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Journal of Physics A Mathematical and General

Tunnelling process for non-Hermitian systems: the complex-frequency inverted oscillator

Sotirios Baskoutas, A. D. Jannussis, R Mignani, V. Papatheou

Published Sep 7, 1993• 22 citations
Publisher

Abstract

We study the tunnel effect for an inverted oscillator with complex frequency. The solution of the corresponding non-Hermitian (NH) Schrodinger equation is found by the evolution operator method, based on the SU(1, 1) structure of the Hamiltonian and the Wei-Norman theorem. We put forward a generalization of dwell time for NH systems built up from their biorthonormal states. The resulting tunnelling time turns out to be complex.

DOI
10.1088/0305-4470/26/17/012

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We study the tunnel effect for an inverted oscillator with complex frequency. The solution of the corresponding non-Hermitian (NH) Schrodinger equation is found by the evolution operator method, based on the SU(1, 1) structure of the Hamiltonian and the Wei-Norman theorem. We put forward a generalization of dwell time for NH systems built up from their biorthonormal states. The resulting tunnelling time turns out to be complex.
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.
Using the modified biorthonormal Heisenberg equations of motion for non-Hermitian (NH) Hamilton operators, in order to imply a consistent Lie-algebraic structure and also the equivalence between the Heisenberg and Schrödinger pictures, we have obtained the analytical form of the Wigner distribution function which is unavoidable complex. Its imaginary part accounts for the influence of additional degrees of freedom, which are always present in the phenomenological representation of dissipative systems through (NH) Hamiltonians. Applications of the above formalism can be found, for instance, in dissipative macroscopic quantum tunneling (MQT) effect for Josephson junctions, and in the dissipative tunneling of trapped atoms in optical crystals.
Using the algebraic method, we derive the exact wavefunction of a driven time-dependent inverted harmonic oscillator (TDIHO). With the Gaussian wave-packet as the initial state, the general expressions of the transmission probability and dwell time of the driven TDIHO are obtained. For a weak driving force, the dwell time can be separated into two terms. The first term is for the unperturbed part, and the second one is that from the perturbation, which is the effect from the driving force. Furthermore, with two examples, the effect of the driving force on the dwell time is explicitly indicated in detail.
Abstract With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent   -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the  ( t ) PT −inner product, where  ( t ) is a obtained from the charge conjugation operator  through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the  ( t ) PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.
Abstract In this study, we introduce a two-dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time-independent Schrödinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian.