Universal correlations and coherence in quasi-two-dimensional trapped Bose gases

We study the quasi-two-dimensional Bose gas in harmonic traps at temperatures above the Kosterlitz-Thouless transition, where the gas is in the normal phase. We show that mean-field theory takes into account the dominant interaction effects for experimentally relevant trap geometries. Comparing with quantum Monte Carlo calculations, we quantify the onset of the fluctuation regime, where correlations beyond mean-field become important. Although the density profile depends on the microscopic parameters of the system, we show that the correlation density (the difference between the exact and the mean-field density) is accurately described by a universal expression, obtained from classical-field calculations of the homogeneous strictly two-dimensional gas. Deviations from universality, due to the finite value of the interaction or to the trap geometry, are shown to be small for current experiments. We further study coherence and pair correlations on a microscopic scale. Finite-size effects in the off-diagonal density matrix allow us to characterize the crossover from Kosterlitz-Thouless to Bose-Einstein behavior for small particle numbers. Bose-Einstein condensation occurs below a characteristic number of particles which rapidly diverges with vanishing interactions.

Figure 1

QMC density profile for the ENS parameters ($\mathrm{g̃}=0.13,{\mathrm{\omega ̃}}_z=0.55$) at temperature $t=T/T_{\mathrm{BEC}}^{2\mathrm{D}}=0.71$ for different values of the particle number, compared with the LDA mean-field solution of Sec. 2c at the same total number of particles. The inset shows the density distribution $n(\mathrm{z̃})$ at different values of the radial distance $\mathrm{r̃}$.

Figure 2

Mean-field equation of state ${\mathrm{ñ}}_{\mathrm{mf}}$, and negative gap $-{\Delta }_{\mathrm{mf}}$, as a function of the chemical potential $\mathrm{\mu ̃}$ for ENS parameters $\mathrm{g̃}=0.13,{\mathrm{\omega ̃}}_z=0.55,t=0.71$. The upper two curves compare the LDA density of the fully quasi-two-dimensional system (“quasi-2D density”) to the corresponding strictly two-dimensional one (“2D density”). The two lower curves compare the gap of the quasi-two-dimensional trap for ENS parameters to a strictly two-dimensional trap. The LDA gap, ${\Delta }_{\mathrm{mf}}={\mathrm{\epsilon ̃}}_0/t-\mathrm{\mu ̃}$, differs from the strictly two-dimensional approximation $\mathrm{g̃}\mathrm{ñ}/\pi -\mathrm{\mu ̃}$ only at high density. The comparisons with the strictly two-dimensional mean-field results [see Eqs. (22) and (23)] illustrates the dependence of the equation of state on microscopic parameters.

Figure 3

Correlation density $\Delta \mathrm{ñ}$ vs rescaled mean-field gap ${\Delta }_{\mathrm{mf}}/\mathrm{g̃}$. QMC data for various interaction strengths and confinements ${\mathrm{\omega ̃}}_z$ are compared with the interpolation Eq. (30) of classical-field results [12].

Figure 4

QMC density profile for the NIST parameters $t=T/T_{\mathrm{BEC}}^{2\mathrm{D}}=0.75,\mathrm{g̃}=0.02$, and ${\mathrm{\omega ̃}}_z=0.5$, for different $N$. To illustrate that the density profile in the normal phase cannot be described well as a single Gaussian, we compare the data with the expansion of Eq. (32) as a sum of $k_{max}$ Gaussians with $k_{max}=1$ and with $k_{max}=4$. The inset shows the QMC cycle weights ${\pi }_k$ for $N=100\hspace{0.5em}000$ (see [21]).

Figure 5

Schematic density profile of a strictly two-dimensional trapped Bose gas at $T_{\mathrm{KT}}$ for $\mathrm{g̃}\to 0$. For $\mathrm{r̃}\gg \sqrt{\mathrm{g̃}},\mathrm{ñ}$ coincides with the ideal gas at $T_{\mathrm{BEC}}^{2\mathrm{D}}$ (inset, the classical Boltzmann distribution $e^{-{\mathrm{r̃}}^2/2}$ is given for comparison). In the fluctuation regime, for $\mathrm{r̃}\lesssim 1.25\sqrt{\mathrm{g̃}}$, mean-field and correlation effects become important. The density diverges as $~\ln (1/\mathrm{g̃})$, yet the correlation contribution $\Delta \mathrm{ñ}$ remains finite.

Figure 6

Off-diagonal coherence $c(r)$ for ENS parameters with $t=0.71$ (main graph, $\mathrm{ñ}>{\mathrm{ñ}}_f$) and $t=0.769$ (inset, $\mathrm{ñ}<{\mathrm{ñ}}_f$) compared with the mean-field prediction $c_{\mathrm{mf}}(r)$ and the gap model of Eq. (37). In the fluctuation regime, finite-size effects for off-diagonal correlations are more pronounced than for the density (see Fig. 1).

Figure 7

Off-diagonal coherence $c(r)$ for NIST parameters with $t=0.74$ (main graph) and $t=0.769$ (inset) in comparison with the mean-field prediction $c_{\mathrm{mf}}(r)$ and the gap model $c_{\Delta }(r)$ defined in Eq. (37). At $t=0.769$, the total central density is $\mathrm{ñ}(0)\simeq 5.1<{\mathrm{ñ}}_f$, and the system is outside the fluctuation regime. At $t=0.74,\mathrm{ñ}(0)\simeq 10.5>{\mathrm{ñ}}_f$, and the system is close to the Kosterlitz-Thouless transition, ${\Delta }_{\mathrm{mf}}/\mathrm{g̃}\simeq 0.08$. Strong finite-size effects are evident in the fluctuation regime.

Figure 8

Condensate fraction $n_0$ (inset) and wave function ${\phi }_0(r)$ (main graph), computed by QMC for ENS parameters with $t=0.71$ for various system sizes. The condensate wave function is modified with respect to the ground-state wave function ${\psi }_0(r)$ of the unperturbed harmonic oscillator, but the Gaussian shape is preserved. The two largest systems are above the scale $N_{\mathrm{fs}}$ [see Eq. (39)].

Figure 9

Central pair correlations, $g(r)=n^{(2)}(r,0)/[n(r)n(0)]$, of the quasi-two-dimensional trapped Bose gas at temperature $T=T_{\mathrm{BEC}}^{2\mathrm{D}}$ (main figure) and $T=0.75T_{\mathrm{BEC}}^{2\mathrm{D}}$ (inset) for $N=100\hspace{0.5em}000$ atoms (ENS parameters), together with the prediction of the mean-field gap model, $g_{\Delta }(r)=n_{\Delta }^{(2)}(r,0)/n(r)n(0)$, and the short-range improved mean-field model, ${\chi }_{2\mathrm{D}}(r)g_{\Delta }(r)$.

Figure 10

Quasi-condensate density $n_{\mathrm{qc}}$ obtained by QMC from the renormalized pair-correlation function at $T=0.71T_{\mathrm{BEC}}^{2\mathrm{D}}$, and $T=0.75T_{\mathrm{BEC}}^{2\mathrm{D}}$ (see Fig. 9) for ENS parameters, plotted as a function of ${\Delta }_{\mathrm{mf}}$, and compared with classical-field (CFT) simulations [12]. The inset shows the boundary of the region with strong finite-size effects [see Eq. (39)].