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Physical Review A

Nonadiabatic effects in two-level systems: A classical analysis

Marc Boiron, M. Lombardi, Laurent Wiesenfeld

Published Aug 1, 1994• 4 citations
Publisher

Abstract

Our aim in this paper is to study classical dynamics in two-level molecular systems. We first derive, through the Wigner phase-space transform, a classical limit that reduces unfortunately to the adiabatic approximation. This lead us to develop a one-center quantal approximation whose variables can be interpreted in a fully classical Hamiltonian scheme. As a prominent feature, this Hamiltonian couples polarization and spatial motion. We apply it successfully to the Rosenthal-St\"uckelberg oscillations. We analyze one-dimensional diffusion similar to a molecular reaction, it appears to exhibit chaotic behavior.

DOI
10.1103/physreva.50.1409

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Our aim in this paper is to study classical dynamics in two-level molecular systems. We first derive, through the Wigner phase-space transform, a classical limit that reduces unfortunately to the adiabatic approximation. This lead us to develop a one-center quantal approximation whose variables can be interpreted in a fully classical Hamiltonian scheme. As a prominent feature, this Hamiltonian couples polarization and spatial motion. We apply it successfully to the Rosenthal-St"uckelberg oscillations. We analyze one-dimensional diffusion similar to a molecular reaction, it appears to exhibit chaotic behavior.
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The Wigner phase space formulation of quantum mechanics is a complete framework for quantum dynamic calculations that elegantly highlights connections with classical dynamics. In this series of two articles, building upon previous efforts, we derive the full hierarchy of approximate semiclassical (SC) dynamic methods for adiabatic and non-adiabatic problems in Wigner phase space. In Paper I, focusing on adiabatic single surface processes, we derive the well-known double Herman-Kluk (DHK) approximation for real-time correlation functions in Wigner phase space and connect it to the linearized SC (LSC) approximation through a stationary phase approximation. We exploit this relationship to introduce a new hybrid SC method, termed Adiabatic Hybrid Wigner Dynamics (AHWD) that allows for a few important "system" degrees of freedom (dofs) to be treated at the DHK level, while treating the rest of the dofs (the "bath") at the LSC level. AHWD is shown to accurately capture quantum interference effects in models of coupled oscillators and the decoherence of vibrational probability density of a model I2 Morse oscillator coupled to an Ohmic thermal bath. We show that AHWD significantly mitigates the sign problem and employs reduced dimensional prefactors bringing calculations of complex system-bath problems within the reach of SC methods. Paper II focuses on extending this hybrid SC dynamics to nonadiabatic processes.
An analysis of the semiclassical regime of the quantum-classical transition is given for open, bounded, one-dimensional chaotic dynamical systems. Environmental fluctuations-characteristic of all realistic dynamical systems-suppress the development of a fine structure in classical phase space and damp nonlocal contributions to the semiclassical Wigner function, which would otherwise invalidate the approximation. This dual regularization of the singular nature of the semiclassical limit is demonstrated by a numerical investigation of the chaotic Duffing oscillator.
We explore the dynamics of mixed classical-quantum Hamiltonians, a class of systems that frequently appears in the study of physical systems characterized by both slow and fast modes. Traditionally such systems are studied within the adiabatic approximation and as a rule they display level crossings. We show that level crossings can have peculiar effects on the dynamics and, in the regime between the adiabatic and diabatic limits in both classical and quantum approaches, the full dynamics has a series of non-intuitive features.
We systematically investigate master equations involving double-bracket dissipators in the classical limit. Dissipators defined by double commutators arise naturally in dephasing dynamics, while those defined by double anticommutators are typically associated with noisy Hamiltonians. The classical limit of such master equations is derived by reformulating the dissipative open dynamics in phase space using the Wigner-Weyl transform and Moyal bracket formalism, and by identifying the leading-order terms in a systematic ℏ\hbarℏ-expansion. We first analyze a double-commutator master equation associated with energy diffusion, and then turn to master equations containing a double anticommutator with the system Hamiltonian, recently derived in the context of noisy non-Hermitian systems. For both classes of double-bracket equations, we establish a gradient-flow representation of the dynamics within both the quantum and semiclassical formalisms. We further study the semiclassical evolution generated by double-bracket master equations for the harmonic (integrable) and driven anharmonic (chaotic) oscillators, considering both classical and quantum initial states. The dynamics are characterized through several observables, including mean position, momentum, and energy. The quantumness of the time-evolved state is assessed via the Wigner logarithmic negativity, and the interplay between double-bracket dissipation and chaotic dynamics is examined in detail. Finally, we extend our analysis to generalized master equations involving higher-order nested brackets, which arise naturally as a time-continuous formulation of spectral filtering techniques widely used in the numerical simulation of quantum systems.
The dynamics of quantum systems under the adiabatic Hamiltonian has attracted attention not only in quantum control but also in a wide range of fields from condensed-matter physics to high-energy physics because of its nonperturbative behavior. Here we analyze the adiabatic dynamics in two-level systems and multilevel systems using exact WKB analysis, which is one of the nonperturbative analysis methods. As a result, we obtain a formula for the transition probability, which is similar to the known formula in the two-level system. Although nonperturbative analysis in the adiabatic limit has rarely been studied for multilevel systems, we show that the same analysis can be applied and also provide a concrete example. The results will serve as a basis for the application of exact WKB analysis in various fields of physics.