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arXiv:2210.00988 · Journal of Physics A Mathematical and Theoretical

Dynamics of mixed quantum–classical spin systems *

François Gay–Balmaz, Cesare Tronci

Published Mar 3, 2023• 10 citations
arXivHTMLPDF

Abstract

Abstract Mixed quantum–classical spin systems have been proposed in spin chain theory and, more recently, in magnon spintronics. However, current models of quantum–classical dynamics beyond mean-field approximations typically suffer from long-standing consistency issues, and, in some cases, invalidate Heisenberg’s uncertainty principle. Here, we present a fully Hamiltonian theory of quantum–classical spin dynamics that appears to be the first to ensure an entire series of consistency properties, including positivity of both the classical and the quantum density, so that Heisenberg’s principle is satisfied at all times. We show how this theory may connect to recent energy-balance considerations in measurement theory and we present its Poisson bracket structure explicitly. After focusing on the simpler case of a classical Bloch vector interacting with a quantum spin observable, we illustrate the extension of the model to systems with several spins, and restore the presence of orbital degrees of freedom.

DOI
10.1088/1751-8121/acc145

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Abstract Mixed quantum–classical spin systems have been proposed in spin chain theory and, more recently, in magnon spintronics. However, current models of quantum–classical dynamics beyond mean-field approximations typically suffer from long-standing consistency issues, and, in some cases, invalidate Heisenberg’s uncertainty principle. Here, we present a fully Hamiltonian theory of quantum–classical spin dynamics that appears to be the first to ensure an entire series of consistency properties, including positivity of both the classical and the quantum density, so that Heisenberg’s principle is satisfied at all times. We show how this theory may connect to recent energy-balance considerations in measurement theory and we present its Poisson bracket structure explicitly. After focusing on the simpler case of a classical Bloch vector interacting with a quantum spin observable, we illustrate the extension of the model to systems with several spins, and restore the presence of orbital degrees of freedom.
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.
We study compound systems with a classical sector and a quantum sector. Among other consistency conditions we require a canonical structure, that is, a Lie bracket for the dynamical evolution of hybrid observables in the Heisenberg picture, interpolating between the Poisson bracket and the commutator. Weak and strong postulates are proposed. We explicitly construct one such hybrid bracket when the Hilbert space of the quantum sector is finite dimensional and show that it is unique if the strong postulates are enforced. The adjoint bracket for the Schrodinger picture version of the dynamics is also obtained. Unfortunately, preservation of the positivity of the density matrix under the evolution is not guaranteed. The case of a particle with classical position and momentum and quantum spin-12\frac{1}{2}21​ is discussed and the spin-orbit dynamics is worked out.
We study compound systems with a classical sector and a quantum sector. Among other consistency conditions we require a canonical structure, that is, a Lie bracket for the dynamical evolution of hybrid observables in the Heisenberg picture, interpolating between the Poisson bracket and the commutator. Weak and strong postulates are proposed. We explicitly construct one such hybrid bracket when the Hilbert space of the quantum sector is finite dimensional and show that it is unique if the strong postulates are enforced. The adjoint bracket for the Schr\"odinger picture version of the dynamics is also obtained. Unfortunately, preservation of the positivity of the density matrix under the evolution is not guaranteed. The case of a particle with classical position and momentum and quantum spin-12\frac{1}{2}21​ is discussed and the spin-orbit dynamics is worked out.
Quantum mechanics is not logically closed with respect to the classical world.Its formalism unfolds as the quantization of a sub-set of classical Hamiltonians.The interpretation of quantum theory in terms of the measurement process inevitably requires to deal with systems composed by a mixture of both classical and quantum degrees of freedom.Moreover, when energy can flow between the quantum and classical degrees of freedom (i.e., in the case of nonadiabatic dynamics), there are more theoretical difficulties in order to obtain a fully consistent quantum-classical formalism.In order to perform calculations, one can renounce to the usual Lie algebraic structure of well-established physical theories, adopt non-Hamiltonian brackets, and obtain a formalism for the dynamics and statistics of quantum-classical systems that has an affordable computational complexity.Recent progress in the algorithms for the sampling of nonadiabatic dynamics of quantum-classical systems at long time is reviewed here.
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.
Mixed quantum-classical models are widely used to reduce the computational cost of fully quantum simulations. However, their general applicability across different classes of problems remains an open question. Here, we address this issue for systems featuring spin-orbit coupling. In particular, we study the interaction dynamics of quantum spin-1/2 and classical orbital momentum in one-dimensional models of Rashba nanowires. We tackle this problem by resorting to a new quantum-classical Hamiltonian model that, unlike conventional approaches, retains the Heisenberg principle and captures correlation effects beyond the common Ehrenfest method. Based on Koopman wavefunctions in classical mechanics, the new model was recently implemented numerically via a particle scheme -- the koopmon method -- which is extended here to treat spin-orbit coupling. We apply the koopmon method to study the quantum-classical dynamics of nanowire models, with and without the presence of a harmonic potential and in both Rashba-dominated (strong coupling) and Zeeman-dominated (weak coupling) regimes. Considering realistic semiconductor parameters, the results are contrasted with both fully quantum and quantum-classical Ehrenfest dynamics. In the absence of external potential, the koopmon method qualitatively reproduces the features of the fully quantum evolution for all coupling regimes. While it exhibits a slight loss in spin accuracy compared to Ehrenfest simulations, the latter fail to capture the orbital dynamics. In the presence of a harmonic potential, the koopmon scheme reproduces several features of full quantum dynamics with accuracy levels that are unachievable by the Ehrenfest model. We conclude by presenting a test case that exhibits the formation of cat-like states.