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arXiv:2502.08044 · Journal of Chemical Theory and Computation

Twin-Space Representation of Classical Mapping Model in the Constraint Phase Space Representation: Numerically Exact Approach to Open Quantum Systems

Jiaji Zhang, Jian Liu, Lipeng Chen

Published Jun 26, 2025• 4 citations
arXivHTMLPDF

Abstract

(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.

DOI
10.1021/acs.jctc.5c00224

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Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.

•Aug 31, 2011•0 citations
(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.
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.
We extend the twin-space formulation of the hierarchical equations of motion approach in combination with the matrix product state representation [R. Borrelli, J. Chem. Phys. 150, 234102 (2019)] to nonequilibrium scenarios where the open quantum system is coupled to a hybrid fermionic and bosonic environment. The key ideas used in the extension are a reformulation of the hierarchical equations of motion for the auxiliary density matrices into a time-dependent Schrödinger-like equation for an augmented multi-dimensional wave function as well as a tensor decomposition into a product of low-rank matrices. The new approach facilitates accurate simulations of non-equilibrium quantum dynamics in larger and more complex open quantum systems. The performance of the method is demonstrated for a model of a molecular junction exhibiting current-induced mode-selective vibrational excitation.
Abstract Discretizing distribution function in a phase space for an efficient quantum dynamics simulation is non-trivial challenge, in particular for a case that a system is further coupled to environmental degrees of freedom. Such open quantum dynamics is described by a reduced equation of motion (REOM) most notably by a quantum Fokker-Planck equation (QFPE) for a Wigner distribution function (WDF). To develop a discretization scheme that is stable for numerical simulations from the REOM approach, we find that a two-dimensional (2D) periodically invariant system-bath (PISB) model with two heat baths is an ideal platform not only for a periodical system but also for a system confined by a potential. We then derive the numerically ''exact'' hierarchical equations of motion (HEOM) for a discrete WDF in terms of periodically invariant operators in both coordinate and momentum spaces. The obtained equations can treat non-Markovian heat-bath in a non-perturbative manner at finite temperatures regardless of the mesh size. The stability of the present scheme is demonstrated in a high-temperature Markovian case by numerically integrating the discrete QFPE with by a coarse mesh for a 2D free rotor and harmonic potential systems for an initial condition that involves singularity.
The twin-formulation of quantum statistical mechanics is employed to describe a new methodology for the solution of the equations of motion of the reduced density matrix in their hierarchical formulation. It is shown that the introduction of tilde operators and of their algebra in the dual space greatly simplifies the application of numerical techniques for the propagation of the density matrix. The application of tensor-train representation of a vector to solve complex quantum dynamical problems within the framework of the twin-formulation is discussed. Next, applications of the hierarchical equations of motion to a dissipative polaron model are presented showing the validity and accuracy of the new approach.
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
Solving quantum dynamics is an exponentially difficult problem. Thus, an exact numerical solution is inaccessible for any condensed matter system.\nA promising approach is to divide the system into a quantum subsystem containing degrees of freedom which are of greater interest or those which have more profound quantum character (e.g., have smaller mass) and a classical bath containing the rest of the system.\nImposing such a partition and treating the bath classically results in quantum-classical dynamics. The quantum-classical Liouville equation is a general equation in the Hilbert space of quantum degrees of freedom while it resides in the phase space of the classical degrees of freedom.\nAny numerical solution to this equation requires representation of the quantum subsystem in some basis. Solutions to this equation have been already proposed in the subsystem, adiabatic and force bases, each with its own cons and pros.\n\nIn this work, the quantum-classical equations of motion are cast in the subsystem basis and subsequently mapped to a number of fictitious harmonic oscillators.\nThe result is quantum-classical dynamics in the mapping basis which treats both quantum and classical degrees of freedom on the same footing, i.e., in phase space. Neglecting a portion of the back reaction of the quantum-subsystem to classical bath results in an expression for the time evolution of an operator (density matrix) equal to its Poisson bracket with the Hamiltonian.\nThis equation can be solved in terms of characteristics to provide a computationally tractable method for calculating quantum-classical dynamical properties. The expressions for expectation values and correlation functions in this formalism are derived.\nCalculations on spin-boson system, barrier crossing models---the so called Tully models---and the Fenna-Mathews-Olson pigments show very good agreement between the results of this method and numerical solutions to the Schrödinger equation.