Mixing of counterpropagating signals in a traveling-wave Josephson device

Domain: physics
Published: 2025-12-12
Editorial outcome: major_revision

Abstract
Light waves do not interact in vacuum but can mix in nonlinear media. A strong pump wave can thus convert the frequency of a weaker signal, provided energy and momentum are conserved. These conditions are typically satisfied when all waves propagate with comparable phase velocity along a given axis. Here, we investigate an alternative scheme by which a microwave signal propagating along a one-dimensional Josephson metamaterial is converted into a counter-propagating wave through interaction with a slower pump. In this regime, the input wave is exponentially attenuated, enabling an on-chip microwave isolator reconfigurable into a reciprocal, tunable coupler. The device’s operating mode and working frequency can be selected in situ over a broad microwave range. We measure isolation exceeding 5 dB in the 5-8.5 GHz range and 10 dB in the 7-8.5 GHz range, with a typical 200 MHz bandwidth. Further improvements are expected through design optimization and reduced fabrication disorder, opening new possibilities for microwave routing and processing in superconducting circuits.

Introduction
Parametric processes are potent and versatile tools in quantum technologies, enabling signal amplification1,2, harmonic generation3, controlled interaction between standing modes4, non-reciprocal signal transmission5,6 and entanglement generation7, both in the optical and microwave domains. In these processes, input waves mix in a nonlinear medium to produce output waves such that energy is conserved:

\sum_{i=1}^{N} \epsilon_i \omega_i = 0 \quad (1)

where \epsilon_i = 1 for input waves \epsilon_i = -1 for output waves, \omega_i is the frequency of each wave, and the number of involved waves N matches the order of the wave-mixing process. Since the process rate scales as the product of wave amplitudes, a strong pump wave (P) is typically employed to enhance interactions between weak signal (S) and idler (I) waves. When N = 4—which is the case for the \chi^{(3)} medium considered in this work—a pump at frequency \omega_P = (\omega_S + \omega_I)/2 is typically employed to activate a two-mode squeezing interaction between the signal and idler waves and amplify the input signal close to the quantum limit. In another mode of operation, a pump at frequency \omega_P = |\omega_S - \omega_I|/2 activates a beam-splitting interaction between the waves, converting an input signal to an output idler and vice-versa.

In the first experimental implementations of these ideas, the nonlinear medium was embedded in a high quality factor resonator in order to enhance interactions8–15, at the expense of limiting the process working bandwidth. Alternatively, one may send waves through a long section of nonlinear medium, which is known as a traveling wave parametric process16–18. For the interactions along this line to add up constructively, phase-matching must be enforced, which results in a second conservation law on momentum

\sum_{i=1}^{N} \epsilon_i k_i = 0, \quad (2)

where k_i is the wave vector corresponding to each wave. While waves naturally propagate at the same speed in a dispersionless linear medium—trivially linking equations (1) and (2) since \omega_i = v_i k_i—phase-matching is difficult to satisfy in a nonlinear medium due to the distinct wave vector shifts induced by the pump on the various waves. Thus, the onset of traveling wave parametric amplifiers (TWPAs) allowed for broadband and high dynamic range amplification, at the cost of increased design complexity19–40 to enforce phase-matching.

Traveling wave parametric converters (TWPCs)41–43, capable of converting waves from signal to idler frequency, have attracted interest as they are non-reciprocal devices: if one reverses the direction of propagation of the signal and idler waves, their propagation velocities v_{S,I} become negative and the now unmatched conversion process is inhibited. Implementing a robust non-reciprocal element with high enough bandwidth and isolation to efficiently route microwave signals is a long standing goal of circuit QED44–57. However, TWPCs suffer from the same design complexity as TWPAs and from limited isolation as the nonlinear medium length and pump amplitude need be precisely adjusted for the conversion from signal to idler to be complete (see Fig. 1a).

In this work, we introduce a different regime of traveling wave conversion where the signal and idler propagate in opposite directions. Phase-matched conversion is achieved in this configuration by lowering significantly the pump wave velocity |v_P| \ll |v_{I,S}|, in a configuration analog to Stimulated Brillouin Scattering in optics58 (In stimulated Brillouin scattering, the low velocity wave is acoustic and interacts with light through electrostriction.), (see Fig 1b). This change of direction of propagation upon conversion drastically changes the waves dynamics: as the idler escapes backward, it cannot be back converted to signal, and the signal amplitude decreases exponentially along the line. Moreover, the device may be reconfigured to enable controllable reciprocal transmission with a high on/off ratio. We now detail these two modes of operation.

On-chip circulation: in this non-reciprocal process, conversion of a signal to an idler photon is enabled by the absorption of 2 photons from a pump wave co-propagating with the idler. The reverse conversion process from idler to signal leads to the emission of 2 photons in the pump wave. The corresponding phase-matching diagram is presented in Fig. 1b. From relations (1) and (2), we find the process to be phase-matched for:

\omega_S = 2\omega_P - \omega_I \quad \omega_S = \omega_I + 2\omega_P \quad (3)

where we recall that v_I, v_P < 0. Crucially, if one reverses the direction of propagation of both the signal and idler waves, this process is no longer phase-matched, and waves propagate unaffected along the transmission line. The device thus circulates signals through 2 frequencies of interest (\omega_S and \omega_I) at either end of the transmission line, implementing an effective 4-port circulator (see Fig. 1c).

Tunable coupling: in this configuration, pump waves of the same frequency are applied in both directions. A signal photon may be reflected back to an idler photon of the same frequency \omega_S by absorbing a photon from one of the pump waves and emitting it back in the other (see Fig. 1d, e). We find that the process is phase-matched for:

\omega_S = \omega_P \quad \omega_I = \omega_S \quad (4)

Since this process is stimulated by pump photons, adjusting the pump amplitudes allows one to continuously tune the effective transmission and reflection coefficients of the line, implementing a tunable coupler. We note that phase-matching of the target process is straightforward in both configurations, as no assumptions are imposed on the relative phase velocities of the propagation modes. It is, however, exactly verified for a precise set of wave frequencies only. In practice, the operating band of the device (typically of order 200 MHz in our experiment, see Supplementary materials) is set by the pump amplitude. Its center frequency is tunable over a wide frequency range by adjusting the pump frequency.

We design and fabricate a microwave transmission line made of lumped circuit elements and embedding two modes of propagation with different velocities (see Fig. 2). Wave-mixing is enabled by Josephson junctions supporting currents from both modes. Note that mixing of counterpropagating signals could also be achieved in a single-mode transmission line if its dispersion relation were properly engineered to slow down the pump wave. Beyond its simple design and versatility, our two-mode architecture allows for straightforward separation of the high-amplitude pump wave from the signals to be processed.

In detail, the device is composed of 400 identical cells arranged in series, forming two inner electrodes and a ground reference electrode. In each cell, the two inner electrodes are interrupted by two Josephson junctions (inductance L_J = 0.94 nH), shunted to ground by two identical parallel-plate capacitors (capacitance C_g = 0.13 pF) and connected to each other by a large capacitor (C_i = 0.57 pF). The line symmetry ensures that the common and differential excitations of the inner electrodes (labeled \Sigma and \Delta, respectively) are eigenmodes of propagation. The large inner capacitor loads the \Delta propagation mode—supporting the pump wave—while not affecting the \Sigma mode—supporting the signal and idler waves of higher velocity. For low-frequency and low-amplitude waves, the mode velocities read

v_{\Sigma,0} = a / \sqrt{L_J C_g}

v_{\Delta,0} = a / \sqrt{L_J (C_g + 2C_i)} \quad (5)

where a is the unit cell length. We choose the line parameters such that v_{\Sigma,0}/v_{\Delta,0} \simeq 3, enabling the target processes in the band 4–10 GHz with a pump tuned from 2.5 GHz to 5 GHz. Due to the discrete nature of the unit cells, the wave velocity of each mode decreases when approaching a cut-off frequency above which no propagation is possible (see Supplementary materials). Remaining parameters are chosen such that propagation modes exhibit a characteristic impedance of order 50 \Omega, allowing us to address them with minimum spurious reflections through a tapered impedance transformer and a hybrid coupler (see Supplementary materials for details on the wiring setup and transformer geometry).

Results
We first probe the \Sigma mode transmission in the presence of a continuous pump wave applied from the right \Delta port. In Fig. 3, the magnitude of the probe transmission coefficient in each direction is reported in color against the pump frequency \omega_{pump} and probe frequency \omega_{probe}, exhibiting gaps (blue lines) when a conversion process is phase-matched. Note that the coherently converted signal may be retrieved by detecting the reflected field at \omega_{probe} \pm 2\omega_{pump}, which we verified for a few values of \omega_{pump} in the Supplementary materials.

The circulation process, denoted by a marker Ci on the maps, corresponds to the leftmost feature in the left to right transmission (signal to idler upconversion in Fig. 1b) and to the bottom feature in right to left transmission (reverse idler to signal downconversion). Due to the discreteness of the device unit cells, these features curve at high frequency, as expected from the dispersion of the propagation modes. Moreover, a supplementary gap corresponding to the same circulation process opens at high frequency through aliasing of a signal whose wave vector exceeds \pi/a where a is the length of a unit cell (top right feature labeled (Al) in right panel, with corresponding phase-matching diagram in inset). Crucially, all these features are mostly visible in one direction of propagation only, demonstrating non-reciprocity. We attribute the opening of residual gaps in the opposite direction to the spurious reflection of the pump within the device, unintentionally activating a conversion process in that direction. Quantitatively, we estimate following an in situ calibration of the device feed lines that the pump is reflected on the \Delta propagation mode at the -2 dB level and leaks to the \Sigma mode at the -4 dB level, which indicates that one of the junctions in the device is open (see Supplementary materials for detailed characterization of wave scattering by the defective junction). Note that the pump reflection could have been efficiently suppressed with a canceling pump applied in the reverse direction if the defective junction were located close to the device ports. Unfortunately, time-domain reflectometry measurements indicate that the defective junction is located close to the middle of the line, making this strategy ineffective. This situation effectively halves the length of our active Josephson line and limits its performance.

Another consequence of pump reflection is that forward and backward propagating pump waves may combine to activate the tunable coupling process. The corresponding gap, denoted by a marker (Co) on the maps, is clearly visible at the center of both panels in Fig. 3, as expected from a reciprocal process. All data pertaining to the tunable coupling process presented in Fig. 4 was acquired by relying on pump reflection rather than by purposely applying a pump in both directions.

The location of all gaps corresponding to the circulation and tunable coupling processes is accurately predicted (black dashed lines for expected gaps, gray dashed lines for residual gaps in the reverse transmission) by solving the phase-matching relations (1) and (2) as detailed in the Supplementary materials. In this model, we take into account the discrete nature of the line unit cells and only leave as free fit parameters the wave velocities at low frequency—found to be within a few percent of their nominal design values—and the Josephson junctions plasma frequency at 38.0 GHz. We further note the presence of a peak in the transmissions at \omega_{probe} = \omega_{pump} and of a thin gap lying close to the circulation process gap (visible left of the gray dashed line on the right panel). Both features are reproduced by numerical simulations based on a full discrete nonlinear model in the Supplementary materials59. The former can be attributed to signal amplification arising from pump waves propagating in both directions, while the origin of the latter remains unclear at this stage.

We further characterize the performance of our device for non-reciprocal isolation and tunable coupling by recording the probe attenuation in both directions as a function of the applied pump amplitude. The attenuation is obtained, for a given pump amplitude and frequency, by smoothing the probe transmission spectrum—referenced at the \Sigma ports of the hybrid couplers—over a 40 MHz window around the gap frequency and extracting its maximum depth (see inset of Fig. 4a for a typical spectrum before smoothing). In Fig. 4b, we report the attenuation measured for each process at a specific probe frequency and for a varying pump amplitude, referred by its voltage at room temperature. The low-frequency circulation process (labeled (Ci) in Fig. 3 forward transmission) is activated by a pump at 4.63 GHz. It yields a desired attenuation from the left to right port (dark red circles) and a residual attenuation in the opposite direction, resulting from pump reflection (light red diamonds). The tunable coupling process is activated by a pump at 2.6 GHz, and the corresponding attenuation is reported with green squares. Strikingly, all attenuations scale exponentially with the applied pump amplitude, in qualitative agreement with the simple three-wave model presented in Fig. 1.

Quantitatively, for a line of length L in which waves propagate without reflection, this model predicts an attenuation 2e^{-\alpha L}/(1+e^{-2\alpha L}), where the attenuation constant \alpha depends on the considered process and scales quadratically with the traveling pump wave amplitude (see Supplementary materials). To capture the effect of wave scattering by the defective junction, we first calibrate the pump amplitude in each direction and on either side of the defect. We then solve the signal and idler wave dynamics in each section with joint boundary conditions set by the defect. We thus obtain the signal attenuation through the whole line (dashed lines in Fig. 4b, with a vertical offset adjusted to match the measured attenuation at zero pump amplitude), in good agreement with the measured attenuation for each activated process. For both processes, the measured attenuations start deviating significantly from the prediction of our model above a critical pump amplitude indicated by green and red arrows in Fig. 4b. In particular, we measure a wideband drop in probe transmission for pump amplitudes above critical value, which is not captured by our model. Note that numerical simulations based on a full discrete nonlinear model do not yield significantly different results at the considered pump frequency (see Supplementary materials59). Over the pump frequency range considered in this work, the critical pump amplitude corresponds to an approximately constant peak amplitude \Phi \simeq 0.12\Phi_0 for the generalized flux across the junctions located right of the defective junction (see Fig. 4d).

In Fig. 4c, we report the attenuations measured at the critical pump amplitude over a wider range of operating frequencies. Overall, our device enables circulation with approximately 10 dB isolation—defined as the ratio of forward to backward attenuation—through the 5-8.5 GHz band, and approximately 20 dB in the 7-8.5 GHz band. Employed as a tunable coupler, it reaches an on/off ratio in transmission from 10 dB to 20 dB in the 7–12 GHz band. The quoted values correspond to the attenuation at the center of each process gap. An attenuation reduced by a factor of two is maintained over a typical 200 MHz bandwidth around this frequency. Note that the operating range in circulation is limited by the modest electrical length of our device—resulting in performance tailing off towards lower frequencies—and by the line cutoff frequencies—curving the dispersion relation towards higher frequencies. This range could thus be extended by increasing the number of cells in the device and by reducing the value of capacitances and inductances in each cell, which would also increase our device dynamic range, currently limited at -110 dBm (see Supplementary materials). Alternatively, one could invert the pump direction and leverage the other circulation processes (labeled Ci and Al in Fig. 3, backward transmission) to enable circulation at higher frequencies. As for the operating range in tunable coupling, it is only limited by the working bandwidth of various microwave components present in our setup (see Supplementary materials).

An important figure of merit for a circulator is its insertion loss, corresponding to the right to left attenuation in Fig. 4c. When no pump is applied, we measure significant insertion loss (gray line) with a value of 8.5 dB at 7 GHz. Through independent calibration, we attribute 4.5 dB of this loss to the hybrid couplers and in-line filters and cables, which we consider as part of the device. We estimate that signal reflection and leakage to the \Delta mode—caused by the defective junction—contribute an additional 2 dB. The remaining 2 dB are attributed to dielectric losses in the alumina, corresponding to a loss tangent of tan \delta \simeq 3 \times 10^{-3} (see Supplementary materials for a detailed account of each source of loss over the device working range). Finally, when turning on the pump at critical value to activate the circulation process, we typically detect a few dB increase in insertion loss (light red diamonds in Fig. 4c above the gray baseline recorded in the absence of pump). This apparent loss is caused by reflected pump waves, which activate the same target process but in the reversed direction. Note that all sources of internal loss could be significantly reduced through simple design changes—such as replacing parallel-plate capacitors with coplanar ones to eliminate lossy dielectric material38,39—and improved fabrication to suppress internal reflections of pump and probe waves.

Methods

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper describes work toward realizing an on-chip microwave circulator (that can also function as a controllable coupler), a device for which there is currently much interest because of it's applications to quantum computing and other experiments detecting signals at the level of quantum noise. The methods used and design of the device are new and should be of high interest to others working on related devices, such as parametric amplifiers. The work presented is technically impressive and is generally well presented and supported by the extensive supplementary sections. Although in its current form, the device may have too much loss to be directly applicable to experiments, there are clear pathways toward improving it. Based on the newness of the techniques and the significant progress toward demonstration, I recommend that the paper be published. I attach a copy of the manuscript with a few grammatical suggestions.", "reviewer_2": "This paper experimentally explores the use of nonlinear processes involving counter-propagating waves in superconducting quantum circuits for tunable isolation and variable-strength coupling. Employing a coupled nonlinear Josephson transmission line architecture with a pair of 180-degree hybrid couplers, the authors experimentally show ~15dB of isolation with ~100MHz of instantaneous bandwidth, 3-GHz tunable bandwidth, and ~-120dBm dynamic range. This performance is similar to those of other on-chip magnetic-free isolators achieved in the field using copropagating nonlinear interactions. Nonlinear interactions involving counter-propagating waves are not new in physics and are often encountered in fiber optics as stimulated Brillouin scattering (SBS). Nevertheless, this work still contains several novelties as it is, to my knowledge, the first experimental work in circuit QED using (1) counter-propagating nonlinear process, and (2) coupled-nonlinear transmission line. Whereas this work may meet the novelty bar and is overall well-written with useful theory details in the supplement, I found its demonstrated performance and degree of completeness challenging to justify publication in Nature Communications as-is. I think this work may be eventually accepted in Nature Communications if further improvements in both measurement calibration and device performance can be made, and the following comments are satisfactorily addressed, preferably by changing the manuscript where needed rather than just in a response document. 1. The insertion loss of the device was not reported in the manuscript. Insertion loss is an important metric for isolators as it should ideally be as small as possible to avoid further degrading the SNR forward signal. The authors only briefly mentioned that they estimate the insertion loss at the end of Section III from the reported dielectric loss tangent of other works, which to me is insufficient and not standard practice. The authors’ experimental setup has a through line in the setup and should be able to take such measurements and report the insertion loss over the quoted frequency range directly. 2. From both Fig. 3 and Fig. 4(a), the insertion loss of the forward signal seems to also increase when the pump is turned on. I would appreciate if the authors could quantify such increase and provide explanations to its cause. Despite the attenuation increases more for the backward signal, the deceased background forward transmission if large could still severely impact its practical use as an isolator. 3. In the introduction section, the authors mentioned that traveling-wave processes have the advantage of larger working bandwidth, power handling and scalability. This is in general true for parametric amplifiers where broadband phase matching is possible and is less interaction-length sensitive, but for the traveling wave parametric converters considered in this work they too suffer from limited bandwidth and dynamic range. This work, along with a prior work utilizing copropagating traveling-wave conversion (Ranzani et al., Phys. Rev. Appl 8, 054035 (2017)), has only ~100MHz instantaneous bandwidth. As a comparison, a recently demonstrated resonator-based device by Beck et al. Phys. Rev. Appl 20, 034054 (2023) showcased 15dB isolation over 600-MHz of instantaneous bandwidth. The level of achieved performance in this work doesn’t seem to be convincing or justify its significant hardware overhead (e.g. hybrid couplers), increased device footprint and complexity, and potentially increased device insertion loss, warranting further improvement in device performance. 4. The authors mentioned that their proposed design of this work has the intrinsic limitation that the impedance of the even and odd modes cannot be simultaneously impedance matched as explained by Eqs. (S1) (S2). Indeed, as the authors observed and described on measurements, significant pump reflection (~-6dB to -3dB) occurred and is a show stopper to the device performance, despite Klopfenstein tapers were used. How well are the Klopfenstein tapers in impedance matching the two modes to 50 ohm? Are there test structures / chips with the tapers only to quantify the resulting reflection and validate that they function properly? Are they able to explain this large pump reflection? Whereas fabrication disorder could certainly be a reason for spurious reflection, it would be helpful to provide further evidence to substantiate/refute those hypotheses. 5. The authors claimed “…the lines are coupled, and it is impossible to have both propagating mode impedances at 50Ω at the same time.” This statement is not entirely accurate. This is indeed possible if both capacitive and inductive couplings are employed; see page 4 of Naghiloo et al., ArXiv: 2103.07793 (2021) for more details. The authors could consider introducing inductive coupling to help mitigate the pump reflection issue which is critical to device performance. 6. The authors mentioned that they use 3dB attenuators to sandwich the device to provide a clean impedance environment for the device. In my opinion, this leads to artificially better performance than it would actually behave in real setups as attenuators won’t be used downstream the information-carrying signal, limiting its practical applicability. I would like to see how the device performs without the sandwiching attenuators, which is more interesting from a practical point of view. 7. I do not fully understand why the baseline attenuation could not be extracted for Fig. 3. From my understanding, the through line in the switch bank can be used for this purpose. I found the authors explanation in the main text on baseline attenuation to be rather unsatisfactory: (1) power calibration can and should be done for circuit QED experiments to justify central results, and it can be done using various methods such as qubits, waveguide-QED devices, SNTJs, variable temperature source etc, (2) the hybrid couplers are a part of the overall circulator/coupler implementation so their effect should be included for reporting its performance metrics such as attenuation, (3) the hybrid coupler attenuation can also be calibrated out in a separate cooldown with only itself in the switch bank using the TRL standards the authors have in the experimental setup, (4) in the worst case the authors can subtract off from the through line response as a fair baseline. If the reasonings above are incorrect or miss something would the authors please clarify. The existing presentation of Fig.3 in my opinion casts doubts on the credibility of the reported results/conclusion and makes it difficult for interpretation. 8. Besides signal isolation performance, how well does the device prevent backward pump propagation into the signal input port or the common port? One common use case of isolators in qubit measurements is to prevent the much-higher-power back-propagating pumps driving the quantum amplifiers from dephasing qubits, and it would be helpful to understand how well this device works for this purpose, and if there are explanations or ways to improve it. 9. Additional comments: a) In the second paragraph, “…activate a two-mode squeezing interaction between the signal and idler waves and induce phase-preserving amplification …” it should be noted that it is only phase preserving when w_s != w_i and considering single-mode phase space. b) In the paragraph right below Eq. (2), “.. range amplification, at the cost of increased design complexity to ensure that all waves propagate at the same velocity v_S = v_I = v_P”. I don’t think this is correct. In fact, this would imply a dispersionless medium which would cause cascaded processes and lead to shock waves forming, preventing efficient nonlinear processes from happening. The phase-matching condition Eq (2) is sufficient. c) The authors have assumed the continuum limit in analyzing the device design in the main text. However, the continuum limit is only valid when k*a << 1, where k is the wave vector of the interacting waves. For this work, the cutoff frequency is ~38 GHz, but the signal and idler frequencies are as high as 14 GHz. This means the continuum assumption does not hold; the analytic dispersion relation in the linear undriven case in the supplement should be directly used instead, and discretized nonlinear model a discrete nonlinear framework should instead be used, such as in Peng et al pp. 331-340, IEEE QCE (2022)."}, "author_response": "Response to referee A\nThe paper describes work toward realizing an on-chip microwave circulator (that can also func;on as a controllable coupler), a device for which there is currently much interest because of its applica;ons to quantum compu;ng and other experiments detec;ng signals at the level of quantum noise. The methods used and design of the device are new and should be of high interest to others working on related devices, such as parametric amplifiers. The work presented is technically impressive and is generally well presented and supported by the extensive supplementary sec;ons. Although in its current form, the device may have too much loss to be directly applicable to experiments, there are clear pathways toward improving it. Based on the newness of the techniques and the significant progress toward demonstra;on, I recommend that the paper be published.\nI aFach a copy of the manuscript with a few gramma;cal sugges;ons.\nWe thank the referee for his/her positive appreciation of our work and for the suggested corrections which we have implemented (changes highlighted in red in the revised manuscript).\nResponse to referee B\nReviewer #2 (Remarks to the Author):\nThis paper experimentally explores the use of nonlinear processes involving counter-propaga;ng waves in superconduc;ng quantum circuits for tunable isola;on and variable-strength coupling. Employing a coupled nonlinear Josephson transmission line architecture with a pair of 180-degree hybrid couplers, the authors experimentally show ~15dB of isola;on with ~100MHz of instantaneous bandwidth, 3-GHz tunable bandwidth, and ~-120dBm dynamic range. This performance is similar to those of other on-chip magne;c-free isolators achieved in the field using copropaga;ng nonlinear interac;ons. Nonlinear interac;ons involving counter-propaga;ng waves are not new in physics and are oWen encountered in fiber op;cs as s;mulated Brillouin scaFering (SBS). Nevertheless, this work s;ll contains several novel;es as it is, to my knowledge, the first experimental work in circuit QED using (1) counter-propaga;ng nonlinear process, and (2) coupled-nonlinear transmission line.\nWe are grateful to the referee for poin;ng out this analogy with SBS, which we had overlooked. We have modified our manuscript to\n- Remove all men;ons to the “novelty” of the regime we explore\n- Explicitly say that the phase-matching configura;on we consider is\n[…] analog to S,mulated Brillouin Sca5ering in op,cs\nand in a footnote that\nIn s,mulated Brillouin sca5ering, the low velocity wave is acous,c and interacts with light through electrostric,on.\nNevertheless, as the referee points out, our work is the first to consider this configura;on in circuitQED. It is also the first ;me phase-matched conversion occurs exclusively between counter-propaga;ng photonic waves, and the first experimental implementa;on of coupled nonlinear transmission lines (similar architectures were proposed in Refs [52, 61]).\nWhereas this work may meet the novelty bar and is overall well-wriFen with useful theory details in the supplement, I found its demonstrated performance and degree of completeness challenging to jus;fy publica;on in Nature Communica;ons as-is. I think this work may be eventually accepted in Nature Communica;ons if further improvements in both measurement calibra;on and device performance can be made, and the following comments are sa;sfactorily addressed, preferably by changing the manuscript where needed rather than just in a response document.\nWe thank the referee for his/her careful review of our manuscript and his/her challenging and relevant comments as we believe that the quality of our manuscript has significantly improved. We provide below a point-by-point response.\nIn terms of completeness:\n- We took a new dataset aWer simplifying our experimental setup, allowing for a clearer read of device performance.\n- We have significantly improved the calibra;on of our experiment and our theore;cal modeling. Our theore;cal model now reproduces experimental data with no fit parameter (the device performance has slightly degraded as we now limit ourselves to this well understood regime of opera;on).\n- We have rephrased several sec;ons of the paper to improve the paper clarity (changes are highlighted in red in the revised manuscript). We have completely re-wriFen the supplemental materials which now include detailed calibra;ons and a more complete theore;cal model.\nIn terms of device performance, we were unfortunately unable to produce a new sample as our cleanroom facili;es have been closed in prepara;on for an upgrade (precisely meant to improve future devices’ performance). Nevertheless\n- The mechanisms limi;ng the current device’s isola;on and inser;on loss are well-understood.\n- One should note that our work is both a proposal and an experimental proof-of-concept. This is the reason we rather centered our paper on novel concepts than on device performance.\n- Our device shows the desired non-reciprocal behavior and expected aFenua;on scaling despite the presence of an open junc;on in the middle of the device, which shows its robustness (see below).\n- Device performance should significantly improve aWer op;mizing the architecture and parameter regime based on the findings of the current work (see paper conclusion).\n1. The inser;on loss of the device was not reported in the manuscript. Inser;on loss is an important metric for isolators as it should ideally be as small as possible to avoid further degrading the SNR forward signal. The authors only briefly men;oned that they es;mate the inser;on loss at the end of Sec;on III from the reported dielectric loss tangent of other works, which to me is insufficient and not standard prac;ce. The authors’ experimental setup has a through line in the setup and should be able to take such measurements and report the inser;on loss over the quoted frequency range directly.\nThis number was reported in our manuscript but was not introduced clearly enough.\nAs suggested by the referee in (7), we take a conserva;ve defini;on of inser;on loss, including losses in the hybrid couplers, cables (with in-line filters) and impedance adapters used to address the device. Moreover, we consider any uninten;onally reflected signal as loss.\nAs can be seen from Fig. 4, inser;on loss in absence of pump ranges from 5 to 11 dB over the 4-10 GHz band.\n- A defec;ve junc;on leads to reduc;on of signal transmission at the 1.5 to 2.5 dB level\n- We measure in a separate cooldown 2.5 to 6.5 dB of aFenua;on for a signal transmiFed through the hybrid couplers and coaxial cables aWer replacing the sample by two short through lines.\n- The remaining loss (1 to 2.5 dB) can be aFributed to dielectric loss in alumina (tan(𝛿) = 2.10!\").\nWe rephrased the corresponding paragraph in Sec;on III to quote these numbers at 7 GHz:\nWhen no pump is applied, we measure significant inser,on loss (gray line) with a value of 8.5 dB at 7 GHz. Through independent calibra,on, we a5ribute 4.5 dB of this loss to the hybrid couplers and in-line filters connec,ng to the device (see supplemental Fig S1). We es,mate that signal reflec,on and leakage to the ∆ mode—caused by the defec,ve junc,on---contribute an addi,onal 2 dB. The remaining 2 dB are a5ributed to dielectric losses in the alumina, corresponding to a loss tangent of tan δ ≈ 2 · 10−3\nAnd explicitly men;on in the cap;on of Fig. 3 that reported aFenua;ons are referenced at the Σ ports of the hybrid couplers.\nWe have further added a sec;on to the Supplemental materials (Sec. S5) detailing an in situ power calibra;on method along with es;mates for reflec;ons and dielectric loss within the device.\nWe note that while such loss level would disqualify our device for many applica;ons in circuitQED, we hope to significantly reduce it in a future experiment through beFer controlled fabrica;on and by removing the need for hybrid couplers and turning to a planar architecture for capacitors (see paper conclusion).\n2. From both Fig. 3 and Fig. 4(a), the inser;on loss of the forward signal seems to also increase when the pump is turned on. I would appreciate if the authors could quan;fy such increase and provide explana;ons to its cause. Despite the aFenua;on increases more for the backward signal, the deceased background forward transmission if large could s;ll severely impact its prac;cal use as an isolator.\nWe aFribute this increase in inser;on loss to internal reflec;on of the pump, mostly due to a single defec,ve junc,on (open junc,on) close to the center of our device. Wave scaFering by the defect causes pump reflec;on at the -2 dB level, and the reflected pump ac;vates a circula;on process in the reverse direc;on (the same process that we are targe;ng with the applied pump).\nWe men;oned this defect in the previous version of our manuscript but made this clearer in the revised manuscript:\n[…] spurious reflec,on of the pump within the device uninten,onally [ac,vates] a conversion process in [the reverse] direc,on. Quan,ta,vely, we es,mate following an in situ calibra,on of the device feedlines that the pump is reflected on the Δ propaga,on mode at the -2dB level and leaks to the Σ mode at the -4 dB level, which indicates that one of the junc,ons making the device is open (see supplemental S5 B). […] ,me-domain reflectometry measurements indicate that the defec,ve junc,on is located close to the middle of the line\n[…]\nwhen turning on the pump at cri,cal value to ac,vate the circula,on process, we typically detect a few dB increase in inser,on loss (light red diamonds in Fig. 4).\nMoreover, we have substan;ally refined calibra;ons and modeling of pump scaFering in supplemental S5\n- We use auto-modula;on and cross-modula;on of microwaves applied from each port of the device to es;mate the aFenua;on of input lines.\n- We fit 12 coefficients of the device scaFering matrix to es;mate the aFenua;on of the 4 output lines.\n- AWer calibra;ng out the aFenua;on of the input and output lines, we confirm the wave scaFering paFern expected from an open junc;on\n- We locate the posi;on of the defec;ve junc;on in ;me-domain reflectometry\nWith these calibra;ons, we now predict the device isola;on and inser;on loss induced by the pump with no fit parameter.\nStrikingly, despite a pump reflec;on at the -2 dB level and only half of the device being ac;ve due to pump scaFering by the defect, our device behaves with the proper aFenua;on scaling and features only moderate pump induced inser;on loss. Indeed, the exponen;al scaling of the signal aFenua;on makes it very robust to spurious pump reflec;on.\n3. In the introduc;on sec;on, the authors men;oned that traveling-wave processes have the advantage of larger working bandwidth, power handling and scalability. This is in general true for parametric amplifiers where broadband phase matching is possible and is less interac;on-length sensi;ve, but for the traveling wave parametric converters considered in this work they too suffer from limited bandwidth and dynamic range. This work, along with a prior work u;lizing copropaga;ng traveling-wave conversion (Ranzani et al., Phys. Rev. Appl 8, 054035 (2017)), has only ~100MHz instantaneous bandwidth. As a comparison, a recently demonstrated resonator-based device by Beck et al. Phys. Rev. Appl 20, 034054 (2023) showcased 15dB isola;on over 600-MHz of instantaneous bandwidth. The level of achieved performance in this work doesn’t seem to be convincing or jus;fy its significant hardware overhead (e.g. hybrid couplers), increased device footprint and complexity, and poten;ally increased device inser;on loss, warran;ng further improvement in device performance.\nWe thank the referee for poin;ng out this reference which we had missed (we now properly cite it, along with a few other missed references).\nIndeed, the working bandwidth is not a par;cularly strong point of our approach. As we hope is now clear from the paper introduc;on and results, its strength lies in the exponen;al isola;on of in-band signal, endowing it with robustness and, hopefully, isola;on beyond state-of-the-art in the near future.\nNevertheless,\n- The current bandwidth ~200 MHz could be increased by modifying the device architecture. Indeed, the bandwidth is set by the aFenua;on constant (𝛼 in Eq S16) which is itself set by the local modula;on of the Josephson inductance induced by the pump. In the present architecture, this modula;on is given by a current-bias and is limited by the junc;ons switching current. By moving to a flux-bias in future experiments, we hope to increase the working bandwidth to more than 500 MHz\n- Our device is accordable and its working frequency can be chosen over a 10 GHz range (using the circula;on process in both direc;ons and/or the aliasing process)\n- For many applica;ons, a bandwidth in the sub-GHz range is actually desirable to avoid ac;va;ng cascaded processes. In par;cular, to use the device as an actual circulator (and not an isolator), one should make sure that signal waves converted to idler waves are not subsequently converted to another frequency.\nUsing a non-reciprocal traveling wave device can also find applica;ons in quantum simula;on (e.g. the Hatano-Nelson model connec;ng par;cles via a chiral line)\nAs for power handling capability, we have revised our es;mate of the device dynamic range to -110 dBm aWer properly calibra;ng the system feedlines. This value could be increased with a device featuring larger junc;ons (and increasing the number of cells to maintain a constant op;cal path).\n4. The authors men;oned that their proposed design of this work has the intrinsic limita;on that the impedance of the even and odd modes cannot be simultaneously impedance matched as explained by Eqs. (S1) (S2). Indeed, as the authors observed and described on measurements, significant pump reflec;on (~-6dB to -3dB) occurred and is a show stopper to the device performance, despite Klopfenstein tapers were used. How well are the Klopfenstein tapers in impedance matching the two modes to 50 ohm? Are there test structures / chips with the tapers only to quan;fy the resul;ng reflec;on and validate that they func;on properly? Are they able to explain this large pump reflec;on?\nWhereas fabrica;on disorder could certainly be a reason for spurious reflec;on, it would be helpful to provide further evidence to substan;ate/refute those hypotheses.\nWe apology for the lack of clarity of our manuscript. As we hope is now clear from our response to ques;on (2) and corresponding revision of the manuscript, reflec;on occurs internally due to an open junc;on, and not at the device ports (through microwave simula;ons, we es;mate the level of pump reflec;on at the impedance adapters at -20 dB, which should not impact the device performances). Moreover, as we now note in our revised manuscript,\nthe pump reflec,on could have been efficiently suppressed with a canceling pump applied in the reverse direc,on if the defec,ve junc,on were located close to the device ports.\nSimilarly, a canceling pump with the proper phase and amplitude (adjusted in situ) could be applied to suppress residual pump wave propaga;ng in the wrong direc;on within the device if reflec;on occurred at the ports.\n5. The authors claimed “…the lines are coupled, and it is impossible to have both propaga;ng mode impedances at 50Ω at the same ;me.” This statement is not en;rely accurate. This is indeed possible if both capaci;ve and induc;ve couplings are employed; see page 4 of Naghiloo et al., ArXiv:2103.07793 (2021) for more details. The authors could consider introducing induc;ve coupling to help mi;gate the pump reflec;on issue which is cri;cal to device performance.\nThe referee is correct and we thank him/her for this sugges;on which may be helpful in the future. We now state in supplemental S2 that\nthe lines are capaci,vely coupled, and it is impossible to have both propaga,ng mode impedances at 50Ω at the same ,me unless addi,onal induc,ve coupling is used [52], which was not a5empted\n6. The authors men;oned that they use 3dB aFenuators to sandwich the device to provide a clean impedance environment for the device. In my opinion, this leads to ar;ficially beFer performance than it would actually behave in real setups as aFenuators won’t be used downstream the informa;on-carrying signal, limi;ng its prac;cal applicability. I would like to see how the device performs without the sandwiching aFenuators, which is more interes;ng from a prac;cal point of view.\nThe new dataset presented in the revised manuscript was acquired aWer removing the 3dB aFenuators. This did not lead to any visible degrada;on of performance.\n7. I do not fully understand why the baseline aFenua;on could not be extracted for Fig. 3. From my understanding, the through line in the switch bank can be used for this purpose. I found the authors explana;on in the main text on baseline aFenua;on to be rather unsa;sfactory: (1) power calibra;on can and should be done for circuit QED experiments to jus;fy central results, and it can be done using various methods such as qubits, waveguide-QED devices, SNTJs, variable temperature source etc, (2) the hybrid couplers are a part of the overall circulator/coupler implementa;on so their effect should be included for repor;ng its performance metrics such as aFenua;on, (3) the hybrid coupler aFenua;on can also be calibrated out in a separate cooldown with only itself in the switch bank using the TRL standards the authors have in the experimental setup, (4) in the worst case the authors can subtract off from the through line response as a fair baseline. If the reasonings above are incorrect or miss something would the authors please clarify. The exis;ng presenta;on of Fig.3 in my opinion casts doubts on the credibility of the reported results/conclusion and makes it difficult for interpreta;on.\nWe agree with the referee and have followed his/her point-by-point recommenda;on. As detailed in our response to comments (1) and (2):\n- we have calibrated power using self and cross-phase modula;on in our device\n- the main results of our paper use a conserva;ve calibra;on including loss in hybrid couplers in the circulator inser;on loss (using the switch bank). This is par;cular the case for Fig 3 and Fig 4\n- we have measured loss in hybrid couplers separately (see response to comment 1)\n8. Besides signal isola;on performance, how well does the device prevent backward pump propaga;on into the signal input port or the common port? One common use case of isolators in qubit measurements is to prevent the much-higher-power back-propaga;ng pumps driving the quantum amplifiers from dephasing qubits, and it would be helpful to understand how well this device works for this purpose, and if there are explana;ons or ways to improve it.\nWe have now properly calibrated such leakage of the pump (and its harmonics) into the signal mode (see Fig. S20). Owing to the scaFering of the pump by the defec;ve junc;on we record inter-mode leakage at the -4dB level (-7dB in each direc;on).\nWe explicitly men;on that\nstrong inter-mode leakage […] would negate any benefit gained from spa,ally separa,ng the pump and signal waves if our device were to be interfaced with sensi,ve quantum systems.\nWe hope to solve this issue in a future experiment.\n9. Addi;onal comments:\na) In the second paragraph, “…ac;vate a two-mode squeezing interac;on between the signal and idler waves and induce phase-preserving amplifica;on …” it should be noted that it is only phase preserving when w_s != w_i and considering single-mode phase space.\nIndeed, this sentence was imprecise. We have removed the reference to “phase-preserving” amplifica;on as it was not needed in that context.\nb) In the paragraph right below Eq. (2), “.. range amplifica;on, at the cost of increased design complexity to ensure that all waves propagate at the same velocity v_S = v_I = v_P”. I don’t think this is correct. In fact, this would imply a dispersionless medium which would cause cascaded processes and lead to shock waves forming, preven;ng efficient nonlinear processes from happening. The phase-matching condi;on Eq (2) is sufficient.\nWe have rephrased this paragraph.\nc) The authors have assumed the con;nuum limit in analyzing the device design in the main text. However, the con;nuum limit is only valid when k*a << 1, where k is the wave vector of the interac;ng waves. For this work, the cutoff frequency is ~38 GHz, but the signal and idler frequencies are as high as 14 GHz. This means the con;nuum assump;on does not hold; the analy;c dispersion rela;on in the linear undriven case in the supplement should be directly used instead, and discre;zed nonlinear model a discrete nonlinear framework should instead be used, such as in Peng et al pp. 331-340, IEEE QCE (2022).\nWe thank the referee for poin;ng this out. We have performed complementary simula;ons using a discre;zed nonlinear model (open source library derived from the reference cited by the referee, now properly cited in our paper) in order to es;mate the accuracy of our con;nuous model (see Fig S14). For the simula;ons used to reproduce data in paper figures, the con;nuous model seems to underes;mate isola;on by up to 5 dB compared to the discre;zed nonlinear model (see Sec S6 A). Nevertheless, as now stated in the paper:\nthis imprecision is not significant compared to that entailed by the finite accuracy with which we calibrate the pump amplitude and to the unpredictability entailed by the defec,ve junc,on.\nWe thank the reviewers for their thoughtful feedback. Our responses appear below in black, and corresponding revisions are highlighted in blue in our manuscript."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The changes to the manuscript made in response mainly to the other reviewer's comments have only made the paper stronger, so I stand by my original recommendation to publish.", "reviewer_2": "I appreciate the authors’ detailed responses and revisions, many of which address my earlier comments well. However, I still have concerns about several claims in the response and manuscript to be addressed, as outlined below. While I empathize the challenges in fabricating new devices, the open junction and significant reflection (~–2 dB) is essentially a show stopper for its intended applications. The presence of this big reflection also very likely masks out the effects of other potential imperfections that could not be accurately extracted. Considering authors’ argument that this work functions mostly as a proof-of-concept, its performance thus marginally meets the bar for Nature Communications. 1. “Our device shows the desired non-reciprocal behavior and expected attenuation scaling despite the presence of an open junction in the middle of the device, which shows its robustness (see below).” I do not agree with the robustness claim. Showing the non-reciprocal behavior is only part of the requirement for the intended functionality – once there is an open junction, the device is essentially unusable for any practical applications due to significant SNR degradation and backaction. 2. “As we hope is now clear from the paper introduction and results, its strength lies in the exponential isolation of in-band signal…” “Our device is accordable and its working frequency can be chosen over a 10 GHz range…” While I agree with the exponential scaling of isolation from frequency conversion and tunable range, the same is true for other existing traveling-wave isolator works in the literature. In the the context of comparing this work to those, the authors’ response did not address my question of how it could be a compelling technology and justify its significant hardware overhead (e.g. hybrid couplers), increased device footprint and complexity, and potentially increased device insertion loss over others. 3. “For many applications, a bandwidth in the sub-GHz range is actually desirable to avoid ac;tivating cascaded processes. In particular, to use the device as an actual circulator (and not an isolator), one should make sure that signal waves converted to idler waves are not subsequently converted to another frequency.” I disagree with this statement. The advantage of narrow bandwidth to avoid cascaded frequency conversion is only specific to the particular implementation type of this work. The end goal of an isolator is to block as broadband the backward noise (at higher effective temperature) as possible, hence why commercial ferrite circulators are made as broadband as possible. 4. “we estimate the level of pump reflection at the impedance adapters at -20 dB, which should not impact the device performances” As mentioned earlier, I don’t think a claim of ~-20dB reflection can be made on this device. The presence of the large -2dB reflection from open junction dominates and masks out the smaller reflections from being accurately extracted from measurement. On the other hand, actual device are are almost always much wores than EM simulations due to imperfect parameter targeting and packaging (wirebonds, etc), especially down to the -20dB level (~1%). It is therefore reaching to claim -20dB and therefore do not impact device performances based solely on EM simulation. 5. “when turning on the pump … detect a few dB increase in insertion loss a few dB increase in insertion loss” “…. only moderate pump induced insertion loss” Please be precise and quantify wherever possible, and explain the implications of such “a few dB increase”. The use of “moderate” is quite subjective and I would argue its misleading if not incorrect. Those “moderate” insertion loss will essentially disqualify its intended use as the authors acknowledged themselves. Moreover, whats the threshold for the level of pump reflection to not lead to appreciable increased loss, say 1dB, in the forward direction from simulation? 6. I found the order of the processes in Fig. 1b to be confusing. Naturally one would expect the direction of the circulator operation to be from left to right, but Fig. 1b does the opposite, having the backward signal to be attenuated going to the right. I would suggest flipping panel b and labelling the positive direction as the direction of circulator. 7. “The remaining loss (1 to 2.5 dB) can be attributed to dielectric loss in alumina (tan(delta) = 2e-3.” Can the authors corroborate this from fitting the slope of the extracted insertion loss in dB scale vs frequency? The inductance or electrical length of the device should not be fitting parameters but rather use those from fitting the nonlinear dynamics. Moreover, the authors should explicitly acknowledge that hybrid couplers and filters are part of the device implementation, and their contribution of insertion loss should be counted."}, "author_response": "We again thank the reviewer for his/her positive appreciation of our work.\nWe thank the reviewer for the time dedicated to evaluating our paper. We would like to reiterate that our experiment is intended as a proof of concept, introducing novel traveling-wave mixing mechanisms that expand the circuit-QED toolbox. In the manuscript, we emphasize how these mechanisms lead to qualitatively different behavior compared to standard traveling-wave converters (also see our response to comment (2)).\nWe do not claim that our device represents a mature technology. We join the reviewer in looking forward to a non-reciprocal device with sufficiently good specs that it can be integrated in cQED experiments and help the field move forward. However, to the best of our knowledge, such a device never was demonstrated.\nIn our current implementation, fabrication imperfections limited the performance of our device. This is standard in quantum technologies where a first paper pioneers an idea (here the mixing of counter propagating signals in a Josephson transmission line) which triggers a series of sequel experiments that optimize the specs. For context, the use of Josephson traveling-wave devices for conversion with the goal of achieving amplifier-isolator functionality was pioneered in 2017 by Ranzani et al. [41]. Despite recent advances [42–43], this approach has not yet enabled integrated qubit readout. Similarly, several other strategies for on-chip circulation have been actively pursued over the past decade [44–57], yet none have reached full maturity to date.\n1. “Our device shows the desired non-reciprocal behavior and expected attenuation scaling despite the presence of an open junction in the middle of the device, which shows its robustness (see below).”\nI do not agree with the robustness claim. Showing the non-reciprocal behavior is only part of the requirement for the intended functionality – once there is an open junction, the device is essentially unusable for any practical applications due to significant SNR degradation and backaction.\nWe made this statement in our response letter to illustrate the benefits of an exponential isolation, but not in the paper. We have removed the sole occurrence of “robust” pertaining to our device from the paper.\n2. “As we hope is now clear from the paper introduction and results, its strength lies in the exponential isolation of in-band signal…”\n“Our device is accordable and its working frequency can be chosen over a 10 GHz range…”\nWhile I agree with the exponential scaling of isolation from frequency conversion and tunable range, the same is true for other existing traveling-wave isolator works in the literature. In the the context of comparing this work to those, the authors’ response did not address my question of how it could be a compelling technology and justify its significant hardware overhead (e.g. hybrid couplers), increased device footprint and complexity, and potentially increased device insertion loss over others.\nTo our understanding, attenuation in a regular TWPC (when the signal and idler propagate in the same direction) does not scale exponentially but rather follows an oscillating pattern (see Fig 1a and References [41-43]). A consequence is that, in a regular TWPC, the pump amplitude should be fine-tuned to ensure full conversion of the signal to the idler frequency, and this can only be perfectly realized at a single frequency. In principle, in our case (when signal and idler counter-propagate), in-band isolation only gets better as the device length or the pump amplitude increases. We try to convey this key difference in Fig. 1. Moreover, our device is reconfigurable into a reciprocal tunable coupler.\nAs for the increased complexity and footprint in our approach:\n- introducing a second traveling mode comes with the added benefit of spatially separating the high-amplitude pump from the low amplitude signal (also proposed in Ref [52]).\n- another advantage is that signal waves propagating backward in the device are converted and reflected (as opposed to converted and transmitted in a regular TWPC), removing the need for reflective bandpass filters when one wants to prevent frequency-converted noise to backpropagate.\n- the choice of a symmetric two-mode line, which requires a hybrid coupler to operate, was made based on considerations of simplicity of fabrication. It is clearly not optimal in terms of insertion loss and robustness. We sketch in the conclusion of the paper an alternative architecture similar to that proposed in [52], which would remove the need for hybrid couplers.\n3. “For many applications, a bandwidth in the sub-GHz range is actually desirable to avoid ac;tivating cascaded processes. In particular, to use the device as an actual circulator (and not an isolator), one should make sure that signal waves converted to idler waves are not subsequently converted to another frequency.”\nI disagree with this statement. The advantage of narrow bandwidth to avoid cascaded frequency conversion is only specific to the particular implementation type of this work. The end goal of an isolator is to block as broadband the backward noise (at higher effective temperature) as possible, hence why commercial ferrite circulators are made as broadband as possible.\nWe agree with the reviewer that this comment in our response letter only applies to devices in which circulation fidelity can be limited by cascaded processes. Indeed, for most applications, high bandwidth is desirable.\n4. “we estimate the level of pump reflection at the impedance adapters at -20 dB, which should not impact the device performances”\nAs mentioned earlier, I don’t think a claim of ~-20dB reflection can be made on this device. The presence of the large -2dB reflection from open junction dominates and masks out the smaller reflections from being accurately extracted from measurement. On the other hand, actual device are are almost always much wores than EM simulations due to imperfect parameter targeting and packaging (wirebonds, etc), especially down to the -20dB level (~1%). It is therefore reaching to claim -20dB and therefore do not impact device performances based solely on EM simulation.\nIndeed, claiming -20 dB reflection without properly calibrating it would be misleading, and we do not make this claim in our paper. We only mentioned the -20 dB estimated from EM simulations in our response letter to argue that the observed level of pump reflection is far above what is expected from imperfect impedance matching at the ports.\nWe can still estimate reflection at the ports of the device from time domain measurements, which we now do in the revised manuscript (see updated Figure S8). Indeed, we observe in the time-domain two peaks in reflection (around 16 and 17 ns) that correspond to reflections right before the taper used to impedance match feedlines (i.e. in the sample holder) and right after the taper (when connecting to the Josephson line). From the height of these peaks, we estimate reflections at the -22 dB and -21 dB respectively, adding up to give at most a ~-15 dB reflection.\nThis level of reflection should not yield significant insertion loss for the signal (below 1 dB), and the reflected pump could not activate any process to a significant level (see response to the next comment). Finally, we note that the reflected pump wave could be efficiently suppressed in situ with a cancelling pump wave applied from the other port (we mention in our paper that “the pump reflection could have been efficiently suppressed with a canceling pump applied in the reverse direction if the defective junction were located close to the device ports.”)\n5. “when turning on the pump … detect a few dB increase in insertion loss a few dB increase in insertion loss”\n“…. only moderate pump induced insertion loss”\nPlease be precise and quantify wherever possible, and explain the implications of such “a few dB increase”. The use of “moderate” is quite subjective and I would argue its misleading if not incorrect. Those “moderate” insertion loss will essentially disqualify its intended use as the authors acknowledged themselves.\nMoreover, whats the threshold for the level of pump reflection to not lead to appreciable increased loss, say 1dB, in the forward direction from simulation?\nWe agree with the referee that figures of merit should be accurately and quantitatively quoted:\n- “detect a few dB increase in insertion loss”. We chose this qualitative description in our paper (as opposed to quoting several numbers covering the whole working range of the device) to avoid drowning the reader in numbers. Nevertheless, we pointed at the end of the sentence to Fig. 4c, in which the increased insertion loss can be read over the whole range. We have moved this reference mid-sentence to make it more visible.\n- “…. only moderate pump induced insertion loss” Indeed this wording used in our response letter was subjective. Factually, the observed increase in insertion loss is typically 4 times smaller than the desired isolation in dB.\nAs for the tolerable level of pump reflection to not lead to appreciable increased loss, the continuous model predicts that, when targeting -20 dB of isolation, a -8dB pump reflection only leads to 1dB of added loss. Such a level of pump reflection can realistically be reached, even considering fabrication disorder and impedance mismatch at the ports.\n6. I found the order of the processes in Fig. 1b to be confusing. Naturally one would expect the direction of the circulator operation to be from left to right, but Fig. 1b does the opposite, having the backward signal to be attenuated going to the right. I would suggest flipping panel b and labelling the positive direction as the direction of circulator.\nWe chose this convention such that the signal (yellow arrow in Fig.~1a-b) is injected from the left port of the device. While we understand the reviewer’s preference for the alternative convention when considering the device used as a circulator, we prefer to keep the current convention.\n7. “The remaining loss (1 to 2.5 dB) can be attributed to dielectric loss in alumina (tan(delta) = 2e-3.”\nCan the authors corroborate this from fitting the slope of the extracted insertion loss in dB scale vs frequency? The inductance or electrical length of the device should not be fitting parameters but rather use those from fitting the nonlinear dynamics. Moreover, the authors should explicitly acknowledge that hybrid couplers and filters are part of the device implementation, and their contribution of insertion loss should be counted.\nWe have added a figure to the Supplemental materials representing the various contributions to insertion loss, over the working range of our device (now Fig S12). Doing so, we have refined our estimate for the tangent loss of alumina, which we now estimate at tan(𝛿) ≃3.10!\".\nThe electric length of the device is independently calibrated and we account for the discrete nature of the line as well as for the renormalization of the junction inductance owing to nonlinearity (negligible at the probe power we are using).\nAs for the contribution of hybrid couplers and filters to insertion loss, it is taken into account as mentioned in the captions of Figures 3 and 4, in the last paragraph of Section III, in the conclusion of the paper as well as multiple times in the Supplemental materials.\nWe have added the comment “which we consider as part of the device” when referring to the hybrid couplers and filters in the last paragraph of Sec. III."}, {"round": 3, "reviewer_comments": {"reviewer_2": "From the authors response, I now understand better that the exponential attenuation feature of this work is rather unique in the circuit QED space and I thank the authors for the explanations. Given this, I agree this work can be accepted for publication. Overall, I found most of the other point-by-point responses acceptable, although I do feel the need to push back on a few issues. “. We join the reviewer in looking forward to a non-reciprocal device with sufficiently good specs that it can be integrated in cQED experiments and help the field move forward. However, to the best of our knowledge, such a device never was demonstrated.” https://arxiv.org/abs/2505.04059 https://arxiv.org/abs/2406.19752 These two works, as cited by the authors in my opinion satisfies authors’ requirement “with sufficiently good specs that it can be integrated in cQED experiments”. As competing works and put on arxiv around the same time as this work, they have > 20dB gain and > 20dB reverse isolation simultaneously over hundred MHz to gigahertz bandwidth. Even in the context of justifying the significance of this work in response letter, it is expected that competing works to be fairly assessed and acknowledged. It is ultimately up to the authors, but I do want to point out again that, in my opinion, it is generally recommended to avoid using imprecise descriptions in scientific publication; even worse sometimes doing so could raise the suspicion of being purposefully vague, especially when the topic in question is a device limitation. “to avoid drowning the reader in numbers” I wouldn’t worry about this, it is important to quantify this drawback/limitation of the device to allow readers to assesse and appreciate themselves how much it matters. “as opposed to quoting several numbers covering the whole working range of the device)” one could simply say “< XdB” or “X – Y dB” within a “AA-BB frequency band”. The scale in Fig 4c is also too large and I would challenge that “in which the increased insertion loss can be read over the whole range.” is instead putting burdens to prevent readers from assessing this information easily, and the large Y axis span does not help either and affects readability. “…. only moderate pump induced insertion loss” Indeed this wording used in our response letter was subjective. Factually, the observed increase in insertion loss is typically 4 times smaller than the desired isolation in dB.” Again, “moderate” only makes sense when a certain context is given. Sure, it could be moderate when compared to the isolation, but this context needs to be provided explicitly whenever these qualitative statements are made. In other contexts, such as using the device as a circulator, moderate pump induced insertion loss would mean only a few fractions of dB, which is an equally valid metric but not the case here. It doesnt seem like the authors use this exact word in the main text, which is good, and I would recommend going through similar claims and making sure contexts were provided accordinlgy to avoid confusion."}, "author_response": ""}]

Decision letter