Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

The Journal of Chemical Physics

Path integral density matrix dynamics: A method for calculating time-dependent properties in thermal adiabatic and non-adiabatic systems

Scott Habershon

Published Sep 12, 2013• 8 citations
Publisher

Abstract

We introduce a new approach for calculating quantum time-correlation functions and time-dependent expectation values in many-body thermal systems; both electronically adiabatic and non-adiabatic cases can be treated. Our approach uses a path integral simulation to sample an initial thermal density matrix; subsequent evolution of this density matrix is equivalent to solution of the time-dependent Schrödinger equation, which we perform using a linear expansion of Gaussian wavepacket basis functions which evolve according to simple classical-like trajectories. Overall, this methodology represents a formally exact approach for calculating time-dependent quantum properties; by introducing approximations into both the imaginary-time and real-time propagations, this approach can be adapted for complex many-particle systems interacting through arbitrary potentials. We demonstrate this method for the spin Boson model, where we find good agreement with numerically exact calculations. We also discuss future directions of improvement for our approach with a view to improving accuracy and efficiency.

DOI
10.1063/1.4819322

Related papers

Accurate calculation of equilibrium reduced density matrix for the system-bath model: A multilayer multiconfiguration time-dependent Hartree approach and its comparison to a multi-electronic-state path integral molecular dynamics approach

Haobin Wang, Xinzijian Liu, Jian Liu

An efficient and accurate method for computing the equilibrium reduced density matrix is presented for treating open quantum systems characterized by the system-bath model. The method employs the multilayer multiconfiguration time-dependent Hartree theory for imaginary time propagation and an importance sampling procedure for calculating the quantum mechanical trace. The method is applied to the spin-boson Hamiltonian, which leads to accurate results in agreement with those produced by the multi-electronic-state path integral molecular dynamics method.

Chinese Journal of Chemical Physics•Aug 1, 2018•11 citations

Time-evolving matrix product operator method and its applications in open quantum system

Ruofan Chen

Open quantum systems play an important role in developing quantum sciences, and therefore the study of corresponding numerical method is of great significance. For the open quantum systems, the quasi-adiabatic propagator path integral invented in 1990s is one of the few numerically exact methods. However, its computational complexity scales exponentially with system size and correlation length, and therefore its application is limited in practical calculation. In recent years, the study and application of tensor network have made rapid progress. Representing the path integral by tensor network makes the computational complexity increase polynomially, thus greatly improving the computational efficiency. Such a new method is called time-evolving matrix product operator. At the very beginning, the reduced density matrix is represented as a matrix product state. Then the time evolution of the system can be achieved by iteratively applying matrix product operators to the matrix product state. The iterative process is amenable to the standard matrix product states compression algorithm, which keeps the computational cost on a polynomial scale. The time-evolving matrix product operator is an efficient, numerically exact and fully non-Markovian method, which has a broad application prospect in the study of quantum open systems. For instance, it is already used in the study of the thermalization, heat statistic, heat transfer and optimal control of the quantum open systems, and conversely it can be also used to investigate the effect of the system on the environment. In addition, the TEMPO method is naturally related to the process tensor, and can be used to calculate the correlation function of the system efficiently. In this article we review this method and its applications. We give a brief introduction of the path integral formalism of Caldeira-Leggett model. According to the path integral formalism, we demonstrate the usage of quasi-adiabatic propagator path integral method. we give the basic idea of matrix product states, and we show how to recast quasi-adiabatic propagator path integral method into time-evolving matrix product operators method by employing the concept of matrix product states and matrix product operators, and give a review of its applications. In addition, we use the calculation results of physical quantities, correlation functions and heat currents in the spin-boson model to illustrate the applications of the time-evolving matrix product operator method.

Acta Physica Sinica•Jan 1, 2023•0 citations

Quantum Molecular Dynamics

Riccardo Fantoni

We formulate a new quantum many body simulation method for a general quantum fluid at any given temperature. Unlike path integral Monte Carlo our method evolves in imaginary time the density matrix from its initial delta function condition to its final thermal form at time time equal to the inverse temperature. It does this with a molecular dynamics scheme on a classical Hamiltonian that has the same functional form as the one of the quantum mechanical Hamiltonian according to the properties of the continuous representation of John R. Klauder. We then end up with the thermal density matrix that can be used to extract thermal averages of observables using the Monte Carlo method equally well in any statistics.

arXiv:2606.01389•Quantum Reports•Feb 6, 2026•0 citations

Reduced density matrix hybrid approach: An efficient and accurate method for adiabatic and non-adiabatic quantum dynamics

Timothy C. Berkelbach, David R. Reichman, Thomas E. Markland

We present a new approach to calculate real-time quantum dynamics in complex systems. The formalism is based on the partitioning of a system's environment into "core" and "reservoir" modes with the former to be treated quantum mechanically and the latter classically. The presented method only requires the calculation of the system's reduced density matrix averaged over the quantum core degrees of freedom which is then coupled to a classically evolved reservoir to treat the remaining modes. We demonstrate our approach by applying it to the spin-boson problem using the noninteracting blip approximation to treat the system and core, and Ehrenfest dynamics to treat the reservoir. The resulting hybrid methodology is accurate for both fast and slow baths, since it naturally reduces to its composite methods in their respective regimes of validity. In addition, our combined method is shown to yield good results in intermediate regimes where neither approximation alone is accurate and to perform equally well for both strong and weak system-bath coupling. Our approach therefore provides an accurate and efficient methodology for calculating quantum dynamics in complex systems.

arXiv:1110.0490•The Journal of Chemical Physics•Jan 19, 2012•69 citations

Fast converging path integrals for time-dependent potentials: II. Generalization to many-body systems and real-time formalism

Antun Balaž, Ivana Vidanović, Aleksandar Bogojević +2 more

Based on a previously developed recursive approach for calculating the short-time expansion of the propagator for systems with time-independent potentials and its time-dependent generalization for simple single-particle systems, in this paper we present a full extension of this formalism to a general quantum system with many degrees of freedom in a time-dependent potential. Furthermore, we also present a recursive approach for the velocity-independent part of the effective potential, which is necessary for calculating diagonal amplitudes and partition functions, as well as an extension from the imaginary-time formalism to the real-time one, which enables to study the dynamical properties of quantum systems. The recursive approach developed here allows an analytic derivation of the short-time expansion to orders that have not been accessible before, using the implemented SPEEDUP symbolic calculation code. The analytically derived results are extensively numerically verified by treating several models in both imaginary and real time.

arXiv:1011.5185•Journal of Statistical Mechanics Theory and Experiment•Mar 4, 2011•8 citations

Nonadiabatic dynamics of electrons and atoms under nonequilibrium conditions

Lev N Kantorovich

An approach to nonadiabatic dynamics of atoms in molecular and condensed matter systems under general nonequilibrium conditions is proposed. In this method interaction between nuclei and electrons is considered explicitly up to the second order in atomic displacements defined with respect to the mean atomic trajectory, enabling one to consider movement of atoms beyond their simple vibrations. Both electrons and nuclei are treated fully quantum-mechanically using a combination of path integrals applied to nuclei and nonequilibrium Green's functions to electrons. Our method is partitionless: initially, the entire system is coupled and assumed to be at thermal equilibrium. Then, the exact application of the Hubbard-Stratonovich transformation in mixed real and imaginary times enables us to obtain, without doing any additional approximations, an exact expression for the reduced density matrix for nuclei and hence an effective quantum Liouville equation for them, both containing Gaussian noises. It is shown that the time evolution of the expectation values for atomic positions is described by an infinite hierarchy of stochastic differential equations for atomic positions and momenta and their various fluctuations. The actual dynamics is obtained by sampling all stochastic trajectories. It is expected that applications of the method may include photoinduced chemical reactions (e.g., dissociation), electromigration, and atomic manipulation in scanning tunneling microscopy, to name just a few.

arXiv:1803.05828•Physical review. B./Physical review. B•Jul 23, 2018•12 citations
We introduce a new approach for calculating quantum time-correlation functions and time-dependent expectation values in many-body thermal systems; both electronically adiabatic and non-adiabatic cases can be treated. Our approach uses a path integral simulation to sample an initial thermal density matrix; subsequent evolution of this density matrix is equivalent to solution of the time-dependent Schrödinger equation, which we perform using a linear expansion of Gaussian wavepacket basis functions which evolve according to simple classical-like trajectories. Overall, this methodology represents a formally exact approach for calculating time-dependent quantum properties; by introducing approximations into both the imaginary-time and real-time propagations, this approach can be adapted for complex many-particle systems interacting through arbitrary potentials. We demonstrate this method for the spin Boson model, where we find good agreement with numerically exact calculations. We also discuss future directions of improvement for our approach with a view to improving accuracy and efficiency.
An efficient and accurate method for computing the equilibrium reduced density matrix is presented for treating open quantum systems characterized by the system-bath model. The method employs the multilayer multiconfiguration time-dependent Hartree theory for imaginary time propagation and an importance sampling procedure for calculating the quantum mechanical trace. The method is applied to the spin-boson Hamiltonian, which leads to accurate results in agreement with those produced by the multi-electronic-state path integral molecular dynamics method.
Open quantum systems play an important role in developing quantum sciences, and therefore the study of corresponding numerical method is of great significance. For the open quantum systems, the quasi-adiabatic propagator path integral invented in 1990s is one of the few numerically exact methods. However, its computational complexity scales exponentially with system size and correlation length, and therefore its application is limited in practical calculation. In recent years, the study and application of tensor network have made rapid progress. Representing the path integral by tensor network makes the computational complexity increase polynomially, thus greatly improving the computational efficiency. Such a new method is called time-evolving matrix product operator. At the very beginning, the reduced density matrix is represented as a matrix product state. Then the time evolution of the system can be achieved by iteratively applying matrix product operators to the matrix product state. The iterative process is amenable to the standard matrix product states compression algorithm, which keeps the computational cost on a polynomial scale. The time-evolving matrix product operator is an efficient, numerically exact and fully non-Markovian method, which has a broad application prospect in the study of quantum open systems. For instance, it is already used in the study of the thermalization, heat statistic, heat transfer and optimal control of the quantum open systems, and conversely it can be also used to investigate the effect of the system on the environment. In addition, the TEMPO method is naturally related to the process tensor, and can be used to calculate the correlation function of the system efficiently. In this article we review this method and its applications. We give a brief introduction of the path integral formalism of Caldeira-Leggett model. According to the path integral formalism, we demonstrate the usage of quasi-adiabatic propagator path integral method. we give the basic idea of matrix product states, and we show how to recast quasi-adiabatic propagator path integral method into time-evolving matrix product operators method by employing the concept of matrix product states and matrix product operators, and give a review of its applications. In addition, we use the calculation results of physical quantities, correlation functions and heat currents in the spin-boson model to illustrate the applications of the time-evolving matrix product operator method.
We formulate a new quantum many body simulation method for a general quantum fluid at any given temperature. Unlike path integral Monte Carlo our method evolves in imaginary time the density matrix from its initial delta function condition to its final thermal form at time time equal to the inverse temperature. It does this with a molecular dynamics scheme on a classical Hamiltonian that has the same functional form as the one of the quantum mechanical Hamiltonian according to the properties of the continuous representation of John R. Klauder. We then end up with the thermal density matrix that can be used to extract thermal averages of observables using the Monte Carlo method equally well in any statistics.
We present a new approach to calculate real-time quantum dynamics in complex systems. The formalism is based on the partitioning of a system's environment into "core" and "reservoir" modes with the former to be treated quantum mechanically and the latter classically. The presented method only requires the calculation of the system's reduced density matrix averaged over the quantum core degrees of freedom which is then coupled to a classically evolved reservoir to treat the remaining modes. We demonstrate our approach by applying it to the spin-boson problem using the noninteracting blip approximation to treat the system and core, and Ehrenfest dynamics to treat the reservoir. The resulting hybrid methodology is accurate for both fast and slow baths, since it naturally reduces to its composite methods in their respective regimes of validity. In addition, our combined method is shown to yield good results in intermediate regimes where neither approximation alone is accurate and to perform equally well for both strong and weak system-bath coupling. Our approach therefore provides an accurate and efficient methodology for calculating quantum dynamics in complex systems.
Based on a previously developed recursive approach for calculating the short-time expansion of the propagator for systems with time-independent potentials and its time-dependent generalization for simple single-particle systems, in this paper we present a full extension of this formalism to a general quantum system with many degrees of freedom in a time-dependent potential. Furthermore, we also present a recursive approach for the velocity-independent part of the effective potential, which is necessary for calculating diagonal amplitudes and partition functions, as well as an extension from the imaginary-time formalism to the real-time one, which enables to study the dynamical properties of quantum systems. The recursive approach developed here allows an analytic derivation of the short-time expansion to orders that have not been accessible before, using the implemented SPEEDUP symbolic calculation code. The analytically derived results are extensively numerically verified by treating several models in both imaginary and real time.
An approach to nonadiabatic dynamics of atoms in molecular and condensed matter systems under general nonequilibrium conditions is proposed. In this method interaction between nuclei and electrons is considered explicitly up to the second order in atomic displacements defined with respect to the mean atomic trajectory, enabling one to consider movement of atoms beyond their simple vibrations. Both electrons and nuclei are treated fully quantum-mechanically using a combination of path integrals applied to nuclei and nonequilibrium Green's functions to electrons. Our method is partitionless: initially, the entire system is coupled and assumed to be at thermal equilibrium. Then, the exact application of the Hubbard-Stratonovich transformation in mixed real and imaginary times enables us to obtain, without doing any additional approximations, an exact expression for the reduced density matrix for nuclei and hence an effective quantum Liouville equation for them, both containing Gaussian noises. It is shown that the time evolution of the expectation values for atomic positions is described by an infinite hierarchy of stochastic differential equations for atomic positions and momenta and their various fluctuations. The actual dynamics is obtained by sampling all stochastic trajectories. It is expected that applications of the method may include photoinduced chemical reactions (e.g., dissociation), electromigration, and atomic manipulation in scanning tunneling microscopy, to name just a few.