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Progress of Theoretical Physics

The Quantum Mechanical Tunneling under the Influence of Environment: Using the Method of Density Matrix

Hiroshi Kawamoto

Published Jun 1, 1984• 2 citations
Publisher

Abstract

The purpose of this paper is to study an effect of an environment on a quantum tunneling. Here, the system we consider is of a symmetric two-level type with “bare” tunneling frequency Δω0, and the correlation time of actions due to the environment, τc, is assumed to be much shorter than Δω0-1. In the investigations, the method of density matrix is used, and a coupling between the system and the environment is included up to lowest nontrivial order. After calculations, we reach the following conclusions: First, the possibility of the quantum tunneling is, in a high temperature range βℏΔω0 ≪1, washed out by the thermal actions from the environment. In the inverse case βℏΔω0 ≪1, where the quantum fluctuation of the environment plays a dominant role, the possibility of the quantum tunneling remains. Secondaly, in the case βℏΔω0 ≪1, the dissipation effect depresses the tunneling probability, while the potential barrier is effectively lowered by the actions due to the environment.

DOI
10.1143/ptp.71.1142

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The purpose of this paper is to study an effect of an environment on a quantum tunneling. Here, the system we consider is of a symmetric two-level type with “bare” tunneling frequency Δω0, and the correlation time of actions due to the environment, τc, is assumed to be much shorter than Δω0-1. In the investigations, the method of density matrix is used, and a coupling between the system and the environment is included up to lowest nontrivial order. After calculations, we reach the following conclusions: First, the possibility of the quantum tunneling is, in a high temperature range βℏΔω0 ≪1, washed out by the thermal actions from the environment. In the inverse case βℏΔω0 ≪1, where the quantum fluctuation of the environment plays a dominant role, the possibility of the quantum tunneling remains. Secondaly, in the case βℏΔω0 ≪1, the dissipation effect depresses the tunneling probability, while the potential barrier is effectively lowered by the actions due to the environment.
The influence of a dissipative environment on scattering of a particle by a barrier is investigated by using the recently introduced bohmian mechanics with complex action [J. Chem. Phys. 125, 231103 (2006)]. An extension of this complex trajectory based formalism to include the interaction of the tunneling particle with an environment of harmonic oscillators with a continuous spectral density and at a certain finite temperature allows us to calculate transmission probabilities beyond the weak system bath coupling regime. The results display an increasing tunneling probability for energies below the barrier and a decreased transmission above the barrier due to the coupling. Furthermore, we demonstrate that solutions of a markovian master equation fail to do so in general.
An investigation as to the possibility of the quantum phase coherence in a circuit containing the Josephson junction is made. In the study the primary part of the system is assumed to be a two-state system, and so the quantum oscillation between two states is expected. For an analysis of the influence exerted by an environment two types of environments are considered. One type has the Lorentzian spectral density and the other has the Ohmic one. In both types a total loss of the phase coherence is found when a strength of a coupling between the primary part of the system and the environment, α ≡φ2 / \hbarRN(φ ≃φ0), satisfies a condition √\mathstrutα ≫Δω0 τc. Here \hbar Δω0 is a level splitting and τc a relaxation time of the environments. Furthermore, we find that the total loss of the phase coherence leads to a quantum tunneling governed by a Poisson process with a tunnelin rate Ω2 / Γ (Ω is the renormalized natural frequency and Γ the damping constant). It must be emphasized that the Ohmic spectral density of the environments gives rise to an instability in the primary part when the coupling strength α (√α ≫ Δω0 τc) exceeds some critical value.
The problem of metastability due to quantum and thermal fluctuations in a weakly damped system is treated within Feynman's influence-functional theory. It is shown that quite generally the nonequilibrium density matrix may be written as a single path integral. For a cubic metastable potential it has the form of a Feynman propagator with a complex Lagrangian containing second-order derivatives in time and terms noninvariant under time reversal. Dissipative tunneling and thermal activation emerge together as different solutions of the associated Lagrange equations. The low-temperature correction to the exponent of the tunneling rate behaves as Tn+1{T}^{n+1}Tn+1 for a spectrum of bath excitations vanishing as ${\ensuremath{\omega}}^{n}$. It is shown, moreover, how this result may be understood in terms of environment-assisted tunneling. Thus the low-temperature behavior of the tunneling rate provides a method to measure the dissipation mechanism in the quantum regime.
A quantum system which can tunnel, at T=0T=0T=0, out of a metastable state and whose interaction with its environment is adequately described in the classically accessible region by a phenomenological friction coefficient $\ensuremath{\eta}$, is considered. By only assuming that the environment response is linear, it is found that dissipation multiplies the tunneling probability by the factor $\mathrm{exp}[\ensuremath{-}\frac{A\ensuremath{\eta}{(\ensuremath{\Delta}q)}^{2}}{\ensuremath{\hbar}}]$, where $\ensuremath{\Delta}q$ is the "distance under the barrier" and AAA is a numerical factor which is generally of order unity.
In many physical situations the behavior of a quantum system is affected by interaction with a larger environment. We develop, using the method of an influence functional, how to deduce the density matrix of the quantum system incorporating the effect of environment. After introducing the characterization of the environment by spectral weight, we first devise schemes to approximate the spectral weight, and then a perturbation method in field theory models, in order to approximately describe the environment. All of these approximate models may be classified as extended Ohmic models of dissipation whose differences are in the high frequency part. The quantum system we deal with in the present work is a general class of harmonic oscillators with an arbitrary time-dependent frequency. The late time behavior of the system is well described by an approximation that employs a localized friction in the dissipative part of the correlation function appearing in the influence functional. The density matrix of the quantum system is then determined in terms of a single classical solution obtained with the time-dependent frequency. With this one can compute the entropy, the energy distribution function, and other physical quantities of the system in a closed form. A specific application is made to the case of a periodically varying frequency. This dynamical system has a remarkable property when the environmental interaction is switched off: The effect of the parametric resonance gives rise to an exponential growth of the populated number in higher excitation levels, or particle production in field theory models. The effect of the environment is investigated for this dynamical system and it is demonstrated that there exists a critical strength of the friction for the parametric effect. In the model of a periodically oscillating field coupled to a system quantum field, it is verified that the parametric effect occurs in medium, with a somewhat diminished rate, if the relaxation time scale of the system field towards thermalization given by the friction term is larger than the time scale of the coherent parametric amplification. The effect persists until the back reaction against the periodic oscillation stops particle production. The resulting energy distribution of produced particles described by a universal function deviates from the thermal one, having an average energy that exponentially increases with time. The dynamical system driven by the parametric oscillator thus maintains and does not lose its quantum nature even in thermal bath. In the present work analytic formulas of how physical quantities behave at late times both in the high and in the low temperature regions are given, along with the results of a numerical computation displaying time evolution.