Quasicondensation and coherence in the quasi-two-dimensional trapped Bose gas

We simulate a trapped quasi-two-dimensional Bose gas using a classical field method. To interpret our results we identify the uniform Berezinskii-Kosterlitz-Thouless (BKT) temperature $T_{\text{BKT}}$ as where the system phase-space density satisfies a critical value. We observe that density fluctuations are suppressed in the system well above $T_{\text{BKT}}$ when a quasicondensate forms as the first occurrence of degeneracy. At lower temperatures, but still above $T_{\text{BKT}}$, we observe the development of appreciable coherence as a prominent finite-size effect, which manifests as bimodality in the momentum distribution of the system. At $T_{\text{BKT}}$ algebraic decay of off-diagonal correlations occurs near the trap center with an exponent of 0.25, as expected for the uniform system. Our results characterize the low-temperature phase diagram for a trapped quasi-two-dimensional Bose gas and are consistent with observations made in recent experiments.

Figure 1

(Color online) Instantaneous planar density of a quasi-2D classical field at a range of temperatures. Inner white (outer black) circle marks the $1/e$ boundary of the condensate (quasicondensate) density. Vortices $(+)$ and antivortices $(-)$ are shown. Results are indicated for various values of reduced temperature $T^{\prime }$ (see Sec. 2D). Other simulation parameters are given in the text.

Figure 2

(Color online) Component densities along the $x$ axis at (a) $T^{\prime }=0.91$ and (b) $T^{\prime }=1.0$: total density $n\left(x\right)$ (dotted line), quasicondensate $n_{\text{qc}}\left(x\right)$ (dashed line), condensate $n_c\left(x\right)$ (solid line), and superfluid $n_{\text{sf}}\left(x\right)$ (shaded region). (c) Fractions of atoms in quasicondensate (dots), condensate (squares), and superfluid (triangles). System (a): $T=30.6\text{nK}$ and $N=9.9\times {10}^3$; system (b): $T=48.5\text{nK}$ and $N=25.9\times {10}^3$. Other parameters as discussed in the text.

Figure 3

Density distributions versus temperature. (a) Position phase-space density and (b) momentum space density distributions. The black (white) curves mark the densities at $T_{\text{BKT}}$ $\left(T_{\text{coh}}\right)$ (also see Fig. 2). Parameters are the same as in Fig. 2.

Figure 4

(Color online) (a) $g^1\left(r\right)$ for various temperatures (solid lines). Fitting region (light-colored segments of curves) and algebraic fits (dashed lines) are also shown. (b) Peak superfluid density determined from fits to algebraic decay (triangles) and from model density, Eq. (3) (circles). Parameters are as in Fig. 2.

Figure 5

Quantities determined from classical field evolution as a function of the classical field energy per particle. (a) Simulation temperature. (b) Average occupation of the least occupied mode. (c) Ratio of interaction energy scale to the energy cutoff.

Figure 6

(Color online) Relevant quantities for our classical field simulations of the quasi-2D trapped Bose gas as a function of temperature. (a) Total particle number of our results and (b) the respective quasi-2D ideal condensation temperature.

Figure 7

Relative thermal activation of $z$ direction of the system as a function of reduced temperature.