Probing matter-field and atom-number correlations in optical lattices by global nondestructive addressing

We show that light scattering from an ultracold gas reveals not only density correlations, but also matter-field interference at its shortest possible distance in an optical lattice, which defines key properties such as tunneling and matter-field phase gradients. This signal can be enhanced by concentrating probe light between lattice sites rather than at density maxima. As addressing between two single sites is challenging, we focus on global nondestructive scattering, allowing probing order parameters, matter-field quadratures, and their squeezing. The scattering angular distribution displays peaks even if classical diffraction is forbidden and we derive generalized Bragg conditions. Light scattering distinguishes all phases in the Mott insulator–superfluid–Bose glass phase transition.

Figure 1

Setup. Atoms in an optical lattice are illuminated by a probe beam. The light scatters in free space or into a cavity and is measured by a detector.

Figure 2

Light intensity scattered into a standing wave mode from a SF in a 3D lattice (units of $R/N_K)$. Arrows denote incoming traveling wave probes. The Bragg condition, $\mathrm{\Delta }\mathbf{k}=\mathbf{G}$, is not fulfilled, so there is no classical diffraction, but intensity still shows multiple peaks, whose heights are tunable by simple phase shifts of the optical beams: (a) ${\phi }_1=0$, (b) ${\phi }_1=\pi /2$. Interestingly, there is also a significant uniform background level of scattering which does not occur in its classical counterpart.

Figure 3

The Wannier function products: (a) $W_0\left(x\right)$ (solid line, right axis), $W_1\left(x\right)$ (dashed line, left axis), and (b) their Fourier transforms $\mathcal{F}\left[W_{0,1}\right]$. The density $J_{i,i}$ and matter-interference $J_{i,i+1}$ coefficients (1) in (c) diffraction maximum and (d) diffraction minimum are shown as functions of standing wave shifts $\phi$ or, if one were to measure the quadrature variance ${\left(\mathrm{\Delta }{X}_{\beta }^F\right)}^2$, the local oscillator phase $\beta$. The black points indicate the positions, where light measures matter interference $\widehat{B}\ne 0$, and the density term is suppressed, $\widehat{D}=0$. The trapping potential depth is approximately five recoil energies.

Figure 4

Photon number scattered in a diffraction minimum, given by Eq. (12), where $\tilde{C}=2{\left|C\right|}^2(K-1){\mathcal{F}}^2\left[W_1\right](\pi /d)$. More light is scattered from a MI than a SF due to the large uncertainty in the phase in the insulator. (a) The variances of quadratures $\mathrm{\Delta }{X}_0^b$ (solid line) and $\mathrm{\Delta }{X}_{\pi /2}^b$ (dashed line) of the matter field across the phase transition. Level 1/4 is the minimal (Heisenberg) uncertainty. There are three important points along the phase transition: the coherent state (SF) at A, the amplitude-squeezed state at B, and the Fock state (MI) at C. (b) The uncertainties plotted in phase space.

Figure 5

(a) The angular dependence of scattered light $R$ for SF (thin black line, left scale, $U/2J^{\text{cl}}=0)$ and MI (thick green line, right scale, $U/2J^{\text{cl}}=10)$. The two phases differ in both their value of $R_{\text{max}}$ as well as $W_R$, showing that density correlations in the two phases differ in magnitude as well as extent. (b),(d) Light-scattering maximum $R_{\text{max}}$ and (c),(e) the width $W_R$. It is very clear that varying chemical potential $\mu$ or density $\langle n\rangle$ sharply identifies the SF-MI transition in both quantities. (b),(c) Cross sections of the phase diagrams (d) and (e) at $U/2J^{\text{cl}}=2$ (thick blue line), 3 (thin purple line), and 4 (dashed blue line). Insets: Density dependencies for the $U/\left(2J^{\text{cl}}\right)=3$ line. $K=M=N=25$.

Figure 6

The MI-SF-BG phase diagrams for (a) light-scattering maximum $R_{\text{max}}/N_K$ and (b) width $W_R$. Measurement of both quantities distinguishes all three phases. Transition lines are shifted due to finite-size effects [49], but it is possible to apply well-known numerical methods to extract these transition lines from such experimental data extracted from $R$ [41]. $K=M=N=35$.