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arXiv:0907.1025 · Physical Review A

Coherence time of a Bose-Einstein condensate

Alice Sinatra, Yvan Castin, Emilia Witkowska

Published Sep 16, 2009• 37 citations
arXivHTMLPDF

Abstract

Temporal coherence is a fundamental property of macroscopic quantum systems, such as lasers in optics and Bose-Einstein condensates in atomic gases and it is a crucial issue for interferometry applications with light or matter waves. Whereas the laser is an ``open'' quantum system, ultracold atomic gases are weakly coupled to the environment and may be considered as isolated. The coherence time of a condensate is then intrinsic to the system and its derivation is out of the frame of laser theory. Using quantum kinetic theory, we predict that the interaction with noncondensed modes gradually smears out the condensate phase, with a variance growing as $\mathcal{A}{t}^{2}+\mathcal{B}t+\mathcal{C}$ at long times $t$, and we give a quantitative prediction for $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$. Whereas the coefficient $\mathcal{A}$ vanishes for vanishing energy fluctuations in the initial state, the coefficients $\mathcal{B}$ and $\mathcal{C}$ are remarkably insensitive to these fluctuations. The coefficient $\mathcal{B}$ describes a diffusive motion of the condensate phase that sets the ultimate limit to the condensate coherence time. We briefly discuss the possibility to observe the predicted phase spreading, also including the effect of particle losses.

DOI
10.1103/physreva.80.033614

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Temporal coherence is a fundamental property of macroscopic quantum systems, such as lasers in optics and Bose-Einstein condensates in atomic gases and it is a crucial issue for interferometry applications with light or matter waves. Whereas the laser is an ``open'' quantum system, ultracold atomic gases are weakly coupled to the environment and may be considered as isolated. The coherence time of a condensate is then intrinsic to the system and its derivation is out of the frame of laser theory. Using quantum kinetic theory, we predict that the interaction with noncondensed modes gradually smears out the condensate phase, with a variance growing as At2+Bt+C\mathcal{A}{t}^{2}+\mathcal{B}t+\mathcal{C}At2+Bt+C at long times ttt, and we give a quantitative prediction for A\mathcal{A}A, B\mathcal{B}B, and C\mathcal{C}C. Whereas the coefficient A\mathcal{A}A vanishes for vanishing energy fluctuations in the initial state, the coefficients B\mathcal{B}B and C\mathcal{C}C are remarkably insensitive to these fluctuations. The coefficient B\mathcal{B}B describes a diffusive motion of the condensate phase that sets the ultimate limit to the condensate coherence time. We briefly discuss the possibility to observe the predicted phase spreading, also including the effect of particle losses.
The central problem of this chapter is temporal coherence of a three-dimensional spatially homogeneous Bose-condensed gas, initially prepared at finite temperature and then evolving as an isolated interacting system. A first theoretical tool is a number-conserving Bogoliubov approach that allows to describe the system as a weakly interacting gas of quasi-particles. This approach naturally introduces the phase operator of the condensate: a central actor since loss of temporal coherence is governed by the spreading of the condensate phase-change. A second tool is the set of kinetic equations describing the Beliaev-Landau processes for the quasi-particles. We find that in general the variance of the condensate phase-change at long times ttt is the sum of a ballistic term ∝t2\propto t^2∝t2 and a diffusive term ∝t\propto t∝t with temperature and interaction dependent coefficients. In the thermodynamic limit, the diffusion coefficient scales as the inverse of the system volume. The coefficient of t2t^2t2 scales as the inverse volume squared times the variance of the energy of the system in the initial state and can also be obtained by a quantum ergodic theory (the so-called eigenstate thermalisation hypothesis).
It is generally assumed that a condensate of paired fermions at equilibrium is characterized by a macroscopic wavefunction with a well-defined, immutable phase. In reality, all systems have a finite size and are prepared at non-zero temperature; the condensate has then a finite coherence time, even when the system is isolated. This fundamental effect, crucial for applications using macroscopic coherence, was scarcely studied. Here, we link the coherence time to the condensate phase dynamics, and show using a microscopic theory that the time derivative of the condensate phase operator θ^0\hat{\theta}_0θ^0​ is proportional to a chemical potential operator which includes both the fermionic pair-breaking and the bosonic pair-motion excitation branches. For a given realization of the number of particle NNN and of the energy EEE, the phase evolves at long times as −2μmc(E,N)t/ℏ-2\mu_{mc}(E,N) t/\hbar−2μmc​(E,N)t/ℏ where μmc(E,N)\mu_{mc}(E,N)μmc​(E,N) is the microcanonical chemical potential; fluctuations of NNN and EEE from one realization to the other then lead to a ballistic spreading of the phase and to a Gaussian decay of the temporal coherence function with a characteristic time ∝N1/2\propto N^{1/2}∝N1/2.On the contrary, in the absence of energy and number fluctuations, the decay of the temporal coherence function is exponential with a characteristic time scaling as NNN due to the diffusive motion of θ^0\hat{\theta}_0θ^0​ in the environnement created by the excited modes. We give an explict expression of this characteristic time at low temperature in the case where the bosonic branch is convex and the phonons undergo 2↔12 \leftrightarrow 12↔1 Beliaev-Landau process. Finally, we propose methods to measure each contribution to the coherence time using ultracold atoms.
The kinetics of the formation of coherent correlation properties associated with Bose condensation is studied in detail. The evolution of a nonequilibrium state with no “condensate-seed” is related to a hierarchy of relaxation times. At the first stage, a particle flux in energy space toward low energies sets in. The evolution in this case is described by a nonlinear Boltzmann equation, with a characteristic time given by interparticle collisions. When the particles which will later form the condensate have a kinetic energy which is less than the potential energy, a quasicondensate starts to form. In this stage, fluctuations of the density (but not of the phase) are suppressed and short-range coherent correlation properties are governed by the equation of motion for a quasiclassical complex field. The next stage is connected with the formation of the long-range order. The time for forming topological order and therefore genuine superfluidity proves to be dependent on the system size. The off-diagonal long-range order, arising after the attenuation of long-wave phase fluctuations, has a size-dependent relaxation time as well. Introduction The problem of Bose–Einstein Condensation (BEC) kinetics, being interesting in itself, has acquired a special significance in connection with experimental efforts to observe this condensation in a number of systems with particles with a finite lifetime. Such systems include spin-polarized atomic hydrogen [1], excitons [2] and biexcitons [3] in semiconductors and, more recently, laser-cooled atomic systems [4].
We study the dynamics of a Bose-Einstein condensate in a double-well potential in the two-mode approximation. The dissipation of energy from the condensate is described by the coupling to a thermal reservoir of noncondensate modes. As a consequence of the coupling, the self-locked population imbalance in the macroscopic quantum self-trapping decays away. We show that a coherent state predicted by spontaneous symmetry breaking is not robust and decoheres rapidly into a statistical mixture due to the interactions between condensate and noncondensate atoms. However, via stochastic simulations we find that with a sufficiently fast measurement rate of the relative phase between the two wells the matter wave coherence is established even in the presence of the decoherence.
In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.
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.