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arXiv:0807.0751 · Physical Review A

Quantum fluctuations in the image of a Bose gas

Antonio Negretti, Carsten Henkel, Klaus Mølmer

Published Aug 25, 2008• 20 citations
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Abstract

We analyze the information content of density profiles for an ultracold Bose gas of atoms and extract resolution limits for observables contained in these images. Our starting point is density correlations that we compute within the Bogoliubov approximation, taking into account quantum and thermal fluctuations beyond mean-field theory. This provides an approximate way to construct the joint counting statistics of atoms in an array of pixels covering the gas. We derive the Fisher information of an image and the associated Cram\'er-Rao sensitivity bound for measuring observables contained in the image. We elaborate on our recent study on position measurements of a dark soliton [Negretti et al., Phys. Rev. A 77, 043606 (2008)] where a sensitivity scaling with the atomic density as ${n}^{\ensuremath{-}3∕4}$ was found. We discuss here a wider class of soliton solutions and present a detailed analysis of the Bogoliubov excitations and the gapless (Goldstone) excitation modes. These fluctuations around the mean field contribute to the noise in the image, and we show how they can actually improve the ability to locate the position of the soliton.

DOI
10.1103/physreva.78.023630

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We analyze the information content of density profiles for an ultracold Bose gas of atoms and extract resolution limits for observables contained in these images. Our starting point is density correlations that we compute within the Bogoliubov approximation, taking into account quantum and thermal fluctuations beyond mean-field theory. This provides an approximate way to construct the joint counting statistics of atoms in an array of pixels covering the gas. We derive the Fisher information of an image and the associated Cram'er-Rao sensitivity bound for measuring observables contained in the image. We elaborate on our recent study on position measurements of a dark soliton [Negretti et al., Phys. Rev. A 77, 043606 (2008)] where a sensitivity scaling with the atomic density as n\ensuremath−3∕4{n}^{\ensuremath{-}3∕4}n\ensuremath−3∕4 was found. We discuss here a wider class of soliton solutions and present a detailed analysis of the Bogoliubov excitations and the gapless (Goldstone) excitation modes. These fluctuations around the mean field contribute to the noise in the image, and we show how they can actually improve the ability to locate the position of the soliton.
We show that the position of a dark matter-wave soliton can be determined with a precision that scales with the atomic density as ${n}^{\ensuremath{-}3/4}$. This surpasses the standard shot-noise detection limit for independent particles, without use of squeezing and entanglement, and it suggests that interactions among particles may present new advantages in high-precision metrology. We also take into account quantum density fluctuations due to phonon and Goldstone modes and we show that they, somewhat unexpectedly, actually improve the resolution. This happens because the fluctuations depend on the soliton position and make a larger amount of information available.
We study, within the framework of the classical-field approximation, the density correlations of a weakly interacting expanding Bose gas for the whole range of temperatures across the Bose-Einstein condensation threshold. We focus on elongated quasi-one-dimensional systems where there is a huge discrepancy between the existing theory and experimental results [A. Perrin et al., Nat. Phys. 8, 195 (2012)]. We find that the density correlation function is not reduced for temperatures below the critical one as it is predicted for the ideal gas or for a weakly interacting system within the Bogoliubov approximation. This behavior of the density correlations agrees with the above-mentioned experiment with the elongated system. Although the system was much larger than that studied here, we believe that the behavior of the density correlation function found there is quite generic. Our theoretical study indicates also large density fluctuations in the trap in the quasicondensate regime where only phase fluctuations were expected. We argue that the enhanced density fluctuations can originate in the presence of interactions in the system, or more precisely in the presence of spontaneous dark solitons in the elongated gas at thermal equilibrium.
We show that the position of a dark matter-wave soliton can be determined with a precision that scales with the atomic density as n(−3/4). This surpasses the standard shot-noise detection limit for independent particles, without use of squeezing and entanglement, and it suggests that interactions among particles may present new advantages in high-precision metrology. We also take into account quantum density fluctuations due to phonon and Goldstone modes and we show that they, somewhat unexpectedly, actually improve the resolution. This happens because the fluctuations depend on the soliton position and make a larger amount of information available.
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.
Ultra-cold spinor Bose gases present rich physics due to the special combination of superfluidity and magnetism in a quantum system. This system was first realized in 1998 by confining a Bose-Einstein condensate in an all-optical trap.\nIn this thesis, we consider the fluctuations of observable quantities in a spin-1 Bose gas. Understanding how fluctuations arise due to the spin excitations of the condensate is important because it offers an insight into how measurement noise reveals the many-body physics of the system.\nTo begin, we present the mean-field and Bogoliubov theory of a uniform spin-1 Bose gas subject to a constant magnetic field, describing the condensate and its low-energy collective excitations.\nWe then develop a formalism to describe the fluctuations in a general density-like observable. We start from the two-point correlation function and cast it in the form of a generalised static structure factor determined by the three Bogoliubov quasiparticle excitation branches. We derive analytic results for the fluctuation amplitudes and the temperature-dependent static structure factors for observables of total density and the three spin densities.\nFor all four magnetic phases, we analyse the spinor order parameter and quasiparticle spectra while numerically mapping out the fluctuation amplitudes and static structure factors for the total and spin density operators.\nWe describe the fluctuations in experimental measurements made within finite cells, which is an important step to making meaningful predictions for experiments. We consider cylindrical cells and gaussian cells as two limiting cases. We apply this analysis to an experimentally realisable system of a quasi-2D spinor gas in a harmonic trap, comparing extensive numeric results with analytic limits.
In this thesis we present a detailed investigation of the role played by quantum and thermal fluctuations in ultracold Bose gases. We begin with a review of several important concepts and analytical tools within a functional integration formalism. We first focus on the so-called zero-range approximation for the interaction potential, by recovering the Bogoliubov results and the Landau two-fluid model from a field-theory perspective. In deriving the beyond-mean-field equation of state, we are going to show that a crucial point concerns the proper regularization of the divergent zero-point energy. Among the alternative approaches to investigate finite-temperatures Bose gases, we apply the kinetic theory to explain some recent results on the sound propagation in two-dimensional Bose gases. We then move to consider the eventual corrections to the thermodynamics of Bose gases due to the finite-range character of the interaction potential. The coupling constants of the finite-range model are related to measurable scattering parameters through an effective-field-theory approach. The role of finite-range corrections is considered not only in three spatial dimensions but also in systems with lower dimensionalities. Our analytical predictions are in good agreement with available Monte Carlo simulations and consistent with other theoretical frames, as the Lieb-Lininger theory for one-dimensional systems. In the third chapter, the relevance of fluctuations is investigated from an alternative point of view. Indeed, for a single-component Bose gas we have actually considered their effect as deviations of thermodynamic quantities from the mean-field and zero-range picture. In the case of collapsing Bose mixtures, we are going to show that zero-temperature fluctuations play a crucial stabilizing role against the collapse instability. Because of this peculiar mechanism, ultracold mixtures can display finite-density configurations also in free space. Inspired by recent experiments, we characterize this novel self-bound state by comparison with bright solitons, following a variational scheme. We also consider the case of binary mixtures where a coherent internal coupling is turned on. In the last chapter, we move to deal with dipolar condensates. In particular, we are interested in beyond-mean-field effects leading to the formation of inhomogeneous ground states. In order to provide a reliable answer to the open issue of superfluid properties of these structures, we present our recent numerical investigation on the phase diagram of dipolar bosons.