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International Journal of Modern Physics E

QUANTUM TUNNELING IN OPEN SYSTEMS

Eliade N. Stefanescu, A. Sãndulescu, Walter Greiner

Published Mar 1, 1993• 12 citations
Publisher

Abstract

We study the barrier penetrability in the frame of the Lindblad theory of open quantum systems. In addition to the diagonal elements of the density matrix, leading to the Gamow’s formula, new terms, describing energy dissipation and spectral line broadening effects are obtained. It is shown that the presence of a dissipative environment increase the barrier penetrability, in accordance with a very simple physical interpretation: for a system initially found in its ground state the dissipation can lead only to transitions to the reaction channels where lower-energy levels exist.

DOI
10.1142/s0218301393000078

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We study the barrier penetrability in the frame of the Lindblad theory of open quantum systems. In addition to the diagonal elements of the density matrix, leading to the Gamow’s formula, new terms, describing energy dissipation and spectral line broadening effects are obtained. It is shown that the presence of a dissipative environment increase the barrier penetrability, in accordance with a very simple physical interpretation: for a system initially found in its ground state the dissipation can lead only to transitions to the reaction channels where lower-energy levels exist.
The damping of the harmonic oscillator is studied in the framework of the Lindblad theory for open quantum systems. A generalization of the fundamental constraints on quantum mechanical diffusion coefficients which appear in the master equation for the damped quantum oscillator is presented; the Schrödinger, Heisenberg and Weyl-Wigner-Moyal representations of the Lindblad equation are given explicitly. On the basis of these representations it is shown that various master equations for the damped quantum oscillator used in the literature are particular cases of the Lindblad equation and that not all of these equations are satisfying the constraints on quantum mechanical diffusion coefficients. Analytical expressions for the first two moments of coordinate and momentum are obtained by using the characteristic function of the Lindblad master equation. The master equation is transformed into Fokker-Planck equations for quasiprobability distributions and a comparative study is made for the Glauber P representation, the antinormal ordering Q representation, and the Wigner W representation. The density matrix is represented via a generating function, which is obtained by solving a timedependent linear partial differential equation derived from the master equation. Illustrative examples for specific initial conditions of the density matrix are provided. The solution of the master equation in the Weyl-Wigner-Moyal representation is of Gaussian type if the initial form of the Wigner function is taken to be a Gaussian corresponding (for example) to a coherent wavefunction. The damped harmonic oscillator is applied for the description of the charge equilibration mode observed in deep inelastic reactions. For a system consisting of two harmonic oscillators the time dependence of expectation values, Wigner function and Weyl operator, are obtained and discussed. In addition models for the damping of the angular momentum are studied. Using this theory to the quantum tunneling through the nuclear barrier, besides Gamow’s transitions with energy conservation, additional transitions with energy loss are found. The tunneling spectrum is obtained as a function of the barrier characteristics. When this theory is used to the resonant atom-field interaction, new optical equations describing the coupling through the environment of the atomic observables are obtained. With these equations, some characteristics of the laser radiation absorption spectrum and optical bistability are described.
Quantum tunnelling induces transitions not only through potential barriers, but also through energy barriers of dynamical origin. The coupling to a noisy environment introduces dissipation which affects tunnelling. This effect is studied by an analytic non-perturbative solution of a problem involving tunnelling through a dynamical barrier in the presence of a noisy environment. The accessibility to the classically forbidden energy region above the dynamical barrier is reduced (enhanced) by the friction(diffusion) effect. The occupation of each level above the dynamical barrier is characterized by the succession in time of two different regimes. The transition between both regimes is ruled by an energy criterion which is discussed.
We discuss some aspects of the time picture of tunneling for open quantum systems described by non-Hermitian (NH) Hamiltonians. The concept of sojourn time for such systems is introduced in the framework of the biorthonormal formalism. Due to the various definitions of probability density in the non-Hermitian case, we get three different sojourn times, two real and one complex. We consider as model of a dissipative NH system the complex, generalized parametric oscillator, for which we derive the exact expressions of the three sojourn times in terms of the Wei-Norman characteristic functions entering the non-unitary evolution operators. The special case of the inverted Caldirola-Kanai oscillator with complex friction parameter is investigated for an initial extended wavepacket. We also discuss the Landau-Zener-like transitions of the NH parametric oscillator, i.e. the dissipative tunneling through a dynamical barrier due to the perturbative effect of the damping.
We develop a theory of adiabatic response for open systems governed by Lindblad evolutions. The theory determines the dependence of the response coefficients on the dephasing rates and allows for residual dissipation even when the ground state is protected by a spectral gap . We give the quantum response a geometric interpretation in terms of Hilbert space projections: for a two-level system and, more generally, for systems with a suitable functional form of the dephasing, the dissipative and non-dissipative parts of the response are linked to a metric and to a symplectic form. The metric is the Fubini–Study metric and the symplectic form is the adiabatic curvature. When the metric and symplectic structures are compatible , the non-dissipative part of the inverse matrix of response coefficients turns out to be immune to dephasing . We give three examples of physical systems whose quantum states induce compatible metric and symplectic structures on control space: qubit, coherent states and a model of the integer quantum Hall effect.
For a tunneling system we consider the coupling to the dissipative environment of the coordinate q, the momentum p and the potential function U(q). In the frame of the quantum theory of open systems we derive a quantum master equation, fundamental constraints and uncertainty relations. We obtain an analytical expression of the tunneling spectrum and show that at thermal equilibrium its shape entirely depends on the barrier characteristics while its amplitude is proportional to the strength of the dissipative coupling.