Temporal coherence, anomalous moments, and pairing correlations in the classical-field description of a degenerate Bose gas

Abstract

The coherence properties of degenerate Bose gases have usually been expressed in terms of spatial correlation functions, neglecting the rich information encoded in their temporal behavior. In this article we show, using a Hamiltonian classical-field formalism, that temporal correlations can be used to characterize familiar properties of a finite-temperature degenerate Bose gas. The temporal coherence of a Bose-Einstein condensate is limited only by the slow diffusion of its phase, and thus the presence of a condensate is indicated by a sharp feature in the temporal power spectrum of the field. We show that the condensate mode can be obtained by averaging the field for a short time in an appropriate phase-rotating frame, and that for a wide range of temperatures, the condensate obtained in this approach agrees well with that defined by the Penrose-Onsager criterion based on one-body (spatial) correlations. For time periods long compared to the phase diffusion time, the field will average to zero, as we would expect from the overall U(1) symmetry of the Hamiltonian. We identify the emergence of the first moment on short time scales with the concept of U(1) symmetry breaking that is central to traditional mean-field theories of Bose condensation. We demonstrate that the short-time averaging procedure constitutes a general analog of the "anomalous" averaging operation of symmetry-broken theories by calculating the anomalous thermal density of the field, which we find to have form and temperature dependence consistent with the results of mean-field theories.

Anomalous density for Bose gases at finite temperature

“…This overall behavior has also been obtained in [36,43]. The anomalous density remains real and negative value whatever the temperature and the position.…”

“…On the other hand, the anomalous density profiles seem to have no structure at the centre of the trap for weak interactions. This is in contradiction with what was found in the literature [36,43] where the HFB-BdG approximation and the classical-field trajectories of the Projected GrossPitaevskii equation were used, and where these densities are found to have a "dip".…”

“…Yukalov [42] adopted a quite different approach by using the notion of representative statistical ensembles. Recently, Wright et al [43,44] have used the classical-field trajectories of the Projected Gross-Pitaevskii equation.…”

C-Field Methods for Non-Equilibrium Bose Gases

“…However, dynamical calculations within a pure PGPE formalism are able to provide useful insights into the dynamics of degenerate Bose-gas systems in situations where a precise identification of the method with the full field theory is impractical [45,58,66]. The PGPE has also been used to establish the connection between c-field methods and more traditional theoretical methods based on U(1) symmetry breaking [61,67].…”

Stochastic projected Gross-Pitaevskii equation for spinor and multicomponent condensates

“…At low to moderate temperatures generalised mean field theories have been developed, and successfully modelled a number of experimental scenarios. The Zaremba-Nikuni-Griffin (ZNG) [28][29][30][31][32][33], projected Gross-Pitaevskii equation (PGPE) [34][35][36][37][38][39][40][41][42][43][44] (including applications to spinor condensates [45,46]), and number conserving [47][48][49] theories each have advantages for describing BEC evolution, namely, relative ease of handling thermal cloud dynamics, inclusion of many appreciably populated coherent modes, and inclusion of off-diagonal long range order, respectively. At temperatures well below the BEC transition (T c ), these effects are essential aspects of finite-temperature BEC physics.…”