%******************************************************************************* %*********************************** Third Chapter ***************************** %******************************************************************************* \chapter{Probing Correlations by Global Nondestructive Addressing} %Title of the Third Chapter \ifpdf \graphicspath{{Chapter3/Figs/Raster/}{Chapter3/Figs/PDF/}{Chapter3/Figs/}} \else \graphicspath{{Chapter3/Figs/Vector/}{Chapter3/Figs/}} \fi %********************************** %First Section ************************************** \section{Introduction} Having developed the basic theoretical framework within which we can treat the fully quantum regime of light-matter interactions we now consider possible applications. There are three prominent directions in which we can proceed: nondestructive probing of the quantum state of matter, quantum measurement backaction induced dynamics and quantum optical lattices. Here, we deal with the first of the three options. In this chapter we develop a method to measure properties of ultracold gases in optical lattices by light scattering. In the previous chapter we have shown that quantum light field couples to the bosons via the operator $\hat{F}$. This is the key element of the scheme we propose as this makes it sensitive to the quantum state of the matter and all of its possible superpositions which will be reflected in the quantum state of the light itself. We have also shown in section \ref{sec:derivation} that this coupling consists of two parts, a density component $\hat{D}$ given by Eq. \eqref{eq:D}, and a phase component $\hat{B}$ given by Eq. \eqref{eq:B}. Therefore, when probing the quantum state of the ultracold gas we can have access to not only density correlations, but also matter-field interference at its shortest possible distance in an optical lattice, i.e.~the lattice period. Previous work on quantum non-demolition (QND) schemes \cite{rogers2014, mekhov2007prl, eckert2008} probe only the density component as it is generally challenging to couple to the matter-field observables directly. Here, we will consider nondestructive probing of both density and interference operators. Firstly, we will consider the simpler and more typical case of coupling to the atom number operators via $\hat{F} = \hat{D}$. However, we show that light diffraction in this regime has several nontrivial characteristics due to the fully quantum nature of the interaction. Firstly, we show that the angular distribution has multiple interesting features even when classical diffraction is forbidden facilitating their experimental observation. We derive new generalised Bragg diffraction conditions which are different to their classical counterpart. Furthermore, due to the fully quantum nature of the interaction our proposal is capable of probing the quantum state beyond mean-field prediction. We demonstrate this by showing that this scheme is capable of distinguishing all three phases in the Mott insulator - superfluid - Bose glass phase transition in a 1D disordered optical lattice which is not very well described by a mean-field treatment. We underline that transitions in 1D are much more visible when changing an atomic density rather than for fixed-density scattering. It was only recently that an experiment distinguished a Mott insulator from a Bose glass \cite{derrico2014} via a series of destructive measurements. Our proposal, on the other hand, is nondestructive and is capable of extracting all the relevant information in a single experiment. Having shown the possibilities created by this nondestructive measurement scheme we move on to considering light scattering from the phase related observables via the operator $\hat{F} = \hat{B}$. This enables in-situ probing of the matter-field coherence at its shortest possible distance in an optical lattice, i.e. the lattice period, which defines key processes such as tunnelling, currents, phase gradients, etc. This is in contrast to standard destructive time-of-flight measurements which deal with far-field interference although a relatively near-field scheme was use in Ref. \cite{miyake2011}. We show how within the mean-field treatment, this enables measurements of the order parameter, matter-field quadratures and squeezing. This can have an impact on atom-wave metrology and information processing in areas where quantum optics already made progress, e.g., quantum imaging with pixellized sources of non-classical light \cite{golubev2010, kolobov1999}, as an optical lattice is a natural source of multimode nonclassical matter waves. \section{Coupling to the Quantum State of Matter} As we have seen in section \ref{sec:a} under certain approximations the scattered light mode, $\a_1$, is linked to the quantum state of matter via \begin{equation} \label{eq:a-3} \a_1 = C \hat{F} = C \left(\hat{D} + \hat{B} \right), \end{equation} where the atomic operators $\hat{D}$ and $\hat{B}$, given by Eq. \eqref{eq:D} and Eq. \eqref{eq:B}, are responsible for the coupling to on-site density and inter-site interference respectively. It is crucial to note that light couples to the bosons via an operator as this makes it sensitive to the quantum state of the matter as this will imprint the fluctuations in the quantum state of the scattered light. Here, we will use this fact that the light is sensitive to the atomic quantum state due to the coupling of the optical and matter fields via operators in order to develop a method to probe the properties of an ultracold gas. Therefore, we neglect the measurement back-action and we will only consider expectation values of light observables. Since the scheme is nondestructive (in some cases, it even satisfies the stricter requirements for a QND measurement \cite{mekhov2012, mekhov2007pra}) and the measurement only weakly perturbs the system, many consecutive measurements can be carried out with the same atoms without preparing a new sample. We will show how the extreme flexibility of the the measurement operator $\hat{F}$ allows us to probe a variety of different atomic properties in-situ ranging from density correlations to matter-field interference. \subsection{On-site Density Measurements} We have seen in section \ref{sec:B} that typically the dominant term in $\hat{F}$ is the density term $\hat{D}$ \cite{LP2009, mekhov2007pra, rist2010, lakomy2009, ruostekoski2009}. This is simply due to the fact that atoms are localised with lattice sites leading to an effective coupling with atom number operators instead of inter-site interference terms. Therefore, we will first consider nondestructive probing of the density related observables of the quantum gas. However, we will focus on the novel nontrivial aspects that go beyond the work in Ref. \cite{mekhov2012, mekhov2007prl, mekhov2007pra} which only considered a few extremal cases. As we are only interested in the quantum information imprinted in the state of the optical field we will simplify our analysis by considering the light scattering to be much faster than the atomic tunnelling. Therefore, our scheme is actually a QND scheme \cite{rogers2014, mekhov2007prl, mekhov2007pra, eckert2008} as normally density-related measurements destroy the matter-phase coherence since it is its conjugate variable, but here we neglect the $\bd_i b_j$ terms. Furthermore, we will consider a deep lattice. Therefore, the Wannier functions will be well localised within their corresponding lattice sites and thus the coefficients $J_{i,i}$ reduce to $u_1^*(\b{r}_i) u_0(\b{r}_i)$ leading to \begin{equation} \label{eq:D-3} \hat{D}=\sum_i^K u_1^*(\b{r}_i) u_0(\b{r}_i) \hat{n}_i, \end{equation} which for travelling [$u_l(\b{r})=\exp(i \b{k}_l \cdot \b{r}+i\varphi_l)$] or standing [$u_l(\b{r})=\cos(\b{k}_l \cdot \b{r}+\varphi_l)$] waves is just a density Fourier transform at one or several wave vectors $\pm(\b{k}_1 \pm \b{k}_0)$. We will now define a new auxiliary quantity to aid our analysis, \begin{equation} \label{eq:R} R = \langle \ad_1 \a_1 \rangle - | \langle \a_1 \rangle |^2, \end{equation} which we will call the ``quantum addition'' to light scattering. By construction $R$ is simply the full light intensity minus the classical field diffraction. In order to justify its name we will show that this quantity depends purely quantum mechanical properties of the ultracold gase. We will substitute $\a_1 = C \hat{D}$ using Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and we will make use of the shorthand notation $A_i = u_1^*(\b{r}_i) u_0(\b{r}_i)$. The result is \begin{equation} R = |C|^2 \sum_{i,j}^K A^*_i A_j \langle \delta \hat{n}_i \delta \hat{n}_j \rangle, \end{equation} where $\delta \hat{n}_i = \hat{n}_i - \langle \hat{n}_i \rangle$. Thus, we can clearly see that $R$ is a result of light scattering from fluctuations in the atom number which is a purely quantum mechanical property of a system. Therefore, $R$, the ``quantum addition'' faithfully represents the new contribution from the quantum light-matter interaction to the diffraction pattern. If instead we are interested in quantities linear in $\hat{D}$, we can measure the quadrature of the light fields which in section \ref{sec:a} we saw that $\hat{X}_\phi = |C| \hat{X}^F_\beta$. For the case when both the scattered mode and probe are travelling waves the quadrature \begin{equation} \hat{X}^F_\beta = \frac{1}{2} \left( \hat{F} e^{-i \beta} + \hat{F}^\dagger e^{i \beta} \right) = \sum_i^K \hat{n}_i\cos[(\b{k}_1 - \b{k}_2) \cdot \b{r}_i - \beta]. \end{equation} Note that different light quadratures are differently coupled to the atom distribution, hence by varying the local oscillator phase, and thus effectively $\beta$, and/or the detection angle one can scan the whole range of couplings. A similar expression exists for $\hat{D}$ for a standing wave probe, where $\beta$ is replaced by $\varphi_0$, and scanning is achieved by varying the position of the wave with respect to atoms. The ``quantum addition'', $R$, and the quadrature variance, $(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$. Therefore, they will havea nontrivial angular dependence, showing more peaks than classical diffraction. Furthermore, these peaks can be tuned very easily with $\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular dependence of $R$ for the case when the scattered mode is a standing wave and the probe is a travelling wave scattering from bosons in a 3D optical lattice. The first noticeable feature is the isotropic background which does not exist in classical diffraction. This background yields information about density fluctuations which, according to mean-field estimates (i.e.~inter-site correlations are ignored), are related by $R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. In Fig. \ref{fig:scattering} we can see a significant signal of $R = |C|^2 N_K/2$, because it shows scattering from an ideal superfluid which has significant density fluctuations with correlations of infinte range. However, as the parameters of the lattice are tuned across the phase transition into a Mott insulator the signal goes to zero. This is because the Mott insulating phase has well localised atoms at each site which suppresses density fluctuations entirely leading to absolutely no ``quantum addition''. \begin{figure}[htbp!] \centering \includegraphics[width=\linewidth]{Ep1} \caption[Light Scattering Angular Distribution]{Light intensity scattered into a standing wave mode from a superfluid in a 3D lattice (units of $R/(|C|^2N_K)$). Arrows denote incoming travelling wave probes. The classical Bragg condition, $\Delta \b{k} = \b{G}$, 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) $\varphi_1=0$; (b) $\varphi_1=\pi/2$. Interestingly, there is also a significant uniform background level of scattering which does not occur in its classical counterpart. } \label{fig:scattering} \end{figure} We can also observe maxima at several different angles in Fig. \ref{fig:scattering}. Interestingly, they occur at different angles than predicted by the classical Bragg condition. Moreover, the classical Bragg condition is actually not satisfied which means there actually is no classical diffraction on top of the ``quantum addition'' shown here. Therefore, these features would be easy to see in an experiment as they wouldn't be masked by a stronger classical signal. We can even derive the generalised Bragg conditions for the peaks that we can see in Fig. \ref{fig:scattering}. % Derive and show these Bragg conditions As $(\Delta X^F_\beta)^2$ and $R$ are quadratic variables, the generalized Bragg conditions for the peaks are $2 \Delta \b{k} = \b{G}$ for quadratures of travelling waves, where $\Delta \b{k} = \b{k}_0 - \b{k}_1$ and $\b{G}$ is the reciprocal lattice vector, and $2 \b{k}_1 = \b{G}$ for standing wave $\a_1$ and travelling $\a_0$, which is clearly different from the classical Bragg condition $\Delta \b{k} = \b{G}$. The peak height is tunable by the local oscillator phase or standing wave shift as seen in Fig. \ref{fig:scattering}b. In section \ref{sec:Efield} we have estimated the mean photon scattering rates integrated over the solid angle for the only two experiments so far on light diffraction from truly ultracold bosons where the measurement object was light \begin{equation} n_{\Phi}= \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma K}{8} (\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2). \end{equation} Therefore, applying these results to the scattering patters in Fig. \ref{fig:scattering} the background signal should reach $n_\Phi \approx 10^6$ s$^{-1}$ in Ref. \cite{weitenberg2011} (150 atoms in 2D), and $n_\Phi \approx 10^{11}$ s$^{-1}$ in Ref. \cite{miyake2011} ($10^5$ atoms in 3D). These numbers show that the diffraction patterns we have seen due to the ``quantum addition'' should be visible using currently available technology, especially since the most prominent features, such as Bragg diffraction peaks, do not coincide at all with the classical diffraction pattern. \subsection{Mapping the quantum phase diagram} \begin{figure}[htbp!] \centering \includegraphics[width=\linewidth]{oph11} \caption[Mapping the Bose-Hubbard Phase Diagram]{(a) The angular dependence of scattered light $R$ for a superfluid (thin black, left scale, $U/2J^\text{cl} = 0$) and Mott insulator (thick green, 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. Light scattering maximum $R_\text{max}$ is shown in (b, d) and the width $W_R$ in (c, e). It is very clear that varying chemical potential $\mu$ or density $\langle n\rangle$ sharply identifies the superfluid-Mott insulator transition in both quantities. (b) and (c) are cross-sections of the phase diagrams (d) and (e) at $U/2J^\text{cl}=2$ (thick blue), 3 (thin purple), and 4 (dashed blue). Insets show density dependencies for the $U/(2 J^\text{cl}) = 3$ line. $K=M=N=25$.} \label{fig:SFMI} \end{figure} We have shown how in mean-field, we can track the order parameter, $\Phi$, by probing the matter-field interference using the coupling of light to the $\hat{B}$ operator. In this case, it is very easy to follow the superfluid to Mott insulator quantum phase transition since we have direct access to the order parameter which goes to zero in the insulating phase. In fact, if we're only interested in the critical point, we only need access to any quantity that yields information about density fluctuations which also go to zero in the MI phase and this can be obtained by measuring $\langle \hat{D}^\dagger \hat{D} \rangle$. However, there are many situations where the mean-field approximation is not a valid description of the physics. A prominent example is the Bose-Hubbard model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, pino2012, pino2013}. Observing the transition in 1D by light at fixed density was considered to be difficult \cite{rogers2014} or even impossible \cite{roth2003}. By contrast, here we propose varying the density or chemical potential, which sharply identifies the transition. We perform these calculations numerically by calculating the ground state using DMRG methods \cite{tnt} from which we can compute all the necessary atomic observables. Experiments typically use an additional harmonic confining potential on top of the optical lattice to keep the atoms in place which means that the chemical potential will vary in space. However, with careful consideration of the full ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the system \cite{batrouni2002}. The 1D phase transition is best understood in terms of two-point correlations as a function of their separation \cite{giamarchi}. In the Mott insulating phase, the two-point correlations $\langle \bd_i b_j \rangle$ and $\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$ ($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay exponentially with $|i-j|$. On the other hand the superfluid will exhibit long-range order which in dimensions higher than one, manifests itself with an infinite correlation length. However, in 1D only pseudo long-range order happens and both the matter-field and density fluctuation correlations decay algebraically \cite{giamarchi}. The method we propose gives us direct access to the structure factor, which is a function of the two-point correlation $\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$, by measuring the light intensity. For two travelling waves maximally coupled to the density (atoms are at light intensity maxima so $\hat{F} = \hat{D}$), the quantum addition is given by \begin{equation} R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0) (\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta \hat{n}_j \rangle, \end{equation} The angular dependence of $R$ for a Mott insulator and a superfluid is shown in Fig. \ref{fig:SFMI}a, and there are two variables distinguishing the states. Firstly, maximal $R$, $R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$, probes the fluctuations and compressibility $\kappa'$ ($\langle \delta \hat{n}^2_i \rangle \propto \kappa' \langle \hat{n}_i \rangle$). The Mott insulator is incompressible and thus will have very small on-site fluctuations and it will scatter little light leading to a small $R_\text{max}$. The deeper the system is in the MI phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the smaller these values will be until ultimately it will scatter no light at all in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a this can be seen in the value of the peak in $R$. The value $R_\text{max}$ in the SF phase ($U/2J^\text{cl} = 0$) is larger than its value in the MI phase ($U/2J^\text{cl} = 10$) by a factor of $\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of $R_\text{max}$ changes across the phase transition. We see that the transition shows up very sharply as $\mu$ is varied. Secondly, being a Fourier transform, the width $W_R$ of the dip in $R$ is a direct measure of the correlation length $l$, $W_R \propto 1/l$. The Mott insulator being an insulating phase is characterised by exponentially decaying correlations and as such it will have a very large $W_R$. However, the superfluid in 1D exhibits pseudo long-range order which manifests itself in algebraically decaying two-point correlations \cite{giamarchi} which significantly reduces the dip in the $R$. This can be seen in Fig. \ref{fig:SFMI}a and we can also see that this identifies the phase transition very sharply as $\mu$ is varied in Figs. \ref{fig:SFMI}(c,e). One possible concern with experimentally measuring $W_R$ is that it might be obstructed by the classical diffraction maxima which appear at angles corresponding to the minima in $R$. However, the width of such a peak is much smaller as its width is proportional to $1/M$. It is also possible to analyse the phase transition quantitatively using our method. Unlike in higher dimensions where an order parameter can be easily defined within the MF approximation there is no such quantity in 1D. However, a valid description of the relevant 1D low energy physics is provided by Luttinger liquid theory \cite{giamarchi}. In this model correlations in the supefluid phase as well as the superfluid density itself are characterised by the Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies the phase transition in the thermodynamic limit at $K_b = 1/2$. This quantity can be extracted from various correlation functions and in our case it can be extracted directly from $R$ \cite{ejima2011}. By extracting this parameter from $R$ for various lattice lengths from numerical DMRG calculations it was even possible to give a theoretical estimate of the critical point for commensurate filling, $N = M$, in the thermodynamic limit to occur at $U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our proposal provides a method to directly measure $R$ in a lab which can then be used to experimentally determine the location of the critical point in 1D. So far both variables we considered, $R_\text{max}$ and $W_R$, provide similar information. Next, we present a case where it is very different. The Bose glass is a localized insulating phase with exponentially decaying correlations but large compressibility and on-site fluctuations in a disordered optical lattice. Therefore, measuring both $R_\text{max}$ and $W_R$ will distinguish all the phases. In a Bose glass we have finite compressibility, but exponentially decaying correlations. This gives a large $R_\text{max}$ and a large $W_R$. A Mott insulator will also have exponentially decaying correlations since it is an insulator, but it will be incompressible. Thus, it will scatter light with a small $R_\text{max}$ and large $W_R$. Finally, a superfluid will have long range correlations and large compressibility which results in a large $R_\text{max}$ and a small $W_R$. \begin{figure}[htbp!] \centering \includegraphics[width=\linewidth]{oph22} \caption[Mapping the Disoredered Phase Diagram]{The Mott-superfluid-glass phase diagrams for light scattering maximum $R_\text{max}/N_K$ (a) and width $W_R$ (b). Measurement of both quantities distinguish all three phases. Transition lines are shifted due to finite size effects \cite{roux2008}, but it is possible to apply well known numerical methods to extract these transition lines from such experimental data extracted from $R$ \cite{ejima2011}. $K=M=N=35$.} \label{fig:BG} \end{figure} We confirm this in Fig. \ref{fig:BG} for simulations with the ratio of superlattice- to trapping lattice-period $r\approx 0.77$ for various disorder strengths $V$ \cite{roux2008}. Here, we only consider calculations for a fixed density, because the usual interpretation of the phase diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane for a fixed ratio $V/U$ becomes complicated due to the presence of multiple compressible and incompressible phases between successive MI lobes \cite{roux2008}. This way, we have limited our parameter space to the three phases we are interested in: superfluid, Mott insulator, and Bose glass. From Fig. \ref{fig:BG} we see that all three phases can indeed be distinguished. In the 1D BHM there is no sharp MI-SF phase transition in 1D at a fixed density \cite{cazalilla2011, ejima2011, kuhner2000, pino2012, pino2013} just like in Figs. \ref{fig:SFMI}(d,e) if we follow the transition through the tip of the lobe which corresponds to a line of unit density. However, despite the lack of an easily distinguishable critical point it is possible to quantitatively extract the location of the transition lines by extracting the Tomonaga-Luttinger parameter from the scattered light, $R$, in the same way it was done for an unperturbed BHM \cite{ejima2011}. Only recently \cite{derrico2014} a Bose glass phase was studied by combined measurements of coherence, transport, and excitation spectra, all of which are destructive techniques. Our method is simpler as it only requires measurement of the quantity $R$ and additionally, it is nondestructive. \subsection{Matter-field interference measurements} We now focus on enhancing the interference term $\hat{B}$ in the operator $\hat{F}$. Firstly, we will use this result to show how one can probe $\langle \hat{B} \rangle$ which in MF gives information about the matter-field amplitude, $\Phi = \langle b \rangle$. Hence, by measuring the light quadrature we probe the kinetic energy and, in MF, the matter-field amplitude (order parameter) $\Phi$: $\langle \hat{X}^F_{\beta=0} \rangle = | \Phi |^2 \mathcal{F}[W_1](2\pi/d) (K-1)$. Secondly, we show that it is also possible to access the fluctuations of matter-field quadratures $\hat{X}^b_\alpha = (b e^{-i\alpha} + \bd e^{i\alpha})/2$, which in MF can be probed by measuring the variance of $\hat{B}$. Across the phase transition, the matter field changes its state from Fock (in MI) to coherent (deep SF) through an amplitude-squeezed state as shown in Fig. \ref{Quads}(a,b). Assuming $\Phi$ is real in MF: \begin{equation} \label{intensity} \langle \ad_1 \a_1 \rangle = 2 |C|^2(K-1)\mathcal{F}^2[W_1](\frac{\pi}{d}) \times [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ] \end{equation} and it is shown as a function of $U/(zJ^\text{cl})$ in Fig. \ref{Quads}. Thus, since measurement in the diffraction maximum yields $\Phi^2$ we can deduce $\langle b^2 \rangle - \Phi^2$ from the intensity. This quantity is of great interest as it gives us access to the quadrature variances of the matter-field \begin{equation} (\Delta X^b_{0,\pi/2})^2 = 1/4 + [(n - \Phi^2) \pm (\langle b^2 \rangle - \Phi^2)]/2, \end{equation} where $n=\langle\hat{n}\rangle$ is the mean on-site atomic density. \begin{figure}[htbp!] \centering \includegraphics[width=\linewidth]{Quads} \captionsetup{justification=centerlast,font=small} \caption[Mean-Field Matter Quadratures]{Photon number scattered in a diffraction minimum, given by Eq. (\ref{intensity}), where $\tilde{C} = 2 |C|^2 (K-1) \mathcal{F}^2 [W_1](\pi/d)$. More light is scattered from a MI than a SF due to the large uncertainty in phase in the insulator. (a) The variances of quadratures $\Delta X^b_0$ (solid) and $\Delta X^b_{\pi/2}$ (dashed) 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.} \label{Quads} \end{figure} Probing $\hat{B}^2$ gives us access to kinetic energy fluctuations with 4-point correlations ($\bd_i b_j$ combined in pairs). Measuring the photon number variance, which is standard in quantum optics, will lead up to 8-point correlations similar to 4-point density correlations \cite{mekhov2007pra}. These are of significant interest, because it has been shown that there are quantum entangled states that manifest themselves only in high-order correlations \cite{kaszlikowski2008}. Surprisingly, inter-site terms scatter more light from a Mott insulator than a superfluid Eq. \eqref{intensity}, as shown in Fig. \eqref{Quads}, although the mean inter-site density $\langle \hat{n}(\b{r})\rangle $ is tiny in a MI. This reflects a fundamental effect of the boson interference in Fock states. It indeed happens between two sites, but as the phase is uncertain, it results in the large variance of $\hat{n}(\b{r})$ captured by light as shown in Eq. \eqref{intensity}. The interference between two macroscopic BECs has been observed and studied theoretically \cite{horak1999}. When two BECs in Fock states interfere a phase difference is established between them and an interference pattern is observed which disappears when the results are averaged over a large number of experimental realizations. This reflects the large shot-to-shot phase fluctuations corresponding to a large inter-site variance of $\hat{n}(\b{r})$. By contrast, our method enables the observation of such phase uncertainty in a Fock state directly between lattice sites on the microscopic scale in-situ. \section{Conclusions} In summary, we proposed a nondestructive method to probe quantum gases in an optical lattice. Firstly, we showed that the density-term in scattering has an angular distribution richer than classical diffraction, derived generalized Bragg conditions, and estimated parameters for the only two relevant experiments to date \cite{weitenberg2011, miyake2011}. Secondly, we proposed how to measure the matter-field interference by concentrating light between the sites. This corresponds to interference at the shortest possible distance in an optical lattice. By contrast, standard destructive time-of-flight measurements deal with far-field interference and a relatively near-field one was used in Ref. \cite{miyake2011}. This defines most processes in optical lattices. E.g. matter-field phase changes may happen not only due to external gradients, but also due to intriguing effects such quantum jumps leading to phase flips at neighbouring sites and sudden cancellation of tunneling \cite{vukics2007}, which should be accessible by our method. In mean-field, one can measure the matter-field amplitude (order parameter), quadratures and squeezing. This can link atom optics to areas where quantum optics has already made progress, e.g., quantum imaging \cite{golubev2010, kolobov1999}, using an optical lattice as an array of multimode nonclassical matter-field sources with a high degree of entanglement for quantum information processing. Thirdly, we demonstrated how the method accesses effects beyond mean-field and distinguishes all the phases in the Mott-superfluid-glass transition, which is currently a challenge \cite{derrico2014}. Based on off-resonant scattering, and thus being insensitive to a detailed atomic level structure, the method can be extended to molecules \cite{LP2013}, spins, and fermions \cite{ruostekoski2009}.