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Revista Mexicana de Física

Dimensional crossover for the Bose-Einstein condensation

Mario I. Molina, Jaime Rösler

Published Jan 1, 1998• 3 citations
OpenAlex

Abstract

We study Bose-Einstein condensation (BEC) for a non-interacting gas in a very anisotropic trap. Therefore, at enough small temperatures some degrees of freedom "freeze", thus reducing the "effective" dimensionality of the system. We mainly focus in quasi-bidimensional traps characterized by a surface S in the (x, y) plane. We consider two confining mechanisms in the z direction: harmonic and rigid wall potentials. There are not forces parallel to S, excepting the rigid wall at the edges. The most relevant results are: (a) The condensate smoothly sets at T ∼ script O sign(Tc/ log N), where Tc is the tridimensional condensation temperature. (b) When BEC is present, also the low-lying excited states have a macroscopic occupation; this effect is also present in a quasi-onedimensional harmonic "cigar-shape" trap. (c) The condensation process is sensitive to the shape of S.

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Multistep Bose-Einstein condensation of an ideal Bose gas in anisotropic harmonic atom traps is studied. In the presence of strong anisotropy realized by the different trap frequency in each direction, finite size effect dictates a series of dimensional crossovers into lower-dimensional excitations. Two-step condensation and the dynamical reduction of the effective dimension can appear in three separate steps. When the multistep behavior occurs, the occupation number of atoms excited in each dimension is shown to behave similarly as a function of the temperature. Multistep behaviors can be easily controlled by changing the degree of anisotropy. 1

•Jun 21, 1999•2 citations
We study Bose-Einstein condensation (BEC) for a non-interacting gas in a very anisotropic trap. Therefore, at enough small temperatures some degrees of freedom "freeze", thus reducing the "effective" dimensionality of the system. We mainly focus in quasi-bidimensional traps characterized by a surface S in the (x, y) plane. We consider two confining mechanisms in the z direction: harmonic and rigid wall potentials. There are not forces parallel to S, excepting the rigid wall at the edges. The most relevant results are: (a) The condensate smoothly sets at T ∼ script O sign(Tc/ log N), where Tc is the tridimensional condensation temperature. (b) When BEC is present, also the low-lying excited states have a macroscopic occupation; this effect is also present in a quasi-onedimensional harmonic "cigar-shape" trap. (c) The condensation process is sensitive to the shape of S.
We study an ideal Bose gas of N atoms contained in a box formed by two identical planar and parallel surfaces S, enclosed by a mantle of height a perpendicular to them. Calling r0 the mean atomic distance, we assume S >> r0^2 while a may be comparable to r0. In the bidimensional limit (a/r0 << 1) we find a macroscopic number of atoms in the condensate at temperatures T ~1/log(N); therefore, condensation cannot be described in terms of intensive quantities; in addition, it occurs at temperatures not too low in comparison to the tridimensional case. When condensation is present we also find a macroscopic occupation of the low--lying excited states. In addition, the condensation phenomenon is sensitive to the shape of S. The former two effects are significant for a nanoscopic system. The tridimensional limit is slowly attained for increasing (a/r0), roughly at (a/r0) ~ 10^{2}-10^{3}.
We discuss Bose-Einstein condensation (BEC) in quasi-2D trapped gases and find that well below the transition temperature T(c) the equilibrium state is a true condensate, whereas at intermediate temperatures T<T(c) one has a quasicondensate (condensate with fluctuating phase). The mean-field interaction in a quasi-2D gas is sensitive to the frequency omega(0) of the (tight) confinement in the "frozen" direction, and one can switch the sign of the interaction by changing omega(0). Variation of omega(0) can also reduce the rates of inelastic processes. This offers promising prospects for tunable BEC in trapped quasi-2D gases.
Multistep Bose-Einstein condensation of an ideal Bose gas in anisotropic harmonic atom traps is studied. In the presence of strong anisotropy realized by the different trap frequency in each direction, the finite size effect dictates a series of dimensional crossovers into lower-dimensional excitations. Two-step condensation and the dynamical reduction of the effective dimension can appear in three separate steps. When the multistep behaviour occurs, the occupation number of atoms excited in each dimension is shown to behave similarly as a function of the temperature. Multistep behaviours can be easily controlled by changing the degree of anisotropy.
We investigate systems of interacting bosonic particles confined within slablike boxes of size ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ with $Z\ensuremath{\ll}L$, at their three-dimensional (3D) Bose-Einstein-condensation (BEC) transition temperature Tc{T}_{c}Tc​, and below Tc{T}_{c}Tc​ where they experience a quasi-two-dimensional (quasi-2D) Berezinskii-Kosterlitz-Thouless (BKT) transition (at TBKT<Tc{T}_{\mathrm{BKT}}<{T}_{c}TBKT​<Tc​ depending on the thickness ZZZ). The low-temperature phase below TBKT{T}_{\mathrm{BKT}}TBKT​ shows quasi-long-range order: the planar correlations decay algebraically as predicted by the 2D spin-wave theory. This dimensional crossover, from a 3D behavior for $T\ensuremath{\gtrsim}{T}_{c}$ to a quasi-2D critical behavior for $T\ensuremath{\lesssim}{T}_{\mathrm{BKT}}$, can be described by a transverse finite-size scaling limit in a slab geometry. Numerical evidence of the $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$ dimensional crossover is presented for the Bose-Hubbard model defined in anisotropic ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ lattices with $Z\ensuremath{\ll}L$. We extend this scaling analysis to the case the slab geometry of the gas is effectively realized by a transverse (inhomogeneous) harmonic trap. Finally, we discuss off-equilibrium behaviors arising from slow time variations of the temperature across the BEC transition of gases confined within a slab geometry. We argue that the system develops an off-equilibrium transverse finite-size scaling under these time-dependent protocols.
We study theoretically the dimensional crossover from a three-dimensional elongated condensate to a one-dimensional condensate as the transverse degrees of freedom get frozen by tight confinement, in the limit of small density fluctuations, i.e. for a strongly degenerate gas. We compute analytically the radially integrated density profile at low temperatures using a local density approximation, and study the behavior of phase fluctuations with the transverse confinement. Previous studies of phase fluctuations in trapped gases have either focused on the 3D elongated regimes or on the 1D regime. The present approach recovers these previous results and is able to interpolate between them. We show in particular that in this strongly degenerate limit the shape of the spatial correlation function is insensitive to the transverse regime of confinement, pointing out to an almost universal behavior of phase fluctuations in elongated traps.
Multistep Bose-Einstein condensation of an ideal Bose gas in anisotropic harmonic atom traps is studied. In the presence of strong anisotropy realized by the different trap frequency in each direction, finite size effect dictates a series of dimensional crossovers into lower-dimensional excitations. Two-step condensation and the dynamical reduction of the effective dimension can appear in three separate steps. When the multistep behavior occurs, the occupation number of atoms excited in each dimension is shown to behave similarly as a function of the temperature. Multistep behaviors can be easily controlled by changing the degree of anisotropy. 1