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arXiv:cond-mat/9812016 · Physical Review A

Testing quantum correlations in a confined atomic cloud by the scattering of fast atoms

Anatoly B. Kuklov, Boris V. Svistunov

Published Aug 1, 1999• 15 citations
arXivHTMLPDF

Abstract

We suggest measuring the one-particle density matrix of a trapped ultracold atomic cloud by the scattering of fast atoms in a pure momentum state off the cloud. The lowest-order probability for the process, resulting in a pair of outcoming fast atoms for each incoming one, as well as of its time-reversed counterpart, turns out to be given by the Fourier transform of the density matrix. Accordingly, important information about quantum correlations can be deduced directly from the differential scattering cross section of these processes. Several most interesting cases of scattering --- from a single condensate containing a vortex and from a split condensate characterized by some phase difference --- are discussed.

DOI
10.1103/physreva.60.r769

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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

•Jan 1, 2007•2 citations
We suggest measuring the one-particle density matrix of a trapped ultracold atomic cloud by the scattering of fast atoms in a pure momentum state off the cloud. The lowest-order probability for the process, resulting in a pair of outcoming fast atoms for each incoming one, as well as of its time-reversed counterpart, turns out to be given by the Fourier transform of the density matrix. Accordingly, important information about quantum correlations can be deduced directly from the differential scattering cross section of these processes. Several most interesting cases of scattering --- from a single condensate containing a vortex and from a split condensate characterized by some phase difference --- are discussed.
We suggest measuring the one-particle density matrix of a trapped ultracold atomic cloud by scattering fast atoms in a pure momentum state off the cloud. The lowest-order probability for the process, resulting in a pair of outcoming fast atoms for each incoming one, as well as of its time reversed counterpart, turns out to be given by the Fourier transform of the density matrix. Accordingly, important information about quantum correlations can be deduced directly from the differential scattering cross-section of these processes. Several most interesting cases of scattering - from a single condensate containing a vortex, and from a split condensate characterized by some phase difference - are discussed.
The first successful experimental realization of a Bose-Einstein condensate (BEC) in 1995 sparked a still growing research activity that in the meantime has led to beautiful and surprising results, such as the interference of matter waves [4], the Mott-superfluid phase transition [49] or the direct observation of Anderson localization of matter waves [10]. The strong interest in BECs arises, among others, from the possibility to investigate known but hard to study quantum systems on a rather macroscopic scale: for example one can load ultra-cold bosons into an optical lattice created by a standing light wave, and thereby simulate a solid state device. When it comes to a measurement on the BEC, however, essentially all measurement techniques result in the destruction of the condensate: for example in the so called “time of flight method” the optical lattice is switched off to observe the expansion of the trapped atoms and thereby directly visualize their momentum distribution, which highlights the condensed component [11]. In this thesis we elaborate on a means to probe a Bose-Einstein condensate from outside, with a scattering particle. The interaction between the probe particle and the bosons allows energy exchange between them, i.e., this setup includes inelastic scattering. After the scattering process, the condensate is still confined in the optical lattice and can therefore be used for further investigations. Of course, this method of probing a BEC is only then a serious alternative if the properties of the BEC can be inferred from the scattering signal. Besides outlining the scattering formalism, the central aim of this work is to show how the properties of the BEC can be deduced from those quantities accessible in a scattering experiment, like for example the transmission or the inelastic cross section. The other, equally strong motivation of this work concerns the signature of the interaction between the bosons in the scattering quantities. The dynamics of an analog classical system depends strongly on the interaction strength, which can induce a complex structure of the underlying phase space like, e.g., the appearance of a separatrix or even the emergence of chaotic dynamics in a globally mixed regular-chaotic phase space. We are interested to which extent such phase-space metamorphosis is reflected in the properties of the scattering particle. Currently light scattering (Bragg spectroscopy) on an optical lattice [78, 77] or rather on an optical lattice embedded into a cavity [62,68,19] is under intense investigation. These efforts predominantly aim at a characterization of the Mott-superfluid transition by specific scattering properties. In the present thesis we focus on condensates loaded in two and three lattice sites, with a large number of bosons. These systems exhibit a diversity of important physical effects (as we will see below) and thus are of paradigmatic relevance for a detailed understanding of multiparticle quantum dynamics in periodic potentials.
The use of temporal correlations in scattered photons to probe the microscopic dynamics of ultracold quantum gases has emerged as a powerful, minimally destructive approach for in situ analysis. Here, we demonstrate that temporal coherence spectroscopy can quantitatively characterize atomic motion in a magnetic trap, despite the perturbative effects of the probing light. By measuring the first-order correlation function g (1) (τττ ) of light scattered by a 87 Rb cloud confined in a quadrupole trap, we identify radiation-pressure-induced acceleration and heating as the origin of the apparent discrepancy between coherence spectra and temperatures inferred from time-of-flight measurements. A simple dynamical model incorporating these effects restores agreement between theory and experiment, establishing coherence spectroscopy as a reliable in situ probe of velocity distributions in trapped atomic ensembles. Our results pave the way for time-resolved studies of nonequilibrium dynamics and thermalization processes in confined cold gases, complementing conventional destructive imaging techniques.
Recently developed techniques allow for simultaneous measurements of the positions of all ultra-cold atoms in a trap with high resolution. Each such single-shot experiment detects one element of the quantum ensemble formed by the cloud of atoms. Repeated single-shot measurements can be used to determine all correlations between particle positions as opposed to standard measurements that determine particle density or two-particle correlations only. In this paper we discuss the possible outcomes of such single-shot measurements in the case of cloud of ultra-cold noninteracting Fermi atoms. We show that the Pauli exclusion principle alone leads to correlations between particle positions that originate from unexpected spatial structures formed by the atoms.
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