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arXiv:1203.3286 · Physical Review C

Symmetry breaking and fluctuations within stochastic mean-field dynamics: Importance of initial quantum fluctuations

Denis Lacroix A, Sakir Ayik B, Bulent Yilmaz

Published Apr 19, 2012• 32 citations
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Abstract

Dynamics of spontaneous symmetry breaking and fluctuations in the Lipkin-Meshkov-Glick model are investigated in a stochastic mean-field approach. Different from the standard mean-field approach, in the stochastic approach, initial state fluctuations are incorporated. In weak coupling, the approach perfectly reproduces the exact quantal dynamics. On the other hand, for increasing coupling strength, above the symmetry breaking threshold, the approach provides descriptions of gross properties (i.e., time-averaged behavior) of the exact quantal evolution.

DOI
10.1103/physrevc.85.041602

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Dissipation and noise in mean field dynamics

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Cambridge University Press eBooks•Nov 4, 2022•0 citations
Dynamics of spontaneous symmetry breaking and fluctuations in the Lipkin-Meshkov-Glick model are investigated in a stochastic mean-field approach. Different from the standard mean-field approach, in the stochastic approach, initial state fluctuations are incorporated. In weak coupling, the approach perfectly reproduces the exact quantal dynamics. On the other hand, for increasing coupling strength, above the symmetry breaking threshold, the approach provides descriptions of gross properties (i.e., time-averaged behavior) of the exact quantal evolution.
In the stochastic mean-field (SMF) approach, an ensemble of initial values for a selected set of one-body observables is formed by stochastic sampling from a phase-space distribution that reproduces the initial quantum fluctuations. Independent mean-field evolutions are performed with each set of initial values followed by averaging over the resulting ensemble. This approach has been recently shown to be rather versatile and accurate in describing the correlated dynamics beyond the independent particle picture. In the original formulation of SMF, it was proposed to use a Gaussian assumption for the phase-space distribution. This assumption turns out to be rather effective when the dynamics of an initially uncorrelated state is considered, which was the case in all applications of this approach up to now. Using the Lipkin-Meshkov-Glick (LMG) model, we show that such an assumption might not be adequate if the quantum system under interest is initially correlated and presents configuration mixing between several Slater determinants. In this case, a more realistic description of the initial phase-space is necessary. We show that the SMF approach can be advantageously combined with standard methods to describe phase-space in quantum mechanics. As an illustration, the Husimi distribution function is used here to obtain a realistic representation of the phase-space of a quantum many-body system. This method greatly improves the description of initially correlated fermionic many-body states. In the LMG model, while the Gaussian approximation failed to describe these systems in all interaction strength range, the novel approach gives a perfect agreement with the exact evolution in the weak coupling regime and significantly improves the description of correlated systems in the strong coupling regime.
By extending the stochastic mean-field model by including pairing, an approach is proposed for describing the evolution 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. The nonlinear evolution of the initial fluctuations provides an approximate description of quantal correlations and fluctuations of collective observables. Since the initial-state fluctuations break particle-number symmetry, the dynamical description in which pairing correlations play a crucial role is greatly improved as compared to the mean-field evolution. The approach is illustrated for a system of particles governed by a pairing Hamiltonian.
In Chapter 6 we presented the main computational schemes to derive the dynamical laws for the mean field, including the back-reaction from quantum fluctuations. These equations may be derived from the variation of the CTPEA. The result of this approach is a semiclassical theory of a c-number condensate interacting with a quantized fluctuation field. This approach developed at this level of sophistication is limited as it offers no description of the fluctuations themselves. In most applications the magnitude of the fluctuations can be comparable and at times dominates the effects of the mean field in the semiclassical description. One possible way to incorporate fluctuations is to use the 2PI formalism, where the propagators describing the fluctuations are considered as dynamical variables evolving along with the mean fields. In this chapter we shall explore a different strategy, which is to allow for a stochastic component in the mean field. This component arises from both the uncertainty of the initial configuration of the mean field, and from the fluctuations in the back-reaction from the quantized excitations. Both sources of randomness combine so that stochastic averages in the noisy theory reproduce suitable quantum averages in the underlying quantum field theory. Formally, this approach lifts the seemingly overladen CTPEA. So far in this generally complex object, only the real part is enlisted in the derivation of the relevant equations of motion of the mean field.
We study the dynamics of the Lipkin-Meshkov-Glick (LMG) model with a finite number of spins. In the thermodynamic limit, the ground state of the LMG model with an isotropic Hamiltonian in the broken phase breaks to a mean-field ground state with a certain direction. However, when the spin number NNN is finite, the exact ground state is always unique and is not given by a classical mean-field ground state. Here, we prove that when NNN is large but finite, through a tiny external perturbation, a localized state which is close to a mean-field ground state can be prepared, which mimics spontaneous symmetry breaking. Also, we find the localized in-plane spin polarization oscillates with two different frequencies $\ensuremath{\sim}O(1/N)$, and the lifetime of the localized state is long enough to exhibit this oscillation. We numerically test the analytical results and find that they agree very well with each other. Finally, we link the phenomena to quantum time crystals and time quasicrystals.
We investigate the superfluid dynamics of a Josephson junction beyond the mean-field description, incorporating the role of thermal fluctuations as well as quantum fluctuations. Using a formalism that accounts for the fluctuations in a homogeneous gas, and under the assumption that the transport of the non-condensed component is negligible, we derive a corrected equation of motion within the two-site approximation. The resulting corrections for the typical dynamical quantities, like the Josephson frequency, the strength of macroscopic quantum self-trapping, and the threshold for spontaneous symmetry breaking, allow us to predict the effects of both types of fluctuations and assess their relative importance in different regimes in a semianalytical fashion. For all the dynamical quantities, the quantum fluctuations are shown to play an opposite role with respect to the thermal fluctuations. Josephson frequency is decreased by thermal fluctuations and both the critical strenghts of macroscopic quantum self trapping and spontaneous symmetry breaking are increased. We assess the experimentally accessible regimes by calculating the relevant parameters of recent experimental realizations of Bosonic Josephson junction and show that the expected regime is dominated by quantum fluctuations.
Dissipation and noise in mean field dynamicsIn Chapter 6 we presented the main computational schemes to derive the dynamical laws for the mean field, including the back-reaction from quantum fluctuations.These equations may be derived from the variation of the CTPEA.The result of this approach is a semiclassical theory of a c-number condensate interacting with a quantized fluctuation field.This approach developed at this level of sophistication is limited as it offers no description of the fluctuations themselves.In most applications the magnitude of the fluctuations can be comparable and at times dominates the effects of the mean field in the semiclassical description.One possible way to incorporate fluctuations is to use the 2PI formalism, where the propagators describing the fluctuations are considered as dynamical variables evolving along with the mean fields.In this chapter we shall explore a different strategy, which is to allow for a stochastic component in the mean field.This component arises from both the uncertainty of the initial configuration of the mean field, and from the fluctuations in the back-reaction from the quantized excitations.Both sources of randomness combine so that stochastic averages in the noisy theory reproduce suitable quantum averages in the underlying quantum field theory.Formally, this approach lifts the seemingly overladen CTPEA.So far in this generally complex object, only the real part is enlisted in the derivation of the relevant equations of motion of the mean field.By regarding the CTPEA as a kind of influence functional, we shall see that the imaginary part contains the information about the stochastic sources.The material in this chapter also clarifies the relationship between the CTP and the influence functional approaches.This issue has been addressed by Su et al. [SCYC88] and the authors [CalHu94].In Chapter 5, we derived a Langevin equation for the long-wavelength modes of a quantum field, viewed as an open system interacting with the environment made up of short-wavelength modes.The system-environment divide we shall assume in this chapter is more elusive, since it depends on the c vs. q-number nature rather than on the value of a "hard" observable such as wavelength.In the end, as we shall discuss in detail, the physics is very much the same in one or the other approach.The stochastic mean field approach we shall discuss in this chapter has the redeeming feature that it does not force us to choose an a priori separation between modes which go into the system and which are relegated to the environment.In this sense, it is more pliable to the demands of a particular application: for example, if