Light scattering from ultracold atoms in optical lattices as an optical probe of quantum statistics

We study off-resonant collective light scattering from ultracold atoms trapped in an optical lattice. Scattering from different atomic quantum states creates different quantum states of the scattered light, which can be distinguished by measurements of the spatial intensity distribution, quadrature variances, photon statistics, or spectral measurements. In particular, angle-resolved intensity measurements reflect global statistics of atoms (total number of radiating atoms) as well as local statistical quantities (single-site statistics even without optical access to a single site) and pair correlations between different sites. As a striking example we consider scattering from transversally illuminated atoms into an optical cavity mode. For the Mott-insulator state, similar to classical diffraction, the number of photons scattered into a cavity is zero due to destructive interference, while for the superfluid state it is nonzero and proportional to the number of atoms. Moreover, we demonstrate that light scattering into a standing-wave cavity has a nontrivial angle dependence, including the appearance of narrow features at angles, where classical diffraction predicts zero. The measurement procedure corresponds to the quantum nondemolition measurement of various atomic variables by observing light.

Figure 1

(Color online) Setup. Atoms in a lattice are illuminated by a pump wave at angle ${\theta }_0$; scattered (probe) light is collected by a cavity at angle ${\theta }_1$ and measured by a detector.

Figure 2

(Color online) Intensity angular distributions for two traveling waves; the pump is transverse to the lattice $\left({\theta }_0=0\right)$. (a) Intensity of classical diffraction of coherent light (curve A), and isotropic intensity of incoherent light scattering, Eq. (14), for coherent atomic state (line B) and MI state (line C). (b) Noise quantity, Eqs. (23a, 23b, 23c), for coherent atomic state (constant value 1, line A), SF state with all sites illuminated $K=M$ (curve B), and MI state (constant value 0, line C). (c) The same as in (b) but for partially illuminated SF state with $K=M∕2$. $N=M=30$.

Figure 3

(Color online) Intensity angular distributions for scattering into a standing-wave cavity. (a) Traveling-wave pump at ${\theta }_0=0.1\pi$; (b) traveling- or standing-wave pump at ${\theta }_0=0$; (c) standing-wave pump at ${\theta }_0=0.1\pi$. Intensities of classical diffraction are shown in (a1), (b1), and (c1); noise quantities for coherent state are shown in (a2), (b2), and (c2) and for SF state in (a3), (b3), and (c3). $N=M=K=30$.

Figure 4

(Color online) Quadrature angular distributions for two traveling waves. (a) Quadrature for classical diffraction (a1), and quadrature variance for coherent (a2) and SF (a3) states; pump-homodyne phase difference $\beta =0$. (b) Quadrature variance for coherent state for $\beta =\pi ∕4$ (b1), $\pi ∕2$ (b2), $3\pi ∕4$ (b3); (c) quadrature variance for SF for $\beta =\pi ∕4$ (c1), $\pi ∕2$ (c2), $3\pi ∕4$ (c3). ${\theta }_0=0$, $N=M=K=30$.

Figure 5

(Color online) Angular distributions of photon number variances for two traveling waves. (a) Intensity of classical diffraction (curve A), and isotropic intensity of incoherent light scattering, Eq. (14), for coherent atomic state (line B) and MI state (line C); (b) normal-ordered photon number variance for coherent atomic state under scattering of coherent, Eq. (24) (curve A), and incoherent (line B) light; (c) normal-ordered photon number variance for SF state under scattering of coherent, Eq. (24) (curve A), and incoherent (line C) light, and variance for coherent (line B) and MI (curve D) states under scattering of incoherent light. Normal-ordered photon number variance for MI state under scattering of coherent light is zero for all angles. ${\theta }_0=0$, $N=M=K=30$.