Unraveling the switching dynamics in a quantum double-well potential

TL;DR

The paper tackles spontaneous switching between the wells of a quantum double-well realized in a two-photon-driven Kerr nonlinear oscillator, explaining the experimentally observed staircase pattern as the drive amplitude $\epsilon_2$ increases. It develops a semi-analytical framework by projecting the density operator onto two-level manifolds, deriving an effective Lindbladian, and solving for inter-well dynamics through a sequence of adiabatic eliminations that yield a closed formula for the switching rate $\Gamma$. The main result is a decomposition $\Gamma = \sum_n \Gamma_n$, with each excited-state manifold contributing via tunneling suppressed by a quantum Zeno effect and controlled by dephasing $\mu_n$ and decay $\lambda_n$, as well as inter-manifold currents $V_{pq}$ and $W_{pq}$; the staircase steps occur at drive values where the branching ratio $f_n$ crosses 1/2, which can be estimated from the condition $\delta_n^2(\alpha)/\mu_n(\alpha) = \lambda_n(\alpha)$ using WKB tunnel splittings $\delta_n(\alpha)$. The analysis further shows that activation into the excited manifolds is dominated by direct and cascaded thermal heating at moderate to low temperatures, while a novel non-Hermitian–induced quantum heating emerges at very low temperatures. Overall, the work deepens understanding of metastable-state switching in quantum multi-stable systems and informs design considerations for Kerr-cat qubits and related dissipative quantum devices.

Abstract

The spontaneous switching of a quantum particle between the wells of a double-well potential is a phenomenon of general interest to physics and chemistry. It was broadly believed that the switching rate decreases steadily with the size of the energy barrier. This view was challenged by a recent experiment on a driven superconducting Kerr nonlinear oscillator (often called the Kerr-cat qubit or the Kerr parametric oscillator), whose energy barrier can be increased by ramping up the drive. Remarkably, as the drive amplitude increases, the switching rate exhibits a step-like decrease termed the "staircase". The view challenged by the experiment demands a deep review of our understanding of quantum effects in double wells. In this work, we derive a semi-analytical formula for the switching rate that resolves a continuous transition between tunneling- and dissipation-dominated dynamics. These two dynamics are observed respectively in the flat and the steep parts of each step in the staircase. Our formula exposes two distinct dissipative processes that limit tunneling: dephasing and decay. This allows us to predict the critical drive amplitudes where steps occur. In addition, we show that in the regime of a few states in the well and under moderate to low temperatures, highly excited states are populated predominantly via cascaded and direct thermal heating rather than quantum heating. At very low temperatures, however, the perturbation induced by the nonhermitian Hamiltonian becomes important and facilitates a new form of quantum heating. We numerically map the activation mechanism as a function of drive amplitude, damping rate, and temperature. Our theory deepens the understanding of switching dynamics between metastable quantum states, highlights the importance of a general interplay between tunneling and dissipation, and identifies a novel quantum regime in activated transitions.

Figures (20)

• Figure 1: The double-well shapes the spectrum of the Hamiltonian $\hat{H}$ defined in Eq. \ref{['eq:hamiltonian']}. (a) The double-well $U(\beta) = -\bra{\beta}\hat{H}\ket{\beta}$, defined in Eq. \ref{['eq:double-well']}, in the phase space of the oscillator in the rotating frame. The depth of the well (black arrows) is $\epsilon_2^2/K$. Spontaneous switching (red arrows) reduces the inter-well population difference. The 2D projection below the double well shows the energy contours. Black dots indicate the minimum points of the double-well, $\beta = \pm \alpha$. (b) The energies $E_n^+$ of even parity excited states $\ket{\psi_n^+}$ (solid lines) and $E_n^-$ of odd parity excited states $\ket{\psi_n^-}$ (dashed lines) as functions of the two-photon drive amplitude $\epsilon_2$, where $E_n^{\pm}$ are defined by Eq. \ref{['eq:energies']}. The shaded region indicates states for which $U(\alpha) \leq -\bra{\psi_n^\pm}\hat{H}\ket{\psi_n^\pm} \leq U(0)$. (c) The excited state manifold tunnel splitting $\delta_n$, defined in Eq. \ref{['eq:tunnel-splitting-definition']}, as a function of the two-photon drive amplitude $\epsilon_2$. $\delta_n$ becomes exponentially small when the manifold $n$ drops into the double well. The states inside the well and the values of the tunnel splittings are important to determining the spontaneous switching rate.
• Figure 2: The switching rate and the staircase. (a) The population difference $\langle\hat{X}\rangle$ [see Eq. \ref{['eq:definition-of-Gamma-0']}] as a function of time, obtained via numerical simulation of the master equation Eq. \ref{['eq:lindblad']}. Insets show the population distribution over position $\hat{x} \propto \hat{a}^\dagger + \hat{a}$ at select times. No matter whether the initial state is $\ket{+\alpha}$ (solid line) or $\ket{-\alpha}$ (dashed line), the inter-well population difference decays exponentially over time. (b) We plot the numerically obtained spontaneous switching rate $\Gamma$ as a function of the two-photon drive amplitude $\epsilon_2$. The plot shows a staircase pattern. As $\epsilon_2$ increases, the double-well captures more quantized states (bottom schematics), and the spontaneous switching rate $\Gamma$ (red curves) decreases abruptly at several critical $\epsilon_2/K$ values that shift with the damping rate $\kappa$ (as well as with $n_{\text{th}}$, not shown in the plot). The red dot indicates the parameters used in (a).
• Figure 3: Approximating the system state as a mixture between two-level manifolds. (a) Illustration of the projection of the system density operator $\hat{\rho}$ onto a series of two-level manifolds, as described by Eq. \ref{['eq:projection']}. The matrix elements of $\hat{\rho}$ under the eigenbasis $\ket{\psi_n^\pm}$, with $\ket{\psi_n^\pm}$ defined by Eq. \ref{['eq:energies']}, are ordered by the manifold number $n$ and the parity $\pm$ as indicated above the matrix. The projection preserves the blue matrix elements (all populations and in-manifold coherences), but removes the light grey ones (cross-manifold coherences) which are perturbatively small. As a consequence of the projection of $\hat{\rho}$, the Lindbladian $\mathcal{L}$ defined in Eq. \ref{['eq:lindblad']} can be approximated by the effective Lindbladian defined in Eq. \ref{['eq:effective-lindbladian']}, which operates on the subspace formed by the blue matrix elements only. (b) Right- and left-well states $\ket{\psi_n^{R/L}}$ defined in Eqs. \ref{['eq:right-wellstate']} and \ref{['eq:left-wellstate']}. (c) The definition of the Bloch sphere for manifold $n$. The axes are consistent with the Pauli operators $\hat{X}_n, \hat{Y}_n, \hat{Z}_n$ defined in Eqs. \ref{['eq:pauli-x']}, \ref{['eq:pauli-z']}, and \ref{['eq:pauli-y']}. Together with the projector $\hat{I}_n$ defined in Eq. \ref{['eq:projector']}, the Pauli operators form a basis for operators supported on manifold $n$.
• Figure 4: Entanglement with the environment measures "which-well" information. (a) The wave function of some inter-well superposition state created by tunneling. The black arrows indicate that under the interaction Hamiltonian Eq. \ref{['eq:interaction-hamiltonian']}, the components of the wavefunction in the right and left wells drive the environment in opposite directions (b) and (c). The two squiggly red arrows indicate orthogonal radiation states of the environment. The straight red arrows and circular red arrows indicate possible transitions that may accompany this radiation. After tracing out the environment, the evolution becomes dissipative, and the wavefunction collapses to either the right (b) or the left well (c).
• Figure 5: Illustration of a manifold transitioning from tunneling-dominated to dissipation-dominated. The quantum Zeno effect prevents dissipation-dominated manifolds from contributing significantly to the total spontaneous switching rate $\Gamma$. As the two-photon drive amplitude $\epsilon_2$ increases, a manifold (in the pictured example, the one with $n = 2$) can abruptly transition from being tunneling-dominated to being dissipation-dominated, leading to a step-like decrease in the spontaneous switching rate. Note that while tunneling typically refers to the penetration of a classically forbidden region, we use tunneling to refer to any coherent inter-well population transfer due to non-zero $\delta_n$ because the boundary between states below and states above the semiclassical energy barrier is fuzzy.