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arXiv:1810.11376 · Optik

A comparative study on different non-Hermitian approaches for modeling open quantum systems

Santiago Echeverri-Arteaga, Herbert Vinck-Posada, Edgar A. Gómez

Published Nov 29, 2018• 5 citations
arXivHTMLPDF
DOI
10.1016/j.ijleo.2018.11.133

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We compare two approaches to open quantum systems, namely, the non-Hermitian dynamics and the Lindblad master equation. In order to deal with more general dissipative phenomena, we propose the unified master equation that combines the characteristics of both of these approaches. This allows us to assess the differences between them as well as to clarify which observed features come from the Lindblad or the non-Hermitian part, when it comes to experiment. Using a generic two-mode single-atom laser system as a practical example, we analytically solve the dynamics of the normalized density matrix operator. We study the two-level model in a number of cases (depending on parameters and types of dynamics), compute different observables and study their physical properties. It turns out that one can not only able to describe the different types of damping in dissipative quantum optical systems but also mimic the undamped anharmonic oscillatory phenomena which happen in quantum systems with more than two levels (while staying within the framework of the analytically simple two-mode approximation).
In this work, taking the most general non-Hermitian Hamiltonian without parity-time symmetry and with non-Hermitian off-diagonal elements into account, we study the exceptional points of the model system. We find that there exists an exceptional ellipse in theory, compared with the exceptional ring studied in non-Hermitian symmetry Hamiltonian, in which the parameters in the model Hamiltonian take particular values. We also report on the dynamics of the model Hamiltonian and symmetry Hamiltonian. We find that they display different dynamical behaviors under different parameters. Moreover, there exists an unstable behavior of the system. The unstable behavior has a surprising property by which the gain always dominates at long times no matter how small the gain is. This phenomenon enables the compensation for losses in dissipative systems and opens a wide range of applications in quantum optics, plasmonics, and optoelectronics, where the loss is inevitable but can be compensated by the gain.
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.