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Review

Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths

1
Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università degli Studi di Messina, Contrada Papardo, 98166 Messina, Italy
2
Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa
3
Istituto Nazionale di Fisica Nucleare, Sez. di Catania,95123 Catania, Italy
4
Department of Chemistry, University of Alberta, 11227 Saskatchewan Drive Edmonton, Edmonton, AB T6G 2G2, Canada
5
Dipartimento di Fisica e Chimica dell’Universitá di Palermo, Via Archirafi 36, I-90123 Palermo, Italy
6
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, I-90123 Palermo, Italy
*
Author to whom correspondence should be addressed.
Received: 13 September 2018 / Revised: 11 October 2018 / Accepted: 12 October 2018 / Published: 16 October 2018
(This article belongs to the Special Issue New Trends in Quantum Electrodynamics)
Many open quantum systems encountered in both natural and synthetic situations are embedded in classical-like baths. Often, the bath degrees of freedom may be represented in terms of canonically conjugate coordinates, but in some cases they may require a non-canonical or non-Hamiltonian representation. Herein, we review an approach to the dynamics and statistical mechanics of quantum subsystems embedded in either non-canonical or non-Hamiltonian classical-like baths which is based on operator-valued quasi-probability functions. These functions typically evolve through the action of quasi-Lie brackets and their associated Quantum-Classical Liouville Equations, or through quasi-Lie brackets augmented by dissipative terms. Quasi-Lie brackets possess the unique feature that, while conserving the energy (which the Noether theorem links to time-translation symmetry), they violate the time-translation symmetry of their algebra. This fact can be heuristically understood in terms of the dynamics of the open quantum subsystem. We then describe an example in which a quantum subsystem is embedded in a bath of classical spins, which are described by non-canonical coordinates. In this case, it has been shown that an off-diagonal open-bath geometric phase enters into the propagation of the quantum-classical dynamics. Next, we discuss how non-Hamiltonian dynamics may be employed to generate the constant-temperature evolution of phase space degrees of freedom coupled to the quantum subsystem. Constant-temperature dynamics may be generated by either a classical Langevin stochastic process or a Nosé–Hoover deterministic thermostat. These two approaches are not equivalent but have different advantages and drawbacks. In all cases, the calculation of the operator-valued quasi-probability function allows one to compute time-dependent statistical averages of observables. This may be accomplished in practice using a hybrid Molecular Dynamics/Monte Carlo algorithms, which we outline herein. View Full-Text
Keywords: quasi-lie brackets; quantum-classical Liouville equation; hybrid quantum-classical systems; breaking of time-translation symmetry; classical spin dynamics; Langevin dynamics; Nosé–Hoover dynamics quasi-lie brackets; quantum-classical Liouville equation; hybrid quantum-classical systems; breaking of time-translation symmetry; classical spin dynamics; Langevin dynamics; Nosé–Hoover dynamics
MDPI and ACS Style

Sergi, A.; Hanna, G.; Grimaudo, R.; Messina, A. Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths. Symmetry 2018, 10, 518. https://0-doi-org.brum.beds.ac.uk/10.3390/sym10100518

AMA Style

Sergi A, Hanna G, Grimaudo R, Messina A. Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths. Symmetry. 2018; 10(10):518. https://0-doi-org.brum.beds.ac.uk/10.3390/sym10100518

Chicago/Turabian Style

Sergi, Alessandro, Gabriel Hanna, Roberto Grimaudo, and Antonino Messina. 2018. "Quasi-Lie Brackets and the Breaking of Time-Translation Symmetry for Quantum Systems Embedded in Classical Baths" Symmetry 10, no. 10: 518. https://0-doi-org.brum.beds.ac.uk/10.3390/sym10100518

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