Echoes in a parametrically perturbed Kerr-nonlinear oscillator

Abstract

We study classical and quantum echoes in a Kerr oscillator driven by a frequency-controlling pulsed perturbation. We consider dynamical response to the perturbation for a single coherent state and for Schrödinger cat states constructed as both balanced and imbalanced superpositions of two coherent states. For individual coherent states, we demonstrate that a weak parametric drive yields a long-lived sequence of classical echoes. Cat states are found to exhibit distinct quantum echoes that are sensitive to the initial relative phase and weights of the coherent states in superposition. We examine the effect of dissipation on quantum echoes and quantum revivals of cat states. We demonstrate that, even when dissipation suppresses quantum revivals, quantum echoes can be recovered by properly tuning the timing and strength of the perturbation. These results may be useful for characterizing and mitigating errors of cat qubits.

Figures (4)

• Figure 1: (a-d) Phase-space snapshots showing the evolution of $2\times10^{5}$ classical oscillators. (a) Early stages of filamentation where the initial distribution is a Gaussian centered at $(q_0,0)$, corresponding to a coherent state $\ket{\alpha_0}$, with $\alpha_0=q_0/\sqrt{2} = 6$. (b) Well-developed filaments at $t=\tau=0.5$. (c) Phase-space distribution of the free oscillators at $t = 2\tau$. (d) Distribution at the echo time ($t = 2\tau$) following a kick at $t = \tau$. The magnified image in (d) highlights thickening filament segments around $q = q_0$ in contrast to (c). (e,f) Average position as a function of time. (e) Free oscillators. (f) Kicked oscillators exhibit echoes at $t = 2n\tau$ ($n\in\mathbb{Z}^+$). Values of the parameters in Eq. \ref{['eq:dimensionless-quantum-H']} are $\chi=1$, $g_0=0.01$. The pulse width in numerical simulations is $\sigma_g=10^{-3}$.
• Figure 2: Quantum position expectation value as a function of time for the (a) free and (b) kicked Kerr oscillator. The initial state is a coherent state, $\ket{\alpha_0}$ with $\alpha_0 = 6$. The values of the parameters for Eqs. \ref{['eq:dimensionless-quantum-H']} and \ref{['eq:quantum-<q>-kicked']} are $\chi=1$, $\tau=2\pi/19\approx0.33$, and $g_0=0.01$. The pulse width in numerical simulations is $\sigma_g=10^{-3}$.
• Figure 3: (a,b) Position expectation values of kicked asymmetric cat states ($\mathcal{N}_+^2 = 1 - \mathcal{N}_-^2 = 0.8$, $\alpha_0=6$). (c,d) Amplitudes of the first classical and quantum echoes as a function of the initial cat state parameters, $\mathcal{N}_+$ and $\theta$. Other fixed parameters in Eq. \ref{['eq:dimensionless-quantum-H']}: $\chi=1$, $\tau=0.27$, $g_0=0.01$.
• Figure 4: Position expectation value as a function of time for (a) a free and (b) a kicked asymmetric cat state ($\mathcal{N}_+^2=1-\mathcal{N}_-^2=0.8$, $\alpha_0=6$, $\theta=\pi/2$) with or without dissipation. Common parameters in Eq. \ref{['eq:master-equation']}: $\chi=1$, damping constant $\gamma=0.03$, and dimensionless temperature $\epsilon^{-1}=1$. Kick parameters in (b): $\tau=0.27$, $g_0=0.03$.