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

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 $D=|{\Phi }_a\rangle \langle {\Phi }_b|/\langle {\Phi }_b|{\Phi }_a\rangle$, where $|{\Phi }_a\rangle$ and $|{\Phi }_b\rangle$ 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 $D=|\Phi \rangle \langle \Phi |$, where $|\Phi \rangle$ is a normalized Slater determinant. The associated stochastic Schrödinger equation for single-particle wave functions is given.

Figure 1

(Color online) Top: Evolution of the rms radius as a function of time. Black circles correspond to the standard TDHF evolution, different lines correspond to different stochastic paths. Bottom: Error bars correspond to the rms evolution obtained by averaging over different paths; black circles correspond to the TDHF case. The stochastic simulation is performed for $g_0=500$ MeV/fm. The average is taken over 200 trajectories. The width of the error bars corresponds to the statistical fluctuations of the rms.

Figure 2

(Color online) Evolution of the dispersion of the rms as a function of time for different values of $g_0$. The different curves from bottom to top correspond, respectively, to $g_0=100,250$, and 500 MeV fm. Solid lines and open circles correspond, respectively, to initial constraints $\lambda =0$ MeV fm${}^{-2}$ and $\lambda =0.25$ MeV fm${}^{-2}$.