Theory of correlations between ultracold bosons released from an optical lattice

In this paper we develop a theoretical description of the correlations between ultracold bosons after free expansion from confinement in an optical lattice. We consider the system evolution during expansion and give criteria for a far-field regime. We develop expressions for first and second order two-point correlations based on a variety of commonly used approximations to the many-body state of the system including Bogoliubov, mean-field decoupling, and particle-hole perturbative solution about the perfect Mott-insulator state. Using these approaches we examine the effects of quantum depletion and pairing on the system correlations. Comparison with the directly calculated correlation functions is used to justify a Gaussian form of our theory from which we develop a general three-dimensional formalism for inhomogeneous lattice systems suitable for numerical calculations of realistic experimental regimes.

Figure 1

Schematic of atomic far-field expansion from a lattice: Consider an atom initially $(t=0)$ located at lattice position ${\mathbf{R}}_{\mathbf{j}}$, and after expansion time $t$ is found at far-field position $\mathbf{x}$ (measured relative to the lattice center indicated by position 0). The initial lattice sites are indicated as gray filled circles.

Figure 2

Far-field expansion of atoms from a lattice: A tight-binding Wannier state at lattice position ${\mathbf{R}}_{\mathbf{j}}$ expands for time $t$.

Figure 3

Evolution of Wannier state width $W\left(t\right)$ in units of the direct lattice vector $a$ for typical optical lattice parameters and two lattice depths: $V_0=5E_R$ (black curve) and $V_0=50E_R$ (gray curve). Results for ${}^{87}\mathrm{Rb}$ in a lattice made from counterpropagating $850\mathrm{nm}$ light fields.

Figure 4

The function $F\left(Q\right)$ for $M=3$ (black line), $M=5$ (medium gray line), and $M=25$ (light gray line).

Figure 5

(Color online) Comparison of expanded density distributions. (a) Perfect Mott-insulator (dotted line) and pure Bose-Einstein condensate (solid line) results. (b) Various superfluid phase results: pure Bose-Einstein condensate (solid line), Bogoliubov result (dashed line), and mean-field decoupling result (dash-dotted line). (c) Various Mott-insulator phase results: perfect Mott-insulator result (dotted line) and including particle-hole corrections (dash-dotted line). Parameters: $N\approx 85$ atoms in a lattice with $M=15$. The Bogoliubov and mean-field decoupling calculations are for $U∕2J\approx 3.2$ and $N_0\approx 65$ with noncondensate atom numbers being $\tilde{N}\approx 20$ (Bogoliubov) and $\tilde{N}\approx 17$ (decoupling). The perfect Mott-insulator and particle-hole results are for $U∕2J\approx 42$ with $N\approx 85$ (see Table ).

Figure 6

(Color online) Two-point first order correlation function $G^{\left(1\right)}$ for (a) pure Bose-Einstein condensate and (b) perfect Mott insulator. Two-point second order correlation function $G^{\left(2\right)}$, for (c) pure Bose-Einstein condensate and (d) perfect Mott insulator. Parameters are the same as for the insulating cases considered in Fig. 5.

Figure 7

(Color online) Second order correlation function $G^{\left(2\right)}$ in the superfluid regime. (a) Bogoliubov result and (b) mean-field decoupling result. (c) Decoupling prediction for the pairing strength, $G^{\left(2\right)}(l_b∕2,-l_b∕2)$ (see the text) as $U∕J$ is varied. The result for the case shown in (b) is indicated by a square. Parameters: Cases (a) and (b) are the same parameters as used in the superfluid calculations in Fig. 5.

Figure 8

(Color online) Behavior of mean-field decoupling parameters as lattice depth changes. (a) Commensurately filled lattice with $n=1$ and (b) incommensurate case with $n=1.2$. Results for $\alpha$ (solid line), $\beta$ (dashed line), $\gamma$ (dash-dotted line), and $\delta$ (dotted line) are shown. A mean-field estimate of the Mott-insulator transition for $n=1$ is indicated with the vertical dotted line.

Figure 9

(Color online) Second order correlation functions in the Mott-insulator regime. (a) Perfect Mott-insulator state and (b) particle-hole corrections to the perfect Mott-insulator state. The same parameters are used as for the Mott-insulator calculations in Fig. 5.

Figure 10

(Color online) Comparison of Gaussian and full decoupling results. (a) Comparison of ${\chi }_A$ for $n=1$ (stars) and $n=1.2$ (crosses), and ${\tilde{\chi }}_A$ for $n=1$ (circles) and $n=1.2$ (dots). (b) Comparison of ${\chi }_B$ for $n=1$ (stars) and $n=1.2$ (crosses), and ${\tilde{\chi }}_B$ for $n=1$ (circles) and $n=1.2$ (dots). The mean-field estimate of the Mott-insulator transition for $n=1$ is indicated with the vertical dotted line.

Figure 11

(Color online) Comparison of $G^{\left(2\right)}$ correlation functions for 1D lattice systems: (a) Superlattice with periodically modulated density, (b) uniformly filled translationally invariant lattice. Both cases are calculated using the inhomogeneous perfect Mott-insulator approach (i.e., tunneling is neglected) with $N=29$ (superlattice), $N=30$ (uniform lattice), and $M=15$.