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arXiv:2308.14440 · The European Physical Journal Plus

Effective nonlinear Ehrenfest hybrid quantum-classical dynamics

José L. Alonso, Carlos Bouthelier-Madre, Jesús Clemente-Gallardo, David Martínez-Crespo, Javier Pomar

Published Jul 25, 2023• 11 citations
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Abstract

Abstract The definition of a consistent evolution equation for statistical hybrid quantum-classical systems is still an open problem. In this paper, we analyze the case of Ehrenfest dynamics on systems defined by a probability density and identify the relations of the nonlinearity of the dynamics with the obstructions to define a consistent dynamics for the first quantum moment of the distribution. This first quantum moment represents the physical states as a family of classically-parametrized density matrices $${\hat{\rho }}(\xi )$$ ρ ^ ( ξ ) , for $$\xi$$ ξ a classical point; and it is the most common representation of hybrid systems in the literature. Due to this obstruction, we consider higher order quantum moments, and argue that only a finite number of them are physically measurable. Because of this, we propose an effective solution for the hybrid dynamics problem based on approximating the distribution by those moments and representing the states by them.

DOI
10.1140/epjp/s13360-023-04266-w

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Abstract The definition of a consistent evolution equation for statistical hybrid quantum-classical systems is still an open problem. In this paper, we analyze the case of Ehrenfest dynamics on systems defined by a probability density and identify the relations of the nonlinearity of the dynamics with the obstructions to define a consistent dynamics for the first quantum moment of the distribution. This first quantum moment represents the physical states as a family of classically-parametrized density matrices ρ^(ξ){\hat{\rho }}(\xi )ρ^​(ξ) ρ ^ ( ξ ) , for ξ\xiξ ξ a classical point; and it is the most common representation of hybrid systems in the literature. Due to this obstruction, we consider higher order quantum moments, and argue that only a finite number of them are physically measurable. Because of this, we propose an effective solution for the hybrid dynamics problem based on approximating the distribution by those moments and representing the states by them.
Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.
We present a consistent formalism to describe the dynamics of hybrid systems with mixed classical and quantum degrees of freedom. The probability function of the system, which, in general, will be a combination of the classical distribution function and the quantum density matrix, is described in terms of its corresponding moments. We then define a hybrid Poisson bracket, such that the dynamics of the moments is ruled by an effective Hamiltonian. In particular, a closed formula for the Poisson brackets between any two moments for an arbitrary number of degrees of freedom is presented, which corrects previous expressions derived in the literature for the purely quantum case. This formula is of special relevance for practical applications of the formalism. Finally, we study the dynamics of a particular hybrid system given by two coupled oscillators, one being quantum and the other classical. Due to the coupling, specific quantum and classical properties are transferred between different sectors. In particular, the quantum sector is allowed to violate the uncertainty relation, though we explicitly show that there exists a minimum positive bound of the total uncertainty of the hybrid system.
General statistical ensembles in the Hamiltonian formulation of hybrid quantum-classical systems are analyzed. It is argued that arbitrary probability densities on the hybrid phase space must be considered as the class of possible physically distinguishable statistical ensembles of hybrid systems. Nevertheless, statistical operators associated with the hybrid system and with the quantum subsystem can be consistently defined. Dynamical equations for the statistical operators representing the mixed states of the hybrid system and its quantum subsystem are derived and analyzed. In particular, these equations irreducibly depend on the total probability density on the hybrid phase space.
Quantum dynamics (e.g., the Schrödinger equation) and classical dynamics (e.g., Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. The difference between both worlds is due to the presence of extra structure in the quantum case, that leads to the appearance of the probabilistic nature of the measurements and the indetermination and superposition principles. In this paper we first show that the quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well defined Poisson bracket allows to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann's distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. This work provides the basis for extending stochastic methods to Ehrenfest dynamics.
Quantum dynamics (i.e. the Schrödinger equation) and classical dynamics (i.e. Hamilton equations) can both be formulated in equal geometric terms: a Poisson bracket defined on a manifold. In this paper, we first show that the hybrid quantum-classical dynamics prescribed by the Ehrenfest equations can also be formulated within this general framework, what has been used in the literature to construct propagation schemes for Ehrenfest dynamics. Then, the existence of a well-defined Poisson bracket allows us to arrive to a Liouville equation for a statistical ensemble of Ehrenfest systems. The study of a generic toy model shows that the evolution produced by Ehrenfest dynamics is ergodic and therefore the only constants of motion are functions of the Hamiltonian. The emergence of the canonical ensemble characterized by the Boltzmann distribution follows after an appropriate application of the principle of equal a priori probabilities to this case. Once we know the canonical distribution of an Ehrenfest system, it is straightforward to extend the formalism of Nosé (invented to do constant temperature molecular dynamics by a non-stochastic method) to our Ehrenfest formalism. This work also provides the basis for extending stochastic methods to Ehrenfest dynamics.
Abstract We consider the interaction dynamics of a classical oscillator and a quantum two-level system for different pure-dephasing Hamiltonians of the type H ˆ ( q , p ) = H C ( q , p ) 1 + H I ( q , p ) σ ˆ z . This type of systems represents a severe challenge for popular hybrid quantum–classical descriptions. For example, in the case of the common Ehrenfest model, the classical density evolution is shown to decouple entirely from the pure-dephasing quantum dynamics. We focus on a recently proposed hybrid wave equation that is based on Koopman’s wavefunction description of classical mechanics. This model retains quantum–classical correlations whenever a coupling potential is present. Here, several benchmark problems are considered and the results are compared with those arising from fully quantum dynamics. A good agreement is found for a series of study cases involving harmonic oscillators with linear and quadratic coupling, as well as time-varying coupling parameters. In all these cases the classical evolution coincides exactly with the oscillator dynamics resulting from the fully quantum description. In the special case of time-independent coupling involving a classical oscillator with varying frequency, the quantum Bloch rotation exhibits peculiar features that escape from the hybrid description. In addition, nonlinear corrections to the harmonic Hamiltonian lead to an overall growth of decoherence at long times, which is absent in the fully quantum treatment.