We present a fully quantum mechanical treatment of collisional correlations beyond mean-field dynamics. Correlations are handled by an ensemble of mean-field states emerging from a stochastic propagation scheme. This is done by first propagating two-body correlations in time-dependent perturbation theory for a certain time span, reducing the coherently correlated state into an incoherent sum of two-particle-two-hole states, and then sampling this state into an ensemble of mean-field states according to the jump probability obtained from perturbation theory. The scheme is applied to a simple 1D test case calibrated to typical scales of molecules and clusters. Even for the small system size and low dimension, the scheme produces robust results. Occupation numbers and entropy show steady relaxation towards thermal equilibrium.
We present a fully quantum mechanical treatment of collisional correlations beyond mean-field dynamics. Correlations are handled by an ensemble of mean-field states emerging from a stochastic propagation scheme. This is done by first propagating two-body correlations in time-dependent perturbation theory for a certain time span, reducing the coherently correlated state into an incoherent sum of two-particle-two-hole states, and then sampling this state into an ensemble of mean-field states according to the jump probability obtained from perturbation theory. The scheme is applied to a simple 1D test case calibrated to typical scales of molecules and clusters. Even for the small system size and low dimension, the scheme produces robust results. Occupation numbers and entropy show steady relaxation towards thermal equilibrium.
We present a formalism to describe collisional correlations responsible for thermalization effects in finite quantum systems . The approach consists in a stochastic extension of time dependent mean field theory. Correlations are treated in time dependent perturbation theory and loss of coherence is assumed at some time intervals allowing a stochastic reduction of the correlated dynamics in terms of a stochastic ensemble of time dependent mean-fields. This theory was formulated long ago in terms of density matrices but never applied in practical cases because of its complexity. We propose here a reformulation of the theory in terms of wave functions and use a simplified 1D model of cluster and molecules allowing to test the theory in a schematic but realistic manner. We illustrate the performance in terms of several observables, in particular global moments of the density matrix and single particle entropy built on occupation numbers. The occupation numbers remain fixed in time dependent mean-field propagation and change when evaluating the correlations, then taking fractional values. They converge asymptotically towards Fermi distributions which is a clear indication of thermalization.
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 40Ca 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.
Abstract We propose a collisional extension of time-dependent mean-field theories on the basis of a recently proposed stochastic extension of mean-field dynamics (stochastic time-dependent Hartree–Fock, STDHF). The latter theory is unfortunately too involved to envision practical applications in realistic systems in the near future and is thus bound to model systems. It is also hard to explore moderate to low energies with STDHF, because of vanishing transition probabilities that are impossible to sample properly. For such moderately excited situations covering small fluctuations, we compactify sampling by employing the same average mean field for all STDHF trajectories. The new approach, coined average STDHF (ASTDHF), ignores the fluctuations of the mean field but still accounts correctly for the collisional correlations responsible for dissipative features on top of mean-field dynamics. We detail the main features of the new approach in relation to existing equations, in particular quantum kinetic theories. The new theory is directly connected to STDHF, both formally and practically. We thus discuss in detail how the two approaches are related to each other. We apply the new scheme to illustrative examples taking as benchmark STDHF dynamics in 1D. ASTDHF provides results that are in remarkable agreement with the more elaborate STDHF. It makes it a promising approach to deal with dissipative dynamics in finite quantum systems, because of its moderate cost allowing applications in realistic systems and the possibility of exploring any excitation energy range where collisional correlations are expected to play a role.
We show that the dynamics of interacting fermions can be exactly replaced by a quantum jump theory in the many-body density matrix space. In this theory, jumps occur between densities formed of pairs of Slater determinants, ${D}_{\mathit{ab}}=|{\ensuremath{\Phi}}_{a}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\Phi}}_{b}|$, where each state evolves according to the stochastic Schr\"odinger equation given by O. Juillet and Ph. Chomaz [Phys. Rev. Lett. 88, 142503 (2002)]. A stochastic Liouville-von Neumann equation is derived as well as the associated. Bogolyubov-Born-Green-Kirwood-Yvon hierarchy. Due to the specific form of the many-body density along the path, the presented theory is equivalent to a stochastic theory in one-body density matrix space, in which each density matrix evolves according to its own mean-field augmented by a one-body noise. Guided by the exact reformulation, a stochastic mean-field dynamics valid in the weak coupling approximation is proposed. This theory leads to an approximate treatment of two-body effects similar to the extended time-dependent Hartree-Fock scheme. In this stochastic mean-field dynamics, statistical mixing can be directly considered and jumps occur on a coarse-grained time scale. Accordingly, numerical effort is expected to be significantly reduced for applications.
We discuss extensions of time-dependent mean-field theories such as time-dependent local density approximation (TDLDA) in order to include incoherent dynamical correlations, which are known to play a key role in far-off equilibrium dynamics. We focus here on the case of irradiation dynamics in clusters and molecules. The field, still largely unexplored, requires quantum approaches which represents a major formal and computational effort. We present several approaches we have investigated to address such an issue. We start with time-dependent current-density functional theory (TDCDFT), known to provide damping in the linear regime and explore its capability far-off equilibrium. We observe difficulties with the scaling of relaxation times with deposited energy. We next briefly discuss semi-classical approaches which deliver kinetic equations applicable at sufficiently large excitation energies. We then consider a first quantum kinetic equation at the level of a simplified, though rather elaborate in its content, relaxation time approximation (RTA). Thanks to its sophistication, the method allows us to address numerous realistic irradiation scenarios beyond the usual domain of reliability of such theories. We demonstrate in particular the key role played by dense spectral regions in the impact of dissipation in the response of the irradiated system. RTA nevertheless remains a phenomenological approach which calls for more fundamental descriptions. This is achieved by a stochastic extension of mean field theory, coined stochastic time dependent Hartree–Fock (STDHF), which provides an ensemble description of far-off equilibrium dynamics. The method is equivalent to a quantum kinetic equation complemented by a stochastic collision term. STDHF clearly leads to proper thermalization behaviors in 1D test systems considered here. It remains limited by its ensemble nature which requires possibly huge ensembles to properly sample small transition rates. An alternative approach, coined average STDHF (ASTDHF), consists in overlooking mean field fluctuations of STDHF. ASTDHF provides a robust tool, properly matching STDHF when possible and allowing extension to realistic dynamical scenarios in full 3D. It can also be used in open systems to explore, as done in RTA, the competition between ionization and dissipation.
A powerful perspective in understanding nonequilibrium quantum dynamics is through the time evolution of its entanglement content. Yet apart from a few guiding principles for the entanglement entropy, to date, much less is known about the refined characteristics of entanglement propagation. Here, we unveil signatures of the entanglement evolving and information propagating out of equilibrium, from the view of the entanglement Hamiltonian. We investigate quantum quench dynamics of prototypical Bose-Hubbard model using state-of-the-art numerical technique combined with conformal field theory. Before reaching equilibrium, it is found that a current operator emerges in the entanglement Hamiltonian, implying that entanglement spreading is carried by particle flow. In the long-time limit the subsystem enters a steady phase, evidenced by the dynamic convergence of the entanglement Hamiltonian to the expectation of a thermal ensemble. Importantly, the entanglement temperature in steady state is spatially independent, which provides an intuitive trait of equilibrium. These findings not only provide crucial information on how equilibrium statistical mechanics emerges in many-body dynamics, but also add a tool to exploring quantum dynamics from the perspective of the entanglement Hamiltonian.