Path-integral formulation of bosonic Markovian open quantum dynamics with Monte Carlo stochastic trajectories using the Glauber–Sudarshan P, Wigner, and Husimi Q functions and hybrids
The Monte Carlo trajectory sampling of stochastic differential equations based on the quasiprobability distribution functions, such as the Glauber–Sudarshan P, Wigner, and Husimi Q functions, enables us to investigate bosonic open quantum many-body dynamics described by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation. In this method, the Monte Carlo samplings for the initial distribution and stochastic noises incorporate quantum fluctuations, and thus, we can go beyond the mean-field approximation. However, description using stochastic differential equations is possible only when the corresponding Fokker–Planck equation has a positive-semidefinite diffusion matrix. In this work, we analytically derive the stochastic differential equations for arbitrary Hamiltonian and jump operators based on the path-integral formula, independently of the derivation of the Fokker–Planck equation. In the course of the derivation, we formulate the path-integral representation of the GKSL equation by using the -ordered quasiprobability distribution function, which systematically describes the aforementioned quasiprobability distribution functions by changing the real parameter . The essential point of this derivation is that we employ the Hubbard–Stratonovich transformation in the path integral, and its application is not always feasible. We find that the feasible condition of the Hubbard–Stratonovich transformation is identical to the positive-semidefiniteness condition of the diffusion matrix in the Fokker–Planck equation. In the benchmark calculations, we confirm that the Monte Carlo simulations of the obtained stochastic differential equations well reproduce the exact dynamics of physical quantities and non-equal time correlation functions of numerically solvable models, including the Bose–Hubbard model. This work clarifies the applicability of the approximation and gives systematic and simplified procedures to obtain the stochastic differential equations to be numerically solved.
The Monte Carlo trajectory sampling of stochastic differential equations based on the quasiprobability distribution functions, such as the Glauber–Sudarshan P, Wigner, and Husimi Q functions, enables us to investigate bosonic open quantum many-body dynamics described by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation. In this method, the Monte Carlo samplings for the initial distribution and stochastic noises incorporate quantum fluctuations, and thus, we can go beyond the mean-field approximation. However, description using stochastic differential equations is possible only when the corresponding Fokker–Planck equation has a positive-semidefinite diffusion matrix. In this work, we analytically derive the stochastic differential equations for arbitrary Hamiltonian and jump operators based on the path-integral formula, independently of the derivation of the Fokker–Planck equation. In the course of the derivation, we formulate the path-integral representation of the GKSL equation by using the -ordered quasiprobability distribution function, which systematically describes the aforementioned quasiprobability distribution functions by changing the real parameter . The essential point of this derivation is that we employ the Hubbard–Stratonovich transformation in the path integral, and its application is not always feasible. We find that the feasible condition of the Hubbard–Stratonovich transformation is identical to the positive-semidefiniteness condition of the diffusion matrix in the Fokker–Planck equation. In the benchmark calculations, we confirm that the Monte Carlo simulations of the obtained stochastic differential equations well reproduce the exact dynamics of physical quantities and non-equal time correlation functions of numerically solvable models, including the Bose–Hubbard model. This work clarifies the applicability of the approximation and gives systematic and simplified procedures to obtain the stochastic differential equations to be numerically solved.
The Monte Carlo sampling of the stochastic differential equations (SDEs) based on the quasiprobability distribution function, such as the Glauber--Sudarshan P, Wigner, and Husimi Q functions provides a powerful framework for investigating bosonic open quantum many-body dynamics described by the Gorini--Kossakowski--Sudarshan--Lindblad (GKSL) equation, while considering the effects of quantum fluctuations beyond the mean-field approximation. However, the stochastic Monte Carlo simulation is possible only when the corresponding Fokker--Planck equation has a positive-semidefinite diffusion matrix, and the general conditions for the diffusion matrix to be positive semidefinite have remained unclear. In this work, starting from the path integral formulation, we first derive the sufficient conditions under which the diffusion matrix is positive semidefinite for an arbitrary Hamiltonian, jump operators, and choice of quasiprobability distribution functions. We also analytically derive the corresponding SDEs to be solved. We then investigate the dynamics of the GKSL equation in the thermodynamic limit and show that, depending on the form of the jump operators, the mean-field approximation may fail to describe the dynamics accurately, making stochastic Monte Carlo simulations indispensable. Furthermore, we derive the sufficient conditions under which the higher-order quantum fluctuation terms beyond the Fokker--Planck description vanish identically, even when the jump operators contain quadratic terms. Under these conditions, whenever the corresponding SDEs can be derived, the stochastic Monte Carlo simulation reproduces the exact dynamics. These results clarify the conditions under which the stochastic Monte Carlo simulations are both feasible and necessary for accurately describing the dynamics governed by the GKSL equation in phase space.
The truncated Wigner approximation (TWA) enables us to investigate bosonic quantum many-body dynamics, including open quantum systems described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. In the TWA, the Weyl-Wigner transformation, a way of mapping from quantum-mechanical operators to c -numbers, of the GKSL equation leads to the Fokker–Planck equation, which we calculate by reducing it to the corresponding stochastic differential equations. However, the Fokker–Planck equation is not always reduced to the stochastic differential equations depending on details of jump operators. In this work, we clarify the condition for obtaining the stochastic differential equations from the Fokker–Planck equation and derive analytical expressions of these equations for a system with an arbitrary Hamiltonian with jump operators that do not couple different states. This result leads to remarkable simplification of the complicated calculations required in the conventional procedures of the TWA. In the course of the derivation, we formulate the GKSL equation by using the path-integral representation based on the Weyl-Wigner transformation, which gives us a clear interpretation of the relation between the TWA and quantum fluctuations and allows us to calculate the non-equal time correlation functions in the TWA. In the benchmark calculations, we numerically confirm that the relaxation dynamics of physical quantities including the non-equal time correlation functions obtained in our formulation agrees well with the exact ones in the numerically solvable models. Since our formulation is widely applicable to bosonic systems such as the Bose-Hubbard model, it facilitates studies of the open quantum many-body phenomenon.
Abstract As a typical quantum many body problem, we consider the time evolution of density matrix elements in the Bose-Hubbard model. For an arbitrary initial state, these quantities can be obtained from an SDE or stochastic differential equation system. To this SDE system, a Girsanov transformation can be applied. This has the effect that all the information from the initial state moves into the drift part, into the mean field part, of the transformed system. In the large N limit with g=UN g = U N fixed, the diffusive part of the transformed system vanishes and as a result, the exact quantum dynamics is given by an ODE system which turns out to be the time dependent discrete Gross Pitaevskii equation. For the two site Bose-Hubbard model, the GP equation reduces to the mathematical pendulum and the difference of expected number of particles at the two lattice sites is equal to the velocity of that pendulum which is either oscillatory or it can have rollovers which then corresponds to the self trapping or insulating phase. As a by-product, we also find an equivalence of the mathematical pendulum with a quartic double well potential. Collapse and revivals are a more subtle phenomenom, in order to see these the diffusive part of the SDE system or quantum corrections have to be taken into account. This can be done with an approximation and collapse and revivals can be reproduced, numerically and also through an analytic calculation. Since expectation values of Fresnel or Wiener diffusion processes, we write the density matrix elements exactly in this way, can be obtained from parabolic second order PDEs, we also obtain various exact PDE representations. The paper has been written with the goal to come up with an efficient calculation scheme for quantum many body systems and as such the formalism is generic and applies to arbitrary dimension, arbitrary hopping matrices and, with suitable adjustments, to fermionic models.
We critically explore the applicability of a recently proposed framework to sample the quantum dynamics of a many-body quantum system interacting with light by stochastic trajectories, applying it to the closed and open Tavis-Cummings model (TCM). The stochastic differential equations (SDEs) sample the positive P phase-space representation by analog complex-valued dynamical variables that are linked to the quantum operators. Statistical average over the stochastic trajectories yields the evolution of the quantum mechanical expectation values. However, numerical implementation of these SDEs for the TCM indicates divergent solutions, also known from other phase-space methods. This limits the applicability of the framework to finite propagation times, that are strongly dependent on the physical parameters and initial conditions of the system. We outline the underlying mathematical reason for these divergences and show that their contribution to the averages are, however, essential. To attempt to regularize the divergences, we transform the SDEs to an equivalent set of SDEs with different noise realisations, thereby pushing the valid time boundary. Quantum collapse and revival of the TCM, however, cannot be recovered by the stochastic trajectory approach, pointing to the general difficulty of the applicability of stochastic phase-space sampling methods to systems with strong quantum features.
For the efficient simulation of open quantum systems, we often use quantum jump trajectories given by pure states that evolve stochastically to unravel the dynamics of the underlying master equation. In the Markovian regime, when the dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, this procedure is known as Monte Carlo wave function approach. However, beyond ultraweak system-bath coupling, the dynamics of the system is not described by an equation of GKSL type, but rather by the Redfield equation, which can be brought into pseudo-Lindblad form. Here, negative dissipation strengths prohibit the conventional approach. To overcome this problem, we propose a pseudo-Lindblad quantum trajectory (PLQT) unraveling. It does not require an effective extension of the state space, like other approaches, except for the addition of a single classical bit. We test the PLQT for the eternal non-Markovian master equation for a single qubit and an interacting Fermi-Hubbard chain coupled to a thermal bath and discuss its computational effort compared to solving the full master equation.
The time evolution of quantum fields is shown to be equivalent to a time-symmetric Fokker-Planck equation.Results are obtained using a Q-function representation, including fermion-fermion, boson-boson, and fermion-boson interactions with linear, quadratic, cubic, and quartic Hamiltonians, typical of QED and many other cases.For local boson-boson coupling, the resulting probability distribution is proved to have a positive, time-symmetric path integral and action principle, leading to a forward-backward stochastic process in both time directions.The solution corresponds to a c-number field equilibrating in an additional dimension.Paths are stochastic trajectories of fields in space-time, which are samples of a statistical mechanical steady state in a higher-dimensional space.We derive numerical methods and examples of solutions to the resulting stochastic partial differential equations in a higher time dimension, giving agreement with examples of simple bosonic quantum dynamics.This approach may lead to useful computational techniques for quantum field theory, as well as to new ontological models of physical reality.