Local atom-number fluctuations in quantum gases at finite temperature

We investigate the number fluctuations in small cells of quantum gases pointing out important deviations from the thermodynamic limit fixed by the isothermal compressibility. Both quantum and thermal fluctuations in weakly as well as highly compressible fluids are considered. For the two-dimensional (2D) superfluid Bose gas we find a significant quenching of fluctuations with respect to the thermodynamic limit, in agreement with recent experimental findings. An enhancement of the thermal fluctuations is instead predicted for the 2D dipolar superfluid Bose gas, which becomes dramatic when the size of the sample cell is of the order of the wavelength of the rotonic excitation induced by the interaction.

Figure 1

Number fluctuation $\langle \delta N^2\rangle$ for a free Fermi gas as a function of $T/T_F$ calculated integrating the density-density correlations (solid line) for $N=100$ and $N=2000$ and using the thermodynamic expression Eq. (1) (dashed line). Inset: $\langle \delta N^2\rangle$ at $T=0$ for a cylinder (solid line) and for a sphere (dashed-dotted line) for $N=100$ as a function of the aspect ratio $L/R$. The fluctuations closely follow the cylinder surface dependence (dashed line) for a fixed volume (see text).

Figure 2

Static structure factor $S(k,T)$ for a quasi-2D Bose gas with (continuous lines) and without (dashed lines) dipolar interaction for $k_{B}T/m{c}^2=0$, 1, 2. Inset: The dispersion relation Eq. (7). For the dipolar case we use values for ${}^{52}$Cr atoms, namely $a=-1.5a_0$, $l_z=0.09\mu \text{m}$ which corresponds to $\beta =-1.083$, $n/\left(\sqrt{2\pi }{l}_z\right)={10}^{14}$ cm${}^{-3}$, and $\xi =2.5\mu$m.

Figure 3

$\langle \delta N^2\rangle /N$ for a quasi-2D Bose gas in the superfluid regime as a function of $k_{B}T/\left(m{c}^2\right)$ for $N=12$, 100 and $g={\hslash }^2/4m$, calculated from Eq. (2) (solid lines) and from Eq. (1) (dashed lines). Inset: $\langle \delta N^2\rangle$ as a function of the inverse of the dimensionless coupling constant $\overline{g}=mg/{\hslash }^2$ for $N=12$, $T=20\text{nK}$ calculated from Eq. (2) (solid lines) and from Eq. (1) (dashed lines). The (red) small boxes are the experimental value extracted from [7] at $T\approx 20$ nK. The $T=0$ result Eq. (11) is also reported for comparison (dashed-dotted line).

Figure 4

$\langle \delta N^2\rangle /N$ as a function of $R/\xi$ at zero temperature (dashed lines) and $k_{B}T/\left(m{c}^2\right)=2$ (solid lines) for a dipolar gas of ${}^{52}$Cr atoms (black lines) and for a nondipolar gas $(\beta =0)$ (gray lines) with the same chemical potential. Parameters as in Fig. 2 and in particular $\xi =2.5\mu$m.

Figure 5

Comparison of $\langle \delta N^2\rangle$ as a function of $R/\xi$ for $\tilde{g}=1/4$ calculated numerically from Eqs. (A1) and (A7). Inset: Zoom onto small $R/\xi$.