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arXiv:0707.0458 · Physical Review A

Fluctuations of the number of particles within a given volume in cold quantum gases

G. E. Astrakharchik, R. Combescot, Лев П. Питаевский

Published Dec 19, 2007• 31 citations
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Abstract

In ultracold gases many experiments use atom imaging as a basic observable. The resulting image is averaged over a number of realizations and mostly only this average is used. Only recently the noise has been measured to extract physical information. In the present paper we investigate the quantum noise arising in these gases at zero temperature. We restrict ourselves to the homogeneous situation and study the fluctuations in particle number found within a given volume in the gas, and more specifically inside a sphere of radius $R$. We show that zero-temperature fluctuations are not extensive and the leading term scales with sphere radius $R$ as ${R}^{2}\text{ }\text{ln}\text{ }R$ (or $\text{ln}\text{ }R$) in three- (or one-) dimensional systems. We calculate systematically the next term beyond this leading order. We consider first the generic case of a compressible superfluid. Then we investigate the whole Bose-Einstein-condensation (BEC) --BCS crossover, and in particular the limiting cases of the weakly interacting Bose gas and of the free Fermi gas.

DOI
10.1103/physreva.76.063616

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In ultracold gases many experiments use atom imaging as a basic observable. The resulting image is averaged over a number of realizations and mostly only this average is used. Only recently the noise has been measured to extract physical information. In the present paper we investigate the quantum noise arising in these gases at zero temperature. We restrict ourselves to the homogeneous situation and study the fluctuations in particle number found within a given volume in the gas, and more specifically inside a sphere of radius RRR. We show that zero-temperature fluctuations are not extensive and the leading term scales with sphere radius RRR as R2 ln R{R}^{2}\text{ }\text{ln}\text{ }RR2 ln R (or ln R\text{ln}\text{ }Rln R) in three- (or one-) dimensional systems. We calculate systematically the next term beyond this leading order. We consider first the generic case of a compressible superfluid. Then we investigate the whole Bose-Einstein-condensation (BEC) --BCS crossover, and in particular the limiting cases of the weakly interacting Bose gas and of the free Fermi gas.
We theoretically investigate the specific heat at constant volume C V in the BCS(Bardeen-Cooper-Schrieffer)-BEC(Bose-Einstein-condensation)-crossover regime of an ultracold Fermi gas, below the superfluid phase transition temperature T c . Within the strong-coupling framework developed by Nozières and Schmitt-Rink, we show that the temperature dependence of C V drastically changes as one passes through the crossover region, and is sensitive to strong fluctuations in the Cooper channel near the unitarity limit. We also compare our results to a recent experiment on a 6 Li unitary Fermi gas. Since fluctuation effects are a crucial key in the BCS-BEC-crossover phenomenon, our results would be helpful in considering how the fermionic BCS superfluid changes into BEC with increasing the interaction strength, from the viewpoint of specific heat.
We present an exact many-body theory of ultracold fermionic gases for the Bose-Einstein condensation (BEC) regime of the BEC-BCS crossover. This is a purely fermionic approach which treats explicitly and systematically the dimers formed in the BEC regime as made of two fermions. We consider specifically the zero temperature case and calculate the first terms of the expansion of the chemical potential in powers of the density nnn. We derive first the mean-field contribution, which has the expected standard expression when it is written in terms of the dimer-dimer scattering length aM{a}_{M}aM​. We go next in the expansion to the Lee-Huang-Yang order, proportional to n3∕2{n}^{3∕2}n3∕2. We find the far less obvious result that it retains also the same expression in terms of aM{a}_{M}aM​ as for elementary bosons. The composite nature of the dimers appears only in the next term proportional to n2{n}^{2}n2.
The mesoscopic effects in the quantum trapped gases of the Bose atoms constitute\nthe main subject of the present thesis. These effects are the most difficult for\nthe theoretical analysis in the quantum statistical physics since they can’t be seen by\nneither a standard quantum mechanics of the simple microscopic systems of one or\nvery few atoms nor a standard statistical physics of the macroscopic systems that are\ninfinite in the bulk (thermodynamic) limit.\nMost of the experiments on the cold quantum gases performed in the last decade,\nstarting from the first demonstration of BEC in 1995, involve the mesoscopic systems\nof a finite number of atoms. The mesoscopic effects should manifest themselves most\nclearly and easily near a critical temperature of BEC; however, they could be observed\nalso above and below the critical temperature.\nHere I study the quantum and thermal fluctuations of the Bose-Einstein condensate\n(BEC) in a box with the periodic boundary conditions under a particle-number\nconstraint. The above constraint is the only reason for the BEC and is crucial for the\nmesoscopic effects in the BEC fluctuations, especially in the vicinity of the critical\ntemperature in the Bose gas.\nI employ the particle-number conserving operator formalism of Girardeau and\nArnowitt introduced in 1959 to analyze the canonical ensemble fluctuations. I present\nanalytical formulas and numerical calculations for the central moments of the ground\nstate occupation fluctuations in an ideal Bose gas in a box with a mesoscopic number of particles.\nI present the analysis of the BEC statistics both on a temperature at a fixed\nnumber of particles and on a number of particles at a fixed temperature. Both analyses\nare valid for the purpose of understanding the important mesoscopic effects near the\ncritical temperature. I emphasize the non-Gaussian nature of the fluctuations.\nThe presented formalism can be generalized to the case of a weakly interacting\nBose gas in a box in the framework of the Bogoliubov approximation. The work in\nthis direction is in progress but is not included in the present thesis.
We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms. Adapting the concept of full counting statistics, well known from quantum optics and mesoscopic electron transport, we study second-order as well as higher-order correlators of density fluctuations. Using the mean-field BCS state to describe the whole interval between the BCS limit and the Bose-Einstein condensate (BEC) limit, we obtain an exact expression for the cumulant-generating function of the density fluctuations of an atomic cloud. In the two-dimensional case, we obtain a closed analytical expression. Poissonian fluctuations of a molecular condensate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We also discuss the BEC-BCS crossover of the third cumulant and the temperature dependence of the second cumulant.
We present a theoretical study of the ground state of the BCS-BEC crossover in dilute two-dimensional Fermi gases. While the mean-field theory provides a simple and analytical equation of state, the pressure is equal to that of a noninteracting Fermi gas in the entire BCS-BEC crossover, which is not consistent with the features of a weakly interacting Bose condensate in the BEC limit and a weakly interacting Fermi liquid in the BCS limit. The inadequacy of the two-dimensional mean-field theory indicates that the quantum fluctuations are much more pronounced than those in three dimensions. In this work, we show that the inclusion of the Gaussian quantum fluctuations naturally recovers the above features in both the BEC and the BCS limits. In the BEC limit, the missing logarithmic dependence on the boson chemical potential is recovered by the quantum fluctuations. Near the quantum phase transition from the vacuum to the BEC phase, we compare our equation of state with the known grand canonical equation of state of two-dimensional Bose gases and determine the ratio of the composite boson scattering length aB{a}_{B}aB​ to the fermion scattering length a2D{a}_{2\mathrm{D}}a2D​. We find ${a}_{B}\ensuremath{\simeq}0.56{a}_{2\mathrm{D}}$, in good agreement with the exact four-body calculation. We compare our equation of state in the BCS-BEC crossover with recent results from the quantum Monte Carlo simulations and the experimental measurements and find good agreements.
We study fluctuations of the number of Bose condensed atoms in a weakly interacting homogeneous and trapped gases. For a homogeneous system we apply the particle-number-conserving formulation of the Bogoliubov theory and calculate the condensate fluctuations within the canonical and the microcanonical ensembles. We demonstrate that, at least in the low-temperature regime, predictions of the particle-number-conserving and traditional, nonconserving theory are identical, and lead to the anomalous scaling of fluctuations. Furthermore, the microcanonical fluctuations differ from the canonical ones by a quantity which scales normally in the number of particles, thus predictions of both ensembles are equivalent in the thermodynamic limit. We observe a similar behavior for a weakly interacting gas in a harmonic trap. This is in contrast to the trapped, ideal gas, where microcanonical and canonical fluctuations are different in the thermodynamic limit.