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Working on Chapter 3

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@ -27,7 +27,11 @@ general Hamiltonian that describes the coupling of atoms with
far-detuned optical beams \cite{mekhov2012}. This will serve as the far-detuned optical beams \cite{mekhov2012}. This will serve as the
basis from which we explore the system in different parameter regimes, basis from which we explore the system in different parameter regimes,
such as nondestructive measurement in free space or quantum such as nondestructive measurement in free space or quantum
measurement backaction in a cavity. measurement backaction in a cavity. Another interesting direction for
this field of research are quantum optical lattices where the trapping
potential is treated quantum mechanically. However this is beyond the
scope of this work.
\mynote{insert our paper citations here}
We consider $N$ two-level atoms in an optical lattice with $M$ We consider $N$ two-level atoms in an optical lattice with $M$
sites. For simplicity we will restrict our attention to spinless sites. For simplicity we will restrict our attention to spinless
@ -43,14 +47,10 @@ capable of describing a range of different experimental setups ranging
from a small number of sites with a large filling factor (e.g.~BECs from a small number of sites with a large filling factor (e.g.~BECs
trapped in a double-well potential) to a an extended multi-site trapped in a double-well potential) to a an extended multi-site
lattice with a low filling factor (e.g.~a system with one atom per lattice with a low filling factor (e.g.~a system with one atom per
site will exhibit the Mott insulator to superfluid quantum phase site which will exhibit the Mott insulator to superfluid quantum phase
transition). \mynote{extra fermion citations, Piazza? Look up Gabi's transition). \mynote{extra fermion citations, Piazza? Look up Gabi's
AF paper.} AF paper.}
\mynote{Potentially some more crap, but come to think of it the
content will strongly depend on what was included in the preceding
section on plain ultracold bosons}
As we have seen in the previous section, an optical lattice can be As we have seen in the previous section, an optical lattice can be
formed with classical light beams that form standing waves. Depending formed with classical light beams that form standing waves. Depending
on the detuning with respect to the atomic resonance, the nodes or on the detuning with respect to the atomic resonance, the nodes or
@ -83,9 +83,9 @@ For simplicity, we will be considering one-dimensional lattices most
of the time. However, the model itself is derived for any number of of the time. However, the model itself is derived for any number of
dimensions and since none of our arguments will ever rely on dimensions and since none of our arguments will ever rely on
dimensionality our results straightforwardly generalise to 2- and 3-D dimensionality our results straightforwardly generalise to 2- and 3-D
systems. This simplification allows us to present a much simpler systems. This simplification allows us to present a much more
picture of the physical setup where we only need to concern ourselves intuitive picture of the physical setup where we only need to concern
with a single angle for each optical mode. As shown in ourselves with a single angle for each optical mode. As shown in
Fig. \ref{fig:LatticeDiagram} the angle between the normal to the Fig. \ref{fig:LatticeDiagram} the angle between the normal to the
lattice and the probe and detected beam are denoted by $\theta_0$ and lattice and the probe and detected beam are denoted by $\theta_0$ and
$\theta_1$ respectively. We will consider these angles to be tunable $\theta_1$ respectively. We will consider these angles to be tunable
@ -199,11 +199,11 @@ equation for the lowering operator
Therefore, by inserting this expression into the Heisenberg equation Therefore, by inserting this expression into the Heisenberg equation
for the light mode $m$ given by for the light mode $m$ given by
\begin{equation} \begin{equation}
\dot{\a}_m = - \sigma^- g^*_m u^*_m(\b{r}) \dot{\a}_m = - \sigma^- g_m u^*_m(\b{r})
\end{equation} \end{equation}
we get the following equation of motion we get the following equation of motion
\begin{equation} \begin{equation}
\dot{\a}_m = \frac{i}{\Delta_a} \sum_l g_l g^*_m u_l(\b{r}) \dot{\a}_m = \frac{i}{\Delta_a} \sum_l g_l g_m u_l(\b{r})
u^*_m(\b{r}) \a_l. u^*_m(\b{r}) \a_l.
\end{equation} \end{equation}
An effective Hamiltonian which results in the same optical equations An effective Hamiltonian which results in the same optical equations
@ -311,30 +311,49 @@ This contribution can be separated into two parts, one which couples
directly to the on-site atomic density and one that couples to the directly to the on-site atomic density and one that couples to the
tunnelling operators. We will define the operator tunnelling operators. We will define the operator
\begin{equation} \begin{equation}
\label{eq:F}
\hat{F}_{l,m} = \hat{D}_{l,m} + \hat{B}_{l,m}, \hat{F}_{l,m} = \hat{D}_{l,m} + \hat{B}_{l,m},
\end{equation} \end{equation}
where $\hat{D}_{l,m}$ is the direct coupling to atomic density where $\hat{D}_{l,m}$ is the direct coupling to atomic density
\begin{equation} \begin{equation}
\label{eq:D}
\hat{D}_{l,m} = \sum_{i}^K J^{l,m}_{i,i} \hat{n}_i, \hat{D}_{l,m} = \sum_{i}^K J^{l,m}_{i,i} \hat{n}_i,
\end{equation} \end{equation}
and $\hat{B}_{l,m}$ couples to the matter-field via the and $\hat{B}_{l,m}$ couples to the matter-field via the
nearest-neighbour tunnelling operators nearest-neighbour tunnelling operators
\begin{equation} \begin{equation}
\label{eq:B}
\hat{B}_{l,m} = \sum_{\langle i, j \rangle}^K J^{l,m}_{i,j} \bd_i b_j, \hat{B}_{l,m} = \sum_{\langle i, j \rangle}^K J^{l,m}_{i,j} \bd_i b_j,
\end{equation} \end{equation}
and we neglect couplings beyond nearest neighbours for the same reason where $K$ denotes a sum over the illuminated sites and we neglect
as before when deriving the matter Hamiltonian. couplings beyond nearest neighbours for the same reason as before when
deriving the matter Hamiltonian.
\mynote{make sure all group papers are cited here} These equations
encapsulate the simplicity and flexibility of the measurement scheme
that we are proposing. The operators given above are entirely
determined by the values of the $J^{l,m}_{i,j}$ coefficients and
despite its simplicity, this is sufficient to give rise to a host of
interesting phenomena via measurement back-action such as the
generation of multipartite entangled spatial modes in an optical
lattice \cite{elliott2015, atoms2015, mekhov2009pra}, the appearance
of long-range correlated tunnelling capable of entangling distant
lattice sites, and in the case of fermions, the break-up and
protection of strongly interacting pairs \cite{mazzucchi2016,
kozlowski2016zeno}. Additionally, these coefficients are easy to
manipulate experimentally by adjusting the optical geometry via the
light mode functions $u_l(\b{r})$.
It is important to note that we are considering a situation where the It is important to note that we are considering a situation where the
contribution of quantized light is much weaker than that of the contribution of quantized light is much weaker than that of the
classical trapping potential. If that was not the case, it would be classical trapping potential. If that was not the case, it would be
necessary to determine thw Wannier functions in a self-consistent way necessary to determine the Wannier functions in a self-consistent way
which takes into account the depth of the quantum poterntial generated which takes into account the depth of the quantum poterntial generated
by the quantized light modes. This significantly complicates the by the quantized light modes. This significantly complicates the
treatment, but can lead to interesting physics. Amongst other things, treatment, but can lead to interesting physics. Amongst other things,
the atomic tunnelling and interaction coefficients will now depend on the atomic tunnelling and interaction coefficients will now depend on
the quantum state of light. the quantum state of light. \mynote{cite Santiago's papers and
\mynote{cite Santiago's papers and Maschler/Igor EPJD} Maschler/Igor EPJD}
Therefore, combining these final simplifications we finally arrive at Therefore, combining these final simplifications we finally arrive at
our quantum light-matter Hamiltonian our quantum light-matter Hamiltonian
@ -346,18 +365,25 @@ our quantum light-matter Hamiltonian
\end{equation} \end{equation}
where we have phenomologically included the cavity decay rates where we have phenomologically included the cavity decay rates
$\kappa_l$ of the modes $a_l$. A crucial observation about the $\kappa_l$ of the modes $a_l$. A crucial observation about the
structure of this Hamiltonian is that in the last term, the light structure of this Hamiltonian is that in the interaction term, the
modes $a_l$ couple to the matter in a global way. Instead of light modes $a_l$ couple to the matter in a global way. Instead of
considering individual coupling to each site, the optical field considering individual coupling to each site, the optical field
couples to the global state of the atoms within the illuminated region couples to the global state of the atoms within the illuminated region
via the operator $\hat{F}_{l,m}$. This will have important via the operator $\hat{F}_{l,m}$. This will have important
implications for the system and is one of the leading factors implications for the system and is one of the leading factors
responsible for many-body behaviour beyong the Bose-Hubbard responsible for many-body behaviour beyond the Bose-Hubbard
Hamiltonian paradigm. Hamiltonian paradigm.
\subsection{Scattered light behaviour} Furthermore, it is also vital to note that light couples to the matter
via an operator, namely $\hat{F}_{l,m}$, which makes it sensitive to
the quantum state of matter. This is a key element of our treatment of
the ultimate quantum regime of light-matter interaction that goes
beyond previous treatments.
Having derived the full quantum light-matter Hamiltonian we will no \subsection{Scattered light behaviour}
\label{sec:a}
Having derived the full quantum light-matter Hamiltonian we will now
look at the behaviour of the scattered light. We begin by looking at look at the behaviour of the scattered light. We begin by looking at
the equations of motion in the Heisenberg picture the equations of motion in the Heisenberg picture
\begin{equation} \begin{equation}
@ -391,6 +417,7 @@ pumping $\eta_l$. We also limit ourselves to only a single scattered
mode, $a_1$. This leads to a simple linear relationship between the mode, $a_1$. This leads to a simple linear relationship between the
light mode and the atomic operator $\hat{F}_{1,0}$ light mode and the atomic operator $\hat{F}_{1,0}$
\begin{equation} \begin{equation}
\label{eq:a}
\a_1 = \frac{U_{1,0} a_0} {\Delta_{p} + i \kappa} \hat{F} = \a_1 = \frac{U_{1,0} a_0} {\Delta_{p} + i \kappa} \hat{F} =
C \hat{F}, C \hat{F},
\end{equation} \end{equation}
@ -400,19 +427,45 @@ cavity. Furthermore, since there is no longer any ambiguity in the
indices $l$ and $m$, we have dropped indices on $\Delta_{1p} \equiv indices $l$ and $m$, we have dropped indices on $\Delta_{1p} \equiv
\Delta_p$, $\kappa_1 \equiv \kappa$, and $\hat{F}_{1,0} \equiv \Delta_p$, $\kappa_1 \equiv \kappa$, and $\hat{F}_{1,0} \equiv
\hat{F}$. We also do the same for the operators $\hat{D}_{1,0} \equiv \hat{F}$. We also do the same for the operators $\hat{D}_{1,0} \equiv
\hat{D}$ and $\hat{B}_{1,0} \equiv \hat{B}$. \hat{D}$, $\hat{B}_{1,0} \equiv \hat{B}$, and the coefficients
$J^{1,0}_{i,j} \equiv J_{i,j}$.
Whilst the light amplitude itself is only linear in atomic operators, The operator $\a_1$ itself is not an observable. However, it is
we can easily have access to higher moments by simply simply possible to combine the outgoing light field with a stronger local
considering higher moments of the $\a_1$ such as the photon number oscillator beam in order to measure the light quadrature
$\ad_1 \a_1$. Additionally, even though we only consider a single \begin{equation}
scattered mode, this model can be applied to free space by simply \hat{X}_\phi = \frac{1}{2} \left( \a_1 e^{-i \phi} + \ad_1 e^{i \phi} \right),
varying the direction of the scattered light mode if multiple \end{equation}
scattering events can be neglected. This is likely to be the case which in turn can expressed via the quadrature of $\hat{F}$,
since the interactions will be dominated by photons scattering from $\hat{X}^F_\beta$, as
the much larger coherent probe. \begin{equation}
\hat{X}_\phi = |C| \hat{X}_\beta^F = \frac{|C|}{2} \left( \hat{F}
e^{-i \beta} + \hat{F}^\dagger e^{i \beta} \right),
\end{equation}
where $\beta = \phi - \phi_C$, $C = |C| \exp(i \phi_C)$, and $\phi$ is
the local oscillator phase.
Whilst the light amplitude and the quadratures are only linear in
atomic operators, we can easily have access to higher moments via
related quantities such as the photon number
$\ad_1 \a_1 = |C|^2 \hat{F}^\dagger \hat{F}$ or the quadrature
variance
\begin{equation}
\label{eq:Xvar}
( \Delta X_\phi )^2 = \langle \hat{X}_\phi^2 \rangle - \langle
\hat{X}_\phi \rangle^2 = \frac{1}{4} + |C|^2 (\Delta X^F_\beta)^2,
\end{equation}
which reflect atomic correlations and fluctuations.
Finally, even though we only consider a single scattered mode, this
model can be applied to free space by simply varying the direction of
the scattered light mode if multiple scattering events can be
neglected. This is likely to be the case since the interactions will
be dominated by photons scattering from the much larger coherent
probe.
\subsection{Density and Phase Observables} \subsection{Density and Phase Observables}
\label{sec:B}
Light scatters due to its interactions with the dipole moment of the Light scatters due to its interactions with the dipole moment of the
atoms which for off-resonant light, like the type that we consider, atoms which for off-resonant light, like the type that we consider,
@ -430,21 +483,22 @@ condensates even though the two components have initially well-defined
atom numbers which is phase's conjugate variable. atom numbers which is phase's conjugate variable.
In our model light couples to the operator $\hat{F}$ which consists of In our model light couples to the operator $\hat{F}$ which consists of
a density opertor part, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a a density component, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a phase
phase operator part, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j} component, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j} \bd_i
\bd_i b_j$. Most of the time the density component dominates, $\hat{D} b_j$. In general, the density component dominates,
\gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$. However, it is $\hat{D} \gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$. However,
possible to engineer an optical geometry in which $\hat{D} = 0$ it is possible to engineer an optical geometry in which $\hat{D} = 0$
leaving $\hat{B}$ as the dominant term in $\hat{F}$. This approach is leaving $\hat{B}$ as the dominant term in $\hat{F}$. This approach is
fundamentally different from the aforementioned double-well proposals fundamentally different from the aforementioned double-well proposals
as it directly couples to the interference terms caused by atoms as it directly couples to the interference terms caused by atoms
tunnelling rather than combining light scattered from different tunnelling rather than combining light scattered from different
sources. sources.
For clarity we will consider a 1D lattice with lattice spacing $d$ For clarity we will consider a 1D lattice as shown in
along the $x$-axis direction, but the results can be applied and Fig. \ref{fig:LatticeDiagram} with lattice spacing $d$ along the
generalised to higher dimensions. Central to engineering the $\hat{F}$ $x$-axis direction, but the results can be applied and generalised to
operator are the coefficients $J_{i,j}$ given by higher dimensions. Central to engineering the $\hat{F}$ operator are
the coefficients $J_{i,j}$ given by
\begin{equation} \begin{equation}
\label{eq:Jcoeff} \label{eq:Jcoeff}
J_{i,j} = \int \mathrm{d} x \,\,\, w(x - x_i) u_1^*(x) u_0(x) w(x - x_j). J_{i,j} = \int \mathrm{d} x \,\,\, w(x - x_i) u_1^*(x) u_0(x) w(x - x_j).
@ -453,156 +507,139 @@ The operators $\hat{B}$ and $\hat{D}$ depend on the values of
$J_{i,i+1}$ and $J_{i,i}$ respectively. These coefficients are $J_{i,i+1}$ and $J_{i,i}$ respectively. These coefficients are
determined by the convolution of the coupling strength between the determined by the convolution of the coupling strength between the
probe and scattered light modes, $u_1^*(x)u_0(x)$, with the relevant probe and scattered light modes, $u_1^*(x)u_0(x)$, with the relevant
Wannier function overlap shown in Fig. \ref{fig:WannierOverlaps}. For Wannier function overlap shown in Fig. \ref{fig:WannierProducts}a. For
the $\hat{B}$ operator we calculate the convolution with the nearest the $\hat{B}$ operator we calculate the convolution with the nearest
neighbour overlap, $W_1(x) \equiv w(x - d/2) w(x + d/2)$ shown in neighbour overlap, $W_1(x) \equiv w(x - d/2) w(x + d/2)$, and for the
Fig. \ref{fig:WannierOverlaps}c, and for the $\hat{D}$ operator we $\hat{D}$ operator we calculate the convolution with the square of the
calculate the convolution with the square of the Wannier function at a Wannier function at a single site, $W_0(x) \equiv w^2(x)$. Therefore,
single site, $W_0(x) \equiv w^2(x)$ shown in in order to enhance the $\hat{B}$ term we need to maximise the overlap
Fig. \ref{fig:WannierOverlaps}b. Therefore, in order to enhance the between the light modes and the nearest neighbour Wannier overlap,
$\hat{B}$ term we need to maximise the overlap between the light modes $W_1(x)$. This can be achieved by concentrating the light between the
and the nearest neighbour Wannier overlap, $W_1(x)$. This can be sites rather than at the positions of the atoms. Ideally, one could
achieved by concentrating the light between the sites rather than at measure between two sites similarly to single-site addressing
the positions of the atoms. Ideally, one could measure between two \cite{greiner2009, bloch2011}, which would measure a single term
sites similarly to single-site addressing, which would measure a $\langle \bd_i b_{i+1}+b_i \bd_{i+1}\rangle$. This could be achieved,
single term $\langle \bd_i b_{i+1}+b_i \bd_{i+1}\rangle$, e.g.~by for example, by superposing a deeper optical lattice shifted by $d/2$
superposing a deeper optical lattice shifted by $d/2$ with respect to with respect to the original one, catching and measuring the atoms in
the original one, catching and measuring the atoms in the new lattice the new lattice sites. A single-shot success rate of atom detection
sites. A single-shot success rate of atom detection will be small. As will be small. As single-site addressing is challenging, we proceed
single-site addressing is challenging, we proceed with the global with the global scattering.
scattering.
\mynote{Fix labels in this figure} \mynote{Potentially expand details of the derivation of these equations}
\begin{figure}[htbp!]
\centering
\includegraphics[width=1.0\textwidth]{Wannier1}
\includegraphics[width=1.0\textwidth]{Wannier2}
\caption[Wannier Function Overlaps]{(a) The Wannier functions
corresponding to four neighbouring sites in a 1D
lattice. $\lambda$ is the wavelength of the trapping beams, thus
lattice sites occur every $\lambda/2$. (Bottom Left) The square of
a single Wannier function - this quantity is used when evaluating
$\hat{D}$. It's much larger than the overlap between two
neighbouring Wannier functions, but it is localised to the
position of the lattice site it belongs to. (Bottom Right) The
overlap of two neighbouring Wannier functions - this quantity is
used when evaluating $\hat{B}$. It is much smaller than the square
of a Wannier function, but since it's localised in between the
sites, thus $\hat{B}$ can be maximised while $\hat{D}$ minimised
by focusing the light in between the sites.}
\label{fig:WannierOverlaps}
\end{figure}
\mynote{show the expansion into an FT}
In order to calculate the $J_{i,j}$ coefficients we perform numerical In order to calculate the $J_{i,j}$ coefficients we perform numerical
calculations using realistic Wannier functions. However, it is calculations using realistic Wannier functions
possible to gain some analytic insight into the behaviour of these \cite{walters2013}. However, it is possible to gain some analytic
values by looking at the Fourier transforms of the Wannier function insight into the behaviour of these values by looking at the Fourier
overlaps, $\mathcal{F}[W_{0,1}](k)$, shown in Fig. transforms of the Wannier function overlaps,
\ref{fig:WannierFT}b. This is because the light mode product, $\mathcal{F}[W_{0,1}](k)$, shown in Fig.
$u_1^*(x) u_0(x)$, can be in general decomposed into a sum of \ref{fig:WannierProducts}b. This is because the for plane and standing
oscillating exponentials of the form $e^{i k x}$ making the integral wave light modes the product $u_1^*(x) u_0(x)$ can be in general
in Eq. \eqref{eq:Jcoeff} a sum of Fourier transforms of decomposed into a sum of oscillating exponentials of the form
$W_{0,1}(x)$. We consider both the detected and probe beam to be $e^{i k x}$ making the integral in Eq. \eqref{eq:Jcoeff} a sum of
standing waves which gives the following expressions for the $\hat{D}$ Fourier transforms of $W_{0,1}(x)$. We consider both the detected and
and $\hat{B}$ operators probe beam to be standing waves which gives the following expressions
\begin{eqnarray} for the $\hat{D}$ and $\hat{B}$ operators
\begin{align}
\label{eq:FTs} \label{eq:FTs}
\hat{D} = \hat{D} = & \frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_m\hat{n}_m\cos(k_-
\frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_m\hat{n}_m\cos(k_- x_m x_m +\varphi_-) \nonumber\\
+\varphi_-) & + \mathcal{F}[W_0](k_+)\sum_m\hat{n}_m\cos(k_+ x_m +\varphi_+)],
\nonumber\\ +\mathcal{F}[W_0](k_+)\sum_m\hat{n}_m\cos(k_+ \nonumber\\
x_m +\varphi_+)], \nonumber\\ \hat{B} = \hat{B} = & \frac{1}{2}[\mathcal{F}[W_1](k_-)\sum_m\hat{B}_m\cos(k_- x_m
\frac{1}{2}[\mathcal{F}[W_1](k_-)\sum_m\hat{B}_m\cos(k_- x_m +\frac{k_-d}{2}+\varphi_-) \nonumber\\
+\frac{k_-d}{2}+\varphi_-) & +\mathcal{F}[W_1](k_+)\sum_m\hat{B}_m\cos(k_+
\nonumber\\ +\mathcal{F}[W_1](k_+)\sum_m\hat{B}_m\cos(k_+
x_m +\frac{k_+d}{2}+\varphi_+)], x_m +\frac{k_+d}{2}+\varphi_+)],
\end{eqnarray} \end{align}
where $k_\pm = k_{0x} \pm k_{1x}$, $k_{(0,1)x} = k_{0,1} where $k_\pm = k_{0x} \pm k_{1x}$,
\sin(\theta_{0,1}$), $\hat{B}_m=b^\dag_mb_{m+1}+b_mb^\dag_{m+1}$, and $k_{(0,1)x} = k_{0,1} \sin(\theta_{0,1}$),
$\hat{B}_m=b^\dag_mb_{m+1}+b_mb^\dag_{m+1}$, and
$\varphi_\pm=\varphi_0 \pm \varphi_1$. The key result is that the $\varphi_\pm=\varphi_0 \pm \varphi_1$. The key result is that the
$\hat{B}$ operator is phase shifted by $k_\pm d/2$ with respect to the $\hat{B}$ operator is phase shifted by $k_\pm d/2$ with respect to the
$\hat{D}$ operator since it depends on the amplitude of light in $\hat{D}$ operator since it depends on the amplitude of light in
between the lattice sites and not at the positions of the atoms, between the lattice sites and not at the positions of the atoms
allowing to decouple them at specific angles. allowing to decouple them at specific angles.
\begin{figure}[htbp!] \begin{figure}[hbtp!]
\begin{center} \centering
\includegraphics[width=\linewidth]{WF_S} \includegraphics[width=0.8\linewidth]{BDiagram}
\end{center} \caption[Maximising Light-Matter Coupling between Lattice
\caption[Wannier Function Fourier Transforms]{The Wannier function Sites]{Light field arrangements which maximise coupling, $u_1^*u_0$,
products: (a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed between lattice sites. The thin black line indicates the trapping
line, left axis) and their (b) Fourier transforms potential (not to scale). (a) Arrangement for the uniform pattern
$\mathcal{F}[W_{0,1}]$. The Density $J_{i,i}$ and $J_{i,i+1} = J_1$. (b) Arrangement for spatially varying pattern
matter-interference $J_{i,i+1}$ coefficients in diffraction $J_{i,i+1}=(-1)^m J_2$; here $u_0=1$ so it is not shown and $u_1$
maximum (c) and minimum (d) as are shown as functions of standing is real thus $u_1^*u_0=u_1$. \label{fig:BDiagram}}
wave shifts $\varphi$ or, if one were to measure the quadrature
variance $(\Delta X^F_\beta)^2$, the local oscillator phase
$\beta$. The black points indicate the positions, where light
measures matter interference $\hat{B} \ne 0$, and the density-term
is suppressed, $\hat{D} = 0$. The trapping potential depth is
approximately 5 recoil energies.}
\label{fig:WannierFT}
\end{figure} \end{figure}
The simplest case is to find a diffraction maximum where $J_{i,i+1} = \begin{figure}[hbtp!]
J_B$. This can be achieved by crossing the light modes such that \centering
$\theta_0 = -\theta_1$ and $k_{0x} = k_{1x} = \pi/d$ and choosing the \includegraphics[width=\linewidth]{WF_S}
light mode phases such that $\varphi_+ = 0$. Fig. \ref{fig:WannierFT}c \caption[Wannier Function Products]{The Wannier function products:
shows the value of the $J_{i,j}$ coefficients under these (a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed line, left
circumstances. In order to make the $\hat{B}$ contribution to light axis) and their (b) Fourier transforms $\mathcal{F}[W_{0,1}]$. The
scattering dominant we need to set $\hat{D} = 0$ which from Density $J_{i,i}$ and matter-interference $J_{i,i+1}$ coefficients
Eq. \eqref{eq:FTs} we see is possible if $\varphi_0 = -\varphi_1 = in diffraction maximum (c) and minimum (d) as are shown as
\arccos[-\mathcal{F}[W_0](2\pi/d)/\mathcal{F}[W_0](0)]/2$. This functions of standing wave shifts $\varphi$ or, if one were to
arrangement of light modes maximizes the interference signal, measure the quadrature variance $(\Delta X^F_\beta)^2$, the local
oscillator phase $\beta$. The black points indicate the positions,
where light measures matter interference $\hat{B} \ne 0$, and the
density-term is suppressed, $\hat{D} = 0$. The trapping potential
depth is approximately 5 recoil energies.}
\label{fig:WannierProducts}
\end{figure}
The simplest case is to find a diffraction maximum where
$J_{i,i+1} = J_1$, where $J_1$ is a constant. This can be achieved by
crossing the light modes such that $\theta_0 = -\theta_1$ and
$k_{0x} = k_{1x} = \pi/d$ and choosing the light mode phases such that
$\varphi_+ = 0$. Fig. \ref{fig:BDiagram}a shows the resulting light
mode functions and their product along the lattice and
Fig. \ref{fig:WannierProducts}c shows the value of the $J_{i,j}$
coefficients under these circumstances. In order to make the $\hat{B}$
contribution to light scattering dominant we need to set $\hat{D} = 0$
which from Eq. \eqref{eq:FTs} we see is possible if
\begin{equation}
\xi \equiv \varphi_0 = -\varphi_1 =
\frac{1}{2}\arccos[-\mathcal{F}[W_0]\left(\frac{2\pi}{d}\right)/\mathcal{F}[W_0](0)].
\end{equation}
This arrangement of light modes maximizes the interference signal,
$\hat{B}$, by suppressing the density signal, $\hat{D}$, via $\hat{B}$, by suppressing the density signal, $\hat{D}$, via
interference compensating for the spreading of the Wannier interference compensating for the spreading of the Wannier functions.
functions.
Another possibility is to obtain an alternating pattern similar Another possibility is to obtain an alternating pattern similar
corresponding to a classical diffraction minimum. We consider an corresponding to a diffraction minimum. We consider an arrangement
arrangement where the beams are arranged such that $k_{0x} = 0$ and where the beams are arranged such that $k_{0x} = 0$ and
$k_{1x} = \pi/d$ which gives the following expressions for the density $k_{1x} = \pi/d$ which gives the following expressions for the density
and interference terms and interference terms
\begin{eqnarray} \begin{align}
\label{eq:DMin} \label{eq:DMin}
\hat{D} = \mathcal{F}[W_0](\pi/d) \sum_m (-1)^m \hat{n}_m \hat{D} = & \mathcal{F}[W_0](\pi/d) \sum_m (-1)^m \hat{n}_m
\cos(\varphi_0) \cos(\varphi_1) \nonumber \\ \hat{B} = \cos(\varphi_0) \cos(\varphi_1) \nonumber \\
-\mathcal{F}[W_1](\pi/d) \sum_m (-1)^m \hat{B}_m \hat{B} = & -\mathcal{F}[W_1](\pi/d) \sum_m (-1)^m \hat{B}_m
\cos(\varphi_0) \sin(\varphi_1). \cos(\varphi_0) \sin(\varphi_1).
\end{eqnarray} \end{align}
The corresponding $J_{i,j}$ coefficients are shown in The corresponding $J_{i,j}$ coefficients are given by
Fig. \ref{fig:WannierFT}d for $\varphi_0=0$. It is clear that for $J_{i,i+1} = -(-1)^i J_2$, where $J_2$ is a constant, and are shown in
$\varphi_1 = \pm \pi/2$, $\hat{D} = 0$, which is intuitive as this Fig. \ref{fig:WannierProducts}d for $\varphi_0=0$. The light mode
places the lattice sites at the nodes of the mode $u_1(x)$. This is a coupling along the lattice is shown in Fig. \ref{fig:BDiagram}b. It is
diffraction minimum as the light amplitude is also zero, $\langle clear that for $\varphi_1 = \pm \pi/2$, $\hat{D} = 0$, which is
\hat{B} \rangle = 0$, because contributions from alternating intuitive as this places the lattice sites at the nodes of the mode
inter-site regions interfere destructively. However, the intensity $u_1(x)$. This is a diffraction minimum as the light amplitude is also
$\langle \ad_1 \a \rangle = |C|^2 \langle \hat{B}^2 \rangle$ is zero, $\langle \hat{B} \rangle = 0$, because contributions from
proportional to the variance of $\hat{B}$ and is non-zero. alternating inter-site regions interfere destructively. However, the
intensity $\langle \ad_1 \a \rangle = |C|^2 \langle \hat{B}^2 \rangle$
is proportional to the variance of $\hat{B}$ and is non-zero.
\mynote{explain quadrature}
Alternatively, one can use the arrangement for a diffraction minimum Alternatively, one can use the arrangement for a diffraction minimum
described above, but use travelling instead of standing waves for the described above, but use travelling instead of standing waves for the
probe and detected beams and measure the light quadrature variance. In probe and detected beams and measure the light quadrature variance.
this case $\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B} In this case
\sin(\beta)$ and by varying the local oscillator phase, one can choose $\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B} \sin(\beta)$ and by
which conjugate operator to measure. varying the local oscillator phase, one can choose which conjugate
operator to measure.
\mynote{fix labels}
\begin{figure}[hbtp!]
\includegraphics[width=\linewidth]{BDiagram}
\caption[Maximising Light-Matter Coupling between Lattice
Sites]{Light field arrangements which maximise coupling,
$u_1^*u_0$, between lattice sites. The thin black line
indicates the trapping potential (not to scale). (a)
Arrangement for the uniform pattern $J_{m,m+1} = J_1$. (b)
Arrangement for spatially varying pattern $J_{m,m+1}=(-1)^m
J_2$; here $u_0=1$ so it is not shown and $u_1$ is real thus
$u_1^*u_0=u_1$. \label{fig:BDiagram}}
\end{figure}
\subsection{Electric Field Stength} \subsection{Electric Field Stength}
\label{sec:Efield}
The Electric field operator at position $\b{r}$ and at time $t$ is The Electric field operator at position $\b{r}$ and at time $t$ is
usually written in terms of its positive and negative components: usually written in terms of its positive and negative components:
@ -634,7 +671,7 @@ atom located at $\b{r}^\prime$ at the observation point $\b{r}$ is
given by given by
\begin{equation} \begin{equation}
\label{eq:Ep} \label{eq:Ep}
\b{\hat{E}}^{(+)}(\b{r},\b{r}^\prime,t) = \frac{\omega_a^2 d \sin \eta}{4 \pi \b{\hat{E}}^{(+)}(\b{r},\b{r}^\prime,t) = \frac{\omega_a^2 d_A \sin \eta}{4 \pi
\epsilon_0 c^2 |\b{r} - \b{r}^\prime|} \hat{\epsilon} \sigma^- \epsilon_0 c^2 |\b{r} - \b{r}^\prime|} \hat{\epsilon} \sigma^-
\left( \b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right), \left( \b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right),
\end{equation} \end{equation}
@ -644,10 +681,9 @@ in the far field, $\eta$ is the angle the dipole makes with
$\b{r} - \b{r}^\prime$, $\hat{\epsilon}$ is the polarization vector $\b{r} - \b{r}^\prime$, $\hat{\epsilon}$ is the polarization vector
which is perpendicular to $\b{r} - \b{r}^\prime$ and lies in the plane which is perpendicular to $\b{r} - \b{r}^\prime$ and lies in the plane
defined by $\b{r} - \b{r}^\prime$ and the dipole, $\omega_a$ is the defined by $\b{r} - \b{r}^\prime$ and the dipole, $\omega_a$ is the
atomic transition frequency, and $d$ is the dipole matrix element atomic transition frequency, and $d_A$ is the dipole matrix element
between the two levels, and $c$ is the speed of light in vacuum. between the two levels, and $c$ is the speed of light in vacuum.
We have already derived an expression for the atomic lowering We have already derived an expression for the atomic lowering
operator, $\sigma^-$, in Eq. \eqref{eq:sigmam} and it is given by operator, $\sigma^-$, in Eq. \eqref{eq:sigmam} and it is given by
\begin{equation} \begin{equation}

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@ -2,8 +2,8 @@
%*********************************** Third Chapter ***************************** %*********************************** Third Chapter *****************************
%******************************************************************************* %*******************************************************************************
\chapter{Probing Matter-Field and Atom-Number Correlations in Optical Lattices \chapter{Probing Correlations by Global
by Global Nondestructive Addressing} %Title of the Third Chapter Nondestructive Addressing} %Title of the Third Chapter
\ifpdf \ifpdf
\graphicspath{{Chapter3/Figs/Raster/}{Chapter3/Figs/PDF/}{Chapter3/Figs/}} \graphicspath{{Chapter3/Figs/Raster/}{Chapter3/Figs/PDF/}{Chapter3/Figs/}}
@ -13,3 +13,481 @@
%********************************** %First Section ************************************** %********************************** %First Section **************************************
\section{Introduction}
Having developed the basic theoretical framework within which we can
treat the fully quantum regime of light-matter interactions we now
consider possible applications. There are three prominent directions
in which we can apply our model: nondestructive probing, quantum
measurement backaction 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
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.
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}
As we have seen in section \ref{sec:a} under certain approximations
the scattered light mode, $\a_1$, is linked to the quantum state of
matter via
\begin{equation}
\label{eq:a-3}
\a_1 = C \hat{F} = C \left(\hat{D} + \hat{B} \right),
\end{equation}
where the atomic operators $\hat{D}$ and $\hat{B}$, given by
Eq. \eqref{eq:D} and Eq. \eqref{eq:B}, are responsible for the
coupling to on-site density and inter-site interference
respectively. It crucial to note that light couples to the bosons via
an operator as this makes it sensitive to the quantum state of the
matter.
Here, we will use this fact that the light is sensitive to the atomic
quantum state due to the coupling of the optical and matter fields via
operators in order to develop a method to probe the properties of an
ultracold gas. Therefore, we neglect the measurement back-action and
we will only consider expectation values of light observables. Since
the scheme is nondestructive (in some cases, it even satisfies the
stricter requirements for a QND measurement \cite{mekhov2007pra,
mekhov2012}) 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
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}
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,
\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.
\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}
\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.
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
lattice vector, and $2 \b{k}_1 = \b{G}$ for standing wave $\a_1$ and
travelling $\a_0$, which is clearly different from the classical Bragg
condition $\Delta \b{k} = \b{G}$. The peak height is tunable by the
local oscillator phase or standing wave shift as seen in Fig.
\ref{fig:Scattering}b.
In section \ref{sec:Efield} we have estimated the mean photon
scattering rates integrated over the solid angle for the only two
experiments so far on light diffraction from truly ultracold bosons
where the measurement object was light
\begin{equation}
n_{\Phi}= \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma K}{8}
(\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2).
\end{equation}
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.
\subsection{Mapping the quantum phase diagram}
\begin{figure}[htbp!]
\centering
\includegraphics[width=\linewidth]{oph11}
\caption[Mapping the Bose-Hubbard Phase Diagram]{(a) The angular
dependence of scattered light $R$ for a superfluid (thin black,
left scale, $U/2J^\text{cl} = 0$) and Mott insulator (thick green,
right scale, $U/2J^\text{cl} =10$). The two phases differ in both
their value of $R_\text{max}$ as well as $W_R$ showing that
density correlations in the two phases differ in magnitude as well
as extent. Light scattering maximum $R_\text{max}$ is shown in (b,
d) and the width $W_R$ in (c, e). It is very clear that varying
chemical potential $\mu$ or density $\langle n\rangle$ sharply
identifies the superfluid-Mott insulator transition in both
quantities. (b) and (c) are cross-sections of the phase diagrams
(d) and (e) at $U/2J^\text{cl}=2$ (thick blue), 3 (thin purple),
and 4 (dashed blue). Insets show density dependencies for the
$U/(2 J^\text{cl}) = 3$ line. $K=M=N=25$.}
\label{fig:SFMI}
\end{figure}
We have shown how in mean-field, we can track the order parameter,
$\Phi$, by probing the matter-field interference using the coupling of
light to the $\hat{B}$ operator. In this case, it is very easy to
follow the superfluid to Mott insulator quantum phase transition since
we have direct access to the order parameter which goes to zero in the
insulating phase. In fact, if we're only interested in the critical
point, we only need access to any quantity that yields information
about density fluctuations which also go to zero in the MI phase and
this can be obtained by measuring
$\langle \hat{D}^\dagger \hat{D} \rangle$. However, there are many
situations where the mean-field approximation is not a valid
description of the physics. A prominent example is the Bose-Hubbard
model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, pino2012,
pino2013}. Observing the transition in 1D by light at fixed density
was considered to be difficult \cite{rogers2014} or even impossible
\cite{roth2003}. By contrast, here we propose varying the density or
chemical potential, which sharply identifies the transition. We
perform these calculations numerically by calculating the ground state
using DMRG methods \cite{tnt} from which we can compute all the
necessary atomic observables. Experiments typically use an additional
harmonic confining potential on top of the optical lattice to keep the
atoms in place which means that the chemical potential will vary in
space. However, with careful consideration of the full
($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in
Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the
system \cite{batrouni2002}.
The 1D phase transition is best understood in terms of two-point
correlations as a function of their separation \cite{giamarchi}. In
the Mott insulating phase, the two-point correlations
$\langle \bd_i b_j \rangle$ and
$\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$
($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay
exponentially with $|i-j|$. On the other hand the superfluid will
exhibit long-range order which in dimensions higher than one,
manifests itself with an infinite correlation length. However, in 1D
only pseudo long-range order happens and both the matter-field and
density fluctuation correlations decay algebraically \cite{giamarchi}.
The method we propose gives us direct access to the structure factor,
which is a function of the two-point correlation $\langle \delta
\hat{n}_i \delta \hat{n}_j \rangle$, by measuring the light
intensity. For two travelling waves maximally coupled to the density
(atoms are at light intensity maxima so $\hat{F} = \hat{D}$), the
quantum addition is given by
\begin{equation}
R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0)
(\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta
\hat{n}_j \rangle,
\end{equation}
The angular dependence of $R$ for a Mott insulator and a superfluid is
shown in Fig. \ref{fig:SFMI}a, and there are two variables
distinguishing the states. Firstly, maximal $R$,
$R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$,
probes the fluctuations and compressibility $\kappa'$
($\langle \delta \hat{n}^2_i \rangle \propto \kappa' \langle \hat{n}_i
\rangle$). The Mott insulator is incompressible and thus will have
very small on-site fluctuations and it will scatter little light
leading to a small $R_\text{max}$. The deeper the system is in the MI
phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the smaller
these values will be until ultimately it will scatter no light at all
in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a this can
be seen in the value of the peak in $R$. The value $R_\text{max}$ in
the SF phase ($U/2J^\text{cl} = 0$) is larger than its value in the MI
phase ($U/2J^\text{cl} = 10$) by a factor of
$\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of
$R_\text{max}$ changes across the phase transition. We see that the
transition shows up very sharply as $\mu$ is varied.
Secondly, being a Fourier transform, the width $W_R$ of the dip in $R$
is a direct measure of the correlation length $l$, $W_R \propto
1/l$. The Mott insulator being an insulating phase is characterised by
exponentially decaying correlations and as such it will have a very
large $W_R$. However, the superfluid in 1D exhibits pseudo long-range
order which manifests itself in algebraically decaying two-point
correlations \cite{giamarchi} which significantly reduces the dip in
the $R$. This can be seen in Fig. \ref{fig:SFMI}a and we can also see
that this identifies the phase transition very sharply as $\mu$ is
varied in Figs. \ref{fig:SFMI}(c,e). One possible concern with
experimentally measuring $W_R$ is that it might be obstructed by the
classical diffraction maxima which appear at angles corresponding to
the minima in $R$. However, the width of such a peak is much smaller
as its width is proportional to $1/M$.
It is also possible to analyse the phase transition quantitatively
using our method. Unlike in higher dimensions where an order parameter
can be easily defined within the MF approximation there is no such
quantity in 1D. However, a valid description of the relevant 1D low
energy physics is provided by Luttinger liquid theory
\cite{giamarchi}. In this model correlations in the supefluid phase as
well as the superfluid density itself are characterised by the
Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies
the phase transition in the thermodynamic limit at $K_b = 1/2$. This
quantity can be extracted from various correlation functions and in
our case it can be extracted directly from $R$ \cite{ejima2011}. By
extracting this parameter from $R$ for various lattice lengths from
numerical DMRG calculations it was even possible to give a theoretical
estimate of the critical point for commensurate filling, $N = M$, in
the thermodynamic limit to occur at $U/2J^\text{cl} \approx 1.64$
\cite{ejima2011}. Our proposal provides a method to directly measure
$R$ in a lab which can then be used to experimentally determine the
location of the critical point in 1D.
So far both variables we considered, $R_\text{max}$ and $W_R$, provide
similar information. Next, we present a case where it is very
different. The Bose glass is a localized insulating phase with
exponentially decaying correlations but large compressibility and
on-site fluctuations in a disordered optical lattice. Therefore,
measuring both $R_\text{max}$ and $W_R$ will distinguish all the
phases. In a Bose glass we have finite compressibility, but
exponentially decaying correlations. This gives a large $R_\text{max}$
and a large $W_R$. A Mott insulator will also have exponentially
decaying correlations since it is an insulator, but it will be
incompressible. Thus, it will scatter light with a small
$R_\text{max}$ and large $W_R$. Finally, a superfluid will have long
range correlations and large compressibility which results in a large
$R_\text{max}$ and a small $W_R$.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\linewidth]{oph22}
\caption[Mapping the Disoredered Phase Diagram]{The
Mott-superfluid-glass phase diagrams for light scattering maximum
$R_\text{max}/N_K$ (a) and width $W_R$ (b). Measurement of both
quantities distinguish all three phases. Transition lines are
shifted due to finite size effects \cite{roux2008}, but it is
possible to apply well known numerical methods to extract these
transition lines from such experimental data extracted from $R$
\cite{ejima2011}. $K=M=N=35$.}
\label{fig:BG}
\end{figure}
We confirm this in Fig. \ref{fig:BG} for simulations with the ratio of
superlattice- to trapping lattice-period $r\approx 0.77$ for various
disorder strengths $V$ \cite{roux2008}. Here, we only consider
calculations for a fixed density, because the usual interpretation of
the phase diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane
for a fixed ratio $V/U$ becomes complicated due to the presence of
multiple compressible and incompressible phases between successive MI
lobes \cite{roux2008}. This way, we have limited our parameter space
to the three phases we are interested in: superfluid, Mott insulator,
and Bose glass. From Fig. \ref{fig:BG} we see that all three phases
can indeed be distinguished. In the 1D BHM there is no sharp MI-SF
phase transition in 1D at a fixed density \cite{cazalilla2011,
ejima2011, kuhner2000, pino2012, pino2013} just like in
Figs. \ref{fig:SFMI}(d,e) if we follow the transition through the tip
of the lobe which corresponds to a line of unit density. However,
despite the lack of an easily distinguishable critical point it is
possible to quantitatively extract the location of the transition
lines by extracting the Tomonaga-Luttinger parameter from the
scattered light, $R$, in the same way it was done for an unperturbed
BHM \cite{ejima2011}.
Only recently \cite{derrico2014} a Bose glass phase was studied by
combined measurements of coherence, transport, and excitation spectra,
all of which are destructive techniques. Our method is simpler as it
only requires measurement of the quantity $R$ and additionally, it is
nondestructive.
\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}.

2
Preamble/preamble.tex
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@ -92,7 +92,7 @@
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% \onehalfspacing % \onehalfspacing
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661
References/references.bib
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@ -1,6 +1,258 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Books, theses, reference material
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@book{foot,
author = {Foot, C. J.},
title = {{Atomic Physics}},
publisher = {Oxford University Press},
year = {2005}
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title = {{Quantum Physics in One Dimension}},
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year = {2011}
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@misc{tnt,
howpublished="\url{http://ccpforge.cse.rl.ac.uk/gf/project/tntlibrary/}"
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Igor's original papers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@article{mekhov2007prl,
title={Cavity-enhanced light scattering in optical lattices to probe
atomic quantum statistics},
author={Mekhov, Igor B and Maschler, Christoph and Ritsch, Helmut},
journal={Physical review letters},
volume={98},
number={10},
pages={100402},
year={2007},
publisher={APS}
}
@article{mekhov2007pra,
title={Light scattering from ultracold atoms in optical lattices as
an optical probe of quantum statistics},
author={Mekhov, Igor B and Maschler, Christoph and Ritsch, Helmut},
journal={Physical Review A},
volume={76},
number={5},
pages={053618},
year={2007},
publisher={APS}
}
@article{mekhov2008,
title = {Dicke quantum phase transition with a superfluid gas in an
optical cavity},
author = {Maschler, C. and Mekhov, I. B. and Ritsch, H.},
journal = {Eur. Phys. J. D},
volume = {146},
pages = {545},
year = {2008},
}
@article{mekhov2009prl,
author = {Mekhov, Igor B and Ritsch, Helmut},
doi = {10.1103/PhysRevLett.102.020403},
journal = {Phys. Rev. Lett.},
month = jan,
number = {2},
pages = {020403},
title = {{Quantum Nondemolition Measurements and State Preparation
in Quantum Gases by Light Detection}},
volume = {102},
year = {2009}
}
@article{mekhov2009pra,
author = {Mekhov, Igor B and Ritsch, Helmut},
doi = {10.1103/PhysRevA.80.013604},
journal = {Phys. Rev. A},
month = jul,
number = {1},
pages = {013604},
title = {{Quantum optics with quantum gases: Controlled state
reduction by designed light scattering}},
volume = {80},
year = {2009}
}
@article{LP2009,
title={Quantum optics with quantum gases},
author={Mekhov, Igor B and Ritsch, Helmut},
journal={Laser physics},
volume={19},
number={4},
pages={610--615},
year={2009},
publisher={Springer}
}
@article{LP2010,
author = {Mekhov, Igor B and Ritsch, Helmut},
doi = {10.1134/S1054660X10050105},
journal = {Laser Phys.},
volume = {20},
pages = {694},
title = {Quantum Optical Measurements in Ultracold Gases:
Macroscopic Bose–Einstein Condensates},
year = {2010}
}
@article{LP2011,
author = {Mekhov, Igor B and Ritsch, Helmut},
doi = {10.1134/S1054660X11150163},
journal = {Laser Phys.},
volume = {21},
pages = {1486},
title = {Atom State Evolution and Collapse in Ultracold Gases during
Light Scattering into a Cavity},
year = {2011}
}
@article{LP2013,
author={Igor B Mekhov},
title={Quantum non-demolition detection of polar molecule complexes:
dimers, trimers, tetramers},
journal={Laser Phys.},
volume={23},
number={1},
pages={015501},
year={2013},
}
@article{mekhov2012,
title={Quantum optics with ultracold quantum gases: towards the full
quantum regime of the light--matter interaction},
author={Mekhov, Igor B and Ritsch, Helmut},
journal={Journal of Physics B: Atomic, Molecular and Optical
Physics},
volume={45},
number={10},
pages={102001},
year={2012},
publisher={IOP Publishing}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Group papers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@article{elliott2015,
author = {Elliott, T. J. and Kozlowski, W. and Caballero-Benitez,
S. F. and Mekhov, I. B.},
doi = {10.1103/PhysRevLett.114.113604},
journal = {Phys. Rev. Lett.},
pages = {113604},
title = {{Multipartite Entangled Spatial Modes of Ultracold Atoms
Generated and Controlled by Quantum Measurement}},
volume = {114},
year = {2015}
}
@article{atoms2015,
title={Probing and Manipulating Fermionic and Bosonic Quantum Gases
with Quantum Light},
author={Elliott, Thomas J and Mazzucchi, Gabriel and Kozlowski,
Wojciech and Caballero-Benitez, Santiago F and
Mekhov, Igor B},
journal={Atoms},
volume={3},
number={3},
pages={392--406},
year={2015},
publisher={Multidisciplinary Digital Publishing Institute}
}
@article{mazzucchi2016,
title = {Quantum measurement-induced dynamics of many-body ultracold
bosonic and fermionic systems in optical lattices},
author = {Mazzucchi, Gabriel and Kozlowski, Wojciech and
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J. and Mekhov, Igor B.},
journal = {Physical Review A},
volume = {93},
issue = {2},
pages = {023632},
numpages = {12},
year = {2016},
month = {Feb},
publisher = {American Physical Society}
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@article{kozlowski2016zeno,
title={Non-hermitian dynamics in the quantum zeno limit},
author={Kozlowski, Wojciech and Caballero-Benitez, Santiago F and Mekhov, Igor B},
journal={arXiv preprint arXiv:1510.04857},
year={2015}
}
@article{mazzucchi2016af,
title={Quantum measurement-induced antiferromagnetic order and
density modulations in ultracold Fermi gases in
optical lattices},
author={Mazzucchi, Gabriel and Caballero-Benitez, Santiago F and
Mekhov, Igor B},
journal={arXiv preprint arXiv:1510.04883},
year={2015}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Other papers
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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author = {Foot, C. J.},
title = {{Atomic Physics}},
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year = {2005}
}
@phdthesis{weitenbergThesis,
author = {Weitenberg, Christof},
number = {April},
title = {{Single-Atom Resolved Imaging and Manipulation in an Atomic Mott Insulator}},
year = {2011}
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title = {Quantum measurement-induced dynamics of many-body ultracold bosonic and fermionic systems in optical lattices}, title = {Mott-Insulator States of Ultracold Atoms in Optical
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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,draft]{Classes/PhDThesisPSnPDF} \documentclass[a4paper,12pt,times,numbered,print,index]{Classes/PhDThesisPSnPDF}
% ****************************************************************************** % ******************************************************************************
% ******************************* Class Options ******************************** % ******************************* Class Options ********************************