Dissipating quartets of excitations in a superconducting circuit

TL;DR

This work demonstrates the autonomous stabilization of a four-component bosonic qubit by engineering a genuine high-order dissipative channel through a Kerr-free Josephson mixer in a high-impedance circuit. By realizing a four-to-one photon exchange between a memory resonator and a lossy buffer, they achieve a four-photon dissipation channel that speeds up decay from the |4⟩ state by about an order of magnitude beyond intrinsic single-photon loss, while largely preserving lower-energy states. The study characterizes relevant circuit non-idealities, notably stray inductances that induce a cos(2φ) nonlinearity and TLS-related TLS-induced frequency shifts, and demonstrates mitigation strategies including off-resonant Stark shifts and potential circuit redesigns. The results substantiate a viable route toward stabilizing four-legged cat qubits and suggest scalable extensions to even more complex bosonic codes, advancing autonomous QEC on superconducting hardware.

Abstract

Over the past decade, autonomous stabilization of bosonic qubits has emerged as a promising approach for hardware-efficient protection of quantum information. However, applying these techniques to more complex encodings than the Schrödinger cat code requires exquisite control of high-order wave mixing processes. The challenge is to enable specific multiphotonic dissipation channels while avoiding unintended non-linear interactions. In this work, we leverage a genuine six-wave mixing process enabled by a near Kerr-free Josephson element to enforce dissipation of quartets of excitations in a high-impedance superconducting resonator. Owing to residual non-linearities stemming from stray inductances in our circuit, this dissipation channel is only effective when the resonator holds a specific number of photons. Applying it to the fourth excited state of the resonator, we show an order of magnitude enhancement of the state decay rate while only marginally impacting the relaxation and coherence of lower energy states. Given that stray inductances could be strongly reduced through simple modifications in circuit design and that our methods can be adapted to activate even higher-order dissipation channels, these results pave the way toward the dynamical stabilization of four-component Schrödinger cat qubits and even more complex bosonic qubits.

Figures (19)

• Figure 1: Dissipation engineering in a Josephson circuit. a). A target oscillator (the memory, blue) is coupled to an auxiliary dissipative mode (the buffer, orange) by a SQUID participating in both modes. When the circuit loops are biased with magnetic fluxes $\Phi_1$ and $\Phi_2$ following Eq. \ref{['eq:fluxes']} with $\epsilon(t)$ a pump oscillating at $4\omega_a-\omega_b$, the SQUID mediates a Kerr-free four-to-one photon exchange interaction, opening a four-photon dissipation channel acting on the memory. The mode impedances $Z_{a,b}$ set the strength of the interaction and are of the order of $R_q/10=650~\Omega$ in our experiment. b) Energy level-diagram featuring the four-to-one photon pump (purple wriggled arrow) and the buffer single-photon dissipation (orange wriggled arrow). In presence of strong non-linearity, energy levels are unevenly spaced and the pump drives a single transition only (dashed purple arrow represents an off-resonantly driven transition). c) Schematic representation of the circuit used in our experiment. A three-pad superconducting structure hosts the memory and buffer modes with respective anti-symmetric and symmetric voltages across the pads. A distributed $\lambda/2$ resonator capacitively coupled to the structure forms a third mode employed for readout. Foster synthesis of the linear circuit connected to the SQUID yields an effective circuit as in (a) but for the presence of a third mode and for series inductances in the arms of the SQUID, responsible for spurious non-linearities (see Fig. \ref{['fig2']}a and text). Magnetic biases are applied through two flux lines (top). A charge line (bottom) is capacitively connected to the circuit and enables linear drives and single-photon dissipation of the buffer. All lines are filtered by periodic loading to limit single-photon dissipation in the memory (see Sec. \ref{['sm:detailcrystal']}). d) Optical microscope image of the sample, featuring all the structures pictured in (c) with matching false colors, surrounded by a ground plane (brown).
• Figure 2: Circuit characterization. a) Effective three-mode circuit obtained by Foster synthesis of the linear environment connected to the SQUID and used to reproduce data presented in (b-d). Fitted parameters are given in Table \ref{['tableparam']}. b) Measured and simulated resonance frequency of the readout mode as a function of applied DC magnetic biases. Black pixels indicate that the resonance frequency could not be extracted, either because it lied out of the probed frequency range (encoded in color, simulation results clipped to the same range) or because the fit was unreliable. Two saddle points, labeled $\mathrm{SP}_1$ and $\mathrm{SP}_2$, are identified by intersecting double-headed arrows. c) Frequency of the memory and buffer low-energy transitions, along cuts of the frequency map intersecting $\mathrm{SP}_1$ and $\mathrm{SP}_2$ (respectively in bright and dim colors). Cut angles, indicated by double-headed arrows with matching colors in (b), either maximize or minimize curvature. All frequencies are near-identical around both saddle points, indicating that the ATS is near-perfectly balanced. d) First seven transition frequencies of the memory measured at $\mathrm{SP}_1$ (empty blue circles). The anharmonicity is positive and decreases with transition index. This spectrum is well captured by our model accounting for stray inductance in series with the junctions of the ATS (filled black circles).
• Figure 3: Two-mode dynamics under the four-to-one photon pump a) Principle of the memory photon-number resolving measurement. Dispersive pull on the readout mode saturates for high energy states, preventing their direct resolution. To estimate the occupation of $|n\rangle_a$, a $\pi$-pulse is applied on the $|n\rangle_a \leftrightarrow |n+3\rangle_a$ transition, modifying the dynamics of its relaxation to the ground state. Subsequently probing the readout mode on resonance, one obtains a signal $s_n^{ON}$ which is time-shifted from the signal $s_n^{OFF}$ acquired in absence of pulse. Assuming that $|n\rangle_a$ is the highest occupied state before the pulse is applied, the integrated differential signal is proportional to its occupation (see text). b) Time-decay of population in state $|4\rangle_a$ (measured up to an unknown proportionality factor) as a function of the pump amplitude (encoded in color and referenced by the voltage $V_p$ applied to the mixer generating the pump wave). The decay gets faster for increasing pump amplitude, and eventually deviates from an exponential law. c) Buffer reflection spectrum acquired when simultaneously pumping the four-to-one photon transition (spectra offset proportionally to the pump amplitude for readability). The buffer line shifts, broadens and eventually splits for increasing pump amplitudes, revealing a near-strong hybridization of the $|4\rangle_a|0\rangle_b$ and $|0\rangle_a|1\rangle_b$ states. Datasets (b) and (c) are reproduced by Lindblad master-equation simulations (lines), whose parameters are given in Table \ref{['tableparam']}. For (c), we fit a displacement rate of $\epsilon_b=2\pi\times175$ kHz yielded by the resonant drive applied to the buffer. d) Four-to-one photon exchange rate against pump amplitude, extracted from these simulations. It slightly deviates from the expected linear scaling.
• Figure 4: Four-photon loss channel. Decay rates $\Gamma_n$ of the first four excited states of the memory ($1\leq n\leq 4$) in presence of a pump of varying amplitude. $\Gamma_4$ is estimated through spectral analysis of the model Lindbladian used to reproduce the datasets presented in Fig. \ref{['fig3']}b-c. Following an initial quadratic increase with respect to pump amplitude, it saturates near its maximum theoretical value at half the buffer damping rate (red dashed line). $\Gamma_n$ for $n\leq 3$ is directly obtained from an exponential fit of the time-decay of population prepared in $|n\rangle_a$ (see Sec. \ref{['sec:gammaslow']}). Their increase at large pump amplitude is not captured by our model.
• Figure 5: Optical microscope pictures of the circuit (false colors). The three-pad structure supporting the memory and buffer modes is visible in blue and orange in the zoomed-in picture on the left (corresponding to the purple rectangle in the zoomed-out picture on the right). The readout resonator is colored in green. The width of the center track of the charge line (light blue) and that of the flux lines (mauve) is sinusoidally modulated to prevent radiative decay of the memory (see Fig. \ref{['fig:crystals']}). The lengths of the flux lines are not exactly matched.