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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
\includegraphics[width=1.0\textwidth]{LatticeDiagram}
\caption[Experimental Setup]{Atoms (green) trapped in an optical
lattice are illuminated by a coherent probe beam (red). The light
scatters (blue) in free space or into a cavity and is measured by
a detector. 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.}
lattice are illuminated by a coherent probe beam (red), $a_0$,
with a mode function $u_0(\b{r})$ which is at an angle $\theta_0$
to the normal to the lattice. The light scatters (blue) into the
mode $\a_1$ in free space or into a cavity and is measured by a
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}
\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
of light modes which can be pumped directly into the cavity or
produced via scattering from other modes. To simplify the equation
slightly we will neglect the cavity resonancy shift, $U_{l,l}
\hat{F}_{l,l}$ which is possible provided the cavity decay rate and/or
probe detuning are large enough. We will also only consider probing
with an external coherent beam, $a_0$, and thus we neglect any cavity
pumping $\eta_l$. We also limit ourselves to only a single scattered
mode, $a_1$. This leads to a simple linear relationship between the
light mode and the atomic operator $\hat{F}_{1,0}$
slightly we will neglect the cavity resonancy shift,
$U_{l,l} \hat{F}_{l,l}$ which is possible provided the cavity decay
rate and/or probe detuning are large enough. We will also only
consider probing with an external coherent beam, $a_0$ with mode
function $u_0(\b{r})$, and thus we neglect any cavity pumping
$\eta_l$. We also limit ourselves to only a single scattered mode,
$a_1$ 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}
\label{eq:a}
\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,
rist2012}. Such measurements establish a relative phase between the
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
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
as it directly couples to the interference terms caused by atoms
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,
mekhov2008, larson2008, chen2009, habibian2013, ivanov2014,
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
between the light modes and the nearest neighbour Wannier overlap,
$W_1(x)$. This can be achieved by concentrating the light between the
sites rather than at the positions of the atoms. Ideally, one could
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. A single-shot success rate of atom detection
will be small. As single-site addressing is challenging, we proceed
with the global scattering.
sites rather than at the positions of the atoms.
\mynote{Potentially expand details of the derivation of these
equations}
\mynote{Potentially expand details of the derivation of these equations}
In order to calculate the $J_{i,j}$ coefficients we perform numerical
calculations using realistic Wannier functions
\cite{walters2013}. However, it is possible to gain some analytic
insight into the behaviour of these values by looking at the Fourier
transforms of the Wannier function overlaps,
$\mathcal{F}[W_{0,1}](k)$, 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
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
@ -598,46 +609,59 @@ allowing to decouple them at specific angles.
\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
$J_{i,i+1} = J^B_\mathrm{max}$, where $J^B_\mathrm{max}$ is a
constant. This results in a diffraction maximum where each bond
(inter-site term) scatters light in phase and the operator is given by
\begin{equation}
\hat{B} = J^B_\mathrm{max} \sum_m^K \hat{B}_m .
\end{equation}
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)].
\frac{1}{2}\arccos\left[\frac{-\mathcal{F}[W_0](2\pi/d)}{\mathcal{F}[W_0](0)}\right].
\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
interference compensating for the spreading of the Wannier functions.
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
$k_{1x} = \pi/d$ which gives the following expressions for the density
and interference terms
\begin{align}
\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 \\
\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).
\end{align}
The corresponding $J_{i,j}$ coefficients are given by
$J_{i,i+1} = -(-1)^i J_2$, where $J_2$ is a constant, and are shown in
Fig. \ref{fig:WannierProducts}d for $\varphi_0=0$. The light mode
coupling along the lattice is shown in Fig. \ref{fig:BDiagram}b. It is
clear that for $\varphi_1 = \pm \pi/2$, $\hat{D} = 0$, which is
intuitive as this places the lattice sites at the nodes of the mode
$u_1(x)$. This is a diffraction minimum as the light amplitude is also
zero, $\langle \hat{B} \rangle = 0$, because contributions from
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.
For $\varphi_0 = 0$ the corresponding $J_{i,j}$ coefficients are given
by $J_{i,i+1} = (-1)^i J^B_\mathrm{min}$, where
$J^B_\mathrm{min} = -\mathcal{F}[W_1](\pi / d)$, and are shown in
Fig. \ref{fig:WannierProducts}d. The light mode coupling along the
lattice is shown in Fig. \ref{fig:BDiagram}b. It is clear that for
$\varphi_1 = \pm \pi/2$, $\hat{D} = 0$, which is intuitive as this
places the lattice sites at the nodes of the mode $u_1(x)$. This is a
diffraction minimum as the light amplitude is also zero,
$\langle \hat{B} \rangle = 0$, because contributions from 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.
Alternatively, one can use the arrangement for a diffraction minimum
described above, but use travelling instead of standing waves for the

516
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
gases in optical lattices by light scattering. In the previous chapter
we have shown that quantum light field couples to the bosons via the
operator $\hat{F}$. This is the key element of the scheme we propose
as this makes it sensitive to the quantum state of the matter and all
of its possible superpositions which will be reflected in the quantum
state of the light itself. We have also shown in section
we have shown that the quantum light field couples to the bosons via
the operator $\hat{F}$. This is the key element of the scheme we
propose as this makes it sensitive to the quantum state of the matter
and all of its possible superpositions which will be reflected in the
quantum state of the light itself. We have also shown in section
\ref{sec:derivation} that this coupling consists of two parts, a
density component $\hat{D}$ given by Eq. \eqref{eq:D}, and a phase
component $\hat{B}$ given by Eq. \eqref{eq:B}. Therefore, when probing
@ -62,7 +62,7 @@ fixed-density scattering. It was only recently that an experiment
distinguished a Mott insulator from a Bose glass \cite{derrico2014}
via a series of destructive measurements. Our proposal, on the other
hand, is nondestructive and is capable of extracting all the relevant
information in a single experiment.
information in a single experiment making our proposal timely.
Having shown the possibilities created by this nondestructive
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
classical field diffraction. In order to justify its name we will show
that this quantity depends purely quantum mechanical properties of the
ultracold gase. We will substitute $\a_1 = C \hat{D}$ using
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
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
\begin{equation}
\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
light-matter interaction to the diffraction pattern.
If instead we are interested in quantities linear in $\hat{D}$, we can
measure the quadrature of the light fields which in section
\ref{sec:a} we saw that $\hat{X}_\phi = |C| \hat{X}^F_\beta$. For the
case when both the scattered mode and probe are travelling waves the
quadrature
Another interesting quantity to measure are the quadratures of the
light fields which we have seen in section \ref{sec:a} are related to
the quadrature of $\hat{F}$ by $\hat{X}_\phi = |C|
\hat{X}^F_\beta$. An interesting feature of quadratures is that the
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}
\label{eq:Xtrav}
\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
\b{r}_i + (\phi_0 - \phi_1) - \beta].
\end{equation}
Note that different light quadratures are differently coupled to the
atom distribution, hence by varying the local oscillator phase, and
thus effectively $\beta$, and/or the detection angle one can scan the
whole range of couplings. A similar expression exists for $\hat{D}$
for a standing wave probe, where instead of varying $\beta$ scanning
is achieved by varying the position of the wave with respect to
atoms. Additionally, the quadrature variance, $(\Delta X^F_\beta)^2$,
will have a similar form to $R$ given in Eq. \eqref{eq:Rfluc},
Different light quadratures are differently coupled to the atom
distribution, hence by varying the local oscillator phase, $\beta$,
and/or the detection angle one can scan the whole range of
couplings. This is similar to the case for $\hat{D}$ for a standing
wave probe, where instead of varying $\beta$ scanning is achieved by
varying the position of the wave with respect to atoms. Additionally,
the quadrature variance, $(\Delta X^F_\beta)^2$, will have a similar
form to $R$ given in Eq. \eqref{eq:Rfluc},
\begin{equation}
(\Delta X^F_\beta)^2 = |C|^2 \sum_{i.j}^K A_i^\beta A_j^\beta
\langle \dn_i \dn_j \rangle,
@ -202,18 +204,18 @@ quantity.
The ``quantum addition'', $R$, and the quadrature variance,
$(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$ and both rely
heavily on the quantum state of the matter. Therefore, they will have
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
$\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular
dependence of $R$ for the case when the scattered mode is a standing
wave and the probe is a travelling wave scattering from an ideal
superfluid in a 3D optical lattice. The first noticeable feature is
the isotropic background which does not exist in classical
dependence of $R$ for the case when the probe is a travelling wave
scattering from an ideal superfluid in a 3D optical lattice into a
standing wave scattered mode. The first noticeable feature is the
isotropic background which does not exist in classical
diffraction. This background yields information about density
fluctuations which, according to mean-field estimates (i.e.~inter-site
correlations are ignored), are related by
$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. In
Fig. \ref{fig:scattering} we can see a significant signal of
$R = |C|^2 K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2
)/2$. In Fig. \ref{fig:scattering} we can see a significant signal of
$R = |C|^2 N_K/2$, because it shows scattering from an ideal
superfluid which has significant density fluctuations with
correlations of infinte range. However, as the parameters of the
@ -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
in an experiment as they wouldn't be masked by a stronger classical
signal. This difference in behaviour is due to the fact that
classical diffraction is ignorant of any quantum correlations. This
signal is given by the square of the light field amplitude squared
classical diffraction is ignorant of any quantum correlations as it is
given by the square of the light field amplitude squared
\begin{equation}
|\langle \a_1 \rangle|^2 = |C|^2 \sum_{i,j} A_i^*
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
not only depends on the diffraction structure due to the distribution
of atoms in the lattice, but also on the quantum correlations between
different lattice sites which will be dependent on the quantum state
of the matter. These correlations are imprinted in $R$ as shown in
Eq. \eqref{eq:Rfluc} and it highlights the key feature of our model,
i.e.~the light couples to the quantum state directly via operators.
different lattice sites which will in turn be dependent on the quantum
state of the matter. These correlations are imprinted in $R$ as shown
in Eq. \eqref{eq:Rfluc} and it highlights the key feature of our
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 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
to find the Bragg condition by considering angles at which the
distance travelled by light scattered from different points in the
lattice is equal to an integer multiple of wavelength. The ``quantum
addition'' is more complicated and less intuitive as we now have to
consider quantum correlations which are not only nonlocal, but can
also be nagative.
lattice is equal to an integer multiple of the wavelength. The
``quantum addition'' is more complicated and less intuitive as we now
have to consider quantum correlations which are not only nonlocal, but
can also be negative.
We will consider scattering from a superfluid, because the Mott
insulator has no ``quantum addition'' due to a lack of density
fluctuations. The wavefunction of a superfluid on a lattice is given
by
\begin{equation}
\frac{1}{\sqrt{M^N N!}} \left( \sum_i^M \bd_i \right)^N |0 \rangle,
\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,
by \textbf{Eq. (??)}. 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.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$
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
@ -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
and thus it has no generalised Bragg condition and it only consists of
a uniform background. This is a consequence of the fact that
travelling waves couple equally strongly with every site as only the
phase differs. Therefore, since superfluid correlations lack structure
as they're uniform we do no get a strong coherent peak. The
contribution from the lattice structure is included in the classical
Bragg peaks which we have subtracted in order to obtain the quantity
$R$. However, if we consider the case where the scattered mode is
collected as a standing wave using a pair of mirrors we get the
diffraction pattern that we saw in Fig. \ref{fig:scattering}. This
time we get strong visible peaks, because at certain angles the
standing wave couples to the atoms maximally at all lattice sites and
thus it uses the structure of the lattice to amplify the signal from
the quantum fluctuations. This becomes clear when we look at
Eq. \eqref{eq:RSF}. We can neglect the second term as it is always
negative and it has the same angular distribution as the classical
diffraction pattern and thus it is mostly zero except when the
classical Bragg condition is satisfied. Since in
Fig. \ref{fig:scattering} we have chosen an angle such that the Bragg
is not satisfied this term is essentially zero. Therefore, we are left
with the first term $\sum_i^K |A_i|^2$ which for a travelling wave
probe and a standing wave scattered mode is
travelling waves couple equally strongly with every atom as only the
phase is different between lattice sites. Therefore, since superfluid
correlations lack structure as they're uniform we do no get a strong
coherent peak. The contribution from the lattice structure is included
in the classical Bragg peaks which we have subtracted in order to
obtain the quantity $R$. However, if we consider the case where the
scattered mode is collected as a standing wave using a pair of mirrors
we get the diffraction pattern that we saw in
Fig. \ref{fig:scattering}. This time we get strong visible peaks,
because at certain angles the standing wave couples to the atoms
maximally at all lattice sites and thus it uses the structure of the
lattice to amplify the signal from the quantum fluctuations. This
becomes clear when we look at Eq. \eqref{eq:RSF}. We can neglect the
second term as it is always negative and it has the same angular
distribution as the classical diffraction pattern and thus it is
mostly zero except when the classical Bragg condition is
satisfied. Since in Fig. \ref{fig:scattering} we have chosen an angle
such that the Bragg is not satisfied this term is essentially
zero. Therefore, we are left with the first term $\sum_i^K |A_i|^2$
which for a travelling wave probe and a standing wave scattered mode
is
\begin{equation}
\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
@ -344,14 +343,15 @@ probe and a standing wave scattered mode is
Therefore, it is straightforward to see that unless
$2 \b{k}_0 = \b{G}$, where $\b{G}$ is a reciprocal lattice vector,
there will be no coherent signal and we end up with the mean uniform
signal of strength $N_k/2$. When this condition is satisifed all the
cosine terms will be equal and they will add up constructively instead
of cancelling each other out. Note that this new Bragg condition is
different from the classical one $\b{k}_0 - \b{k}_1 = \b{G}$. This
result makes it clear that the uniform background signal is not due to
any coherent scattering, but rather due to the lack of structure in
the quantum correlations. Furthermore, we see that the peak height is
actually tunable via the phase, $\phi_0$, which is illustrated in
signal of strength $|C|^2 N_k/2$. When this condition is satisifed all
the cosine terms will be equal and they will add up constructively
instead of cancelling each other out. Note that this new Bragg
condition is different from the classical one
$\b{k}_0 - \b{k}_1 = \b{G}$. This result makes it clear that the
uniform background signal is not due to any coherent scattering, but
rather due to the lack of structure in the quantum
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.
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
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
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 $\beta$.
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,
pino2012, pino2013}. Observing the transition in 1D by light at
fixed density was considered to be difficult \cite{rogers2014} or even
impossible \cite{roth2003}. This is, because the one-dimensional
impossible \cite{roth2003}. This is because the one-dimensional
quantum phase transition is in a different universality class than its
higher dimensional counterparts. The energy gap, which is the order
parameter, decays exponentially slowly across the phase transition
@ -453,26 +453,25 @@ addition is given by
This alone allows us to analyse the phase transition quantitatively
using our method. Unlike in higher dimensions where an order parameter
can be easily defined within the mean-field approximation as a simple
expectation value of some quantity, the situation in 1D is more
complex as it is difficult to directly access the excitation energy
gap which defines this phase transition. However, a valid description
of the relevant 1D low energy physics is provided by Luttinger liquid
theory \cite{giamarchi}. In this model correlations in the supefluid
phase as well as the superfluid density itself are characterised by
the Tomonaga-Luttinger parameter, $K_b$. This parameter also
identifies the critical point in the thermodynamic limit at $K_b =
1/2$. This quantity can be extracted from various correlation
functions and in our case it can be extracted directly from $R$
\cite{ejima2011}. This quantity was used in numerical calculations
that used highly efficient density matrix renormalisation group (DMRG)
methods to calculate the ground state to subsequently fit the
Luttinger theoru to extract this parameter $K_b$. These calculations
yield a theoretical estimate of the critical point in the
thermodynamic limit for commensurate filling in 1D to be at
$U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our proposal provides
a method to directly measure $R$ nondestructively in a lab which can
then be used to experimentally determine the location of the critical
point in 1D.
expectation value, the situation in 1D is more complex as it is
difficult to directly access the excitation energy gap which defines
this phase transition. However, a valid description of the relevant 1D
low energy physics is provided by Luttinger liquid theory
\cite{giamarchi}. In this model correlations in the supefluid phase as
well as the superfluid density itself are characterised by the
Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies
the critical point in the thermodynamic limit at $K_b = 1/2$. This
quantity can be extracted from various correlation functions and in
our case it can be extracted directly from $R$ \cite{ejima2011}. This
quantity was used in numerical calculations that used highly efficient
density matrix renormalisation group (DMRG) methods to calculate the
ground state to subsequently fit the Luttinger theory to extract this
parameter $K_b$. These calculations yield a theoretical estimate of
the critical point in the thermodynamic limit for commensurate filling
in 1D to be at $U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our
proposal provides a method to directly measure $R$ nondestructively in
a lab which can then be used to experimentally determine the location
of the critical point in 1D.
However, whilst such an approach will yield valuable quantitative
results we will instead focus on its qualitative features which give a
@ -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
aspects. However, our method is much more general than the
Bose-Hubbard model as it can be easily applied to many other systems
such as fermions, photonic circuits, optical lattices qith quantum
such as fermions, photonic circuits, optical lattices with quantum
potentials, etc. Therefore, by providing a better physical picture of
what information is carried by the ``quantum addition'' it should be
easier to see its usefuleness in a broader context.
@ -527,22 +526,23 @@ system \cite{batrouni2002}.
\end{figure}
We then consider probing these ground states using our optical scheme
and we calculate the ``quantum addition'', $R$. The angular dependence
of $R$ for a Mott insulator and a superfluid is shown in
Fig. \ref{fig:SFMI}a, and we note that 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 insulating 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 superfluid phase
($U/2J^\text{cl} = 0$) is larger than its value in the Mott insulating
phase ($U/2J^\text{cl} = 10$) by a factor of
and we calculate the ``quantum addition'', $R$, based on these ground
states. The angular dependence of $R$ for a Mott insulator and a
superfluid is shown in Fig. \ref{fig:SFMI}a, and we note that 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
insulating 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 superfluid phase ($U/2J^\text{cl} = 0$) is
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
$R_\text{max}$ changes across the phase transition. There are a few
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
slowly which makes precise identification of the critical point
extremely difficult. The best option at this point would be to fit
Tomonage-Luttinger theory to the results in order to find this
Tomonaga-Luttinger theory to the results in order to find this
critical point. However, we note that there is a drastic change in
signal as the chemical potential (and thus the density) is
varied. This is highlighted in Fig. \ref{fig:SFMI}b which shows how
@ -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
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
a superfluid and an insulating state is fairly significant $\sim
20^\circ$ which should make the two phases easy to identify using this
measure. In this particular case, measuring $W_R$ in the Mott phase is
not very practical as the insulating phase does not scatter light
(small $R_\mathrm{max}$). The phase transition information is easier
extracted from $R_\mathrm{max}$. However, this is not always the case
and we will shortly see how certain phases of matter scatter a lot of
light and can be distinguished using measurements of $W_R$. Another
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 $1/M$.
a superfluid and an insulating state is fairly significant
$\sim 20^\circ$ which should make the two phases easy to identify
using this measure. In this particular case, measuring $W_R$ in the
Mott phase is not very practical as the insulating phase does not
scatter light (small $R_\mathrm{max}$). The phase transition
information is easier extracted from $R_\mathrm{max}$. However, this
is not always the case and we will shortly see how certain phases of
matter scatter a lot of light and can be distinguished using
measurements of $W_R$ where $R_\mathrm{max}$ is not
sufficient. Another 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 $1/M$.
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
@ -601,34 +603,33 @@ Hamiltonian can be shown to be
where $V$ is the strength of the superlattice potential, $r$ is the
ratio of the superlattice and trapping wave vectors and $\phi$ is some
phase shift between the two lattice potentials \cite{roux2008}. The
first two terms are the standard Bose-Hubbard Hamiltonian. The only
modification is an additionally spatially varying potential shift. We
first two terms are the standard Bose-Hubbard Hamiltonian and the only
modification is an additional spatially varying potential shift. We
will only consider the phase diagram at fixed density as the
introduction of disorder makes the usual interpretation of the phase
diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane for a
fixed ratio $V/U$ complicated due to the presence of multiple
compressible and incompressible phases between successive MI lobes
\cite{roux2008}. Therefore, the chemical potential no longer appears
in the Hamiltonian as we are no longer considering the grand canonical
ensemble.
compressible and incompressible phases between successive Mott
insulator lobes \cite{roux2008}. Therefore, the chemical potential no
longer appears in the Hamiltonian as we are no longer considering the
grand canonical ensemble.
The reason for considering such a system is that it introduces a
third, competing phase, the Bose glass into our phase diagram. It is
an insulating phase like the Mott insulator, but it has local
superfluid susceptibility making it compressible. Therefore this
localized insulating phase will have exponentially decaying
correlations just like the Mott phase, but it will have large on-site
fluctuations just like the compressible superfluid phase. As these are
the two physical variables encoded in $R$ measuring both
$R_\text{max}$ and $W_R$ will provide us with enough information to
distinguish all three phases. In a Bose glass we have finite
compressibility, but exponentially decaying correlations. This gives a
large $R_\text{max}$ and a large $W_R$. A Mott insulator will also
have exponentially decaying correlations since it is an insulator, but
it will be incompressible. Thus, it will scatter light with a small
$R_\text{max}$ and large $W_R$. Finally, a superfluid will have long
range correlations and large compressibility which results in a large
$R_\text{max}$ and a small $W_R$.
localized insulating phase has exponentially decaying correlations
just like the Mott phase, but it has large on-site fluctuations just
like the compressible superfluid phase. As these are the two physical
variables encoded in $R$ measuring both $R_\text{max}$ and $W_R$ will
provide us with enough information to distinguish all three phases. In
a Bose glass we have finite compressibility, but exponentially
decaying correlations. This gives a large $R_\text{max}$ and a large
$W_R$. A Mott insulator also has exponentially decaying correlations
since it is an insulator, but it is incompressible. Thus, it will
scatter light with a small $R_\text{max}$ and large $W_R$. Finally, a
superfluid has long range correlations and large compressibility which
results in a large $R_\text{max}$ and a small $W_R$.
\begin{figure}[htbp!]
\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
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
$R$, in the same way it was done for an unperturbed Bose-Hubbard model
\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}
We now focus on enhancing the interference term $\hat{B}$ in the
operator $\hat{F}$.
We have shown in section \ref{sec:B} that certain optical arrangements
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
$\langle \hat{B} \rangle$ which in MF gives information about the
matter-field amplitude, $\Phi = \langle b \rangle$.
% I mention mean-field here, but do not explain it. That should be
% done in Chapter 2.1}
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:
Unlike in the previous sections, here we will use the mean-field
description of the Bose-Hubbard model in order to obtain a simple
physical picture of what information is contained in the quantum
light. In the mean-field approximation the inter-site interference
terms become
\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
\bd_i b_j = \Phi \bd_i + \Phi^* b_j - |\Phi|^2,
\end{equation}
where $\langle b_i \rangle = \Phi$ which we assume is uniform across
the whole lattice. This approach has the advantage that the quantity
$\Phi$ is the mean-field order parameter of the superfluid to Mott
insulator phase transition and is effectively a good measure of the
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}
\label{eq:Bmax}
\hat{B} = J^B_\mathrm{max} \sum_i \left( \bd_i b_{i+1} + b_i \bd_{i+1} \right),
\end{equation}
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 = 1/4 + [(n - \Phi^2) \pm
(\langle b^2 \rangle - \Phi^2)]/2,
(\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 $n=\langle\hat{n}\rangle$ is the mean on-site atomic density.
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!]
\centering
@ -729,32 +774,58 @@ where $n=\langle\hat{n}\rangle$ is the mean on-site atomic density.
\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}.
We now consider the alternative arrangement in which we probe the
diffraction minimum and the matter operator is given by
\begin{equation}
\label{eq:Bmin}
\hat{B} = J^B_\mathrm{min} \sum_i (-1)^i \left( \bd_i b_{i+1} b_i
\bd_{i+1} \right),
\end{equation}
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
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.
A surprising feature seen in Fig. \ref{Quads} is that the inter-site
terms scatter more light from a Mott insulator than a superfluid
Eq. \eqref{intensity}, although the mean inter-site density
$\langle \hat{n}(\b{r})\rangle $ is tiny in the Mott insulating
phase. 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.
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}
@ -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
interference at the shortest possible distance in an optical
lattice. This is in contrast to standard destructive time-of-flight
measurements which deal with far-field interference and a relatively
near-field one was used in Ref. \cite{miyake2011}. This defines most
processes in optical lattices. E.g. matter-field phase changes may
happen not only due to external gradients, but also due to intriguing
effects such quantum jumps leading to phase flips at neighbouring
sites and sudden cancellation of tunneling \cite{vukics2007}, which
should be accessible by our method. We showed how in mean-field, one
can measure the matter-field amplitude (order parameter), quadratures
and squeezing. This can link atom optics to areas where quantum optics
has already made progress, e.g., quantum imaging \cite{golubev2010,
kolobov1999}, using an optical lattice as an array of multimode
nonclassical matter-field sources with a high degree of entanglement
for quantum information processing. Since our scheme is based on
off-resonant scattering, and thus being insensitive to a detailed
atomic level structure, the method can be extended to molecules
\cite{LP2013}, spins, and fermions \cite{ruostekoski2009}.
measurements which deal with far-field interference. This quantity
defines most processes in optical lattices. E.g. matter-field phase
changes may happen not only due to external gradients, but also due to
intriguing effects such quantum jumps leading to phase flips at
neighbouring sites and sudden cancellation of tunneling
\cite{vukics2007}, which should be accessible by our method. We showed
how in mean-field, one can measure the matter-field amplitude (order
parameter), quadratures and squeezing. This can link atom optics to
areas where quantum optics has already made progress, e.g., quantum
imaging \cite{golubev2010, kolobov1999}, using an optical lattice as
an array of multimode nonclassical matter-field sources with a high
degree of entanglement for quantum information processing. Since our
scheme is based on off-resonant scattering, and thus being insensitive
to a detailed atomic level structure, the method can be extended to
molecules \cite{LP2013}, spins, and fermions \cite{ruostekoski2009}.