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End for the day - rewriting chapter 3 content to make more sense within the scope of the thesis

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11
Chapter2/chapter2.tex
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@ -95,6 +95,7 @@ variable angles in our model as this lends itself to a simpler
geometrical representation.
\subsection{Derivation of the Hamiltonian}
\label{sec:derivation}
A general many-body Hamiltonian coupled to a quantized light field in
second quantized can be separated into three parts,
@ -358,6 +359,7 @@ the quantum state of light. \mynote{cite Santiago's papers and
Therefore, combining these final simplifications we finally arrive at
our quantum light-matter Hamiltonian
\begin{equation}
\label{eq:fullH}
\H = \H_f -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j +
\frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) +
\frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m} -
@ -492,7 +494,14 @@ leaving $\hat{B}$ as the dominant term in $\hat{F}$. This approach is
fundamentally different from the aforementioned double-well proposals
as it directly couples to the interference terms caused by atoms
tunnelling rather than combining light scattered from different
sources.
sources. Such a counter-intuitive configuration may affect works on
quantum gases trapped in quantum potentials \cite{mekhov2012,
mekhov2008, larson2008, chen2009, habibian2013, ivanov2014,
caballero2015} and quantum measurement-induced preparation of
many-body atomic states \cite{mazzucchi2016, mekhov2009prl,
pedersen2014, elliott2015}.
\mynote{add citiations above if necessary}
For clarity we will consider a 1D lattice as shown in
Fig. \ref{fig:LatticeDiagram} with lattice spacing $d$ along the

476
Chapter3/chapter3.tex
View File

@ -19,53 +19,69 @@ Nondestructive Addressing} %Title of the Third Chapter
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 apply our model: nondestructive probing, quantum
measurement backaction and quantum optical lattices. Here, we deal
with the first of the three options.
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.
\mynote{adjust for the fact that the derivation of B operator has been moved}
\mynote{update some outdated mentions to previous experiments}
In this chapter we develop a method to measure properties of ultracold
gases in optical lattices by light scattering. We show that such
measurements can reveal not only density correlations, but also
matter-field interference. Recent quantum non-demolition (QND)
schemes \cite{rogers2014, mekhov2007prl, eckert2008} probe density
fluctuations and thus inevitably destroy information about phase,
i.e.~the conjugate variable, and as a consequence destroy matter-field
coherence. In contrast, we focus on probing the atom interference
between lattice sites. Our scheme is nondestructive in contrast to
totally destructive methods such as time-of-flight measurements. It
gases in optical lattices by light scattering. In the previous chapter
we have shown that quantum light field couples to the bosons via the
operator $\hat{F}$. This is the key element of the scheme we propose
as this makes it sensitive to the quantum state of the matter and all
of its possible superpositions which will be reflected in the quantum
state of the light itself. We have also shown in section
\ref{sec:derivation} that this coupling consists of two parts, a
density component $\hat{D}$ given by Eq. \eqref{eq:D}, and a phase
component $\hat{B}$ given by Eq. \eqref{eq:B}. Therefore, when probing
the quantum state of the ultracold gas we can have access to not only
density correlations, but also matter-field interference at its
shortest possible distance in an optical lattice, i.e.~the lattice
period. Previous work on quantum non-demolition (QND) schemes
\cite{rogers2014, mekhov2007prl, eckert2008} probe only the density
component as it is generally challenging to couple to the matter-field
observables directly. Here, we will consider nondestructive probing of
both density and interference operators.
Firstly, we will consider the simpler and more typical case of
coupling to the atom number operators via $\hat{F} =
\hat{D}$. However, we show that light diffraction in this regime has
several nontrivial characteristics due to the fully quantum nature of
the interaction. Firstly, we show that the angular distribution has
multiple interesting features even when classical diffraction is
forbidden facilitating their experimental observation. We derive new
generalised Bragg diffraction conditions which are different to their
classical counterpart. Furthermore, due to the fully quantum nature of
the interaction our proposal is capable of probing the quantum state
beyond mean-field prediction. We demonstrate this by showing that this
scheme is capable of distinguishing all three phases in the Mott
insulator - superfluid - Bose glass phase transition in a 1D
disordered optical lattice which is not very well described by a
mean-field treatment. We underline that transitions in 1D are much
more visible when changing an atomic density rather than for
fixed-density scattering. It was only recently that an experiment
distinguished a Mott insulator from a Bose glass \cite{derrico2014}
via a series of destructive measurements. Our proposal, on the other
hand, is nondestructive and is capable of extracting all the relevant
information in a single experiment.
Having shown the possibilities created by this nondestructive
measurement scheme we move on to considering light scattering from the
phase related observables via the operator $\hat{F} = \hat{B}$. This
enables in-situ probing of the matter-field coherence at its shortest
possible distance in an optical lattice, i.e. the lattice period,
which defines key processes such as tunnelling, currents, phase
gradients, etc. This is achieved by concentrating light between the
sites. 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}. Such a counter-intuitive configuration
may affect works on quantum gases trapped in quantum potentials
\cite{mekhov2012, mekhov2008, larson2008, chen2009, habibian2013,
ivanov2014, caballero2015} and quantum measurement-induced
preparation of many-body atomic states \cite{mazzucchi2016,
mekhov2009prl, pedersen2014, elliott2015}. 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.
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.
Furthermore, the scattering angular distribution is nontrivial, even
when classical diffraction is forbidden and we derive generalized
Bragg conditions for this situation. The method works beyond
mean-field, which we demonstrate by distinguishing all three phases in
the Mott insulator - superfluid - Bose glass phase transition in a 1D
disordered optical lattice. 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}.
\section{Global Nondestructive Measurement}
\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
@ -77,9 +93,10 @@ matter via
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 crucial to note that light couples to the bosons via
an operator as this makes it sensitive to the quantum state of the
matter.
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
@ -87,97 +104,149 @@ 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{mekhov2007pra,
mekhov2012}) and the measurement only weakly perturbs the system,
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. Again, we will show how the extreme
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}
\subsection{On-site Density Measurements}
Typically, the dominant term in $\hat{F}$ is the density term
$\hat{D}$, rather than inter-site matter-field interference $\hat{B}$
\cite{mekhov2007pra, rist2010, lakomy2009, ruostekoski2009,
LP2009}. However, before we move onto probing the interference
terms, $\hat{B}$, we will first discuss typical light scattering. We
start with a simpler case when scattering is faster than tunneling and
$\hat{F} = \hat{D}$. This corresponds to a QND scheme
\cite{mekhov2007prl, mekhov2007pra, eckert2008, rogers2014}. The
density-related measurement destroys some matter-phase coherence in
the conjugate variable \cite{mekhov2009pra, LP2010, LP2011}
$\bd_i b_{i+1}$, but this term is neglected. For this purpose we will
define an auxiliary quantity,
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. $R$
is simply the full light intensity minus the classical field intensity
and thus it faithfully represents the new contribution from the
quantum light-matter interaction to the diffraction pattern.
which we will call the ``quantum addition'' to light scattering. By
construction $R$ is simply the full light intensity minus the
classical field diffraction. In order to justify its name we will show
that this quantity depends purely quantum mechanical properties of the
ultracold gase. We will substitute $\a_1 = C \hat{D}$ using
Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and
we will make use of the shorthand notation
$A_i = u_1^*(\b{r}_i) u_0(\b{r}_i)$. The result is
\begin{equation}
R = |C|^2 \sum_{i,j}^K A^*_i A_j \langle \delta \hat{n}_i \delta
\hat{n}_j \rangle,
\end{equation}
where $\delta \hat{n}_i = \hat{n}_i - \langle \hat{n}_i
\rangle$. Thus, we can clearly see that $R$ is a result of light
scattering from fluctuations in the atom number which is a purely
quantum mechanical property of a system. Therefore, $R$, the ``quantum
addition'' faithfully represents the new contribution from the quantum
light-matter interaction to the diffraction pattern.
If instead we are interested in quantities linear in $\hat{D}$, we can
measure the quadrature of the light fields which in section
\ref{sec:a} we saw that $\hat{X}_\phi = |C| \hat{X}^F_\beta$. For the
case when both the scattered mode and probe are travelling waves the
quadrature
\begin{equation}
\hat{X}^F_\beta = \frac{1}{2} \left( \hat{F} e^{-i \beta} +
\hat{F}^\dagger e^{i \beta} \right) = \sum_i^K \hat{n}_i\cos[(\b{k}_1 - \b{k}_2) \cdot
\b{r}_i - \beta].
\end{equation}
Note that different light quadratures are differently coupled to the
atom distribution, hence by varying the local oscillator phase, and
thus effectively $\beta$, and/or the detection angle one can scan the
whole range of couplings. A similar expression exists for $\hat{D}$
for a standing wave probe, where $\beta$ is replaced by $\varphi_0$,
and scanning is achieved by varying the position of the wave with
respect to atoms.
The ``quantum addition'', $R$, and the quadrature variance,
$(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$. Therefore, they
will havea nontrivial angular dependence, showing more peaks than
classical diffraction. Furthermore, these peaks can be tuned very
easily with $\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows
the angular dependence of $R$ for the case when the scattered mode is
a standing wave and the probe is a travelling wave scattering from
bosons in a 3D optical lattice. The first noticeable feature is the
isotropic background which does not exist in classical
diffraction. This background yields information about density
fluctuations which, according to mean-field estimates (i.e.~inter-site
correlations are ignored), are related by
$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. In
Fig. \ref{fig:scattering} we can see a significant signal of
$R = |C|^2 N_K/2$, because it shows scattering from an ideal
superfluid which has significant density fluctuations with
correlations of infinte range. However, as the parameters of the
lattice are tuned across the phase transition into a Mott insulator
the signal goes to zero. This is because the Mott insulating phase has
well localised atoms at each site which suppresses density
fluctuations entirely leading to absolutely no ``quantum addition''.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\linewidth]{Ep1}
\caption[Light Scattering Angular Distribution]{Light intensity
scattered into a standing wave mode from a superfluid in a 3D
lattice (units of $R/N_K$). Arrows denote incoming travelling wave
probes. The 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}
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}
In a deep lattice,
\begin{equation}
\hat{D}=\sum_i^K u_1^*({\bf r}_i) u_0({\bf 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
$\pm(\b{k}_1 \pm \b{k}_0)$. The quadrature, as defined in section
\ref{sec:a}, for two travelling waves is reduced to
\begin{equation}
\hat{X}^F_\beta = \sum_i^K \hat{n}_i\cos[(\b{k}_1 - \b{k}_2) \cdot
\b{r}_i - \beta].
\end{equation}
Note that different light quadratures are differently coupled to the
atom distribution, hence varying local oscillator phase and detection
angle, one scans the coupling from maximal to zero. An identical
expression exists for $\hat{D}$ for a standing wave, where $\beta$ is
replaced by $\varphi_l$, and scanning is achieved by varying the
position of the wave with respect to atoms. Thus, variance
$(\Delta X^F_\beta)^2$ and quantum addition $R$, have a non-trivial
angular dependence, showing more peaks than classical diffraction and
the peaks can be tuned by the light-atom coupling.
We can also observe maxima at several different angles in
Fig. \ref{fig:scattering}. Interestingly, they occur at different
angles than predicted by the classical Bragg condition. Moreover, the
classical Bragg condition is actually not satisfied which means there
actually is no classical diffraction on top of the ``quantum
addition'' shown here. Therefore, these features would be easy to see
in an experiment as they wouldn't be masked by a stronger classical
signal. We can even derive the generalised Bragg conditions for the
peaks that we can see in Fig. \ref{fig:scattering}.
Fig. \ref{fig:Scattering} shows the angular dependence of $R$ for
standing and travelling waves scattering from bosons in a 3D optical
lattice. The isotropic background gives the density fluctuations
[$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$ in
mean-field with inter-site correlations neglected]. The radius of the
sphere changes from zero, when it is a Mott insulator with suppressed
fluctuations, to half the atom number at $K$ sites, $N_K/2$, in the
deep superfluid. There exist peaks at angles different than the
classical Bragg ones and thus, can be observed without being masked by
classical diffraction. Interestingly, even if 3D diffraction
\cite{miyake2011} is forbidden as seen in Fig. \ref{fig:Scattering},
the peaks are still present. As $(\Delta X^F_\beta)^2$ and $R$ are
quadratic variables, the generalized Bragg conditions for the peaks
are $2 \Delta \b{k} = \b{G}$ for quadratures of travelling waves,
where $\Delta \b{k} = \b{k}_0 - \b{k}_1$ and $\b{G}$ is the reciprocal
% Derive and show these Bragg conditions
As $(\Delta X^F_\beta)^2$ and $R$ are quadratic variables,
the generalized Bragg conditions for the peaks are
$2 \Delta \b{k} = \b{G}$ for quadratures of travelling waves, where
$\Delta \b{k} = \b{k}_0 - \b{k}_1$ and $\b{G}$ is the reciprocal
lattice vector, and $2 \b{k}_1 = \b{G}$ for standing wave $\a_1$ and
travelling $\a_0$, which is clearly different from the classical Bragg
condition $\Delta \b{k} = \b{G}$. The peak height is tunable by the
local oscillator phase or standing wave shift as seen in Fig.
\ref{fig:Scattering}b.
\ref{fig:scattering}b.
In section \ref{sec:Efield} we have estimated the mean photon
scattering rates integrated over the solid angle for the only two
@ -187,97 +256,15 @@ where the measurement object was light
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}
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{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.
Therefore, applying these results to the scattering patters in
Fig. \ref{fig:scattering} the background signal should reach
$n_\Phi \approx 10^6$ s$^{-1}$ in Ref. \cite{weitenberg2011} (150
atoms in 2D), and $n_\Phi \approx 10^{11}$ s$^{-1}$ in
Ref. \cite{miyake2011} ($10^5$ atoms in 3D). These numbers show that
the diffraction patterns we have seen due to the ``quantum addition''
should be visible using currently available technology, especially
since the most prominent features, such as Bragg diffraction peaks, do
not coincide at all with the classical diffraction pattern.
\subsection{Mapping the quantum phase diagram}
@ -461,6 +448,89 @@ all of which are destructive techniques. Our method is simpler as it
only requires measurement of the quantity $R$ and additionally, it is
nondestructive.
\subsection{Matter-field interference measurements}
We now focus on enhancing the interference term $\hat{B}$ in the
operator $\hat{F}$.
Firstly, we will use this result to show how one can probe
$\langle \hat{B} \rangle$ which in MF gives information about the
matter-field amplitude, $\Phi = \langle b \rangle$.
Hence, by measuring the light quadrature we probe the kinetic energy
and, in MF, the matter-field amplitude (order parameter) $\Phi$:
$\langle \hat{X}^F_{\beta=0} \rangle = | \Phi |^2
\mathcal{F}[W_1](2\pi/d) (K-1)$.
Secondly, we show that it is also possible to access the fluctuations
of matter-field quadratures $\hat{X}^b_\alpha = (b e^{-i\alpha} + \bd
e^{i\alpha})/2$, which in MF can be probed by measuring the variance
of $\hat{B}$. Across the phase transition, the matter field changes
its state from Fock (in MI) to coherent (deep SF) through an
amplitude-squeezed state as shown in Fig. \ref{Quads}(a,b).
Assuming $\Phi$ is real in MF:
\begin{equation}
\label{intensity}
\langle \ad_1 \a_1 \rangle = 2 |C|^2(K-1)\mathcal{F}^2[W_1](\frac{\pi}{d})
\times [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ]
\end{equation}
and it is shown as a function of $U/(zJ^\text{cl})$ in
Fig. \ref{Quads}. Thus, since measurement in the diffraction maximum
yields $\Phi^2$ we can deduce $\langle b^2 \rangle - \Phi^2$ from the
intensity. This quantity is of great interest as it gives us access to
the quadrature variances of the matter-field
\begin{equation}
(\Delta X^b_{0,\pi/2})^2 = 1/4 + [(n - \Phi^2) \pm
(\langle b^2 \rangle - \Phi^2)]/2,
\end{equation}
where $n=\langle\hat{n}\rangle$ is the mean on-site atomic density.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\linewidth]{Quads}
\captionsetup{justification=centerlast,font=small}
\caption[Mean-Field Matter Quadratures]{Photon number scattered in a
diffraction minimum, given by Eq. (\ref{intensity}), where
$\tilde{C} = 2 |C|^2 (K-1) \mathcal{F}^2 [W_1](\pi/d)$. More
light is scattered from a MI than a SF due to the large
uncertainty in phase in the insulator. (a) The variances of
quadratures $\Delta X^b_0$ (solid) and $\Delta X^b_{\pi/2}$
(dashed) of the matter field across the phase transition. Level
1/4 is the minimal (Heisenberg) uncertainty. There are three
important points along the phase transition: the coherent state
(SF) at A, the amplitude-squeezed state at B, and the Fock state
(MI) at C. (b) The uncertainties plotted in phase space.}
\label{Quads}
\end{figure}
Probing $\hat{B}^2$ gives us access to kinetic energy fluctuations
with 4-point correlations ($\bd_i b_j$ combined in pairs). Measuring
the photon number variance, which is standard in quantum optics, will
lead up to 8-point correlations similar to 4-point density
correlations \cite{mekhov2007pra}. These are of significant interest,
because it has been shown that there are quantum entangled states that
manifest themselves only in high-order correlations
\cite{kaszlikowski2008}.
Surprisingly, inter-site terms scatter more light from a Mott
insulator than a superfluid Eq. \eqref{intensity}, as shown in
Fig. \eqref{Quads}, although the mean inter-site density
$\langle \hat{n}(\b{r})\rangle $ is tiny in a MI. This reflects a
fundamental effect of the boson interference in Fock states. It indeed
happens between two sites, but as the phase is uncertain, it results
in the large variance of $\hat{n}(\b{r})$ captured by light as shown
in Eq. \eqref{intensity}. The interference between two macroscopic
BECs has been observed and studied theoretically
\cite{horak1999}. When two BECs in Fock states interfere a phase
difference is established between them and an interference pattern is
observed which disappears when the results are averaged over a large
number of experimental realizations. This reflects the large
shot-to-shot phase fluctuations corresponding to a large inter-site
variance of $\hat{n}(\b{r})$. By contrast, our method enables the
observation of such phase uncertainty in a Fock state directly between
lattice sites on the microscopic scale in-situ.
\section{Conclusions}
In summary, we proposed a nondestructive method to probe quantum gases

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