Stochastic quantum dynamics beyond mean-field

Mean-field approaches where a complex fermionic many-body problem is replaced by an ensemble of independent particles in a self-consistent mean-field can describe many static and dynamical aspects. It generally provides a rather good approximation for the average properties of one-body degrees of freedom. However, the mean-field approximation generally fails to produce quantum fluctuations of collective motion. To overcome this difficulty, noise can be added to the mean-field theory leading to a stochastic description of the many-body problem. In the present work, we summarize recent progress in this field and discuss approaches where fluctuations have been added either to the initial time, like in the Stochastic Mean-Field theory or continuously in time as in the Stochastic Time-Dependent Hartree-Fock. In some cases, the initial problem can even be re-formulated exactly by introducing Quantum Monte-Carlo methods in real-time. The possibility to describe superfluid systems is also invoked. Successes and shortcomings of the different beyond mean-field theories are discussed and illustrated.Comment: 34 pages, submitted to EPJA-Review sectio

Quantal corrections to mean-field dynamics including pairing

Extending the stochastic mean-field model by including pairing, an approach is proposed for describing evolutions of complex many-body systems in terms of an ensemble of Time-Dependent Hartree-Fock Bogoliubov trajectories which is determined by incorporating fluctuations in the initial state. Non-linear evolution of the initial fluctuations provides an approximate description of quantal correlations and fluctuations of collective observables. Since the initial-state fluctuations break the particle-number symmetry, the dynamical description in which pairing correlations play a crucial role is greatly improved as compare to the mean-field evolution. The approach is illustrated for a system of particles governed by a pairing Hamiltonian.Comment: 5 pages, 2 figures, To appear in Phys. Rev. C (Rapid communication

Nucleus-nucleus potential, energy dissipation and mass dispersion in fusion and transfer reactions

The nucleus-nucleus potential and energy dissipation in fusion reactions are obtained from microscopic mean-field dynamics. The deduced potentials nicely reproduce the one extracted from experimental data. Energy dissipation shows a universal behaviour between different reactions. Also, the dispersion of mass distribution in transfer reaction is investigated in a stochastic mean-field dynamics. By including initial fluctuations in collective space, the description of the dispersion is much improved compared to that of mean field only. The result is consistent with the macroscopic phenomenological analysis of the experimental data.Comment: 4 pages, 4 figures. Proceedings of Second International Workshop on Compound Nuclear Reactions and Related Topics (CNR*09), October 5-8, 2009, Bordeaux, Franc

Collective response of nuclei: Comparison between experiments and extended mean-field calculations

The giant monopole, dipole and quadrupole responses in $^{40}$Ca, $^{90}$Zr, $^{120}$Sn and $^{208}$Pb are investigated using linear response treatment based on a stochastic one-body transport theory. Effects of the coupling to low-lying surface modes (coherent mechanism) and the incoherent mechanism due to nucleon-nucleon collisions are included beyond the usual mean-field description. We emphasize the importance of both mechanism in the fragmentation and damping of giant resonance. Calculated spectra are compared with experiment in terms of percentage of Energy-Weighted Sum-Rules in various energy regions. We obtained reasonable agreement in all cases. A special attention as been given to the fragmentation of the Giant Quadrupole Resonance in calcium and lead. In particular, the equal splitting of the $2^{+}$ in $^{40}$Ca is correctly reproduced. In addition, the appearance of fine structure in the response $^{208}$Pb is partly described by the calculations in which the coherent mechanism play an important role.Comment: 44 pages, 8 figures, submitted to Physical Review

Finite Temperature Nuclear Response in Extended Random-Phase Approximation

The nuclear collective response at finite temperature is investigated for the first time in the quantum framework of the small amplitude limit of the extended TDHF approach, including a non-Markovian collision term. It is shown that the collision width satisfies a secular equation. By employing a Skyrme force, the isoscalar monopole, isovector dipole and isoscalar quadrupole excitations in $^{40}Ca$ are calculated and important quantum features are pointed out. The collisional damping due to decay into incoherent 2p-2h states is small at low temperatures but increases rapidly at higher temperatures.Comment: 22 Latex pages including 9 figures. Phys. Rev. C (in press

Multi-nucleon transfer in 58Ni+60Ni{}^{58}\text{Ni}+{}^{60}\text{Ni} and 60Ni+60Ni{}^{60} \text{Ni}+{}^{60}\text{Ni} in stochastic mean-field approach

The multi-nucleon exchange mechanism in ${}^{58} \text{Ni}+{}^{60} \text{Ni}$ and ${}^{60} \text{Ni}+{}^{60}\text{Ni}$ collisions is analyzed in the framework of the stochastic mean-field approach. The results of calculations are compared with the TDRPA calculations and the recent data of ${}^{58} \text{Ni}+{}^{60} \text{Ni}$. A good description of the data and a relatively good agreement with the TDRPA calculations are found.Comment: 6 pages, 6 figure

A simplified BBGKY hierarchy for correlated fermionic systems from a Stochastic Mean-Field approach

The stochastic mean-field (SMF) approach allows to treat correlations beyond mean-field using a set of independent mean-field trajectories with appropriate choice of fluctuating initial conditions. We show here, that this approach is equivalent to a simplified version of the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy between one-, two-, ..., N-body degrees of freedom. In this simplified version, one-body degrees of freedom are coupled to fluctuations to all orders while retaining only specific terms of the general BBGKY hierarchy. The use of the simplified BBGKY is illustrated with the Lipkin-Meshkov-Glick (LMG) model. We show that a truncated version of this hierarchy can be useful, as an alternative to the SMF, especially in the weak coupling regime to get physical insight in the effect beyond mean-field. In particular, it leads to approximate analytical expressions for the quantum fluctuations both in the weak and strong coupling regime. In the strong coupling regime, it can only be used for short time evolution. In that case, it gives information on the evolution time-scale close to a saddle point associated to a quantum phase-transition. For long time evolution and strong coupling, we observed that the simplified BBGKY hierarchy cannot be truncated and only the full SMF with initial sampling leads to reasonable results.Comment: 10 pages, 4 figure

One-body energy dissipation in fusion reaction from mean-field theory

Information on dissipation in the entrance channel of heavy-ion collisions is extracted by macroscopic reduction procedure of Time-Dependent Hartree-Fock theory. The method gives access to a fully microscopic description of the friction coefficient associated with transfer of energy from the relative motion towards intrinsic degrees of freedom. The reduced friction coefficient exhibits a universal behavior, i.e. almost independent of systems investigated, whose order of magnitude is comparable with the calculations based on linear response theory. Similarly to nucleus-nucleus potential, especially close to the Coulomb barrier, there are sizable dynamical effects on the magnitude and form factor of friction coefficient.Comment: 7 pages, 10 figure

Stochastic Semi-Classical Description of Fusion at Near-Barrier Energies

Fusion reactions of heavy ions are investigated by employing a simple stochastic semi-classical model which includes the coupling between relative motion and low frequency collective surface modes of colliding ions similarly to the quantal coupled-channels description. The quantal effect enters into the calculation through the initial zero-point fluctuations of the surface vibrations. Good agreement with the result of coupled-channels calculations as well as data is obtained for the fusion cross sections of nickel isotopes. The internal excitations in non-fusing events as well as the fusion time are investigated.Comment: 8 pages, 8 figures, Published in Phys. Rev.