Bose condensation of interacting gases in traps with and without optical lattice

We discuss effects of particle interaction on Bose condensation in inhomogeneous traps with and without optical lattice. Interaction pushes normal particles away from the condensate droplet, which is located in the center of the trap, towards the periphery of the trap where the trapping potential is large. In the end, the remaining normal particles are squeezed to a quasi-2D shell around the condensate droplet thus changing the effective dimensionality of the system. In the absence of the optical lattice the index in the temperature dependence of the condensate density at the later stages of the process is close to 2 with a weak dependence on the number of trapped particles. In the presence of the lattice inside the trap this index acquires a strong dependence on the number of particles inside the trap and gradually falls from a 3D to a 2D value with an increase in the number of particles. This change in index is explained by the lattice-driven spread of the condensate droplet and the localization of the narrow band particles by the trap potential.

The spectacular experimental discovery of Bose condensation (BEC) made the study of alkali gases in traps the focal point in atomic, low temperature, and condensed matter physics.
For the first time, it became possible to observe some of the phenomena that have been discussed earlier only within theoretical models (see review [1]).The phenomena in ultracold alkali gases are incredibly rich and combine features inherent to diverse condensed matter and low temperature systems (Refs.[2] and references therein) from "classical" superfluid or superconducting systems [2] to spin-polarized quantum gases [3] to Mott transition in the optical lattice [4].
The trap potential is always inhomogeneous.The interplay between the repulsive interaction and the trapping potential complicates BEC [5,6] since these two factors have opposite effects on condensation: while the trap tends to concentrate the condensate in a narrow region of space around the particle ground state in the trap, the repulsion is responsible for the widening of this condensate droplet.The analytical description of the combined effects tends to be rather elusive and our previous experience with condensation in homogeneous systems is not very helpful.The problem becomes even more complex in the presence of the optical lattice inside the trap which adds two different localization processes -Mott transition and localization of narrow band particles by an inhomogeneous potential.
Below we investigate a situation in which it is possible to get an accurate picture of the condensation in trapped interacting gases.The main attention is paid to the index in the temperature dependence of the condensate fraction and to the size of the condensate droplet.
It turns out that this index is not universal even for a low density gas.What is more, the effective dimensionality of the problem changes with condensation and the later stages of BEC are different from initial.
We start from BEC in trapped gases without the optical lattice, and add the complications associated with the optical lattice later on.We assume that the density is sufficiently low to neglect the interaction before the onset of condensation even in the center of the trap.This means that T c is unaffected by the interaction.The interaction is brought into play only with the onset of condensation since the particles condensate in the center of the trap making the density in the center large.This makes the interaction, which is proportional to the particle density, large only in and around the condensate droplet.This also means that the normal particles are pushed out by the dense condensate towards the periphery of the trap where the interaction is negligible.The further particles move away from the center the higher is the gradient of the trapping potential which is responsible for the force pushing the normal particles back towards the trap center.Thus, at the later stages of BEC, the majority of remaining normal particles are located in an almost two-dimensional shell around the condensate droplet and the dimensionality of the problem changes from the three-dimensional in the beginning of the condensation to quasi-2D later on.
We consider a 3D harmonic trap with a single-particle ground state of frequency ω and spacial size σ 0 (axial asymmetry of real traps is irrelevant in our context).Without interaction, BEC starts at T c = 0.941 ωN 1/3 [7] and the size of the condensate droplet is σ 0 .
Repulsion increases the size of the droplet with N c (T ) particles to σ (T ).Then the potential well for normal particles U (r) has a shell-type structure, where N 0 = ( √ π/8) ωmσ 3 0 / a s and we assume that the condensate density is Gaussian.The number of normal particles N n (T ) = N − N c (T ) is determined from the condition µ = 0.The size of the condensate droplet σ (T ) can be obtained from minimization of the condensate energy, including repulsion, similarly to Ref. [5].The interaction between the normal particles can often be excluded from Eq. ( 1).First, for less than 10 5 particles in a trap, the density of the normal particles is negligible even in the trap center.For larger N, the number of the normal particles on the later stages of the condensation is small.Finally, the density of the normal particles is suppressed even more by repulsion from the condensate droplet which spreads them through a large shell around the droplet 4πσ 2 σ 0 instead of concentrating them near the center in the volume (4π/3) σ 3 0 .This gives N at least an extra order of magnitude for which we can neglect the interaction of normal particles.
N 0 in Eq. ( 1) is the minimal number of particles in the condensate that is sufficient to create a strong repulsive core in the center of the trap.When N > N c ≫ N 0 the normal particles are pushed away from the center by the repulsive core (1) into a potential valley surrounding the condensate droplet.For Rb in a trap with ω = 24 Hz, the values a s = 58.2Å, σ 0 = 2.2 × 10 −6 m, and the critical number N 0 that changes the topology of the normal cloud is N 0 ≈ 84.The center of the trap becomes inaccessible for normal particles when T is much smaller than the repulsion from the core.Using T c instead of T and N instead of N c , one gets σ 3 N 0 /σ 3 0 ≪ N 2/3 and the critical value of N c is around 10 5 .All this means that our results are applicable for N in the range 10 4 ÷ 10 6 .
We are able to obtain an analytical description of the situation (cf.Ref. [6]).At the later stages of the condensation, the potential (1) forms a distinct valley away from the center of the trap as soon as N c ≫ N 0 and equations for N c (T ) and σ (T ) reduce to The summation provides the temperature dependencies N c (T ) and σ (T ).
We found that the condensate fraction at the later stages of condensation can be given as with a relatively high accuracy.The important feature of Eq.( 3) is that the temperature is normalized not by the critical temperature T c for the onset of condensation but by a different value T * c .Since the squeezing of the normal particles towards the fringes of the trap accelerates with the number of particles in the condensate N c , the normal shell narrows with increasing N c , and, therefore, N. As a result, the effective temperature T * c should be higher than T c and increase with increasing N. Dependence of T * c , or, more precisely, T * c / ωN 1/3 , on N is presented in Figure 1.For comparison, the critical temperature T c for non-interacting particles in a 3D harmonic trap is T c = 0.9 ωN 1/3 [7].
The striking change in behavior of T * c (N) in Figure 1 occurs at N for which T c ∼ 1 2 ω (N c σ 3 0 /N 0 σ 3 ).At higher densities the repulsion from the condensate droplet keeps the normal particles near the bottom of the potential valley around the droplet; at lower densities, the normal particles spread out and can even reach the center of the trap.An anomaly at the same threshold density is also observed in α (N), Figure 2, though the index α remains very close to the value 2 and is practically independent of N, α = 2.02 ± 1%, in a wide range of N from 10 4 to 10 6 .This weak dependence α (N) is surprising for a nonlinear problem of this nature.The residual temperature dependence α (T ) is within the same error bars.3) .For a non-interacting gas in a 3D harmonic trap this index should be 3.
These results confirm the evolution of the effective dimensionality from 3D, for which α = 3, to quasi-2D and the effective narrowing of the trap during condensation.
The situation with an optical lattice (Refs.[8] and references therein) with a period a 0 inside the trap is more complex.Here one deals with the Hubbard Hamiltonian, modified by the trap potential, and can encounter the Mott transition [9] which requires full occupancy of the lattice sites.The latter can occur with lowering of the temperature when particles gravitate towards the bottom (center) of the trap.With sufficiently strong on-site repulsion, the localization is practically inevitable for the condensate in the center of the trap though, of course, the Mott transition is sensitive to the trap profile [9,10].The increased size of the condensate droplet in comparison to the system without the lattice changes the normal cloud surrounding the condensate for which it is possible to disregard the Mott transition.
In the case of low initial density of particles na 3 0 ≪ 1 and strong on-site repulsion, the condensation starts at the same temperature T c as without the interaction.The condensate forms in the center of the trap and rapidly expands in size because of the strong on-site repulsion which tends to keep the density n c a 3 0 ≈ 1.As a result, the size of the condensate droplet σ ∼ a 0 N 1/3 c becomes larger than σ max ∼ (2 ÷ 5) σ 0 for traps without the optical lattice.We will not dwell on potential freezing of the condensate resulting from the Mott transition and will concentrate on the condensation of the normal gas outside the condensate droplet.
The main changes are associated with the band nature of the energy spectrum for particles in the optical lattice and a more complicated form of the wave functions.For the sake of comparison, in computations we use the same trap potential and particle scattering length.
For the particle effective mass we use in computations the value [4,9] m * = 16m.
The single-particle spectrum in the optical lattice ǫ (p) has a band structure with a bandwidth ∆.The effect of the trap potential U tr (r) = 1  2 ω (r/σ 0 ) 2 on the particles with narrow bands results in localization of particles with energy E in 2D shells ǫ (p) + U tr (r) = E of the thickness ℓ (r) ∼ (∆/ ω) σ 2 0 /r.[An exception is the center of the trap, where the gradient of the potential is small].The particle wave function consists of three regions: rapid oscillations within this classically accessible shell and two attenuating tails beyond the classical turning points.The wave function for a particle with the energy E decays relatively slowly beyond the turning points, often as the Airy function of the type Ai − where x is the distance from the "center" of the classically accessible shell for the particle with the energy E in the direction of the gradient, ν = 2 ω (E − ∆/2)/σ 0 , and m * is the particle effective mass at the turning point.The spatial distribution of particles should be calculated taking into account all three regions since for relatively shallow traps the contribution from the tails of the wave function can be large.Since such localization suppresses the accessibility range of narrow-band particles, the density in each point contains the contributions from the particles in a finite range of energies that are localized close to this point.
For example, since only the particles with very low energies, E < ∆, can reach the center of the trap, the density in the center is suppressed in comparison with the trap without the optical lattice inside.
As above, we start from the situation when the particle density above condensation is low and the (Hubbard) repulsion in the normal phase is negligible.The condition of low density allows us also to disregard the Mott transition in the normal phase [11].Since the particles in the optical lattice are located mostly on the lattice sites of the size a 0 rather than spread uniformly, the repulsion is more effective than without the lattice.This means that the size of the condensate droplet σ (T ) should be larger than in the absence of the lattice.This is illustrated in Figure 3 which presents the ratio σ (T = 0) /σ 0 for identical traps with (curve 1) and without (curve 2) the optical lattice.The scattering amplitude a s , which is responsible for repulsion, is the same in both cases.This seemingly innocuous lattice-driven increase in σ leads to major effects and can eliminate a repulsive bump (1) in the center of the trap (at N c σ 5 0 /N 0 σ 5 = 1) thus restoring Eq. ( 3), is even more dramatic than the change in σ, Figure 4.
In Fig. 4, α starts from a 3D value at small density of particles which is understandable since there is no repulsive core in the center.With increasing number of particles the size of the condensate droplet grows leaving fewer normal particles in the central area and gradually reducing α to its quasi-2D value.What is not clear is why does α continue to decline with a further increase in N; however, since our approach loses accuracy beyond N = 10 6 , we do not present these data in the Figure .In general, the decrease in α (N) is accompanied by an increase in T * c (N), which in the presence of the optical lattice grows much faster than N 1/3 -dependence inherent to a free gas in a trap.
In summary, we calculated the index for a temperature dependence of the condensate fraction for interacting gas inside harmonic trap.The results for traps without the optical lattice inside are quite clear: the repulsion from the condensate droplet pushes normal particles away from the center of the trap and concentrates them in a relatively thin shell around this droplet.Then the condensation becomes almost quasi-2D with the index α ≈ 2.
The presence of the optical lattice inside the trap changes the situation.The index α acquires a strong dependence on the number of particles inside the trap and gradually falls from a 3D to a 2D value with an increase in the number of particles.This change in the index,

FIG. 3 :
FIG. 3: (Color online) Size of the condensate droplet σ (T = 0) relative to the size of the trap σ 0 , σ/σ 0 , with (curve 1) and without (curve 2) the optical lattice as a function of the number of particles in the trap.The scattering lengths and effective masses are identical in both cases.Parameters of the lattice and the trap are given in the text.

FIG. 4 :
FIG. 4: Index α, Eq.(3), as a function of the number of the trapped particles in the presence of the optical lattice.Parameters of the lattice used in the computation are given in the text.