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arXiv:1606.00579 · Physical Review A

Shape dependence and anisotropic finite-size scaling of the phase coherence of three-dimensional Bose-Einstein-condensed gases

Giacomo Ceccarelli, Francesco Delfino, Michele Mesiti, Ettore Vicari

Published Nov 10, 2016• 7 citations
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Abstract

We investigate the equilibrium phase-coherence properties of Bose-condensed particle systems, focusing on their shape dependence and finite-size scaling (FSS). We consider three-dimensional (3D) homogeneous systems confined to anisotropic $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{a}$ boxes, below the Bose-Einstein-condensate (BEC) transition temperature ${T}_{c}$. We show that the phase correlations develop peculiar anisotropic FSS for any $T<{T}_{c}$, in the large-$L$ limit keeping the ratio $\ensuremath{\lambda}\ensuremath{\equiv}{L}_{a}/{L}^{2}$ and the particle density fixed. This phenomenon is effectively described by the 3D spin-wave theory. Its universality is confirmed by quantum Monte Carlo simulations of the 3D Bose-Hubbard model in the BEC phase. The phase-coherence correlations of very elongated BEC systems, $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$, are characterized by the coherence length ${\ensuremath{\xi}}_{a}\ensuremath{\sim}{A}_{t}{\ensuremath{\rho}}_{s}/T$, where ${A}_{t}$ is the transverse area and ${\ensuremath{\rho}}_{s}$ is the superfluid density.

DOI
10.1103/physreva.94.053609

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We investigate the equilibrium phase-coherence properties of Bose-condensed particle systems, focusing on their shape dependence and finite-size scaling (FSS). We consider three-dimensional (3D) homogeneous systems confined to anisotropic L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiLaL\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}{L}_{a}L\ifmmode×\else\texttimes\fiL\ifmmode×\else\texttimes\fiLa​ boxes, below the Bose-Einstein-condensate (BEC) transition temperature Tc{T}_{c}Tc​. We show that the phase correlations develop peculiar anisotropic FSS for any T<TcT<{T}_{c}T<Tc​, in the large-LLL limit keeping the ratio \ensuremathλ\ensuremath≡La/L2\ensuremath{\lambda}\ensuremath{\equiv}{L}_{a}/{L}^{2}\ensuremathλ\ensuremath≡La​/L2 and the particle density fixed. This phenomenon is effectively described by the 3D spin-wave theory. Its universality is confirmed by quantum Monte Carlo simulations of the 3D Bose-Hubbard model in the BEC phase. The phase-coherence correlations of very elongated BEC systems, \ensuremathλ\ensuremath→\ensuremath∞\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}\ensuremathλ\ensuremath→\ensuremath∞, are characterized by the coherence length \ensuremathξa\ensuremath∼At\ensuremathρs/T{\ensuremath{\xi}}_{a}\ensuremath{\sim}{A}_{t}{\ensuremath{\rho}}_{s}/T\ensuremathξa​\ensuremath∼At​\ensuremathρs​/T, where At{A}_{t}At​ is the transverse area and \ensuremathρs{\ensuremath{\rho}}_{s}\ensuremathρs​ is the superfluid density.
We investigate systems of interacting bosonic particles confined within slablike boxes of size ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ with $Z\ensuremath{\ll}L$, at their three-dimensional (3D) Bose-Einstein-condensation (BEC) transition temperature Tc{T}_{c}Tc​, and below Tc{T}_{c}Tc​ where they experience a quasi-two-dimensional (quasi-2D) Berezinskii-Kosterlitz-Thouless (BKT) transition (at TBKT<Tc{T}_{\mathrm{BKT}}<{T}_{c}TBKT​<Tc​ depending on the thickness ZZZ). The low-temperature phase below TBKT{T}_{\mathrm{BKT}}TBKT​ shows quasi-long-range order: the planar correlations decay algebraically as predicted by the 2D spin-wave theory. This dimensional crossover, from a 3D behavior for $T\ensuremath{\gtrsim}{T}_{c}$ to a quasi-2D critical behavior for $T\ensuremath{\lesssim}{T}_{\mathrm{BKT}}$, can be described by a transverse finite-size scaling limit in a slab geometry. Numerical evidence of the $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$ dimensional crossover is presented for the Bose-Hubbard model defined in anisotropic ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ lattices with $Z\ensuremath{\ll}L$. We extend this scaling analysis to the case the slab geometry of the gas is effectively realized by a transverse (inhomogeneous) harmonic trap. Finally, we discuss off-equilibrium behaviors arising from slow time variations of the temperature across the BEC transition of gases confined within a slab geometry. We argue that the system develops an off-equilibrium transverse finite-size scaling under these time-dependent protocols.
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We consider the evolution of superfluid properties of a three-dimensional p-wave Fermi gas from a weak coupling Bardeen-Cooper-Schrieffer (BCS) to strong coupling Bose-Einstein condensation (BEC) limit as a function of scattering volume. At zero temperature, we show that a quantum phase transition occurs for p-wave systems, unlike the s-wave case where the BCS to BEC evolution is just a crossover. Near the critical temperature, we derive a time-dependent Ginzburg-Landau (GL) theory and show that the GL coherence length is generally anisotropic due to the p-wave nature of the order parameter, and becomes isotropic only in the BEC limit.
We investigate the phase fluctuations of a Bose-Einstein condensate in one- (1D) and three-dimensional (3D) elongated harmonic traps with the help of the numerical simulation using classical field approximation. By calculating the eigenvalues of the one-dimensional single-particle density matrix we show that the phase fluctuations of a Bose-Einstein Condensate in a 1D system are due to the thermal low-energy excitations. We find that the phase fluctuations depend on the temperature as has been predicted [Phys. Rev. Lett. 85, 3745 (2000), Phys. Rev. Lett. 87, 050404 (2001)]. Finally we show that the phase coherence length of a condensate in a harmonic trap depends on the aspect ratio of anisotropy (geometry) of the trap. We determine the border between the 3D and quasi-one-dimensional systems by calculating the phase coherence length of the condensate.
When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomly-behaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a Bose-Einstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phase-coherent wavefunction of macroscopic size. But how exactly does this harmonious behaviour arise within a collection of incoherent atoms as their temperature decreases? Does the condensate wavefunction grow from a single central nucleus, or does it emerge simultaneously throughout the system? And how does the situation differ when the gas is cooled suddenly, away from thermodynamic equilibrium? Can non-equilibrium behaviour be inherited by the condensate that forms? 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This thesis is composed of four principal studies, each constituting a separate chapter.In Chapter 3 we study the formation of a Bose-Einstein condensate in a cigar-shaped 3D harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis in a recent experiment [Garrett et al., Phys. Rev. A 83, 013630 (2011)]. In this manner condensation is induced without cooling due to a localized increase in the phase space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semi-classical Hartree-Fock theory describing the condensate and thermal cloud. We observe there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the non-equilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory. In Chapter 4 we study the onset of phase coherence during the formation of a BEC following a sudden quench to below the transition temperature. Using the SPGPE, we simulate a recent experiment performed at ANU in which a sudden RF quench is applied to a dilute gas of metastable Helium atoms confined in an elongated 3D harmonic magnetic trap. At regularly-spaced intervals following the quench, broadband RF pulses are applied to homogeneously outcouple small (<5%) pulses of atoms, which expand and fall to a microchannel plate detector. The second-order correlation function at the detector, postulated to provide a measure of phase coherence of the in-situ cloud from which they were outcoupled, is measured versus time and compared with the condensate growth curve. A delay was observed in the relaxation of the second-order correlation function, relative to the peak in the number of condensate atoms. In our simulations we incorporate the effects of periodic outcoupling both the condensate and thermal cloud, and monitor the time-dependence of first- and second-order correlation functions in both position and momentum space. We observe long-range phase coherence in the final equilibrium state, though mild persistent phase fluctuations are present. We also observe the curious phenomenon of ``superbunching'' - an enhancement of density fluctuations - at specific values of the momentum, which we relate to anomalous correlations of fluctuations. Despite obtaining a good quantitative fit to the experimentally-measured condensate growth curve in our SPGPE simulations, we do not observe any evidence of the delay in coherence observed in the experiment, which remains unexplained. In Chapter 5 we use the SPGPE to study the crossover from a true BEC with long-range phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highly-elongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a non-equilibrium condensate, the dynamics of which we study in the chapter that follows. Sweeping through a range of temperatures and chemical potentials within the crossover region, we reveal the behaviour of phase fluctuations through analysis of correlation functions. We compare the ideal-gas approximation for the transition temperature with the more accurate value obtained via the Binder cumulant. 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Rev. Lett. 89, 270404 (2002)]. We simulate this experiment by modifying the SPGPE to incorporate a hydrodynamic thermal cloud, parametrized by axially and temporally-dependent temperature and chemical potential. Comparison against simulations without this axial dependence suggests the hydrodynamic behaviour of the thermal cloud plays a crucial role in the formation of a non-equilibrium condensate. Analyses of the first-order correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.
We study the equilibrium correlations of a Bose gas in an elongated three-dimensional harmonic trap using a grand-canonical classical-field method. We focus in particular on the progressive transformation of the gas from the normal phase, through a phase-fluctuating quasicondensate regime to the so-called true-condensate regime, with decreasing temperature. Choosing realistic experimental parameters, we quantify the density fluctuations and phase coherence of the atomic field as functions of the system temperature. We identify the onset of Bose condensation through analysis of both the generalized Binder cumulant appropriate to the inhomogeneous system, and the suppression of the effective many-body TTT matrix that characterizes interactions between condensate atoms in the finite-temperature field. We find that the system undergoes a second-order transition to condensation near the critical temperature for an ideal Bose gas in the strongly anisotropic three-dimensional geometry but remains in a strongly phase-fluctuating quasicondensate regime until significantly lower temperatures. We characterize the crossover from a quasicondensate to a true condensate by a qualitative change in the form of the nonlocal first-order coherence function of the field and compare our results to those of previous works employing a density-phase Bogoliubov--de Gennes analysis.