Bandpass josephson traveling wave parametric amplifier

Abstract

A bandpass parametric amplifier circuit includes a plurality of unit cells. At least one unit cell includes a first inductor having a first node coupled to a center conductor and a second node coupled to ground. A first capacitor is present having a first node coupled to the center conductor and a second node coupled to ground. A second inductor is present having a first node coupled to the center conductor. A second capacitor has a first node coupled to a second node of the second inductor. The second capacitor and the second inductor are in series with the center conductor.

Description

Technical Field

[0001] This disclosure generally relates to superconducting devices, and more specifically to amplifiers that can be used for qubit readout. Background Technology

[0002] Superconducting quantum computing is the realization of a quantum computer within superconducting electronic circuits. Quantum computing studies the application of quantum phenomena in information processing and communication. Different models of quantum computing exist, with the most popular models including the concepts of qubits and quantum gates. A qubit is a generalization of a bit having two possible states, but can be in a quantum superposition of both states. A quantum gate is a generalization of logic gates; however, it describes the transformation that one or more qubits will undergo after the gate is applied to them, given their initial states.

[0003] The electromagnetic energy associated with a qubit can be stored in a so-called Josephson junction and in capacitive and inductive elements used to form the qubit. In one instance, to read out the qubit state, a microwave signal is applied to a microwave readout cavity coupled to the qubit at that cavity frequency. The emitted (or reflected) microwave signal passes through multiple thermal isolation stages and low-noise amplifiers, which are used to block or reduce noise and improve the signal-to-noise ratio. The amplitude and / or phase of the returned / output microwave signal carry information about the qubit state, such as whether the qubit is in a grounded state, an excited state, or a superposition of both. The microwave signal carrying quantum information about the qubit state is typically weak (e.g., on the order of a few microwave photons). To measure this weak signal, a low-noise Josephson amplifier can be used as a preamplifier (i.e., the first amplification stage) at the output of the quantum system to amplify the quantum signal and improve the signal-to-noise ratio of the output chain. In addition to Josephson amplifiers, certain Josephson microwave components that use Josephson amplifiers or Josephson mixers (such as Josephson circulators, Josephson isolators, and Josephson mixers) can be used in scalable quantum processors.

[0004] One type of Josephson amplifier is the Josephson Traveling Wave Parametric Amplifier (JTWPA), which is a Josephson parametric amplifier based on distributed elements. The device is formed by a nonlinear transmission line whose central conductor comprises a large array of Josephson junctions, which are periodically shunt to ground by lumped-element capacitors. Because it is not a resonant structure, the amplification bandwidth in a JTWPA is typically very large (e.g., 3 to 5 GHz). This amplification range can be too large for qubit readouts, which may be concentrated around 7 GHz. JTWPAs are known to amplify unrelated signals, such as qubit pulses, during qubit readouts, which can lead to instability. Furthermore, they amplify quantum noise over a wide bandwidth. Further, the wide amplification bandwidth of known JTWPAs is highly sensitive to impedance mismatch, which causes multiple reflections and consequently ripple in their gain curves. To minimize the effects of impedance mismatch at the input and output ports of a JTWPA, broadband devices with 50-ohm matching, such as magnetically based isolators, are typically added at the input and output. Adding these broadband magnetic-based devices at the inputs and outputs of the JTWPA limits its integrability with other microwave devices, which is significant for the operation of quantum processors. This application has been prepared to address these and other issues. Summary of the Invention

[0005] According to various embodiments, a bandpass parameter amplifier circuit and method are provided. The bandpass parameter amplifier circuit includes a plurality of unit cells. At least one unit cell includes a first inductor having a first node coupled to a center conductor and a second node coupled to ground. A first capacitor is present, having a first node coupled to the center conductor and a second node coupled to ground. A second inductor is present, having a first node coupled to the center conductor. The second capacitor has a first node coupled to the second node of the second inductor. The second capacitor and the second inductor are connected in series with the center conductor.

[0006] In one embodiment, the first and second inductors are nonlinear.

[0007] In one embodiment, the plurality of unit cells are part of a stepped circuit structure that creates the bandpass parameter amplifier circuit. The number of unit cells in the stepped circuit structure may be odd. Each plurality of unit cells may further include a resonant structure comprising: a third capacitor having a first node coupled to a central conductor; a third inductor having a first node coupled to a second node of the third capacitor and a second node coupled to ground; and a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground. In one embodiment, the multiplier is 3 to 9.

[0008] In one embodiment, the resonant structure in each of the plurality of unit cells defines the stopband in the gain curve of the bandpass parameter amplifier.

[0009] In one embodiment, the resonant structure in each of the plurality of unit cells provides phase matching between the pump driver and the propagation band microwave signal amplified by the bandpass parameter amplifier.

[0010] In one embodiment, there is a third capacitor having a first node coupled to a central conductor. There is a third inductor having a first node coupled to a second node of the third capacitor and a second node coupled to ground. There is a fourth capacitor having a first node coupled to a second node of the third capacitor and a second node coupled to ground.

[0011] In one embodiment, the characteristic impedance of the bandpass filter is higher than 50 ohms.

[0012] In one embodiment, the bandpass parameter amplifier is part of a circulator circuit.

[0013] In one embodiment, each of the first and second inductors includes a Josephson junction array.

[0014] In one embodiment, each Josephson junction includes an aluminum (Al) or niobium (Nb) superconducting electrode.

[0015] In one embodiment, each of the first and second capacitors is constructed on a low-loss dielectric substrate within a coplanar waveguide geometry.

[0016] In one embodiment, the lumped element capacitor in the bandpass parameter amplifier is a parallel plate capacitor using a low-loss dielectric material.

[0017] In one embodiment, the bandpass parameter amplifier is a directional amplifier.

[0018] According to one embodiment, a Josephson-based directional parametric amplifier includes a plurality of nonlinearly distributed bandpass filters positioned along a transmission line, and a linear element that applies a periodic resonant structural loading to a pump driver.

[0019] In one embodiment, each nonlinear distributed bandpass filter includes a first inductor having a first node coupled to a center conductor and a second node coupled to ground. A first capacitor is present, having a first node coupled to the center conductor and a second node coupled to ground. A second capacitor is present, having a first node coupled to the first node of the center conductor. A second inductor is present, having a first node coupled to the second node of the second capacitor. The second capacitor and the second inductor are connected in series with the center conductor.

[0020] In one embodiment, each periodic resonant structure includes: a third capacitor having a first node coupled to the central conductor; a third inductor having a first node coupled to a second node of the third capacitor; and a second node coupled to ground. A fourth capacitor exists, having a first node coupled to the second node of the third capacitor and a second node coupled to ground.

[0021] In one embodiment, a periodic resonant structure is added to the nonlinear unit cell of every 3-9 lumped element bandpass filter.

[0022] These and other features will become apparent from the following detailed description of illustrative embodiments thereof, which will be read in conjunction with the accompanying drawings. Attached Figure Description

[0023] The accompanying drawings are illustrative embodiments. They do not show all embodiments. Other embodiments may be used alternatively or as an alternative. Details that may be obvious or unnecessary may be omitted to save space or for more efficient illustration. Some embodiments may be practiced with additional components or steps and / or without all components or steps shown. When the same number appears in different drawings, it refers to the same or similar parts or steps.

[0024] Figure 1 A known JTWPA circuit is shown.

[0025] Figure 2 This is a block diagram of a bandpass Josephson traveling-wave parametric amplifier system according to an illustrative embodiment.

[0026] Figure 3 An example circuit of a nonlinear dispersion filter medium, consistent with the illustrative embodiment, is shown. This nonlinear dispersion filter medium can be used to implement... Figure 2 A bandpass Josephson-based directional parametric amplifier block. Detailed Implementation

[0027] Overview

[0028] In the following detailed description, numerous specific details are illustrated by way of examples to provide a thorough understanding of the relevant teachings. However, it should be clear that the teachings can be practiced without such details. In other cases, well-known methods, processes, components, and / or circuits have been described at a relatively high level without detail to avoid unnecessarily obscuring aspects of this teaching.

[0029] This disclosure generally relates to superconducting devices, and more specifically to efficient multiple readouts of qubits. The electromagnetic energy associated with a qubit can be stored in a so-called Josephson junction and in capacitive and inductive elements used to form the qubit. In one example, to read out the qubit state, a microwave signal is applied to a microwave readout cavity coupled to the qubit at that cavity frequency. The emitted (or reflected) microwave signal passes through multiple thermal isolation stages and low-noise amplifiers, which are used to block or reduce noise and improve the signal-to-noise ratio. The amplitude and / or phase of the returned / output microwave signal carry information about the qubit state, such as whether the qubit is in a grounded state, an excited state, or a superposition of both states. The microwave signal carrying quantum information about the qubit state is typically weak (e.g., on the order of a few microwave photons).

[0030] The readout of the qubit state is typically facilitated by a JTWPA, which is a Josephson parametric amplifier based on distributed elements. The device is formed by a nonlinear transmission line whose central conductor comprises an array of Josephson junctions periodically shunt to ground using lumped-element capacitors. Directional amplification of the weak microwave signal propagating in the device is achieved by injecting a strong microwave driver (sometimes referred to as a pump) at a frequency near the center of the desired amplification band, with its propagation direction aligned with that of the input weak microwave signal.

[0031] Pumping the nonlinear Kerr medium (i.e., a Josephson junction array) with a strongly coherent microwave driver generates a four-wave mixing process that results in parametric amplification of the propagated weak microwave signal. In other words, the distributed nonlinear medium enables the amplification of weak microwave signals through coherent energy exchange with the propagating strongly driven driver.

[0032] To ensure phase matching between the pump driver and the detuned weak microwave signal propagating in the distributed nonlinear medium, the center conductor is loaded with a resonant structural capacitor every few unit cells, which resonates near the pump frequency. This resonant load creates a bandgap in the structure, which in turn facilitates phase matching.

[0033] Figure 1 A known JTWPA circuit 100 is shown. The JTWPA circuit is implemented as a nonlinear lumped element transmission line 102 connected between input 110 and output 112. The unit cell 120 of the JTWPA includes a Josephson junction having a critical current I0 and an intrinsic capacitance C. J It also has a capacitor shunt to ground C. For example, every third unit cell, including a Josephson junction and a grounded parallel capacitor C, also includes a lumped element resonator connected to a capacitor C. rand inductor L t The specified coupling strength is determined by the coupling capacitor C. c set up.

[0034] Figure 1 The JTWPA 102 can suffer from significant insertion losses (e.g., 4 dB or greater). This loss is primarily caused by dielectric losses in the parallel-plate capacitor, which is part of the device. As previously mentioned, due to the distributed nature of traveling-wave amplifiers and their lack of resonant structures for weak signals, the JTWPA 102 can have a large amplification bandwidth (e.g., 3 to 5 GHz). Consequently, due to the wide amplification bandwidth, the JTWPA 102 amplifies unwanted quantum noise in the frequency band, such as at qubit frequencies. Some of this amplified noise across the entire frequency band can propagate back into the quantum system due to impedance mismatch in the output lines.

[0035] Another drawback of the JTWPA 102 is that small impedance mismatches within the JTWPA 102, or at input 110 or output 112, can cause multiple reflections, resulting in ripple in the JTWPA 102's gain curve. Furthermore, due to the wide bandwidth, impedance matching between the amplifier and the 50-ohm input and output circuitry requires the use of broadband cryogenic isolators (e.g., 4 to 12 GHz) at input 110 and output 112. It should be noted that these isolators protect the quantum system from noise returned from the JTWPA or high electron mobility transistor (HEMT) when a well-defined 50-ohm environment is provided over a wide bandwidth. However, these isolators are typically bulky, expensive, lossy, and use strong magnetic fields, which negatively impact JTWPA operation. Furthermore, these isolators are difficult to integrate onto a single chip, hindering scalability.

[0036] Therefore, in one aspect, the teachings herein provide a Josephson-based directional parametric amplifier with a medium-range bandwidth (e.g., 500 MHz–2000 MHz) centered at a frequency close to the qubit readout frequency (e.g., 7 GHz). Instead of the nonlinear transmission line discussed above, a nonlinear distributed bandpass filter architecture is implemented. A linear resonator resonating at the center of the band is capacitively loaded into the nonlinear distributed bandpass filter. These resonators are integrated in every few unit cells (e.g., 3 to 9) to ensure phase matching between the pump driver and the propagating weak microwave signal within the band.

[0037] In one embodiment, the teachings herein employ two strategies to reduce the insertion loss of the bandpass filter (due to dielectric losses in the parallel-plate capacitor). First, the bandpass filter is configured to have a relatively high characteristic impedance, greater than 50 ohms (e.g., 80 ohms). This increase in characteristic impedance typically reduces capacitance in the amplifier device. To ensure proper impedance matching between the filter's characteristic impedance and the 50-ohm input and output environment, impedance matching networks (e.g., adiabatic tapered transmission lines) are added at the input and output of the device.

[0038] Secondly, the lumped element capacitance, which is part of the linear resonant circuit used for phase matching in the amplifier device, can be reduced by implementing a parallel LC circuit with a high characteristic impedance (i.e., greater than 50 ohms). This can be achieved, for example, by using a large inductor and a small parallel capacitor to generate a characteristic impedance greater than 50 ohms while maintaining the same resonant frequency of the linear resonant circuit. In one example, this high inductance can be achieved using a thin, narrow superconductor layer with a large dynamic inductance.

[0039] With the aid of the bandpass-parameter amplifier discussed herein, directional amplifiers can amplify weak microwave signals with very little additional noise (i.e., operating near the quantum limit). Furthermore, it can be used to perform fast, high-fidelity quantum non-decomposition (QND) measurements of qubits. This near-quantum-finite directional amplifier with a moderate bandwidth offers several advantages. For example, it significantly relaxes the requirement for a 50-ohm matching environment at wide-bandwidth matched input and output ports. This flexibility from the tight matching environment allows the bandpass-parameter amplifier to be compatible with microwave devices with similar moderate bandwidths, such as circulators, isolators, and passive filters. Furthermore, when amplifying the readout signal of diffusely coupled qubits, the moderate bandwidth of this bandpass-parameter amplifier ensures that only the qubit readout signal is amplified, not the qubit signal itself. This focused readout reduces unwanted backlash on the qubit (due to small impedance mismatch) and increases the stability of the device by rejecting out-of-band qubit pulses. Furthermore, the narrower bandwidth of the bandpass parameter amplifier (compared to a conventional JTWPA) reduces ripple in the gain curve and reduces the pump power required for device operation.

[0040] Example diagram

[0041] Figure 2This is a block diagram of a bandpass Josephson traveling-wave parametric amplifier system 200 according to an illustrative embodiment. System 200 includes a nonlinear dispersive filter medium (NDFM) 206 coupled between external transmission lines 202 and 210. For example, the first external transmission line 202 may be an input and the second external transmission line 210 may be an output. External transmission lines 202 and 210 may each have a characteristic impedance of Z0, which may be 50 ohms.

[0042] In various embodiments, an intermediate stage may exist between at least one of the external transmission lines 202 and 210, in Figure 2 The intermediate stages are represented by 204 and 208. In one embodiment, the intermediate stage is an impedance transformer between Z1 and Z0. For example, these intermediate impedance matching networks / circuits are useful if the characteristic impedance of the nonlinear dispersive filter medium located at the heart of the bandpass Josephson traveling-wave parametric amplifier is Z1 (different from Z0 (of these external ports)).

[0043] Example circuit implementation of NDFM

[0044] Now refer to Figure 3 It illustrates that, according to the illustrative embodiments, it can be used to implement Figure 2 An example of a nonlinear distributed filter medium (NDFM) 300 in the NDFM 260 module. The NDFM 300 is a distributed element directional amplifier comprising a stepped structure of lumped element capacitors and inductors. The NDFM 300 includes a distributed nonlinear bandpass filter located on the transmission line. Several linear elements are present for the pump driver to achieve periodic resonant structural loading. In unit cell 302, a capacitor C is included... c 320. Inductor L r 322 and capacitor C r The linear element 324 illustrates an example of a periodic resonant structure. In one embodiment, a periodic resonant structure is added every 3-9 nonlinear filter elements, defining a narrow stopband in the gain curve of the NDFM 300, sometimes referred to as the amplifier's dispersion characteristic.

[0045] In one embodiment, the unit cell 302 of the NDFM 300 includes a first inductor 304 having a first node coupled to a center conductor 330 and a second node coupled to ground 340 (e.g., at an external portion of the NDFM 300 structure). A first capacitor 306 is present in parallel with the first inductor 304. In other words, the first capacitor 306 has a first node coupled to the center conductor 330 and a second node coupled to ground 340. A second capacitor 308 is present, having a first node coupled to the center conductor 330. A second inductor 310 is present, having a first node coupled to the second node of the second capacitor 308, wherein the second capacitor and the second inductor are connected in series with the center conductor 330. A series of unit cells 302 creates a ladder circuit to create the NDFM 300.

[0046] In one embodiment, the unit cell 302 of the NDFM300 further includes a periodic resonant structure comprising a third capacitor 320 having a first node coupled to a center conductor 330. A third inductor 322 is present, having a first node coupled to a second node of the third capacitor 320 and a second node coupled to ground 340. A fourth capacitor 324 is present, having a first node coupled to a second node of the third capacitor 320 and a second node coupled to ground 340. As previously described, in one embodiment, the resonant structure repeats every 3 to 9 nonlinear filter elements.

[0047] In some embodiments, at least one of the first inductor 304 and the second inductor 310 is nonlinear, as indicated by the arrows pointing through the inductors. For example, inductors 304 and 310 may comprise an array of Josephson junctions. The number of unit cells 302, sometimes referred to herein as the bandpass filter order N, is odd (e.g., 101 to 1001). The filter order is odd to maintain symmetry of the device with respect to its two external ports.

[0048] The filtering characteristics of the NDFM 300 can be designed using the insertion loss method. In different embodiments, each cell 302 of the NDFM 300 can be designed based on maximum flatness (e.g., Butterworth response) or equiripple (Chebyshev polynomial). A portion of the bandwidth of the NDFM 300 is provided by the following Equation 1:

[0049]

[0050] in,

[0051] ω0 is the center angular frequency of the bandpass filter medium;

[0052] ω1 is the lower angular cutoff frequency of the bandpass filter medium;

[0053] ω2 is the upper corner cutoff frequency of the bandpass filter medium; and ω2 > ω0 > ω1.

[0054] In one embodiment, the center frequency of the bandpass amplifier 300 (e.g., the geometric mean of the angular cutoff frequencies ω2 and ω1) is provided by the following Equation 2:

[0055]

[0056] Based on the NDFM 300 level and the selected ripple amplitude (i.e., the latter related to an equal-ripple filter configuration), the appropriate component value g corresponding to the normalized source impedance and normalized cutoff frequency can be calculated using known insertion loss methods used for filter design. i .

[0057] Utilize the calculated or listed (in the literature) value g i The characteristic impedance Z1, center angular frequency ω0, and desired fractional bandwidth Δ of the filter medium can be used to calculate the values ​​of the lumped elements, inductors and capacitors, using the following equation.

[0058] For series components:

[0059]

[0060]

[0061] For parallel elements:

[0062]

[0063]

[0064] The linear inductance of the Josephson junction (JJ) is provided by the following equation 7:

[0065]

[0066] Φ0=h / (2e) (Equation 9)

[0067] in,

[0068] I o This is the critical current of the Josephson junction;

[0069] It is a reduced flux quantum;

[0070] h is Planck's constant; and

[0071] e is the magnitude of the electron charge.

[0072] In one embodiment, a lossless nonlinear inductor and Both (310 and 304) are implemented using arrays of JJ, where each array has a linear inductance L. J0 Capacitive elements are implemented using superconducting plate capacitors with low-loss dielectric materials (such as amorphous silicon, crystalline silicon, or silicon nitride). and (308 and 306). NDFM 300 can be implemented on a low-loss dielectric substrate in a coplanar waveguide geometry. The substrate can be high-resistivity silicon or sapphire.

[0073] In various embodiments, JJ may include aluminum (Al) or niobium (Nb). The linear inductor Lr (e.g., 322) may be a thin, high-dynamic inductor. A second capacitor 320 (C) is included in each cell 302. c ) and / or the third capacitor 324 (C r It can be a superconducting plate capacitor with low-loss dielectric material. Each unit 302 has a linear inductor 322L. r Second capacitor 320 (C) c ) and the third capacitor 324 (C r The periodic resonant structure loading for the pump driver is implemented together. In one embodiment, these elements are added every 3 to 9 filter elements. These elements define the stopband in the gain curve, also known as the dispersion characteristic of the amplifier. The center frequency of the NDFM 300 is provided by Equation 10 below.

[0074]

[0075] in conclusion

[0076] Various embodiments of this teaching have been described for illustrative purposes, but are not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terminology used herein has been chosen to best explain the principles of the embodiments, their practical application, or technical improvements to technologies found in the market, or to enable those skilled in the art to understand the embodiments disclosed herein.

[0077] While the content considered to be the best state and / or other instances has been described above, it should be understood that various modifications may be made therein, and the subject matter disclosed herein can be implemented in different forms and instances, and the teachings can be applied to many applications, of which only some have been described herein. The appended claims are intended to claim protection for any and all applications, modifications, and variations falling within the true scope of this teaching.

[0078] The components, steps, features, purposes, benefits, and advantages discussed herein are illustrative only. They, and the discussions associated with them, are not intended to limit the scope of protection. While various advantages have been discussed herein, it will be understood that not all embodiments are necessarily required to include all advantages. Unless otherwise stated, all measurements, values, ratings, positions, amplitudes, sizes, and other specifications set forth in this specification (including in the following claims) are approximate and imprecise. They are intended to have a reasonable range of functionality associated with them and consistent with functionality customary in the art to which they pertain.

[0079] Many other embodiments are also conceived. These include embodiments with fewer, additional, and / or different components, steps, features, purposes, benefits, and advantages. They also include embodiments where components and / or steps are arranged and / or ordered differently. For example, any signal discussed herein may be scaled, buffered, scaled and cached, transformed into another state (e.g., voltage, current, charge, time, etc.), or transformed into another state (e.g., from high to low and from low to high) without substantially altering the underlying control method.

[0080] Although the foregoing has been described in conjunction with exemplary embodiments, it should be understood that the term "exemplary" means only as an example, and not the best or optimal. Nothing stated or shown beyond what is immediately stated above is intended or should be construed as causing any contribution or public equivalent of any component, step, feature, object, benefit, advantage, or advantage, whether or not it is stated in the claims.

[0081] It should be understood that the terms and expressions used herein have their general meanings as assigned to their respective corresponding queries and fields of study, unless otherwise specified herein. Relational terms such as "first" and "second" may be used merely to distinguish one entity or action from another, without necessarily requiring or implying any actual such relationship or order between these entities or actions. The terms "comprises," "comprising," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but may also include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element beginning with "an" or "a" does not exclude the presence of additional identical elements in the process, method, article, or apparatus that includes that element.

[0082] An abstract of this disclosure is provided to allow the reader to quickly determine the nature of this technical disclosure. It is submitted under the understanding that it is not intended to interpret or limit the scope or meaning of the claims. Furthermore, as can be seen from the above detailed description, various features are combined in various embodiments for the purpose of simplification. The method of this disclosure should not be construed as reflecting an intention that the claimed embodiments require more features than those expressly stated in each claim. Rather, as reflected in the following claims, the inventive subject matter lies in all features of fewer than those of a single disclosed embodiment. Therefore, the following claims are thereby incorporated into the detailed description, wherein each claim is independently claimed as a separate subject matter.

Claims

1. A bandpass parametric amplifier circuit, comprising: a plurality of unit cells, at least one of the unit cells comprising: a first inductor having a first node coupled to a center conductor and a second node coupled to ground; a first capacitor having a first node coupled to the center conductor and a second node coupled to ground; a second inductor having a first node coupled to the center conductor; a second capacitor having a first node coupled to a second node of the second inductor, wherein the second capacitor and the second inductor are in series with the center conductor; wherein the plurality of unit cells are part of a ladder circuit structure creating the bandpass parametric amplifier circuit, and each multiple number of unit cells further comprises a resonant structure; wherein the resonant structure comprises: a third capacitor having a first node coupled to the center conductor; a third inductor having a first node coupled to a second node of the third capacitor and a second node coupled to ground; and a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground.

2. The bandpass parametric amplifier of claim 1, wherein, the first inductor and the second inductor are non-linear.

3. The bandpass parametric amplifier of claim 1, wherein, a number of unit cells in the ladder circuit structure is odd.

4. The bandpass parametric amplifier of claim 1, wherein, the multiple number is 3 to 9.

5. The bandpass parametric amplifier of claim 1, wherein, the resonant structure in each of the plurality of unit cells defines a stopband in a gain curve of the bandpass parametric amplifier.

6. The bandpass parametric amplifier of any one of claims 1-5, wherein, the resonant structure in each of the plurality of unit cells provides a phase match between a pump driver and a propagating in-band microwave signal amplified by the bandpass parametric amplifier.

7. The bandpass parametric amplifier of any one of claims 1-5, further comprising: a third capacitor having a first node coupled to the center conductor; a third inductor having a first node coupled to a second node of the third capacitor and a second node coupled to ground; a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground.

8. The bandpass parametric amplifier of any one of claims 1-5, wherein, a characteristic impedance of the bandpass filter is higher than 50 ohms.

9. The bandpass parametric amplifier of any one of claims 1-5, wherein, the bandpass parametric amplifier is part of a circulator circuit.

10. The bandpass parametric amplifier of any one of claims 1-5, wherein, each of the first inductor and the second inductor comprises an array of Josephson junctions.

11. The bandpass parametric amplifier of claim 10, wherein, each Josephson junction comprises aluminum (Al) or niobium (Nb) superconducting electrodes.

12. The bandpass parametric amplifier of any one of claims 1-5, wherein, each of the first capacitor and the second capacitor is constructed on a low-loss dielectric substrate in a co-planar waveguide geometry.

13. The bandpass parametric amplifier of any one of claims 1-5, wherein, the bandpass parametric amplifier is a directional amplifier.

14. A method, comprising: providing a plurality of unit cells; for at least one of the unit cells: providing a first inductor having a first node coupled to a center conductor and a second node coupled to ground; providing a first capacitor having a first node coupled to the center conductor and a second node coupled to ground; providing a second inductor having a first node coupled to the center conductor; and providing a second capacitor having a first node coupled to a second node of the second inductor, ​ wherein the second capacitor and the second inductor are in series with the center conductor; creating a bandpass parametric amplifier by using the plurality of unit cells as part of a ladder circuit structure; and including a resonant structure for each multiple number of the unit cells; wherein the resonant structure includes: a third capacitor having a first node coupled to the center conductor; a third inductor having a first node coupled to a second node of the third capacitor and a second node coupled to ground; and a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground.

15. The method of claim 14, wherein, the first inductor and the second inductor are non-linear.

16. The method of claim 14, wherein a number of unit cells in the ladder circuit structure is odd.

17. The method of claim 14, further comprising defining a stopband in a gain curve by the resonant structure implemented in the plurality of unit cells.

18. The method of any one of claims 14 to 17, wherein the first and second inductors each include an array of Josephson junctions.

19. A Josephson-based directional parametric amplifier, comprising: a plurality of non-linear distributed bandpass filters positioned along a transmission line; and linear elements implemented for pump drivers with periodic resonant structure loading; wherein each non-linear distributed bandpass filter includes: a first inductor having a first node coupled to a center conductor and a second node coupled to ground; a first capacitor having a first node coupled to the center conductor and a second node coupled to ground; a second inductor having a first node coupled to the center conductor; and a second capacitor having a first node coupled to a second node of the second inductor, wherein the second capacitor and the second inductor are in series with the center conductor; wherein each periodic resonant structure includes: a third capacitor having a first node coupled to the center conductor; a third inductor having a first node coupled to a second node of the third capacitor and a second node coupled to ground; and a fourth capacitor having a first node coupled to the second node of the third capacitor and a second node coupled to ground.

20. The amplifier of claim 19, wherein every 3 to 9 of the non-linear distributed bandpass filters, the periodic resonant structure is added.

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