Microwave measurement beyond the quantum limit with a nonreciprocal amplifier
F. Lecocq L. Ranzani G. A. Peterson K. Cicak A. Metelmann S. Kotler R. W. Simmonds J. D. Teufel J. Aumentado








^1National Institute of Standards and Technology, 325 Broadway, 
	Boulder, CO 80305, USA
^2Raytheon BBN Technologies, Cambridge, Massachusetts 02138, USA
^3Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universitat Berlin, 14195 Berlin, Germany

             


The measurement of a quantum system is often performed by encoding its state in a single observable of a light field. The measurement efficiency of this observable can be reduced by loss or excess noise on the way to the detector. Even a quantum-limited detector that simultaneously measures a second non-commuting observable would double the output noise, therefore limiting the efficiency to 50%. At microwave frequencies, an ideal measurement efficiency can be achieved by noiselessly amplifying the information-carrying quadrature of the light field, but this has remained an experimental challenge. Indeed, while state-of-the-art Josephson-junction based parametric amplifiers can perform an ideal single-quadrature measurement, they require lossy ferrite circulators in the signal path, drastically decreasing the overall efficiency. In this paper, we present a nonreciprocal parametric amplifier that combines single-quadrature measurement and directionality without the use of strong external magnetic fields. We extract a measurement efficiency of 62_-9^+17% that exceeds the quantum limit and that is not limited by fundamental factors. The amplifier can be readily integrated with superconducting devices, creating a path for ideal measurements of quantum bits and mechanical oscillators.


Valid PACS appear here
                             

                              
Microwave measurement beyond the quantum limit with a nonreciprocal amplifier
    F. Lecocq^1, L. Ranzani^2, G. A. Peterson^1, K. Cicak^1, A. 
Metelmann^3, S. Kotler^1, R. W. Simmonds^1, J. D. Teufel^1 and J. 
Aumentado^1
    December 30, 2023
===============================================================================================================================================




  




§ INTRODUCTION




To measure the state of a quantum system, one typically entangles it with an auxiliary quantum system, a `meter', that in turn can be easily measured with a classical apparatus <cit.>. Non-idealities in the meter measurement, such as dissipation and excess noise, are quantified by a single parameter: the measurement efficiency, 0≤η≤1. While state estimation can be achieved with high fidelity even with limited measurement efficiency <cit.>, a highly efficient measurement is critical for controlling the quantum system, enabling, for example, analog quantum feedback <cit.> and measurement-based entanglement <cit.>.

In superconducting microwave devices, the quantum information  — the state of a quantum bit or the position of a mechanical oscillator — is typically encoded in a single quadrature of a microwave field <cit.>. The measurement of this meter takes the form of amplification, necessary to overcome the noise of room temperature electronics. State-of-the-art measurement schemes rely on Josephson-junction based parametric amplifiers as the first stage of amplification <cit.>. While these amplifiers can in principle have ideal noise performance, in practice they require additional components in the signal path that significantly reduce the overall system measurement efficiency. Most importantly, these amplifiers are reciprocal and therefore require multiple microwave circulators to route amplified microwave signals away from the quantum system and toward the measurement apparatus. Microwave circulators break reciprocity using the Faraday effect induced by large magnetic fields<cit.> that are incompatible with superconducting devices, thus requiring additional wiring with long coaxial cables and multiple connectors to physically separate them from both the quantum system and the amplifier. Highly optimized measurement chains have demonstrated efficiencies around 0.5 to 0.7 <cit.>, leaving little room for further improvement.

Recently, amplifiers breaking reciprocity without the use of strong magnetic fields have been developed, in an effort to directly integrate them with superconducting quantum devices to maximize measurement efficiency. Among these efforts are traveling wave amplifiers <cit.> and multi-mode amplifiers <cit.>. While all of these approaches meet the requirements for gain, bandwidth, dynamic range, and integrability, they also amplify, i.e. measure, both non-commuting quadratures of the microwave field (known as phase-preserving amplification). As a result, the output noise doubles <cit.>, effectively putting a quantum limit on the measurement efficiency at 0.5 <cit.>. Realizing a single quadrature measurement (known as phase-sensitive amplification) with a nonreciprocal amplifier has been a long-standing challenge and has, so far, only been proposed theoretically <cit.>.

Here we report on the novel theory and first experimental implementation of a phase-sensitive nonreciprocal amplifier. Based on a Field-Programmable Josephson Amplifier (FPJA)<cit.>, it utilizes four parametric pumps to achieve ≈24 dB of quadrature gain with a 7 MHz bandwidth, near-unity transmission in the reverse direction, with both input and output impedance matched to the 50 Ω environment. Using a self-calibrated noise source we extract a measurement efficiency of η= 0.62_-0.09^+0.17, exceeding the quantum limit of 0.5 for phase-preserving amplification. Straightforward improvements could enable efficiency of more that 0.95.








§ CONCEPT




Conceptually, the device in this work can be understood by analogy with the typical combination of a Josephson Parametric Amplifier (JPA) and a traditional 3-port circulator, see Fig. <ref>. Both systems implement the same generic scattering parameters: a signal incident at the input gets phase-sensitively amplified toward the output, whereas a signal incident at the output is simply routed with unity transmission to the input, see Fig. <ref>(a).

A JPA consists of a single resonant circuit whose frequency ω_a is tunable by the flux applied to a SQUID, Fig. <ref>(b) and (c). A strong pump modulates the flux at a frequency ω_aa^p≈2ω_a, mediating the mixing process between a signal and an idler of frequency ω_a^S and ω_a^I, with conservation of energy dictating ω_aa^p=ω_a^S+ω_a^I. Physically, pump photons enable the creation and annihilation of correlated pairs of signal and idler photons, leading to direct and idler gain <cit.>. Importantly, as the signal and idler tones exist within the same resonance of the system, it is natural to consider their linear combinations, quadratures of the resonator's field. The correlations between signal and idler inherent to the mixing process lead to quadrature-sensitive gain <cit.>. However, this amplifier is reciprocal and therefore requires a microwave circulator to control the signal flow. In a typical microwave circulator, three ports are coupled by a circular ferrite resonator biased by a large magnetic field. Input signals acquire a phase that, due to the Faraday effect, depends on whether they propagate clockwise or counterclockwise, leading to constructive interference at the output port and destructive interference at the isolated port <cit.>.

By contrast, here we engineer this behavior by programming a single FPJA with four parametric pumps to realize both circulation and phase sensitive amplification, without a strong magnetic field, see Fig. <ref>(d) and (e). The FPJA is a circuit with three resonances (`modes') whose frequencies ω_a, ω_b, and ω_c are tunable by the flux applied to a single SQUID. As in a JPA, a strong pump of frequency ω_bb^p≈2ω_b couples a signal at ω_b^S to its idler at ω_b^I, inducing quadrature-sensitive gain. Circulation is ensured by three pumps, of frequencies ω_ab^p ≈ω_b-ω_a, ω_bc^p ≈ω_c-ω_b and ω_ac^p ≈ω_c-ω_a. These pumps provide the energy needed for the coherent exchange of signal photons between all three modes (frequencies ω_j^S, with j∈{a,b,c}), as well as for the coherent exchange of idler photons between all three modes (frequencies ω_j^I, with j∈{a,b,c}). Importantly these pumps must satisfy the condition ω_ab^p + ω_bc^p = ω_ac^p, forming a loop in frequency space. As in a ferrite circulator, signals traveling clockwise or counter clockwise acquire a different phases, now set by the pump phases ϕ_loop=ϕ_ab+ϕ_bc-ϕ_ac <cit.>. As an example, for ϕ_loop=π/2, an input signal near mode c is converted toward mode b, amplified, and converted to the output mode a. In the reverse direction an input signal near mode a is scattered to mode c without gain. Finally, input and output signals are routed to different physical ports using on-chip frequency filters. A complete theoretical description and details about device fabrication and amplifier tuning can be found in appendix.





§ SCATTERING PARAMETERS




To calculate the scattering parameters of the device we solve the coupled equations of motion (EoM) for the field amplitudes in the Fourier domain, following the graph-based method discussed in <cit.>. We define the vector of intra-cavity field amplitudes 𝐀=(a^S,b^S,c^S,a^I*,b^I*,c^I*)^⊤, input field amplitudes 𝐀_in=(a_in^S,b_in^S,c_in^S,a_in^I*,b_in^I*,c_in^I*)^⊤, ouput field amplitudes 𝐀_out=(a_out^S,b_out^S,c_out^S,a_out^I*,b_out^I*,c_out^I*)^⊤, diagonal matrices for the total loss rates 𝐊=diag(√(κ_a),√(κ_b),√(κ_c),√(κ_a),√(κ_b),√(κ_c)), external couplings 𝐊^ext=diag(√(κ_a^ext),√(κ_b^ext),√(κ_c^ext),√(κ_a^ext),√(κ_b^ext),√(κ_c^ext)) and coupling efficiencies 𝐇=diag(√(η_a),√(η_b),√(η_c),√(η_a),√(η_b),√(η_c)), where  η_j=κ_j^ext/κ_j. The EoM can be expressed as 𝐊𝐌𝐊𝐀 = i𝐊^ext𝐀_in, where  𝐌 is the mode-coupling matrix:


    𝐌=
    ([    Δ_a^S     β_ab     β_ac        0        0        0;   β_ab^*    Δ_b^S     β_bc        0     β_bb        0;   β_ac^*   β_bc^*    Δ_c^S        0        0        0;        0        0        0 -Δ_a^I^*  -β_ab^*  -β_ac^*;        0  -β_bb^*        0    -β_ab -Δ_b^I^*  -β_bc^*;        0        0        0    -β_ac    -β_bc -Δ_c^I^* ]).


The diagonal elements are the normalized complex detuning terms Δ_j^S,I=(ω_j^S,I-ω_j)/κ_j+i/2 and the off-diagonal elements are the normalized coupling terms 
β_jk=g_jk/(2√(κ_jκ_k)) between modes j and k, with j,k∈{a,b,c}. Each g_jk=g_jke^iϕ_jk is the parametrically induced coupling rate, proportional to the amplitudes of the pump with frequency ω_jk^p and phase ϕ_jk. We emphasize the block structure of 𝐌, which reflects the coupling networks and frequency space diagram in Fig. <ref>(d): the diagonal blocks correspond to circulation between the signals and between the idlers, and the anti-diagonal blocks correspond to amplification between signal and idler in mode b. Solving these EoM yields the scattering matrix 𝐒, defined as 𝐀_out=𝐒𝐀_in:

    𝐒= i𝐇𝐌^-1𝐇-1.


The device behavior is programmed in four steps. First, each frequency conversion pump is individually calibrated to match the conversion rates to the  dissipation rates g_jk=√(κ_jκ_k), β_jk=1/2. Second, all three frequency conversion pumps are turned on, forming an interferometer that completely cancels propagation in the clockwise or counter-clockwise direction depending on sum of the pump phases  ϕ_loop=ϕ_ab+ϕ_bc-ϕ_ac=±π/2 <cit.>. Third, the frequency conversion rates to mode b, β_ab and β_bc, are increased in order to overwhelm the loss rate of mode b, while maintaining |β_ab|=|β_bc|. This yields high reflection off of mode b and the scattering parameters resemble the case of a JPA on a circulator in the absence of gain. As we will discuss later, this step is critical to obtain low added noise, preventing the signal of interest from dissipating in mode b and being replaced by uncorrelated vacuum noise. Finally, the amplification pump is turned on. In the ideal resonant case, neglecting internal loss and for ϕ_loop=π/2, the simplified scattering matrix 𝐒_𝐚𝐜 between field amplitudes a^S and c^S is:


    𝐒_𝐚𝐜=
    [      0 √(G_S);      1      0 ],


where √(G_S)=(2s+r^2-1)/(1-r^2) and 


    s     =4|β_ab|^2/1+4|β_ab|^2,
    
    r     =2|β_bb|/1+4|β_ab|^2,

with 0≤ r,s≤1. Physically, s represents the ratio between the conversion rates to mode b and the total dissipation rate of mode b, while r represents the ratio between the amplification rate of mode b and the total dissipation rate of mode b. 




Experimentally we start by fixing a flux bias point and measure the modes frequencies (ω_a,ω_b,ω_c)/2π= (6.876,7.932,10.782) GHz, linewidths (κ_a,κ_b,κ_c)/2π= (83,15,45) MHz and coupling parameters η_a=0.99 and η_c=0.99. We then proceed to the four programming steps described above, by comparing the measured scattering parameters to the solutions of Eq.<ref>. The results are shown in Fig. <ref>, for |β_ac|=0.5, |β_ab|=|β_bc|=1 (s=0.8), and r=0.91, in good agreement with the solutions of Eq.<ref>. The device exhibits a direct gain of 18 dB in the forward direction (with a 7 MHz bandwidth), unity transmission in the reverse direction and is impedance matched to the 50 Ω environment, with return loss exceeding  10 dB over a 4 MHz bandwidth. At this gain, the input power at the 1 dB compression point is -125 dBm (See Appendix C and Fig. <ref>).





§ QUADRATURE SENSITIVITY AND NOISE PERFORMANCE




Importantly, note that 𝐒_𝐚𝐜 is only a subset of the complete scattering matrix 𝐒, the latter being necessary to understand the quadrature sensitivity of the gain, as well as the added noise of the amplifier. By defining the quadratures of mode a
as X_a= (a^S+a^I*)/√(2), Y_a= i(a^I*-a^S)/√(2), and similarly for modes b and c, we compute the simplified scattering matrix 𝐐_𝐚𝐜 in the quadrature basis ( X_a,Y_a,X_c,Y_c ), still in the resonant and lossless case:


    𝐐_𝐚𝐜=
    [      0      0      0 √(G_X);      0      0 √(G_Y)      0;      0      1      0      0;      1      0      0      0 ],


where the forward quadrature gains are given by:


    √(G_X)   =(2s/1-r-1),
    √(G_Y)   =(2s/1+r-1).




The phase-sensitivity is evident, with G_X≠ G_Y. Ideal squeezing, defined as G_XG_Y=1 <cit.>, is achieved in the limit s=1, β_ab=β_bc→∞, corresponding to frequency conversion rates to mode b overwhelming its loss rate. Experimentally, starting from the same pump configuration as in Fig. <ref>, we measure the quadrature gain by driving a single quadrature at the frequency ω_bb^p/2 + ω_bc^p≈ω_c and demodulating the output using a mixer and a local oscillator (LO) at the frequency ω_bb^p/2 - ω_ab^p≈ω_a. The results are shown in Fig. <ref>(a) and we observe a quadrature gain G_X≈24 dB and anti-gain G_Y≈-11 dB, in good agreement with the theory.

To understand the noise properties of the amplifier, one needs to consider the full scattering matrix 𝐒 and the noise at each port. Indeed, the added noise of a parametric amplifier always originates from the noise in additional degrees of freedom. In this work, these degrees of freedom are the signal and idler in mode b (or equivalently the quadratures X_b and Y_b). Intuitively, for finite values of β_ab the frequency conversion rates to mode b do not fully overwhelm its loss rate, and noise entering mode b gets amplified alongside signals entering mode c, contributing to added noise and limiting squeezing. Formally, one can express the added noise of the amplifier in photon units, n_add^FPJA, in the resonant and lossless case, as


    n_add^FPJA=1/8|β_ab|^2(1+G_X^-1/2)^2.


Noiseless amplification is achieved, together with pure squeezing, in the limit of high conversion rate to mode b, β_ab→∞. Experimentally, the system-added noise referred to the input of the amplifier is n_add=n_add^FPJA+n_add^chain/G_X, were n_add^chain=19.8_-3.3^+3.2 is the noise added by the measurement chain following the amplifier, calibrated in a separate experiment using the shot noise emitted by a metallic tunnel junction <cit.>. By comparing the measured noise rise and quadrature gain we extract the improvement in signal-to-noise ratio and convert it into an equivalent system-added noise n_add and measurement efficiency η^-1=1+2n_add. In Fig. <ref>(b) we show the measured efficiency η as a function of the quadrature gain (dots). At low gain, the efficiency is limited by the following stages of amplification. At high gain the efficiency plateaus, revealing the intrinsic efficiency of the amplifier, η= 0.62_-0.09^+0.17, exceeding the quantum limit for phase-preserving amplification. Data are in reasonable agreement with the prediction from Eq. <ref> (solid line), within the error bars dominated by the uncertainty of the calibration of n_add^chain.





§ DISCUSSION AND CONCLUSION




Looking forward, lower added noise and larger squeezing can be obtained by reducing the dissipation of mode b, κ_b→ 0, or by increasing the coupling rates g_ab and g_bc. The first option is technically limited by the dielectric loss in mode b, here estimated to be 1-2 MHz. The second option is fundamentally limited by the stability of the amplifier (See Appendix B). To prevent free-oscillation of the amplifier, the amplification rate is limited to g_bb < κ_a+κ_b+κ_c, or equivalently 4β_ab^2 = 4β_bc^2 < (κ_a+κ_c)/κ_b . This puts bounds on the amount of squeezing, added noise, and efficiency of the amplifier:


    √(G_Y) > κ_b/κ_a+κ_b+κ_c,



    n_add^FPJA > κ_b/2(κ_a+κ_c),



    η < κ_a+κ_c/κ_a+κ_b+κ_c.


One can see that the performance of the amplifier only depends on the ratio between the dissipation rate of mode b and the dissipation rates of mode a and c. Straightforward improvement of the device design would increase κ_a and κ_c by a factor of two and decrease κ_b by a factor of five, leading to a tenfold decrease in added noise and an efficiency above 0.95. Additionally larger bandwidth could be obtained by operating at a lower gain, inserting the device as a pre-amplifier in front of a typical wideband and reciprocal parametric amplifier.

In conclusion, we have proposed and experimentally demonstrated, for the first time, a microwave amplifier that is both nonreciprocal and phase-sensitive, demonstrating an efficiency beyond the quantum limit. As a consequence, this amplifier no longer requires a magnetic circulator at its input and can be directly integrated on-chip with a superconducting quantum device. At a quadrature gain of G_X=20 dB, this first generation amplifier has a bandwidth of 10 MHz and an input power of -120 dBm at the 1 dB compression point, already meeting the requirements for a number of applications. For example, this is sufficient for the measurement of a transmon qubit using a coherent state of more than 100 photons in a 1 MHz wide readout cavity <cit.>. In addition, this amplifier could enhance quantum state tomography of itinerant microwave fields <cit.>, directly validate continuous variable entanglement <cit.>, and enable position measurements beyond the quantum limit <cit.>.





§ ACKNOWLEDGMENT

This work was supported by the NIST Quantum Information Program. This article is a contribution of the U.S. government, not subject to copyright.







§ APPENDIX A: DEVICE DESCRIPTION




An optical micrograph of the device and the corresponding circuit schematic are shown in Fig. <ref>. The device is fabricated using a Nb/Al-AlO_x/Nb trilayer process with amorphous silicon (a-Si) interlayer dielectric. This process is similar to the one described in earlier work <cit.>, with one difference: the top 300 nm Nb wiring layer is replaced by a Nb/Al/Nb trilayer, with respective thicknesses of 8 nm, 8 nm and 300 nm. The thin Al layer acts as an etch stop for the plasma etch of the Nb wiring layer, while the thin Nb layer prevents intermixing between the (a-Si) and Al layers. The patterning of the wiring layer is finalized by ion milling to mechanically remove the exposed thin layers of Nb and Al.

The central element is the FPJA, a lumped element circuit with three resonance frequencies that all depend on the inductance of a single SQUID. The resonance frequencies as a function of the flux through the SQUID are shown in Fig. <ref>. Resonators a and c are routed to different physical ports using low-pass and high-pass filters, forming an on-chip diplexer with a cut-off frequency around 7.6 GHz.



The mode frequencies were carefully chosen so that all possible first-order modulation frequencies ω_j±ω_k (j,k∈{a,b,c}) were well separated. This ensures  that a good rotating wave approximation can be made for each parametric process. For example, a pump of frequency ω_c-ω_a only couples via frequency conversion the signals a^S↔ c^S and the signals a^I*↔ c^I*, but no other signals.








§ APPENDIX B: DETAILED THEORY





In this section we described the calculations of the scattering parameters, output noise and stability of the amplifier, following concepts and style of <cit.>.



  §.§.§ Scattering parameters in the mode basis


The device can be described as three modes with natural frequencies ω_a, ω_b and ω_c, coupled to each other via a tunable inductor (the SQUID). We apply four pumps to modulate the inductance at the frequencies ω_ab^p ≈ω_b-ω_a, ω_bc^p ≈ω_c-ω_b, ω_ac^p ≈ω_c-ω_a and ω_bb^p ≈ω_b+ω_b, effectively coupling signals at six different frequencies, ω_a^S, ω_b^S, ω_c^S, ω_a^I, ω_b^I and ω_c^I.

We define the vector of intra-cavity field amplitudes 𝐀, input field amplitudes 𝐀_in, output field amplitudes 𝐀_out, diagonal matrices for the total loss rates 𝐊, external couplings 𝐊^ext and coupling efficiencies 𝐇: 


    𝐀   =(a^S,b^S,c^S,a^I*,b^I*,c^I*)^⊤,
    𝐀_in   =(a_in^S,b_in^S,c_in^S,a_in^I*,b_in^I*,c_in^I*)^⊤,
    𝐀_out   =(a_out^S,b_out^S,c_out^S,a_out^I*,b_out^I*,c_out^I*)^⊤,
    𝐊   =diag(√(κ_a),√(κ_b),√(κ_c),√(κ_a),√(κ_b),√(κ_c)),
    𝐊^ext   =diag(√(κ_a^ext),√(κ_b^ext),√(κ_c^ext),√(κ_a^ext),√(κ_b^ext),√(κ_c^ext)),
    𝐇   =diag(√(η_a),√(η_b),√(η_c),√(η_a),√(η_b),√(η_c)),


where  η_j=κ_j^ext/κ_j. Note that, as in <cit.>, we study here the dynamics of the expectation values of the field operators, defining a^S≡< â^S> and a^I*≡< â^I†> (and similarly for b^S, b^I*,c^S and c^I*).

The Equations of Motion (EoM) can be expressed as 𝐊𝐌𝐊𝐀 = i𝐊^ext𝐀_in, where  𝐌 is the mode-coupling matrix:


    𝐌=
    ([    Δ_a^S     β_ab     β_ac        0        0        0;   β_ab^*    Δ_b^S     β_bc        0     β_bb        0;   β_ac^*   β_bc^*    Δ_c^S        0        0        0;        0        0        0 -Δ_a^I^*  -β_ab^*  -β_ac^*;        0  -β_bb^*        0    -β_ab -Δ_b^I^*  -β_bc^*;        0        0        0    -β_ac    -β_bc -Δ_c^I^* ]).





The diagonal elements are the normalized complex detuning terms Δ_j^S,I=(ω_j^S,I-ω_j)/κ_j+i/2 and the off-diagonal elements are the normalized coupling terms 
β_jk=g_jk/(2√(κ_jκ_k)) between modes j and k, with j,k∈{a,b,c}. Each g_jk=g_jke^iϕ_jk is the parametrically induced coupling rate, proportional to the amplitudes of the pump with frequency ω_jk^p and phase ϕ_jk.  

We emphasize the block structure of 𝐌, which reflects the coupling networks in Fig. <ref>a and frequency space diagram in Fig. <ref>(c,d): the diagonal blocks correspond to circulation between signals or between idlers, and the anti-diagonal blocks correspond to amplification between signal and idler of mode b.

Solving these EoM yields the scattering matrix 𝐒, defined as 𝐀_out=𝐒𝐀_in:

    𝐒= i𝐇𝐌^-1𝐇-1.


We now derive analytical expressions for the scattering parameters when all four pumps are on resonance (that is ω_ab^p=ω_b-ω_a, ω_bc^p=ω_c-ω_b, ω_ac^p=ω_c-ω_a, and ω_bb^p=2ω_b). The resonance conditions mean -Δ_j^I*=Δ_j^S, so the matrix 𝐌 from Eq. <ref> becomes much simpler. To simplify notations we will omit in the following the signal and idler superscript for the detuning terms Δ_j^S,I→Δ_j.

We are most interested in the 4×4 scattering matrix between
the signal and idler modes in the “input” mode c and “output” mode a. We can write
some of these scattering elements compactly as

    S_a^Sa^S    =iη_aC/|𝐌|(Δ_b^effΔ_c-|β_bc|^2)-1,
    
    S_c^Sc^S    =iη_cC/|𝐌|(Δ_b^effΔ_a-|β_ab|^2)-1,
    
    S_a^I*a^S    =iη_a/|𝐌|β_bb^*(β_bc^2β_ac^*2-β_ab^*2Δ_c^2),
    
    S_c^I*c^S    =iη_c/|𝐌|β_bb^*(β_ac^2β_ab^*2-β_bc^2Δ_a^2),
    
    S_a^Sc^S    =i√(η_aη_c)C/|𝐌|(β_abβ_bc-β_acΔ_b^eff),
    
    S_c^Sa^S    =i√(η_aη_c)C/|𝐌|(β_ab^*β_bc^*-β_ac^*Δ_b^eff),
    
    S_a^I*c^S    =i√(η_aη_c)/|𝐌|β_bb(β_abΔ_c-β_acβ_bc^*)(β_bc^*Δ_a+β_abβ_ac^*).

where

    C = Δ_aΔ_bΔ_c-|β_bc|^2Δ_a-|β_ac|^2Δ_b-|β_ab|^2Δ_c
    
    -β_abβ_bcβ_ac^*-β_ab^*β_bc^*β_ac

is the determinant of the a^I*→ b^I*→ c^I*
loop, and Δ_b^eff is an effective detuning for the field amplitude b^S  that includes the dynamics of all the idler fields (reducing out the upper triangle in Fig. <ref>a to obtain Fig. <ref>b): 

    Δ_b^eff=Δ_b+|β_bb|^2/C(Δ_aΔ_c-|β_ac|^2).


S_a^Sa^S and S_c^Sc^S are the reflection coefficient off of mode a and c respectively. S_a^Sc^S and S_c^Sa^S are respectively the forward and reverse gain and S_a^I*c^S is the forward idler gain.

Imposing the condition S_a^I*a^S=S_c^I*c^S=0 on resonance
(that is, no transmission from signal to idler of the same resonance),
we find |β_ab|=|β_bc|, |β_ac|=1/2 and ϕ_loop=ϕ_ab+ϕ_bc-ϕ_ac=±π/2.
In the following, we choose ϕ_loop=π/2 to fix the direction of
circulation from c^S→ b^S→ a^S. Substituting these conditions into the scattering parameters above, and evaluating on resonance (Δ_j=i/2), we can calculate the scattering matrix for the reduced basis (a^S,c^S,a^I*,c^I*) as 

    𝐒_𝐚𝐜=([           η_a-1        -i√(G_S)               0  -e^iϕ_bb√(G_I);      i√(η_aη_c)           η_c-1               0               0;               0 -e^-iϕ_bb√(G_I)           η_a-1         i√(G_S);               0               0     -i√(η_aη_c)           η_c-1 ]),

where

    √(G_S)    =√(η_aη_c)2s+r^2-1/1-r^2,
    √(G_I)    =√(η_aη_c)2rs/1-r^2,

and 

    s     =4|β_ab|^2/1+4|β_ab|^2,
    
    r     =2|β_bb|/1+4|β_ab|^2,

with 0≤ r,s≤1.

Note the differences between Eq. <ref> and the corresponding equation Eq. 3 in the main text. In the main text we simplified 𝐒_𝐚𝐜 by writing only the magnitude of the matrix elements, and used a reduced basis that excludes the idlers.



  §.§.§ Scattering parameters in the quadrature basis


We rotate from the mode basis 𝐀, 𝐀_in, 𝐀_out to the quadrature basis 𝐗=𝐔𝐀, 𝐗^in=𝐔𝐀_in, 𝐗^out=𝐔𝐀_out where 𝐔 is the unitary matrix 


    𝐔=1/√(2)([  1  0  0  1  0  0; -i  0  0  i  0  0;  0  1  0  0  1  0;  0 -i  0  0  i  0;  0  0  1  0  0  1;  0  0 -i  0  0  i ])

and 

    𝐗   =(X_a,X_b,X_c,Y_a,Y_b,Y_c)^⊤,
    𝐗_in   =(X_a,in,X_b,in,X_c,in,Y_a,in,Y_b,in,Y_c,in)^⊤,
    𝐗_out   =(X_a,out,X_b,out,X_c,out,Y_a,out,Y_b,out,Y_c,out)^⊤.


The scattering matrix in the quadrature basis, 𝐐, defined as 𝐗_out=𝐐𝐗_in, is 𝐐=𝐔𝐒𝐔^-1. To line up the quadrature definitions with the phase-sensitive gain,
we choose ϕ_bb=-π/2. The scattering matrix for the reduced quadrature basis
(X_a,Y_a,X_c,Y_c) is then:


    𝐐_𝐚𝐜=([      η_a-1          0          0     √(G_X);          0      η_a-1    -√(G_Y)          0;          0 -√(η_aη_c)      η_c-1          0;  √(η_aη_c)          0          0      η_c-1 ]),

where 

    √(G_X)=√(G_S)+√(G_I)=√(η_aη_c)(2s/1-r-1),

and 

    √(G_Y)=√(G_S)-√(G_I)=√(η_aη_c)(2s/1+r-1).


Note that in the main text we simplified 𝐐_𝐚𝐜 (Eq.6) by writing only the magnitude of the matrix elements of Eq. <ref>.

The product of the two quadrature gains is 

    √(G_XG_Y)=η_aη_c(1-4s1-s/1-r^2).


The two free parameters, r and s, are determined by the two remaining free coupling rates |β_bb| and |β_ab|=|β_bc|. Ideal squeezing, corresponding to G_XG_Y=1 <cit.>, is achieved in the limit η_a=η_c=1 and s=1, β_ab=β_bc→∞.



  §.§.§ Amplifier output noise


To compute the output noise of the amplifier, we apply the scattering matrix to general input states that includes vacuum fluctuations and thermal contributions at every port. To that end we redefine here the vectors of field and quadrature amplitudes to be vectors of operators, ensuring the proper commutation relations. The output covariance matrix in terms of the field quadratures is 

    ⟨𝐗_out^†𝐗_out^⊤⟩=𝐔^*𝐒^*⟨𝐀_in^†𝐀_in^⊤⟩𝐒𝐔,

where the input covariance matrix ⟨𝐀_in^†𝐀_in^⊤⟩
is diagonal with elements determined by the input thermal state occupancies. While general analytical solutions
exist, it is more useful to look at relevant limits. Evaluating the
quadrature output noise on resonance for lossless modes with vacuum-state
inputs, we find

    ⟨ X_a,out^†X_a,out⟩    =G_X[1/2+n_add^FPJA],

where the added noise n_add^FPJA can be written in terms of
G_X and |β_ab| as 

    n_add^FPJA=1/8|β_ab|^2(1+G_X^-1/2)^2.

For large gain, the added noise is determined by the coupling strength
to the b mode. For the minimal added noise, |β_ab|=|β_bc|
must be maximized either by taking κ_b→0 or g_ab,g_bc→∞. However, g_ab and g_bc are limited by the stability of the amplifier.

As described in Appendix B.4 and B.5 of <cit.>, a more complete model can be obtained by generalizing
the scattering matrix to a 12×12 matrix that includes internal ports for each mode.



  §.§.§ Amplifier stability


For the Langevin equations of motion to be stable, all the eigenvalues of the Langevin matrix 𝐌 need to have a negative real part corresponding to damping. In the ideal resonant case, neglecting internal loss, and under the conditions |β_ac|=1/2 and |β_ab|=|β_bc|, the characteristic polynomial  P(λ)=|𝐌(ω=iλ)| takes on a rather simple form:


    P =P_+P_-+b_ϕ, b_ϕ = κ_a^2κ_b^2κ_c^2|β_ab|^2cos^2ϕ_loop


where the polynomials P± are of third order and independent of the loop phase:

    P± = λ^3 + b_1^±λ^3 + b_2^±λ^2 + b_3^±


with 


    b_1^±   = κ_b/2(1 ∓ 2β_bb)+κ_a+κ_c/2,
    
    b_2^±   = κ_b(κ_a+κ_c)/4(4|β_ab|^2+1∓ 2β_bb)+κ_aκ_c/2,
    
    b_3^±   = κ_aκ_bκ_c/4(4|β_ab|^2+1∓ 2β_bb).


The Langevin equations of motion are stable if all these coefficients are positive <cit.>. We notice that b_1,2,3^->0 and b_ϕ=0 for ϕ_loop=±π/2. Solving b_1,2,3^+>0 gives the following conditions:


    β_bb    <1/2+ 2|β_ab|^2,
    β_bb    <1/2+κ_a+κ_c/2κ_b,
    β_bb    <1/2+2|β_ab|^2+κ_aκ_c/κ_b(κ_a+κ_c).


The first condition corresponds to the pole r=1 in the gain equations (Eq. <ref>), therefore the second and last conditions must be less restrictive than the first one. This implies |β_ab|^2<(κ_a+κ_c)/4κ_b and therefore g_bb<κ_a+κ_b+κ_c. The latter condition has a simple physical interpretation: the parametric amplification rate cannot be greater than the total dissipation rate of the three modes.

The upper limit on |β_ab| sets the limits on the amount of squeezing, added noise and efficiency:


    √(G_Y) > κ_b/κ_a+κ_b+κ_c



    n_add^FPJA > κ_b/2(κ_a+κ_c)



    η < κ_a+κ_c/κ_a+κ_b+κ_c




We observe experimentally this instability by measuring the scattering parameters of the amplifier as a function of the loop phase ϕ_loop and the direct gain. The region of instability is shown in Fig. <ref>. When the direct gain of the amplifier exceeds ≈6 dB, a region of parametric oscillations appears around ϕ_loop=0. This region grows with the gain, approaching at high gain the loop phases ϕ_loop=±π/2 at which the amplifier is directional.
Comparison with theory remains a work in progress and involves finding the roots of the characteristic polynomial P in the more general case of arbitrary pump strengths, frequencies, and phases.  





§ APPENDIX C: DYNAMIC RANGE





The dynamic range of the amplifier is similar to that of a regular Josephson Parametric Amplifier of similar bandwidth and non-linearity. In Fig. <ref>, we compare the input power at the 1 dB compression point when the FPJA is programmed as a directional phase-sensitive amplifier (in red) or a regular phase-sensitive amplifier (in green). The latter is achieved by using a single pump at ω_aa^p = 2ω_a, leading to phase-sensitive gain around mode a. The 1 dB compression point is shown as a function of G_s(r)-G_s(0), of the direct gain relative to the scattering amplitude when the amplifier's gain pump is off (pump at ω_aa^p or ω_bb^p). This accounts for internal losses and frequency conversion losses for s<1. The solid green and red lines are linear fits, with respective slope of -1.5 dBm/dB and -1.3 dBm/dB.







§ APPENDIX D: AMPLIFIER TUNING





As discussed in the main text, the device is programmed in four steps. We describe here in more detail these steps and we show the corresponding measured scattering parameters in Fig. <ref>.


	
  * First each frequency conversion pump is individually calibrated to produce ideal conversion, β_jk=1/2. In Fig. <ref>(a) we show the example of the scattering parameters for the case of frequency conversion between modes a and c. We observe near-ideal transmission and low reflection. 
	
  * Second, all three frequency conversion pumps are turned on. As discussed in detail in <cit.>, this pump configuration creates a parametric circulator, whose circulation direction is set by the sum of the pump phases, i.e. loop phase, ϕ_loop=ϕ_ab+ϕ_bc-ϕ_ac. In Fig. <ref>(b), we show the scattering parameters between modes a and c for ϕ_loop=π/2. We observe unity transmission from a to c and isolation up to 30 dB in the reverse direction. Both modes are still impedance matched, with low reflection coefficients.
	
  * Third, the strength of the frequency conversion pumps to mode b, β_ab and β_bc, are increased in order to overwhelm the loss of mode b, maintaining |β_ab|=|β_bc|, see Fig. <ref>(c). This effectively yields high reflection off of mode b (not shown) and the scattering parameters resemble the case of the lossless JPA on a circulator in the absence of gain. We still observe unity transmission from  a to c, but the device is almost reciprocal again, with only about 3 dB of isolation in the reverse direction (set by the value of s=0.8). Importantly, this pump configuration sets the gain-bandwidth product of the amplifier, set by the width of modes a and c, and mostly independent of the width of mode b (in the limit κ_b≪κ_a,κ_c)
	
  * Finally, the amplification pump is turned on and corresponding scattering are show in Fig. <ref>(d) (same data than Fig. 2, over a wider frequency span). Gain from c to a grows, while unity transmission is preserved from mode a to c. Impedance matching is maintained, over a narrower bandwidth.


 




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