Quasi-Lie brackets and the breaking of time-translation symmetry for quantum systems embedded in classical baths

Roberto Grimaudo, Antonino Messina, Alessandro Sergi, Roberto Grimaudo, Antonino Messina, Gabriel Hanna

Risultato della ricerca: Article

4 Citazioni (Scopus)

Abstract

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.
Lingua originaleEnglish
pagine (da-a)518-
Numero di pagine28
RivistaSymmetry
Volume10
Stato di pubblicazionePublished - 2018

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Lie Brackets
brackets
Embedded systems
Quantum Systems
baths
Symmetry
Subsystem
symmetry
Probability function
Mathematical operators
degrees of freedom
Liouville equation
Thermostats
operators
Degree of freedom
thermostats
Statistical mechanics
Liouville equations
stochastic processes
Geometric Phase

All Science Journal Classification (ASJC) codes

  • Computer Science (miscellaneous)
  • Chemistry (miscellaneous)
  • Mathematics(all)
  • Physics and Astronomy (miscellaneous)

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Quasi-Lie brackets and the breaking of time-translation symmetry for quantum systems embedded in classical baths. / Grimaudo, Roberto; Messina, Antonino; Sergi, Alessandro; Grimaudo, Roberto; Messina, Antonino; Hanna, Gabriel.

In: Symmetry, Vol. 10, 2018, pag. 518-.

Risultato della ricerca: Article

Grimaudo, Roberto ; Messina, Antonino ; Sergi, Alessandro ; Grimaudo, Roberto ; Messina, Antonino ; Hanna, Gabriel. / Quasi-Lie brackets and the breaking of time-translation symmetry for quantum systems embedded in classical baths. In: Symmetry. 2018 ; Vol. 10. pagg. 518-.
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T1 - Quasi-Lie brackets and the breaking of time-translation symmetry for quantum systems embedded in classical baths

AU - Grimaudo, Roberto

AU - Messina, Antonino

AU - Sergi, Alessandro

AU - Grimaudo, Roberto

AU - Messina, Antonino

AU - Hanna, Gabriel

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - Breaking of time-translation symmetry

KW - Chemistry (miscellaneous)

KW - Classical spin dynamics

KW - Computer Science (miscellaneous)

KW - Hybrid quantum-classical systems

KW - Langevin dynamics

KW - Mathematics (all)

KW - Nosé-Hoover dynamics

KW - Physics and Astronomy (miscellaneous)

KW - Quantum-classical Liouville equation

KW - Quasi-lie brackets

UR - http://hdl.handle.net/10447/328093

UR - https://www.mdpi.com/2073-8994/10/10/518/pdf

M3 - Article

VL - 10

SP - 518-

JO - Symmetry

JF - Symmetry

SN - 2073-8994

ER -