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APS

Phase of a Bose-Einstein Condensate.

Juha Javanainen

Published May 1, 1996• 0 citations
OpenAlex

Abstract

Summary form only given. According to the conventional argument, Bose-Einstein condensate (BEG) has a macroscopic wave function with a definite phase. This phase is analogous to the phase of laser light. Here we describe a novel theoretical approach to the study of phase, coherence, and potential of interference of a BEG. We consider a thought experiment in which two BECs are dropped on top of each other. We adapt the standard theory of photon counting to atom detection, and calculate the joint probability density for atom detection.

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Cambridge University Press eBooks•Sep 11, 2008•0 citations
Summary form only given. According to the conventional argument, Bose-Einstein condensate (BEG) has a macroscopic wave function with a definite phase. This phase is analogous to the phase of laser light. Here we describe a novel theoretical approach to the study of phase, coherence, and potential of interference of a BEG. We consider a thought experiment in which two BECs are dropped on top of each other. We adapt the standard theory of photon counting to atom detection, and calculate the joint probability density for atom detection.
We study the interference of two Bose-Einstein condensates within an elementary model. The detection of the atoms is modeled by adapting the standard theory of photon detection. Even though the condensates are taken to be in number states with no phases whatsoever, our stochastic simulations of atom detection produce interference patterns as would also be predicted on the basis of the phases of the macroscopic wave functions describing the condensates. In statistical mechanics terms, we have devised a method to analyze spontaneous symmetry breaking for an arbitrary (not necessarily larger) number of particles.
Summary form only given. Bose-Einstein condensation (BEG) has been observed in a dilute gas of sodium atoms. A Bose-Einstein condensate consists of a macroscopic population of the ground state of the system and is a coherent state of matter. In an ideal gas, this phase transition is purely quantum statistical. The study of BEC in weakly interacting systems that can be controlled and observed with precision holds the promise of revealing new macroscopic quantum phenomena that can be understood from first principles.
Summary form only given.We consider two overlapping and optically thin dilute gas Bose-Einstein condensates occupying two different Zeeman sublevels. We propose an optical method for increasing the number of condensate atoms. Two laser beams drive Raman transitions between the Zeeman states and induce a coherent quantum tunneling between the Bose-Einstein condensates. The scattering of photons is dominantly coherent and in the forward direction due to the Bose enhancement. This corresponds to the scattering of atoms between the two condensates. Some laser-driven Raman transitions also scatter atoms between the condensate and noncondensate atom fractions. For a range of condensate phase differences there is destructive quantum interference of the amplitudes for scattering atoms out of the condensates. Because the total atom scattering rate into the condensates is unaffected the condensates grow. This mechanism is analogous to that responsible for optical lasing without inversion, and it may find application as a pump for an atom laser.
This thesis features fundamental experiments on the coherence properties of Bose-Einstein condensates (BEC) and atom lasers, made possible by a newly developed apparatus combining a Bose-Einstein condensate with an ultrahigh nesse optical cavity.Using the cavity as a single atom detector, atoms extracted from a trapped cloud are detected with unprecedented sensitivity, and the rst time-resolved counting of single atoms from a quantum degenerate source was demonstrated.O-diagonal long-range order is a dening property of a BEC and equivalent to longrange phase coherence.Its formation is a complex process accompanied by spontaneous symmetry breaking.We witnessed this formation in an ultracold cloud of 87 Rb atoms crossing the phase transition of BEC.Two overlapping atomic beams originating from dierent positions inside the condensate propagated downwards into the cavity.They showed a matter wave interference pattern with a visibility determined by the coherence of the probed regions.The modulation of the ux of atoms, recorded with single-atom resolution, yields the visibility, whereas its mean value gives information about the density of the sample.By shock cooling, a cloud of atoms was prepared in a highly nonequilibrium state slightly above the phase transition.While the ensemble subsequently relaxed, it crossed the phase transition into an equilibrium with a small condensate fraction.Employing our minimally invasive measurement method, the growth of both the rst-order coherence and the density of the sample was tracked in real time while the system crossed the phase transition.The growth of the coherence was found to start later, proceed faster and nish earlier than the increase in density.The coherent region increased at one fth of the speed of sound.Employing time-resolved single atom counting with the cavity detector, the full counting statistics of an atom laser beam was recorded.In a Hanbury Brown and Twiss type experiment, the second-order correlation function was found to be constant g (2) (τ ) = 1.00±0.01,proving the atom laser's second-order coherence and the absence of excess intensity uctuations that were observed in a pseudothermal beam.An analysis of the time interval distributions of atoms in the beam conrmed these ndings.The iii
We consider the interference of two overlapping ideal Bose-Einstein condensates. The usual description of this phenomenon involves the introduction of a condensate wave function with a definite phase. We investigate the origin of this phase and the theoretical basis of treating interference. It is possible to construct a phase state for which the particle number is uncertain, but the phase is known. How such a state would be prepared before an experiment is not obvious. We show that a phase can also arise from experiments using condensates with known particle numbers. The analysis of measurements in such states also gives us a prescription for preparing phase states. The connection of this procedure to questions of spontaneously broken gauge symmetry and to hidden variables is discussed.
Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. In Sec. 13.1 we discuss Josephson tunnelling of a condensate between two wells and the role of fluctuations in particle number and phase. The number and phase variables play a key role in the description of the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap (Sec. 13.2). Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. Density fluctuations in a Bose gas are studied in Sec. 13.3, where we relate atomic clock shifts to the two-particle correlation function. The ability to manipulate gaseous Bose–Einstein condensates by lasers has made possible the study of coherent matter wave optics and in Sec. 13.4 we describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves. How to characterize Bose–Einstein condensation in terms of the density matrix is the subject of Sec. 13.5, where we also consider fragmented condensates.