where/d α is 0 and c4 is 0. Here too c₃Not equal to 0. The optimal parameters for n-2 and n-3 are as follows:

n	α	Φext/Φ0	c3	c4
					2	0.27	0.40	-0.030	0
3	0.12	0.36	-0.016	0

fig. 3 illustrates an example Josephson element in accordance with at least some embodiments of the present invention. In the upper half of the figure, a Josephson element is shown, with two large Josephson junctions on one half of the loop and one smaller Josephson junction on the other half of the loop.

In the lower part of the figure, a gradient measuring Josephson element is shown, with n-2 larger Josephson junctions I₁And a smaller Josephson junction I₂As in the upper part of the figure. The superconducting portion 301 and the tunnel junction 302 are included in a Josephson element. Showing two critical currents I₁And a critical current I₂Their Josephson energies are related to each other by α, as in the upper part of the figure.

The loops are in fact relatively easy to manufacture so that they are symmetrical. An example value of a junction is I₁13.7 μ a and I₂3.7 μ a. The element being at optimum phi_extThe Josephson inductance series extension of (A) will reach 5 μ A times [1+0,50(Ip/Ic) +0,00(Ip/Ic)²+…]. Shown element pair small junction size I₂The error in (b) is first order insensitive.

FIG. 4 is a flow chart of a method according to at least some embodiments of the inventions. The stages of the method shown may be performed in, for example, a plant, an auxiliary device or a personal computer, or in a control device configured to control its functions (when installed therein).

Stage 410 includes providing a waveguide transmission line including at least ten Josephson elements therein, wherein each of the at least ten Josephson elements includes a loop having one junction of a first size on one half of the loop and at least two junctions of a second size on a second half of the loop, the second size being greater than the first size. Stage 420 includes providing a flux bias line configured to generate a magnetic field across each of the at least one loop. Stage 430 includes providing a set of resistors coupled with the flux bias lines.

As described above, the knot may comprise a Josephson knot. Once a direct current is applied to pass through the flux bias lines, the flux bias lines can generate the desired magnetic field gradient. Exactly one means one and not more than one, and exactly two means two and not more than two.

It is to be understood that the disclosed embodiments of the invention are not limited to the particular structures, process steps, or materials disclosed herein, but extend to equivalents thereof as would be recognized by those ordinarily skilled in the relevant arts. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting.

Reference throughout this specification to one embodiment or an embodiment means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. Thus, the appearances of the phrases "in one embodiment" or "in an embodiment" in various places throughout this specification are not necessarily all referring to the same embodiment. Where a numerical value is referred to using a term such as, for example, about or substantially, the exact numerical value is also disclosed.

As used herein, a plurality of items, structural elements, compositional elements, and/or materials may be presented in a common list for convenience. However, these lists should be construed as though each member of the list is individually identified as a separate and unique member. Thus, no individual member of such list should be construed as being in fact equivalent to any other member of the same list solely based on their presentation in a common population without indications to the contrary. Furthermore, embodiments and examples of the invention may be mentioned herein along with alternatives to the various components thereof. It should be understood that such embodiments, examples, and alternatives are not to be construed as being in fact equivalent to each other, but are to be regarded as separate and autonomous representations of the invention.

Furthermore, the described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the previous descriptions, numerous specific details are provided, such as examples of lengths, widths, shapes, etc., to provide a thorough understanding of embodiments of the invention. One skilled in the relevant art will recognize, however, that the invention may be practiced without one or more of the specific details, or with other methods, components, materials, and so forth. In other instances, well-known structures, materials, or operations are not shown or described in detail to avoid obscuring aspects of the invention.

While the foregoing embodiments illustrate the principles of the invention in one or more particular applications, it will be apparent to those of ordinary skill in the art that numerous modifications in form, usage and details of implementation can be made without the exercise of inventive faculty, and without departing from the principles and concepts of the invention. Accordingly, the invention is not intended to be limited except as by the appended claims.

The verbs "comprise" and "comprise" are used herein as open-ended limitations that neither exclude nor require the presence of unrecited features. The features recited in the dependent claims may be freely combined with each other, unless explicitly stated otherwise. Furthermore, it should be understood that the use of "a" or "an" throughout this document, i.e., the singular forms, does not exclude the plural forms.

INDUSTRIAL APPLICABILITY

At least some embodiments of the invention are industrially applicable to the amplification of low amplitude signals.

List of abbreviations

f1 idle frequency

fP oscillator/pump frequency

frequency of fS signal

Critical current of Ic Josephson junction

Ip pumping current amplitude

JTWPA Josephson traveling wave parametric amplifier

SQUID superconducting quantum interference device

TWPA travelling wave parametric amplifier

List of reference numerals

110	Josephson element
		120	Branch capacitor (parallel plate capacitor)
130	Magnetic flux bias line
		140	Resistor with a resistor element
150	Contact hole
		101	Wiring layer element
301	Superconducting part
		302	Tunnel junction
410–420	Stages of the method of FIG. 4

List of cited documents

[1] Zorin, "Josephson tracking-wave parameter amplifier with thread-wave mixing", arXiv 1602.026550v3,2016, 9/19 days.

[2] Frattini, U.Vool, S.Shankar, A.Narla, K.M.Sliwa and M.H.Dedevet, "3-wave missing Josephson dipole element", arXiv:1702.00869v3,2017, 6 months and 1 day.