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Finished chapter 3

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Wojciech Kozlowski 2016-07-18 18:59:07 +01:00
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Chapter2/chapter2.tex
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@ -71,11 +71,15 @@ quantum potential in contrast to the classical lattice trap.
\centering \centering
\includegraphics[width=1.0\textwidth]{LatticeDiagram} \includegraphics[width=1.0\textwidth]{LatticeDiagram}
\caption[Experimental Setup]{Atoms (green) trapped in an optical \caption[Experimental Setup]{Atoms (green) trapped in an optical
lattice are illuminated by a coherent probe beam (red). The light lattice are illuminated by a coherent probe beam (red), $a_0$,
scatters (blue) in free space or into a cavity and is measured by with a mode function $u_0(\b{r})$ which is at an angle $\theta_0$
a detector. If the experiment is in free space light can scatter to the normal to the lattice. The light scatters (blue) into the
in any direction. A cavity on the other hand enhances scattering mode $\a_1$ in free space or into a cavity and is measured by a
in one particular direction.} detector. Its mode function is given by $u_1(\b{r})$ and it is at
an angle $\theta_1$ relative to the normal to the lattice. If the
experiment is in free space light can scatter in any direction. A
cavity on the other hand enhances scattering in one particular
direction.}
\label{fig:LatticeDiagram} \label{fig:LatticeDiagram}
\end{figure} \end{figure}
@ -411,13 +415,15 @@ adiabatically follows the quantum state of matter.
The above equation is quite general as it includes an arbitrary number The above equation is quite general as it includes an arbitrary number
of light modes which can be pumped directly into the cavity or of light modes which can be pumped directly into the cavity or
produced via scattering from other modes. To simplify the equation produced via scattering from other modes. To simplify the equation
slightly we will neglect the cavity resonancy shift, $U_{l,l} slightly we will neglect the cavity resonancy shift,
\hat{F}_{l,l}$ which is possible provided the cavity decay rate and/or $U_{l,l} \hat{F}_{l,l}$ which is possible provided the cavity decay
probe detuning are large enough. We will also only consider probing rate and/or probe detuning are large enough. We will also only
with an external coherent beam, $a_0$, and thus we neglect any cavity consider probing with an external coherent beam, $a_0$ with mode
pumping $\eta_l$. We also limit ourselves to only a single scattered function $u_0(\b{r})$, and thus we neglect any cavity pumping
mode, $a_1$. This leads to a simple linear relationship between the $\eta_l$. We also limit ourselves to only a single scattered mode,
light mode and the atomic operator $\hat{F}_{1,0}$ $a_1$ with a mode function $u_1(\b{r})$. This leads to a simple linear
relationship between the light mode and the atomic operator
$\hat{F}_{1,0}$
\begin{equation} \begin{equation}
\label{eq:a} \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} =
@ -482,7 +488,16 @@ that one can couple to the interference term between two condensates
\cite{cirac1996, castin1997, ruostekoski1997, ruostekoski1998, \cite{cirac1996, castin1997, ruostekoski1997, ruostekoski1998,
rist2012}. Such measurements establish a relative phase between the rist2012}. Such measurements establish a relative phase between the
condensates even though the two components have initially well-defined 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 a lattice
geometry, one would ideally measure between two sites similarly to
single-site addressing \cite{greiner2009, bloch2011}, which would
measure a single term $\langle \bd_i b_{i+1}+b_i
\bd_{i+1}\rangle$. This could be achieved, for example, by superposing
a deeper optical lattice shifted by $d/2$ with respect to the original
one, catching and measuring the atoms in the new lattice
sites. However, a single-shot success rate of atom detection will be
small and as single-site addressing is challenging, we proceed with
our global scattering scheme.
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 component, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a phase a density component, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a phase
@ -494,7 +509,9 @@ 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. Such a counter-intuitive configuration may affect works on sources. Furthermore, it is not limited to a double-wellsetup and
naturally extends to a lattice structure which is a key
advantage. Such a counter-intuitive configuration may affect works on
quantum gases trapped in quantum potentials \cite{mekhov2012, quantum gases trapped in quantum potentials \cite{mekhov2012,
mekhov2008, larson2008, chen2009, habibian2013, ivanov2014, mekhov2008, larson2008, chen2009, habibian2013, ivanov2014,
caballero2015} and quantum measurement-induced preparation of caballero2015} and quantum measurement-induced preparation of
@ -524,24 +541,18 @@ Wannier function at a single site, $W_0(x) \equiv w^2(x)$. Therefore,
in order to enhance the $\hat{B}$ term we need to maximise the overlap in order to enhance the $\hat{B}$ term we need to maximise the overlap
between the light modes and the nearest neighbour Wannier overlap, between the light modes and the nearest neighbour Wannier overlap,
$W_1(x)$. This can be achieved by concentrating the light between the $W_1(x)$. This can be achieved by concentrating the light between the
sites rather than at the positions of the atoms. Ideally, one could sites rather than at the positions of the atoms.
measure between two sites similarly to single-site addressing
\cite{greiner2009, bloch2011}, which would measure a single term \mynote{Potentially expand details of the derivation of these
$\langle \bd_i b_{i+1}+b_i \bd_{i+1}\rangle$. This could be achieved, equations}
for example, by superposing a deeper optical lattice shifted by $d/2$
with respect to the original one, catching and measuring the atoms in
the new lattice sites. A single-shot success rate of atom detection
will be small. As single-site addressing is challenging, we proceed
with the global scattering.
\mynote{Potentially expand details of the derivation of these equations}
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 calculations using realistic Wannier functions
\cite{walters2013}. However, it is possible to gain some analytic \cite{walters2013}. However, it is possible to gain some analytic
insight into the behaviour of these values by looking at the Fourier insight into the behaviour of these values by looking at the Fourier
transforms of the Wannier function overlaps, transforms of the Wannier function overlaps,
$\mathcal{F}[W_{0,1}](k)$, shown in Fig. $\mathcal{F}[W_{0,1}](k)$, shown in Fig.
\ref{fig:WannierProducts}b. This is because the for plane and standing \ref{fig:WannierProducts}b. This is because for plane and standing
wave light modes the product $u_1^*(x) u_0(x)$ can be in general wave light modes the product $u_1^*(x) u_0(x)$ can be in general
decomposed into a sum of oscillating exponentials of the form decomposed into a sum of oscillating exponentials of the form
$e^{i k x}$ making the integral in Eq. \eqref{eq:Jcoeff} a sum of $e^{i k x}$ making the integral in Eq. \eqref{eq:Jcoeff} a sum of
@ -598,46 +609,59 @@ allowing to decouple them at specific angles.
\end{figure} \end{figure}
The simplest case is to find a diffraction maximum where 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 $J_{i,i+1} = J^B_\mathrm{max}$, where $J^B_\mathrm{max}$ is a
crossing the light modes such that $\theta_0 = -\theta_1$ and constant. This results in a diffraction maximum where each bond
$k_{0x} = k_{1x} = \pi/d$ and choosing the light mode phases such that (inter-site term) scatters light in phase and the operator is given by
$\varphi_+ = 0$. Fig. \ref{fig:BDiagram}a shows the resulting light \begin{equation}
mode functions and their product along the lattice and \hat{B} = J^B_\mathrm{max} \sum_m^K \hat{B}_m .
Fig. \ref{fig:WannierProducts}c shows the value of the $J_{i,j}$ \end{equation}
coefficients under these circumstances. In order to make the $\hat{B}$ This can be achieved by crossing the light modes such that
contribution to light scattering dominant we need to set $\hat{D} = 0$ $\theta_0 = -\theta_1$ and $k_{0x} = k_{1x} = \pi/d$ and choosing the
which from Eq. \eqref{eq:FTs} we see is possible if 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} \begin{equation}
\xi \equiv \varphi_0 = -\varphi_1 = \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)]. \frac{1}{2}\arccos\left[\frac{-\mathcal{F}[W_0](2\pi/d)}{\mathcal{F}[W_0](0)}\right].
\end{equation} \end{equation}
This arrangement of light modes maximizes the interference signal, Under these conditions, the coefficient $J^B_\mathrm{max}$ is simply
given by $J^B_\mathrm{max} = \mathcal{F}[W_1](2 \pi / d)$. 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 functions. interference compensating for the spreading of the Wannier functions.
Another possibility is to obtain an alternating pattern similar Another possibility is to obtain an alternating pattern similar
corresponding to a diffraction minimum. We consider an arrangement corresponding to a diffraction minimum where each bond scatters light
in anti-phase with its neighbours giving
\begin{equation}
\hat{B} = J^B_\mathrm{min} \sum_m^K (-1)^m \hat{B}_m,
\end{equation}
where $J^B_\mathrm{min}$ is a constant. We consider an 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{align} \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]\left(\frac{\pi}{d}\right) \sum_m (-1)^m \hat{n}_m
\cos(\varphi_0) \cos(\varphi_1) \nonumber \\ \cos(\varphi_0) \cos(\varphi_1) \nonumber \\
\hat{B} = & -\mathcal{F}[W_1](\pi/d) \sum_m (-1)^m \hat{B}_m \hat{B} = & -\mathcal{F}[W_1]\left(\frac{\pi}{d}\right) \sum_m (-1)^m \hat{B}_m
\cos(\varphi_0) \sin(\varphi_1). \cos(\varphi_0) \sin(\varphi_1).
\end{align} \end{align}
The corresponding $J_{i,j}$ coefficients are given by For $\varphi_0 = 0$ the corresponding $J_{i,j}$ coefficients are given
$J_{i,i+1} = -(-1)^i J_2$, where $J_2$ is a constant, and are shown in by $J_{i,i+1} = (-1)^i J^B_\mathrm{min}$, where
Fig. \ref{fig:WannierProducts}d for $\varphi_0=0$. The light mode $J^B_\mathrm{min} = -\mathcal{F}[W_1](\pi / d)$, and are shown in
coupling along the lattice is shown in Fig. \ref{fig:BDiagram}b. It is Fig. \ref{fig:WannierProducts}d. The light mode coupling along the
clear that for $\varphi_1 = \pm \pi/2$, $\hat{D} = 0$, which is lattice is shown in Fig. \ref{fig:BDiagram}b. It is clear that for
intuitive as this places the lattice sites at the nodes of the mode $\varphi_1 = \pm \pi/2$, $\hat{D} = 0$, which is intuitive as this
$u_1(x)$. This is a diffraction minimum as the light amplitude is also places the lattice sites at the nodes of the mode $u_1(x)$. This is a
zero, $\langle \hat{B} \rangle = 0$, because contributions from diffraction minimum as the light amplitude is also zero,
alternating inter-site regions interfere destructively. However, the $\langle \hat{B} \rangle = 0$, because contributions from alternating
intensity $\langle \ad_1 \a \rangle = |C|^2 \langle \hat{B}^2 \rangle$ inter-site regions interfere destructively. However, the intensity
is proportional to the variance of $\hat{B}$ and is non-zero. $\langle \ad_1 \a \rangle = |C|^2 \langle \hat{B}^2 \rangle$ is
proportional to the variance of $\hat{B}$ and is non-zero.
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

514
Chapter3/chapter3.tex
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@ -25,11 +25,11 @@ optical lattices. Here, we deal with the first of the three options.
In this chapter we develop a method to measure properties of ultracold In this chapter we develop a method to measure properties of ultracold
gases in optical lattices by light scattering. In the previous chapter gases in optical lattices by light scattering. In the previous chapter
we have shown that quantum light field couples to the bosons via the we have shown that the quantum light field couples to the bosons via
operator $\hat{F}$. This is the key element of the scheme we propose the operator $\hat{F}$. This is the key element of the scheme we
as this makes it sensitive to the quantum state of the matter and all propose as this makes it sensitive to the quantum state of the matter
of its possible superpositions which will be reflected in the quantum and all of its possible superpositions which will be reflected in the
state of the light itself. We have also shown in section quantum state of the light itself. We have also shown in section
\ref{sec:derivation} that this coupling consists of two parts, a \ref{sec:derivation} that this coupling consists of two parts, a
density component $\hat{D}$ given by Eq. \eqref{eq:D}, and a phase density component $\hat{D}$ given by Eq. \eqref{eq:D}, and a phase
component $\hat{B}$ given by Eq. \eqref{eq:B}. Therefore, when probing component $\hat{B}$ given by Eq. \eqref{eq:B}. Therefore, when probing
@ -62,7 +62,7 @@ fixed-density scattering. It was only recently that an experiment
distinguished a Mott insulator from a Bose glass \cite{derrico2014} distinguished a Mott insulator from a Bose glass \cite{derrico2014}
via a series of destructive measurements. Our proposal, on the other via a series of destructive measurements. Our proposal, on the other
hand, is nondestructive and is capable of extracting all the relevant hand, is nondestructive and is capable of extracting all the relevant
information in a single experiment. information in a single experiment making our proposal timely.
Having shown the possibilities created by this nondestructive Having shown the possibilities created by this nondestructive
measurement scheme we move on to considering light scattering from the measurement scheme we move on to considering light scattering from the
@ -155,9 +155,9 @@ which we will call the ``quantum addition'' to light scattering. By
construction $R$ is simply the full light intensity minus the construction $R$ is simply the full light intensity minus the
classical field diffraction. In order to justify its name we will show classical field diffraction. In order to justify its name we will show
that this quantity depends purely quantum mechanical properties of the that this quantity depends purely quantum mechanical properties of the
ultracold gase. We will substitute $\a_1 = C \hat{D}$ using ultracold gas. We substitute $\a_1 = C \hat{D}$ using
Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and
we will make use of the shorthand notation we make use of the shorthand notation
$A_i = u_1^*(\b{r}_i) u_0(\b{r}_i)$. The result is $A_i = u_1^*(\b{r}_i) u_0(\b{r}_i)$. The result is
\begin{equation} \begin{equation}
\label{eq:Rfluc} \label{eq:Rfluc}
@ -171,25 +171,27 @@ quantum mechanical property of a system. Therefore, $R$, the ``quantum
addition'' faithfully represents the new contribution from the quantum addition'' faithfully represents the new contribution from the quantum
light-matter interaction to the diffraction pattern. light-matter interaction to the diffraction pattern.
If instead we are interested in quantities linear in $\hat{D}$, we can Another interesting quantity to measure are the quadratures of the
measure the quadrature of the light fields which in section light fields which we have seen in section \ref{sec:a} are related to
\ref{sec:a} we saw that $\hat{X}_\phi = |C| \hat{X}^F_\beta$. For the the quadrature of $\hat{F}$ by $\hat{X}_\phi = |C|
case when both the scattered mode and probe are travelling waves the \hat{X}^F_\beta$. An interesting feature of quadratures is that the
quadrature coupling strength at different sites can be tuned using the local
oscillator phase $\beta$. To see this we consider the case when both
the scattered mode and probe are travelling waves the quadrature
\begin{equation} \begin{equation}
\label{eq:Xtrav} \label{eq:Xtrav}
\hat{X}^F_\beta = \frac{1}{2} \left( \hat{F} e^{-i \beta} + \hat{X}^F_\beta = \frac{1}{2} \left( \hat{F} e^{-i \beta} +
\hat{F}^\dagger e^{i \beta} \right) = \sum_i^K \hat{n}_i\cos[(\b{k}_0 - \b{k}_1) \cdot \hat{F}^\dagger e^{i \beta} \right) = \sum_i^K \hat{n}_i\cos[(\b{k}_0 - \b{k}_1) \cdot
\b{r}_i + (\phi_0 - \phi_1) - \beta]. \b{r}_i + (\phi_0 - \phi_1) - \beta].
\end{equation} \end{equation}
Note that different light quadratures are differently coupled to the Different light quadratures are differently coupled to the atom
atom distribution, hence by varying the local oscillator phase, and distribution, hence by varying the local oscillator phase, $\beta$,
thus effectively $\beta$, and/or the detection angle one can scan the and/or the detection angle one can scan the whole range of
whole range of couplings. A similar expression exists for $\hat{D}$ couplings. This is similar to the case for $\hat{D}$ for a standing
for a standing wave probe, where instead of varying $\beta$ scanning wave probe, where instead of varying $\beta$ scanning is achieved by
is achieved by varying the position of the wave with respect to varying the position of the wave with respect to atoms. Additionally,
atoms. Additionally, the quadrature variance, $(\Delta X^F_\beta)^2$, the quadrature variance, $(\Delta X^F_\beta)^2$, will have a similar
will have a similar form to $R$ given in Eq. \eqref{eq:Rfluc}, form to $R$ given in Eq. \eqref{eq:Rfluc},
\begin{equation} \begin{equation}
(\Delta X^F_\beta)^2 = |C|^2 \sum_{i.j}^K A_i^\beta A_j^\beta (\Delta X^F_\beta)^2 = |C|^2 \sum_{i.j}^K A_i^\beta A_j^\beta
\langle \dn_i \dn_j \rangle, \langle \dn_i \dn_j \rangle,
@ -202,18 +204,18 @@ quantity.
The ``quantum addition'', $R$, and the quadrature variance, The ``quantum addition'', $R$, and the quadrature variance,
$(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$ and both rely $(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$ and both rely
heavily on the quantum state of the matter. Therefore, they will have heavily on the quantum state of the matter. Therefore, they will have
a nontrivial angular dependence, showing more peaks than classical a nontrivial angular dependence showing more peaks than classical
diffraction. Furthermore, these peaks can be tuned very easily with diffraction. Furthermore, these peaks can be tuned very easily with
$\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular $\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular
dependence of $R$ for the case when the scattered mode is a standing dependence of $R$ for the case when the probe is a travelling wave
wave and the probe is a travelling wave scattering from an ideal scattering from an ideal superfluid in a 3D optical lattice into a
superfluid in a 3D optical lattice. The first noticeable feature is standing wave scattered mode. The first noticeable feature is the
the isotropic background which does not exist in classical isotropic background which does not exist in classical
diffraction. This background yields information about density diffraction. This background yields information about density
fluctuations which, according to mean-field estimates (i.e.~inter-site fluctuations which, according to mean-field estimates (i.e.~inter-site
correlations are ignored), are related by correlations are ignored), are related by
$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. In $R = |C|^2 K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2
Fig. \ref{fig:scattering} we can see a significant signal of )/2$. In Fig. \ref{fig:scattering} we can see a significant signal of
$R = |C|^2 N_K/2$, because it shows scattering from an ideal $R = |C|^2 N_K/2$, because it shows scattering from an ideal
superfluid which has significant density fluctuations with superfluid which has significant density fluctuations with
correlations of infinte range. However, as the parameters of the correlations of infinte range. However, as the parameters of the
@ -246,8 +248,8 @@ actually is no classical diffraction on top of the ``quantum
addition'' shown here. Therefore, these features would be easy to see addition'' shown here. Therefore, these features would be easy to see
in an experiment as they wouldn't be masked by a stronger classical in an experiment as they wouldn't be masked by a stronger classical
signal. This difference in behaviour is due to the fact that signal. This difference in behaviour is due to the fact that
classical diffraction is ignorant of any quantum correlations. This classical diffraction is ignorant of any quantum correlations as it is
signal is given by the square of the light field amplitude squared given by the square of the light field amplitude squared
\begin{equation} \begin{equation}
|\langle \a_1 \rangle|^2 = |C|^2 \sum_{i,j} A_i^* |\langle \a_1 \rangle|^2 = |C|^2 \sum_{i,j} A_i^*
A_j \langle \n_i \rangle \langle \n_j \rangle, A_j \langle \n_i \rangle \langle \n_j \rangle,
@ -263,10 +265,11 @@ correlations
Therefore, we see that in the fully quantum picture light scattering Therefore, we see that in the fully quantum picture light scattering
not only depends on the diffraction structure due to the distribution not only depends on the diffraction structure due to the distribution
of atoms in the lattice, but also on the quantum correlations between of atoms in the lattice, but also on the quantum correlations between
different lattice sites which will be dependent on the quantum state different lattice sites which will in turn be dependent on the quantum
of the matter. These correlations are imprinted in $R$ as shown in state of the matter. These correlations are imprinted in $R$ as shown
Eq. \eqref{eq:Rfluc} and it highlights the key feature of our model, in Eq. \eqref{eq:Rfluc} and it highlights the key feature of our
i.e.~the light couples to the quantum state directly via operators. model, i.e.~the light couples to the quantum state directly via
operators.
We can even derive the generalised Bragg conditions for the peaks that We can even derive the generalised Bragg conditions for the peaks that
we can see in Fig. \ref{fig:scattering}. The exact conditions under we can see in Fig. \ref{fig:scattering}. The exact conditions under
@ -278,22 +281,17 @@ not the lattice itself as seen in Eq. \eqref{eq:Rfluc}. For classical
light it is straightforward to develop an intuitive physical picture light it is straightforward to develop an intuitive physical picture
to find the Bragg condition by considering angles at which the to find the Bragg condition by considering angles at which the
distance travelled by light scattered from different points in the distance travelled by light scattered from different points in the
lattice is equal to an integer multiple of wavelength. The ``quantum lattice is equal to an integer multiple of the wavelength. The
addition'' is more complicated and less intuitive as we now have to ``quantum addition'' is more complicated and less intuitive as we now
consider quantum correlations which are not only nonlocal, but can have to consider quantum correlations which are not only nonlocal, but
also be nagative. can also be negative.
We will consider scattering from a superfluid, because the Mott We will consider scattering from a superfluid, because the Mott
insulator has no ``quantum addition'' due to a lack of density insulator has no ``quantum addition'' due to a lack of density
fluctuations. The wavefunction of a superfluid on a lattice is given fluctuations. The wavefunction of a superfluid on a lattice is given
by by \textbf{Eq. (??)}. This state has infinte range correlations and
\begin{equation} thus has the convenient property that all two-point density
\frac{1}{\sqrt{M^N N!}} \left( \sum_i^M \bd_i \right)^N |0 \rangle, fluctuation correlations are equal regardless of their separation,
\end{equation}
where $| 0 \rangle$ denotes the vacuum state. This state has infinte
range correlations and thus has the convenient property that all
two-point density fluctuation correlations are equal regardless of
their separation,
i.e.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$ i.e.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$
for all $(i \ne j)$, where the right hand side is a constant for all $(i \ne j)$, where the right hand side is a constant
value. This allows us to extract all correlations from the sum in value. This allows us to extract all correlations from the sum in
@ -316,26 +314,27 @@ The ``quantum addition'' for the case when both the scattered and
probe modes are travelling waves is actually trivial. It has no peaks probe modes are travelling waves is actually trivial. It has no peaks
and thus it has no generalised Bragg condition and it only consists of and thus it has no generalised Bragg condition and it only consists of
a uniform background. This is a consequence of the fact that a uniform background. This is a consequence of the fact that
travelling waves couple equally strongly with every site as only the travelling waves couple equally strongly with every atom as only the
phase differs. Therefore, since superfluid correlations lack structure phase is different between lattice sites. Therefore, since superfluid
as they're uniform we do no get a strong coherent peak. The correlations lack structure as they're uniform we do no get a strong
contribution from the lattice structure is included in the classical coherent peak. The contribution from the lattice structure is included
Bragg peaks which we have subtracted in order to obtain the quantity in the classical Bragg peaks which we have subtracted in order to
$R$. However, if we consider the case where the scattered mode is obtain the quantity $R$. However, if we consider the case where the
collected as a standing wave using a pair of mirrors we get the scattered mode is collected as a standing wave using a pair of mirrors
diffraction pattern that we saw in Fig. \ref{fig:scattering}. This we get the diffraction pattern that we saw in
time we get strong visible peaks, because at certain angles the Fig. \ref{fig:scattering}. This time we get strong visible peaks,
standing wave couples to the atoms maximally at all lattice sites and because at certain angles the standing wave couples to the atoms
thus it uses the structure of the lattice to amplify the signal from maximally at all lattice sites and thus it uses the structure of the
the quantum fluctuations. This becomes clear when we look at lattice to amplify the signal from the quantum fluctuations. This
Eq. \eqref{eq:RSF}. We can neglect the second term as it is always becomes clear when we look at Eq. \eqref{eq:RSF}. We can neglect the
negative and it has the same angular distribution as the classical second term as it is always negative and it has the same angular
diffraction pattern and thus it is mostly zero except when the distribution as the classical diffraction pattern and thus it is
classical Bragg condition is satisfied. Since in mostly zero except when the classical Bragg condition is
Fig. \ref{fig:scattering} we have chosen an angle such that the Bragg satisfied. Since in Fig. \ref{fig:scattering} we have chosen an angle
is not satisfied this term is essentially zero. Therefore, we are left such that the Bragg is not satisfied this term is essentially
with the first term $\sum_i^K |A_i|^2$ which for a travelling wave zero. Therefore, we are left with the first term $\sum_i^K |A_i|^2$
probe and a standing wave scattered mode is which for a travelling wave probe and a standing wave scattered mode
is
\begin{equation} \begin{equation}
\sum_i^K |A_i|^2 = \sum_i^K \cos^2(\b{k}_0 \cdot \b{r}_i + \phi_0) = \sum_i^K |A_i|^2 = \sum_i^K \cos^2(\b{k}_0 \cdot \b{r}_i + \phi_0) =
\frac{1}{2} \sum_i^K \left[1 + \cos(2 \b{k}_0 \cdot \b{r}_i + 2 \frac{1}{2} \sum_i^K \left[1 + \cos(2 \b{k}_0 \cdot \b{r}_i + 2
@ -344,14 +343,15 @@ probe and a standing wave scattered mode is
Therefore, it is straightforward to see that unless Therefore, it is straightforward to see that unless
$2 \b{k}_0 = \b{G}$, where $\b{G}$ is a reciprocal lattice vector, $2 \b{k}_0 = \b{G}$, where $\b{G}$ is a reciprocal lattice vector,
there will be no coherent signal and we end up with the mean uniform there will be no coherent signal and we end up with the mean uniform
signal of strength $N_k/2$. When this condition is satisifed all the signal of strength $|C|^2 N_k/2$. When this condition is satisifed all
cosine terms will be equal and they will add up constructively instead the cosine terms will be equal and they will add up constructively
of cancelling each other out. Note that this new Bragg condition is instead of cancelling each other out. Note that this new Bragg
different from the classical one $\b{k}_0 - \b{k}_1 = \b{G}$. This condition is different from the classical one
result makes it clear that the uniform background signal is not due to $\b{k}_0 - \b{k}_1 = \b{G}$. This result makes it clear that the
any coherent scattering, but rather due to the lack of structure in uniform background signal is not due to any coherent scattering, but
the quantum correlations. Furthermore, we see that the peak height is rather due to the lack of structure in the quantum
actually tunable via the phase, $\phi_0$, which is illustrated in correlations. Furthermore, we see that the peak height is actually
tunable via the phase, $\phi_0$, which is illustrated in
Fig. \ref{fig:scattering}b. Fig. \ref{fig:scattering}b.
For light field quadratures the situation is different, because as we For light field quadratures the situation is different, because as we
@ -362,7 +362,7 @@ $\beta$. The rest is similar to the case we discussed for $R$ with a
standing wave mode and we can show that the new Bragg condition in standing wave mode and we can show that the new Bragg condition in
this case is $2 (\b{k}_0 - \b{k}_1) = \b{G}$ which is different from this case is $2 (\b{k}_0 - \b{k}_1) = \b{G}$ which is different from
the condition we had for $R$ and is still different from the classical the condition we had for $R$ and is still different from the classical
condition $2 (\b{k}_0 - \b{k}_1) = \b{G}$. Furthermore, just like in condition $\b{k}_0 - \b{k}_1 = \b{G}$. Furthermore, just like in
Fig. \ref{fig:scattering}b the peak height can be tuned using $\beta$. Fig. \ref{fig:scattering}b the peak height can be tuned using $\beta$.
A quantum signal that isn't masked by classical diffraction is very A quantum signal that isn't masked by classical diffraction is very
@ -411,7 +411,7 @@ valid description of the physics. A prominent example is the
Bose-Hubbard model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, Bose-Hubbard model in 1D \cite{cazalilla2011, ejima2011, kuhner2000,
pino2012, pino2013}. Observing the transition in 1D by light at pino2012, pino2013}. Observing the transition in 1D by light at
fixed density was considered to be difficult \cite{rogers2014} or even fixed density was considered to be difficult \cite{rogers2014} or even
impossible \cite{roth2003}. This is, because the one-dimensional impossible \cite{roth2003}. This is because the one-dimensional
quantum phase transition is in a different universality class than its quantum phase transition is in a different universality class than its
higher dimensional counterparts. The energy gap, which is the order higher dimensional counterparts. The energy gap, which is the order
parameter, decays exponentially slowly across the phase transition parameter, decays exponentially slowly across the phase transition
@ -453,26 +453,25 @@ addition is given by
This alone allows us to analyse the phase transition quantitatively This alone allows us to analyse the phase transition quantitatively
using our method. Unlike in higher dimensions where an order parameter using our method. Unlike in higher dimensions where an order parameter
can be easily defined within the mean-field approximation as a simple can be easily defined within the mean-field approximation as a simple
expectation value of some quantity, the situation in 1D is more expectation value, the situation in 1D is more complex as it is
complex as it is difficult to directly access the excitation energy difficult to directly access the excitation energy gap which defines
gap which defines this phase transition. However, a valid description this phase transition. However, a valid description of the relevant 1D
of the relevant 1D low energy physics is provided by Luttinger liquid low energy physics is provided by Luttinger liquid theory
theory \cite{giamarchi}. In this model correlations in the supefluid \cite{giamarchi}. In this model correlations in the supefluid phase as
phase as well as the superfluid density itself are characterised by well as the superfluid density itself are characterised by the
the Tomonaga-Luttinger parameter, $K_b$. This parameter also Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies
identifies the critical point in the thermodynamic limit at $K_b = the critical point in the thermodynamic limit at $K_b = 1/2$. This
1/2$. This quantity can be extracted from various correlation quantity can be extracted from various correlation functions and in
functions and in our case it can be extracted directly from $R$ our case it can be extracted directly from $R$ \cite{ejima2011}. This
\cite{ejima2011}. This quantity was used in numerical calculations quantity was used in numerical calculations that used highly efficient
that used highly efficient density matrix renormalisation group (DMRG) density matrix renormalisation group (DMRG) methods to calculate the
methods to calculate the ground state to subsequently fit the ground state to subsequently fit the Luttinger theory to extract this
Luttinger theoru to extract this parameter $K_b$. These calculations parameter $K_b$. These calculations yield a theoretical estimate of
yield a theoretical estimate of the critical point in the the critical point in the thermodynamic limit for commensurate filling
thermodynamic limit for commensurate filling in 1D to be at in 1D to be at $U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our
$U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our proposal provides proposal provides a method to directly measure $R$ nondestructively in
a method to directly measure $R$ nondestructively in a lab which can a lab which can then be used to experimentally determine the location
then be used to experimentally determine the location of the critical of the critical point in 1D.
point in 1D.
However, whilst such an approach will yield valuable quantitative However, whilst such an approach will yield valuable quantitative
results we will instead focus on its qualitative features which give a results we will instead focus on its qualitative features which give a
@ -481,7 +480,7 @@ $R$. This is because the superfluid to Mott insulator phase transition
is well understood, so there is no reason to dwell on its quantitative is well understood, so there is no reason to dwell on its quantitative
aspects. However, our method is much more general than the aspects. However, our method is much more general than the
Bose-Hubbard model as it can be easily applied to many other systems Bose-Hubbard model as it can be easily applied to many other systems
such as fermions, photonic circuits, optical lattices qith quantum such as fermions, photonic circuits, optical lattices with quantum
potentials, etc. Therefore, by providing a better physical picture of potentials, etc. Therefore, by providing a better physical picture of
what information is carried by the ``quantum addition'' it should be what information is carried by the ``quantum addition'' it should be
easier to see its usefuleness in a broader context. easier to see its usefuleness in a broader context.
@ -527,22 +526,23 @@ system \cite{batrouni2002}.
\end{figure} \end{figure}
We then consider probing these ground states using our optical scheme We then consider probing these ground states using our optical scheme
and we calculate the ``quantum addition'', $R$. The angular dependence and we calculate the ``quantum addition'', $R$, based on these ground
of $R$ for a Mott insulator and a superfluid is shown in states. The angular dependence of $R$ for a Mott insulator and a
Fig. \ref{fig:SFMI}a, and we note that there are two variables superfluid is shown in Fig. \ref{fig:SFMI}a, and we note that there
distinguishing the states. Firstly, maximal $R$, $R_\text{max} \propto are two variables distinguishing the states. Firstly, maximal $R$,
\sum_i \langle \delta \hat{n}_i^2 \rangle$, probes the fluctuations $R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$,
and compressibility $\kappa'$ ($\langle \delta \hat{n}^2_i \rangle probes the fluctuations and compressibility $\kappa'$
\propto \kappa' \langle \hat{n}_i \rangle$). The Mott insulator is ($\langle \delta \hat{n}^2_i \rangle \propto \kappa' \langle \hat{n}_i
incompressible and thus will have very small on-site fluctuations and \rangle$). The Mott insulator is incompressible and thus will have
it will scatter little light leading to a small $R_\text{max}$. The very small on-site fluctuations and it will scatter little light
deeper the system is in the insulating phase (i.e. that larger the leading to a small $R_\text{max}$. The deeper the system is in the
$U/2J^\text{cl}$ ratio is), the smaller these values will be until insulating phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the
ultimately it will scatter no light at all in the $U \rightarrow smaller these values will be until ultimately it will scatter no light
\infty$ limit. In Fig. \ref{fig:SFMI}a this can be seen in the value at all in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a
of the peak in $R$. The value $R_\text{max}$ in the superfluid phase this can be seen in the value of the peak in $R$. The value
($U/2J^\text{cl} = 0$) is larger than its value in the Mott insulating $R_\text{max}$ in the superfluid phase ($U/2J^\text{cl} = 0$) is
phase ($U/2J^\text{cl} = 10$) by a factor of larger than its value in the Mott insulating phase
($U/2J^\text{cl} = 10$) by a factor of
$\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of $\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of
$R_\text{max}$ changes across the phase transition. There are a few $R_\text{max}$ changes across the phase transition. There are a few
things to note at this point. Firstly, if we follow the transition things to note at this point. Firstly, if we follow the transition
@ -552,7 +552,7 @@ 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 critical point. This is due to the energy gap closing exponentially
slowly which makes precise identification of the critical point slowly which makes precise identification of the critical point
extremely difficult. The best option at this point would be to fit extremely difficult. The best option at this point would be to fit
Tomonage-Luttinger theory to the results in order to find this Tomonaga-Luttinger theory to the results in order to find this
critical point. However, we note that there is a drastic change in critical point. However, we note that there is a drastic change in
signal as the chemical potential (and thus the density) is signal as the chemical potential (and thus the density) is
varied. This is highlighted in Fig. \ref{fig:SFMI}b which shows how varied. This is highlighted in Fig. \ref{fig:SFMI}b which shows how
@ -570,18 +570,20 @@ the dip in the $R$. This can be seen in
Fig. \ref{fig:SFMI}a. Furthermore, just like for $R_\text{max}$ we see Fig. \ref{fig:SFMI}a. Furthermore, just like for $R_\text{max}$ we see
that the transition is much sharper as $\mu$ is varied. This is shown that the transition is much sharper as $\mu$ is varied. This is shown
in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle between in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle between
a superfluid and an insulating state is fairly significant $\sim a superfluid and an insulating state is fairly significant
20^\circ$ which should make the two phases easy to identify using this $\sim 20^\circ$ which should make the two phases easy to identify
measure. In this particular case, measuring $W_R$ in the Mott phase is using this measure. In this particular case, measuring $W_R$ in the
not very practical as the insulating phase does not scatter light Mott phase is not very practical as the insulating phase does not
(small $R_\mathrm{max}$). The phase transition information is easier scatter light (small $R_\mathrm{max}$). The phase transition
extracted from $R_\mathrm{max}$. However, this is not always the case information is easier extracted from $R_\mathrm{max}$. However, this
and we will shortly see how certain phases of matter scatter a lot of is not always the case and we will shortly see how certain phases of
light and can be distinguished using measurements of $W_R$. Another matter scatter a lot of light and can be distinguished using
possible concern with experimentally measuring $W_R$ is that it might measurements of $W_R$ where $R_\mathrm{max}$ is not
be obstructed by the classical diffraction maxima which appear at sufficient. Another possible concern with experimentally measuring
angles corresponding to the minima in $R$. However, the width of such $W_R$ is that it might be obstructed by the classical diffraction
a peak is much smaller as its width is proportional to $1/M$. maxima which appear at angles corresponding to the minima in
$R$. However, the width of such a peak is much smaller as its width is
proportional to $1/M$.
So far both variables we considered, $R_\text{max}$ and $W_R$, provide So far both variables we considered, $R_\text{max}$ and $W_R$, provide
similar information. They both take on values at one of its extremes similar information. They both take on values at one of its extremes
@ -601,34 +603,33 @@ Hamiltonian can be shown to be
where $V$ is the strength of the superlattice potential, $r$ is the where $V$ is the strength of the superlattice potential, $r$ is the
ratio of the superlattice and trapping wave vectors and $\phi$ is some ratio of the superlattice and trapping wave vectors and $\phi$ is some
phase shift between the two lattice potentials \cite{roux2008}. The phase shift between the two lattice potentials \cite{roux2008}. The
first two terms are the standard Bose-Hubbard Hamiltonian. The only first two terms are the standard Bose-Hubbard Hamiltonian and the only
modification is an additionally spatially varying potential shift. We modification is an additional spatially varying potential shift. We
will only consider the phase diagram at fixed density as the will only consider the phase diagram at fixed density as the
introduction of disorder makes the usual interpretation of the phase introduction of disorder makes the usual interpretation of the phase
diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane for a 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 fixed ratio $V/U$ complicated due to the presence of multiple
compressible and incompressible phases between successive MI lobes compressible and incompressible phases between successive Mott
\cite{roux2008}. Therefore, the chemical potential no longer appears insulator lobes \cite{roux2008}. Therefore, the chemical potential no
in the Hamiltonian as we are no longer considering the grand canonical longer appears in the Hamiltonian as we are no longer considering the
ensemble. grand canonical ensemble.
The reason for considering such a system is that it introduces a The reason for considering such a system is that it introduces a
third, competing phase, the Bose glass into our phase diagram. It is third, competing phase, the Bose glass into our phase diagram. It is
an insulating phase like the Mott insulator, but it has local an insulating phase like the Mott insulator, but it has local
superfluid susceptibility making it compressible. Therefore this superfluid susceptibility making it compressible. Therefore this
localized insulating phase will have exponentially decaying localized insulating phase has exponentially decaying correlations
correlations just like the Mott phase, but it will have large on-site just like the Mott phase, but it has large on-site fluctuations just
fluctuations just like the compressible superfluid phase. As these are like the compressible superfluid phase. As these are the two physical
the two physical variables encoded in $R$ measuring both variables encoded in $R$ measuring both $R_\text{max}$ and $W_R$ will
$R_\text{max}$ and $W_R$ will provide us with enough information to provide us with enough information to distinguish all three phases. In
distinguish all three phases. In a Bose glass we have finite a Bose glass we have finite compressibility, but exponentially
compressibility, but exponentially decaying correlations. This gives a decaying correlations. This gives a large $R_\text{max}$ and a large
large $R_\text{max}$ and a large $W_R$. A Mott insulator will also $W_R$. A Mott insulator also has exponentially decaying correlations
have exponentially decaying correlations since it is an insulator, but since it is an insulator, but it is incompressible. Thus, it will
it will be incompressible. Thus, it will scatter light with a small scatter light with a small $R_\text{max}$ and large $W_R$. Finally, a
$R_\text{max}$ and large $W_R$. Finally, a superfluid will have long superfluid has long range correlations and large compressibility which
range correlations and large compressibility which results in a large results in a large $R_\text{max}$ and a small $W_R$.
$R_\text{max}$ and a small $W_R$.
\begin{figure}[htbp!] \begin{figure}[htbp!]
\centering \centering
@ -664,52 +665,96 @@ unit density. However, despite the lack of an easily distinguishable
critical point, as we have already discussed, it is possible to critical point, as we have already discussed, it is possible to
quantitatively extract the location of the transition lines by quantitatively extract the location of the transition lines by
extracting the Tomonaga-Luttinger parameter from the scattered light, extracting the Tomonaga-Luttinger parameter from the scattered light,
$R$, in the same way it was done for an unperturbed BHM $R$, in the same way it was done for an unperturbed Bose-Hubbard model
\cite{ejima2011}. \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.
\subsection{Matter-field interference measurements} \subsection{Matter-field interference measurements}
We now focus on enhancing the interference term $\hat{B}$ in the We have shown in section \ref{sec:B} that certain optical arrangements
operator $\hat{F}$. lead to a different type of light-matter interaction where coupling is
maximised in between lattice sites rather than at the sites themselves
yielding $\a_1 = C\hat{B}$. This leads to the optical fields
interacting directly with the interference terms $\bd_i b_{i+1}$ via
the operator $\hat{B}$ given by Eq. \eqref{eq:B}. This opens up a
whole new way of probing and interacting with a quantum gas trapped in
an optical lattice as this gives an in-situ method for probing the
inter-site interference terms at its shortest possible distance,
i.e.~the lattice period.
Firstly, we will use this result to show how one can probe % I mention mean-field here, but do not explain it. That should be
$\langle \hat{B} \rangle$ which in MF gives information about the % done in Chapter 2.1}
matter-field amplitude, $\Phi = \langle b \rangle$.
Hence, by measuring the light quadrature we probe the kinetic energy Unlike in the previous sections, here we will use the mean-field
and, in MF, the matter-field amplitude (order parameter) $\Phi$: description of the Bose-Hubbard model in order to obtain a simple
$\langle \hat{X}^F_{\beta=0} \rangle = | \Phi |^2 physical picture of what information is contained in the quantum
\mathcal{F}[W_1](2\pi/d) (K-1)$. light. In the mean-field approximation the inter-site interference
terms become
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} \begin{equation}
\label{intensity} \bd_i b_j = \Phi \bd_i + \Phi^* b_j - |\Phi|^2,
\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} \end{equation}
and it is shown as a function of $U/(zJ^\text{cl})$ in where $\langle b_i \rangle = \Phi$ which we assume is uniform across
Fig. \ref{Quads}. Thus, since measurement in the diffraction maximum the whole lattice. This approach has the advantage that the quantity
yields $\Phi^2$ we can deduce $\langle b^2 \rangle - \Phi^2$ from the $\Phi$ is the mean-field order parameter of the superfluid to Mott
intensity. This quantity is of great interest as it gives us access to insulator phase transition and is effectively a good measure of the
the quadrature variances of the matter-field superfluid character of the quantum ground state. This greatly
simplifies the physical interpretation of our results.
Firstly, we will show that our nondestructive measurement scheme
allows one to probe the mean-field order parameter, $\Phi$,
directly. Normally, this is achieved by releasing the trapped gas and
performing a time-of-flight measurement. Here, this can be achieved
in-situ. In section \ref{sec:B} we showed that one of the possible
optical arrangement leads to a diffraction maximum with the matter
operator
\begin{equation} \begin{equation}
(\Delta X^b_{0,\pi/2})^2 = 1/4 + [(n - \Phi^2) \pm \label{eq:Bmax}
(\langle b^2 \rangle - \Phi^2)]/2, \hat{B} = J^B_\mathrm{max} \sum_i \left( \bd_i b_{i+1} + b_i \bd_{i+1} \right),
\end{equation} \end{equation}
where $n=\langle\hat{n}\rangle$ is the mean on-site atomic density. where $J^B_\mathrm{max} = \mathcal{F}[W_1](2\pi/d)$. Therefore, by measuring the
expectation value of the quadrature we obtain the following quantity
\begin{equation}
\langle \hat{X}^F_{\beta=0} \rangle = J^B_\mathrm{max} (K-1) | \Phi |^2 .
\end{equation}
This quantity is directly proportional to square of the order
parameter $\Phi$ and thus lets us very easily follow this quantity
across the phase transition with a very simple quadrature measurement
setup.
In the mean-field treatment, the order parameter also lets us deduce a
different quantity, namely matter-field quadratures
$\hat{X}^b_\alpha = (b e^{-i\alpha} + \bd e^{i\alpha})/2$. Quadrature
measurements of optical fields are a standard and common tool in
quantum optics. However, this is not the case for matter-fields as
normally most interactions lead to an effective coupling with the
density as we have seen in the previous sections. Therefore, such
measurements provide us with new opportunities to study the quantum
matter state which was previously unavailable. We will take $\Phi$ to
be real which in the standard Bose-Hubbard Hamiltonian can be selected
via an inherent gauge degree of freedom in the order parameter. Thus,
the quadratures themself straightforwardly become
\begin{equation}
\hat{X}^b_\alpha = \frac{\Phi}{2} (e^{-i\alpha} + e^{i\alpha}) =
\Phi \cos(\alpha).
\end{equation}
From our measurement of the light field quadrature we have already
obtained the value of $\Phi$ and thus we immediately also know the
values of the matter-field quadratures. Unfortunately, the variance of
the quadrature is a more complicated quantity given by
\begin{equation}
(\Delta X^b_{0,\pi/2})^2 = \frac{1}{4} + [(n - \Phi^2) \pm
\frac{1}{2}(\langle b^2 \rangle - \Phi^2)],
\end{equation}
where we have arbitrarily selected two orthogonal quadratures
$\alpha =0 $ and $\alpha = \pi / 2$. This quantity cannot be estimated
from light quadrature measurements alone as we do not know the value
of $\langle b^2 \rangle$. To obtain the value of
$\langle b^2 \rangle$this quantity we need to consider a second-order
light observable such as light intensity. However, in the diffraction
maximum this signal will be dominated by a contribution proportional
to $K^2 \Phi^4$ whereas the terms containing information on
$\langle b^2 \rangle$ will only scale as $K$. Therefore, it would be
difficult to extract the quantity that we need by measuring in the
difraction maximum.
\begin{figure}[htbp!] \begin{figure}[htbp!]
\centering \centering
@ -729,32 +774,58 @@ where $n=\langle\hat{n}\rangle$ is the mean on-site atomic density.
\label{Quads} \label{Quads}
\end{figure} \end{figure}
Probing $\hat{B}^2$ gives us access to kinetic energy fluctuations We now consider the alternative arrangement in which we probe the
with 4-point correlations ($\bd_i b_j$ combined in pairs). Measuring diffraction minimum and the matter operator is given by
the photon number variance, which is standard in quantum optics, will \begin{equation}
lead up to 8-point correlations similar to 4-point density \label{eq:Bmin}
correlations \cite{mekhov2007pra}. These are of significant interest, \hat{B} = J^B_\mathrm{min} \sum_i (-1)^i \left( \bd_i b_{i+1} b_i
because it has been shown that there are quantum entangled states that \bd_{i+1} \right),
manifest themselves only in high-order correlations \end{equation}
\cite{kaszlikowski2008}. where $J^B_\mathrm{min} = - \mathcal{F}[W_1](\pi / d)$ and which
unlike the previous case is no longer proportional to the Bose-Hubbard
kinetic energy term. Unlike in the diffraction maximum the light
intensity in the diffraction minimum does not have the term
proportional to $K^2$ and thus we obtain the following quantity
\begin{equation}
\label{intensity}
\langle \ad_1 \a_1 \rangle = 2 |C|^2(K-1) \mathcal{F}^2[W_1]
(\frac{\pi}{d}) [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ],
\end{equation}
This is plotted in Fig. \ref{Quads} as a function of
$U/(zJ^\text{cl})$. Now, we can easily deduce the value of
$\langle b^2 \rangle$ since we will already know the mean density,
$n$, from our experimental setup and we have seen that we can obtain
$\Phi^2$ from the diffraction maximum. Thus, we now have access to the
quadrature variance of the matter-field as well giving us a more
complete picture of the matter-field amplitude than previously
possible.
Surprisingly, inter-site terms scatter more light from a Mott A surprising feature seen in Fig. \ref{Quads} is that the inter-site
insulator than a superfluid Eq. \eqref{intensity}, as shown in terms scatter more light from a Mott insulator than a superfluid
Fig. \eqref{Quads}, although the mean inter-site density Eq. \eqref{intensity}, although the mean inter-site density
$\langle \hat{n}(\b{r})\rangle $ is tiny in a MI. This reflects a $\langle \hat{n}(\b{r})\rangle $ is tiny in the Mott insulating
fundamental effect of the boson interference in Fock states. It indeed phase. This reflects a fundamental effect of the boson interference in
happens between two sites, but as the phase is uncertain, it results Fock states. It indeed happens between two sites, but as the phase is
in the large variance of $\hat{n}(\b{r})$ captured by light as shown uncertain, it results in the large variance of $\hat{n}(\b{r})$
in Eq. \eqref{intensity}. The interference between two macroscopic captured by light as shown in Eq. \eqref{intensity}. The interference
BECs has been observed and studied theoretically between two macroscopic BECs has been observed and studied
\cite{horak1999}. When two BECs in Fock states interfere a phase theoretically \cite{horak1999}. When two BECs in Fock states interfere
difference is established between them and an interference pattern is a phase difference is established between them and an interference
observed which disappears when the results are averaged over a large pattern is observed which disappears when the results are averaged
number of experimental realizations. This reflects the large over a large number of experimental realizations. This reflects the
shot-to-shot phase fluctuations corresponding to a large inter-site large shot-to-shot phase fluctuations corresponding to a large
variance of $\hat{n}(\b{r})$. By contrast, our method enables the inter-site variance of $\hat{n}(\b{r})$. By contrast, our method
observation of such phase uncertainty in a Fock state directly between enables the observation of such phase uncertainty in a Fock state
lattice sites on the microscopic scale in-situ. directly between lattice sites on the microscopic scale in-situ.
More generally, beyond mean-field, probing $\hat{B}^\dagger \hat{B}$
via light intensity measurements gives us access with 4-point
correlations ($\bd_i b_j$ combined in pairs). Measuring the photon
number variance, which is standard in quantum optics, will lead up to
8-point correlations similar to 4-point density correlations
\cite{mekhov2007pra}. These are of significant interest, because it
has been shown that there are quantum entangled states that manifest
themselves only in high-order correlations \cite{kaszlikowski2008}.
\section{Conclusions} \section{Conclusions}
@ -771,20 +842,19 @@ at measuring the matter-field interference via the operator $\hat{B}$
by concentrating light between the sites. This corresponds to probing by concentrating light between the sites. This corresponds to probing
interference at the shortest possible distance in an optical interference at the shortest possible distance in an optical
lattice. This is in contrast to standard destructive time-of-flight lattice. This is in contrast to standard destructive time-of-flight
measurements which deal with far-field interference and a relatively measurements which deal with far-field interference. This quantity
near-field one was used in Ref. \cite{miyake2011}. This defines most defines most processes in optical lattices. E.g. matter-field phase
processes in optical lattices. E.g. matter-field phase changes may changes may happen not only due to external gradients, but also due to
happen not only due to external gradients, but also due to intriguing intriguing effects such quantum jumps leading to phase flips at
effects such quantum jumps leading to phase flips at neighbouring neighbouring sites and sudden cancellation of tunneling
sites and sudden cancellation of tunneling \cite{vukics2007}, which \cite{vukics2007}, which should be accessible by our method. We showed
should be accessible by our method. We showed how in mean-field, one how in mean-field, one can measure the matter-field amplitude (order
can measure the matter-field amplitude (order parameter), quadratures parameter), quadratures and squeezing. This can link atom optics to
and squeezing. This can link atom optics to areas where quantum optics areas where quantum optics has already made progress, e.g., quantum
has already made progress, e.g., quantum imaging \cite{golubev2010, imaging \cite{golubev2010, kolobov1999}, using an optical lattice as
kolobov1999}, using an optical lattice as an array of multimode an array of multimode nonclassical matter-field sources with a high
nonclassical matter-field sources with a high degree of entanglement degree of entanglement for quantum information processing. Since our
for quantum information processing. Since our scheme is based on scheme is based on off-resonant scattering, and thus being insensitive
off-resonant scattering, and thus being insensitive to a detailed to a detailed atomic level structure, the method can be extended to
atomic level structure, the method can be extended to molecules molecules \cite{LP2013}, spins, and fermions \cite{ruostekoski2009}.
\cite{LP2013}, spins, and fermions \cite{ruostekoski2009}.