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arXiv:1806.03658 · New Journal of Physics

The Casimir-like effect in a one-dimensional Bose gas

Benjamin Reichert, Zoran Ristivojević, Aleksandra Petković

Published Apr 23, 2019• 43 citations
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Abstract

Abstract The electromagnetic Casimir effect manifests as the interaction between uncharged conducting objects that are placed in a vacuum. More generally, the Casimir-like effect denotes an induced interaction between external bodies in a fluctuating medium. We study the Casimir-like interaction between two impurities embedded in a weakly interacting one-dimensional Bose gas. We develop a theory based on the Gross–Pitaevskii equation that accounts for the effect of quantum fluctuations. At small separations, the induced interaction between the impurities decays exponentially with the distance. This is a classical result that can be understood using the mean-field Gross–Pitaevskii equation. We find that at larger distances, the induced interaction crosses over into a power law dependence due to the quantum fluctuations. We obtain an analytic expression for the interaction that interpolates between the two limiting behaviors. The obtained result does not require any regularization.

DOI
10.1088/1367-2630/ab1b8e

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Abstract The electromagnetic Casimir effect manifests as the interaction between uncharged conducting objects that are placed in a vacuum. More generally, the Casimir-like effect denotes an induced interaction between external bodies in a fluctuating medium. We study the Casimir-like interaction between two impurities embedded in a weakly interacting one-dimensional Bose gas. We develop a theory based on the Gross–Pitaevskii equation that accounts for the effect of quantum fluctuations. At small separations, the induced interaction between the impurities decays exponentially with the distance. This is a classical result that can be understood using the mean-field Gross–Pitaevskii equation. We find that at larger distances, the induced interaction crosses over into a power law dependence due to the quantum fluctuations. We obtain an analytic expression for the interaction that interpolates between the two limiting behaviors. The obtained result does not require any regularization.
We study the fluctuation-induced interaction between two impurities in a weakly interacting one-dimensional Bose gas using the field-theoretical approach. At separations between impurities shorter and of the order of the healing length of the system, the induced interaction has a classical origin and behaves exponentially. At separations longer than the healing length, the interaction is of a quantum origin and scales as the third power of the inverse distance. Finite temperature destroys the quasi-long-range order of the Bose gas and, accordingly, the induced interaction becomes exponentially suppressed beyond the thermal length. We obtain analytical expressions for the induced interaction at zero and finite temperature that are valid at arbitrary distances. We discuss experimental realizations as well as possible formation of bound states of two impurities, known as bipolarons.
We consider an impurity in a semi-infinite one-dimensional system of weakly-interacting bosons. We calculate the interaction potential for the impurity due to the end of the system, i.e., the wall. For local repulsive (attractive) interaction between the impurity and the Bose gas, the interaction potential is attractive (repulsive). At short distances from the wall it decays exponentially crossing over into a universal 1/r21/{r}^{2}1/r2 behavior at separations rrr above the healing length. Our results can also be interpreted as a Casimir-like interaction between two impurities, where one of them is infinitely strongly coupled to the Bose gas. We discuss various scenarios for the induced interaction between the impurities using the scattering approach. We finally address the phenomenon of localization of the impurity near the wall. In the paper we mainly study the case of a static impurity, however the universal 1/r21/{r}^{2}1/r2 interaction also holds for a slowly moving impurity.
We consider an impurity in a one-dimensional weakly interacting Bose gas and analytically calculate the density profile of the Bose gas. Within the mean-field approximation, by increasing the distance from the impurity, the Bose gas density saturates exponentially fast to its mean thermodynamic-limit value at distances beyond the healing length. The effect of quantum fluctuations drastically changes this behavior, leading to a power-law decay of the density deviation from the mean density. At distances longer than the healing length and shorter than a new length scale proportional to the impurity coupling strength, the power-law exponent is 2, while at the longest distances the corresponding exponent becomes 3. The latter crossover does not exist in two special cases. The first one is realized for infinitely strongly coupled impurity; then the density deviation always decays with the exponent 2. The second special case occurs when the new length scale is smaller than the healing length, i.e., at weak impurity coupling; then the density deviation always decays with the exponent 3. The obtained results are exact in the impurity coupling strength and account for the leading order in the interaction between the particles of the Bose gas.
We present a detailed investigation of one of the cleanest examples where it is possible to detect the “analog” Dynamical Casimir Effect in a Bose–Einstein condensate: an ultracold atom gas in toroidal confinement. The analytical solution of the time dependent Gross–Pitaevskii equation allows to follow the time evolution of the phonon spectrum and shows that periodic oscillations of the ring radius do not induce modulations in the density profile but give rise to the mixing of clockwise and anticlockwise modes, leading to the creation of pairs of entangled phonons in a squeezed vacuum state, if the drive frequency equals twice the frequency of the phonon mode. The Dynamical Casimir Effect is predicted to occur in the weakly interacting regime, where the Gross–Pitaevskii equation provides a faithful description of the many body dynamics. In the strong coupling limit, when the ultracold gas behaves as hard core bosons, the effect disappears and no amplification occurs. The presence of symmetry-breaking perturbations and finite temperature effects are also considered, as well as the comparison with the classical phenomenon of parametric amplification.
We study a weakly interacting one-dimensional Bose gas with two impurities coupled locally to the boson density. We derive analytical results for the induced interaction between the impurities at arbitrary positive coupling and separation rrr. At $r\ensuremath{\lesssim}\ensuremath{\xi}$, where $\ensuremath{\xi}$ denotes the healing length of the Bose gas, the interaction is well described by the mean-field contribution. Its form changes as the coupling is increased, approaching a linear function of rrr at short distances in the regime of strong coupling. The mean-field contribution decays exponentially at arbitrary coupling for $r\ensuremath{\gg}\ensuremath{\xi}$. At such long distances, however, the effect of quantum fluctuations becomes important, giving rise to a long-ranged quantum contribution to the induced interaction. At longest distances it behaves as 1/r31/{r}^{3}1/r3, while at strong coupling we find an intermediate distance regime with a slower decay, 1/r1/r1/r. The quantum contribution in the crossover regime is also calculated. The induced interaction between impurities (i.e., polarons) is attractive and leads to the formation of their bound state, known as bipolaron. We discuss its binding energy.
We consider the Casimir force between two vortices due to the presence of density fluctuations induced by turbulent modes in a Bose–Einstein condensate. We discuss the cases of unbounded and finite condensates. Turbulence is described as a superposition of elementary excitations (phonons or BdG modes) in the medium. Expressions for the Casimir force between two identical vortex lines are derived, assuming that the vortices behave as point particles. Our analytical model of the Casimir force is confirmed by numerical simulations of the Gross–Pitaevskii equation, where the finite size of the vortices is retained. Our results are valid in the mean-field description of the turbulent medium. However, the Casimir force due to quantum fluctuations can also be estimated, assuming the particular case where the occupation number of the phonon modes in the condensed medium is reduced to zero and only zero-point fluctuations remain.