Atom counting in expanding ultracold clouds

We study the counting statistics of ultracold bosonic atoms that are released from an optical lattice. We show that the counting probability distribution of the atoms collected at a detector located far away from the optical lattice can be used to characterize the initially trapped states. We consider trapped superfluid and insulating states with different occupation patterns. We analyze how the correlations between the modes that develop during the expansion in the gravitational field appear in the counting distribution and find that the ratio of the detector size with respect to the expanded wave function determines whether short-range or long-range correlations of the initial state are reflected in the counting statistics. We find that detectors which are large compared to the size of the expanded wave function distinguish insulating and superfluid phases irrespective of the occupation pattern. We show that using detectors that are small compared to the size of the expanded wave function, occupation patterns in insulating and supersolid states can be distinguished. Finally, we show how the magnetic phase patterns are dramatically reflected in the number distribution.

Figure 1

The ratio between the autocorrelations and cross-correlation terms that contribute to the counting distribution depend on the size of the detector. For a 1D system, we plot the following: (a) wide detector limit ${\Delta }_z\simeq {\omega }_{t_d}$; (b) narrow detector limit ${\Delta }_z\ll {\omega }_{t_d}$. We plot $|A_{ii}|$ (blue squares) $|A_{i,i+1}|$ (green circles), $|A_{i,i+2}|$ (red diamonds), and $|A_{i,i+3}|$ (light blue triangles). Parameters used: $z_0=1$ cm, ${\Delta }_x={\Delta }_y=1$ cm, and $\kappa =1$, $a_0=266$ nm.

Figure 2

MI vs. SF as a function of distance from detector. Probability distribution for MI (black bars) and superfluid (white bars) states in a $3\times 3\times 3$ lattice. The counting distribution of the SF state is Poissonian, whereas the counting distribution of the MI state is sub-Poissonian. For increasing distance between the detector, the difference between the states is no longer visible. ${\Delta }_x={\Delta }_y=2$ mm; ${\Delta }_z=2$ cm; $\kappa =1$. (a) $z_0=1$ cm; (b) $z_0=3$ cm; (c) $z_0=5$ cm.

Figure 3

Variance ${\sigma }^2/N$ of the counting distribution for a MI (blue squares) and SF (green circles) state in a $3\times 3\times 3$ lattice with respect to the distance from the detector $z_0$. ${\Delta }_x={\Delta }_y=2$ mm; ${\Delta }_z=2$ cm; $\kappa =1$.

Figure 4

Joint probability distribution of an expanded MI (upper row) and a SF (lower row) in a $4\times 4$ lattice in the $xy$ plane with two symmetrically placed detectors. In (a) and (c) $x_d=0$. In (b) and (d) $x_d=1$ cm. Parameters used: $z_0=1$ cm; ${\Delta }_z=2$ mm; ${\Delta }_x={\Delta }_y=2$ cm; $\kappa =0.5$.

Figure 5

Correlations of the joint probability distribution for an expanded MI state in a $4\times 4$ lattice in the $xy$ plane registered at two symmetrically placed detectors. For short distances an antibunching of the particles is observed. As the distance between the detectors increases, the counting events at the two detectors are no longer correlated. Parameters used: $z_0=1$ cm; ${\Delta }_z=2$ mm; ${\Delta }_x={\Delta }_y=2$ cm; $\kappa =0.5$.

Figure 6

The counting distributions of an expanded 1D checkerboard state, where every second site is occupied (black bars), and striped insulating pattern, where the first half of the sites is occupied and the remaining sites are empty (white bars), are clearly distinguishable. Parameters used: $z_0=1$ cm; $\Delta =0.1$ cm; ${\Delta }_z=0.02$ mm; $\kappa =1$, $N=12$.

Figure 7

The counting distributions of an expanded 1D supersolid state with ${\left|\beta \right|}^2=0.5$ and ${\left|\gamma \right|}^2=1.5$ (black bars) and a superfluid state ${\left|\alpha \right|}^2=1$ (white bars) are clearly distinguishable. Parameters used: $z_0=1$ cm; $\Delta =1$ cm; ${\Delta }_z=0.02$ mm; $\kappa =1$, $N=40$.

Figure 8

The counting distribution of the particles released from a 2D square lattice in the $xy$ plane with positive tunneling rate (black bars) and negative tunneling rate (white bars). As the sign in the tunneling rate changes, the mean of the counting distribution decreases significantly. Parameters used: ${\Delta }_x={\Delta }_y=1$ cm; ${\Delta }_z=0.1$ mm; $z_0=1$ cm; $a_0=415$ nm; $N=100$; $d_{\max }=5$.

Figure 9

The counting distribution of the particles released from a 2D triangular lattice in the $xy$ plane with positive tunneling rate (black bars) and negative tunneling rate (white bars). As the sign in the tunneling rate changes, the mean of the counting distribution decreases significantly. Parameters used: ${\Delta }_x={\Delta }_y=1$ cm; ${\Delta }_z=0.1$ mm; $z_0=1$ cm; $a_0=415$ nm; $N=100$; $d_{\max }=5$.