The atom-light coupling in the Hamiltonian

${\mathcal{H}}^{ml}$ introduces correlations between light and matter. As a consequence, it is possible to characterise the quantum state of the atoms by performing measurements of the light observables. Specifically, probing the light amplitude

${\widehat{a}}_{1}$ or photon number

${\widehat{a}}_{1}^{\u2020}{\widehat{a}}_{1}$ describes a quantum nondemolition (QND) measurement of the observables related to

${\widehat{D}}_{lm}$ [

36]. We focus on the regime where the atomic dynamics (

i.e., tunnelling) is much slower than the light scattering: the tunnelling amplitude

${t}_{0}$ and the interaction

U determine the quantum properties of the atomic state, but such properties do not change during the measurement time. The stationary solution of the Heisenberg equations for the scattered light operator

${\widehat{a}}_{1}$ (neglecting the small dispersive frequency shift

${\Omega}_{11}{\widehat{D}}_{11}$) is given by:

where

${\Delta}_{p}={\omega}_{p}-{\omega}_{c}$ is the probe-cavity detuning. Analogously to classical physics, the light amplitude

$\langle {\widehat{a}}_{1}\rangle $ depends on the mean density at each lattice site and describes the scattering of the probe beam

${\alpha}_{0}$ from a periodic grating of atoms. Its angular distribution is determined by the position of the atoms:

$\langle {\widehat{a}}_{1}\rangle $ reaches maximum value if they scatter light with the same phase, while it vanishes if the contributions from each lattice site interfere destructively. Moreover, quantum density-density correlations of the atomic state are imprinted on the expectation value of the photon number operator:

**Figure 2.**
Classical diffraction patterns (normalised to ${N}^{2}$) and quantum additions (normalised to N) for N fermions in a one-dimensional optical lattice (45 sites, half filling) in the non-interacting regime (first row) and for $U/{t}_{0}=10$ (second row). The lattice extends into the vertical direction, and the probe is along the horizontal, i.e., the angle marked with ${0}^{\circ}$ indicates the light intensity in the forward direction. The classical diffraction patterns (**a**,**d**) do not depend on the interaction while the quantum additions ${R}_{y}$ (**b**,**e**) and ${R}_{x}$ (**c**,**f**) distinguish between different ground states. Note that the scale in (f) is different from (a–e): the fluctuations in the atom number increase as the attraction between the atoms favours doubly-occupied sites. Red circles indicate the orientation of the lattice.

This quantity depends on the scattering angle via the coefficients

${J}_{ii}$ and is directly related to the fluctuations of the atom number. Therefore, quantum light scattering distinguishes between different quantum states, such as the superfluid and Mott insulator states [

24]. As mentioned in the previous section, we extend the bosonic model [

24,

37,

38] to also describe spin-1/2 fermionic atoms and define the quantum additions

${R}_{x}$ and

${R}_{y}$, related to the fluctuations in density and magnetisation, respectively. If the mode functions of the probe and scattered light are travelling waves with wave-vectors

${\mathbf{k}}_{\mathrm{in}}$ and

${\mathbf{k}}_{\mathrm{out}}$, respectively, the coefficients

${J}_{jj}$ are proportional to

$exp\left[i({\mathbf{k}}_{\mathrm{in}}-{\mathbf{k}}_{\mathrm{out}})\xb7{\mathbf{r}}_{j}\right]$, and the quantum addition

R is proportional to a structure factor,

i.e., the Fourier transform of the density-density correlations.

Figure 2 compares the classical diffraction pattern and quantum additions

${R}_{x}$ and

${R}_{y}$ for fermions in a one-dimensional optical lattice at half filling. The scattering patterns were calculated for the ground state, obtained via imaginary time evolution using the TNTlibrary [

39]. In particular, we describe the system using the Hubbard model and focus on the ground state of the system in two different regimes: non-interacting (

$U/{t}_{0}=0$) and strongly-attractive interactions (

$U/{t}_{0}=10$). Note that in both cases, the local magnetisation of the system is zero (

$\langle {\widehat{m}}_{i}\rangle =0$), and as a consequence, the classical diffraction pattern for linear-

y polarised light vanishes,

i.e.,

$\langle {\widehat{a}}_{1y}\rangle =0$. Nevertheless, the quantum addition

${R}_{y}$ is non-zero and depends on the quantum state of the system. In particular, we find that

${R}_{y}$ decreases with increasing values of

$U/{t}_{0}$, as the attraction between the atoms favours doubly-occupied sites,

i.e., the formation of pairs of fermions with opposite spin, which suppress the fluctuations in the magnetisation. Furthermore, the classical diffraction pattern due to the atomic density does not depend on

$U/{t}_{0}$, as the local density is independent of the interaction for a translationally-invariant optical lattice (

$\langle {\widehat{\rho}}_{i}\rangle =1$). However, the presence of doubly-occupied sites increases the fluctuations in the atom number, leading to a stronger signal for

${R}_{x}$. In general, the scheme we propose allows direct probing of the phase diagram for the superfluid-Mott insulator-Bose glass phases in the one-dimensional Bose–Hubbard model [

8] and the detection of different quantum states of atomic systems such as dimers [

14], density waves and magnetic order.