Bose gas in a single-beam optical dipole trap

We study an ultracold Bose gas in an optical dipole trap consisting of one single focused laser beam. An analytical expression for the corresponding density of states beyond the usual harmonic approximation is obtained. We are thus able to discuss the existence of a critical temperature for Bose-Einstein condensation and find that the phase transition must be enabled by a cutoff near the threshold. Moreover, we study the dynamics of evaporative cooling and observe significant deviations from the findings for the well-established harmonic approximation. Furthermore, we investigate Bose-Einstein condensates in such a trap in Thomas-Fermi approximation and determine analytical expressions for chemical potential, internal energy, and Thomas-Fermi radii beyond the usual harmonic approximation.

Figure 1

Dipole trap potential (black solid) and its harmonic approximation (blue dashed) at $z=y=0$ (left) and $x=y=0$ (right).

Figure 2

Comparison of the densities of states of the dipole trap potential (black solid line) and its harmonic approximation (blue dashed line), along with the simple approximate formula (7) (dotted line).

Figure 3

Comparison of the number density $g(\epsilon )f(\epsilon )$ of the dipole trap potential (black, solid line) and its harmonic approximation (blue, dashed line) for $\eta =U_0/k_{\mathrm{B}}T=2.5$, 5, 10, 20. The areas below the graphs must be finite to allow for Bose-Einstein condensation.

Figure 4

The dipole trap potential without gravity (black solid), including gravity (red, dashed), and the pure gravitational potential (green dashed-dotted) at $y=z=0$. Due to gravity, atoms may escape from the trap as soon as their energy is $\epsilon >1-\Delta \epsilon$. Thus, the real potential is cut off at ${\epsilon }_{\mathrm{t}}=1-\Delta \epsilon$.

Figure 5

Time evolution of normalized distribution function $f(\epsilon )$ during plain evaporative cooling in a dipole trap (a) and a harmonic trap (c); time evolution of the number density $g(\epsilon )f(\epsilon )$ for the dipole (b) and the harmonic trap (d). Initially, $f(\epsilon )$ is chosen to correspond to a Boltzmann distribution with $k_{\mathrm{B}}T=U_0$, normalized to 1. The arrows indicate increasing time; time steps grow exponentially.

Figure 6

Fraction of atoms $N/N_0$ remaining in the trap as a function of scaled time $\tau /G$ after starting evaporation from a scaled temperature $\mathrm{T̃}=k_{\mathrm{B}}T/U_0=1$. The black, solid line is the result for the dipole trap and the blue dashed for its harmonic approximation obtained by integration of the differential equations resulting from the truncated Boltzmann approximation. The squares and circles are the results from the fully numerical treatment for dipole and harmonic trap, respectively.

Figure 7

Scaled temperature $\mathrm{T̃}=k_{\mathrm{B}}T/U_0$ of the remaining atoms as a function of scaled time $\tau /G$ after starting evaporation from a temperature $\mathrm{T̃}=1$. The black solid line is the result for the dipole trap and the blue dashed line for its harmonic approximation obtained by integration of the differential equations resulting from the truncated Boltzmann approximation. The squares and circles result from the fully numerical treatment for dipole and harmonic trap, respectively.

Figure 8

Chemical potential versus atom number of a condensate in a dipole trap (black solid) and a harmonic trap (blue dashed) in the Thomas-Fermi approximation.

Figure 9

Thomas-Fermi radius along the propagation direction of the laser versus chemical potential for dipole trap (black solid) and its harmonic approximation (blue dashed).