We present a study of the Casimir effect in an imperfect (mean-field) Bose gas contained between two infinite parallel plane walls. The derivation of the Casimir force follows from the calculation of the excess grand canonical free energy density under periodic, Dirichlet, and Neumann boundary conditions with the use of the steepest descent method. In the one-phase region the force decays exponentially fast when distance $D$ between the walls tends to infinity. When Bose-Einstein condensation point is approached the decay length in the exponential law diverges with critical exponent $ν_{IMP}=1$, which differs from the perfect gas case where $ν_{P}=1/2$. In the two-phase region the Casimir force is long-range, and decays following the power law $D^{-3}$, with the same amplitude as in the perfect gas.
We have presented an analytic theory for the Casimir force on a Bose-Einstein condensate (BEC) which is confined between two parallel plates. We have considered Dirichlet boundary conditions for the condensate wave function as well as for the phonon field. We have shown that, the condensate wave function (which obeys the Gross-Pitaevskii equation) is responsible for the mean field part of Casimir force, which usually dominates over the quantum (fluctuations) part of the Casimir force.
The relation between the bulk correlation length and the decay length of thermodynamic Casimir forces is investigated microscopically in two three-dimensional systems undergoing Bose-Einstein condensation: the perfect Bose gas and the imperfect mean-field Bose gas. For each of these systems, both lengths diverge upon approaching the corresponding condensation point from the one-phase side, and are proportional to each other. We determine the proportionality factors and discuss their dependence on the boundary conditions. The values of the corresponding critical exponents for the decay length a…
Recently it has been suggested that relativistic Bose gas of some type can be playing role in issues like dark matter, dark energy, and in some cosmological problems. In the current article we investigate one known exactly solvable model of three-dimensional statistical-mechanical model of relativistic Bose gas that takes into account the existence of both particles and antiparticles. We derive exact expressions for the behavior of the Casimir force for the system subjected to film geometry under periodic boundary conditions. We show that the Casimir force between the plates is attractive, mon…
We analyze the thermodynamic Casimir effect occurring in a gas of non-interacting bosons confined by two parallel walls with a strongly anisotropic dispersion inherited from an underlying lattice. In the direction perpendicular to the confining walls the standard quadratic dispersion is replaced by the term $|{\bf p}|^α$ with $α\geq 2$ treated as a parameter. We derive a closed, analytical expression for the Casimir force depending on the dimensionality $d$ and the exponent $α$, and analyze it for thermodynamic states in which the Bose-Einstein condensate is present. For $α\in\{4,6,8,\dots\}$ …
We study the Casimir effect for a 3-d system of ideal Bose gas in a slab geometry with Dirichlet boundary condition. We calculate the temperature($T$) dependence of the Casimir force below and above the Bose-Einstein condensation temperature($T_c$). At $T\le T_c$ the Casimir force vanishes as $[\frac{T}{T_c}]^{3/2}$. For $T\gtrsim T_c$ it weakly depends on temperature. For $T\gg T_c$ it vanishes exponentially. At finite temperatures this force for thermalized photons in between two plates has a classical expression which is independent of $\hbar$. At finite temperatures the Casimir force for o…
We consider the Bose gas on a $d$-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature $T$. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this…
We study the Bose-Einstein condensation for a 3-d system of ideal Bose gas which is harmonically trapped along two perpendicular directions and is confined in between two slabs along the other perpendicular direction. We calculate the Casimir force between the two slabs for this system of trapped Bose gas. At finite temperatures this force for thermalized photons in between two plates has a classical expression which is independent of $\hbar$. At finite temperatures the Casimir force for our system depends on $\hbar$. For the calculation of Casimir force we consider only the Dirichlet boundary…
We consider two-dimensional imperfect attractive Fermi and repulsive Bose gases consisting of spinless point particles whose total interparticle interaction energy is represented by $a N^2/2 V$ with $a=-a_{F}\leq 0$ for fermions, and $a=a_{B}\geq 0$ for bosons. We show that in spite of the attraction the thermodynamics of $d=2$ imperfect Fermi gas remains well defined for $0 \leq a_{F}\leq a_{0}=h^2/2πm$, and is exactly the same as the one of the repulsive imperfect Bose gas with $a_{B}=a_{0}-a_{F}$. In particular, for $a_{F}=a_{0}$ one observes the thermodynamic equivalence of the attractive …
The Casimir effect of an ideal Bose gas trapped in a generic power-law potential and confined between two slabs with Dirichlet, Neumann, and periodic boundary conditions is investigated systematically, based on the grand potential of the ideal Bose gas, the Casimir potential and force are calculated. The scaling function is obtained and discussed. The special cases of free and harmonic potentials are also discussed. It is found that when T<Tc (where Tc is the critical temperature of Bose-Einstein condensation), the Casimir force is a power-law decay function; when T>Tc, the Casimir force is an…
In a recent letter, Martin and Zagrebnov [Europhys. Lett., 73 (2006) 1] discussed the thermodynamic Casimir effect for the ideal Bose gas confined in a thin film. We point out that their findings can be expressed in terms of previous general results for the Casimir effect induced by confined critical fluctuations. This highlights the links between the Casimir effect in the contexts of critical phenomena and Bose-Einstein condensation.
We consider the d-dimensional imperfect (mean-field) Bose gas confined in a slit-like geometry and subject to periodic boundary conditions. Within an exact analytical treatment we first extract the bulk critical properties of the system at Bose-Einstein condensation and identify the bulk universality class to be the one of the classical d-dimensional spherical model. Subsequently we consider finite slit width D and analyze the excess surface free energy and the related Casimir force acting between the slit boundaries. Above the bulk condensation temperature (T>T_c) the Casimir force decays exp…
Some properties of an ideal gas of massive bosons in a mean field potential and, confined between two infinite parallel slabs in a d-dimensional configuration space are investigated systematically. Here, one particle density of states approach is employed to study the critical temperature, shift of density, Casimir effects and critical exponents, starting from the evaluation of the grand canonical free energy in d-dimension. We have found that, the shift of density, Casimir force and the critical temperature depend on the space dimensionality. But the Casimir force decays as an inverse power l…
We study the Casimir effect for the perfect Bose-gase in the slab geometry for various boundary conditions. We show that the grand canonical potential per unit area at the bulk critical chemical potential $μ=0$ has the standard asymptotic form with universal Casimir terms.
We explore Casimir effect on an interacting Bose-Einstein condensate (BEC) inside a cylindrical tube. The Casimir force for the confined BEC comprises of (i) a mean-field part arising from the spatial inhomogeneity of the condensate order parameter, and (ii) a quantum fluctuation part arising from the confinement of Bogoliubov excitations in the condensate. Our analytical result predicts Casimir force on a cylindrical shallow of $^4$He well below the $λ$-point, and can be tested experimentally.
In this study, we have derived close form of the Casimir force for the non-interacting ideal Bose gas between two slabs in harmonic-optical lattice potential by using Ketterle and van Druten approximation. We find that Bose-Einstein condensation temperature $T_{c}$ is a critical point for different physical behavior of the Casimir force. We have shown that Casimir force of confined Bose gas in the presence of the harmonic-optical potential decays with inversely proportional to $d^{5}$ when $T\leq T_{c}$. However, in the case of $T>T_{c}$, it decays exponentially depends on separation $d$ of th…
In this study, we consider motion of the massive and charged Bose gas trapped in electric and magnetic fields between two parallel plates in the $x-y$ plane that are separated by a distance $d$ in the $z$ direction. We derive analytical expression of the grand canonical potential of the quantum particles by using of Ketterle and van Druten approximation. By evaluating canonical potential, we obtain Casimir potential and force at the Bose-Einstein condensation temperature $T_{c}$. We also show that Casimir force oscillates depends on distance $d$ for fixed parameters.
We study the low-temperature limit of the d-dimensional imperfect Bose gas. Relying on an exact analysis of the microscopic model, we establish the existence of a second-order quantum phase transition to a phase involving the Bose-Einstein condensate. The transition is triggered by varying the chemical potential and persists at non-zero temperatures T for d>2. We extract the exact phase diagram and identify the scaling regimes in the vicinity of the quantum critical point focusing on the behavior of the correlation length ξ. The length ξdevelops an essential singularity exclusively for d=2. We…
The shift of density of Bose gas due to the mean field potential (MFP) and the Casimir effect is systematically investigated in the $d$-dimensional configuration space from the point of thermodynamic consideration. We show that, for $d=3$, the shift of density arises completely due to the Casimir effect and, the MFP remains totally ineffective regardless of the state, condensate or non-condensate. But for dimension $d>3$ the MFP plays an active role in shifting the density of Bose gas along with the Casimir interaction. The sign of density shift becomes positive in the present case. So, the co…
We consider an ideal Bose gas enclosed in a $d$-dimensional slab of thickness $D$. Using the grand canonical ensemble we calculate the variance of the thermal Casimir force acting on the slab's walls. The variance evaluated per unit wall area is shown to decay like $ Δ_{var}/D$ for large $D$. The amplitude $ Δ_{var}$ is a non-universal function of two scaling variables $λ/ ξ$ and $D/ ξ$, where $λ$ is the thermal de Broglie wavelength and $ξ$ is the bulk correlation length. It can be expressed via the bulk pressure, the Casimir force per unit wall area, and its derivative with respect to chemic…
We analyze the Casimir forces for an ideal Bose gas enclosed between two infinite parallel walls separated by the distance D. The walls are characterized by the Dirichlet boundary conditions. We show that if the thermodynamic state with Bose-Einstein condensate present is correctly approached along the path pertinent to the Dirichlet b.c. then the leading term describing the large-distance decay of thermal Casimir force between the walls is 1/D^2 with a non-universal amplitude. The next order correction is lnD/D^3. These observations remain in contrast with the decay law for both the periodic …
We study the Casimir effect for a three dimensional system of ideal free massive Bose gas in a slab geometry with Zaremba and anti-periodic boundary conditions. It is found that for these type of boundary conditions the resulting Casimir force is repulsive in nature, in contrast with usual periodic, Dirichlet or Neumann boundary condition where the Casimir force is attractive (Martin P. A. and Zagrebnov V. A., Europhys. Lett., 73 (2006) 15.). Casimir forces in these boundary conditions also maintain a power law decay function below condensation temperature and exponential decay function above …
The Casimir force due to {\it thermal} fluctuations (or pseudo-Casimir force) was previously calculated for the perfect Bose gas in the slab geometry for various boundary conditions. The Casimir pressure due to {\it quantum} fluctuations in a weakly-interacting dilute Bose-Einstein condensate (BEC) confined to a parallel plate geometry was recently calculated for Dirichlet boundary conditions. In this paper we calculate the Casimir energy and pressure due to quantum fluctuations in a zero-temperature homogeneous weakly-interacting dilute BEC confined to a parallel plate geometry with periodic …
We study the Casimir effect in the vicinity of a quantum critical point. As a prototypical system we analyze the $d$-dimensional imperfect (mean-field) Bose gas enclosed in a slab of extension $L^{d-1}\times D$ and subject to periodic boundary conditions. The thermodynamic state is adjusted so that $L\ggλ\gg D\ggł_{mic}$, where $λ\sim T^{-1/2}$ is the thermal de Broglie length, and $l_{mic}$ denotes microscopic lengthscales. Our exact analysis indicates that the Casimir force in the above specified regime is generically repulsive and decays either algebraically or exponentially, with a non-uni…
Fluctuation-induced forces are investigated for ideal and imperfect Bose gases confined to $d$-dimensional films of size $\infty^{d-1}\times D$ under periodic (P), antiperiodic (A), Dirichlet-Dirichlet (DD), Neumann-Neumann (NN), and Robin (R) boundary conditions (BCs). The full scaling functions $Υ^{\text{BC}}_d(x_λ=D/λ_{th},{x_ξ=D/ξ})$ of the residual reduced grand potential per area, $φ_{\text{res},d}^{\text{BC}}(T,μ,D)=D^{-(d-1)}Υ_d^{\text{BC}}(x_λ,x_ξ)$, are determined for the ideal gas case with these BCs, where $λ_{th}$ and $ξ$ are the thermal de-Broglie wavelength and the bulk correlat…
We calculate a force due to zero-temperature quantum fluctuations on a stationary object in a moving superfluid flow. We model the object by a localized potential varying only in the flow direction and model the flow by a three-dimensional weakly interacting Bose-Einstein condensate at zero temperature. We show that this force exists for any arbitrarily small flow velocity and discuss the implications for the stability of superfluid flow.
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 details 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…
We study an anisotropic variant of the $d$-dimensional imperfect Bose gas, where the asymptotic behaviour of the dispersion $ε_{\bf k}$ at vanishing momentum $\bf{k}$ may differ from the standard quadratic form. The analysis reveals the key role of the shift exponent $ψ$ governing the asymptotic behaviour of the critical temperature $T_c(μ)$ as a function of the chemical potential $μ$ at $T_c\to 0$. We argue that the universality classes of Bose-Einstein condensation admitted by the model may be classified according to the allowed values of $ψ$ so that spatial dimensionality has only an indire…
We study the acoustic Casimir pressure between imperfectly reflecting plates immersed in various isotropic noise backgrounds. Unlike the case of perfect reflectors, the force tends to strong repulsion at small plate separations due to the existence of modes between the plates. Additionally, in a noise field of constant spectral intensity, the large-bandwidth limit of the pressure has a reflectivity-dependent sensitivity to the infrared cutoff; we uncover the associated singular behavior as the product of the reflection coefficients approaches 1, which is intimately connected to a tendency of c…
We discuss a force arising from zero-temperature quantum fluctuations in a weakly interacting Bose-Einstein condensate (BEC). We argue that there is a drag force on an object moving at speeds well below Landau's critical velocity, and motivate our argument by an analogy to the Casimir force in an electromagnetic vacuum. We sketch the calculation of this force for an object described by a general potential. We comment about the relevance of our results to the stability of superfluid flow, arguing that this effect is consistent with the observation of persistent currents and that this effect int…
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 $1/r^2$ behavior at separations $r$ above the healing length. Our results can also be interpreted as a Casimir-like interaction between two impurities, where one of them is infi…
After a review of the standard calculation of the Casimir force between two metallic plates at zero and non-zero temperatures, we present the study of microscopic models to determine the large-distance asymptotic force in the high-temperature regime. Casimir's conducting plates are modelized by plasmas of interacting charges at temperature T. The charges are either classical, or quantum-mechanical and coupled to a (classical) radiation field. In these models, the force obtained is twice weaker than that arising from standard treatments neglecting the microscopic charge fluctutations inside the…
In an earlier paper (J. Phys. A: Math. Gen. 26 (1993) 4689) we introduced the notion of cycle percolation in the Bose gas and conjectured that it occurs if and only if there is Bose-Einstein condensation. Here we give a complete proof of this statement for the perfect and the imperfect (mean-field) Bose gas and also show that in the condensate there is an infinite number of macroscopic cycles.
We carry out a mathematically rigorous investigation into the equilibrium thermodynamics of massless and massive bosons confined in generalized Sierpinski carpets (GSCs), a class of infinitely ramified fractals having non-integer Hausdorff dimensions $d_h$. Due to the anomalous walk dimension $d_w>2$ associated with Brownian motion on GSCs, all extensive thermodynamic quantities are shown to scale with the spectral volume with dimension $d_s = 2(d_h/d_w)$ rather than the Hausdorff volume. We prove that for a low-temperature, high-density ideal massive Bose gas in an unbounded GSC, Bose-Einstei…
We study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly influences the boson density which is completely suppressed at the boundary position. Away from it, the density is depleted over the distances on the order of the healing length at the mean-field level. Quantum fluctuations modify the density profile considerably. The local density approaches the average one as an inverse square of the distance from the boundary. We calculate an analytic expression for the density profile at arbitrary separations from the boundary. We then consider the pr…
Adding a gauge symmetry breaking field -ν\sqrt{V}(a_0+a_0^*) to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis \lim_{V\to\infty}< a_0>/\sqrt{V}={\rm sgn}ν\lim_{V\to\infty}\sqrt{< a_0^*a_0>/V} where the averages are taken in the ground state or in thermal equilibrium states. Letting νtend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The…
We study the stochastic behavior of a set of chaotic vortex loops appeared in imperfect Bose gas. Dynamics of Bose-gas is supposed to obey Gross-Pitaevskii equation with additional noise satisfying fluctuation-dissipation relation. The corresponding Fokker-Planck equation for probability functional has solution ${\cal P}(\{ψ({\bf r})\})={\cal N}\exp (-H\{ψ({\bf r)}\} /T),$ where $H\{ψ({\bf r})\} $ is the Ginzburg-Landau free energy. Considering vortex filaments as topological defects of field $ψ({\bf r})$ we derive a Langevin-type equation of motion of the line with the correspondingly transfo…
The surface-atom Casimir-Polder-Lifshitz force out of thermal equilibrium is investigated in the framework of macroscopic electrodynamics. Particular attention is devoted to its large distance limit that shows a new, stronger behaviour with respect to the equilibrium case. The frequency shift produced by the surface-atom force on the the center-of-mass oscillations of a harmonically trapped Bose-Einstein condensate and on the Bloch oscillations of an ultra-cold fermionic gas in an optical lattice are discussed for configurations out of thermal equilibrium.
Disordered systems show deviations from the standard Debye theory of specific heat at low temperatures. These deviations are often attributed to two-level systems of uncertain origin. We find that a source of excess specific heat comes from correlations between quanta of energy if phonon-like excitations are localized on an intermediate length scale. We use simulations of a simplified Creutz model for a system of Ising-like spins coupled to a thermal bath of Einstein-like oscillators. One feature of this model is that energy is quantized in both the system and its bath, ensuring conservation o…
