Particle number fluctuations in a cloven trapped Bose gas at finite temperature

We study fluctuations in the atom number difference between two halves of a harmonically trapped Bose gas in three dimensions. We solve the problem analytically for noninteracting atoms. In the interacting case we find an analytical solution in the Thomas-Fermi and high temperature limit in good agreement with classical field simulations. In the large system size limit, fluctuations in the number difference are maximal for a temperature $T\simeq 0.7T_c$ where $T_c$ is the critical temperature, independently of the trap anisotropy. The occurrence of this maximum is due to an interference effect between the condensate and the noncondensed fields.

Figure 1

We consider fluctuations of the particle number difference $N_L-N_R$ between the left and the right halves of a three-dimensional harmonic potential. The direction $x$ is a trap eigenaxis.

Figure 2

Normalized variance of the particle number difference $N_L-N_R$ as a function of temperature in a cigar-shaped trap with ${\omega }_y={\omega }_z=2{\omega }_x$. The number of particles is $\langle N\rangle =6000$ (black, lower lines) and $\langle N\rangle =13,000$ (red, upper lines). The inset is a magnification of the $T>T_c$ region. Solid lines: exact result (9) together with (11). Dashed line for $T>T_c$: approximate result (17) together with (15). Dashed line for $T<T_c$: approximate result resulting from the improved estimates (26) and (25). The temperature $T$ is expressed in units of the critical temperature $T_c$ defined in Eq. (16).

Figure 3

Symbols: Classical field simulations for $N=17,000$ (circles) and $N=6000$ (triangles) ${}^{87}\mathrm{Rb}$ atoms with interactions (blue and black, lower) and without interactions for comparison (red, upper). (Red solid, upper, line) Analytical prediction (9) in the noninteracting case. (Blue and black solid, lower, lines) Analytical prediction (54) in the interacting case. The oscillation frequencies are ${\omega }_x/(2\pi )=0.234$ kHz, ${\omega }_y/(2\pi )=1.120$ kHz and ${\omega }_z/(2\pi )=1.473$ kHz. For the interacting case the $s$-wave scattering length is $a=100.4$ Bohr radii.

Figure 4

The measurement can be seen as a balanced homodyne detection of the noncondensed field where the condensate acts as a local oscillator.