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Finished for the day. Chapter 3 still needs an update to the B operator section and possibly another look at the conclusion

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@ -188,23 +188,23 @@ for a standing wave probe, where $\beta$ is replaced by $\varphi_0$,
and scanning is achieved by varying the position of the wave with and scanning is achieved by varying the position of the wave with
respect to atoms. respect to atoms.
The ``quantum addition'', $R$, and the quadrature variance, The ``quantum addition'', $R$, and the quadrature variance, $(\Delta
$(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$. Therefore, they X^F_\beta)^2$, are both quadratic in $\a_1$ and both rely heavily on
will havea nontrivial angular dependence, showing more peaks than the quantum state of the matter. Therefore, they will have a
classical diffraction. Furthermore, these peaks can be tuned very nontrivial angular dependence, showing more peaks than classical
easily with $\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows diffraction. Furthermore, these peaks can be tuned very easily with
the angular dependence of $R$ for the case when the scattered mode is $\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular
a standing wave and the probe is a travelling wave scattering from dependence of $R$ for the case when the scattered mode is a standing
bosons in a 3D optical lattice. The first noticeable feature is the wave and the probe is a travelling wave scattering from bosons in a 3D
isotropic background which does not exist in classical optical lattice. The first noticeable feature is the isotropic
diffraction. This background yields information about density background which does not exist in classical diffraction. This
fluctuations which, according to mean-field estimates (i.e.~inter-site background yields information about density fluctuations which,
correlations are ignored), are related by according to mean-field estimates (i.e.~inter-site correlations are
$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. In ignored), are related by $R = K( \langle \hat{n}^2 \rangle - \langle
Fig. \ref{fig:scattering} we can see a significant signal of \hat{n} \rangle^2 )/2$. In Fig. \ref{fig:scattering} we can see a
$R = |C|^2 N_K/2$, because it shows scattering from an ideal significant signal of $R = |C|^2 N_K/2$, because it shows scattering
superfluid which has significant density fluctuations with from an ideal superfluid which has significant density fluctuations
correlations of infinte range. However, as the parameters of the with correlations of infinte range. However, as the parameters of the
lattice are tuned across the phase transition into a Mott insulator lattice are tuned across the phase transition into a Mott insulator
the signal goes to zero. This is because the Mott insulating phase has the signal goes to zero. This is because the Mott insulating phase has
well localised atoms at each site which suppresses density well localised atoms at each site which suppresses density
@ -236,7 +236,7 @@ in an experiment as they wouldn't be masked by a stronger classical
signal. We can even derive the generalised Bragg conditions for the signal. We can even derive the generalised Bragg conditions for the
peaks that we can see in Fig. \ref{fig:scattering}. peaks that we can see in Fig. \ref{fig:scattering}.
% Derive and show these Bragg conditions \mynote{Derive and show these Bragg conditions below}
As $(\Delta X^F_\beta)^2$ and $R$ are quadratic variables, As $(\Delta X^F_\beta)^2$ and $R$ are quadratic variables,
the generalized Bragg conditions for the peaks are the generalized Bragg conditions for the peaks are
@ -248,18 +248,22 @@ condition $\Delta \b{k} = \b{G}$. The peak height is tunable by the
local oscillator phase or standing wave shift as seen in Fig. 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 A quantum signal that isn't masked by classical diffraction is very
scattering rates integrated over the solid angle for the only two useful for future experimental realisability. However, it is still
experiments so far on light diffraction from truly ultracold bosons unclear whether this signal would be strong enough to be
where the measurement object was light 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} \begin{equation}
n_{\Phi}= \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma K}{8} n_{\Phi}= \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma K}{8}
(\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2). (\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2).
\end{equation} \end{equation}
Therefore, applying these results to the scattering patters in These results can be applied directly to the scattering patters in
Fig. \ref{fig:scattering} the background signal should reach Fig. \ref{fig:scattering}. Therefore, the background signal should
$n_\Phi \approx 10^6$ s$^{-1}$ in Ref. \cite{weitenberg2011} (150 reach $n_\Phi \approx 10^6$ s$^{-1}$ in Ref. \cite{weitenberg2011}
atoms in 2D), and $n_\Phi \approx 10^{11}$ s$^{-1}$ in (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 Ref. \cite{miyake2011} ($10^5$ atoms in 3D). These numbers show that
the diffraction patterns we have seen due to the ``quantum addition'' the diffraction patterns we have seen due to the ``quantum addition''
should be visible using currently available technology, especially should be visible using currently available technology, especially
@ -268,6 +272,123 @@ not coincide at all with the classical diffraction pattern.
\subsection{Mapping the quantum phase diagram} \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 of some quantity, 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 theoru 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 qith 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!] \begin{figure}[htbp!]
\centering \centering
\includegraphics[width=\linewidth]{oph11} \includegraphics[width=\linewidth]{oph11}
@ -288,121 +409,106 @@ not coincide at all with the classical diffraction pattern.
\label{fig:SFMI} \label{fig:SFMI}
\end{figure} \end{figure}
We have shown how in mean-field, we can track the order parameter, We then consider probing these ground states using our optical scheme
$\Phi$, by probing the matter-field interference using the coupling of and we calculate the ``quantum addition'', $R$. The angular dependence
light to the $\hat{B}$ operator. In this case, it is very easy to of $R$ for a Mott insulator and a superfluid is shown in
follow the superfluid to Mott insulator quantum phase transition since Fig. \ref{fig:SFMI}a, and we note that there are two variables
we have direct access to the order parameter which goes to zero in the distinguishing the states. Firstly, maximal $R$, $R_\text{max} \propto
insulating phase. In fact, if we're only interested in the critical \sum_i \langle \delta \hat{n}_i^2 \rangle$, probes the fluctuations
point, we only need access to any quantity that yields information and compressibility $\kappa'$ ($\langle \delta \hat{n}^2_i \rangle
about density fluctuations which also go to zero in the MI phase and \propto \kappa' \langle \hat{n}_i \rangle$). The Mott insulator is
this can be obtained by measuring incompressible and thus will have very small on-site fluctuations and
$\langle \hat{D}^\dagger \hat{D} \rangle$. However, there are many it will scatter little light leading to a small $R_\text{max}$. The
situations where the mean-field approximation is not a valid deeper the system is in the insulating phase (i.e. that larger the
description of the physics. A prominent example is the Bose-Hubbard $U/2J^\text{cl}$ ratio is), the smaller these values will be until
model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, pino2012, ultimately it will scatter no light at all in the $U \rightarrow
pino2013}. Observing the transition in 1D by light at fixed density \infty$ limit. In Fig. \ref{fig:SFMI}a this can be seen in the value
was considered to be difficult \cite{rogers2014} or even impossible of the peak in $R$. The value $R_\text{max}$ in the superfluid phase
\cite{roth2003}. By contrast, here we propose varying the density or ($U/2J^\text{cl} = 0$) is larger than its value in the Mott insulating
chemical potential, which sharply identifies the transition. We
perform these calculations numerically by calculating the ground state
using DMRG methods \cite{tnt} from which we can compute all the
necessary atomic observables. Experiments typically use an additional
harmonic confining potential on top of the optical lattice to keep the
atoms in place which means that the chemical potential will vary in
space. However, with careful consideration of the full
($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in
Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the
system \cite{batrouni2002}.
The 1D phase transition is best understood in terms of two-point
correlations as a function of their separation \cite{giamarchi}. In
the Mott insulating phase, the two-point correlations
$\langle \bd_i b_j \rangle$ and
$\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$
($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay
exponentially with $|i-j|$. On the other hand the superfluid will
exhibit long-range order which in dimensions higher than one,
manifests itself with an infinite correlation length. However, in 1D
only pseudo long-range order happens and both the matter-field and
density fluctuation correlations decay algebraically \cite{giamarchi}.
The method we propose gives us direct access to the structure factor,
which is a function of the two-point correlation $\langle \delta
\hat{n}_i \delta \hat{n}_j \rangle$, by measuring the light
intensity. For two travelling waves maximally coupled to the density
(atoms are at light intensity maxima so $\hat{F} = \hat{D}$), the
quantum addition is given by
\begin{equation}
R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0)
(\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta
\hat{n}_j \rangle,
\end{equation}
The angular dependence of $R$ for a Mott insulator and a superfluid is
shown in Fig. \ref{fig:SFMI}a, and there are two variables
distinguishing the states. Firstly, maximal $R$,
$R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$,
probes the fluctuations and compressibility $\kappa'$
($\langle \delta \hat{n}^2_i \rangle \propto \kappa' \langle \hat{n}_i
\rangle$). The Mott insulator is incompressible and thus will have
very small on-site fluctuations and it will scatter little light
leading to a small $R_\text{max}$. The deeper the system is in the MI
phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the smaller
these values will be until ultimately it will scatter no light at all
in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a this can
be seen in the value of the peak in $R$. The value $R_\text{max}$ in
the SF phase ($U/2J^\text{cl} = 0$) is larger than its value in the MI
phase ($U/2J^\text{cl} = 10$) by a factor of phase ($U/2J^\text{cl} = 10$) by a factor of
$\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of $\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of
$R_\text{max}$ changes across the phase transition. We see that the $R_\text{max}$ changes across the phase transition. There are a few
transition shows up very sharply as $\mu$ is varied. 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
Tomonage-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 $R$ 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 is a direct measure of the correlation length $l$, $W_R \propto
1/l$. The Mott insulator being an insulating phase is characterised by 1/l$. The Mott insulator being an insulating phase is characterised by
exponentially decaying correlations and as such it will have a very exponentially decaying correlations and as such it will have a very
large $W_R$. However, the superfluid in 1D exhibits pseudo long-range large $W_R$. On the other hand, the superfluid in 1D exhibits pseudo
order which manifests itself in algebraically decaying two-point long-range order which manifests itself in algebraically decaying
correlations \cite{giamarchi} which significantly reduces the dip in two-point correlations \cite{giamarchi} which significantly reduces
the $R$. This can be seen in Fig. \ref{fig:SFMI}a and we can also see the dip in the $R$. This can be seen in
that this identifies the phase transition very sharply as $\mu$ is Fig. \ref{fig:SFMI}a. Furthermore, just like for $R_\text{max}$ we see
varied in Figs. \ref{fig:SFMI}(c,e). One possible concern with that the transition is much sharper as $\mu$ is varied. This is shown
experimentally measuring $W_R$ is that it might be obstructed by the in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle between
classical diffraction maxima which appear at angles corresponding to a superfluid and an insulating state is fairly significant $\sim
the minima in $R$. However, the width of such a peak is much smaller 20^\circ$ which should make the two phases easy to identify using this
as its width is proportional to $1/M$. measure. In this particular case, measuring $W_R$ in the Mott phase is
not very practical as the insulating phase does not scatter light
It is also possible to analyse the phase transition quantitatively (small $R_\mathrm{max}$). The phase transition information is easier
using our method. Unlike in higher dimensions where an order parameter extracted from $R_\mathrm{max}$. However, this is not always the case
can be easily defined within the MF approximation there is no such and we will shortly see how certain phases of matter scatter a lot of
quantity in 1D. However, a valid description of the relevant 1D low light and can be distinguished using measurements of $W_R$. Another
energy physics is provided by Luttinger liquid theory possible concern with experimentally measuring $W_R$ is that it might
\cite{giamarchi}. In this model correlations in the supefluid phase as be obstructed by the classical diffraction maxima which appear at
well as the superfluid density itself are characterised by the angles corresponding to the minima in $R$. However, the width of such
Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies a peak is much smaller as its width is proportional to $1/M$.
the phase transition in the thermodynamic limit at $K_b = 1/2$. This
quantity can be extracted from various correlation functions and in
our case it can be extracted directly from $R$ \cite{ejima2011}. By
extracting this parameter from $R$ for various lattice lengths from
numerical DMRG calculations it was even possible to give a theoretical
estimate of the critical point for commensurate filling, $N = M$, in
the thermodynamic limit to occur at $U/2J^\text{cl} \approx 1.64$
\cite{ejima2011}. Our proposal provides a method to directly measure
$R$ in a lab which can then be used to experimentally determine the
location of the critical point in 1D.
So far both variables we considered, $R_\text{max}$ and $W_R$, provide So far both variables we considered, $R_\text{max}$ and $W_R$, provide
similar information. Next, we present a case where it is very similar information. They both take on values at one of its extremes
different. The Bose glass is a localized insulating phase with in the Mott insulating phase and they change drastically across the
exponentially decaying correlations but large compressibility and phase transition into the superfluid phase. Next, we present a case
on-site fluctuations in a disordered optical lattice. Therefore, where it is very different. We will again consider ultracold bosons in
measuring both $R_\text{max}$ and $W_R$ will distinguish all the an optical lattice, but this time we introduce some disorder. We do
phases. In a Bose glass we have finite compressibility, but this by adding an additional periodic potential on top of the exisitng
exponentially decaying correlations. This gives a large $R_\text{max}$ setup that is incommensurate with the original lattice. The resulting
and a large $W_R$. A Mott insulator will also have exponentially Hamiltonian can be shown to be
decaying correlations since it is an insulator, but it will be \begin{equation}
incompressible. Thus, it will scatter light with a small \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. The only
modification is an additionally 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 MI 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 will have exponentially decaying
correlations just like the Mott phase, but it will have 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 will also
have exponentially decaying correlations since it is an insulator, but
it will be incompressible. Thus, it will scatter light with a small
$R_\text{max}$ and large $W_R$. Finally, a superfluid will have long $R_\text{max}$ and large $W_R$. Finally, a superfluid will have long
range correlations and large compressibility which results in a large range correlations and large compressibility which results in a large
$R_\text{max}$ and a small $W_R$. $R_\text{max}$ and a small $W_R$.
@ -423,24 +529,26 @@ $R_\text{max}$ and a small $W_R$.
We confirm this in Fig. \ref{fig:BG} for simulations with the ratio of 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 superlattice- to trapping lattice-period $r\approx 0.77$ for various
disorder strengths $V$ \cite{roux2008}. Here, we only consider disorder strengths $V$ \cite{roux2008}. From Fig. \ref{fig:BG} we see
calculations for a fixed density, because the usual interpretation of that all three phases can indeed be distinguished. From the
the phase diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane $R_\mathrm{max}$ plot we can distinguish the two compressible phases,
for a fixed ratio $V/U$ becomes complicated due to the presence of the superfluid and Bose glass, from the incompressible phase, the Mott
multiple compressible and incompressible phases between successive MI insulator. We can now also distinguish the Bose glass phase from the
lobes \cite{roux2008}. This way, we have limited our parameter space superfluid, because only one of them is an insulator and thus the
to the three phases we are interested in: superfluid, Mott insulator, angular width of their scattering patterns will reveal this
and Bose glass. From Fig. \ref{fig:BG} we see that all three phases information. However, unlike the Mott insulator, the Bose glass
can indeed be distinguished. In the 1D BHM there is no sharp MI-SF scatters a lot of light (large $R_\mathrm{max}$) enabling such
phase transition in 1D at a fixed density \cite{cazalilla2011, measurements. Since we performed these calculations at a fixed density
ejima2011, kuhner2000, pino2012, pino2013} just like in the Mott to superfluid phase transition is not particularly sharp
Figs. \ref{fig:SFMI}(d,e) if we follow the transition through the tip \cite{cazalilla2011, ejima2011, kuhner2000, pino2012, pino2013} just
of the lobe which corresponds to a line of unit density. However, like we have seen in Figs. \ref{fig:SFMI}(d,e) if we follow the
despite the lack of an easily distinguishable critical point it is transition through the tip of the lobe which corresponds to a line of
possible to quantitatively extract the location of the transition unit density. However, despite the lack of an easily distinguishable
lines by extracting the Tomonaga-Luttinger parameter from the critical point, as we have already discussed, it is possible to
scattered light, $R$, in the same way it was done for an unperturbed quantitatively extract the location of the transition lines by
BHM \cite{ejima2011}. 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 Only recently \cite{derrico2014} a Bose glass phase was studied by
combined measurements of coherence, transport, and excitation spectra, combined measurements of coherence, transport, and excitation spectra,
@ -533,31 +641,33 @@ lattice sites on the microscopic scale in-situ.
\section{Conclusions} \section{Conclusions}
In summary, we proposed a nondestructive method to probe quantum gases In this chapter we explored the possibility of nondestructively
in an optical lattice. Firstly, we showed that the density-term in probing a quantum gas trapped in an optical lattice using quantized
scattering has an angular distribution richer than classical light. Firstly, we showed that the density-term in scattering has an
diffraction, derived generalized Bragg conditions, and estimated angular distribution richer than classical diffraction, derived
parameters for the only two relevant experiments to date generalized Bragg conditions, and estimated parameters for two
\cite{weitenberg2011, miyake2011}. Secondly, we proposed how to relevant experiments \cite{weitenberg2011, miyake2011}. Secondly, we
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 demonstrated how the method accesses effects beyond mean-field and
distinguishes all the phases in the Mott-superfluid-glass transition, distinguishes all the phases in the Mott-superfluid-glass transition,
which is currently a challenge \cite{derrico2014}. Based on 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 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. 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 off-resonant scattering, and thus being insensitive to a detailed
atomic level structure, the method can be extended to molecules atomic level structure, the method can be extended to molecules
\cite{LP2013}, spins, and fermions \cite{ruostekoski2009}. \cite{LP2013}, spins, and fermions \cite{ruostekoski2009}.

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% ******************************* PhD Thesis Template ************************** % ******************************* PhD Thesis Template **************************
% Please have a look at the README.md file for info on how to use the template % Please have a look at the README.md file for info on how to use the template
\documentclass[a4paper,12pt,times,numbered,print,index,chapter]{Classes/PhDThesisPSnPDF} \documentclass[a4paper,12pt,times,numbered,print,index]{Classes/PhDThesisPSnPDF}
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% ******************************* Class Options ******************************** % ******************************* Class Options ********************************