Flux-pump induced degradation of $T_1$ for dissipative cat qubits

TL;DR

The paper addresses pump-induced degradation of T1 in dissipative cat qubits realized with ATS-based circuits. It develops an effective master equation via time-dependent Schrieffer-Wolff perturbation theory and validates it against Floquet simulations, revealing drive-dependent parity-breaking single-photon losses and higher-order processes that threaten phase-flip protection. The study identifies dominant loss channels (e.g., linear drive on the memory mode) and demonstrates good agreement between SWPT and Floquet theory for moderate pump powers, providing a spectral and impedance-based understanding of the decoherence pathways. It then proposes mitigation via canceling linear drive terms through careful ATS flux control and by designing the mode frequencies and filtering to suppress pump-induced decays, offering practical guidance for preserving the cat-qubit error bias in experiments and extending the approach to other circuit implementations. The work highlights how to maintain κ_1 ≪ κ_2 to keep the error-correction threshold viable and informs experimental design and parameter tuning for robust dissipative quantum information processing.

Abstract

Dissipative stabilization of cat qubits autonomously corrects for bit flip errors by ensuring that reservoir-engineered two-photon losses dominate over other mechanisms inducing phase flip errors. To describe the latter, we derive an effective master equation for an asymmetrically threaded SQUID based superconducting circuit used to stabilize a dissipative cat qubit. We analyze the dressing of relaxation processes under drives in time-dependent Schrieffer-Wolff perturbation theory for weakly anharmonic bosonic degrees of freedom, and in numerically exact Floquet theory. We find that spurious single-photon decay rates can increase under the action of the parametric pump that generates the required interactions for cat-qubit stabilization. Our analysis feeds into mitigation strategies that can inform current experiments, and the methods presented here can be extended to other circuit implementations.

Figures (13)

• Figure 1: a) Abstract cat-qubit system: a high-Q mode (dark blue) and a non-linear low-Q mode (green) driven to implement a 4-wave coupling (light blue arrow). Wavy lines illustrate parametrically activated 2-1 photon exchange interaction driven at $\omega_p$. Additionally, the 'buffer' $b$ mode (green) might be driven through a weak resonant drive (black arrow). Decaying wavy lines on the right represent the strong dissipation of the buffer mode, along with spurious decays in red at frequency $\omega^*$. b) Galvanically coupled circuit for cat-qubit implementation: green branches form the two flux-driven loops of the ATS (two identical Josephson junctions shunted by a superinductance lescanne_exponential_2020), with colors corresponding to a). The dominant dissipation channel of the buffer is represented by a capacitive coupling to a transmission line.
• Figure 2: Analysis of drive-induced collapse operators in $O(\lambda^8)$ SWPT: Absolute value of the prefactor of the monomial written on the $y$-axis in the collapse operator $C(\omega_i)$ of the effective master equation corresponding to Liouvillian \ref{['eq:L_eff']}, whose frequency $\omega_i$ is given on the $x$-axis. The two solid lines indicate the scaling in $g_2$ or $g_2^2$ of the prefactor, as discussed in the main text. The pump power is defined with the magnitude of the leading parametrically activated term $g_2$ with respect to its maximum value $g_{2}^\textit{max}/2\pi=50.8\text{MHz}$ from \ref{['eq:gs_RWA']}. The $\hat{a}$ coupling features a second-order dependence on pump power. We used an experimental parameter set $\omega_a/2\pi = 4\;\mathrm{ GHz}, \omega_b/2\pi = 7.05\;\mathrm{ GHz}, \varphi_a = 0.11, \varphi_b = 0.2, E_J/h = 37\;\mathrm{ GHz}, E_L/h = 62.4\;\mathrm{ GHz}$ and $\epsilon_d = 5 g_2$ corresponding to a cat state with $|\alpha|=\sqrt{5}$ [\ref{['eq:H2pho']}]. The $x$-axis positions of the bars are set by the frequencies of the charge drive and the flux pump $\omega_d/2\pi = 7.05\;\mathrm{ GHz}$ and $\omega_p/2\pi=0.95\;\mathrm{ GHz}$. The minimum of the $y$-axis is $10^{-7}$. For simplicity, we set $u=0$. We find that for $\epsilon_d = 0$ the difference of the absolute value of the prefactor of the monomials with the presented ones are smaller than $10^{-4}$, as we detail in \ref{['app:extra_collapse:ed']}
• Figure 3: Transition rate matrices $\Gamma_{i \to f }^{(F)}/ \kappa_b$ in the Born-Markov approximation \ref{['eq:gamma_fl']} versus initial and final state for various pump amplitudes. The Floquet eigenstates are sorted by their mean-value of $\hat{N}_d$,\ref{['eq:Nd']}, on the x and y-axis. a) At zero pump power, the dissipation of the mode $b$ generates transitions that change excitation number $\hat{N}_d$ by $\pm 2$. The photon-number parity of mode $a$ is conserved, since transitions happen only between $N_d$ and $N_d-2$ sectors. b) Increasing the pump power strongly hybridizes the modes within a given $N_d$ sector but approximately conserves photon-number parity of mode $a$. c) At large pump powers the parity-breaking transitions, such as those connecting $\hat{N}_d$ and $\hat{N}_d\pm1$, start to be non-negligible. We further analyze the sectors highlighted in c) in \ref{['fig:G_a-0_eps']}. Parameters as in \ref{['fig:Collapse_monom']}.
• Figure 4: Transition rates corresponding to leading parity-breaking monomials identified in \ref{['fig:Collapse_monom']}$\hat{a}\hat{b}$, $\hat{a}$ and $\hat{a}^\dagger \hat{b}$ from right to left. As explained in the text, each column corresponds to one or multiple transitions between a pair of $N_d$ sectors [colored rectangles in \ref{['fig:Fl-tran-matrix']}c)]. The top line compares the transition rates computed within Floquet-Markov theory \ref{['eq:gamma_fl']} and the effective transition rates from \ref{['eq:gamma_rwa']} when increasing the pump power. The rates are computed between the tracked eigenstates of the system. The orange dotted line in the panel $N_d:1\to0$ corresponds to the threshold of the repetition code \ref{['eq:tsep']}. The panels on the bottom show the partial impedance associated with the transitions on the above panels (see \ref{['app:abs_spect']}). For comparison with the spectral features of \ref{['fig:Collapse_monom']}, the $y$-axis was labeled with the predicted frequencies of the collapse operators.
• Figure 5: Leading effective rates identified in \ref{['fig:Collapse_monom']} versus $\eta_p/\epsilon_p$. Sweeping the ratio $\eta_p/\epsilon_p$ is equivalent to sweeping $\mathrm{g}_{1,1}$. At the cancellation point the effective rate in $\hat{a}$ is significantly reduced. This is not true for all the effective rates identified in \ref{['fig:Collapse_monom']}. The dashed line represents the cancellation condition \ref{['eq:cancel_cond']} in the limit $\epsilon_p \ll \pi$. The parameters are $E_{L\eta}^\textit{eff}/h = 62.4~\mathrm{GHz}$ and $E_{L\epsilon}^\textit{eff}=0$