WO2019018034A2

WO2019018034A2 - System and method for unidirectional routing of signals

Info

Publication number: WO2019018034A2
Application number: PCT/US2018/027880
Authority: WO
Inventors: Benjamin Chapman; Eric Rosenthal; Konrad LEHNERT; Joseph KERCKHOFF
Original assignee: The Regents Of The University Of Colorado, A Body Corporate
Priority date: 2017-05-22
Filing date: 2018-04-17
Publication date: 2019-01-24
Also published as: WO2019018034A3

Abstract

One aspect of the invention provides a non-reciprocal element having an operating bandwidth and an operating frequency. The non-reciprocal element includes: a first port; a second port; a first frequency converter coupled with the first port and configured to convert a first signal having a first frequency to a second signal having a second frequency; a first delay element coupled with the first frequency converter and configured to introduce a time delay into the second signal, thereby producing a first time-delayed signal having the second frequency; and a second frequency converter coupled with the first delay element and the second port and configured to convert the first time-delayed signal to a third signal having the first frequency. The second frequency is different from the first frequency. The first delay element is a non-resonant element. Another aspect of the invention provides an integrated circuit including a non-reciprocal element as described herein.

Description

SYSTEM AND METHOD FOR UNIDIRECTIONAL ROUTING OF SIGNALS

CROSS-REFERENCE TO RELATED APPLICATION(S) This application claims the benefit of priority of U.S. Patent Application Serial

No. 62/509,565, filed May 22, 2017. The entire content of this application is hereby

incorporated by reference herein.

FIELD OF INVENTION

The disclosure relates generally to unidirectional routing of signals for communication networks, and, more specifically, non-reciprocal elements configured for unidirectional routing of signals.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

OR DEVELOPMENT

This invention was made with government support under grant numbers W91 INF- 14-1- 0079 awarded by the U.S. Army Research Office, and 1125844, awarded by the National Science Foundation. The government has certain rights in the invention.

BACKGROUND OF THE INVENTION

Unidirectional routing of information is commonly used within electronic communication networks. As such, non-reciprocal devices (e.g., gyrators, circulators, and isolators) have been developed across the electromagnetic spectrum for applications including radar,

telecommunication, and reflection amplifiers. In typical implementation, commercial ferrite circulators make use of the Faraday effect, reliant upon permanent magnets; however, ferrite circulators are undesirable due to bulk and loss, and they are not suited for applications requiring on-chip scalability or extreme sensitivity to magnetic fields.

Various non-reciprocal devices employ actively modulated circuits that generate non- reciprocity through parametric coupling between resonant modes. These actively modulated circuits create a synthetic gauge for itinerant modes, which may be changed in-situ by changing the relative phase of the couplings. However, active devices are often constrained to narrowband performance due to reliance on high-Q modes. SUMMARY OF THE INVENTION

One aspect of the invention provides a non-reciprocal element having an operating bandwidth and an operating frequency. The non-reciprocal element includes: a first port; a second port; a first frequency converter coupled with the first port and configured to convert a first signal having a first frequency to a second signal having a second frequency; a first delay element coupled with the first frequency converter and configured to introduce a time delay into the second signal, thereby producing a first time-delayed signal having the second frequency; and a second frequency converter coupled with the first delay element and the second port and configured to convert the first time-delayed signal to a third signal having the first frequency. The second frequency is different from the first frequency. The first delay element is a non- resonant element.

This aspect of the invention can have a variety of embodiments. The operating bandwidth can be equal to the operating frequency. The operating bandwidth is less than or equal to the operating frequency. The operating bandwidth can be a multiple of the operating frequency.

The first frequency converter can include a first single-sideband modulator and the second frequency converter can include a second single-sideband modulator.

The first frequency converter can be a down frequency converter and the second frequency converter can be an up frequency converter. The first frequency converter can be an up frequency converter and the second frequency converter can be a down frequency converter.

The first frequency converter can be configured to modulate the first signal with a first modulation signal. The second frequency converter can be configured to modulate the first time- delayed signal with a second modulation signal. The first modulation signal and second modulation signal can differ in phase by π/2.

The non-reciprocal element can further include: a third frequency converter coupled with the first port and configured to convert the first signal to a fourth signal having a third frequency; a second delay element coupled with the third frequency converter and configured to introduce a time delay into the fourth signal producing a second time-delayed signal having the third frequency; and a fourth frequency converter coupled with the second delay element and the second port and configured to convert the second time-delayed signal to a fifth signal having the first frequency. The third frequency is different from the first frequency. The second delay element is a non-resonant element.

The non-reciprocal element can further include: a first summation element coupled between the first port and the first frequency converter and the third frequency converter; and a second summation element coupled between the second port and the second frequency converter and the fourth frequency converter.

The first time delay and the second time delay can be equivalent.

The first frequency converter can be configured to modulate the first signal with a first modulation signal. The third frequency converter can be configured to modulate the first signal with a third modulation signal. The first modulation signal and third modulation signal can differ in phase by π/2. The second frequency converter can be configured to modulate the first time-delayed signal with a second modulation signal. The fourth frequency converter can be configured to modulate the second time-delayed signal with a fourth modulation signal. The second modulation signal and fourth modulation signal can differ in phase by -π/2.

In one embodiment, at least one of the first frequency converter and the second frequency converter does not include a Wheatstone bridge of Superconducting Quantum Interference Devices (SQUIDs). At least one of the first frequency converter, the second frequency converter and the delay element can be free from ferromagnetic materials. Each of the first frequency converter, the second frequency converter and the delay element can be free from ferromagnetic materials. In one embodiment, the non-reciprocal element does not include a permanent magnet. In one embodiment, the non-reciprocal element does not include a physical inteferometer. At least one of the first frequency converter and the second frequency converter can be non-resonant elements. The delay element can be free from electrical resonance.

Another aspect of the invention provides an integrated circuit including a non-reciprocal element as described herein.

This aspect of the invention can have a variety of embodiments. The integrated circuit can further include a plurality of impedance-tunable circuit elements forming the non-reciprocal element as described herein.

The integrated circuit can further include a plurality of diodes forming the gyrator as described herein. Another aspect of the invention provides a circulator including the non-reciprocal element as described herein.

Another aspect of the invention provides a method for unidirectional signal routing. The method includes: converting a first signal having a first frequency to a second signal having a second frequency different from the first frequency; introducing a time delay into the second signal to produce a first time-delayed signal having the second frequency, wherein the time delay is free from resonance; and converting the first time-delayed signal to a third signal having the first frequency. The first time delay element is coupled between the first frequency converter and the second frequency converter.

Another aspect of the invention provides a non-reciprocal microwave or radio wave element including: a first port; a second port; a third port; a fourth port; a first delay element; a second delay element; a first cross-over switch coupled to the first port, the third port, the first delay element, and the second delay element; and a second cross-over switch coupled to the second port, the fourth port, the first delay element, and the second delay element. The first cross-over switch is driven to oscillate between: a first state coupling the first port and the first delay element and coupling the third port and the second delay element; and a second state coupling the first port and the second delay element and coupling the third port and the first delay element. The second cross-over switch is driven to oscillate between: a first state coupling the second port and the first delay element and coupling the fourth port and the second delay element; and a second state: coupling the second port and the second delay element and coupling the fourth port and the first delay element. The second cross-over switch is driven to oscillate at a 90° offset from the first cross-over switch.

BRIEF DESCRIPTION OF THE DRAWINGS

For a fuller understanding of the nature and desired objects of the present invention, reference is made to the following detailed description taken in conjunction with the

accompanying drawing figures wherein like reference characters denote corresponding parts throughout the several views.

Figure 1 illustrates a non-reciprocal element according to an embodiment of the disclosure. Figure 2 illustrates an frequency -time diagram according to an embodiment of the disclosure.

Figure 3 illustrates a non-reciprocal element according to an embodiment of the disclosure.

Figure 4 illustrates measured phased difference for non-reciprocal elements according to embodiments of the disclosure.

Figure 5 illustrates a non-reciprocal element according to an embodiment of the disclosure.

Figure 6 illustrates a phasor representation of signal propagating through a non-reciprocal element according to an embodiment of the disclosure.

Figure 7 illustrates a non-reciprocal element according to an embodiment of the disclosure.

Figure 8 illustrates a phasor representation of signal propagating through a non-reciprocal element according to an embodiment of the disclosure.

Figure 9 depicts a graph illustrating the amplitude of the forward and reverse

transmission of non-reciprocal elements according to embodiments of the disclosure.

Figures 10 and 11 depict graphs illustrating sidebands in output signals according to embodiments of the disclosure.

Figure 12 is a graph illustrating phase difference between non-reciprocal elements according to an embodiment of the disclosure.

Figure 13 illustrates a circulator according to an embodiment of the disclosure.

Figure 14 depicts a graph illustrating scattering parameters for a circulator according to an embodiment of the disclosure.

Figure 15 illustrates a flowchart for a method of unidirectional routing of signals according an embodiment of the disclosure.

Figure 16 depicts lossless, broadband circulation according to an embodiment of the invention. Panel (a) is a schematic for a lossless, broadband, and superconducting circulator. Circulation arises from interference between the reciprocal even-mode and the gyrating odd- mode of the circuit. Panels (b)-(g) depict how non-reciprocity arises in the odd-mode. Panel (b) depicts modulation profiles of the two square-wave-biased bridge circuits when the delay separating the bridges is τ= 0. Panel (c) depicts the product of the modulation profiles in Panel (b) . Transmission is reciprocal, and power is scattered into modulation sidebands. Panel (d) depicts the modulation profile for a signal exciting the differential mode of the upper port (odd excitation of ports 1 & 3) when the delay separating the bridges is a quarter of the modulation period τ= 774. (Graphically, this shifts the sine trace back by a quarter-period.) Panel (e) depicts the product of the modulation profiles in Panel (d). As the product is 1 for the entire modulation period, the signal is effectively unmodulated. Panel (f) depicts the modulation profile for a signal exciting the differential mode of the lower port (odd excitation of ports 2 & 4) when the delay separating the bridges is a quarter of the modulation period, τ= 774.

(Graphically, this shifts the cosine trace back by a quarter-period.) Panel (g) depicts the product of the modulation profiles in Panel (f). The product is constant in time, as in Panel (e), but the minus sign imparts a ττ-phase shift on the incident signal. The odd-mode of the circuit therefore realizes a gyrator.

Figure 17 depicts direct circulation with transfer switches and delays according to an embodiment of the invention. Panel (a) depicts a four-port circulator constructed from two dynamically modulated transfer switches and a pair of delay lines. The delay is set to one quarter of the modulation period τ= 774. In Panel (b), the left (1702, blue) and right (1704, red) switches are modulated with period J and a relative phase of π=2. For visual clarity, the modulation period is divided into quarter-period intervals indicated by the background. The schematics in Panel (c) depict the way the switches will route a left-propagating signal, depending on the portion of the modulation in which it arrives. Signals incident on port 1 are always routed to port 4 and signals incident on port 3 are routed to port 2. Note that these are not "snap-shots" of the switches' configurations, as the signal is delayed by a quarter-period of the modulation as it propagates through the network. Panel (d) provides the same depiction as Panel

(c) , but for left-travelling waves incident on ports 2 or 4. To keep the device's orientation the same, the arrow of time is drawn from right-to-left.

DEFINITIONS

The instant invention is most clearly understood with reference to the following definitions.

As used herein, the singular form "a," "an," and "the" include plural references unless the context clearly dictates otherwise. Unless specifically stated or obvious from context, as used herein, the term "about" is understood as within a range of normal tolerance in the art, for example within 2 standard deviations of the mean. "About" can be understood as within 10%, 9%, 8%, 7%, 6%, 5%, 4%, 3%, 2%, 1%, 0.5%, 0.1%, 0.05%, or 0.01% of the stated value. Unless otherwise clear from context, all numerical values provided herein are modified by the term about.

As used in the specification and claims, the terms "comprises," "comprising,"

"containing," "having," and the like can have the meaning ascribed to them in U.S. patent law and can mean "includes," "including," and the like.

Unless specifically stated or obvious from context, the term "or," as used herein, is understood to be inclusive.

Ranges provided herein are understood to be shorthand for all of the values within the range. For example, a range of 1 to 50 is understood to include any number, combination of numbers, or sub-range from the group consisting 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, or 50 (as well as fractions thereof unless the context clearly dictates otherwise).

DETAILED DESCRIPTION OF THE INVENTION

Without being bound by theory, Applicants assert that conventional approaches of ferrite- free non-reciprocal devices are limited as ferrite circulators are bulky and require magnetic shielding, and, as such, are not easily integrated within on-chip superconductive devices.

Applicants have identified that through the use of a combination of frequency conversion and delay elements, a broadband non-reciprocal device may be formed. Such an approach requires no resonant physics, and allows for the operation of broadband circuits, where the bandwidth is comparable to an operating frequency of the device. In various embodiments, the operational bandwidth of the device is limited only by the bandwidth of non-resonant subcomponents.

The embodiment of Figure 1 illustrates a non-reciprocal element 100. As illustrated, the non-reciprocal element 100 includes a port 102, a port 104, a frequency converter 106, a delay element 108, and a frequency converter 110. The frequency converter 106 is coupled to the port 102 and the delay element 108 and the second frequency converter 110 is coupled to the delay element 108 and port 104. In various embodiments, non-reciprocal element 100 may be configured as a gyrator and/or a circulator. These elements are described below in greater detail.

The frequency converter 106 may be configured as either an up frequency converter or a down frequency converter. In one embodiment, the frequency converter 106 may be configured to convert a signal having a first frequency to a second signal having a second frequency, where the second frequency is either higher than or lower than the first frequency. For example, the frequency converter 106 may be configured to convert the frequency of a received signal either up or down by Ω. In other embodiments, the frequency converter 106 may be configured to create a first sideband and a second sideband shifted about the same first frequency. For example, the frequency converter may create first and second sidebands at ± Ω . In one embodiment, the frequency converter 106 comprises a single-sideband modulator. In various embodiments, the frequency converter 106 may include an In-Band/Quadrature (IQ) mixer.

The frequency converter 110 may be configured as either an up frequency converter or a down frequency converter. In one embodiment, the frequency converter 110 may be configured to convert a signal having a first frequency to a second signal having a second frequency, where the second frequency is either higher than or lower than the first frequency. For example, the frequency converter 110 may be configured to convert the frequency of a received signal either up or down by Ω. In other embodiments, the frequency converter 110 may be configured to create a first sideband and a second sideband shifted about the first frequency. For example, the frequency converter 110 may create a first and second sidebands are created at ± Ω . In one embodiment, the frequency converter 110 comprises a single-sideband modulator. In various embodiments, the frequency converter 110 may include an In-Band/Quadrature (IQ) mixer.

In one embodiment, the frequency converter 106 is configured as an up frequency converter and frequency converter 110 is configured as a down frequency converter. In other embodiments, the frequency converter 106 is configured as a down frequency converter and frequency converter 110 is configured as an up frequency converter. In one or more

embodiments, the frequency converters 106 and 110 are configured to modulate a receive signal with an equal but opposite frequency shift. In such embodiments, the frequency of the output signal is the same as the frequency of the input signal. Further, the frequency converters 106 and 110 may differ in phase by π/2. Further yet, the frequency converters 106 and 110 may be referred to as being frequency symmetric modulators. Performance of the non-reciprocal element 100 may be related to the match between the frequency converters 106 and 1 10. For example, the frequency converter 106 and the frequency converter 1 10 may be matched elements such that each frequency converter has substantially no reflection, thereby improving performance of the non-reciprocal element 100. Further, by configuring the frequency converter 106 and the frequency converter 110 to have zero reflection, the performance of the non-reciprocal element 100 may be improved.

Delay element 108 may be configured to introduce a time delay into a signal provided by either frequency converter 106 or 1 10. As illustrated in Figure 1, the delay element 108 can be coupled between the frequency converter 106 and the frequency converter 1 10 such that it imparts a delay on a signal received from either frequency converter 106 or 1 10. In one embodiment, the delay is a time delay. In one embodiment, the delay is a non-reciprocal dynamical phase. In various embodiments, the delay element 108 is a non-resonant element. Further, the delay element 108 may be free from resonant physics. In one or more

embodiments, the delay element 108 may be an RC filter. In other embodiments, the delay element 108 is formed from one or more looped SMA cables. In one embodiment, cable length may be defined as I >— wherein c is the speed of light in the cable and ω is the maximum operation frequency of the device. (The biasing rate can exceed the signal frequency.) For example, for device that operates up to 10 GHz, the length of the cable will be about 16 mm. In various embodiment, the cable may be made longer and the modulation frequency Ω may be made smaller.

In various embodiments, the non-reciprocal element 100 has an operating bandwidth at least partially based on the bandwidth of frequency converter 106, frequency converter 1 10 and/or delay element 108.. For example, the operating bandwidth can be as large as one or more of the bandwidth of the first frequency converter 106, the bandwidth of first delay element 108, and the bandwidth of secondary converter 1 10. Further, the operating bandwidth can be a multiple of one or more of the bandwidth of the first frequency converter 106, the bandwidth of first delay element 108, and the bandwidth of secondary converter 1 10. In one or more embodiments, the operating bandwidth is less than or equal to an operating frequency of the non- reciprocal element 100. For example, if the operating frequency of the non-reciprocal element 100 is 1 GHz, the operating bandwidth of the non-reciprocal element 100 can be between 1 Hz to 1 GHz. In other embodiments, the operating bandwidth of the non-reciprocal element 100 is a multiple of the operating frequency of the non-reciprocal element 100. For example, if the operating frequency of the non-reciprocal element 100 is 1 GHz, the operating bandwidth of the non-reciprocal element 100 can be between 1 Hz to lxM GHz.

In one or more embodiments, non-reciprocal element 100 is free from electrical resonance. For example, one or more of the frequency converters 106 and 110 and the delay element 108 can be free from resonant physics.

In various embodiments, non-reciprocal element 100 is a gyrator. In such embodiment, the non-reciprocal element 100 may be combined with one or more linear, reciprocal elements to form a linear, non-reciprocal circuit element. In one or more embodiments, the non-reciprocal element 100 is a gyrator that is configured to convert received signals up and down in frequency symmetrically such that an output signal has the same frequency as the received signal. In various embodiments, the non-reciprocal element 100 is configured to eliminate sidebands created during the frequency conversion process. In one embodiment, the frequency converters of the non-reciprocal element 100 generate signals having sidebands out of phase with each other such that when signals are combined to form an output signal, the sidebands are not present in the output signal. In other embodiments, the frequency converters of the non-reciprocal element 100 generate signals having sidebands out of phase with each other such that when signals are combined to form an output signal, the sidebands do not significantly affect the output signal. In such embodiments, the sidebands may still be present in the output signal.

In various embodiments, the non-reciprocal element 100 may be configured as a ladder gyrator, as incident signals are converted up or down on a ladder of frequencies. The incident signals can be shelved on a rung of the ladder of frequencies while accumulating a delay, and then converted back to their original frequency.

In other embodiments, the non-reciprocal element 100 is configured as a balance gyrator, including one or more arms formed of a pair of frequency converters coupled to a delay element.

Figure 2 illustrates an energy diagram 200 of one embodiment of non-reciprocal element 100. As is illustrated, as a signal, signal 220, moves left to right, the frequency of the signal is equally decreased and then increased while a time delay is inserted. Further, as a signal, signal 210, moves from the right to the left, the frequency of the signal is equally increased and decreased while a time delay is inserted. In both instances, the output signal frequency is the same as the input signal frequency. In one or more embodiments, non-reciprocal element 100 may be implemented as a circulator. In one embodiment, the circulator is configured as an isolator. In various embodiments, the non-reciprocal element 100 may be implemented as part of an integrated circuit. The non-reciprocal element may be configured to route microwave signals for quantum information processing.

Equation 1 illustrates a non-reciprocal scattering matrix for a two-port circuit element such as non-reciprocal element 100.

Equation 1

In one embodiment, transmitted signals received by the non-reciprocal element 100 are applied with a phase shift of either 0 or π depending on their direction of propagation. Non- reciprocity may be conceptually understood as the lack of commutation between cascading operations of the non-reciprocal element 100. In one embodiment, the frequency conversion and delay scheme of the non-reciprocal element 100 is generated by an ambi-directional frequency converter (e.g., frequency converter 106 and/or 110) followed by a non-commuting time delay (e.g., delay element 108).

The operations performed on an incoming signal by the elements of the non-reciprocal element 100 may be realized with frequency and phase shifts. In one or more embodiments, the transfer operator E for a down-converting frequency converter translates a phasor a in frequency Ω, as shown by:

£α[ω] = α[ω— Ω] . Equation 2

In various embodiments, a frequency converter (e.g., frequency converter 106 and/or 110) is non-reciprocal even when used in isolation. As such, time reversal maps the down-converter onto an up-converter Ω→— Ω, in the same way that time-reversal alters magnetic fields, B→— B, in the Faraday effect. As such, by pairing the frequency converter 106 with the frequency converter 110, the frequency of the input signal may be preserved. For example, a down frequency converter having a transfer operator E may be paired with an up frequency converter having transfer operator £^"†.

In one embodiment, frequency converter 106 is configured to modulated a received signal by β^~ίίϊτ and frequency converter 110 is configured to modulate an incident signal by e^+lSlT. Further, cascading complimentary frequency converters make a reciprocal network, and as such, [E, £^"†] = 0. However, non-reciprocity can be restored to the network by inserting a time delay (e.g., time delay element 108) between the frequency converters 106 and 110. The transfer operator a for a delay, D, translates a phasor aft) forward in time by a time shown as: τ: Da(t) = a(t + τ). Equation 3

As time and frequency are Fourier duels, consecutive translations in these variables do not generally commute. As such, with further reference to Equation 1, the scattering parameters of the non-reciprocal element 100 are S₁₂ = E^DE and S₂₁ = EDE^ . These parameters are non- reciprocal when E and D do not commute as shown in:

Si2 - ^s2i = E^f [D, E] + [D, E]E^f . Equation 4

In the expression of Equation 4, non-reciprocity is seen to result from the non- commutation of D and E, and, as shown in Equation 5, non-reciprocity can be maximized for a choice of Ωτ = π/2.

\S₂₁— S₁₂1 °c sin(nr). Equation 5

In various embodiments, non-commutation with time delay may be used to construct a non-reciprocal element having multiple arms (or paths), such as non-reciprocal element 300. In the illustrated embodiment, the non-reciprocal element 300 include a first and second arm.

However, in other embodiments, the non-reciprocal element 300 may have any number of arms including a delay element coupled between up and down frequency converters. For example, the non-reciprocal element 300 may have 3, 4, 5 or more arms. As illustrated, the first arm can include frequency converters 306 and 310 and delay element 308 and the second arm can include frequency converters 312 and 316 and delay element 314. The first and second arms can be coupled between ports 302 and 304.

In one embodiment, the frequency converters 306 and 312 convert the frequency of a signal received at port 302 to a second signal having a different frequency. This conversion generates sidebands in the second signal centered about the frequency of the received signal.

The delay elements 308 and 314 can introduce a time delay into the frequency-converted signals, and output the time-delayed signal to frequency converters 310 and 316. The frequency converters 310 and 316 frequency convert the time-delayed signals opposite to that of the frequency converters 306 and 312 such that the output signal has the same frequency has the received signal. The sidebands in the converter signals can be further offset from the original frequency after conversion by frequency converters 310 and 316. In one embodiment, one of frequency converter 310 and 316 generates a signal having sidebands opposite to that of the signal generated by the other frequency converter such that when these two signals are combined and provided to the output port (port 304), the sidebands are cancelled. For example, one of frequency converter 310 and 316 can be configured to output a signal that differs in phase by π/2 from the other.

In one embodiment, the frequency converters 310 and 316 convert the frequency of a signal received at port 304 to a second signal having a different frequency. This conversion generates sidebands in the second signal centered about the frequency of the received signal. The delay elements 308 and 314 introduce a time delay into the frequency converted signals, and output the time-delayed signal to frequency converters 306 and 312. The frequency

converters 306 and 312 frequency convert the time delayed signals opposite to that of the frequency converters 310 and 316 such that the output signal has the same frequency has the received signal. The sidebands in the converted signals are further offset from the original frequency after conversion by frequency converters 306 and 312. In one embodiment, one of frequency converter 306 and 312 generates a signal having sidebands opposite to that of the signal generated by the other frequency converter such that when these two signals are combined and provided to the output port (port 304), the sidebands are cancelled. For example, one of frequency converter 306 and 312 is configured to output a signal that differs in phase by π/2 from the other.

In embodiments where non-reciprocal element 300 includes more than two arms, the signals outputted by the final frequency converter of each arm may be differ by 45 degrees, 60 degrees and/or 120 degrees, such that when these signals are combined, the sidebands are eliminated. In various embodiments, the outputted signals may differ by any amount such that when they are combined, any sidebands are eliminated.

In various embodiment the frequency converters 306, 310, 312 and 316 are frequency- symmetric and may be represented in the transfer operation formalism by X = cos(nr) = j (£^"† + E) and Y = sin(nr) = (£^"†— E). As such, time reversal alters the phase rather the frequency of modulation of a signal. The phases of frequency converters 312 and 316 may be delayed by ττ/2 as compared to corresponding frequency converters 306 and 310. Such a phase delay coherently erasing the second harmonics of the modulation frequency Ω/2π . Further, the difference between the forward and backward scattering parameters S₂₁ - S₁₂ of the non- reciprocal element 300 is a commutator of transfer operators, which is described by Equation 5. As described with regard to the non-reciprocal element 100, maximal gyration of non-reciprocal element 300 is realized when Ωτ = π/2. In various embodiments, the non-reciprocal element 300 may be configured as a gyrator and/or circulator. In one embodiment, the circulator maybe an isolator. Further, in one or more embodiments, the non-reciprocal element 300 may be referred to as a balanced gyrator, for the symmetrical way in which signals are converted up and down in frequency.

In various embodiments, the entire bandwidth of either the non-reciprocal element 100 or the non-reciprocal element 300 can be utilized simultaneously due to the linear relationship between input and output signal in all device components. For example, a single wave-packet, with a spectral width of several GHz, will gyrate in both non-reciprocal elements 100 and 300. In one embodiment, any one of the frequency converters 306, 310, 312 and 316 may be formed from IQ mixers.

Figure 4 illustrates non-reciprocal phase shift (S₂₁)— ^-(S₁₂) implemented at microwave frequencies. Trace 402 illustrates the non-reciprocal phase shift (5₂₁)— (5₁₂) implemented at microwave frequencies for the non-reciprocal element 100. Further, trace 404 illustrates the non-reciprocal phase shift (5₂₁)— (5₁₂) implemented at microwave frequencies for the non-reciprocal element 300. The mean value of the phase difference for trace 402, as illustrated in graph 400 of Figure 4, measured at a point between about 4 GHz to 8 GHz, is 180.05°±1.46°. Further, the mean value of the phase difference for trace 404, as illustrated in graph 400 of Figure 4, measured at a point between about 4 GHz to 8 GHz, is 180.41° ± 3.72°.

Figure 5 illustrates an embodiment of a non-reciprocal element 500. In the embodiment of Figure 5, the non-reciprocal element 500 includes ports 502 and 504, a frequency

converter 510, a frequency converter 520 and a delay element 530. As is illustrated, frequency converters 510 and 520 are coupled between ports 502 and 504 and the delay element 530 is coupled between frequency converters 510 and 520. The frequency converter 510 and the frequency converter 520 may be single-sideband mixers. Further, the frequency converter 510 and the frequency converter 520 may be IQ mixers. As is illustrated, the frequency

converter 510 modulates a received signal with sin(nr) or cos(nr), and the frequency converter 520 modulates a received signal with cos(nr) or -sin(nr), respectively. In one embodiment, a signal is received by frequency converter 510 via port 502. The frequency converter 510 shifts the frequency of the received signal, the delay element 530 inserts a delay into the frequency signal, and the frequency converter 520 shifts the frequency of the time-delayed signal back to the frequency of the originally received signal. In another embodiment, a signal is received by frequency converter 520 via port 504. The frequency converter 520 shifts the frequency of the received signal, the delay element 530 inserts a delay into the frequency signal, and the frequency converter 510 shifts the frequency of the time- delayed signal back to the frequency of the originally received signal.

In one embodiment, the frequency converter 510 is an up frequency converter, increasing the frequency of the signal, and the frequency converter 520 is a down frequency converter, decreasing the frequency of the signal. In another embodiment, the frequency converter 510 is a down frequency converter, decreasing the frequency of the signal, and the frequency converter 520 is an up frequency converter, increasing the frequency of the signal.

Figure 6 is a phasor representation of transmission of a signal through the non-reciprocal element 500 in a forward (a) and reverse (b) direction. The forward direction may refer to a signal received at port 502 and the reverse direction may refer to a signal received at port 504. As can be seen, after a signal received at the port 502 (1), the frequency of the signal is increased and then a delay is inserted. For example, the frequency converter 510 modulates the received signal by sin(nr) and delay element 530 inserts a delay of τ. After the delay, the frequency of the signal is decreased before it is output via port 504 (2). For example, the frequency converter 520 modulates the time delayed signal by cos(nr) . Further, after a signal received at port 504 (2), the frequency of the signal is decreased and then a delay is inserted. For example, frequency converter 520 modulates the received signal by -sin(nr) and delay element 530 delays the frequency converter signal by τ. After the delay is inserted, the frequency of the signal is increased before it is output via port 504 (2). For example, the frequency

modulator 510 modulates the time-delayed signal by cos(nr) . In both instances, the frequency of the outputted signal is the same as the frequency of the received signal.

Figure 6 additionally illustrates the effects of scattering parameters S₁₂ = E^DE and 5₂₁ = EDE^†on a received signal along the forward and reverse direction of the non-reciprocal element 500. Further, the non-reciprocal element 500 may be configured as a gyrator and/or circulator. In one particular embodiment, non-reciprocal element 500 is a ladder gyrator. In the embodiment of Figure 7, a non-reciprocal element 700 includes ports 702 and 704, frequency converters 710, 720, 740 and 750, and delay elements 730 and 760. The non- reciprocal element 700 may further include summers 770 and 780 (e.g., summing amplifiers or summing circuits). As illustrated, the non-reciprocal element 700 can include a first and second arm. The first arm includes the delay element 730 coupled between frequency converters 710 and 720 and the second arm includes the delay element 760 coupled between frequency converter 740 and 750. Further, the summer 770 can be coupled between the first and second arms and the port 702 and the summer 780 can be coupled between the first and second paths and the port 704. The non-reciprocal element 700 may be a gyrator and/or a circulator. In one embodiment, the non-reciprocal element 700 may be referred to as a balanced gyrator.

Figure 8 is a phasor representation of signals propagating through the non-reciprocal element 700. As is illustrated in Figure 8, a signal propagating through the forward direction (c) of the non-reciprocal element 700 a signal is split along the first path and second path and then recombined at summer 780 to coherently eliminate the upper and lower second harmonics that are produced and modified by the frequency converters 710, 720, 740 and 750. Further, a signal propagating through the reverse direction (d) is split along the first path and second path and then recombined at summer 770 to coherently eliminate the upper and lower second harmonics (sidebands) that are produced and modified by the frequency converters 710, 720, 740 and 750.

In the illustrated embodiment, the frequency converter 710 is configured to apply a modulation signal of cos(nr), the frequency converter 720 is configured to apply a modulation signal of— sin(nr), the frequency converter 740 is configured to apply a modulation signal of sin(nr), and the frequency converter 750 is configured to apply a modulation signal of cos(nr). Further, the frequency converter 710 and the frequency converter 740 create sidebands at ω ± Ω for a signal received at port 702. These sidebands are further shifted by ω ± Ω by frequency converters 720 and 750 to create sidebands at ω ± 2Ω.

The frequency converters may be configured such that when the signals are combined before being output sidebands created in the signal are eliminated. For example, for a signal received at port 702, one frequency converter pair (e.g., frequency converters 710 and 720 or frequency converters 740 and 750) is configured to generate a signal whose sidebands are 180 degrees out of phase with the other. While specific modulation signals are discussed in relation to Figure 7, in other embodiments, other modulation signals may be used such that the sidebands are eliminated. Further, while Figure 7 illustrates the non-reciprocal element 700 as having 2 arms, in other embodiments, the non-reciprocal element 700 may comprise more than 2 arms. In such configurations, the modulations signals may be selected such that when

summer 770 or 780 combines the signals, the sidebands are eliminated. For example, the paths may generate signals that when combined, the combined sidebands are 180 degrees out of phase.

In one embodiment, one more arms are configured to generate a signal having sidebands that are 20, 40, 60, 90, and/or 120 degrees out of phase with the others.

In various embodiments, the performance of non-reciprocal elements 500 and 700 is improved by setting the relative dynamical phase Ωτ acquired in the delay lines. Figure 9 illustrates the forward and scattering parameters for gyrators 500 and 700 plotted as a function of the delay phase Ωτ. In various embodiment, Ωτ may be tuned by varying the modulation frequency of Ω. For example, the frequency of Ω may be modulated between 4 and 33 MHz.

In various embodiments, the amplitude of transmission for the non-reciprocal

elements 200 and 700 is reciprocal with a magnitude that depends on the delay phase. The magnitude may be maximized when Ωτ = π/2, and minimized when up and down converted signals interfere destructively at Ωτ = 0 or Ωτ = .

The non-reciprocal elements 100 and 500 may have non-reciprocal ripples in the magnitude of transmission, as the reciprocal transmission is independent of the delay phase Ωτ.

For example, the scattering matrix for a gyrator, such as non-reciprocal elements 100 and/or 500, may be calculated using transfer matrix operators for delay and frequency conversion. Acting on a phasor, a = a₀e^{l 0t}, these operators can be described by translations in time or in frequency, as shown by:

Da(t) = a[t + τ], and Equation 6

£α[ω] = α[ω— Ω] . Equation 7

Up-conversion is the inverse of down conversion and may be shown by:

£^"†a[6)] = α[ω + Ω] . Equation 8

As is described above, in various embodiments, τ is the time delay of a delay line and Ω is the angular frequency of the modulation signal applied by the frequency converters.

As is shown in Figures 1, 3, 5 and 7, a down-frequency conversion, a delay and an up- frequency conversion can be cascaded in series to create a non-reciprocal element, such as non- reciprocal element 100, 300, 500 and/or 700. Further, forward and reverse transmissions may be shown by:

Ε^ϋΕα[ω] = β ^_ιΩτα[ω], Equation 9

DEE*a[a>] = β ^ίίϊτα[ω] . Equation 10

The scattering matrix for a non-reciprocal element can be calculated based on the transfer operators. Further, off-diagonal scattering matrix elements can be calculated using these transfer operators: off-diagonal scattering matrix elements are the eigenvalue of forward and reverse transmission shown in Equations 9 and 10. As such, the scattering matrix for the non-reciprocal elements 500 becomes: s = {_e -iar o ) Equation 1 1

Transmission magnitude of a the non-reciprocal elements 500 may be independent of Ωτ, where the phase difference between forward and reverse transmission is maximized for Ωτ = TT/2. Further, the difference between off-diagonal scattering matrix elements may be shown as:

S₁₂— S₂₁ = 2i sin(nr). Equation 12

The non-reciprocal elements 700 may be described by cosinusoidal and sinusoidal linear combinations of the frequency conversion operations, where:

X = ± (£t + E), Equation 13

Y = i (£t - E). Equation 14

In one embodiment, to calculate transmission through a non-reciprocal element, such as non-reciprocal element 700, the transmission through each arm (or path) is summed. For example, each arm itself is non-reciprocal at the carrier frequency, but creates reciprocal second sidebands, which are canceled upon recombination. Further, forward and reverse transmissions may be given by the following equations:

^ (XD (-Y) + ΥϋΧ)α[ω] = - ^ίη(Ωτ) α[ω], Equation 15

± ((-Y)DX + XDY)a[a>] = ^ίη(Ωτ) α [ω] . Equation 16

For such embodiments, the scattering matrix of off-diagonal elements are the eigenvalues of the transfer operators shown in Equations 15 and 16, and is shown as: - sin r) '^ΐη(0^Ωτ))· Equation 17 Further, the magnitude of transmission is proportional to sin(nr) and the forward and reverse transmission are π out of phase, as shown by:

S₁₂— S₂₁ = sin(nr) . Equation 18

The non-reciprocity of a non-reciprocal element (e.g., non-reciprocal elements 500 and/or 700) is maximized when Ωτ = π/2. Further, second harmonics of the modulation frequency Ω/2π are created by each path and are coherently canceled upon recombination of the two arms.

In various embodiments, the non-reciprocal element 500 is configured to suppress sidebands by at least 15 dB. In one example, the non-reciprocal element 500 is configured to suppress sidebands by 15.3 dB. As is illustrated by graph 1000 of Figure 10, the sidebands generated by non-reciprocal elements 100 and 500 are a function of 6_L. Further, Figure 10 shows that there is a 15.3 dB suppression of sidebands at 6_L = 0.

In other embodiments, the non-reciprocal element 700 is configured to suppress sidebands by at least 14 dB. In one example, the non-reciprocal element 700 is configured to suppress sidebands by 14.7 dB. As illustrated by graph 1 100 of Figure 1 1, sidebands generated by non-reciprocal elements 200 or 700 are a function of Θ_Β . As can be seen, there is a 14.7 dB suppression of sidebands at Θ_Β = ±ττ/2. Further, at Θ_Β = 0, power is split evenly between second harmonics.

Graph 1200 of Figure 12 illustrates the phase difference of a non -reciprocal element, z(S₂₁)— z(S₁₂), as a function of Ωτ. In a balanced gyrator (e.g., non-reciprocal element 700), the phase difference z(S₂₁)— z(S₁₂) is independent of Ωτ. Fitting the data to a constant function yields z(S₂₁)— z(S₁₂) = 180.8° + 5.3°. In a ladder gyrator (e.g., non-reciprocal element 500), the phase different depends on Ωτ. Further, the linear relationship between z(S₂₁) - z(S₁₂) and Ωτ, where z(S₂₁) - z(S₁₂) is about 2Ωτ, having a slope of 2.010±0.003. In various embodiments, as a ladder gyrator is dependent on delay phase, it may be operated as a gyrator when Ωτ = π/2, or as a reciprocal device with the same frequency-dependent attenuation as a gyrator when Ωτ = 0 or TT.

In one or more embodiments, a non-reciprocal element, such as non-reciprocal element 700, exhibits substantially delay-independent transmission, but delay-dependent non- reciprocal phase. Further, a non-reciprocal element, such as non-reciprocal element 700, can exhibit substantially delay-independent non-reciprocal phase, but delay-dependent transmission. This amplitude-phase duality reflects that the gyrators may be interferometric duals, in the sense that the non-reciprocal element 700 is equivalent to an interferometer with non-reciprocal elements in each arm. As interferometers translate phase difference into transmission amplitude, the two gyrators necessarily exhibit a dual amplitude-phase relationship.

In one or more embodiments, the non-reciprocal element (non-reciprocal element 100,

300, 500 and/or 700) may be a circulator. For example, the non-reciprocal device maybe used to create a four-port circulator using a Hogan-like construction. The embodiment of Figure 13 illustrates a schematic of the circulator made from a ladder gyrator (e.g., non-reciprocal device 500) and a reciprocator connecting to hybrid couplers. The reciprocator ensures equality of the electrical length and the frequency-dependent insertion loss in each path between the hybrids. Further, the reciprocator prevents transmission through the circulator (e.g., between isolated ports) outside of the operation band of the circulator.

Figure 14 illustrates circulator scattering parameters S₂₁ (solid line) and S₁₂ (solid line with circles), when the device is configured for both clockwise and counterclockwise circulation. The direction of the circulation may be configured in-situ either by changing the modulation frequency of the two devices (exchanging the reciprocator and the gyrator) or inverting the phase of the gyrator (reversing its gyration direction). In one embodiment, at least 15 dB of relative isolation may be achieved between about 5 GHz and 9 GHz. In various embodiments, the reverse isolation may be as great as 30 dB over a 500 MHz bandwidth.

The non-reciprocal elements 500 and/or 700 may be configured as lossless modulation elements. Further, the non-reciprocal elements 500 and/or 700 may be implemented as part of a superconducting implementation. In various embodiments, non-reciprocal devices 100, 200, 500 and/or 700 may be used in the construction of a broadband ambi-directional converter that may be made to have conversion gain as high as -3 dB. Further, integrated circuit elements comprising a non-reciprocal devices 100, 300, 500 and/or 700 may have an insertion loss as low as 6 dB. In various embodiments, non-reciprocal devices 100, 300, 500 and/or 700 may be formed from delay operations with frequency tunable resonances, providing lossless gyration in a restricted (but tunable) frequency band.

In various embodiments, non-reciprocal elements 100, 300, 500 and/or 700 are part of an integrated circuit. For example, each frequency converter of a non-reciprocal element may be comprised of one or more impedance-tunable circuit elements. For example, the impedance- tunable circuit elements may be diodes. In one embodiment, the diodes may be formed as a Wheatstone bridge. Further, the impedance tunable circuit elements may be varactors (voltage- variable capacitors). In one embodiment, the varactors may form a Wheatstone bridge. Further yet, each element of a non-reciprocal element may be formed from transistors and each transistor may comprise an adjusted bias voltage.

In one or more embodiments, the delay elements of non-reciprocal element may be an artificial transmission line within an integrated circuit. The artificial transmission line may include one or more circuit elements, such as inductors and/or capacitors. Further, in one or more embodiments, the delay elements may include a baffle filter. In other embodiments, the elements of a non-reciprocal element do not include a Wheatstone Bridge of Superconducting Quantum Interference Devices (SQUIDS) and ferromagnetic material.

Figure 15 illustrates a method for unidirectional signal routing. At block 1502, a first signal having a first frequency is converted to a second signal having a second frequency different from the first frequency. The first signal may be received by a first port of a non- reciprocal element (e.g., non-reciprocal element 100, 300, 500 and/or 700) and converted in frequency by one or more frequency converters. In one or more embodiments, converting the first signal to a second signal having a second frequency includes one of up-converting or down- converting the frequency of the first signal. In other embodiments, converting the first signal to a second signal having a second frequency includes generating a second signal having at least two sidebands centered about the first frequency.

At block 1504, a time delay is introduced into the second signal to produce a first time- delayed signal having the second frequency. The time delay may be inserted by a delay element of a non-reciprocal element (e.g., non-reciprocal element 100, 300, 500 and/or 700).

At block 1506, the time-delayed signal having the second frequency is converted to a third signal having the first frequency. The time-delayed signal may be converted in frequency by one or more frequency converters of a non-reciprocal element (e.g., non-reciprocal element 100, 300, 500 and/or 700). Converting the time-delayed signal back to the first frequency may include one of down-converting or up-converting the frequency of the time- delayed signal. In other embodiments, converting the time-delayed signal back to the first frequency includes generating a signal having sidebands out of phase with the sidebands of the second signal. In one or more embodiment, the sidebands may be eliminated by combining signals.

Figure 16 illustrates one possible design for a broadband, lossless circuit 1600. The device uses bridge-circuit multipliers, but only two bridges. The element Z_m is an impedance included to impedance match the bridge circuits because of the finite-tunability of the SQUTD- array inductors. As the Bode-Fano criterion is relatively lenient for low quality-factor circuits, Applicant estimates that appropriate choice of Z_m would allow reflections to be limited to less than -20 dB over a bandwidth of 8 GHz.

The delays, rather than realized with a resonant mode, are transmission lines connecting the two bridges. The bias signals are square wave, rather than sinusoidal. Biased in this way, the impedance-matched bridge circuits function as cross-over switches.

The circuit's operation is greatly simplified by transforming to a basis of even and odd excitations of the top (ports 1 & 3) and bottom (ports 2 & 4) ports. Viewed in that light, circulation again arises from a "virtual" Hogan construction, e.g., the interference of a reciprocal common mode and a gyrating differential mode. To show how non-reciprocity arises in the differential mode, the square-wave modulation of transmission realized by the two bridge circuits is plotted in Figure 16, Panel (b). If no delay is included between them, a differential signal on the upper (odd excitation of ports 1 & 3) or lower (odd excitation of ports 2 & 4) is modulated in time according to Figure 16, Panel (c), and the network is reciprocal (and scatters the incident signal into sidebands spaced by odd multiples of the modulation rate). If, however, a delay τ of one-quarter the modulation period T= 774 = ττ/(2Ω) is inserted between the two bridge circuits, reciprocity is broken, and the incident signal's frequency is unchanged. A signal incident on the upper port is modulated by

sign(cos[nt]) x sign(sin[nt + ττ/4]) = 1 Equation 19

as depicted in Figure 16, Panels (d) and (e), whereas a signal incident on the lower port is modulated by

sign(cos[nt]) x sign(sin[nt + ττ/4]) = 1 Equation 20

as depicted in Figure 16, Panels (f) and (g). The circuit's odd-mode, therefore, realizes a gyrator, and the network, as a whole, forms a four-port circulator. Circuit 1600 is in-principle matched, meaning that it does not reflect any part of an incident signal. The switches 1602, 1604 are given a square-wave bias so that each switch is either in a first or a second state— never in between.

Circuit 1600 is also low-loss, in principle, meaning that the only loss comes from the delay lines 1606a, 1606b.

Referring now to Figure 17, Panel (a), another embodiments of the invention provides a circulator 1700 including two transfer switches 1702, 1704 connected by a pair of delay lines 1706a, 1706b. The transfer switches 1702, 1704 have four ports, and are dynamically modulated with period T, toggling between their "through" and "crossed" states. Figure 17, Panel (b) shows how the two switches 1702, 1704 are periodically actuated with a relative phase of ττ/2. The delays are chosen to have duration τ= 774.

To see how circulation is created by this network, first consider how a left-propagating wave (e.g., a signal incident on ports 1 or 3) will be scattered by the network. Figure 17, Panel (c) shows how the two switches will route the incident signal, depending on the portion of the modulation period in which it arrives. For example, if the signal impinges in the first quarter- period of the modulation (time 0 to 774), the transfer switch 1702 will be in its through state. But by the time the signal exits the relevant delay line 1706a, 1706b, the transfer switch 1704 will be in its crossed state. Signals incident on port 1 are, therefore, routed to port 4, and signals incident on port 3 are routed to port 2. If the signal arrives in the second quarter-interval of the modulation (time 774 to 772), the same procedure reveals that the switches will remain in this configuration. Note that these are not snapshots of the switches in time, as the signal is delayed as it propagates through the network.

This process may be continued for the entire modulation period. One observes that the port-to-port signal-routing is independent of the incident signal's arrival time, despite the fact that the signal may travel along different paths before exiting the network.

The non-reciprocal character of the network may be verified by repeating this analysis for right-travelling waves (e.g., signals incident on ports 2 or 4), as depicted in Figure 17, Panel (d). Again, the port-to-port signal-routing is independent of the signal's arrival time: signals incident on port 2 are always routed to port 1 and signals incident on port 4 are always routed to port 3. The scattering of the network, however, is not invariant upon exchange of source and detector: it forms a four-port clockwise circulator. This graphical representation of the device's circulation may be complemented by a quantitative analysis. Let HXt) and H-(t) represent the periodically modulated states of the left and right transfer switches 1702 and 1704, respectively, which take values 1 and -1 when the switches are in their through or crossed states, respectively. For this reason,

In addition, due to the ττ/2 relative phase difference between the two modulation profiles, H_t{t -

TI4) = H_r(t).

For left-travelling waves, signals are delayed before encountering the right switch 1704. A signal that impinges on the network's 1 st or 3rd ports at time t is therefore routed by the two switches 1702, 1704 according to the product H_t(t) ^χ H_r{t - T/4) = Hi(t†= -1. The negative value of this product indicates the odd-parity of the switches: one and only one of the switches 1702, 1704 is in its crossed state, so port 1 routes to port 4 and port 3 routes to port 2.

In contrast, right-travelling waves are delayed before they encounter the left switch 1702. Signals incident on the 2nd or 4th port of the network are, therefore, routed according to the product H_r(t) x Hi(t - 774) =

1. The positive value of this product indicates the even parity of the switches 1702, 1704: either zero or both of the switches are in their crossed states, so port 4 routes to port 3 and port 2 routes to port 1.

More involved analytical methods, in which every odd sideband of the modulation frequency Ω = 2Tr/Jis treated as a port of the network, reveal how deviations from the ideal operation condition τ= 774 = ττ/(2Ω) affect performance. The circulation C and isolation I of the network are then seen to be

Equation 21

Equation 22

where the sums in / run over all odd integers. When Ωτ = π/2, C = 1 and 7 = 0. Such expressions are useful in quantifying the implications of finite bandwidth modulation or dispersion in the delay lines. They also make clear the "perfect" bandwidth of the circulator: the frequency of the input signal does not appear in Equations 21 and 22.

Embodiments of the invention are extremely attractive given the high power-handling of SQUID-array based devices, as their integration with a broadband low-noise amplifier could enable scalable frequency-domain multiplexing of many-qubit systems with near-unit measurement efficiency. Given the current state of quantum error-correction and architectures for quantum information processing with superconducting circuits, such signal-processing innovations will almost certainly be necessary for construction of any superconducting quantum computer intended to be more than a proof-of-principle.

Embodiments of the invention are particularly useful for transmission of microwave signals having frequencies between 300 MHz and 300 GHz and radio waves having frequencies between 300 GHz and 3 kHz. For example, embodiments of the invention can process signals within one or more of the following sub-bands:

The operative frequency of a device can be the result of the transmission window of the components of a device, by one or more filters, and the like.

Some embodiments of the invention are non-optical elements that process signals having frequencies below 430 THz, below 300 GHz, and the like.

Claims

1. A non-reciprocal element having an operating bandwidth and an operating frequency, the non-reciprocal element comprising:

a first port;

a second port;

a first frequency converter coupled with the first port and configured to convert a first signal having a first frequency to a second signal having a second frequency, wherein the second frequency is different from the first frequency;

a first delay element coupled with the first frequency converter and configured to introduce a time delay into the second signal, thereby producing a first time-delayed signal having the second frequency, wherein the first delay element is a non-resonant element; and a second frequency converter coupled with the first delay element and the second port and configured to convert the first time-delayed signal to a third signal having the first frequency.

2. The non-reciprocal element of claim 1, wherein the operating bandwidth is equal to the operating frequency.

3. The non-reciprocal element of claim 1, wherein the operating bandwidth is less than or equal to the operating frequency.

4. The non-reciprocal element of claim 1, wherein the operating bandwidth is a multiple of the operating frequency.

5. The non-reciprocal element of claim 1, wherein:

the first frequency converter comprises a first single-sideband modulator and

the second frequency converter comprises a second single-sideband modulator.

6. The non-reciprocal element of claim 1, wherein the first frequency converter is a down frequency converter and the second frequency converter is an up frequency converter.

7. The non-reciprocal element of claim 1, wherein the first frequency converter is an up frequency converter and the second frequency converter is a down frequency converter.

8. The non-reciprocal element of claim 1, wherein:

the first frequency converter is configured to modulate the first signal with a first modulation signal;

the second frequency converter is configured to modulate the first time-delayed signal with a second modulation signal; and

the first modulation signal and second modulation signal differ in phase by π/2.

9. The non-reciprocal element of claim 1, further comprising:

a third frequency converter coupled with the first port and configured to convert the first signal to a fourth signal having a third frequency, wherein the third frequency is different from the first frequency;

a second delay element coupled with the third frequency converter and configured to introduce a time delay into the fourth signal producing a second time-delayed signal having the third frequency, wherein the second delay element is a non-resonant element; and

a fourth frequency converter coupled with the second delay element and the second port and configured to convert the second time-delayed signal to a fifth signal having the first frequency.

10. The non-reciprocal element of claim 9, further comprising:

a first summation element coupled between the first port and the first frequency converter and the third frequency converter; and

a second summation element coupled between the second port and the second frequency converter and the fourth frequency converter.

11. The non-reciprocal element of claim 9, wherein the first time delay and the second time delay are equivalent.

12. The non-reciprocal element of claim 9, wherein:

the third frequency converter is configured to modulate the first signal with a third modulation signal;

the first modulation signal and third modulation signal differ in phase by π/2;

the second frequency converter is configured to modulate the first time-delayed signal with a second modulation signal;

the fourth frequency converter is configured to modulate the second time-delayed signal with a fourth modulation signal; and

the second modulation signal and fourth modulation signal differ in phase by -π/2.

13. The non-reciprocal element of claim 1, wherein at least one of the first frequency converter and the second frequency converter does not comprise a Wheatstone bridge of Superconducting Quantum Interference Devices (SQUIDs).

14. The non-reciprocal element of claim 1, wherein at least one of the first frequency converter, the second frequency converter and the delay element are free from ferromagnetic materials.

15. The non-reciprocal element of claim 1, wherein each of the first frequency converter, the second frequency converter and the delay element are free from ferromagnetic materials.

16. The non-reciprocal element of claim 1, wherein the non-reciprocal element does not include a permanent magnet.

17. The non-reciprocal element of claim 1, wherein the non-reciprocal element does not include a physical inteferometer.

18. The non-reciprocal element of claim 1, wherein at least one of the first frequency converter and the second frequency converter are non-resonant elements.

19. The non-reciprocal element of claim 1, wherein the delay element is free from electrical resonance.

20. An integrated circuit comprising:

the non-reciprocal element any one of claims 1-19.

21. The integrated circuit of claim 20, further comprising a plurality of impedance-tunable circuit elements forming the non -reciprocal element of any one of claims 1-18.

22. The integrated circuit of claim 20, further comprising a plurality of diodes forming the gyrator of any one of claims 1-19.

23. A circulator comprising:

the non-reciprocal element of any one of claims 1-19.

24. A method for unidirectional signal routing, the method comprising:

converting a first signal having a first frequency to a second signal having a second frequency different from the first frequency;

introducing a time delay into the second signal to produce a first time-delayed signal having the second frequency, wherein the time delay is free from resonance; and

converting the first time-delayed signal to a third signal having the first frequency, wherein the first time delay element is coupled between the first frequency converter and the second frequency converter.

25. A non-reciprocal microwave or radio wave element comprising:

a first port;

a second port;

a third port;

a fourth port;

a first delay element;

a second delay element; a first cross-over switch coupled to the first port, the third port, the first delay element, and the second delay element, the first cross-over switch driven to oscillate between:

a first state:

coupling the first port and the first delay element; and coupling the third port and the second delay element; and

a second state:

coupling the first port and the second delay element; and

coupling the third port and the first delay element; and a second cross-over switch coupled to the second port, the fourth port, the first delay element, and the second delay element, the second cross-over switch driven to oscillate between:

a first state:

coupling the second port and the first delay element; and

coupling the fourth port and the second delay element; and

a second state:

coupling the second port and the second delay element; and

coupling the fourth port and the first delay element;

wherein the second cross-over switch is driven to oscillate at a 90° offset from the first cross-over switch.