Finished for the day

Chapter2/chapter2.tex

@@ -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,
+slightly we will neglect the cavity resonancy shift, $U_{l,l}
-$U_{l,l} \hat{F}_{l,l}$ which is possible provided the cavity decay
+\hat{F}_{l,l}$ which is possible provided the cavity decay rate and/or
-rate and/or probe detuning are large enough. We will also only
+probe detuning are large enough. We will also only consider probing
-consider probing with an external coherent beam and thus we neglect
+with an external coherent beam, $a_0$, and thus we neglect any cavity
-any cavity pumping $\eta_l$. This leads to a linear relationship
+pumping $\eta_l$. We also limit ourselves to only a single scattered
-between the light mode and the atomic operator $\hat{F}_{l,m}$
+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} 
+  \a_1 = \frac{U_{1,0} a_0} {\Delta_{p} + i \kappa} \hat{F} =
-  {\Delta_{lp} + i \kappa_l} \hat{F}_{l,m} = C \hat{F}_{l,m},
+  C \hat{F},
 \end{equation}
-where we have defined
+where we have defined $C = U_{1,0} a_0 / (\Delta_{p} + i \kappa)$
-$C = \sum_{m \ne l} U_{l,m} \a_m / (\Delta_{lp} + i \kappa_l)$ which
+which is essentially the Rayleigh scattering coefficient into the
-is essentially the Rayleigh scattering coefficient into the
+cavity. Furthermore, since there is no longer any ambiguity in the
-cavity. Whilst the light amplitude itself is only linear in atomic
+indices $l$ and $m$, we have dropped indices on $\Delta_{1p} \equiv
-operators, we can easily have access to higher moments by simply
+\Delta_p$, $\kappa_1 \equiv \kappa$, and $\hat{F}_{1,0} \equiv
-simply considering higher moments of the $\a_l$ such as the photon
+\hat{F}$. We also do the same for the operators $\hat{D}_{1,0} \equiv
-number $\ad_l \a_l$.
+\hat{D}$ and $\hat{B}_{1,0} \equiv \hat{B}$.
 
-\subsection{Simplifications for Two Modes and Linear Coupling}
+Whilst the light amplitude itself is only linear in atomic operators,
-
+we can easily have access to higher moments by simply simply
-So far we have derived a general Hamiltonian for an arbitrary number
+considering higher moments of the $\a_1$ such as the photon number
-of light modes. However, most of the time we will be considering a
+$\ad_1 \a_1$. Additionally, even though we only consider a single
-much simpler setup, namely the case with a coherent probe beam and a
+scattered mode, this model can be applied to free space by simply
-single scattered mode as shown in Fig. \ref{fig:LatticeDiagram}. We
+varying the direction of the scattered light mode if multiple
-can use this fact to simplify our equations. Even though we consider
+scattering events can be neglected. This is likely to be the case
-only one scattered mode, this treatment is also applicable to free
+since the interactions will be dominated by photons scattering from
-space scattering where the atoms can emit light in any direction. We
+the much larger coherent probe.
-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}
 
 \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}
 

References/references.bib

@@ -1,3 +1,89 @@
+@article{cirac1996,
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+  number={5},
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+  publisher={APS}
+}
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+  publisher={APS}
+}
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+}
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+}
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+}
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+  year={2015}
+}
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+}
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+}
 @book{foot,
   author = {Foot, C. J.},
   title = {{Atomic Physics}},