%******************************************************************************* %*********************************** 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 the 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 making our proposal timely. 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 gas. We substitute $\a_1 = C \hat{D}$ using Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and we make use of the shorthand notation $A_i = u_1^*(\b{r}_i) u_0(\b{r}_i)$. The result is \begin{equation} \label{eq:Rfluc} 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. Another interesting quantity to measure are the quadratures of the light fields which we have seen in section \ref{sec:a} are related to the quadrature of $\hat{F}$ by $\hat{X}_\phi = |C| \hat{X}^F_\beta$. An interesting feature of quadratures is that the coupling strength at different sites can be tuned using the local oscillator phase $\beta$. To see this we consider the case when both the scattered mode and probe are travelling waves the quadrature \begin{equation} \label{eq:Xtrav} \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}_0 - \b{k}_1) \cdot \b{r}_i + (\phi_0 - \phi_1) - \beta]. \end{equation} Different light quadratures are differently coupled to the atom distribution, hence by varying the local oscillator phase, $\beta$, and/or the detection angle one can scan the whole range of couplings. This is similar to the case for $\hat{D}$ for a standing wave probe, where instead of varying $\beta$ scanning is achieved by varying the position of the wave with respect to atoms. Additionally, the quadrature variance, $(\Delta X^F_\beta)^2$, will have a similar form to $R$ given in Eq. \eqref{eq:Rfluc}, \begin{equation} (\Delta X^F_\beta)^2 = |C|^2 \sum_{i.j}^K A_i^\beta A_j^\beta \langle \dn_i \dn_j \rangle, \end{equation} where $A_i^\beta = (A_i e^{-i\beta} + A_i^* e^{i \beta})/2$. However, this has the advantage that unlike in the case of $R$ there is no need to subtract a spatially varying classical signal to obtain this quantity. The ``quantum addition'', $R$, and the quadrature variance, $(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$ and both rely heavily on the quantum state of the matter. Therefore, they will have a 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 probe is a travelling wave scattering from an ideal superfluid in a 3D optical lattice into a standing wave scattered mode. 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 $R = |C|^2 K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. 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. This difference in behaviour is due to the fact that classical diffraction is ignorant of any quantum correlations as it is given by the square of the light field amplitude squared \begin{equation} |\langle \a_1 \rangle|^2 = |C|^2 \sum_{i,j} A_i^* A_j \langle \n_i \rangle \langle \n_j \rangle, \end{equation} which is idependent of any two-point correlations unlike $R$. On the other hand the full light intensity (classical signal plus ``quantum addition'') of the quantum light does include higher-order correlations \begin{equation} \langle \ad_1 \a_1 \rangle = |C|^2 \sum_{i,j} A_i^* A_j \langle \n_i \n_j \rangle. \end{equation} Therefore, we see that in the fully quantum picture light scattering not only depends on the diffraction structure due to the distribution of atoms in the lattice, but also on the quantum correlations between different lattice sites which will in turn be dependent on the quantum state of the matter. These correlations are imprinted in $R$ as shown in Eq. \eqref{eq:Rfluc} and it highlights the key feature of our model, i.e.~the light couples to the quantum state directly via operators. We can even derive the generalised Bragg conditions for the peaks that we can see in Fig. \ref{fig:scattering}. The exact conditions under which diffraction peaks emerge for the ``quantum addition'' will depend on the optical setup as well as on the quantum state of the matter as it is the density fluctuation correlations, $\langle \dn_i \dn_j \rangle$, that provide the structure for $R$ and not the lattice itself as seen in Eq. \eqref{eq:Rfluc}. For classical light it is straightforward to develop an intuitive physical picture to find the Bragg condition by considering angles at which the distance travelled by light scattered from different points in the lattice is equal to an integer multiple of the wavelength. The ``quantum addition'' is more complicated and less intuitive as we now have to consider quantum correlations which are not only nonlocal, but can also be negative. We will consider scattering from a superfluid, because the Mott insulator has no ``quantum addition'' due to a lack of density fluctuations. The wavefunction of a superfluid on a lattice is given by \textbf{Eq. (??)}. This state has infinte range correlations and thus has the convenient property that all two-point density fluctuation correlations are equal regardless of their separation, i.e.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$ for all $(i \ne j)$, where the right hand side is a constant value. This allows us to extract all correlations from the sum in Eq. \eqref{eq:Rfluc} to obtain \begin{equation} \label{eq:RSF} \frac{R}{|C|^2} = (\langle \dn^2 \rangle - \langle \dn_a \dn_b \rangle) \sum_i^K |A_i|^2 + \langle \dn_a \dn_b \rangle |A|^2 = \frac{N}{M} \sum_i^K |A_i|^2 - \frac{N}{M^2} |A|^2, \end{equation} where $A = \sum_i^K A_i$, and we have dropped the index from $\langle \dn^2 \rangle$ as it is equal at every site. The second equality follows from the fact that for a superfluid state $\langle \dn^2 \rangle = N/M - N/M^2$ and $\langle \dn_a \dn_b \rangle = -N/M^2$. Naturally, we get an identical expression for the quadrature variance $(\Delta X^F_\beta)^2$ with the coefficients $A_i$ replaced by the real $A_i^\beta$. The ``quantum addition'' for the case when both the scattered and probe modes are travelling waves is actually trivial. It has no peaks and thus it has no generalised Bragg condition and it only consists of a uniform background. This is a consequence of the fact that travelling waves couple equally strongly with every atom as only the phase is different between lattice sites. Therefore, since superfluid correlations lack structure as they're uniform we do no get a strong coherent peak. The contribution from the lattice structure is included in the classical Bragg peaks which we have subtracted in order to obtain the quantity $R$. However, if we consider the case where the scattered mode is collected as a standing wave using a pair of mirrors we get the diffraction pattern that we saw in Fig. \ref{fig:scattering}. This time we get strong visible peaks, because at certain angles the standing wave couples to the atoms maximally at all lattice sites and thus it uses the structure of the lattice to amplify the signal from the quantum fluctuations. This becomes clear when we look at Eq. \eqref{eq:RSF}. We can neglect the second term as it is always negative and it has the same angular distribution as the classical diffraction pattern and thus it is mostly zero except when the classical Bragg condition is satisfied. Since in Fig. \ref{fig:scattering} we have chosen an angle such that the Bragg is not satisfied this term is essentially zero. Therefore, we are left with the first term $\sum_i^K |A_i|^2$ which for a travelling wave probe and a standing wave scattered mode is \begin{equation} \sum_i^K |A_i|^2 = \sum_i^K \cos^2(\b{k}_0 \cdot \b{r}_i + \phi_0) = \frac{1}{2} \sum_i^K \left[1 + \cos(2 \b{k}_0 \cdot \b{r}_i + 2 \phi_0) \right]. \end{equation} Therefore, it is straightforward to see that unless $2 \b{k}_0 = \b{G}$, where $\b{G}$ is a reciprocal lattice vector, there will be no coherent signal and we end up with the mean uniform signal of strength $|C|^2 N_k/2$. When this condition is satisifed all the cosine terms will be equal and they will add up constructively instead of cancelling each other out. Note that this new Bragg condition is different from the classical one $\b{k}_0 - \b{k}_1 = \b{G}$. This result makes it clear that the uniform background signal is not due to any coherent scattering, but rather due to the lack of structure in the quantum correlations. Furthermore, we see that the peak height is actually tunable via the phase, $\phi_0$, which is illustrated in Fig. \ref{fig:scattering}b. For light field quadratures the situation is different, because as we have seen in Eq. \eqref{eq:Xtrav} even for travelling waves we can tune the contributions to the signal from different sites by changing the angle of measurement or tuning the local oscillator signal $\beta$. The rest is similar to the case we discussed for $R$ with a standing wave mode and we can show that the new Bragg condition in this case is $2 (\b{k}_0 - \b{k}_1) = \b{G}$ which is different from the condition we had for $R$ and is still different from the classical condition $\b{k}_0 - \b{k}_1 = \b{G}$. Furthermore, just like in Fig. \ref{fig:scattering}b the peak height can be tuned using $\beta$. A quantum signal that isn't masked by classical diffraction is very useful for future experimental realisability. However, it is still unclear whether this signal would be strong enough to be visible. After all, a classical signal scales as $N_K^2$ whereas here we have only seen a scaling of $N_K$. 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} These results can be applied directly to the scattering patters in Fig. \ref{fig:scattering}. Therefore, 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} We have shown that scattering from atom number operators leads to a purely quantum diffraction pattern which depends on the density fluctuations and their correlations. We have also seen that this signal should be strong enough to be visible using currently available technology. However, so far we have not looked at what this can tell us about the quantum state of matter. We have briefly mentioned that a deep superfluid will scatter a lot of light due to its infinite range correlations and a Mott insulator will not contriute any ``quantum addition'' at all, but we have not look at the quantum phase transition between these two phases. In two or higher dimensions this has a rather simple answer as the Bose-Hubbard phase transition is described well by mean-field theories and it has a sharp transition at the critical point. This means that the ``quantum addition'' signal would drop rapidly at the critical point and go to zero as soon as it was crossed. However, $R$ given by Eq. \eqref{eq:R} clearly contains much more information. 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}. This is because the one-dimensional quantum phase transition is in a different universality class than its higher dimensional counterparts. The energy gap, which is the order parameter, decays exponentially slowly across the phase transition making it difficult to identify the phase transition even in numerical simulations. Here, we will show the avaialable tools provided by the ``quantum addition'' that allows one to nondestructively map this phase transition and distinguish the superfluid and Mott insulator phases. 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 $|i-j|$. This is a characteristic of insulators. 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 $\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$. This quantity can be extracted from the measured light intensity bu considering the ``quantum addition''. We will consider the case when both the probe and scattered modes are plane waves which can be easily achieved in free space. We will again consider the case of light being maximally coupled to the density ($\hat{F} = \hat{D}$). Therefore, 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} \mynote{can put in more detail here with equations} This alone allows us to analyse the phase transition quantitatively using our method. Unlike in higher dimensions where an order parameter can be easily defined within the mean-field approximation as a simple expectation value, the situation in 1D is more complex as it is difficult to directly access the excitation energy gap which defines this phase transition. 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 critical point 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}. This quantity was used in numerical calculations that used highly efficient density matrix renormalisation group (DMRG) methods to calculate the ground state to subsequently fit the Luttinger theory to extract this parameter $K_b$. These calculations yield a theoretical estimate of the critical point in the thermodynamic limit for commensurate filling in 1D to be at $U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our proposal provides a method to directly measure $R$ nondestructively in a lab which can then be used to experimentally determine the location of the critical point in 1D. However, whilst such an approach will yield valuable quantitative results we will instead focus on its qualitative features which give a more intuitive understanding of what information can be extracted from $R$. This is because the superfluid to Mott insulator phase transition is well understood, so there is no reason to dwell on its quantitative aspects. However, our method is much more general than the Bose-Hubbard model as it can be easily applied to many other systems such as fermions, photonic circuits, optical lattices with quantum potentials, etc. Therefore, by providing a better physical picture of what information is carried by the ``quantum addition'' it should be easier to see its usefuleness in a broader context. We calculate the phase diagram of the Bose-Hubbard Hamiltonian given by \begin{equation} \hat{H}_\mathrm{dis} = -J^\mathrm{cl} \sum_{\langle i, j \rangle} \bd_i b_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) - \mu \sum_i \hat{n}_i, \end{equation} where the $\mu$ is the chemical potential. We have introduced the last term as we are interested in grand canonical ensemble calculations as we want to see how the system's behaviour changes as density is varied. We perform numerical calculations of 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}. \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 then consider probing these ground states using our optical scheme and we calculate the ``quantum addition'', $R$, based on these ground states. The angular dependence of $R$ for a Mott insulator and a superfluid is shown in Fig. \ref{fig:SFMI}a, and we note that 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 insulating 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 superfluid phase ($U/2J^\text{cl} = 0$) is larger than its value in the Mott insulating 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. There are a few things to note at this point. Firstly, if we follow the transition along the line corresponding to commensurate filling (i.e.~any line that is in between the two white lines in Fig. \ref{fig:SFMI}d) we see that the transition is very smooth and it is hard to see a definite critical point. This is due to the energy gap closing exponentially slowly which makes precise identification of the critical point extremely difficult. The best option at this point would be to fit Tomonaga-Luttinger theory to the results in order to find this critical point. However, we note that there is a drastic change in signal as the chemical potential (and thus the density) is varied. This is highlighted in Fig. \ref{fig:SFMI}b which shows how the Mott insulator can be easily identified by a dip in the quantity $R_\text{max}$. 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 $W_R$. On the other hand, 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. Furthermore, just like for $R_\text{max}$ we see that the transition is much sharper as $\mu$ is varied. This is shown in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle between a superfluid and an insulating state is fairly significant $\sim 20^\circ$ which should make the two phases easy to identify using this measure. In this particular case, measuring $W_R$ in the Mott phase is not very practical as the insulating phase does not scatter light (small $R_\mathrm{max}$). The phase transition information is easier extracted from $R_\mathrm{max}$. However, this is not always the case and we will shortly see how certain phases of matter scatter a lot of light and can be distinguished using measurements of $W_R$ where $R_\mathrm{max}$ is not sufficient. Another 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 $1/M$. So far both variables we considered, $R_\text{max}$ and $W_R$, provide similar information. They both take on values at one of its extremes in the Mott insulating phase and they change drastically across the phase transition into the superfluid phase. Next, we present a case where it is very different. We will again consider ultracold bosons in an optical lattice, but this time we introduce some disorder. We do this by adding an additional periodic potential on top of the exisitng setup that is incommensurate with the original lattice. The resulting Hamiltonian can be shown to be \begin{equation} \hat{H}_\mathrm{dis} = -J^\mathrm{cl} \sum_{\langle i, j \rangle} \bd_i b_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) + \frac{V}{2} \sum_i \left[ 1 + \cos (2 r \pi m + 2 \phi) \right] \hat{n}_i, \end{equation} where $V$ is the strength of the superlattice potential, $r$ is the ratio of the superlattice and trapping wave vectors and $\phi$ is some phase shift between the two lattice potentials \cite{roux2008}. The first two terms are the standard Bose-Hubbard Hamiltonian and the only modification is an additional spatially varying potential shift. We will only consider the phase diagram at fixed density as the introduction of disorder makes the usual interpretation of the phase diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane for a fixed ratio $V/U$ complicated due to the presence of multiple compressible and incompressible phases between successive Mott insulator lobes \cite{roux2008}. Therefore, the chemical potential no longer appears in the Hamiltonian as we are no longer considering the grand canonical ensemble. The reason for considering such a system is that it introduces a third, competing phase, the Bose glass into our phase diagram. It is an insulating phase like the Mott insulator, but it has local superfluid susceptibility making it compressible. Therefore this localized insulating phase has exponentially decaying correlations just like the Mott phase, but it has large on-site fluctuations just like the compressible superfluid phase. As these are the two physical variables encoded in $R$ measuring both $R_\text{max}$ and $W_R$ will provide us with enough information to distinguish all three 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 also has exponentially decaying correlations since it is an insulator, but it is incompressible. Thus, it will scatter light with a small $R_\text{max}$ and large $W_R$. Finally, a superfluid has 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}. From Fig. \ref{fig:BG} we see that all three phases can indeed be distinguished. From the $R_\mathrm{max}$ plot we can distinguish the two compressible phases, the superfluid and Bose glass, from the incompressible phase, the Mott insulator. We can now also distinguish the Bose glass phase from the superfluid, because only one of them is an insulator and thus the angular width of their scattering patterns will reveal this information. However, unlike the Mott insulator, the Bose glass scatters a lot of light (large $R_\mathrm{max}$) enabling such measurements. Since we performed these calculations at a fixed density the Mott to superfluid phase transition is not particularly sharp \cite{cazalilla2011, ejima2011, kuhner2000, pino2012, pino2013} just like we have seen 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, as we have already discussed, 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 Bose-Hubbard model \cite{ejima2011}. \subsection{Matter-field interference measurements} We have shown in section \ref{sec:B} that certain optical arrangements lead to a different type of light-matter interaction where coupling is maximised in between lattice sites rather than at the sites themselves yielding $\a_1 = C\hat{B}$. This leads to the optical fields interacting directly with the interference terms $\bd_i b_{i+1}$ via the operator $\hat{B}$ given by Eq. \eqref{eq:B}. This opens up a whole new way of probing and interacting with a quantum gas trapped in an optical lattice as this gives an in-situ method for probing the inter-site interference terms at its shortest possible distance, i.e.~the lattice period. % I mention mean-field here, but do not explain it. That should be % done in Chapter 2.1} Unlike in the previous sections, here we will use the mean-field description of the Bose-Hubbard model in order to obtain a simple physical picture of what information is contained in the quantum light. In the mean-field approximation the inter-site interference terms become \begin{equation} \bd_i b_j = \Phi \bd_i + \Phi^* b_j - |\Phi|^2, \end{equation} where $\langle b_i \rangle = \Phi$ which we assume is uniform across the whole lattice. This approach has the advantage that the quantity $\Phi$ is the mean-field order parameter of the superfluid to Mott insulator phase transition and is effectively a good measure of the superfluid character of the quantum ground state. This greatly simplifies the physical interpretation of our results. Firstly, we will show that our nondestructive measurement scheme allows one to probe the mean-field order parameter, $\Phi$, directly. Normally, this is achieved by releasing the trapped gas and performing a time-of-flight measurement. Here, this can be achieved in-situ. In section \ref{sec:B} we showed that one of the possible optical arrangement leads to a diffraction maximum with the matter operator \begin{equation} \label{eq:Bmax} \hat{B} = J^B_\mathrm{max} \sum_i \left( \bd_i b_{i+1} + b_i \bd_{i+1} \right), \end{equation} where $J^B_\mathrm{max} = \mathcal{F}[W_1](2\pi/d)$. Therefore, by measuring the expectation value of the quadrature we obtain the following quantity \begin{equation} \langle \hat{X}^F_{\beta=0} \rangle = J^B_\mathrm{max} (K-1) | \Phi |^2 . \end{equation} This quantity is directly proportional to square of the order parameter $\Phi$ and thus lets us very easily follow this quantity across the phase transition with a very simple quadrature measurement setup. In the mean-field treatment, the order parameter also lets us deduce a different quantity, namely matter-field quadratures $\hat{X}^b_\alpha = (b e^{-i\alpha} + \bd e^{i\alpha})/2$. Quadrature measurements of optical fields are a standard and common tool in quantum optics. However, this is not the case for matter-fields as normally most interactions lead to an effective coupling with the density as we have seen in the previous sections. Therefore, such measurements provide us with new opportunities to study the quantum matter state which was previously unavailable. We will take $\Phi$ to be real which in the standard Bose-Hubbard Hamiltonian can be selected via an inherent gauge degree of freedom in the order parameter. Thus, the quadratures themself straightforwardly become \begin{equation} \hat{X}^b_\alpha = \frac{\Phi}{2} (e^{-i\alpha} + e^{i\alpha}) = \Phi \cos(\alpha). \end{equation} From our measurement of the light field quadrature we have already obtained the value of $\Phi$ and thus we immediately also know the values of the matter-field quadratures. Unfortunately, the variance of the quadrature is a more complicated quantity given by \begin{equation} (\Delta X^b_{0,\pi/2})^2 = \frac{1}{4} + [(n - \Phi^2) \pm \frac{1}{2}(\langle b^2 \rangle - \Phi^2)], \end{equation} where we have arbitrarily selected two orthogonal quadratures $\alpha =0 $ and $\alpha = \pi / 2$. This quantity cannot be estimated from light quadrature measurements alone as we do not know the value of $\langle b^2 \rangle$. To obtain the value of $\langle b^2 \rangle$this quantity we need to consider a second-order light observable such as light intensity. However, in the diffraction maximum this signal will be dominated by a contribution proportional to $K^2 \Phi^4$ whereas the terms containing information on $\langle b^2 \rangle$ will only scale as $K$. Therefore, it would be difficult to extract the quantity that we need by measuring in the difraction maximum. \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} We now consider the alternative arrangement in which we probe the diffraction minimum and the matter operator is given by \begin{equation} \label{eq:Bmin} \hat{B} = J^B_\mathrm{min} \sum_i (-1)^i \left( \bd_i b_{i+1} b_i \bd_{i+1} \right), \end{equation} where $J^B_\mathrm{min} = - \mathcal{F}[W_1](\pi / d)$ and which unlike the previous case is no longer proportional to the Bose-Hubbard kinetic energy term. Unlike in the diffraction maximum the light intensity in the diffraction minimum does not have the term proportional to $K^2$ and thus we obtain the following quantity \begin{equation} \label{intensity} \langle \ad_1 \a_1 \rangle = 2 |C|^2(K-1) \mathcal{F}^2[W_1] (\frac{\pi}{d}) [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ], \end{equation} This is plotted in Fig. \ref{Quads} as a function of $U/(zJ^\text{cl})$. Now, we can easily deduce the value of $\langle b^2 \rangle$ since we will already know the mean density, $n$, from our experimental setup and we have seen that we can obtain $\Phi^2$ from the diffraction maximum. Thus, we now have access to the quadrature variance of the matter-field as well giving us a more complete picture of the matter-field amplitude than previously possible. A surprising feature seen in Fig. \ref{Quads} is that the inter-site terms scatter more light from a Mott insulator than a superfluid Eq. \eqref{intensity}, although the mean inter-site density $\langle \hat{n}(\b{r})\rangle $ is tiny in the Mott insulating phase. 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. More generally, beyond mean-field, probing $\hat{B}^\dagger \hat{B}$ via light intensity measurements gives us access 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}. \section{Conclusions} In this chapter we explored the possibility of nondestructively probing a quantum gas trapped in an optical lattice using quantized light. 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 two relevant experiments \cite{weitenberg2011, miyake2011}. Secondly, 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}. Finally, we looked at measuring the matter-field interference via the operator $\hat{B}$ by concentrating light between the sites. This corresponds to probing interference at the shortest possible distance in an optical lattice. This is in contrast to standard destructive time-of-flight measurements which deal with far-field interference. This quantity 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. We showed how 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. Since our scheme is 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}.