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Lecture notes in physics

Irreversibility in the Framework of Hermitian and Non-Hermitian Treatments of Resonance States

eanthes A. Nicolaides

Published Jan 1, 2003• 3 citations
Publisher
DOI
10.1007/3-540-44874-8_19

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The many-mode Floquet theory (MMFT) recently developed by authors is extended to incorporate the irreversible damping mechanisms for the nonperturbative treatment of the dynamical evolution of dissipative three-level systems at two-photon or multiphoton coherent resonance trapping conditions induced by two strong linearly polarized monochromatic fields. It has been recently shown by several workers that under the rotating-wave approximation (RWA), population may be permanently trapped in the three-level system if the coherent monochromatic fields are exactly two-photon resonant with the initial and final states, decoupled from the intermediate decaying level. In practice, the inclusion of the non-RWA terms necessarily modifies the resonant trapping conditions and behavior. In this paper we extend the generalized Van Vleck (GVV) nearly degenerate perturbation theory to an analytical treatment of the non-Hermitian two-mode Floquet Hamiltonian. This reduces the infinite-dimensional time-independent non-Hermitian Floquet Hamiltonian to a 3\ifmmode\times\else\texttimes\fi{}3 effective Hamiltonian, from which essential properties of the coherent population-trapping behavior as well as the dynamical evolution of the dissipative SU(3) coherence vector S\ensuremath{\rightarrow}(t) can be readily obtained and expressed in terms of only three complex quasienergy eigenvalues and eigenvectors. The MMFT-GVV studies show that the RWA two-photon resonant trapping condition is substantially modified by the effects of non-RWA terms, and that the system can be ``quasitrapped'' for only a finite amount of time characterized by a small imaginary energy (width) associated with a coherent superposition state of the initial and final levels. Furthermore, it is found that the initially eight-dimensional coherence vector S\ensuremath{\rightarrow}(t) evolves predominantly to a one-dimensional scalar at the two-photon or multiphoton resonant quasitrapping conditions. Detailed results and pictorial representations of the population trapping and SU(3) dissipative dynamical evolution are presented.

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It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical", they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for t > 0 ; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamic ending at a time-reversal symmetric final state is dominated by the anti-resonant states for t < 0 .

arXiv:1903.05227•Entropy•Apr 8, 2019•10 citations
The many-mode Floquet theory (MMFT) recently developed by authors is extended to incorporate the irreversible damping mechanisms for the nonperturbative treatment of the dynamical evolution of dissipative three-level systems at two-photon or multiphoton coherent resonance trapping conditions induced by two strong linearly polarized monochromatic fields. It has been recently shown by several workers that under the rotating-wave approximation (RWA), population may be permanently trapped in the three-level system if the coherent monochromatic fields are exactly two-photon resonant with the initial and final states, decoupled from the intermediate decaying level. In practice, the inclusion of the non-RWA terms necessarily modifies the resonant trapping conditions and behavior. In this paper we extend the generalized Van Vleck (GVV) nearly degenerate perturbation theory to an analytical treatment of the non-Hermitian two-mode Floquet Hamiltonian. This reduces the infinite-dimensional time-independent non-Hermitian Floquet Hamiltonian to a 3\ifmmode\times\else\texttimes\fi{}3 effective Hamiltonian, from which essential properties of the coherent population-trapping behavior as well as the dynamical evolution of the dissipative SU(3) coherence vector S\ensuremath{\rightarrow}(t) can be readily obtained and expressed in terms of only three complex quasienergy eigenvalues and eigenvectors. The MMFT-GVV studies show that the RWA two-photon resonant trapping condition is substantially modified by the effects of non-RWA terms, and that the system can be ``quasitrapped'' for only a finite amount of time characterized by a small imaginary energy (width) associated with a coherent superposition state of the initial and final levels. Furthermore, it is found that the initially eight-dimensional coherence vector S\ensuremath{\rightarrow}(t) evolves predominantly to a one-dimensional scalar at the two-photon or multiphoton resonant quasitrapping conditions. Detailed results and pictorial representations of the population trapping and SU(3) dissipative dynamical evolution are presented.
Abstract We analytically study zero-energy eigenstates for two non-Hermitian models based on the Su–Schrieffer–Heeger (SSH) chain. The first model is an infinite non-Hermitian SSH chain with nonreciprocal hoppings to a neighboring site and two barriers. We find analytical expressions for the eigenfunctions, including those that describe resonant states. For resonant states, an analogue of the non-Hermitian skin effect is observed. For the second model, a finite non-Hermitian SSH chain with nonreciprocal hoppings with an even or odd number of sites, we also find the eigenfunctions and observe the non-Hermitian skin effect. All results are obtained using the derived Green’s function of the infinite non-Hermitian SSH chain.
It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical", they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for t > 0 ; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamic ending at a time-reversal symmetric final state is dominated by the anti-resonant states for t < 0 .