We employ a recently developed formalism for describing open quantum systems, to device efficient ab initio numerical techniques for the time propagation of the Wigner function. The propagator's complexity is $O(N\log N)$, where N is the length of the array containing the Wigner function. We illustrate the developed method by propagating single particle systems undergoing decoherence and draw a faithful comparison with the classical evolution governed by both the Fokker-Planck and the Koopman-von Neumann equations. These simulations allow us to witness the emergence of the classical world by observing how the Wigner function converges to the solution of the Fokker-Planck equation with sufficient classical diffusion. As a second example we simulate for first time a numerically challenging quantum system made of two coupled particles and observe the loss of decoherence on one of the particles as a result of continuous measurements performed on the other one, therefore obtaining quantitative predictions for the loss of coherence of both particles as a function of time.
We employ a recently developed formalism for describing open quantum systems, to device efficient ab initio numerical techniques for the time propagation of the Wigner function. The propagator's complexity is O(NlogN), where N is the length of the array containing the Wigner function. We illustrate the developed method by propagating single particle systems undergoing decoherence and draw a faithful comparison with the classical evolution governed by both the Fokker-Planck and the Koopman-von Neumann equations. These simulations allow us to witness the emergence of the classical world by observing how the Wigner function converges to the solution of the Fokker-Planck equation with sufficient classical diffusion. As a second example we simulate for first time a numerically challenging quantum system made of two coupled particles and observe the loss of decoherence on one of the particles as a result of continuous measurements performed on the other one, therefore obtaining quantitative predictions for the loss of coherence of both particles as a function of time.
The Wigner function is a useful tool for exploring the transition between quantum and classical dynamics, as well as the behavior of quantum chaotic systems. Evolving the Wigner function for open systems has proved challenging, however; a variety of methods have been devised but suffer from being cumbersome and resource intensive. Here we present an efficient fast-Fourier method for evolving the Wigner function that has a complexity of O(NlogN) where N is the size of the array storing the Wigner function. The efficiency, stability, and simplicity of this method allows us to simulate open-system dynamics previously thought to be prohibitively expensive. As a demonstration we simulate the dynamics of both one-particle and two-particle systems under various environmental interactions. For a single particle we also compare the resulting evolution with that of the classical Fokker--Planck and Koopman--von Neumann equations and show that the environmental interactions induce the quantum-to-classical transition as expected. In the case of two interacting particles we show that an environment interacting with one of the particles leads to the loss of coherence of the other.
We develop a dynamical tensor-train method for the Wigner--Fokker--Planck equation arising in continuous-variable open quantum systems. The Wigner function is evolved directly in tensor-train form using a projector-splitting integrator, while the nonlocal Wigner pseudo-differential operator is treated through low-rank representations of the potential difference and matrix-free exponential actions. The resulting method avoids full tensor-product computations in high-dimensional phase space. Numerical experiments in one, two, and three spatial dimensions demonstrate the accuracy and scalability of the method, including simulations in a six-dimensional phase space.
Recent progress in our understanding of quantum effects on the Brownian motion in an external potential is reviewed. This problem is ubiquitous in physics and chemistry, particularly in the context of decay of metastable states, for example, the reversal of the magnetization of a single domain ferromagnetic particle, kinetics of a superconducting tunnelling junction, etc. Emphasis is laid on the establishment of master equations describing the diffusion process in phase space analogous to the classical Fokker-Planck equation. In particular, it is shown how Wigner's [E. P. Wigner, Phys. Rev., 1932, 40, 749] method of obtaining quantum corrections to the classical equilibrium Maxwell-Boltzmann distribution may be extended to the dissipative non-equilibrium dynamics governing the quantum Brownian motion in an external potential V(x), yielding a master equation for the Wigner distribution function W(x,p,t) in phase space (x,p). The explicit form of the master equation so obtained contains quantum correction terms up to o(h(4)) and in the classical limit, h --> 0, reduces to the classical Klein-Kramers equation. For a quantum oscillator, the method yields an evolution equation coinciding in all respects with that of Agarwal [G. S. Agarwal, Phys. Rev. A, 1971, 4, 739]. In the high dissipation limit, the master equation reduces to a semi-classical Smoluchowski equation describing non-inertial quantum diffusion in configuration space. The Wigner function formulation of quantum Brownian motion is further illustrated by finding quantum corrections to the Kramers escape rate, which, in appropriate limits, reduce to those yielded via quantum generalizations of reaction rate theory.
A study is provided of dynamics and dissipative tunneling in a symmetric quartic double well potential. The numerical solution for the position autocorrelation function obtained through the Wigner–Fokker–Planck equation is compared with numerically exact results of Stockburger and Mak [J. Chem. Phys. 110, 4983 (1999)]. We find that the Wigner–Fokker–Planck dynamics agree well with the numerically exact computations, they account for both quantum coherences as well as quantum tunneling phenomena. This, in contrast to the mixed quantum classical approximation, which does not perform as well.
We introduce a method to solve nonlinear open quantum dynamics of a particle in situations where its state undergoes significant expansion in phase space while generating small quantum features at the phase-space Planck scale. Our approach involves simulating two steps. First, we transform the Wigner function into a time-dependent frame that leverages information from the classical trajectory to efficiently represent the quantum state in phase space. Next, we simulate the dynamics in this frame using a numerical method that implements this time-dependent nonlinear change of variables. To demonstrate the capabilities of our method, we examine the open quantum dynamics of a particle evolving in a one-dimensional weak quartic potential after initially being ground-state cooled in a tight harmonic potential. This approach is particularly relevant to ongoing efforts to design, optimize, and understand experiments targeting the preparation of macroscopic quantum superposition states of massive particles through nonlinear quantum dynamics. Published by the American Physical Society 2024
ABSTRACT: The concept of open quantum system is very broad and it is related to the ability of measuring only certain degree of freedoms of a particular system. Although this idea is relatively simple, the separation between the system of interest, the degree of freedoms that are accesible experimentally, and the reservoir, the degree of freedoms that are not accesible experimentally, is not always clear. For instance, in molecular systems, electronic spectroscopy has access only to electronic degree of freedoms so that the nuclear and vibrational degree of freedom become the reservoir, this makes its description not trivial. This scenario leads to have non-trivial and structured reservoirs and to develop powerful tools to analyze them. The hallmark of non-trivial and structured reservoirs is the non-Markovian dynamics. By translating the Feynman and Vernon influence functional approach into phase-space representation, we develop two theories of semiclassical evolution of the Wigner function of the system of interest that incorporate non-Markovian dynamics and highly non-trivial quantum effects such as non-locality of quantum mechanics: (i) We translate the Caldeira-Leggett model into phase-space representation of quantum mechanics and (ii) we consider the possibility of having non-linear baths and therefore, truly quantum reservoirs. During the last forty years the study of energy loss and coherence in quantum systems has been based on the Ullersma-Caldeira-Leggett model, a model that describes the environment of quantum systems of interest as a collection of harmonic oscillators with classical evolution. We constructed this model in the Wigner-Weyl representation of quantum mechanics and discuss the classical nature of the evolution of the bath modes and the semiclassical evolution of the central system. As an application of the semiclassical Wigner propagator, the non-Markovian time evolution under the Morse potencial is analyzed. There, it is clear how decohering processes shrink the propagator to smaller regions of phase space implying that the dynamics become more local, i.e., more classical. The current level of experimentation and control of physical systems have called into question the validity of the model in, one hand, molecular systems (e.g. photosynthetic complexes immersed in solvents, chemical systems in liquid phase or gas and manipulated with intense laser pulses) and, in the other hand, solid state systems (e.g. Josephson junctions, spins in quantum dots or "spinning ice"). Given the importance of these systems in the development of new quantum technologies and in the understanding of quantum phenomena in mesoscopic systems, it has become necessary to develop new models of the environment and efficient methodologies with quantitative prediction power. However, some of them are artificial modifications of the Ullersma-Caldeira-Leggett model without solid and clean physical support. Here we formulated a general framework that allows the study of quantum correlations in quantum systems in the presence of non-harmonic thermal baths (e.g. baths formed by strongly coupled diatomic molecules). This formulation will allow a more precise and quantitative description of processes such as the transport of excitons in photosynthetic complexes, the transfer of heat in solid state devices, among others. Results clearly show the non-classical time dynamics of the bath modes. The implementation of this particular theory remains, however, as a challenge.