The reconfigurable Josephson circulator/directional amplifier

Circulators and directional amplifiers are crucial non-reciprocal signal routing and processing components involved in microwave readout chains for a variety of applications. They are particularly important in the field of superconducting quantum information, where the devices also need to have minimal photon losses to preserve the quantum coherence of signals. Conventional commercial implementations of each device suffer from losses and are built from very different physical principles, which has led to separate strategies for the construction of their quantum-limited versions. However, as recently proposed theoretically, by establishing simultaneous pairwise conversion and/or gain processes between three modes of a Josephson-junction based superconducting microwave circuit, it is possible to endow the circuit with the functions of either a phase-preserving directional amplifier or a circulator. Here, we experimentally demonstrate these two modes of operation of the same circuit. Furthermore, in the directional amplifier mode, we show that the noise performance is comparable to standard non-directional superconducting amplifiers, while in the circulator mode, we show that the sense of circulation is fully reversible. Our device is far simpler in both modes of operation than previous proposals and implementations, requiring only three microwave pumps. It offers the advantage of flexibility, as it can dynamically switch between modes of operation as its pump conditions are changed. Moreover, by demonstrating that a single three-wave process yields non-reciprocal devices with reconfigurable functions, our work breaks the ground for the development of future, more-complex directional circuits, and has excellent prospects for on-chip integration.


I. INTRODUCTION
Connecting elements so that information flows only one way is a key requirement in signal processing. Directional elements route information from source to destination, while simultaneously preventing interference from signals passing in reverse. Typically, the separation of input from output is accomplished in microwave circuits by the circulator 1 . This device has three or more input ports in which signals pass, for instance, from port 1 to 2, but not from 2 to 1, being routed instead to a third port. Circulators based on the non-reciprocal propagation of microwaves in magnetically-biased ferrite materials are widely available commercially. A second crucial directional device is the two-port amplifier. Its role is to increase signal power levels that are otherwise too weak to be processed by subsequent elements. The directionality of an amplifier is specified by the reverse gain, i.e. the gain experienced by a signal traversing the amplifier in reverse, which is usually much less than unity for commercially available components. For both circulators and amplifiers, it is also required that the devices have well-matched input and output impedances, typically 50 Ω, which allows devices to be cascaded without reflections at their interfaces.
In the field of superconducting quantum information 2-4 , cryogenic microwave circulators and amplifiers are regularly used to process signals at the few photon level. The most widely chosen amplifiers for this task are superconducting parametric amplifiers which we abbreviate as paramps. These amplifiers operate by parametrically pumping the non-linear inductance of Josephson junctions that are embedded in microwave resonators. They perform phase-preserving amplification, in which both quadratures of the input signal are amplified equally 5,6 , as well as phase-sensitive amplification, in which one signal quadrature is amplified while the other is de-amplified 7,8 . They operate near the so-called quantum limit 9 , allowing efficient qubit measurement [10][11][12][13] .
However, standard paramp designs are not directional, instead amplifying in reflection and requiring external circulators to separate incoming and outgoing signals. Circulators also play a more general role in signal processing, for instance enabling the remote entanglement of superconducting qubits via cascaded measurement 14 . Commercially available circulators are bulky, magnetic, and somewhat lossy, adding noise to the signal and preventing the on-chip integration of paramps with qubits. Josephson junction-based circulators would alleviate many of these concerns, and thus several theoretical proposals exist [15][16][17] , though none have as yet been constructed. In the classical regime proof-of-principle non-ferrite based circulators, based on lossy varactor diodes, have recently been realized 18 . In addition, several Josephson junction-based amplifiers have been proposed and built which seek to achieve quantum-limited directional amplification without the need for external circulators. One approach uses voltage-biased DC-SQUIDs (Superconducting QUantum Inteference Devices) 19 , which amplify near the quantum-limit. However these devices cannot be simultaneously optimized for directionality and noise performance 20 , and hence are typically operated with circulators between signal source and amplifier [21][22][23] . Another approach, the Traveling Wave Parametric Amplifier (TWPA) seeks to provide broadband amplification by replacing resonant structures with a non-linear superconducting transmission lines. The signal and a microwave pump co-propagate, nominally producing broad-band phase-preserving amplification in the forward direction and de-amplification of backward propagating signals [24][25][26] .
These devices have made rapid progress in recent years, but, to our knowledge, have not yet demonstrated directional amplification of a quantum system. In another recent report, a prototype directional amplifier was realized based on interference between two identical phase-preserving amplifiers and also used in qubit readout 27 .
While many or all of these approaches to circulators and directional amplifiers may eventually succeed, practically speaking, a premium is placed on simple, flexible circuits and pump configurations which are easy to design and operate. To this end, Ranzani and Aumentado have produced a general graph-theoretic treatment of multi-mode parametrically coupled systems 28 . In particular, they consider the case of three coupled modes, explicitly demonstrating the potential for realizing directional amplification and circulation. A closely related subject is directionality in up/down-frequency conversion from bi-harmonic pumping 29 . Additionally, a recent paper treats directionality using the Lindblad master equation formalism, proposing similar devices 30 .
In this article, we adapt the general three-coupled mode model of Ref. 28 to the three modes of the Josephson Parametric Converter (JPC) 31 , a simple and robust, however nondirectional, circuit in wide use as a non-degenerate phase-preserving paramp 5,11,[32][33][34] . We demonstrate both the first experimental realization of a Josephson circulator and a directional amplifier within the same device, achieved by changing only the frequency, power, and relative phase of three microwave pumps. The methodology we demonstrate has excellent prospects for on-chip integration with standard circuit QED systems. It also has excellent prospects in other cryogenic measurements using microwave signals, such as kinetic inductance detectors 35 , dispersive magnetometers 8 , and quantum nano-mechanical resonators 36 .
Furthermore, it makes possible the construction of on-the-fly routing of quantum microwave signals, thereby making possible novel and scalable quantum signal-processing systems.

ON THE JPC
In this section, we consider parametric processes that consist of one or more 2-body interactions between standing microwave modes, each of which are in addition accessible via a "port", i.e. a semi-infinite transmission line giving the mode a finite energy decay rate. In the JPC, the coupling between modes is accomplished via the Josephson Ring Modulator (JRM) at its center, consisting of four Josephson junctions connected to form a closed loop.
The spatial excitation pattern of three orthogonal modes of the JRM are shown schematically in Fig. 1A, and are labeled as a, b, and c. The full JPC circuit is formed by embedding the JRM at the central current antinodes of two crossed λ/2-resonators, which form resonant modes a and b, with frequencies ω a , ω b and energy decay rates κ a , κ b respectively. The third microwave mode c, with frequency ω c , is formed by the common-mode excitation of the a and b modes, and has an energy decay rate κ c . A general tri-linear three-body interaction between these three orthogonal modes is produced by applying a flux through the JRM close to half a flux quantum. Furthermore, by off-resonant pumping of one mode by an RF drive, we couple the other two modes in one of two possible ways, as detailed below.
The first kind of 2-body interaction, yielding conventional phase-preserving amplification, is achieved by pumping one spatial mode (e.g. b) at the sum frequency of the other two (ω p b = ω a + ω c ). Provided the pump frequency is sufficiently detuned from any harmonic of the c-mode, the pump can be approximated as a classical drive and the interaction Hamiltonian can be written as H G int = |g ac |(e +iφp a † c † +e −iφp ac), where a,c are the annihilation operators for their respective modes, g ac is the pump-power-dependent coupling strength, and φ p is the pump phase. The gain of the resulting amplification process at zero detuning can be written as √ G = (1 + |g ac | 2 /κ a κ c )/(1 − |g ac | 2 /κ a κ c ). High gain is achieved for |g ac | 2 → κ a κ c .
This process is denoted graphically as a 'G' bi-directionally connecting two modes, up to a phase factor, as shown schematically in Fig. 1B. We also give a graphical representation of the scattering matrix in Fig. 1B. Signals entering one port are amplified in reflection with voltage gain √ G, and in transmission with gain √ G − 1, together with frequency translation and a pump-phase dependent non-reciprocal phase shift φ p . Phase conjugation is also taking place in the frequency conversion process, as indicated by a white (rather than black) arrowhead. A signal incident on one mode will be combined with amplified vacuum-fluctuations from the other mode, achieving phase-preserving amplification. Due to the symmetry of the amplification process, the signal can be collected from either output port.
The second form of 2-body interaction we employ is gain-less photon conversion, which is achieved by pumping the third mode at the difference frequency of the other two (ω p c = ω a − ω b ). Again, provided the pump frequency is sufficiently detuned from any mode, the interaction Hamiltonian can be written as The process schematic is shown in Fig. 1C, together with a graphical representation of the scattering matrix. Signals incident on either mode are either reflected (with coefficient √ C), As in the gain process, signals transmitted through the device experience frequency translation and a pump-phase dependent non-reciprocal shift (note that here all arrowheads are black, as no phase conjugation occurs). At the full conversion working point, the device resembles a gyrator 1 but with the complication that it also performs a frequency translation.
As shown in Fig. 1D, the three modes of the JPC can be connected with up to six simultaneous gain and conversion processes. A particular function can be realized by identifying the appropriate pumping configuration. Here, we will focus on configurations based only on one process per pair of modes, calculating the scattering matrices following the method of Ref. 28. We first consider the case of three simultaneous gain-less photon conversion processes, which produces a circulator. A schematic for the device coupling configuration and a graphical representation of the scattering matrix are shown in Fig. 2A. In the ideal case, the circulator uses three conversion processes, which would each individually achieve full conversion (C = 1). The final important control variable is the algebraic sum of the three Positive or negative interference occurs as signals travel around the device, their phase being controlled by φ circ tot , which acts here as a gauge flux, and plays the role of the magnetic field in a conventional circulator. As shown in Fig. 2B, for φ circ tot = ±π/2, a matched circulator with clockwise/counter-clockwise circulation is created. We note that this circulator is somewhat different from a ferrite-based circulator in that it translates the frequencies of signals passing through it, but this is generally of little practical consequence as we can freely shift the center frequency of our microwave signals without degrading their information content.
Directional amplification is achieved by combining two gain processes and one conversion process, as shown schematically in Fig. 3A. In the ideal case, we set the pairwise processes so that we have two equal gains G and one full conversion (C = 1). Again, the interference within the device is controlled by a total pump phase φ damp tot , here now given by φ a − φ b + φ c , with directional amplification occurring at φ damp tot = ±π/2. Unlike the circulator, this schematic has a pronounced asymmetry in signal flow through the device, as shown by the graphical scattering matrix in Fig. 3. We therefore label the roles of the three ports in the directional amplifier by the roles played by their inputs as the Signal (S) input, Idler (I) input, and Vacuum (V) input. The S port is matched (no power reflects), and incident power is instead transmitted with gain to the I and V ports. Vacuum fluctuations incident on I are responsible for the additional amplified quantum fluctuations necessarily associated with phase-preserving amplification. Either port I or V can be considered to be the directional amplifier output. The V port is noiselessly and directionally transmitted through the device to the S port with unity gain. This follows from the combined requirements of quantumlimited amplification (sending it to either of the other 2 output ports would degrade the noise performance of the device) and the information conserving nature of the device (no entropy produced, we assume the pump to be perfectly stiff) 38 . Changing φ damp tot by π flips which physical port plays the role of S and V, with I remaining unchanged. The roles can be further re-mapped by changing which pair of modes is linked via conversion and thus, in general, each of the three physical ports can play each role. We note that combining two gain processes with gain G yields a directional amplifier with gain G, not G 2 . The combined operation should be thought of as rerouting the outputs of one port of a non-directional amplifier (from S to V) rather than as two independent stages of amplification.
Unlike the circulator, which is relatively forgiving of imperfect conversion (C < 1), the directional amplifier is much less tolerant of non-idealities. Its directionality is only achieved when the conversion process is adjusted so that 1 − C < 1/G 2 . The device can be thought of as having a 'loop gain' (1 − C)G 2 which needs to be much less than one for the interference process which gives directionality to be dominant. This behavior is demonstrated theoretically in Fig. 4A. The gains have been set to a finite value of ∼ 12 dB, and the input match, initially perfect, degrades as the conversion coefficient decreases. For any pair of finite gains, there exists a conversion coefficient below which any semblance of directional amplification is lost and all nine scattering parameters show positive gain.

III. EXPERIMENTAL APPARATUS AND RESULTS
In the experiment, our JPC is slightly modified from that described above, having an  Fig. 1A). Pumps and probe tones are applied to each mode via the weakly coupled port of a directional coupler connected to each mode.

A. Circulator
We realize a Josephson circulator by coupling all 3 modes pairwise via conversion processes as previously described. When the total pump phase is set to φ circ tot = π/2 this realizes clockwise circulation from mode a to b to c as shown in Fig. 2A. When φ circ tot is changed by π, with no other changes to pump parameters, the direction of circulation is reversed (Fig. 2A).
The pumps are ω p c /2π = 3.928 GHz, ω p a /2π = 1.9291 GHz, and ω p b /2π = 1.9989 GHz, corresponding to conversion coefficients C of 0.97, 0.98, and 0.99, respectively. These processes were applied singly, and the resultant conversion processes were compared to collectively maximize the conversion coefficients while simultaneously matching the single mode fre-quency responses of all three modes (supp Fig. 1). The magnitude of the conversions were then fine tuned by maximizing the magnitude and symmetry of both the input match and the reverse isolation of all three ports of the circulator. Figure 2B shows the complete set of measured scattering parameters (S ij , i, j = a, b, c) for the circulator as a function of probe frequency. These are measured by applying a probe tone to the weakly coupled directional coupler input for each mode. For the diagonal components of the scattering matrix, the reflected output is directed to an amplifier chain and recorded at room temperature by a 2-port vector network analyzer (VNA). Off-diagonal components of the scattering matrix involve inputs and outputs which are at different frequencies, and therefore the output tones are mixed with a local oscillator at the corresponding difference frequency to translate them back to the probe tone frequency prior to being acquired by the VNA.
We identify the unknown offset in the total phase φ circ tot by finding the two values, differing by π, for which S bb is minimized. Following the convention in the theory section, we assign φ circ tot = π/2 to clockwise circulation. We note that it suffices to vary only one pump phase to achieve a desired φ circ tot , and so only the phase of the pump applied to c was varied for all data shown in this paper, although we verified in a separate experiment that the response to all three pump phases was equivalent.
As seen in Fig. 2B, on resonance we have a matched device (with reflection better than −10 dB) having more than 18.5 dB reverse isolation, and less than 0.5 dB of insertion loss. The insertion loss is calibrated relative to the three individual conversion processes, which have been previously demonstrated to be efficient within 0.1 dB 37 . Off resonance, the bandwidth of the individual conversion processes which comprise the circulator, together with the rolling of φ circ tot combine to give an 11 MHz bandwidth over which the input match of all ports is better than −10 dB and the insertion loss is better than 1 dB.
Simply flipping the pump phase by π to φ circ tot = −π/2, without any other variation of pump parameters, switches the direction of circulation, as shown. We see no degradation in overall device performance, and good agreement with theoretical calculations for the scattering parameters in both directions. The theory uses as its inputs the three mode bandwidths and the conversion coefficients of the three individually pumped conversion processes. We note that most deviations are associated with signals input to mode c. We attribute these to the degradation in the spatial mode matching due to phase mismatches in the three cascaded hybrids versus a and b which each pass through a single hybrid. The overall device performance is limited by imperfections in the pairwise conversion processes and drift in the overall pump phase.
We also characterized the device by measuring two representative scattering parameters, S bb and S cb , as a continuous function of pump phase (Fig. 2C). The data are in excellent agreement with theory, showing three working points with alternating circulation directions at points separated by π in phase (−3π/2, −π/2, π/2), with smooth transitions in the scattering parameters versus frequencies in between. Further experimental and theoretical work are required to predict and characterize the effect of higher order nonlinearities on the fine details of the device performance. This is especially vital for determining how many probe photons the device can process without degradation of performance.

B. Directional amplifier
As mentioned earlier, we realize a directional amplifier by coupling two pairs of modes via gain processes and the third pair via a gain-less conversion process (Fig. 3A). Modes a and b are coupled so that C = 0.998 via a pump at ω p c /2π = 3.927 GHz, modes a and c with G = 13 dB via a pump at ω p b /2π = 16.339 GHz, and modes b and c with G = 12 dB via a pump at ω p a /2π = 12.412 GHz. These values were chosen experimentally both to approach perfect conversion and to minimize frequency offsets in the single pump mode responses subject to the constraint ω p c = ω p b − ω p a (Supp Fig. 2). Here, as with the circulator we remove offsets in the pump phases by finding φ damp tot values for which S cc is minimized.
We again define φ damp tot to be ±π/2 at these points, the sign being set by the direction of amplification.
In this mode of operation, physical ports can take on different roles depending on which modes are coupled via conversion and on the value of φ damp tot . Given our pump frequency configuration, when we set φ damp tot = −π/2, mode a is the signal input S, mode b the vacuum input V, and mode c the idler input I as shown in the Fig. 3A. The scattering parameters are plotted in Fig. 3B, showing all the hallmarks of directional amplification. First, the input ports S and V show a reflection coefficient of −16 dB or greater, indicating the device is matched, while the third port shows gain in reflection. Next, signals input at S are amplified and transmitted to I and V (gain of 14 dB). Third, signals incident on I are isolated from S (with isolation of 8 dB), and are instead reflected from I and transmitted to V with gain.
Finally, signals incident on V are transmitted with near unity gain to S (S ba = 0.2 dB). In normal operation, port V will not be driven (hence its name) and can be seen as providing the necessary vacuum fluctuations which must be emitted from S. The directional gain falls off with probe frequency as a lorentzian lineshape with a 3-dB bandwidth of 11 MHz, though we note that other bandwidths can be defined based on the required input match or reverse isolation.
Changing the total pump phase to φ damp tot = π/2 switches the roles of mode a and mode b. This is most directly seen by comparing S ab and S ba , in which the direction of the gain reverses. As shown by the correspondence with the theory curves, all the other scattering parameters also change as expected. Again, the theory curves are calculated with the 3 mode bandwidths and the individual gain and conversion coefficients. In general, the agreement is not as good as for the circulator which we attribute to the fact that there is now gain in the system and therefore misalignments of the pairwise processes and phase drifts can more drastically affect the amplifier performance. In practical implementations, interferometric techniques could be used to stabilize φ damp tot , or several matched, low-gain stages could be cascaded to achieve high net gain without requiring extreme pump precision. As with the circulator, more sophisticated theoretical analysis is needed to understand the effects of higher-order nonlinearities. The effect of such nonlinearities can be seen, for instance, in distortions of the conversion process shown in Supp. Fig. 2. These are also crucial to understanding the dynamic range of the amplifier.
One of the most important characteristics of such paramps is the noise performance. This is characterized by the Noise Visibility Ratio (NVR) of the device, defined as the excess noise visible in a spectrum analyzer at room temperature when the amplifier is turned on versus off. This technique offers a proxy for more difficult direct measurements of the noise temperature. Nevertheless, we can infer the directional amplifier noise performance by comparing to the single gain process of the JPC, which has been previously shown to be nearly quantum-limited 11 . As shown in Fig. 3C, we observe NVR only at the outputs which have gain (I and V) and not at the isolated input port (S). The measured NVR for directional gain agrees to within 1 dB with the associated single pairwise coupling with the same gain, although there are slight shifts of the center frequencies. This indicates that the noise performance of the directional amplifier is essentially as quantum-limited as a JPC acting as a non-directional phase-preserving amplifier.
Finally, we examine the behavior of the device as a function of the conversion coefficient ( Fig. 4). As detailed in the theory section, the conversion process must dominate for the amplifier to be directional. The dependence of two representative scattering parameters S bb and S ab corresponding to input match and directional gain, respectively, are shown for selected conversion coefficients. As expected, the magnitude of all scattering parameters rises as the conversion coefficient decreases, with complete loss of input match and even reflection gain being observed once C falls below a certain threshold (here 0.95). This threshold rises with the amplifier gain; we have chosen a directional gain of 14 dB in order to retain sufficient input match. In general, to achieve a single-stage directional amplifier with high forward gain while retaining a matched input, one requires, surprisingly, a nearly perfect converter as the key element.

IV. CONCLUSIONS
We have successfully realized a Josephson circuit that performs both as a circulator and a directional amplifier. Each function is determined by a specific pump configuration consisting of 3 pump frequencies, amplitudes and phases. The device can be switched on-the-fly between the different signal processing roles. To our knowledge, this work represents the first successful implementation of a Josephson circulator. Our results are in good qualitative agreement with theory but some discrepancies remain which we attribute to neglect of higher order terms in the theory as well as imperfection in our control of the relative phase of the pumps. Further work is needed to analyze the dynamic range characteristics and optimize the device performance. Our present implementation depends on cascaded microwave hybrids; their elimination by further microwave engineering can render the device truly twodimensional and therefore more easily integrated with superconducting qubits/cavities. It will also remove potential sources of loss and mismatch which degrade the device performance.
One avenue for further development consists of constructing directional devices from modes with widely different bandwidths, as suggested in Ref. 30. Still more exotic amplifiers such as a directional phase-sensitive amplifier can also potentially be realized along the same lines as in our experiment. Finally, we believe these in-situ switchable directional Nature Physics 11, 37 (2015).

ACKNOWLEDGMENTS
We are indebted to Leonardo Ranzani for his in-depth presentation at Yale of the graph theory of non-reciprocity. We also acknowledge useful discussions with José Aumentado, Alexander Blais, Aashish Clerk and Archana Kamal.       Figure S2. Single pump curves for directional amplifier The measured single pump conversion and gain curves for all three processes used in the directional amplifier.
The symbol G ij indicates a gain process linking modes i and j. For each port (a b c), the pump parameters are chosen to match the center frequencies for both processes involving that mode. Additionally, the two gains are chosen to match and the conversion is set as close to unity as possible.