MULTI-PATH INTERFEROMETRIC JOSEPHSON ISOLATOR BASED ON NONDEGENERATE THREE-WAVE MIXING JOSEPHSON DEVICES

Abstract

A microwave isolator device includes two nondegenerate microwave mixer devices, each mixer configured to receive a microwave input of an input frequency via a first port and to generate an idler signal of an idler frequency at a second port. The second ports of both mixers are coupled together. A first input/output (I/O) port is coupled to the first ports of the two mixers, and a second I/O port is also coupled to the first ports of the two mixers. A microwave signal (signal) communicated between the first I/O port and the second I/O port is transmitted while propagating in a first direction between the first I/O port to the second I/O port through the first mixer and the second mixer and to be blocked while propagating in a second direction between the second I/O port to the first I/O through the first mixer and the second mixer.

Claims

1. A microwave isolator device comprising: a first nondegenerate microwave mixer device (first mixer) having a first port and a second port, the first mixer configured to receive a microwave input of an input frequency via a first port, and further configured to generate an idler signal of an idler frequency at a second port; a second nondegenerate microwave mixer device (second mixer) having a different first port and a different second port, the second mixer configured to receive the microwave input of the input frequency via a different first port, and to generate the idler signal of the idler frequency at a different second port; a first input/output (I/O) port coupled to the first port and the different first port; and a second I/O port coupled to the first port and the different first port, wherein a microwave signal (signal) communicated between the first I/O port and the second I/O port is transmitted while propagating in a first direction between the first I/O port to the second I/O port through the first mixer and the second mixer and to be blocked while propagating in a second direction between the second I/O port to the first I/O through the first mixer and the second mixer.

2. The microwave isolator device of claim 1, further comprising: a first microwave pump injecting a first microwave drive into the first mixer at a pump frequency and a first pump phase, wherein the idler frequency of the idler signal at the second port is dependent upon the pump frequency and the input frequency; a second microwave pump injecting a second microwave drive into the second mixer at the pump frequency and a second pump phase, wherein the idler frequency of the idler signal at the different second port is dependent upon the pump frequency and the input frequency.

3. The microwave isolator device of claim 2, wherein the first microwave pump is configured to cause the first mixer to operate at a 50:50 beamsplitting working point, and wherein the second microwave pump is configured to cause the second mixer to operate at the 50:50 beamsplitting working point.

4. The microwave isolator device of claim 2, wherein changing a difference between the first pump phase and the second pump phase causes the signal to be transmitted while propagating in the second direction and blocked while propagating in a first direction.

5. The microwave isolator device of claim 1, further comprising: a four-port microwave hybrid device (pump hybrid); and a microwave pump coupled to the pump hybrid such that a first output port of the pump hybrid injects a first microwave drive into the first mixer at a pump frequency and a first pump phase and a second output port of the pump hybrid injects a second microwave drive into the second mixer at the pump frequency and a second pump phase, wherein the pump hybrid fixes a relative phase between the first microwave drive and the second microwave drive to 90 degrees.

6. The microwave isolator device of claim 1, wherein the second port and the different second port are coupled via a transmission line, the transmission line having a negligible energy loss.

7. The microwave isolator device of claim 1, wherein the second port and the different second port are coupled together using a coupling component, wherein the coupling component introduces an attenuation of a signal that is propagated between the first mixer and the second mixer.

8. The microwave isolator of claim 1, wherein the second port of the first mixer comprises a first feedline and a second feedline, wherein the different second port of the second mixer comprises a different first feedline and a different second feedline, wherein the second feedline and the different second feedline are coupled via a transmission line, the transmission line having a negligible energy loss, wherein the first feedline is coupled to a 50 Ohm termination, and wherein the different first feedline is coupled to a different 50 Ohm termination.

9. The microwave isolator device of claim 1, wherein the first and second I/O ports are part of a ninety-degree hybrid coupler, and a function of the first port and the different first port are configured to be equivalent in the first mixer and the second mixer.

10. The microwave isolator device of claim 1, wherein the first mixer and the second mixer are each a nondegenerate three-wave mixer.

11. The microwave isolator device of claim 1, wherein the first mixer and the second mixer are each a Josephson parametric converter (JPC), and wherein the first mixer and the second mixer are nominally identical.

12. A method to form a microwave isolator device, the method comprising: configuring a first nondegenerate microwave mixer device (first mixer) having a first port and a second port, the first mixer configured to receive a microwave input of an input frequency via a first port, and further configured to generate an idler signal of an idler frequency at a second port; configuring a second nondegenerate microwave mixer device (second mixer) having a different first port and a different second port, the second mixer configured to receive the microwave input of the input frequency via a different first port, and to generate the idler signal of the idler frequency at a different second port; coupling a first input/output (I/O) port to the first port and the different first port; and coupling a second I/O port to the first port and the different first port, wherein a microwave signal (signal) communicated between the first I/O port and the second I/O port is transmitted while propagating in a first direction between the first I/O port to the second I/O port through the first mixer and the second mixer and to be blocked while propagating in a second direction between the second I/O port to the first I/O through the first mixer and the second mixer.

13. The method of claim 12, further comprising: injecting, from a first microwave pump, a first microwave drive into the first mixer at a pump frequency and a first pump phase, wherein the idler frequency of the idler signal at the second port is dependent upon the pump frequency and the input frequency; injecting, from a second microwave pump, a second microwave drive into the second mixer at the pump frequency and a second pump phase, wherein the idler frequency of the idler signal at the second port is dependent upon the pump frequency and the input frequency.

14. The method of claim 13, wherein the first microwave pump is configured to cause the first mixer to operate at a 50:50 beamsplitting working point, and wherein the second microwave pump is configured to cause the second mixer to operate at the 50:50 beamsplitting working point.

15. The method of claim 13, wherein changing a difference between the first pump phase and the second pump phase causes the signal to be transmitted while propagating in the second direction and blocked while propagating in a first direction.

16. The method of claim 12, further comprising: coupling a four-port microwave hybrid device (pump hybrid) to a microwave pump such that a first output port of the pump hybrid injects a first microwave drive into the first mixer at a pump frequency and a first pump phase and a second output port of the pump hybrid injects a second microwave drive into the second mixer at the pump frequency and a second pump phase, wherein the pump hybrid fixes a relative phase between the first microwave drive and the second microwave drive to 90 degrees.

17. The method of claim 12, wherein the second port and the different second port are coupled via a transmission line, the transmission line having a negligible energy loss.

18. The method of claim 12, wherein the second port and the different second port are coupled together using a coupling component, wherein the coupling component introduces an attenuation of a signal that is propagated between the first mixer and the second mixer.

19. The method of claim 12, wherein the second port of the first mixer comprises a first feedline and a second feedline, wherein the different second port of the second mixer comprises a different first feedline and a different second feedline, wherein the second feedline and the different second feedline are coupled via a transmission line, the transmission line having a negligible energy loss, wherein the first feedline is coupled to a 50 Ohm termination, and wherein the different first feedline is coupled to a different 50 Ohm termination.

20. The method of claim 12, wherein the first and second I/O ports are part of a ninety-degree hybrid coupler, and a function of the first port and the different first port are configured to be equivalent in the first mixer and the second mixer.

21. The method of claim 12, wherein the first mixer and the second mixer are each a nondegenerate three-wave mixer.

22. The method of claim 12, wherein the first mixer and the second mixer are each a Josephson parametric converter (JPC), and wherein the first mixer and the second mixer are nominally identical.

23. A superconductor fabrication system which when operated to fabricate a microwave isolator device performing operations comprising: configuring a first nondegenerate microwave mixer device (first mixer) having a first port and a second port, the first mixer configured to receive a microwave input of an input frequency via a first port, and further configured to generate an idler signal of an idler frequency at a second port; configuring a second nondegenerate microwave mixer device (second mixer) having a different first port and a different second port, the second mixer configured to receive the microwave input of the input frequency via a different first port, and to generate the idler signal of the idler frequency at a different second port; coupling a first input/output (I/O) port to the first port and the different first port; and coupling a second I/O port to the first port and the different first port, wherein a microwave signal (signal) communicated between the first I/O port and the second I/O port is transmitted while propagating in a first direction between the first I/O port to the second I/O port through the first mixer and the second mixer and to be blocked while propagating in a second direction between the second I/O port to the first I/O through the first mixer and the second mixer.

24. The superconductor fabrication system of claim 23, further comprising: injecting, from a first microwave pump, a first microwave drive into the first mixer at a pump frequency and a first pump phase, wherein the idler frequency of the idler signal at the second port is dependent upon the pump frequency and the input frequency; injecting, from a second microwave pump, a second microwave drive into the second mixer at the pump frequency and a second pump phase, wherein the idler frequency of the idler signal at the second port is dependent upon the pump frequency and the input frequency.

25. The superconductor fabrication system of claim 24, wherein the first microwave pump is configured to cause the first mixer to operate at a 50:50 beamsplitting working point, and wherein the second microwave pump is configured to cause the second mixer to operate at the 50:50 beamsplitting working point.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0031] The novel features believed characteristic of the invention are set forth in the appended claims. The invention itself, however, as well as a preferred mode of use, further objectives and advantages thereof, will best be understood by reference to the following detailed description of the illustrative embodiments when read in conjunction with the accompanying drawings, wherein:

[0032] FIG. 1 depicts a block diagram of a conceptual isolator function achieved by an MPIJIS in accordance with an illustrative embodiment;

[0033] FIG. 2 depicts a block diagram of a signal flow graph for a nondegenerate three-wave mixing device operated in frequency conversion (no photon gain) in accordance with an illustrative embodiment;

[0034] FIG. 3 depicts a block diagram of an example configuration of an MPIJIS in accordance with an illustrative embodiment;

[0035] FIG. 4 depicts a schematic of a nondegenerate three-wave mixer configured in the manner of a nondegenerate three-wave mixer used in an MPIJIS according to an illustrative embodiment;

[0036] FIG. 5 depicts the modes and operations of a four-port 90-degree microwave hybrid device in accordance with an illustrative embodiment;

[0037] FIG. 6 depicts an alternate configuration for an MPIJIS in accordance with an illustrative embodiment;

[0038] FIG. 7 depicts another alternate configuration for an MPIJIS in accordance with an illustrative embodiment;

[0039] FIG. 8 depicts another alternate configuration for an MPIJIS in accordance with an illustrative embodiment;

[0040] FIG. 9A depicts one signal path in an MPIJIS according to an illustrative embodiment;

[0041] FIG. 9B depicts a second signal path in an MPIJIS according to an illustrative embodiment;

[0042] FIG. 9C depicts a third signal path in an MPIJIS according to an illustrative embodiment;

[0043] FIG. 9D depicts a fourth signal path in an MPIJIS according to an illustrative embodiment;

[0044] FIG. 10A depicts one opposite signal path in an MPIJIS according to an illustrative embodiment;

[0045] FIG. 10B depicts a second opposite signal path in an MPIJIS according to an illustrative embodiment;

[0046] FIG. 10C depicts a third opposite signal path in an MPIJIS according to an illustrative embodiment;

[0047] FIG. 10D depicts a fourth opposite signal path in an MPIJIS according to an illustrative embodiment;

[0048] FIG. 11 depicts a generalized signal flow graph of an MPIJIS in accordance with an illustrative embodiment;

[0049] FIG. 12 depicts a simplified signal flow graph of an MPIJIS in accordance with an illustrative embodiment;

[0050] FIG. 13 depicts device equations for MPIJIS 900 based on the simplified signal flow graph of FIG. 12;

[0051] FIG. 14 depicts device equations for the special case of nominally identical mixers operated at the same working point;

[0052] FIG. 15 depicts the device equations when the mixers are operated at the 50:50 beam splitter point;

[0053] FIG. 16 depicts the device equations when the mixers operated at the 50:50 beam splitter point but the transmitting direction is changed; and

[0054] FIG. 17 depicts the device equations when the mixers operated at the 50:50 beam splitter point but the phase acquired by the internal-mode signals propagating between the two mixers is different.

DETAILED DESCRIPTION

[0055] The illustrative embodiments used to describe the invention generally address and solve the above-described needs for a multi-path interferometric Josephson isolator based on nondegenerate three-wave mixing Josephson devices. A multi-path interferometric Josephson isolator based on nondegenerate three-wave mixing Josephson devices is hereinafter compactly and interchangeably referred to as Multi-Path Interferometric Josephson ISolator (MPIJIS).

[0056] An MPIJIS 1) can be integrated on chip or printed circuit board (PCB), 2) does not use ferrites, 3) does not require strong magnets, 4) can be thermalized well, 5) can be made small/compact, and 6) has lighter weight as compared to presently available isolators. Furthermore, a commercial isolator always works to gate the flow in a single direction that is fixed for the isolator. In contrast, the direction of isolation in an MPIJIS can be reversed in situ by negating the phase difference between the two pump tones driving the device as described herein.

[0057] An embodiment provides an MPIJIS device, which can be implemented as a microwave isolator in a superconducting quantum circuit. Another embodiment provides fabrication method for the MPIJIS device, such that the method can be implemented as a software application. The application implementing a fabrication method embodiment can be configured to operate in conjunction with an existing semiconductor fabrication systemsuch as a lithography system.

[0058] For the clarity of the description, and without implying any limitation thereto, the illustrative embodiments are described using some example configurations. From this disclosure, those of ordinary skill in the art will be able to conceive many alterations, adaptations, and modifications of a described configuration for achieving a described purpose, and the same are contemplated within the scope of the illustrative embodiments.

[0059] Furthermore, simplified diagrams of the example mixers, hybrids, and other circuit components are used in the figures and the illustrative embodiments. In an actual fabrication or circuit, additional structures or component that are not shown or described herein, or structures or components different from those shown but for the purpose described herein may be present without departing the scope of the illustrative embodiments.

[0060] Furthermore, the illustrative embodiments are described with respect to specific actual or hypothetical components only as examples. The steps described by the various illustrative embodiments can be adapted for fabricating a circuit using a variety of components that can be purposed or repurposed to provide a described function within an MPIJIS, and such adaptations are contemplated within the scope of the illustrative embodiments.

[0061] The illustrative embodiments are described with respect to certain types of materials, electrical properties, steps, numerosity, frequencies, circuits, components, and applications only as examples. Any specific manifestations of these and other similar artifacts are not intended to be limiting to the invention. Any suitable manifestation of these and other similar artifacts can be selected within the scope of the illustrative embodiments.

[0062] The examples in this disclosure are used only for the clarity of the description and are not limiting to the illustrative embodiments. Any advantages listed herein are only examples and are not intended to be limiting to the illustrative embodiments. Additional or different advantages may be realized by specific illustrative embodiments. Furthermore, a particular illustrative embodiment may have some, all, or none of the advantages listed above.

[0063] With reference to FIG. 1, this figure depicts a block diagram of a conceptual isolator function achieved by an MPIJIS in accordance with an illustrative embodiment. Symbol 102 represents an isolator, whether a prior-art cryogenic isolator or an MPIJIS operating as an isolator. Isolator 102 has two ports 104 and 106, either of which can receive an input signal. Operation 108 shows two scenarios. In one scenario, signal S is input from port 104. According to the direction of the arrow in symbol 102, isolator 102 is configured to pass signal S substantially unattenuated from port 1 to port 2. Accordingly, signal S appears at port 106 substantially unchanged in amplitude), as indicated by the substantially unchanged thickness of the arrow showing output from port 2 as compared to the input arrow at port 1. In the other scenario, signal S is input from port 2 (106). Isolator 102 blocks signal S in the direction from port 2 to port 1. Accordingly, signal S input from port 2 is heavily attenuated such that only a signal of an insignificant amplitude, ideally zero amplitude, appears at port 1 (104), as indicated by the insignificant thickness of the arrow showing output from port 1 as compared to the input arrow at port 2.

[0064] Scattering matrix 110 is a representation of a transmission parameter of ideal isolator 102 in four possible directionssignal input at port 1 and reflected back through port 1 (S.sub.11), signal input at port 1 and transmitted to port 2 (S.sub.21), signal input at port 2 and reflected back through port 2 (S.sub.22), and signal input at port 2 and transmitted to port 1 (S.sub.12).

[0065] Scattering matrix 110 is represented as [S]

[00001]$\left[S\right]=\left(\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right)$

where ideally

[00002]$\left[S\right]=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)$

[0066] Meaning, isolator 102 reflects zero of signal input at port 1 back through port 1 (S.sub.11=0), transmits unity, i.e., entirety of, signal input at port 1 to port 2 (S.sub.21=1), reflects zero of signal input at port 2 back through port 2 (S.sub.22=0), and transmits zero of signal input at port 2 to port 1 (S.sub.12=0).

[0067] With reference to FIG. 2, this figure depicts a signal flow graph for a nondegenerate three-wave mixing device operated in frequency conversion (no photon gain) in accordance with an illustrative embodiment. This nondegenerate three-wave mixing device could be used as part of the MPIJIS in a manner described herein. The nondegenerate three-wave mixer can be a Josephson parametric converter (JPC).

[0068] Superconducting nondegenerate three-wave mixer 202 has 3 ports, which are Signal port 204 (S) through which a microwave signal of frequency f.sub.S can be input, Idler port 206 (I) through which an idler microwave signal of frequency f.sub.I can be input, and pump port 208 (P) through which microwave signal of frequency f.sub.P and phase .sub.P can be input. In one configuration (without loss of generality), f.sub.I is a high frequency, f.sub.P is a low frequency, and f.sub.S is a medium frequency, when f.sub.P, f.sub.S, and f.sub.T are compared relative to each other (i.e., f.sub.I>f.sub.S>f.sub.P). Superconducting nondegenerate three-wave mixer 202 is characterized as nondegenerate because it has two inputsnamely S and I.

[0069] Superconducting nondegenerate three-wave mixer 202 has transmission parameter t from Idler port to Signal port and transmission parameter t from Signal port to Idler port. From Idler to Signal port, the Idler microwave signal enters the Idler port at frequency f.sub.2, is down converted, and exits the Signal port at frequency f.sub.1. From Signal to Idler port, the Signal microwave signal enters the Signal port at frequency f.sub.1, is up converted, and exits the Idler port at frequency f.sub.2. The pump microwave signal provides the energy for frequency up conversion and frequency down conversion. The pump frequency is f.sub.P, where f.sub.P=f.sub.If.sub.S=f.sub.2f.sub.1.

[0070] On resonance, the nondegenerate three-wave mixer (e.g., JPC) satisfies the following scattering matrix when operated in noiseless frequency conversion:

[00003]$\left[S\right]=\left(\begin{array}{cc}r& t\\ t^{}& r\end{array}\right)$

[0071] Since the scattering matrix is unitary, the following relation holds |r|.sup.2+|t|.sup.2=1, where r is the reflection parameter (how much of the signal is reflected back through the same port where the signal is input), t is the transmission parameter (how much of the signal input at one port is transmitted to another port), and t=t* (where t* is the conjugate of t). Unitary means that the nondegenerate three wave mixer preserves the energy and the coherence of the phase.

[0072] nondegenerate three-wave mixer 202 can be operated in three working modes, which are of special interest

[0073] Working mode 1: Perfect mirror working point (pump is off):

[00004]${.Math.r.Math.}^2=1,{.Math.t.Math.}^2=0.Math..Math.\mathrm{and}.Math.\left[S\right]=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$

[0074] This means that the nondegenerate three-wave mixer reflects substantially all of the signal back through the input port (reflection parameters S11 and S22 are unity), and the nondegenerate three-wave mixer transmits substantially zero of the signal through to another port (transmission parameters S.sub.21 and S.sub.12 are 0).

[0075] Working mode 2: Full conversion working point:

[00005]${.Math.r.Math.}^2=0,{.Math.t.Math.}^2=1.Math..Math.\mathrm{and}.Math.\left[S\right]=\left(\begin{array}{cc}0& {\mathrm{ie}}^{-i.Math..Math.{}_P}\\ {\mathrm{ie}}^{i.Math..Math.{}_P}& 0\end{array}\right)$

[0076] In this mode, no reflection occurs and full transmission occurs with frequency conversion. This mode is dependent upon the pump amplitude which has to be set appropriately for this mode to become operational. The phase of the transmission parameters depends on the phase of the pump in a nonreciprocal manner (i.e., the phase acquired by signals propagating from the idler port to the signal port is .sub.P, whereas the phase acquired by signals propagating in the opposite direction is .sub.P).

[0077] Working mode 3: 50:50 Beam splitter working point:

[00006]${.Math.r.Math.}^2=\frac{1}{2},{.Math.t.Math.}^2=\frac{1}{2}.Math..Math.\mathrm{and}.Math.\left[S\right]=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}& \frac{{\mathrm{ie}}^{-i.Math..Math.{}_P}}{\sqrt{2}}\\ \frac{{\mathrm{ie}}^{i.Math..Math.{}_P}}{\sqrt{2}}& \frac{1}{\sqrt{2}}\end{array}\right)$

[0078] This mode is dependent upon the pump amplitude which has to be set appropriately for this mode to become operational. The phase of the pump .sub.P (which are later denoted as .sub.1 and .sub.2 for two pump signals into two different nondegenerate three-wave mixers) will be utilized in accordance with the embodiments described herein. Phase .sub.P is added to the signal propagating from port S to port I, and phase .sub.P is subtracted from signal propagating from port I to port S.

[0079] Two suitable manifestations of nondegenerate three-wave mixer 202, each operating in mode 3 are used as one component in an MPIJIS according to the illustrative embodiments. JPC is one such non-limiting manifestation, which is a low noise microwave amplifier that is driven by a microwave signal, can be integrated on a chip, is light in weight, thermalizes well, is either loss-less or has insignificant energy loss, and employs no magnetic materials or strong magnetic flux in its operation, as compared to prior-art isolators.

[0080] With reference to FIG. 3, this figure depicts a block diagram of an example configuration of an MPIJIS in accordance with an illustrative embodiment. MPIJIS configuration 300 comprises pair 302 of nondegenerate three-wave mixer 302A and nondegenerate three-wave mixer 302B. Each of nondegenerate three-wave mixer 302A and nondegenerate three-wave mixer 302B is an example of nondegenerate three-wave mixer 202 in FIG. 2 with some variations as shown and described herein. nondegenerate three-wave mixers 302A and 302B are both operating in mode 3 at the beam splitter working point as described with respect to FIG. 2.

[0081] nondegenerate three-wave mixer 302A is configured with ports a1, b1, p1, and b1. Physical port a1 corresponds to port S of nondegenerate three-wave mixer 202. Physical ports b1 and b1 correspond to port I of nondegenerate three-wave mixer 202 (or vice versa). Port p1 corresponds to port P of nondegenerate three-wave mixer 202. Physical port b1 is terminated as depicted in configuration 300, e.g., using a cold terminator. For example, the cold termination of port b1 may be a 50 Ohm termination. This configuration of nondegenerate three-wave mixer is described in FIG. 4.

[0082] FIG. 4 shows a schematic of a nondegenerate three-wave mixer configured in the manner of a nondegenerate three-wave mixer used in an MPIJIS according to an illustrative embodiment. For example, schematic 400 corresponds nondegenerate three-wave mixers 302A and 302B in FIG. 3. In one embodiment, nondegenerate three-wave mixers 302A and 302B are nominally identical.

[0083] nondegenerate three-wave mixers 302A and 302B are coupled with each other via transmission line 303, which connects ports b1 and b2 of nondegenerate three-wave mixers 302A and 302B, respectively, as shown. Ideally, nondegenerate three-wave mixers 302A and 302B are coupled back-to-back with the length of transmission line 303 being substantially zero. Preferably, the length of transmission line 303 is as short as an implementation will allow on chip or in the printed circuit board (PCB).

[0084] A signal entering from port a (corresponding to port al of nondegenerate three-wave mixer 302A) can be split as followssome part of the signal is reflected back out of port a, some part of the signal is transmitted out from port b, and some part of the signal is sent to port b where that signal energy is dissipated. A microwave pump inputs microwave of a certain frequency and phase from port p.

[0085] Returning to FIG. 3, component 304 is a device called a hybrid. Hybrid 304 is a four-port 90-degree microwave device which operates as a microwave splitter or a microwave combiner (depending on the input and output ports used) while also phase-shifting along some paths and not others. FIG. 5 depicts and describes the operations of a hybrid device.

[0086] With reference to FIG. 5, this figure depicts the modes and operations of a four-port 90-degree microwave hybrid device in accordance with an illustrative embodiment. Four-port schematic 500 represents a microwave hybrid device that forms a component in MPIJIS 300 of FIG. 3. Ports 1 and 2 in hybrid 500 are ports where a signal S is either input or output to or from the hybrid, from or to a superconducting circuit (not shown). Each of ports 3 and 4 is connected to port a of a nondegenerate three-wave mixer, e.g., port a of nondegenerate three-wave mixer 400, or port a1 of nondegenerate three-wave mixer 302A and port a2 of nondegenerate three-wave mixer 302B. Ports 3 and 4 in hybrid 500 are ports where a portion of signal S is either input from the hybrid to a nondegenerate three-wave mixer (not shown), or an output signal is received from a nondegenerate three-wave mixer into the hybrid.

[0087] Splitter operation 502 shows that signal S when input at port 1 is split into two portionsone portion which is approximately fifty percent of the signal is delivered to port 3 without any phase shift (0-degree phase shift), and the other portion which is approximately fifty percent of the signal is delivered to port 4 with a 90-degree phase shift.

[0088] Splitter operation 504 shows that signal S when input at port 2 is split into two portionsone portion which is approximately fifty percent of the signal is delivered to port 4 without any phase shift (0-degree phase shift), and the other portion which is approximately fifty percent of the signal is delivered to port 3 with a 90-degree phase shift.

[0089] Operation 506 occurs in configuration 300 of the proposed MPIJIS. In such a case, operation 506 forms a combiner operation of hybrid 500. Suppose that the signal entering port a of a nondegenerate three-wave mixer is being reflected, at least in part, back out of port a. Combiner operation 506 shows that a reflected signal when received at port hybrid 3 from port a of one nondegenerate three-wave mixer of the MPIJIS, is split into two portionsone portion which is approximately fifty percent of the signal is delivered to port 1 without any phase shift (0-degree phase shift), and the other portion which is approximately fifty percent of the signal is delivered to port 2 with a 90-degree phase shift. Similarly, combiner operation 506 further shows that a reflected signal when received at port hybrid 4 from port a of the other nondegenerate three-wave mixer of the MPIJIS, is split into two portionsone portion which is approximately fifty percent of the signal is delivered to port 2 without any phase shift (0-degree phase shift), and the other portion which is approximately fifty percent of the signal is delivered to port 1 with a 90-degree phase shift. The portions reaching port 1 from ports 3 and 4 are combined (via an interference phenomenon described herein) by hybrid 500 to be output from port 1. The portions reaching port 2 from ports 3 and 4 are combined (via an interference phenomenon described herein) by hybrid 500 to be output from port 2.

[0090] Back to FIG. 3, signal f.sub.1 (f.sub.1 is the signal frequency) is in resonance with the resonator of nondegenerate three-wave mixer 302A on port a1. Signal f2 (f.sub.2 is the signal frequency) is a signal on a transmission line connecting nondegenerate three-wave mixer 302A with nondegenerate three-wave mixer 302B. signal f.sub.2 is in resonance with the resonator of nondegenerate three-wave mixer 302A and/or nondegenerate three-wave mixer 302B on port b1 and/or port b2.

[0091] Pump input f.sub.p is a non-resonant tone whose frequency satisfies the condition

f.sub.p=f.sub.2f.sub.1

[0092] Pump phase .sub.1 is a phase .sub.p of pump 1 input to nondegenerate three-wave mixer 302A. Pump phase .sub.2 is a phase .sub.p of pump 2 input to nondegenerate three-wave mixer 302B. When .sub.1 and .sub.2 are selected according to an embodiment, MPIJIS 300 operates as an isolator whose direction can be reversed in situ. The isolator operation of MPIJIS 500 is a function of (.sub.1.sub.2) and the amplitudes of the pump signals from pump 1 and pump 2. In one embodiment, the amplitude of pump 1 input into nondegenerate three-wave mixer 302A is substantially the same as the amplitude of pump 2 input into nondegenerate three-wave mixer 302B.

[0093] When operating as an isolator, MPIJIS 300 causes signal f1 to pass from port 1 of hybrid 304 to port 2 of hybrid 304 without any significant attenuation, but causes a very high attenuation of the signal if input at port 2 of hybrid 304 such that an insignificant (ideally zero) portion of that signal can be observed as output at port 1 of hybrid 304. The lack of attenuation in one direction and substantial attenuation in the opposite direction are depicted by the thicknesses of the dark patterned arrows adjacent to ports 1 and 2 of hybrid 304.

[0094] With reference to FIG. 6, this figure depicts an alternate configuration for an MPIJIS in accordance with an illustrative embodiment. Pair 302 of nondegenerate three-wave mixers 302A and 302B is configured as in FIG. 3. 90-degree hybrid 304 is configured as described in FIGS. 3 and 5.

[0095] In configuration 600, a single pump is configured for use with both nondegenerate three-wave mixer 302A and nondegenerate three-wave mixer 302B via a second 90-degree hybridhybrid 606. Pump signal f.sub.p is applied to port 1 of hybrid 606. Port 2 of hybrid 606 is suitably terminated. Port 3 of hybrid 606 is coupled with port p1 of nondegenerate three-wave mixer 302A. Port 4 of hybrid 606 is coupled with port p2 of nondegenerate three-wave mixer 302B.

[0096] With reference to FIG. 7, this figure depicts another alternate configuration for an MPIJIS in accordance with an illustrative embodiment. Pair 702 of nondegenerate three-wave mixers 702A and 702B is configured substantially as pair 302 of nondegenerate three-wave mixers 302A and 302B as in FIG. 3, except that nondegenerate three-wave mixers 702A and 702B are each a hybrid-less JPC manifestation of a nondegenerate three-wave mixer. 90-degree hybrid 304 is configured as described in FIGS. 3 and 5. In configuration 700, hybrid-less JPC 702A and hybrid-less JPC 702B each has an additional a port, namely a1 and a2, respectively, which is suitably terminated (e.g., grounded).

[0097] With reference to FIG. 8, this figure depicts another alternate configuration for an MPIJIS in accordance with an illustrative embodiment. Configuration 800 is a combination of configurations 600 and 700, whereby a single pump is used in conjunction with hybrid 606 to provide the pump input to hybrid-less JPC 702A and hybrid-less JPC 702B.

[0098] With reference to FIG. 9A, this figure depicts one signal path in an MPIJIS according to an illustrative embodiment. MPIJIS 900 is representative of MPIJIS 300 or any of its configuration variants 600, 700, or 800. This path, (path 1), in MPIJIS 900 contributes to S.sub.21 in the scatter matrix [S] of the MPIJIS. Recall that S.sub.21 is the transmission parameter for the transmission from port 1 to port 2.

[0099] A signal at frequency f.sub.1 and relative phase 0 is input to port 1 of MPIJIS 900 and this input is denoted as 120, which means that the signal's normalized amplitude is 1 and the signal's relative phase is 0. Path 1 is illustrated by the dotted line traversing MPIJIS 900 from port 1 to port 2. The contribution of this path to the output at port 2 is denoted as

[00007]$\frac{1}{3.Math.\sqrt{2}}.Math.\mathrm{90},$

meaning that the signal's normalized amplitude is

[00008]$\frac{1}{3.Math.\sqrt{2}}$

and the signal's relative phase is 90.

[0100] With reference to FIG. 9B, this figure depicts a second signal path in an MPIJIS according to an illustrative embodiment. This path, (path 2), of signal f.sub.1 120 in MPIJIS 900 also contributes to S.sub.21 in the scatter matrix [S] of the MPIJIS. Path 2 is illustrated by the dotted line traversing MPIJIS 900 from port 1 to port 2. The contribution of this path to the output at port 2 is denoted as

[00009]$\frac{1}{3.Math.\sqrt{2}}.Math.\mathrm{90},$

meaning that the signal's normalized amplitude is

[00010]$\frac{1}{3.Math.\sqrt{2}}$

and the signal's relative phase is 90.

[0101] With reference to FIG. 9C, this figure depicts a third signal path in an MPIJIS according to an illustrative embodiment. This path, (path 3), of signal f.sub.1 120 in MPIJIS 900 also contributes to S.sub.21 in the scatter matrix [S] of the MPIJIS. Path 3 is illustrated by the dotted line traversing MPIJIS 900 from port 1 to port 2. The contribution of this path to the output at port 2 is denoted as

[00011]$\frac{1}{3.Math.\sqrt{2}}.Math.\mathrm{90},$

meaning that the signal's normalized amplitude is

[00012]$\frac{1}{3.Math.\sqrt{2}}$

and the signal's relative phase is 90.

[0102] With reference to FIG. 9D, this figure depicts a fourth signal path in an MPIJIS according to an illustrative embodiment. This path, (path 4), of signal f.sub.1 120 in MPIJIS 900 also contributes to S.sub.21 in the scatter matrix [S] of the MPIJIS. Path 4 is illustrated by the dotted line traversing MPIJIS 900 from port 1 to port 2. The contribution of this path to the output at port 2 is denoted as

[00013]$\frac{1}{3.Math.\sqrt{2}}.Math.\mathrm{90},$

meaning that the signal's normalized amplitude is

[00014]$\frac{1}{3.Math.\sqrt{2}}$

and the signal's relative phase is 90.

[0103] Thus, the constructive interference between the four principal pathspaths 1, 2, 3, and 4provides:

[00015]${.Math.S_{21}.Math.}^2={.Math.4.Math..Math.\frac{1}{3.Math.\sqrt{2}}.Math.}^2=0.8889$

[0104] With reference to FIG. 10A, this figure depicts one opposite signal path in an MPIJIS according to an illustrative embodiment. MPIJIS 900 is representative of MPIJIS 300 or any of its configuration variants 600, 700, or 800. This path, (path 1), in MPIJIS 900 contributes to S12 in the scatter matrix [S] of the MPIJIS. Recall that S.sub.12 is the transmission parameter for the transmission from port 2 to port 1.

[0105] A signal at frequency f.sub.1 and relative phase 0 is input to port 2 of MPIJIS 900 and this input is denoted as 120, which means that the signal's normalized amplitude is 1 and the signal's relative phase is 0 . Path 1 is illustrated by the dotted line traversing MPIJIS 900 from port 2 to port 1. The contribution of this path to the output at port 1 is denoted as

[00016]$\frac{1}{3.Math.\sqrt{2}}.Math.\mathrm{90},$

meaning that the signal's normalized amplitude is

[00017]$\frac{1}{3.Math.\sqrt{2}}$

and the signal's relative phase is 90.

[0106] With reference to FIG. 10B, this figure depicts a second opposite signal path in an MPIJIS according to an illustrative embodiment. This path, (path 2), of signal f.sub.1 120 in MPIJIS 900 also contributes to S.sub.12 in the scatter matrix [S] of the MPIJIS. Path 2 is illustrated by the dotted line traversing MPIJIS 900 from port 2 to port 1. The contribution of this path to the output at port 1 is denoted as

[00018]$\frac{1}{3.Math.\sqrt{2}}.Math.\mathrm{90},$

meaning that the signal's normalized amplitude is

[00019]$\frac{1}{3.Math.\sqrt{2}}$

and the signal's relative phase is 90.

[0107] With reference to FIG. 10C, this figure depicts a third opposite signal path in an MPIJIS according to an illustrative embodiment. This path, (path 3), of signal f.sub.1 120 in MPIJIS 900 also contributes to S.sub.12 in the scatter matrix [S] of the MPIJIS. Path 3 is illustrated by the dotted line traversing MPIJIS 900 from port 2 to port 1. The contribution of this path to the output at port 2 is denoted as

[00020]$\frac{1}{3.Math.\sqrt{2}}.Math.\mathrm{270},$

meaning that the signal's normalized amplitude is

[00021]$\frac{1}{3.Math.\sqrt{2}}$

and the signal's relative phase is 270 (180 degrees out of phase with the contributions of paths 1 and 2).

[0108] With reference to FIG. 10D, this figure depicts a fourth opposite signal path in an MPIJIS according to an illustrative embodiment. This path, (path 4), of signal f.sub.1 120 in MPIJIS 900 also contributes to S.sub.12 in the scatter matrix [S] of the MPIJIS. Path 4 is illustrated by the dotted line traversing MPIJIS 900 from port 2 to port 1. The contribution of this path to the output at port 2 is denoted as

[00022]$\frac{1}{3.Math.\sqrt{2}}.Math.\mathrm{270},$

meaning that the signal's normalized amplitude is

[00023]$\frac{1}{3.Math.\sqrt{2}}$

and the signal's relative phase is 270 (180 degrees out of phase with the contributions of paths 1 and 2).

[0109] Thus, the destructive interference between the four principal opposite pathspaths 1, 2, 3, and 4provides:

[00024]${.Math.S_{12}.Math.}^2={.Math.2.Math..Math.\frac{1}{3.Math.\sqrt{2}}-2.Math..Math.\frac{1}{3.Math.\sqrt{2}}.Math.}^2=0$

[0110] With reference to FIG. 11, this figure depicts a generalized signal flow graph of an MPIJIS in accordance with an illustrative embodiment.

[0111] With reference to FIG. 12, this figure depicts a simplified signal flow graph of an MPIJIS in accordance with an illustrative embodiment.

[0112] In the signal flow graphs of FIGS. 11 and 12, the two ports bl and b2 of the nondegenerate three-wave mixers 302A and 302B are connected together at ports b1 and b2, respectively, through an effective symmetric coupler 1102 (e.g., 90 degree hybrid), which couples these ports and cold terminations. In one implementation, as in the depiction of FIG. 12, the effective coupler 1102 is replaced by a lossy transmission line with amplitude attenuation ||. The phase acquired by signals at frequency f.sub.2 propagating along the transmission line is Argument() (i.e., the phase of the complex number ).

[0113] The signal flow graph shows the signal flow through ports 1 and 2, 90 hybrid coupler 304, nondegenerate three-wave mixer 302A, nondegenerate three-wave mixer 302B, and the effective coupler 1102. The scheme for the signal flow is lossless. The effective coupler 1102 has a real transmission parameter for signals propagating at frequency f.sub.2 between ports b1 and b2 that are connected together (the real parameter means there is no acquired phase by signals propagating between ports b1 and b2). The two nondegenerate three-wave mixers are (assumed to be) nominally identical. The scattering parameters of MPIJIS 900 at a certain working point are determined by wave interference between 4 principal forward paths depicted in FIGS. 9A-D and 4 principal opposite paths depicted in FIGS. 10A-D (although there are multiple (or infinite) possible paths in each direction due to infinite number of possible beam traversals between b1 and b2).

[0114] The scattering parameter on resonance from port i to j (where i,j{1,2}) vanishes if the waves that propagate from i to j interfere destructively, as in FIGS. 10A-D. The scattering parameter on resonance from port i to j (where i,j{1,2}) is almost unity if the waves that propagate from i to j interfere constructively, as in FIGS. 9A-D.

[0115] The pump is assumed to be enabled where

[00025]$.Math.r.Math.=.Math.t.Math.=\frac{1}{\sqrt{2}}$

and the phase difference is .sub.1.sub.2. The amplitude attenuation coefficient of the lossy transmission line 1102, shown in FIG. 12, is defined as =||e.sup.id. Phase shift acquired by signals propagating along the transmission line 1102 (between ports b1 and b2) at f.sub.2 is .sub.d=.sub.2.sub.d. The angular frequency of microwave signals at f.sub.2 is .sub.2=2f.sub.2. The delay time of the delay/transmission line 1102 is

[00026]${}_d=\frac{l_d.Math.\sqrt{\mathrm{.Math.}}}{c},$

where c is the speed of light, l.sub.d is the effective electrical length of the transmission line (1102), and is the effective dielectric constant of the transmission line 1102.

[0116] In the signal graph of FIGS. 11, r.sub.1 and r.sub.2 are the reflection coefficients, t.sub.1, t.sub.1, t.sub.2, and t.sub.2 are the transmission parameters of nondegenerate three-wave mixers 302A and 302B, respectively, where t.sub.1=t*.sub.1 (where t*.sub.1 is the conjugate of t.sub.1) and where t.sub.2=t*.sub.2 (where t*.sub.2 is the conjugate of t.sub.2). It is presumed that t is equal to or about equal t.sub.2 and r.sub.2 is equal to or about equal to r.sub.2.

[0117] It should be appreciated that there are numerous paths for the signal at f.sub.1 entering the port 1 to follow in MPIJIS 900 and/or for the signal at f.sub.1 entering the port 2 to follow. For any signal at f.sub.1 entering the port 1 or port 2, wave interference results in 4 forward or opposite principal paths, respectively, that determine transmission and/or reflection, based on the phase difference .

[0118] For ease of understanding and explanation, headings are utilized below. It should be appreciated that the headings are not meant to be limiting.

[0119] I. FIG. 13 depicts device equations for MPIJIS 900 based on the simplified signal flow graph of FIG. 12

[0120] First, using the scattering matrix

[00027]$\left[S\right]=\left(\begin{array}{cc}{S}_{11}& S_{12}\\ S_{21}& S_{22}\end{array}\right),$

the scattering parameters on the diagonal are S.sub.21 and S.sub.22 which define the reflection parameters. The scattering parameters on the off diagonal are S.sub.21 and S.sub.22 which define the transmission parameters. The results of the S matrix are (Equations A)

S.sub.11=1/2(S.sub.a1a1S.sub.a2a2+iS.sub.a2a1+iS.sub.a1a2)

S.sub.22=1/2(S.sub.a2a2S.sub.a1a1+iS.sub.a2a1+iS.sub.a1a2)

S.sub.21=1/2(iS.sub.a1a1+iS.sub.a2a2+S.sub.a2a1Sa.sub.1a2)

S.sub.12=1/2(iS.sub.a1a1+iS.sub.a2a2+S.sub.a1a2S.sub.a2a1)

[0121] Intermediate results are

[00028]$S_{a.Math..Math.1.Math.a.Math..Math.1}=r_1+\frac{r_2.Math.t_1.Math.t_1^{}.Math.{}^2}{1-r_1.Math.r_2.Math.{}^2}$$S_{a.Math..Math.1.Math.a.Math..Math.2}=\frac{t_1.Math.t_2^{}.Math.}{1-r_1.Math.r_2.Math.{}^2}$$S_{a.Math..Math.2.Math.a.Math..Math.2}=r_2+\frac{r_1.Math.t_2.Math.t_2^{}.Math.{}^2}{1-r_1.Math.r_2.Math.{}^2}$$S_{a.Math..Math.2.Math.a.Math..Math.1}=\frac{t_2.Math.t_1^{}.Math.}{1-r_1.Math.r_2.Math.{}^2}$

[0122] which can be rewritten as (Equations B)

[00029]$S_{a.Math..Math.1.Math.a.Math..Math.1}=r_1-\frac{e^{2.Math.i.Math..Math.{}_d}.Math.r_2.Math.{.Math.t_1.Math.}^2.Math.{.Math..Math.}^2}{1-r_1.Math.r_2.Math.e^{2.Math.i.Math..Math.{}_d}.Math.{.Math..Math.}^2}$$S_{a.Math..Math.1.Math.a.Math..Math.2}=-\frac{.Math.t_1.Math..Math..Math.t_2^{}.Math..Math..Math..Math..Math.e^{i\left({}_d-\right)}}{1-r_1.Math.r_2.Math.e^{2.Math.i.Math..Math.{}_d}.Math.{.Math..Math.}^2}$$S_{a.Math..Math.2.Math.a.Math..Math.2}=r_2-\frac{e^{2.Math.i.Math..Math.{}_d}.Math.r_1.Math.{.Math.t_2.Math.}^2.Math.{.Math..Math.}^2}{1-r_1.Math.r_2.Math.e^{2.Math.i.Math..Math.{}_d}.Math.{.Math..Math.}^2}$$S_{a.Math..Math.2.Math.a.Math..Math.1}=-\frac{.Math.t_2.Math..Math..Math.t_1^{}.Math..Math..Math..Math..Math.e^{i\left({}_d+\right)}}{1-r_1.Math.r_2.Math.e^{2.Math.i.Math..Math.{}_d}.Math.{.Math..Math.}^2}$$\mathrm{where}.Math..Math.{}_1-{}_2.$

[0123] It should be noted that, for example, the nomenclature S.sub.21 represents a scattering parameter (transmission parameter) for a signal (at frequency f.sub.1) entering from port 1 and exiting port 2, as understood by one skilled in the art. Similarly, for example, the nomenclature S.sub.a1a2 represents a scattering parameter (transmission parameter) for a signal entering port a2 and exiting port a1. As can be seen in the equations above, S.sub.11, S.sub.22, S.sub.21, and S.sub.12 each have 4 principal paths given the wave interference as understood by one skilled in the art, although the actual different possible paths between port 1 and 2 (and vice versa) are numerous or infinite because of the self-loop between the idler resonators (e.g., between coupled ports b1 and b2). However, the scattering parameters in Equations B account for all possible paths and subsequently the scattering parameters for the whole device in Equations A. Additionally, the scattering parameters S.sub.11 and S.sub.21 define the operation of MPIJIS 900 for a signal (at frequency f.sub.1) entering port 1, which then exits port 1 or exits port 2, respectively. Likewise, scattering parameters S.sub.22 and S.sub.12 define the operation of MPIJIS 900 for a signal (at frequency f.sub.1) entering port 2, which then exits port 2 or exits port 1, respectively.

[0124] II. FIG. 14 depicts device equations for the special case of nominally identical mixers operated at the same working point.

[0125] Now, turning to a special case of nominally identical mixers 302A and 302B operated at the same working point, the following relations are satisfied r.sub.1=r.sub.2=r, |t.sub.1|=|t.sub.1|=|t.sub.2|=|t.sub.2|=t (which means that the mixers 302A and 302B are nominally identical and biased at the same working point), ||=, .sub.1.sub.2, and |r|.sup.2+|t|.sup.2=1. In this special case, the results of the [S] matrix are

[00030]$S_{a.Math..Math.1.Math.a.Math..Math.1}=S_{a.Math..Math.2.Math.a.Math..Math.2}=r.Math.\frac{1-{}^2.Math.e^{2.Math.i.Math..Math.{}_d}}{1-r^2.Math.{}^2.Math.e^{2.Math.i.Math..Math.{}_d}}$$S_{a.Math..Math.1.Math.a.Math..Math.2}=-\frac{t^2.Math.{}^{.Math..Math.e^{i\left({}_d-\right)}}}{1-r^2.Math.{}^2.Math.e^{2.Math.i.Math..Math.{}_d}}$$S_{a.Math..Math.2.Math.a.Math..Math.1}=-\frac{t^2.Math.{}^{.Math..Math.e^{i\left({}_d+\right)}}}{1-r^2.Math.{}^2.Math.e^{2.Math.i.Math..Math.{}_d}}$

[0126] which results (finally) in

[00031]$S_{11}=S_{22}=-\frac{{\mathrm{it}}^2.Math.{}^{}.Math.e^{i.Math..Math.{}_d}}{1-r^2.Math.{}^2.Math.e^{2.Math.i.Math..Math.{}_d}}.Math.\cos \left(\right)$$S_{21}=\frac{i}{1-r^2.Math.{}^2.Math.e^{2.Math.i.Math..Math.{}_d}}.Math.\left(r\left(1-{}^2.Math.e^{2.Math.i.Math..Math.{}_d}\right)-t^2.Math.{}^{}.Math.e^{i.Math..Math.{}_d}.Math.\sin \left(\right)\right)$$S_{12}=\frac{i}{1-r^2.Math.{}^2.Math.e^{2.Math.i.Math..Math.{}_d}}.Math.\left(r\left(1-{}^2.Math.e^{2.Math.i.Math..Math.{}_d}\right)+t^2.Math.{}^{}.Math.e^{i.Math..Math.{}_d}.Math.\sin \left(\right)\right)$

[0127] III. FIG. 15 depicts the device equations when the mixers are operated at the 50:50 beam splitter point.

[0128] For MPIJIS 900 to operate as an isolator, mixers 302A and 302B operate at the 50:50 beam splitting working point, the following relations are satisfied

[00032]$r_1=r_2=r=\frac{1}{\sqrt{2}}$$.Math.t_1.Math.=.Math.t_1^{}.Math.=.Math.t_2.Math.=.Math.t_2^{}.Math.=.Math.t.Math.=\frac{1}{\sqrt{2}}$$.Math..Math.=\frac{1}{\sqrt{2}}$

[0129] Phase shifts acquired by on-resonance signals (for the idler resonators) at f.sub.2 propagating along the transmission line .sub.d=k, where k=0,1,2, . . . ,

[0130] For .sub.d=0 and on-resonance, the scattering parameters are as follows (Equations C):

[00033]$S_{a.Math..Math.1.Math.a.Math..Math.1}=S_{a.Math..Math.2.Math.a.Math..Math.2}=\frac{\sqrt{2}}{3}$$S_{a.Math..Math.1.Math.a.Math..Math.2}=-\frac{\sqrt{2}}{3}.Math.e^{-i.Math..Math.}$$S_{a.Math..Math.2.Math.a.Math..Math.1}=-\frac{\sqrt{2}}{3}.Math.e^{i.Math..Math.}$$\mathrm{Where}.Math..Math.{}_1-{}_2$

[0131] which then results in

[00034]$S_{11}=S_{22}=-i.Math.\frac{\sqrt{2}}{3}.Math.\cos .Math..Math.$$S_{21}=i.Math.\frac{\sqrt{2}}{3}.Math.\left(1-\sin .Math..Math.\right)$$S_{12}=i.Math.\frac{\sqrt{2}}{3}.Math.\left(1+\sin .Math..Math.\right)$

[0132] As seen in this figure, for =/2 the isolator transmits from port 1 to port 2 but blocks propagation from port 2 to port 1. Equations D show:

[00035]$S_{11}=0.Math..Math.S_{22}=0$$S_{12}=0.Math..Math.S_{21}=i.Math.\frac{2.Math.\sqrt{2}}{3}$${.Math.S_{21}.Math.}^2=0.8889,$

[0133] which in terms of decibels is |S.sub.21|(dB)=0.51 dB

[0134] The S matrix then is

[00036]$\left[S\right]=\left(\begin{array}{cc}0& 0\\ i.Math.\frac{2.Math.\sqrt{2}}{3}& 0\end{array}\right)$

which shows transmission from port 1 to port 2 and isolation in the direction of port 2 to port 1.

[0135] IV. FIG. 16 depicts the device equations when the mixers operated at the 50:50 beam splitter point but the transmitting direction is changed.

[0136] Now suppose is changed to /2 instead of =/2 as in equations D. As see in this figure, for =/2 the isolator transmits from port 2 to port 1 but blocks propagation from port 1 to port 2. Equations E show:

[00037]$S_{11}=0.Math..Math.S_{22}=0$$S_{21}=0.Math..Math.S_{12}=i.Math.\frac{2.Math.\sqrt{2}}{3}$${.Math.S_{12}.Math.}^2=0.8889,$

[0137] which in terms of decibels is |S.sub.12|(dB)=0.51 dB

[0138] The S matrix then is

[00038]$\left[S\right]=\left(\begin{array}{cc}0& i.Math.\frac{2.Math.\sqrt{2}}{3}\\ 0& 0\end{array}\right)$

which shows transmission from port 2 to port 1 and isolation in the direction of port 1 to port 2. This type of in situ changing of isolation direction is unavailable in state-of-the-art commercial cryogenic isolators.

[0139] V. FIG. 17 depicts the device equations when the mixers operated at the 50:50 beam splitter point but the phase acquired by the internal-mode signals propagating between the two mixers is different. The working point set by the pumps' amplitude, phase difference, and frequency, is the same for the mixers but the phase acquired by the internal-mode signals at frequency f2 propagating between the two mixers is different pi versus 0. So the change is in the device design or the transmission line connecting the two mixers.

[0140] For MPIJIS 900 to operate as an isolator, mixers 302A and 302B operate at the 50:50 beam splitting working point, now at .sub.d= and on-resonance and =/2. The following relations are satisfied

[00039]$r_1=r_2=r=\frac{1}{\sqrt{2}}$$.Math.t_1.Math.=.Math.t_1^{}.Math.=.Math.t_2.Math.=.Math.t_2^{}.Math.=.Math.t.Math.=\frac{1}{\sqrt{2}}$$.Math..Math.=\frac{1}{\sqrt{2}}$

[0141] Now for .sub.d= and on-resonance, the scattering parameters are as follows (Equations F):

[00040]$S_{a.Math..Math.1.Math.a.Math..Math.1}=S_{a.Math..Math.2.Math.a.Math..Math.2}=\frac{\sqrt{2}}{3}$$S_{a.Math..Math.1.Math.a.Math..Math.2}=\frac{\sqrt{2}}{3}.Math.e^{-i.Math..Math.}$$S_{a.Math..Math.2.Math.a.Math..Math.1}=\frac{\sqrt{2}}{3}.Math.e^{i.Math..Math.}$$\mathrm{Where}.Math..Math.{}_1-{}_2$

[0142] which then results in

[00041]$S_{11}=S_{22}=i.Math.\frac{\sqrt{2}}{3}.Math.\cos .Math..Math.$$S_{21}=i.Math.\frac{\sqrt{2}}{3}.Math.\left(1+\sin .Math..Math.\right)$$S_{12}=i.Math.\frac{\sqrt{2}}{3}.Math.\left(1-\sin .Math..Math.\right)$

[0143] As seen in this figure, for =/2 the isolator transmits from port 1 to port 2 again and blocks propagation from port 2 to port 1. Equations G show:

[00042]$S_{11}=0.Math..Math.S_{22}=0$$S_{12}=0.Math..Math.S_{21}=i.Math.\frac{2.Math.\sqrt{2}}{3}$${.Math.S_{21}.Math.}^2=0.8889,$

[0144] which in terms of decibels is |S.sub.21|(dB)=0.51 dB

[0145] The S matrix then is

[00043]$\left[S\right]=\left(\begin{array}{cc}0& 0\\ i.Math.\frac{2.Math.\sqrt{2}}{3}& 0\end{array}\right)$

which shows transmission again from port 1 to port 2 and isolation again in the direction of port 2 to port 1. From this disclosure, those of ordinary skill in the art will be able to understand that such isolation operation and the reversal of the isolation direction are possible using the proposed MPIJIS device at numerous working points by varying .sub.d (by design/layout) and (by changing the phase difference between the drives) in the manner shown in FIGS. 15-17.

[0146] The circuit elements of the MPIJIS device and connections thereto can be made of superconducting material. The respective resonators and transmission/feed/pump lines can be made of superconducting materials. The hybrid couplers can be made of superconducting materials. Examples of superconducting materials (at low temperatures, such as about 10-100 millikelvin (mK), or about 4 K) include Niobium, Aluminum, Tantalum, etc. For example, the Josephson junctions are made of superconducting material, and their tunnel junctions can be made of a thin tunnel barrier, such as an aluminum oxide. The capacitors can be made of superconducting material separated by low-loss dielectric material. The transmission lines (i.e., wires) connecting the various elements can be made of a superconducting material.

[0147] As understood by one skilled in the art, there are many different technical advantages and benefits of the MPIJIS device. The MPIJIS device is substantially lossless (no greater loss than nominal losses in transmission or propagation), is easy to engineer, design, and fabricate, can be realized on chip using superconducting circuits or integrated into a PCB. The MPIJIS device can be made broadband by engineering the impedance of the JPC feedlines, and implementing lumped-element JPCs. The MPIJIS device can be made compact using lumped-element design of the hybrids and JPCs or other manifestations of nondegenerate three-wave mixers and requires no flux control or flux pulses. Additionally, because r and t do not need to be r=0 and t=1, the MPIJIS device can be very stable over a long period of time (limited mainly by the dc-flux bias of the two nondegenerate three-wave mixers) and it is easy to tune up. Also, because r and t do not need to be r=0 and t=1, the MPIJIS device can have a large dynamic range (maximum input power). The pump power feeding the device can be relatively small (less heating of the mixing chamber in a dilution refrigerator). Furthermore, other than having nominally identical nondegenerate three-wave mixers (which is within the standard fabrication capabilities), there are no stringent requirements on uniformity or homogeneity. The MPIJIS device can use one pump for all modes of operation. The MPIJIS device can be made using niobium (Nb) junctions and operated at 4K.

[0148] Various embodiments of the present invention are described herein with reference to the related drawings. Alternative embodiments can be devised without departing from the scope of this invention. Although various connections and positional relationships (e.g., over, below, adjacent, etc.) are set forth between elements in the following description and in the drawings, persons skilled in the art will recognize that many of the positional relationships described herein are orientation-independent when the described functionality is maintained even though the orientation is changed. These connections and/or positional relationships, unless specified otherwise, can be direct or indirect, and the present invention is not intended to be limiting in this respect. Accordingly, a coupling of entities can refer to either a direct or an indirect coupling, and a positional relationship between entities can be a direct or indirect positional relationship. As an example of an indirect positional relationship, references in the present description to forming layer A over layer B include situations in which one or more intermediate layers (e.g., layer C) is between layer A and layer B as long as the relevant characteristics and functionalities of layer A and layer B are not substantially changed by the intermediate layer(s).

[0149] The following definitions and abbreviations are to be used for the interpretation of the claims and the specification. As used herein, the terms comprises, comprising, includes, including, has, having, contains or containing, or any other variation thereof, are intended to cover a non-exclusive inclusion. For example, a composition, a mixture, process, method, article, or apparatus that comprises a list of elements is not necessarily limited to only those elements but can include other elements not expressly listed or inherent to such composition, mixture, process, method, article, or apparatus.

[0150] Additionally, the term illustrative is used herein to mean serving as an example, instance or illustration. Any embodiment or design described herein as illustrative is not necessarily to be construed as preferred or advantageous over other embodiments or designs. The terms at least one and one or more are understood to include any integer number greater than or equal to one, i.e. one, two, three, four, etc. The terms a plurality are understood to include any integer number greater than or equal to two, i.e. two, three, four, five, etc. The term connection can include an indirect connection and a direct connection.

[0151] References in the specification to one embodiment, an embodiment, an example embodiment, etc., indicate that the embodiment described can include a particular feature, structure, or characteristic, but every embodiment may or may not include the particular feature, structure, or characteristic. Moreover, such phrases are not necessarily referring to the same embodiment. Further, when a particular feature, structure, or characteristic is described in connection with an embodiment, it is submitted that it is within the knowledge of one skilled in the art to affect such feature, structure, or characteristic in connection with other embodiments whether or not explicitly described.

[0152] The terms about, substantially, approximately, and variations thereof, are intended to include the degree of error associated with measurement of the particular quantity based upon the equipment available at the time of filing the application. For example, about can include a range of 8% or 5%, or 2% of a given value.

[0153] The descriptions of the various embodiments of the present invention have been presented for purposes of illustration, but are not intended to be exhaustive or limited to the embodiments disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The terminology used herein was chosen to best explain the principles of the embodiments, the practical application or technical improvement over technologies found in the marketplace, or to enable others of ordinary skill in the art to understand the embodiments described herein.