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arXiv:2503.16062

Constraint Phase Space Formulations for Finite-State Quantum Systems: The Relation between Commutator Variables and Complex Stiefel Manifolds

Youhao Shang, Xiangsong Cheng, Jian Liu

Published Mar 20, 2025• 1 citation
arXivHTMLPDF

Abstract

We have recently developed the \textit{constraint} coordinate-momentum \textit{phase space} (CPS) formulation for finite-state quantum systems. It has been implemented for the electronic subsystem in nonadiabatic transition dynamics to develop practical trajectory-based approaches. In the generalized CPS formulation for the mapping Hamiltonian of the classical mapping model with commutator variables (CMMcv) method [\textit{J. Phys. Chem. A} \textbf{2021}, 125, 6845-6863], each {connected} component of the generalized CPS is the \textit{complex Stiefel manifold} labeled by the eigenvalue set of the mapping kernel. Such a phase space structure allows for exact trajectory-based dynamics for pure discrete (electronic) degrees of freedom (DOFs), where the equations of motion of each trajectory are isomorphic to the time-dependent Schrödinger equation. We employ covariant kernels {within the generalized CPS framework} to develop two approaches that naturally yield exact evaluation of time correlation functions (TCFs) for pure discrete (electronic) DOFs. In addition, we briefly discuss the phase space mapping formalisms where the contribution of each trajectory to the integral expression of the {TCF} of population dynamics is strictly positive semi-definite. The generalized CPS formulation also indicates that the equations of motion in phase space mapping model I of our previous work [\textit{J. Chem. Phys.} \textbf{2016}, 145, 204105; \textbf{2017}, 146, 024110; \textbf{2019}, 151, 024105] lead to a complex Stiefel manifold $\mathrm{U}(F)/\mathrm{U}(F-2)$. It is expected that the generalized CPS formulation has implications for simulations of both nonadiabatic transition dynamics and many-body quantum dynamics for spins/bosons/fermions.

DOI
10.48550/arxiv.2503.16062

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We have recently developed the \textit{constraint} coordinate-momentum \textit{phase space} (CPS) formulation for finite-state quantum systems. It has been implemented for the electronic subsystem in nonadiabatic transition dynamics to develop practical trajectory-based approaches. In the generalized CPS formulation for the mapping Hamiltonian of the classical mapping model with commutator variables (CMMcv) method [\textit{J. Phys. Chem. A} \textbf{2021}, 125, 6845-6863], each {connected} component of the generalized CPS is the \textit{complex Stiefel manifold} labeled by the eigenvalue set of the mapping kernel. Such a phase space structure allows for exact trajectory-based dynamics for pure discrete (electronic) degrees of freedom (DOFs), where the equations of motion of each trajectory are isomorphic to the time-dependent Schrödinger equation. We employ covariant kernels {within the generalized CPS framework} to develop two approaches that naturally yield exact evaluation of time correlation functions (TCFs) for pure discrete (electronic) DOFs. In addition, we briefly discuss the phase space mapping formalisms where the contribution of each trajectory to the integral expression of the {TCF} of population dynamics is strictly positive semi-definite. The generalized CPS formulation also indicates that the equations of motion in phase space mapping model I of our previous work [\textit{J. Chem. Phys.} \textbf{2016}, 145, 204105; \textbf{2017}, 146, 024110; \textbf{2019}, 151, 024105] lead to a complex Stiefel manifold U(F)/U(F−2)\mathrm{U}(F)/\mathrm{U}(F-2)U(F)/U(F−2). It is expected that the generalized CPS formulation has implications for simulations of both nonadiabatic transition dynamics and many-body quantum dynamics for spins/bosons/fermions.
(CPS) has recently been developed to study nonadiabatic dynamics in gas-phase and condensed-phase molecular systems. Although the CPS formulation is exact for describing the discrete (electronic/vibrational/spin) state degrees of freedom (DOFs), when system-bath models in condensed phase are studied, previous works often employ the approximation by discretizing environmental bath DOFs. In this paper, we develop an exact trajectory-based phase space approach by adopting the twin-space (TS) formulation of quantum statistical mechanics, in which the density operator of the reduced system is transformed to the wave function of an expanded system with twice the DOFs. The classical mapping model (CMM) is then used to map the Hamiltonian of the expanded system to its equivalent classical counterpart on CPS. To demonstrate the applicability of the TS-CMM approach, we compare simulated population dynamics and nonlinear spectra for a few benchmark condensed phase system-bath models with those obtained from the hierarchical equations of motion method, which shows that our approach yields accurate dynamics of open quantum systems.
Abstract We report recent progress on the phase space formulation of quantum mechanics with coordinate‐momentum variables, focusing more on new theory of (weighted) constraint coordinate‐momentum phase space for discrete‐variable quantum systems. This leads to a general coordinate‐momentum phase space formulation of composite quantum systems, where conventional representations on infinite phase space are employed for continuous variables. It is convenient to utilize (weighted) constraint coordinate‐momentum phase space for representing the quantum state and describing nonclassical features. Various numerical tests demonstrate that new trajectory‐based quantum dynamics approaches derived from the (weighted) constraint phase space representation are useful and practical for describing dynamical processes of composite quantum systems in the gas phase as well as in the condensed phase. This article is categorized under: Molecular and Statistical Mechanics > Molecular Dynamics and Monte‐Carlo Methods Theoretical and Physical Chemistry > Reaction Dynamics and Kinetics Theoretical and Physical Chemistry > Statistical Mechanics
Isomorphism of the two-state system is heuristic in understanding the dynamical or statistical behavior of the simplest yet most quantum system that has no classical counterpart. We use the constraint phase space developed in J. Chem. Phys.145, 204105 (2016); 151, 024105 (2019); J. Phys. Chem. Lett.12, 2496 (2021), non-covariant phase space functions, time-dependent weight functions, and time-dependent normalization factors to construct a novel class of phase space representations of the exact population dynamics of the two-state quantum system. The equations of motion of the trajectory on constraint phase space are isomorphic to the time-dependent Schrodinger equation. The contribution of each trajectory to the integral expression for the population dynamics is always positive semi-definite. We also prove that the triangle window function approach, albeit proposed as a heuristic empirical model in J. Chem. Phys.145, 144108 (2016), is related to a special case of the novel class and leads to an isomorphic representation of the exact population dynamics of the two-state quantum system.
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This paper deals with certain aspects of the phase-space formulation of quantum mechanics. Noting the ambiguities involved in quantizing the classical phase-space dynamics, a generalized quantum-mechanical phase space is introduced in terms of eigenvalues of two mutually incompatible complete sets of operators. General definitions are given for the phase-space distribution function corresponding to a given abstract state of the system and the phase-space function for an arbitrary operator. Our formulation goes over to the Wigner formulation when the complete sets correspond to position and momentum operators in Cartesian coordinates. Explicit examples are discussed with use of action-angle and polar coordinates for simple one-, two-, and three-dimensional systems. Symmetry changes in phase space are discussed by applying Gel'fand-Levitan transformations and it is shown that, in general, the degeneracies of motion in phase space are not reflected in the energy eigenvalues.