Bose-Einstein Condensation into Nonequilibrium States Studied by Condensate Focusing
I. Shvarchuck, Ch. Buggle, Dmitry S. Petrov, Kai Dieckmann, M. Zielonkowski, M. Kemmann, T. G. Tiecke, Wolf von Klitzing, Georgy V. Shlyapnikov, Jook T. M. Walraven
We report the formation of Bose-Einstein condensates into nonequilibrium states. Our condensates are much longer than equilibrium condensates with the same number of atoms, show strong phase fluctuations, and have a dynamical evolution similar to that of quadrupole shape oscillations of regular condensates. The condensates emerge in elongated traps as the result of local thermalization when the nucleation time is short compared to the axial oscillation time. We introduce condensate focusing as a new method to extract the phase-coherence length of Bose-Einstein condensates.
We report the formation of Bose-Einstein condensates into nonequilibrium states. Our condensates are much longer than equilibrium condensates with the same number of atoms, show strong phase fluctuations, and have a dynamical evolution similar to that of quadrupole shape oscillations of regular condensates. The condensates emerge in elongated traps as the result of local thermalization when the nucleation time is short compared to the axial oscillation time. We introduce condensate focusing as a new method to extract the phase-coherence length of Bose-Einstein condensates.
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? In this thesis we aim to answer these and other questions by studying the formation dynamics of Bose-Einstein condensates and the development of phase coherence. The research described herein makes use of several theoretical tools, which we lay out on the workbench in Chapter 2: Semi-classical Hartree-Fock theory provides a mean-field description of a condensate and thermal cloud at equilibrium; Quantum kinetic theory is used to calculate the dynamics of condensate growth into an equilibrium state, from a non-equilibrium thermal cloud; The stochastic projected Gross-Pitaevskii equation (SPGPE) employs a classical-field description to encapsulate the non-equilibrium dynamics of the condensate and low-energy region of the atom cloud, coupled to an equilibrium thermal cloud. 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. By comparing the phase coherence length with the spatial extent across which density fluctuations are suppressed, we identify the BEC/quasicondensate crossover point. We compare directly with predictions of mean-field theory [Petrov et al., Phys. Rev. Lett. 87, 050404 (2001)] and find that phase fluctuations persist to lower temperatures in our SPGPE simulations. We calculate anomalous correlations of the fluctuation field and show that the resulting correction to the mean-field condensate self-interaction energy is significant within the quasicondensate regime. In Chapter 6 we study the formation of a BEC into a non-equilibrium state by simulating an experiment in which a dilute gas of Rubidium-87 atoms in a highly-elongated 3D harmonic magnetic trap is rapidly quenched to below the transition temperature. Following the quench, a condensate is observed to form with an excessively elongated shape, and subsequently undergo damped quadrupole shape oscillations [Shvarchuck et al., Phys. 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 growth of an elongated phase-fluctuating condensate from a nonequilibrium thermal cloud in the axially hydrodynamic regime, obtained by shock cooling. Quantitative measurements using momentum Bragg spectroscopy reveal the evolution of the phase coherence as the condensate grows to equilibrium. We find no delay between the population growth and the development of the phase coherence. We also compare the growth of the condensate with numerical simulations based on kinetic theory, revealing quantitative agreement except for an unexplained time delay.
Background 1 1.22This thesis 2 Chapterr 2 Theoretical background 5 2.11 Rubidium 5 2.22 Magnetic trapping 5 2.33 Bose-Einstein condensation 8 2.44 BEC formation and phase coherence 2.55 Evaporative cooling 2.5.11Evaporative cooling Lensing Chapterr 4 Broad-area diode laser system 4.11 Abstract 4.22 Introduction 4.33 Experimental setup 4.44 Characteristics 4.55 MOT application 4.66 Summary Chapterr 5 Hydrodynamic properties of dense atomic clouds Vlll l 5.11 Introduction 5.22 Density and temperature analysis 5.33 Anisotropic expansion of hydrodynamic clouds 5.44 Frequency shifts and damping of shape oscillations Chapterr 6 Condensation into non-equilibrium states and focusing of a condensate 6.11 Introduction 6.22 Preparation of degenerate samples out of equilibrium 6.33 Kinetics of the condensate growth 6.44 Formation of a non-equilibrium BEC 6.4.11Local critical temperature 6.4.22 Truncation and local temperature 6.55 Oscillation modes 6.66 Non-equilibrium phase fluctuations 6.77 Focusing of a condensate 6.7.11Focusing principle 6.7.22 Focal broadening 6.7.33 Applications of BEC focusing 6.88 High condensate fractions in non-equilibrium states Bibliographyy Summaryy Samenvattingg AcknowledgementssMostt of the theoretical groundwork on the interacting quantum gases has been developedd in the 50's and the 60's in the context of superfluidity of 4 He.However, a detailedd comparison between theory and experiment is extremely difficult in the case of liquidd helium because its density is rather high and can be varied only within a narrow range..In the 70's the observation of BEC in dilute atomic gases under equilibrium conditionss was known to be impossible.Therefore, the efforts shifted towards investigationss of metastable systems.Thee first attempts to reach BEC were done in spin-polarised atomic hydrogen.Foundationss for many techniques and methods were laid in the course of that work.Hydrogenn quantum gas was first stabilised in a cryogenic environment by Silvera and Walravenn [93] and by Cline et al. [23].Magnetic trapping was first demonstrated in sodium [79]] and in hydrogen by Hess et al. [50] and van Roijen et al. [87].Another critically importantt technique, evaporative cooling, was first experimentally demonstrated in hydrogenn [51] and further developed in [74,97, 105].Bose-Einsteinn condensation in alkali systems was achieved in magnetic traps throughh the combination of optical cooling methods with the evaporative cooling technique.iss obtained by observation of the anisotropic character of the expansion.Finally, we measuree frequency shifts and damping of shape oscillations.Inn the final part of the Thesis, Chapter 6, we describe the results produced in the experimentss on formation of condensates far from equilibrium.We compare our work with previouss experiments on condensate formation and describe how the process of formation iss triggered in our system.A brief section deals with the growth of the condensate fraction.Further,, we show how the concepts of local sample temperature and the critical temperature arisee in elongated clouds with high elastic collision rates.We present a simple model, whichh illustrates how the non-equilibrium character of the condensates leads to the quadrupolee oscillations.We also discuss non-equilibrium phase fluctuations, which manifestt themselves in the form of stripes in the time-of-flight absorption images.Condensatee focusing is introduced as a novel method for investigation of Bose-Einstein condensates..The focusing of a condensate in free flight arises from axial contraction of the expandingg cloud when the gas is released from the trap during the inward phase of a shape oscillation.. Possible applications of BEC focusing are discussed, with an estimate of the coherencee length given as an example.The last part of the chapter covers condensation into non-equilibriumm states with high condensate fractions.The situations of large and small condensatee fractions are compared.
The problem of understanding how a coherent, macroscopic Bose- Einstein condensate (BEC) emerges from the cooling of a thermal Bose gas has attracted significant theoretical and experimental interest over several decades. The pioneering achievement of BEC in weakly interacting dilute atomic gases in 1995 was followed by a number of experimental studies examining the growth of the BEC number, as well as the development of its coherence. More recently, there has been interest in connecting such experiments to universal aspects of nonequilibrium phase transitions, in terms of both static and dynamical critical exponents. Here, the spontaneous formation of topological structures such as vortices and solitons in quenched cold-atom experiments has enabled the verification of the Kibble-Zurek mechanism predicting the density of topological defects in continuous phase transitions, first proposed in the context of the evolution of the early universe. This chapter reviews progress in the understanding of BEC formation and discusses open questions and future research directions in the dynamics of phase transitions in quantum gases. Introduction The equilibrium phase diagram of the dilute Bose gas exhibits a continuous phase transition between condensed and noncondensed phases. The order parameter characteristic of the condensed phase vanishes above some critical temperature T c and grows continuously with decreasing temperature below this critical point. However, the dynamical process of condensate formation has proved to be a challenging phenomenon to address both theoretically and experimentally. This formation process is a crucial aspect of Bose systems and of direct relevance to all condensates discussed in this book, despite their evident system-specific properties. Important questions leading to intense discussions in the early literature include the time scale for condensate formation and the role of inhomogeneities and finite-size effects in “closed” systems. These issues are related to the concept of spontaneous symmetry breaking, its causes, and implications for physical systems (see, for example, Chapter 5 by Snoke and Daley). In this chapter, we give an overview of the dynamics of condensate formation and describe the present understanding provided by increasingly well-controlled cold-atom experiments and corresponding theoretical advances over the past twenty years. We focus on the growth of BECs in cooled Bose gases, which, from a theoretical standpoint, requires a suitable nonequilibrium formalism.
The occurrence of phase fluctuations due to thermal excitations in Bose-Einstein condensates (BECs) is studied for a variety of temperatures and trap geometries. We observe the statistical nature of the appearance of phase fluctuations and characterize the dependence of their average value on temperature, number of particles, and the trapping potential. We find pronounced phase fluctuations for condensates in very elongated traps in a broad temperature range. The results are of great importance for the realization of BEC in quasi-1D geometries, for matter wave interferometry with BECs, as well as for coherence properties of guided atom laser beams.
Evolution and collapse of a trapped Bose condensate with negative scattering length are predetermined by 3-body recombination of Bose-condensed atoms and by feeding of the condensate from the nonequilibrium thermal cloud. The collapse, starting once the number of condensate atoms reaches the critical value, ceases and turns to expansion when the density becomes so high that the recombination losses dominate over attractive interparticle interaction. As a result, we obtain a sequence of collapses, each of them followed by dynamic oscillations of the condensate. In every collapse the 3-body recombination burns only a part of the Bose-condensed atoms.