Observation of Strong Quantum Depletion in a Gaseous Bose-Einstein Condensate

We studied quantum depletion in a gaseous Bose-Einstein condensate. An optical lattice enhanced the atomic interactions and modified the dispersion relation resulting in strong quantum depletion. The depleted fraction was directly observed as a diffuse background in the time-of-flight images. Bogoliubov theory provides a semiquantitative description for our observations of depleted fractions in excess of 50%.

Figure 1

Interference patterns in time of flight: The ramping sequence was interrupted as the lattice is ramped up ($11,18,22E_R$) and back down ($11,0E_R$). The time of flight was 10 ms; the field of view is $861\mu \mathrm{m}\times 1075\mu \mathrm{m}$.

Figure 2

Quantum depletion of a ${}^{23}\mathrm{Na}$ BEC confined in a one-, two-, and three-dimensional optical lattice: the data points with statistical error bars are measured $N_{\mathrm{qd}}/N$; the three thick curves are the theoretical calculation of $N_{\mathrm{qd}}/N$ using Bogoliubov theory and local density approximation. For comparison, also shown are (thin curves): (i) the (smoothed out) Mott-insulator fraction $N_{\mathrm{MI}}/N$ based on a mean-field theory; (ii) the calculated quantum depletion for a homogeneous system of per-site occupancy number $n=1$ and (iii) $n=7$.

Figure 3

On-site interaction $U$ and tunneling rate $J$ for a ${}^{23}\mathrm{Na}$ BEC in an optical lattice at 1064 nm: $U_{d\mathrm{D}}$ is for a $d$-dimensional lattice. In a three-dimensional lattice, the superfluid to Mott-insulator transition for occupancy number $n$ occurs when $J_n=6(2n+1+2\sqrt{n(n+1)})J$ equals $U_{3\mathrm{D}}$ [see Eq. (3)]. The horizontal locations of the crossing points where $J_n=U_{3\mathrm{D}}$ are the critical lattice depths.