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@ -383,55 +383,224 @@ 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 and thus we neglect
any cavity pumping $\eta_l$. This leads to a linear relationship
between the light mode and the atomic operator $\hat{F}_{l,m}$
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 $\hat{F}_{1,0}$
\begin{equation}
\a_l = \frac{\sum_{m \ne l} U_{l,m} \a_m}
{\Delta_{lp} + i \kappa_l} \hat{F}_{l,m} = C \hat{F}_{l,m},
\a_1 = \frac{U_{1,0} a_0} {\Delta_{p} + i \kappa} \hat{F} =
C \hat{F},
\end{equation}
where we have defined
$C = \sum_{m \ne l} U_{l,m} \a_m / (\Delta_{lp} + i \kappa_l)$ which
is essentially the Rayleigh scattering coefficient into the
cavity. Whilst the light amplitude itself is only linear in atomic
operators, we can easily have access to higher moments by simply
simply considering higher moments of the $\a_l$ such as the photon
number $\ad_l \a_l$.
where we have defined $C = U_{1,0} a_0 / (\Delta_{p} + i \kappa)$
which is essentially the Rayleigh scattering coefficient into the
cavity. Furthermore, since there is no longer any ambiguity in the
indices $l$ and $m$, we have dropped indices on $\Delta_{1p} \equiv
\Delta_p$, $\kappa_1 \equiv \kappa$, and $\hat{F}_{1,0} \equiv
\hat{F}$. We also do the same for the operators $\hat{D}_{1,0} \equiv
\hat{D}$ and $\hat{B}_{1,0} \equiv \hat{B}$.
\subsection{Simplifications for Two Modes and Linear Coupling}
So far we have derived a general Hamiltonian for an arbitrary number
of light modes. However, most of the time we will be considering a
much simpler setup, namely the case with a coherent probe beam and a
single scattered mode as shown in Fig. \ref{fig:LatticeDiagram}. We
can use this fact to simplify our equations. Even though we consider
only one scattered mode, this treatment is also applicable to free
space scattering where the atoms can emit light in any direction. We
will simply neglect multiple scattering events, thus we can simply
vary the scattered mode's angle to obtain results for all possible
directions.
We consider the two modes indexed with $l = 0$ and $1$, $\a_0$ and $\a_1$. We
will take $\a_0$ to be the external probe beam which is incident on the
quantum gas (it is not pumped into the cavity via $\eta_0$).
Furthermore, we will consider it to be in a coherent state. Therefore,
we can treat $a_0$ in our equations like a complex number instead of a
full operator. This leaves $\a_1$ as our scattered light mode. We can
now simplify the Hamiltonian to
\begin{align}
\H = & \hbar \left( \omega_1 + U_{1,1} \hat{F}_{1,1} \right) \ad_1
\a_1 -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j +
\frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) \nonumber \\ &
+ \hbar U_{0,0} |a_0|^2 \hat{F}_{0,0} + \hbar U_{0,1} \left[ a_0^*
\a_1 \hat{F}_{0,1} + \ad_1 a_0 \hat{F}_{1,0} \right] - i \kappa
\ad_1 \a_1,
\end{align}
Whilst the light amplitude itself is only linear in atomic operators,
we can easily have access to higher moments by simply simply
considering higher moments of the $\a_1$ such as the photon number
$\ad_1 \a_1$. Additionally, even though we only consider a single
scattered mode, this model can be applied to free space by simply
varying the direction of the scattered light mode if multiple
scattering events can be neglected. This is likely to be the case
since the interactions will be dominated by photons scattering from
the much larger coherent probe.
\subsection{Density and Phase Observables}
Light scatters due to its interactions with the dipole moment of the
atoms which for off-resonant light, like the type that we consider,
results in an effective coupling with atomic density, not the
matter-wave amplitude. Therefore, it is challenging to couple light to
the phase of the matter-field as is typical in quantum optics for
optical fields. Most of the exisiting work on measurement couples
directly to atomic density operators \cite{mekhov2012, LP2009,
rogers2014, ashida2015, ashida2015a}. However, it has been shown
that one can couple to the interference term between two condensates
(e.g.~a BEC in a double-well) by using interference measurements
\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.
In our model light couples to the operator $\hat{F}$ which consists of
a density opertor part, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a
phase operator part, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j}
\bd_i b_j$. Most of the time the density component dominates, $\hat{D}
\gg \hat{B}$, and thus $\hat{F} \approx \hat{D}$. However, it is
possible to engineer an optical geometry in which $\hat{D} = 0$
leaving $\hat{B}$ as the dominant term in $\hat{F}$. This approach is
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.
For clarity we will consider a 1D lattice with lattice spacing $d$
along the $x$-axis direction, but the results can be applied and
generalised to higher dimensions. Central to engineering the $\hat{F}$
operator are the coefficients $J_{i,j}$ given by
\begin{equation}
\label{eq:Jcoeff}
J_{i,j} = \int \mathrm{d} x \,\,\, w(x - x_i) u_1^*(x) u_0(x) w(x - x_j).
\end{equation}
The operators $\hat{B}$ and $\hat{D}$ depend on the values of
$J_{i,i+1}$ and $J_{i,i}$ respectively. These coefficients are
determined by the convolution of the coupling strength between the
probe and scattered light modes, $u_1^*(x)u_0(x)$, with the relevant
Wannier function overlap shown in Fig. \ref{fig:WannierOverlaps}. For
the $\hat{B}$ operator we calculate the convolution with the nearest
neighbour overlap, $W_1(x) \equiv w(x - d/2) w(x + d/2)$ shown in
Fig. \ref{fig:WannierOverlaps}c, and for the $\hat{D}$ operator we
calculate the convolution with the square of the Wannier function at a
single site, $W_0(x) \equiv w^2(x)$ shown in
Fig. \ref{fig:WannierOverlaps}b. 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, which would measure a
single term $\langle \bd_i b_{i+1}+b_i \bd_{i+1}\rangle$, e.g.~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{Fix labels in this figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=1.0\textwidth]{Wannier1}
\includegraphics[width=1.0\textwidth]{Wannier2}
\caption[Wannier Function Overlaps]{(a) The Wannier functions
corresponding to four neighbouring sites in a 1D
lattice. $\lambda$ is the wavelength of the trapping beams, thus
lattice sites occur every $\lambda/2$. (Bottom Left) The square of
a single Wannier function - this quantity is used when evaluating
$\hat{D}$. It's much larger than the overlap between two
neighbouring Wannier functions, but it is localised to the
position of the lattice site it belongs to. (Bottom Right) The
overlap of two neighbouring Wannier functions - this quantity is
used when evaluating $\hat{B}$. It is much smaller than the square
of a Wannier function, but since it's localised in between the
sites, thus $\hat{B}$ can be maximised while $\hat{D}$ minimised
by focusing the light in between the sites.}
\label{fig:WannierOverlaps}
\end{figure}
\mynote{show the expansion into an FT}
In order to calculate the $J_{i,j}$ coefficients we perform numerical
calculations using realistic Wannier functions. 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:WannierFT}b. This is because the light mode 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 Fourier transforms of
$W_{0,1}(x)$. We consider both the detected and probe beam to be
standing waves which gives the following expressions for the $\hat{D}$
and $\hat{B}$ operators
\begin{eqnarray}
\label{eq:FTs}
\hat{D} =
\frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_m\hat{n}_m\cos(k_- x_m
+\varphi_-)
\nonumber\\ +\mathcal{F}[W_0](k_+)\sum_m\hat{n}_m\cos(k_+
x_m +\varphi_+)], \nonumber\\ \hat{B} =
\frac{1}{2}[\mathcal{F}[W_1](k_-)\sum_m\hat{B}_m\cos(k_- x_m
+\frac{k_-d}{2}+\varphi_-)
\nonumber\\ +\mathcal{F}[W_1](k_+)\sum_m\hat{B}_m\cos(k_+
x_m +\frac{k_+d}{2}+\varphi_+)],
\end{eqnarray}
where $k_\pm = k_{0x} \pm k_{1x}$, $k_{(0,1)x} = k_{0,1}
\sin(\theta_{0,1}$), $\hat{B}_m=b^\dag_mb_{m+1}+b_mb^\dag_{m+1}$, and
$\varphi_\pm=\varphi_0 \pm \varphi_1$. The key result is that the
$\hat{B}$ operator is phase shifted by $k_\pm d/2$ with respect to the
$\hat{D}$ operator since it depends on the amplitude of light in
between the lattice sites and not at the positions of the atoms,
allowing to decouple them at specific angles.
\begin{figure}[htbp!]
\begin{center}
\includegraphics[width=\linewidth]{WF_S}
\end{center}
\caption[Wannier Function Fourier Transforms]{The Wannier function
products: (a) $W_0(x)$ (solid line, right axis), $W_1(x)$ (dashed
line, left axis) and their (b) Fourier transforms
$\mathcal{F}[W_{0,1}]$. The Density $J_{i,i}$ and
matter-interference $J_{i,i+1}$ coefficients in diffraction
maximum (c) and minimum (d) as are shown as functions of standing
wave shifts $\varphi$ or, if one were to measure the quadrature
variance $(\Delta X^F_\beta)^2$, the local oscillator phase
$\beta$. The black points indicate the positions, where light
measures matter interference $\hat{B} \ne 0$, and the density-term
is suppressed, $\hat{D} = 0$. The trapping potential depth is
approximately 5 recoil energies.}
\label{fig:WannierFT}
\end{figure}
The simplest case is to find a diffraction maximum where $J_{i,i+1} =
J_B$. 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:WannierFT}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 $\varphi_0 = -\varphi_1 =
\arccos[-\mathcal{F}[W_0](2\pi/d)/\mathcal{F}[W_0](0)]/2$. 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 classical diffraction minimum. 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{eqnarray}
\label{eq:DMin}
\hat{D} = \mathcal{F}[W_0](\pi/d) \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
\cos(\varphi_0) \sin(\varphi_1).
\end{eqnarray}
The corresponding $J_{i,j}$ coefficients are shown in
Fig. \ref{fig:WannierFT}d for $\varphi_0=0$. 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.
\mynote{explain quadrature}
Alternatively, one can use the arrangement for a diffraction minimum
described above, but use travelling instead of standing waves for the
probe and detected beams and measure the light quadrature variance. In
this case $\hat{X}^F_\beta = \hat{D} \cos(\beta) + \hat{B}
\sin(\beta)$ and by varying the local oscillator phase, one can choose
which conjugate operator to measure.
\mynote{fix labels}
\begin{figure}[hbtp!]
\includegraphics[width=\linewidth]{BDiagram}
\caption[Maximising Light-Matter Coupling between Lattice
Sites]{Light field arrangements which maximise coupling,
$u_1^*u_0$, between lattice sites. The thin black line
indicates the trapping potential (not to scale). (a)
Arrangement for the uniform pattern $J_{m,m+1} = J_1$. (b)
Arrangement for spatially varying pattern $J_{m,m+1}=(-1)^m
J_2$; here $u_0=1$ so it is not shown and $u_1$ is real thus
$u_1^*u_0=u_1$. \label{fig:BDiagram}}
\end{figure}
\subsection{Electric Field Stength}

86
References/references.bib
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@ -1,3 +1,89 @@
@article{cirac1996,
title={Continuous observation of interference fringes from Bose condensates},
author={Cirac, J I and Gardiner, C W and Naraschewski, M and Zoller, P},
journal={Physical Review A},
volume={54},
number={5},
pages={R3714},
year={1996},
publisher={APS}
}
@article{castin1997,
title={Relative phase of two Bose-Einstein condensates},
author={Castin, Yvan and Dalibard, Jean},
journal={Physical Review A},
volume={55},
number={6},
pages={4330},
year={1997},
publisher={APS}
}
@article{ruostekoski1997,
title={Nondestructive optical measurement of relative phase between two Bose-Einstein condensates},
author={Ruostekoski, Janne and Walls, Dan F},
journal={Physical Review A},
volume={56},
number={4},
pages={2996},
year={1997},
publisher={APS}
}
@article{ruostekoski1998,
title={Macroscopic superpositions of Bose-Einstein condensates},
author={Ruostekoski, Janne and Collett, M J and Graham, Robert and Walls, Dan F},
journal={Physical Review A},
volume={57},
number={1},
pages={511},
year={1998},
publisher={APS}
}
@article{ashida2015,
title={Diffraction-Unlimited Position Measurement of Ultracold Atoms in an Optical Lattice},
author={Ashida, Yuto and Ueda, Masahito},
journal={Physical review letters},
volume={115},
number={9},
pages={095301},
year={2015},
publisher={APS}
}
@article{ashida2015a,
title={Multi-Particle Quantum Dynamics under Continuous Observation},
author={Ashida, Yuto and Ueda, Masahito},
journal={arXiv preprint arXiv:1510.04001},
year={2015}
}
@article{rogers2014,
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author={Rogers, B and Paternostro, M and Sherson, J F and De Chiara, G},
journal={Physical Review A},
volume={90},
number={4},
pages={043618},
year={2014},
publisher={APS}
}
@article{LP2009,
title={Quantum optics with quantum gases},
author={Mekhov, Igor B and Ritsch, Helmut},
journal={Laser physics},
volume={19},
number={4},
pages={610--615},
year={2009},
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}
@article{rist2012,
title={Homodyne detection of matter-wave fields},
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pages={053635},
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}
@book{foot,
author = {Foot, C. J.},
title = {{Atomic Physics}},