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%*******************************************************************************
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%*********************************** Third Chapter *****************************
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%*******************************************************************************
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\chapter{Probing Correlations by Global
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Nondestructive Addressing} %Title of the Third Chapter
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%********************************** %First Section **************************************
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\section{Introduction}
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Having developed the basic theoretical framework within which we can
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treat the fully quantum regime of light-matter interactions we now
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consider possible applications. There are three prominent directions
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in which we can proceed: nondestructive probing of the quantum state
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of matter, quantum measurement backaction induced dynamics and quantum
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optical lattices. Here, we deal with the first of the three options.
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In this chapter we develop a method to measure properties of ultracold
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gases in optical lattices by light scattering. In the previous chapter
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we have shown that quantum light field couples to the bosons via the
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operator $\hat{F}$. This is the key element of the scheme we propose
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as this makes it sensitive to the quantum state of the matter and all
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of its possible superpositions which will be reflected in the quantum
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state of the light itself. We have also shown in section
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\ref{sec:derivation} that this coupling consists of two parts, a
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density component $\hat{D}$ given by Eq. \eqref{eq:D}, and a phase
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component $\hat{B}$ given by Eq. \eqref{eq:B}. Therefore, when probing
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the quantum state of the ultracold gas we can have access to not only
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density correlations, but also matter-field interference at its
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shortest possible distance in an optical lattice, i.e.~the lattice
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period. Previous work on quantum non-demolition (QND) schemes
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\cite{rogers2014, mekhov2007prl, eckert2008} probe only the density
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component as it is generally challenging to couple to the matter-field
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observables directly. Here, we will consider nondestructive probing of
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both density and interference operators.
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Firstly, we will consider the simpler and more typical case of
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coupling to the atom number operators via $\hat{F} =
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\hat{D}$. However, we show that light diffraction in this regime has
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several nontrivial characteristics due to the fully quantum nature of
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the interaction. Firstly, we show that the angular distribution has
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multiple interesting features even when classical diffraction is
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forbidden facilitating their experimental observation. We derive new
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generalised Bragg diffraction conditions which are different to their
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classical counterpart. Furthermore, due to the fully quantum nature of
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the interaction our proposal is capable of probing the quantum state
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beyond mean-field prediction. We demonstrate this by showing that this
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scheme is capable of distinguishing all three phases in the Mott
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insulator - superfluid - Bose glass phase transition in a 1D
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disordered optical lattice which is not very well described by a
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mean-field treatment. We underline that transitions in 1D are much
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more visible when changing an atomic density rather than for
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fixed-density scattering. It was only recently that an experiment
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distinguished a Mott insulator from a Bose glass \cite{derrico2014}
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via a series of destructive measurements. Our proposal, on the other
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hand, is nondestructive and is capable of extracting all the relevant
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information in a single experiment.
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Having shown the possibilities created by this nondestructive
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measurement scheme we move on to considering light scattering from the
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phase related observables via the operator $\hat{F} = \hat{B}$. This
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enables in-situ probing of the matter-field coherence at its shortest
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possible distance in an optical lattice, i.e. the lattice period,
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which defines key processes such as tunnelling, currents, phase
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gradients, etc. This is in contrast to standard destructive
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time-of-flight measurements which deal with far-field interference
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although a relatively near-field scheme was use in
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Ref. \cite{miyake2011}. We show how within the mean-field treatment,
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this enables measurements of the order parameter, matter-field
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quadratures and squeezing. This can have an impact on atom-wave
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metrology and information processing in areas where quantum optics
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already made progress, e.g., quantum imaging with pixellized sources
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of non-classical light \cite{golubev2010, kolobov1999}, as an optical
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lattice is a natural source of multimode nonclassical matter waves.
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\section{Coupling to the Quantum State of Matter}
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As we have seen in section \ref{sec:a} under certain approximations
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the scattered light mode, $\a_1$, is linked to the quantum state of
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matter via
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\begin{equation}
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\label{eq:a-3}
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\a_1 = C \hat{F} = C \left(\hat{D} + \hat{B} \right),
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\end{equation}
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where the atomic operators $\hat{D}$ and $\hat{B}$, given by
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Eq. \eqref{eq:D} and Eq. \eqref{eq:B}, are responsible for the
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coupling to on-site density and inter-site interference
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respectively. It is crucial to note that light couples to the bosons
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via an operator as this makes it sensitive to the quantum state of the
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matter as this will imprint the fluctuations in the quantum state of
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the scattered light.
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Here, we will use this fact that the light is sensitive to the atomic
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quantum state due to the coupling of the optical and matter fields via
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operators in order to develop a method to probe the properties of an
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ultracold gas. Therefore, we neglect the measurement back-action and
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we will only consider expectation values of light observables. Since
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the scheme is nondestructive (in some cases, it even satisfies the
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stricter requirements for a QND measurement \cite{mekhov2012,
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mekhov2007pra}) and the measurement only weakly perturbs the system,
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many consecutive measurements can be carried out with the same atoms
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without preparing a new sample. We will show how the extreme
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flexibility of the the measurement operator $\hat{F}$ allows us to
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probe a variety of different atomic properties in-situ ranging from
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density correlations to matter-field interference.
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\subsection{On-site Density Measurements}
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We have seen in section \ref{sec:B} that typically the dominant term
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in $\hat{F}$ is the density term $\hat{D}$ \cite{LP2009,
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mekhov2007pra, rist2010, lakomy2009, ruostekoski2009}. This is
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simply due to the fact that atoms are localised with lattice sites
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leading to an effective coupling with atom number operators instead of
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inter-site interference terms. Therefore, we will first consider
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nondestructive probing of the density related observables of the
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quantum gas. However, we will focus on the novel nontrivial aspects
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that go beyond the work in Ref. \cite{mekhov2012, mekhov2007prl,
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mekhov2007pra} which only considered a few extremal cases.
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As we are only interested in the quantum information imprinted in the
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state of the optical field we will simplify our analysis by
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considering the light scattering to be much faster than the atomic
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tunnelling. Therefore, our scheme is actually a QND scheme
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\cite{rogers2014, mekhov2007prl, mekhov2007pra, eckert2008} as
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normally density-related measurements destroy the matter-phase
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coherence since it is its conjugate variable, but here we neglect the
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$\bd_i b_j$ terms. Furthermore, we will consider a deep
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lattice. Therefore, the Wannier functions will be well localised
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within their corresponding lattice sites and thus the coefficients
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$J_{i,i}$ reduce to $u_1^*(\b{r}_i) u_0(\b{r}_i)$ leading to
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\begin{equation}
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\label{eq:D-3}
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\hat{D}=\sum_i^K u_1^*(\b{r}_i) u_0(\b{r}_i) \hat{n}_i,
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\end{equation}
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which for travelling
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[$u_l(\b{r})=\exp(i \b{k}_l \cdot \b{r}+i\varphi_l)$] or standing
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[$u_l(\b{r})=\cos(\b{k}_l \cdot \b{r}+\varphi_l)$] waves is just a
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density Fourier transform at one or several wave vectors
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$\pm(\b{k}_1 \pm \b{k}_0)$.
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We will now define a new auxiliary quantity to aid our analysis,
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\begin{equation}
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\label{eq:R}
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R = \langle \ad_1 \a_1 \rangle - | \langle \a_1 \rangle |^2,
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\end{equation}
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which we will call the ``quantum addition'' to light scattering. By
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construction $R$ is simply the full light intensity minus the
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classical field diffraction. In order to justify its name we will show
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that this quantity depends purely quantum mechanical properties of the
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ultracold gase. We will substitute $\a_1 = C \hat{D}$ using
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Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and
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we will make use of the shorthand notation
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$A_i = u_1^*(\b{r}_i) u_0(\b{r}_i)$. The result is
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\begin{equation}
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R = |C|^2 \sum_{i,j}^K A^*_i A_j \langle \delta \hat{n}_i \delta
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\hat{n}_j \rangle,
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\end{equation}
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where $\delta \hat{n}_i = \hat{n}_i - \langle \hat{n}_i
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\rangle$. Thus, we can clearly see that $R$ is a result of light
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scattering from fluctuations in the atom number which is a purely
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quantum mechanical property of a system. Therefore, $R$, the ``quantum
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addition'' faithfully represents the new contribution from the quantum
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light-matter interaction to the diffraction pattern.
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If instead we are interested in quantities linear in $\hat{D}$, we can
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measure the quadrature of the light fields which in section
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\ref{sec:a} we saw that $\hat{X}_\phi = |C| \hat{X}^F_\beta$. For the
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case when both the scattered mode and probe are travelling waves the
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quadrature
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\begin{equation}
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\hat{X}^F_\beta = \frac{1}{2} \left( \hat{F} e^{-i \beta} +
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\hat{F}^\dagger e^{i \beta} \right) = \sum_i^K \hat{n}_i\cos[(\b{k}_1 - \b{k}_2) \cdot
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\b{r}_i - \beta].
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\end{equation}
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Note that different light quadratures are differently coupled to the
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atom distribution, hence by varying the local oscillator phase, and
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thus effectively $\beta$, and/or the detection angle one can scan the
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whole range of couplings. A similar expression exists for $\hat{D}$
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for a standing wave probe, where $\beta$ is replaced by $\varphi_0$,
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and scanning is achieved by varying the position of the wave with
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respect to atoms.
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The ``quantum addition'', $R$, and the quadrature variance,
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$(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$. Therefore, they
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will havea nontrivial angular dependence, showing more peaks than
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classical diffraction. Furthermore, these peaks can be tuned very
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easily with $\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows
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the angular dependence of $R$ for the case when the scattered mode is
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a standing wave and the probe is a travelling wave scattering from
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bosons in a 3D optical lattice. The first noticeable feature is the
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isotropic background which does not exist in classical
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diffraction. This background yields information about density
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fluctuations which, according to mean-field estimates (i.e.~inter-site
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correlations are ignored), are related by
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$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. In
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Fig. \ref{fig:scattering} we can see a significant signal of
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$R = |C|^2 N_K/2$, because it shows scattering from an ideal
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superfluid which has significant density fluctuations with
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correlations of infinte range. However, as the parameters of the
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lattice are tuned across the phase transition into a Mott insulator
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the signal goes to zero. This is because the Mott insulating phase has
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well localised atoms at each site which suppresses density
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fluctuations entirely leading to absolutely no ``quantum addition''.
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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=\linewidth]{Ep1}
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\caption[Light Scattering Angular Distribution]{Light intensity
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scattered into a standing wave mode from a superfluid in a 3D
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lattice (units of $R/(|C|^2N_K)$). Arrows denote incoming
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travelling wave probes. The classical Bragg condition,
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$\Delta \b{k} = \b{G}$, is not fulfilled, so there is no classical
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diffraction, but intensity still shows multiple peaks, whose
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heights are tunable by simple phase shifts of the optical beams:
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(a) $\varphi_1=0$; (b) $\varphi_1=\pi/2$. Interestingly, there is
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also a significant uniform background level of scattering which
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does not occur in its classical counterpart. }
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\label{fig:scattering}
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\end{figure}
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We can also observe maxima at several different angles in
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Fig. \ref{fig:scattering}. Interestingly, they occur at different
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angles than predicted by the classical Bragg condition. Moreover, the
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classical Bragg condition is actually not satisfied which means there
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actually is no classical diffraction on top of the ``quantum
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addition'' shown here. Therefore, these features would be easy to see
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in an experiment as they wouldn't be masked by a stronger classical
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signal. We can even derive the generalised Bragg conditions for the
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peaks that we can see in Fig. \ref{fig:scattering}.
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% Derive and show these Bragg conditions
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As $(\Delta X^F_\beta)^2$ and $R$ are quadratic variables,
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the generalized Bragg conditions for the peaks are
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$2 \Delta \b{k} = \b{G}$ for quadratures of travelling waves, where
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$\Delta \b{k} = \b{k}_0 - \b{k}_1$ and $\b{G}$ is the reciprocal
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lattice vector, and $2 \b{k}_1 = \b{G}$ for standing wave $\a_1$ and
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travelling $\a_0$, which is clearly different from the classical Bragg
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condition $\Delta \b{k} = \b{G}$. The peak height is tunable by the
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local oscillator phase or standing wave shift as seen in Fig.
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\ref{fig:scattering}b.
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In section \ref{sec:Efield} we have estimated the mean photon
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scattering rates integrated over the solid angle for the only two
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experiments so far on light diffraction from truly ultracold bosons
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where the measurement object was light
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\begin{equation}
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n_{\Phi}= \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma K}{8}
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(\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2).
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\end{equation}
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Therefore, applying these results to the scattering patters in
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Fig. \ref{fig:scattering} the background signal should reach
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$n_\Phi \approx 10^6$ s$^{-1}$ in Ref. \cite{weitenberg2011} (150
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atoms in 2D), and $n_\Phi \approx 10^{11}$ s$^{-1}$ in
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Ref. \cite{miyake2011} ($10^5$ atoms in 3D). These numbers show that
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the diffraction patterns we have seen due to the ``quantum addition''
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should be visible using currently available technology, especially
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since the most prominent features, such as Bragg diffraction peaks, do
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not coincide at all with the classical diffraction pattern.
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\subsection{Mapping the quantum phase diagram}
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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=\linewidth]{oph11}
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\caption[Mapping the Bose-Hubbard Phase Diagram]{(a) The angular
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dependence of scattered light $R$ for a superfluid (thin black,
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left scale, $U/2J^\text{cl} = 0$) and Mott insulator (thick green,
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right scale, $U/2J^\text{cl} =10$). The two phases differ in both
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their value of $R_\text{max}$ as well as $W_R$ showing that
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density correlations in the two phases differ in magnitude as well
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as extent. Light scattering maximum $R_\text{max}$ is shown in (b,
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d) and the width $W_R$ in (c, e). It is very clear that varying
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chemical potential $\mu$ or density $\langle n\rangle$ sharply
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identifies the superfluid-Mott insulator transition in both
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quantities. (b) and (c) are cross-sections of the phase diagrams
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(d) and (e) at $U/2J^\text{cl}=2$ (thick blue), 3 (thin purple),
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and 4 (dashed blue). Insets show density dependencies for the
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$U/(2 J^\text{cl}) = 3$ line. $K=M=N=25$.}
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\label{fig:SFMI}
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\end{figure}
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We have shown how in mean-field, we can track the order parameter,
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$\Phi$, by probing the matter-field interference using the coupling of
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light to the $\hat{B}$ operator. In this case, it is very easy to
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|
follow the superfluid to Mott insulator quantum phase transition since
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|
we have direct access to the order parameter which goes to zero in the
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|
insulating phase. In fact, if we're only interested in the critical
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point, we only need access to any quantity that yields information
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|
about density fluctuations which also go to zero in the MI phase and
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this can be obtained by measuring
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|
$\langle \hat{D}^\dagger \hat{D} \rangle$. However, there are many
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|
situations where the mean-field approximation is not a valid
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|
description of the physics. A prominent example is the Bose-Hubbard
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model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, pino2012,
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|
pino2013}. Observing the transition in 1D by light at fixed density
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|
was considered to be difficult \cite{rogers2014} or even impossible
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\cite{roth2003}. By contrast, here we propose varying the density or
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chemical potential, which sharply identifies the transition. We
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perform these calculations numerically by calculating the ground state
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using DMRG methods \cite{tnt} from which we can compute all the
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|
necessary atomic observables. Experiments typically use an additional
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harmonic confining potential on top of the optical lattice to keep the
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atoms in place which means that the chemical potential will vary in
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space. However, with careful consideration of the full
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($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in
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Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the
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system \cite{batrouni2002}.
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The 1D phase transition is best understood in terms of two-point
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correlations as a function of their separation \cite{giamarchi}. In
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the Mott insulating phase, the two-point correlations
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$\langle \bd_i b_j \rangle$ and
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$\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$
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($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay
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exponentially with $|i-j|$. On the other hand the superfluid will
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exhibit long-range order which in dimensions higher than one,
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|
manifests itself with an infinite correlation length. However, in 1D
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only pseudo long-range order happens and both the matter-field and
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density fluctuation correlations decay algebraically \cite{giamarchi}.
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The method we propose gives us direct access to the structure factor,
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which is a function of the two-point correlation $\langle \delta
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|
\hat{n}_i \delta \hat{n}_j \rangle$, by measuring the light
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|
intensity. For two travelling waves maximally coupled to the density
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(atoms are at light intensity maxima so $\hat{F} = \hat{D}$), the
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quantum addition is given by
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\begin{equation}
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R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0)
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(\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta
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|
\hat{n}_j \rangle,
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|
\end{equation}
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The angular dependence of $R$ for a Mott insulator and a superfluid is
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shown in Fig. \ref{fig:SFMI}a, and there are two variables
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distinguishing the states. Firstly, maximal $R$,
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$R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$,
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probes the fluctuations and compressibility $\kappa'$
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|
($\langle \delta \hat{n}^2_i \rangle \propto \kappa' \langle \hat{n}_i
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|
\rangle$). The Mott insulator is incompressible and thus will have
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|
very small on-site fluctuations and it will scatter little light
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|
leading to a small $R_\text{max}$. The deeper the system is in the MI
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|
phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the smaller
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these values will be until ultimately it will scatter no light at all
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|
in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a this can
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|
be seen in the value of the peak in $R$. The value $R_\text{max}$ in
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the SF phase ($U/2J^\text{cl} = 0$) is larger than its value in the MI
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phase ($U/2J^\text{cl} = 10$) by a factor of
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$\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of
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$R_\text{max}$ changes across the phase transition. We see that the
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|
transition shows up very sharply as $\mu$ is varied.
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Secondly, being a Fourier transform, the width $W_R$ of the dip in $R$
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|
is a direct measure of the correlation length $l$, $W_R \propto
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|
1/l$. The Mott insulator being an insulating phase is characterised by
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|
exponentially decaying correlations and as such it will have a very
|
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|
large $W_R$. However, the superfluid in 1D exhibits pseudo long-range
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|
order which manifests itself in algebraically decaying two-point
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|
correlations \cite{giamarchi} which significantly reduces the dip in
|
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|
the $R$. This can be seen in Fig. \ref{fig:SFMI}a and we can also see
|
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|
that this identifies the phase transition very sharply as $\mu$ is
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|
varied in Figs. \ref{fig:SFMI}(c,e). One possible concern with
|
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|
experimentally measuring $W_R$ is that it might be obstructed by the
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|
classical diffraction maxima which appear at angles corresponding to
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|
the minima in $R$. However, the width of such a peak is much smaller
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|
as its width is proportional to $1/M$.
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|
It is also possible to analyse the phase transition quantitatively
|
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|
using our method. Unlike in higher dimensions where an order parameter
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|
can be easily defined within the MF approximation there is no such
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|
quantity in 1D. However, a valid description of the relevant 1D low
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|
energy physics is provided by Luttinger liquid theory
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|
\cite{giamarchi}. In this model correlations in the supefluid phase as
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|
well as the superfluid density itself are characterised by the
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|
Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies
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|
the phase transition in the thermodynamic limit at $K_b = 1/2$. This
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|
quantity can be extracted from various correlation functions and in
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|
our case it can be extracted directly from $R$ \cite{ejima2011}. By
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|
extracting this parameter from $R$ for various lattice lengths from
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|
numerical DMRG calculations it was even possible to give a theoretical
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|
estimate of the critical point for commensurate filling, $N = M$, in
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|
the thermodynamic limit to occur at $U/2J^\text{cl} \approx 1.64$
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|
\cite{ejima2011}. Our proposal provides a method to directly measure
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|
|
$R$ in a lab which can then be used to experimentally determine the
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|
|
location of the critical point in 1D.
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|
So far both variables we considered, $R_\text{max}$ and $W_R$, provide
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|
similar information. Next, we present a case where it is very
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|
|
different. The Bose glass is a localized insulating phase with
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|
exponentially decaying correlations but large compressibility and
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|
on-site fluctuations in a disordered optical lattice. Therefore,
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|
measuring both $R_\text{max}$ and $W_R$ will distinguish all the
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|
phases. In a Bose glass we have finite compressibility, but
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|
|
exponentially decaying correlations. This gives a large $R_\text{max}$
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|
and a large $W_R$. A Mott insulator will also have exponentially
|
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|
|
decaying correlations since it is an insulator, but it will be
|
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|
|
incompressible. Thus, it will scatter light with a small
|
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|
|
$R_\text{max}$ and large $W_R$. Finally, a superfluid will have long
|
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|
range correlations and large compressibility which results in a large
|
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|
$R_\text{max}$ and a small $W_R$.
|
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|
|
|
|
|
\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
|
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|
|
shifted due to finite size effects \cite{roux2008}, but it is
|
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|
|
possible to apply well known numerical methods to extract these
|
|
|
|
transition lines from such experimental data extracted from $R$
|
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|
|
\cite{ejima2011}. $K=M=N=35$.}
|
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|
|
\label{fig:BG}
|
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|
|
\end{figure}
|
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|
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|
|
We confirm this in Fig. \ref{fig:BG} for simulations with the ratio of
|
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|
|
superlattice- to trapping lattice-period $r\approx 0.77$ for various
|
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|
|
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.
|
|
|
|
|
2016-07-15 20:16:19 +02:00
|
|
|
\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.
|
|
|
|
|
2016-07-15 14:21:59 +02:00
|
|
|
\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}.
|