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arXiv:1004.0218 · Physical Review A

Bose-Einstein condensate near a surface: Quantum field theory of the Casimir-Polder interaction

Jürgen Schiefele, Carsten Henkel

Published Aug 12, 2010• 10 citations
arXivHTMLPDF

Abstract

We derive an expression for the collective Casimir-Polder interaction of a trapped gas of condensed bosons with a plane surface through coupling of the condensate atoms with the electromagnetic field. A systematic perturbation theory is developed based on a diagrammatic expansion of the electromagnetic self-energy. In the leading order, the result for the interaction energy is proportional to the number of atoms and involves the one-body density matrix of the trapped gas. At this order, atom-atom interactions and recoil effects lead to corrections compared to the single-atom theory, through shifts of the atomic transition energies. We also discuss the impact of spatial delocalization of the condensate mode.

DOI
10.1103/physreva.82.023605

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We derive an expression for the collective Casimir-Polder interaction of a trapped gas of condensed bosons with a plane surface through coupling of the condensate atoms with the electromagnetic field. A systematic perturbation theory is developed based on a diagrammatic expansion of the electromagnetic self-energy. In the leading order, the result for the interaction energy is proportional to the number of atoms and involves the one-body density matrix of the trapped gas. At this order, atom-atom interactions and recoil effects lead to corrections compared to the single-atom theory, through shifts of the atomic transition energies. We also discuss the impact of spatial delocalization of the condensate mode.
We calculate the effect of the interaction between an optically active material and a Bose-Einstein condensate on the collective oscillations of the condensate. We provide explicit expressions for the frequency shift of the center-of-mass oscillation in terms of the potential generated by the substrate and of the density profile of the gas. The form of the potential is discussed in detail and various regimes (van der Waals--London, Casimir-Polder, and thermal regimes) are identified as a function of the distance of atoms from the surface. Numerical results for the frequency shifts are given for the case of a sapphire dielectric substrate interacting with a harmonically trapped condensate of 87Rb^{87}\mathrm{R}\mathrm{b}87Rb atoms. We find that at distances of $4--8\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$, where thermal effects become visible, the relative frequency shifts produced by the substrate are of the order ${10}^{\ensuremath{-}4}$ and hence accessible experimentally. The effects of nonlinearities due to the finite amplitude of the oscillation are explicitly discussed. Predictions are also given for the radial breathing mode.
Understanding the Casimir-Polder (CP) attraction between ultracold atoms and dielectric surfaces is of fundamental importance to the design and miniaturization of atom chips [1] and for understanding quantum reflection from, and sticking to, the surface [2 - 6]. So far, calculations have focussed on planar dielectrics, either semi-infinite or of finite width [7-9], and non-planar geometries including corrugated surfaces [10], spheres and carbon nanotubes [11-12]. Here, we present a simple scheme, based on the Pairwise-Summation (PWS) approximation, for estimating the CP interaction between a ground-state alkali atom and a dielectric volume of arbitrary shape. The PWS approximation integrates the van der Waals interaction between two ground-state atoms over the many atoms in the surface volume. We relate the microscopic polarizability of atoms in the surface to the bulk dielectric constant via the Clausius-Mossotti relation within the Lorentz local-field theory. We find that the CP potential depends strongly on both the geometry and density of the surface, which modify the fluctuation spectrum of the electromagnetic field that couples the alkali atoms to the dielectric. We compare our results with known analytical expressions for the CP potential near planar surfaces and use them to investigate how BECs interact with, and may quantum reflect from, curvilinear surface structures.
Considering the Casimir effect due to phononic excitations of a weakly interacting dilute Bose–Einstein condensate (BEC), we derive a renormalized expression for the zero-temperature Casimir energy of a BEC confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC. For the weakly interacting case, reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.
Atom-surface dispersion has become a subject of studies involving the process of efficient trapping and manipulating ultracold alkali atoms, Bose-Einstein Condensation (BEC) near the carbon nanotubes (CNT). Known as Casimir-Polder (CP) interaction, it has been approximated as analytical expression using Hamaker's approach of pairwise summation (PWS) by summing up every contribution imposed by ground state BEC treated as single atom over densely packed atoms inside CNT. At larger atom-surface distances, the CP interaction is found to be strikingly weaker on a cylindrical shell compared to a solid cylinder, with this weakening increasing further as the alkali atoms become much lighter. We examine how these results will be beneficial to BEC experiment especially when cold atoms are magnetically or optically trapped above a relatively few microns-distance CNT atom chip, fabricated as a pristine graphene and sustained under high vacuum conditions.
We report on the first measurement of a temperature dependence of the Casimir-Polder force. This measurement was obtained by positioning a nearly pure 87Rb Bose-Einstein condensate a few microns from a dielectric substrate and exciting its dipole oscillation. Changes in the collective oscillation frequency of the magnetically trapped atoms result from spatial variations in the surface-atom force. In our experiment, the dielectric substrate is heated up to 605 K, while the surrounding environment is kept near room temperature (310 K). The effect of the Casimir-Polder force is measured to be nearly 3 times larger for a 605 K substrate than for a room-temperature substrate, showing a clear temperature dependence in agreement with theory.