Non-Hermitian Tunneling of Open Quantum Systems

Abstract

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.

Embedding quantum systems with a non-conserved probability in classical environments

“…A first general book on the topic has appeared [6]; applications of nonHermitian quantum mechanics involve the study of scattering by complex potentials and quantum transport [7][8][9][10][11][12][13][14][15][16][17], description of metastable states [18][19][20][21][22][23], optical waveguides [24][25][26], multi-photon ionization [27][28][29], and nano-photonic and plasmonic waveguides [30]. The theoretical investigations are also undergoing rapid developments: non-Hermitian quantum mechanics has been investigated within a relativistic framework [31] and it has been adopted by various researchers as a means to describe open quantum systems [32][33][34][35][36][37][38][39][40][41][42]. Moreover, it seems that a few theoretical studies have been dedicated to the statistical mechanics and dynamics of systems with non-Hermitian Hamiltonians [43][44][45][46][47][48][49]...…”

Quantum entropy of systems described by non-Hermitian Hamiltonians

“…It is easy to check that this state is pure at p = 0, 1 and mixed otherwise. Solving the evolution equations (5), with the initial condition (27), we obtain the following expression for the non-normalized density operator…”

Time as Quantum Observable, Canonical Conjugated to Energy

“…(Angelopoulon et al, 1995)), other rather simple approaches, based of course on the Schrödinger equation…”