Active protection of a superconducting qubit with an interferometric Josephson isolator

Nonreciprocal microwave devices play critical roles in high-fidelity, quantum-nondemolition (QND) measurement schemes. They impose unidirectional routing of readout signals and protect the quantum systems from unwanted noise originated by the output chain. However, cryogenic circulators and isolators are disadvantageous in scalable superconducting architectures because they use magnetic materials and strong magnetic fields. Here, we realize an active isolator formed by coupling two nondegenerate Josephson mixers in an interferometric scheme and driving them with phase-shifted, same-frequency pumps. By incorporating our Josephson-based isolator into a superconducting qubit setup, we demonstrate fast, high-fidelity, QND measurements of the qubit while providing 20 dB of protection within a bandwidth of 10 MHz against amplified noise reflected off the Josephson amplifier in the output chain. A moderate reduction of 35% is observed in T2E when the Josephson-based isolator is turned on. Such a moderate degradation can be mitigated by minimizing heat dissipation in the pump lines.


Supplementary Note 1: MPIJIS theory
To calculate the scattering parameters of the MPIJIS and demonstrate its isolation operation, we use the effective signal-flow graph exhibited in Supplementary Figure 1a. The graph includes signal-flow graphs for the two coupled JPCs operated in frequency-conversion mode. On-resonance signals at f 1 = f a or f 2 = f b input on port a (e.g., 1 or 2 ) or b (e.g., b1 or b2) are reflected off by a reflection-parameter r and transmitted with frequency-conversion by a transmission-parameter t, where r and t are determined by the pump drive amplitude and satisfy the energy conservation condition r 2 + t 2 = 1. In this calculation, we assume that the two JPCs are balanced, i.e., their reflection and transmission parameters are equal. It is worth noting that although we consider the on-resonance case here, it is straightforward to generalize the device response for signals that lie within the JPC dynamical bandwidth as we show below. The nonreciprocal phases ϕ 1 and ϕ 2 acquired by the frequency-converted transmitted signals between ports a and b, as indicated in the graph, correspond to the phases of the pump drives at frequency f p feeding JPC 1 and JPC 2 , respectively. Supplementary Figure 1a also includes flow-graphs for two couplers coupling the a and b ports of the JPCs; one represents the 90 • hybrid, which couples between the a ports of the JPCs, while the other is a fictitious one coupling the b ports. The main role of the latter coupler is to model the amplitude attenuation present on the b port α, due to signal absorption in the 50 Ω cold loads and the insertion loss of the normal-metal transmission line coupling the two stages. Because of the structural symmetry of our device, we consider a symmetric coupler with real coefficients α and β, which satisfy the condition α 2 + β 2 = 1. For an ideal symmetric coupler (i.e., 90 • hybrid), α = β = 1/ √ 2 [1].
In the stiff pump approximation, the JPC reflection and transmission parameter amplitudes can be written as [2] r = 1 − ρ 2 1 + ρ 2 , where 0 ≤ ρ ≤ 1 is a dimensionless pump amplitude. The lower bound ρ = 0 corresponds to the case of no applied pump in which the JPC is OFF and acts as a perfect mirror, whereas the upper bound ρ = 1 corresponds to the case of full frequency conversion mode between ports a and b. By inspection [1], the scattering matrix of the inner device defined by the ports 1 , 2 , 3, 4, i.e., excluding the first 90 • hybrid, can be written in the form [3] where ϕ ≡ ϕ 1 − ϕ 2 . As we show below, it is this phase difference between the modulation phases of the two pumps feeding the two parametric active devices (i.e., the JPCs), which induces the on resonance versus the transmission parameter t of the balanced JPCs. In both cases a and b, the MPIJIS is biased in the forward direction (ϕ = −π/2). c Periodic response of the various scattering parameters on resonance versus the pump phase difference ϕ: |S 11 | 2 and |S 22 | 2 (red), |S 21 | 2 (blue), |S 21 | 2 (orange), |S 31 | 2 (dashed yellow), |S 23 | 2 (dashed magenta), |S 13 | 2 (dashed black), and |S 32 | 2 (dashed green). In the calculations a and c, the JPCs are operated at the 50:50 beam splitter working point. In the calculations a, b, and c, the effective coupler is assumed to be a hybrid with equal real coefficients α = β = 1/ √ 2. d |S 21 | 2 (blue), |S 13 | 2 (red), and |S 23 | 2 (magenta) on resonance versus the effective coupler coefficient α, which is varied between 0 and 1. In this calculation, β = √ 1 − α 2 and t is calculated for each given α to yield a fixed isolation |S 12 | 2 = 0.01 (−20 dB) on resonance.
nonreciprocal response of the MPIJIS [4][5][6][7]. The common coefficient 1/(1 − α 2 r 2 ) that appears in the scattering parameters of Eq. (2) represents the sum over all possible reflections that the internal signals can experience in the self-loop formed between the two b ports of the device. Unlike the directional amplification case [3,8], where the reflection-gain amplitude needs to be bounded to ensure stability, in the case of frequency conversion, with no photon gain, the scattering parameters of Eq. (2) are stable for all values of 0 ≤ r ≤ 1. In this simplified model, we assume that the phase acquired by signals at frequency f 2 , propagating along the short transmission line between the two JPCs, is 2πk in each direction, where k is an integer. In our device, the electrical length of the short transmission line is designed to give a phase of about 2π at f 2 .
It is straightforward to verify that the scattering matrix of Eq. (2) is unitary (energy preserving).
For example, it satisfies the condition Next, we derive the scattering matrix for the whole device, defined by ports 1, 2, 3, 4, which take into account the signal flow through the 90 • hybrid, whose matrix elements are given by, Note that [S] is unitary because [s] is unitary and the 90 • hybrid is a unitary device. One prominent property seen from Eqs. (5)- (11), is the interferometric nature of the device, manifested in its scattering parameters, which represent the sum over all possible paths that the waves can propagate in it.
By substituting the scattering parameters of the inner device, listed in Eq. (2) into Eqs. (5)- (11), and by writing the resulting expressions in terms of the parameter t, we obtain the scattering parameters of the MPIJIS in an explicit form where In what follows, we examine a few special cases of interest and outline a few important properties of the device.
Without applied pump, i.e., t = 0, the MPIJIS scattering matrix reduces into This result shows that when the device is OFF, the MPIJIS is transparent for propagating signals and effectively behaves as a lossless transmission line with an added reciprocal phase shift of π/2 for transmitted signals within the bandwidth of the 90 • hybrid. In the special case, where the MPIJIS is ON and the phase difference between the pumps is ϕ = −π/2, the scattering matrix can be written in the form In the derivation of Eq. (26), we assume, without loss of generality, that ϕ s = π/2.
To generalize the on-resonance-derived scattering parameters listed in Eqs. (12)-(24) for signals within the device bandwidth, we substitute [2] t where χ s are the bare response functions of modes a and b (whose inverses depend linearly on f 1 and f 2 ): Since the applied pump frequency satisfies the relations Obviously, this generalization holds under the assumption that the bandwidth of the 90 • hybrid is much larger than the dynamical bandwidths of the JPCs, which is generally the case because transmission-line-based hybrids exhibit bandwidths of a few hundreds of megahertz [9].
(dashed magenta), |S 13 | 2 (dashed black), and |S 32 | 2 (dashed green). Note that some of the absent scattering parameters are equal in magnitude to ones that are present, i.e., |S 22 | 2 = |S 11 | 2 , case corresponds to the input power for which the isolation degrades by +1 dB (denoted as P +1dB ).
In our device, we find P +1dB = −108 dBm, which is indicated by a red circle in Supplementary   Figure 4. It is worth noting that this figure is significantly larger than P −1dB of microstrip-based JPCs, when operated in the amplification mode, which is on the order of −130 dBm [2,10]. This is because in the isolator case the JPCs are operated in the frequency conversion mode (without photon gain) [11].
It is worth pointing out that although the MPIJIS employs two JPCs, two external coils, and two input pumps, the device can be quite stable over time provided it is positioned at the bottom of a deep cryoperm magnetic shield can and the microwave generators feeding the pumps are phaselocked to the 10 MHz reference oscillator of a rubidium atomic clock. By using two phase-locked generators, as done in this work, the set working point can stay stable over a few hours at most. To further extend the device stability over days, one can feed the two pumps using the same generator by splitting its output into two arms and incorporating a tunable attenuator and phase shifter into one arm, as was demonstrated in the directional amplifier experiment [8].

Supplementary Note 3: Added noise
To calculate the added noise by the MPIJIS operated in the forward direction, we compare the signal-to-noise ratio at the output S 2 /N 2 to the signal-to-noise ratio at the input S 1 /N 1 , where S i and N i represent the number of signal and noise-equivalent photons per mode per unit time per unit bandwidth at port 'i', respectively [2]. Using the full scattering matrix of the MPIJIS (Eq. (4)), we write S 2 = |S 21 | 2 S 1 , N 2 = |S 21 | 2 N 1 + |S 23 | 2 N 3 + |S 24 | 2 N 4 = N 1 , where the last equality holds because the scattering matrix is unitary and S 22 = 0. We also assume here that the dominant noise entering the system is vacuum noise. Using these relations, we obtain for the noise factor NF = (S 2 /N 2 )/(S 1 /N 1 ) = |S 21 | 2 . Alternatively, NF can be expressed in terms of the number of noise-equivalent photons added by the MPIJIS to the input n add , where S 2 = |S 21 | 2 S 1 and N 2 = |S 21 | 2 (N 1 + n add ). In this representation, NF = N 1 /(N 1 + n add ). Solving for n add , gives and measuring the qubit, one for outputting the measured quantum signal, which contains wideband isolators and a low-noise semiconductor-based amplifier (i.e., HEMT), and three input lines for feeding the pump drives to the JPC (one line) and the MPIJIS (two lines). It is worth noting that the pump input lines include 20 dB of resistive-based attenuation at the 4 K stage and another 20 dB at the base-temperature stage, where the qubit and MPIJIS are mounted, which take the form of either a resistive-based attenuator installed on pump 1 or a dispersive-based attenuator, i.e., commercial broadband directional coupler, installed on pumps 2 and 3. The main advantage of using the dispersive-based attenuator, i.e., the directional coupler, at the base-temperature stage is that, unlike the resistive-based attenuator, which attenuates the pump by dissipating a portion of its power, the directional coupler attenuates the pump by routing a portion of its power to a separate port, which, in turn, can be carried by via a coaxial line and dissipated at a 50 Ohm termination located at a higher-temperature stage with a larger cooling power, i.e., 4 K stage.
Following these results, we have run several experiments (three of which are outlined here) to uncover the main cause for the T 2E degradation due to the operation of the MPIJIS. In one experiment, we substituted the 20 dB resistive-based attenuator at the base-temperature stage on Furthermore, to rule out the possibility that the decrease in T 2E is caused by the MPIJIS operation rather than due to heat generated in the pump lines, we applied frequency-detuned pump drives to the MPIJIS, whose frequency is 380 MHz lower than that of Supplementary Figure 6i  Additional steps that can be taken to further reduce the dissipated power at the basetemperature stage and consequently eliminate the degradation of the qubit coherence, when the MPIJIS is ON, include: 1) replacing the normal-metal directional couplers on the pump lines with superconducting versions, 2) routing and dissipating the reflected pump power off the MPIJIS at a higher-temperature stage, e.g., the 100 mK stage, and 3) redesigning the on-chip pump feedline, illustrated in Fig. 2c in the main text, to allow the JPC to operate in frequency conversion mode at lower pump powers.