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Journal de physique

Excited-state-induced subbarrier tunnelling in the double-well model

Vladislav Čápek

Published Jan 1, 1989• 6 citations
Publisher

Abstract

Standard minimal two-state Hamiltonian (spin-boson model) for a particle in a symmetric double-well potential, interacting with a bath, is complemented by a third particle-excited state. This state is dynamically coupled (via coupling to the bath) with the lowest two states. In addition to the standard over-barrier thermally-activated transfer 1 ↔ 3 ↔ 2 (which might be easily suppressed by choosing a sufficiently low T), the third state is shown to cause excited-state-induced subbarrier tunnelling which persists even at T = 0. This new channel shows a tendency to coherent oscillations if standard (direct-tunnelling and over-barrier-transfer) channels get suppressed. These conclusions challenge the standard theory of the asymptotic symmetry breaking.

DOI
10.1051/jphys:01989005008088700

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Standard minimal two-state Hamiltonian (spin-boson model) for a particle in a symmetric double-well potential, interacting with a bath, is complemented by a third particle-excited state. This state is dynamically coupled (via coupling to the bath) with the lowest two states. In addition to the standard over-barrier thermally-activated transfer 1 ↔ 3 ↔ 2 (which might be easily suppressed by choosing a sufficiently low T), the third state is shown to cause excited-state-induced subbarrier tunnelling which persists even at T = 0. This new channel shows a tendency to coherent oscillations if standard (direct-tunnelling and over-barrier-transfer) channels get suppressed. These conclusions challenge the standard theory of the asymptotic symmetry breaking.
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We identify a non-Hermitian counterpart of dynamical tunnelling induced by a localised absorbing region in a symmetric confining potential. The absorbing region, described by an imaginary potential barrier, induces an effective double-well structure with a pair of long-lived states that have little population in the barrier. In contrast to the periodic tunnel oscillations through a real barrier, the system slowly evolves towards equal population of the two sides, while the absorbing region remains only weakly populated. We demonstrate this generic mechanism in two continuous potentials, and capture its essence analytically in a minimal 3×33\times 33×3 matrix model. We propose a realistic optical waveguide setup for the experimental observation of the effect.
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 as a function of 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 continuously increased by ramping up the drive. Remarkably, as the drive amplitude increases, the switching rate exhibits a steplike decrease termed the ``staircase.'' The view challenged by the experimental staircase demands a deep review of our understanding of the role of quantum effects in double wells. In this work, we use a Lindbladian model of dissipation to derive a semianalytical formula for the switching rate, resolving a continuous transition between tunneling-dominated dynamics and dissipation-dominated dynamics. These two dynamics are observed respectively in the flat part and the steep part of each step in the staircase. Our formula exposes two distinct dissipative processes that limit tunneling: the dephasing of interwell superpositions freezes tunneling via the quantum Zeno effect, and the decay from one excited state to another limits the time during which tunneling can be attempted. This physical understanding allows us to pinpoint the critical drive amplitude that separates the flat and steep part of each step using an equation involving the rates of the two dissipative processes and the tunnel splitting. In addition, analyzing the transition matrix elements in the formula shows that in the regime of a few ($\ensuremath{\lesssim}10$) 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, we find that the perturbation induced by the non-Hermitian Hamiltonian part of the Lindbladian model becomes increasingly important and facilitates a form of quantum heating that has not been identified before. 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.
There has been considerable interest recently in the low temperature dynamics of condensed phase tunneling phenomena. In this paper we consider the interplay between quasiparticle transport and vibrational relaxation; the former taking place via tunneling in a double well potential, and the latter occurring due to interactions of the tunneling system with a harmonic bath. Taking the system–bath interactions to be linear in the bath coordinates, and explicitly allowing for a vibrationally excited well, we present a unified treatment of the weak and strong coupling regimes and obtain reduced equations of motion for the tunneling particle position operator. Solutions are obtained for several important limiting cases. In particular, we find that at sufficiently low temperatures, the dynamical behavior strongly resembles that of multisite spin jump model.
We study the dynamics of a particle tunneling between the ground states of a symmetric double potential well system, in the presence of simultaneous diagonal and non-diagonal couplings to an Ohmic oscillator bath. We use the noninteracting-blip approximation to investigate coherence effects across a wide range of system-environment coupling strengths at physiological temperatures. We show how the presence of a non-diagonal coupling mechanism significantly alters the dynamics of the tunneling particle, producing coherent oscillations despite strong thermal fluctuations in the environment. Fluctuations in both the particle's polarization, as well as tunneling energies, lead to competing influences on the coherent nature of tunneling particle dynamics, with non-diagonal couplings introducing a relatively long-lived oscillatory mode in an otherwise incoherent setting.
We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well). We start from the full many-body Schr{ö}dinger Hamiltonian in a large-filling situation where the on-site interaction and kinetic energies are comparable. When tunneling is negligible against interaction energy, we prove a localization estimate showing that the particle number fluctuations in each well are strongly suppressed. The modes in which the particles condense are minimizers of nonlinear Schr{ö}dinger-type functionals.