Quantum Fokker-Planck Equation Research Articles

Thermodynamic quantum Fokker–Planck equations and their application to thermostatic Stirling engine

We developed a computer code for the thermodynamic quantum Fokker–Planck equations (T-QFPE), derived from a thermodynamic system–bath model. This model consists of an anharmonic subsystem coupled to multiple Ohmic baths at different temperatures, which are connected to or disconnected from the subsystem as a function of time. The code numerically integrates the T-QFPE and their classical expression to simulate isothermal, isentropic, thermostatic, and entropic processes in both quantum and classical cases. The accuracy of the results was verified by comparing the analytical solutions of the Brownian oscillator. In addition, we illustrated a breakdown of the Markovian Lindblad-master equation in the pure quantum regime. As a demonstration, we simulated a thermostatic Stirling engine employed to develop non-equilibrium thermodynamics [S. Koyanagi and Y. Tanimura, J. Chem. Phys. 161, 114113 (2024)] under quasi-static conditions. The quasi-static thermodynamic potentials, described as intensive and extensive variables, were depicted as work diagrams. In the classical case, the work done by the external field is independent of the system–bath coupling strength. In contrast, in the quantum case, the work decreases as the coupling strength increases due to quantum entanglement between the subsystem and bath. The codes were developed for multicore processors using Open Multi-Processing (OpenMP) and for graphics processing units using the Compute Unified Device Architecture. These codes are provided in the supplementary material.

Classical and quantum thermodynamics described as a system-bath model: The dimensionless minimum work principle.

We formulate a thermodynamic theory applicable to both classical and quantum systems. These systems are depicted as thermodynamic system-bath models capable of handling isothermal, isentropic, thermostatic, and entropic processes. Our approach is based on the use of a dimensionless thermodynamic potential expressed as a function of the intensive and extensive thermodynamic variables. Using the principles of dimensionless minimum work and dimensionless maximum entropy derived from quasi-static changes of external perturbations and temperature, we obtain the Massieu-Planck potentials as entropic potentials and the Helmholtz-Gibbs potentials as free energy. These potentials can be interconverted through time-dependent Legendre transformations. Our results are verified numerically for an anharmonic Brownian system described in phase space using the low-temperature quantum Fokker-Planck equations in the quantum case and the Kramers equation in the classical case, both developed for the thermodynamic system-bath model. Thus, we clarify the conditions for thermodynamics to be valid even for small systems described by Hamiltonians and establish a basis for extending thermodynamics to non-equilibrium conditions.

Collective bath coordinate mapping of "hierarchy" in hierarchical equations of motion.

The theory of hierarchical equations of motion (HEOM) is one of the standard methods to give exact evaluations of the dynamics as coupled to harmonic oscillator environments. However, the theory is numerically demanding due to its hierarchy, which is the set of auxiliary elements introduced to capture the non-Markovian and non-perturbative effects of environments. When system-bath coupling becomes relatively strong, the required computational resources and precision move beyond the regime that can be currently handled. This article presents a new representation of HEOM theory in which the hierarchy is mapped into a continuous space of a collective bath coordinate and several auxiliary coordinates as the form of the quantum Fokker-Planck equation. This representation gives a rigorous time evolution of the bath coordinate distribution and is more stable and efficient than the original HEOM theory, particularly when there is a strong system-bath coupling. We demonstrate the suitability of this approach to treat vibronic system models coupled to environments.

A low-temperature quantum Fokker-Planck equation that improves the numerical stability of the hierarchical equations of motion for the Brownian oscillator spectral density.

We investigate the numerical stability of the hierarchical equations of motion (HEOM) method applied to systems with the Brownian oscillator (BO) and multimode BO (MBO) spectral densities. It is shown that, with the increase in the system-bath coupling strength, the standard HEOM may become unstable, and a simple increase in the truncation depth of the HEOM cannot remove the instability at long times. To solve this problem, we first show that the high-temperature approximation of the HEOM with the BO spectral density is equivalent to the celebrated quantum Fokker-Planck equation (QFPE). By starting from the HEOM, we then derive a new multidimensional phase space differential equation that generalizes the QFPE to arbitrary temperature. It is further shown that the numerical instability can be removed if the new low-temperature QFPE is expanded in a basis set different than the one that leads to the conventional HEOM. The matrix product state method is also employed to propagate the new equation based on the low-temperature QFPE and to resolve the numerical instability problem for an electron transfer model with the MBO spectral density presented in the recent literature.

Open quantum dynamics theory on the basis of periodical system-bath model for discrete Wigner function

Discretizing a distribution function in a phase space for an efficient quantum dynamics simulation is a 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 employ a two-dimensional (2D) periodically invariant system-bath (PISB) model with two heat baths. This model is an ideal platform not only for a periodic system but also for a non-periodic 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. As demonstrations, we numerically integrate the discrete QFPE for a 2D free rotor and harmonic potential systems in a high-temperature Markovian case using a coarse mesh with initial conditions that involve singularity.

Landau equation for self-gravitating classical and quantum particles: application to dark matter

We develop the kinetic theory of classical and quantum particles (fermions and bosons) in gravitational interaction. The kinetic theory of quantum particles may have applications in the context of dark matter. For simplicity, we consider an infinite and spatially homogeneous system (or make a local approximation) and neglect collective effects. This leads to the quantum Landau equation derived heuristically in (Physica A 332: 89, 2004). We establish its main properties: conservation laws, H-theorem, equilibrium state, relaxation time, quantum diffusion and friction coefficients, quantum Rosenbluth potentials, self-consistent evolution, (thermal) bath approximation, quantum Fokker–Planck equation, quantum King model $$\ldots $$ For bosonic particles, the Landau equation can describe the process of Bose–Einstein condensation. We discuss the relation of our study with the works of Levkov et al. (Phys Rev Lett 121: 151301, 2018) and Bar-Or et al. (Astrophys J 871: 28, 2019) on fuzzy dark matter halos and the formation of Bose stars and solitons.

Quantum Fokker-Planck modeling of degenerate electrons

An implicit and conservative numerical scheme is proposed for the isotropic quantum Fokker-Planck equation describing the evolution of degenerate electrons subject to elastic collisions with other electrons and ions. The electron-ion and electron-electron collision operators are discretized using a discontinuous Galerkin method, and the electron energy distribution is updated by an implicit time integration method. The numerical scheme is designed to satisfy all conservation laws exactly. Numerical tests and comparisons with other modeling approaches are shown to demonstrate the accuracy and conservation properties of the proposed method.

A Nonlocal Approach to the Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

The Accardi-Boukas quantum Black-Scholes equation (Luigi Accardi, Andreas Boukas: The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, (2006) vol.2, no.2, pp. 155-170.) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus (Hudson, R.L.; Parthasarathy, K.R: Quantum Ito's Formula and Stochastic Evolutions. Commun Math. Phys. 1984, 93, 301--323.). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

Low-Temperature Quantum Fokker-Planck and Smoluchowski Equations and Their Extension to Multistate Systems.

Simulating electron-nucleus coupled dynamics poses a nontrivial challenge and an important problem in the investigation of ultrafast processes involving coupled electronic and vibrational dynamics. Because irreversibility of the system dynamics results from thermal activation and dissipation caused by the environment, in dynamical studies, it is necessary to include heat bath degrees of freedom in the total system. When the system dynamics involves high-energy electronic transitions, the environment is regarded to be in a low-temperature regime and we must treat it quantum mechanically. In this Article, we present rigorous and versatile approaches for investigating the dynamics of open systems with coupled electronic and vibrational degrees of freedom within a fully quantum mechanical framework. These approaches are based on a quantum Fokker-Planck equation and a quantum Smoluchowski equation employing a heat bath with an Ohmic spectral density, with non-Markovian low-temperature correction terms, and extensions of these equations to the case of multistate systems. The accuracy of these equations was numerically examined for a single-state Brownian system, while their applicability was examined for multistate double-well systems by comparing their results with those of the fewest-switch surface hopping and Ehrenfest methods with a classical Markovian Langevin force. Comparison of the transient absorption spectra obtained using these methods clearly reveals the importance of the quantum low-temperature correction terms. These equations allow us to treat nonadiabatic dynamics in an efficient way, while maintaining numerical accuracy. The C++ source codes that we developed, which allow for the treatment of the phase and coordinate space dynamics with any single-state or multistate potential forms, are provided as Supporting Information.

A Nonlocal Approach to The Quantum Kolmogorov Backward Equation and Links to Noncommutative Geometry

The Accardi-Boukas quantum Black-Scholes equation (Luigi Accardi, Andreas Boukas: The Quantum Black-Scholes Equation, Global Journal of Pure and Applied Mathematics, (2006) vol.2, no.2, pp. 155-170.) can be used as an alternative to the classical approach to finance, and has been found to have a number of useful benefits. The quantum Kolmogorov backward equations, and associated quantum Fokker-Planck equations, that arise from this general framework, are derived using the Hudson-Parthasarathy quantum stochastic calculus (Hudson, R.L.; Parthasarathy, K.R: Quantum Ito's Formula and Stochastic Evolutions. Commun Math. Phys. 1984, 93, 301--323.). In this paper we show how these equations can be derived using a nonlocal approach to quantum mechanics. We show how nonlocal diffusions, and quantum stochastic processes can be linked, and discuss how moment matching can be used for deriving solutions.

Phase-space wavepacket dynamics of internal conversion via conical intersection: Multi-state quantum Fokker-Planck equation approach

We theoretically investigate internal conversion processes of a photoexcited molecule in a condensed phase. The molecular system is described by two-dimensional adiabatic ground and excited potential energy surfaces that are coupled to heat baths. We quantify the role of conical intersection (CI) and avoided crossing (AC) in the PESs in dissipative environments by simulating the time evolution of wavepackets to compute the lifetime of the excited wavepacket, yield of the product, and adiabatic electronic coherence. For this purpose, we employ the multi-state quantum Fokker-Planck equation (MSQFPE) for a two-dimensional Wigner space utilizing the Wigner–Moyal expansion for the potential term and the Brinkman hierarchy expression for the momentum. We find that the calculated results are significantly different between the CI and AC cases due to the transition in the tuning mode and vibrational motion in the coupling mode.

Efficient numerical integrator based on Fer expansion: Application to solid-state NMR experiments and to solve quantum Liouville equation and quantum Fokker-Planck equation

There are two important related research areas that I propose to investigate. First, we plan to develop an efficient numerical integrator based on Fer expansion for solid-state NMR simulation of experiments. Second, we intend to extend the method to solve quantum Liouville equation and quantum Fokker-Planck equation in order to improve the understanding of the dynamics of quantum systems subject to dissipation due to its relation to macroscopic quantum phenomena. The goal of the proposed research is to study a numerical integrator based on Fer expansion (Fer integrators of higher orders) in the integration of the time-dependent Schrodinger equation (TDSE) which is a central problem to nuclear magnetic resonance in general and solid-state NMR (SSNMR) in particular. The Fer integrator will provide to experts in quantum mechanics, NMR spectroscopy, and spin dynamics researchers, additional means for controlling spin dynamics in SSNMR. The efficient diagram will be used to compare the different orders of the Fer integrators obtained.

Generalized quantum Fokker-Planck equation for photoinduced nonequilibrium processes with positive definiteness condition.

This work provides a detailed derivation of a generalized quantum Fokker-Planck equation (GQFPE) appropriate for photo-induced quantum dynamical processes. The path integral method pioneered by Caldeira and Leggett (CL) [Physica A 121, 587 (1983)] is extended by utilizing a nonequilibrium influence functional applicable to different baths for the ground and the excited electronic states. Both nonequilibrium and non-Markovian effects are accounted for consistently by expanding the paths in the exponents of the influence functional up to the second order with respect to time. This procedure results in approximations involving only single time integrations for the exponents of the influence functional but with additional time dependent boundary terms that have been ignored in previous works. The boundary terms complicate the derivation of a time evolution equation but do not affect position dependent physical observables or the dynamics in the steady state limit. For an effective density operator with the boundary terms factored out, a time evolution equation is derived, through short time expansion of the effective action and Gaussian integration in analytically continued complex domain of space. This leads to a compact form of the GQFPE with time dependent kernels and additional terms, which renders the resulting equation to be in the Dekker form [Phys. Rep. 80, 1 (1981)]. Major terms of the equation are analyzed for the case of Ohmic spectral density with Drude cutoff, which shows that the new GQFPE satisfies the positive definiteness condition in medium to high temperature limit. Steady state limit of the GQFPE is shown to approach the well-known expression derived by CL in the high temperature and Markovian bath limit and also provides additional corrections due to quantum and non-Markovian effects of the bath.

On Quantum Fokker–Planck Equation

The quantum Fokker–Planck equation (QFPE) is revisited. Provided that the molecule is the Maxwellian molecule, the quantum Landau–Fokker–Planck equation is divided into characteristic four terms. The characteristics of three terms among four terms are investigated on the basis of Grad’s method, whereas the characteristics of the remained term, which is attributed to the collisional term of the QFPE proposed by Kaniadakis–Quarati, when the distribution function of the colliding partner is under the equilibrium state, are numerically investigated. The numerical result indicates that the time evolution of the distribution function obtained using such a remained term is instable, when the equilibrium or nonequilibrium state is given as initial data of the distribution function. Such an instability of the distribution function can be described by analyzing the propagation of the plane harmonic wave in one dimensional velocity space.

Quantum Suppression of Ratchet Rectification in a Brownian System Driven by a Biharmonic Force

We rigorously investigate the quantum dissipative dynamics of a ratchet system described by a periodic potential model based on the Caldeira-Leggett Hamiltonian with a biharmonic force. In this model, we use the reduced hierarchy equations of motion in the Wigner space representation. These equations represent a generalization of the Gaussian-Markovian quantum Fokker-Planck equation introduced by Tanimura and Wolynes (1991), which was formulated to study non-Markovian and nonperturbative thermal effects at finite temperature. This formalism allows us to treat both the classical limit and the tunneling regimes, and it is helpful for identifying purely quantum mechanical effects through the time evolution of the Wigner distribution. We carried out extensive calculations of the classical and quantum currents for various temperatures, coupling strengths, and barrier heights. Our results reveal that at low temperature, while the quantum current is larger than the classical current in the case of a high barrier, the opposite is true in the case of a low barrier. We find that this behavior results from the fact that the tunneling enhances the current in the case of a high barrier, while it suppresses the current in the case of a low barrier. This is because the effect of the ratchet potential is weak in the case of a low barrier due to the large dispersion of the distribution introduced by tunneling. This causes the spatiotemporal asymmetry, which is necessary for ratchet current, to be weak, and as a result, the net current is suppressed.

Gaussian channel in a quantum information network

Starting from the quantum Fokker–Planck equation, a viable quantum Gaussian channel is constructed in a quantum information network. It is shown that the solution of this equation satisfies a Gaussian distribution channel. Transforming the solution of the Fokker–Planck equation and substituting it into the information formula, a quantum dynamical mutual information equation is obtained in the coherent state representation. On the basis of this equation, a new scheme is proposed to implement parallel quantum information processing in the quantum Gaussian channel. The proposed scheme takes the coefficients related to the encryption states of photons as the signal, and encodes the information that will be passed in the quantum Gaussian channel from the input terminal. Information is then accessed by extracting and decoding the coefficients from the output terminal. Compared with a classical Gaussian channel, this approach offers the advantage of quantum parallelism.

External-noise-driven bath and the generalized semiclassical Kramers theory

We address the issue of a system that has been tacitly made thermodynamically open by externally driving the associated heat bath in an attempt to gain better insight regarding many physical situations that are akin to this problem. This work embodies the study of the quantum effects in the rate of decay from a metastable state of a Brownian particle which is in contact with a correlated noise-driven bath. We do this by initiating from a suitable system-reservoir model to derive the operator-valued Langevin equation. This further leads us to the corresponding c-number analog that includes quantum effects in leading order. Suitable mathematical treatment culminates in the quantum Fokker-Planck equation, the solution to which yields the rate expression. Finally, we put this to thorough numerical analysis.

A finite-difference implementation of the Caldeira–Leggett master equation

The Caldeira-Leggett master equation for dissipative quantum dynamics has predominantly been implemented in phase space, where it plays the role of a quantum Fokker-Planck equation. Here we demonstrate the feasibility of a pure coordinate space implementation in the case of intermediate damping strength and temperature, for times long enough to observe thermal equilibration. After a thorough numerical investigation of the analytically solvable harmonic oscillator case, a Morse oscillator model is studied. This is a case in which a quantal phase space approach would be difficult to implement and we show how analytical results for thermal averages are reproduced to a surprisingly high degree.

Towards Nonlinear Quantum Fokker-Planck Equations

It is demonstrated how the equilibrium semiclassical approach of Coffey et al. can be improved to describe more correctly the evolution. As a result a new semiclassical Klein-Kramers equation for the Wigner function is derived, which remains quantum for a free quantum Brownian particle as well. It is transformed to a semiclassical Smoluchowski equation, which leads to our semiclassical generalization of the classical Einstein law of Brownian motion derived before. A possibility is discussed how to extend these semiclassical equations to nonlinear quantum Fokker-Planck equations based on the Fisher information.

Variational principle for mixed classical–quantum systems

An extended variational principle providing the equations of motion for a system consisting of interacting classical, quasiclassical, and quantum components is presented, and applied to the model of bilinear coupling. The relevant dynamical variables are expressed in the form of a quantum state vector that includes the action of the classical subsystem in its phase factor. It is shown that the statistical ensemble of Brownian state vectors for a quantum particle in a classical thermal environment can be described by a density matrix evolving according to a nonlinear quantum Fokker–Planck equation. Exact solutions of this equation are obtained for a two-level system in the limit of high temperatures, considering both stationary and nonstationary initial states. A treatment of the common time shared by the quantum system and its classical environment as a collective variable, rather than as a parameter, is presented in the Appendix. PACS Nos.: 03.65.–w, 03.65.Sq, 05.30.–d, 45.10.Db