In the stochastic mean-field approach, an ensemble of initial conditions is considered to incorporate correlations beyond the mean-field. Then each starting point is propagated separately using the Time-Dependent Hartree-Fock equation of motion. This approach provides a rather simple tool to better describe fluctuations compared to the standard TDHF. Several illustrations are presented showing that this theory can be rather effective to treat the dynamics close to a quantum phase transition. Applications to fusion and transfer reactions demonstrate the great improvement in the description of mass dispersion.In the stochastic mean-field approach, an ensemble of initial conditions is considered to incorporate correlations beyond the mean-field. Then each starting point is propagated separately using the Time-Dependent Hartree-Fock equation of motion. This approach provides a rather simple tool to better describe fluctuations compared to the standard TDHF. Several illustrations are presented showing that this theory can be rather effective to treat the dynamics close to a quantum phase transition. Applications to fusion and transfer reactions demonstrate the great improvement in the description of mass dispersion.

arXiv:1210.2519EPJ Web of ConferencesJan 1, 20120 citations

Stochastic Representation of Quantum Dynamics

Zbigniew Haba

Springer eBooksJan 1, 19950 citations

Exact Stochastic Mean-Field dynamics

Denis Lacroix A, Guillaume Hupin, Karl Ernst Rehm +3 more

The exact evolution of a system coupled to a complex environment can be described by a stochastic mean-field evolution of the reduced system density. The formalism developed in Ref. [D.Lacroix, Phys. Rev. E77, 041126 (2008)] is illustrated in the Caldeira-Leggett model where a harmonic oscillator is coupled to a bath of harmonic oscillators. Similar exact reformulation could be used to extend mean-field transport theories in Many-body systems and incorporate two-body correlations beyond the mean-field one. The connection between open quantum system and closed many-body problem is discussed.The exact evolution of a system coupled to a complex environment can be described by a stochastic mean-field evolution of the reduced system density. The formalism developed in Ref. [D.Lacroix, Phys. Rev. E77, 041126 (2008)] is illustrated in the Caldeira-Leggett model where a harmonic oscillator is coupled to a bath of harmonic oscillators. Similar exact reformulation could be used to extend mean-field transport theories in Many-body systems and incorporate two-body correlations beyond the mean-field one. The connection between open quantum system and closed many-body problem is discussed.

arXiv:0812.3650AIP conference proceedingsJan 1, 20090 citations

Stochastic mean-field dynamics for fermions in the weak-coupling limit

Denis Lacroix A

Assuming that the effect of the residual interaction beyond the mean field is weak and has a short memory time, two approximate treatments of correlation in fermionic systems by means of the Markovian quantum jump are presented. A simplified scenario for the introduction of fluctuations beyond the mean field is presented first. In this theory, part of the quantum correlations between the residual interaction and the one-body density matrix are neglected and jumps occur between many-body densities formed of pairs of states $D=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|/\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$, where $|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$ and $|{\ensuremath{\Phi}}_{b}\ensuremath{\rangle}$ are antisymmetrized products of single-particle states. The underlying stochastic mean-field theory is discussed and is applied to the monopole vibration of a spherical $^{40}\mathrm{Ca}$ nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.Assuming that the effect of the residual interaction beyond the mean field is weak and has a short memory time, two approximate treatments of correlation in fermionic systems by means of the Markovian quantum jump are presented. A simplified scenario for the introduction of fluctuations beyond the mean field is presented first. In this theory, part of the quantum correlations between the residual interaction and the one-body density matrix are neglected and jumps occur between many-body densities formed of pairs of states $D=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|/\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$, where $|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}$ and $|{\ensuremath{\Phi}}_{b}\ensuremath{\rangle}$ are antisymmetrized products of single-particle states. The underlying stochastic mean-field theory is discussed and is applied to the monopole vibration of a spherical $^{40}\mathrm{Ca}$ nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.

arXiv:quant-ph/0509038Physical Review CApr 17, 20067 citations

Criteria for Feasible Monte Carlo Stochastic Simulations of Bosonic Markovian Open Quantum Dynamics

Toma Yoneya, Kazuya Fujimoto, Yuki Kawaguchi

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 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.

cond-mat.quant-gasCross-listedarXiv:2608.06056Annals of PhysicsAug 7, 20260 citations

^{40}\mathrm{Ca}

nucleus under the influence of a statistical ensemble of two-body contact interactions. This framework is however too simplistic to account for both fluctuation and dissipation. In the second part of this work, an alternative quantum jump method is obtained without making the approximation on quantum correlations. By restricting to two-particle--two-hole residual interactions, the evolution of the one-body density matrix of a correlated system is transformed into a Lindblad equation. The associated dissipative dynamics can be simulated by quantum jumps between densities written as $D=|\ensuremath{\Phi}\ensuremath{\rangle}\ensuremath{\langle}\ensuremath{\Phi}|$, where $|\ensuremath{\Phi}\ensuremath{\rangle}$ is a normalized Slater determinant. The associated stochastic Schr\"odinger equation for single-particle wave functions is given.