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%*******************************************************************************
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%*********************************** Second Chapter ****************************
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%Title of the Second Chapter
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\chapter{Quantum Optics of Quantum Gases}
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%********************************** % First Section ****************************
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\section{Ultracold Atoms in Optical Lattices}
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%********************************** % Second Section ***************************
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\section{Quantum Optics of Quantum Gases}
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Having introduced and described the behaviour of ultracold bosons
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trapped and manipulated using classical light, it is time to extend
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the discussion to quantized optical fields. We will first derive a
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general Hamiltonian that describes the coupling of atoms with
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far-detuned optical beams \cite{mekhov2012}. This will serve as the
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basis from which we explore the system in different parameter regimes,
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such as nondestructive measurement in free space or quantum
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measurement backaction in a cavity.
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We consider $N$ two-level atoms in an optical lattice with $M$
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sites. For simplicity we will restrict our attention to spinless
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bosons, although it is straightforward to generalise to fermions,
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which yields its own set of interesting quantum phenomena
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\cite{atoms2015, mazzucchi2016, mazzucchi2016af}, and other spin
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particles. The theory can be also be generalised to continuous
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systems, but the restriction to optical lattices is convenient for a
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variety of reasons. Firstly, it allows us to precisely describe a
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many-body atomic state over a broad range of parameter values due to
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the inherent tunability of such lattices. Furthermore, this model is
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capable of describing a range of different experimental setups ranging
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from a small number of sites with a large filling factor (e.g.~BECs
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trapped in a double-well potential) to a an extended multi-site
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lattice with a low filling factor (e.g.~a system with one atom per
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site will exhibit the Mott insulator to superfluid quantum phase
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transition). \mynote{extra fermion citations, Piazza? Look up Gabi's
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AF paper.}
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2016-06-10 19:16:26 +02:00
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\mynote{Potentially some more crap, but come to think of it the
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content will strongly depend on what was included in the preceding
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section on plain ultracold bosons}
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2016-06-10 19:16:26 +02:00
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As we have seen in the previous section, an optical lattice can be
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formed with classical light beams that form standing waves. Depending
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on the detuning with respect to the atomic resonance, the nodes or
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antinodes form the lattice sites in which atoms accumulate. As shown
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in Fig. \ref{fig:LatticeDiagram} the trapped bosons (green) are
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illuminated with a coherent probe beam (red) and scatter light into a
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different mode (blue) which is then measured with a detector. The most
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straightforward measurement is to simply count the number of photons
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with a photodetector, but it is also possible to perform a quadrature
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measurement by using a homodyne detection scheme. The experiment can
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be performed in free space where light can scatter in any
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direction. The atoms can also be placed inside a cavity which has the
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advantage of being able to enhance light scattering in a particular
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direction. Furthermore, cavities allow for the formation of a fully
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quantum potential in contrast to the classical lattice trap.
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2016-06-10 19:16:26 +02:00
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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=1.0\textwidth]{LatticeDiagram}
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\caption[LatticeDiagram]{Atoms (green) trapped in an optical lattice
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are illuminated by a coherent probe beam (red). The light scatters
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(blue) in free space or into a cavity and is measured by a
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detector. If the experiment is in free space light can scatter in
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any direction. A cavity on the other hand enhances scattering in
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one particular direction.}
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\label{fig:LatticeDiagram}
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\end{figure}
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For simplicity, we will be considering one-dimensional lattices most
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of the time. However, the model itself is derived for any number of
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dimensions and since none of our arguments will ever rely on
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dimensionality our results straightforwardly generalise to 2- and 3-D
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systems. This simplification allows us to present a much simpler
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picture of the physical setup where we only need to concern ourselves
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with a single angle for each optical mode. As shown in
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Fig. \ref{fig:LatticeDiagram} the angle between the normal to the
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lattice and the probe and detected beam are denoted by $\theta_0$ and
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$\theta_1$ respectively. We will consider these angles to be tunable
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although the same effect can be achieved by varying the wavelength of
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the light modes. However, it is much more intuitive to consider
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variable angles in our model as this lends itself to a simpler
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geometrical representation.
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\subsection{Derivation of the Hamiltonian}
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A general many-body Hamiltonian coupled to a quantized light field in
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second quantized can be separated into three parts,
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\begin{equation}
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\label{eq:FullH}
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\H = \H_f + \H_a + \H_{fa}.
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\end{equation}
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The term $\H_f$ represents the optical part of the Hamiltonian,
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\begin{equation}
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\label{eq:Hf}
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\H_f = \sum_l \hbar \omega_l \ad_l \a_l -
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i \hbar \sum_l \left( \eta_l^* \a_l - \eta_l \ad \right).
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\end{equation}
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The operators $\a_l$ ($\ad$) are the annihilation (creation) operators
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of light modes with frequencies $\omega_l$, wave vectors $\b{k}_l$,
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and mode functions $u_l(\b{r})$, which can be pumped by coherent
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fields with amplitudes $\eta_l$. The second part of the Hamiltonian,
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$\H_a$, is the matter-field component given by
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\begin{equation}
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\label{eq:Ha}
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\H_a = \int \mathrm{d}^3 \b{r} \Psi^\dagger(\b{r}) \H_{1,a}
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\Psi(\b{r}) + \frac{2 \pi a_s \hbar^2}{m} \int \mathrm{d}^3 \b{r}
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\Psi^\dagger(\b{r}) \Psi^\dagger(\b{r}) \Psi(\b{r}) \Psi(\b{r}).
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\end{equation}
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Here, $\Psi(\b{r})$ ($\Psi^\dagger(\b{r})$) are the matter-field
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operators that annihilate (create) an atom at position $\b{r}$, $a_s$
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is the $s$-wave scattering length characterising the interatomic
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interaction, and $\H_{1,a}$ is the atomic part of the single-particle
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Hamiltonian $\H_1$. The final component of the total Hamiltonian is
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the interaction given by
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\begin{equation}
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\label{eq:Hfa}
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\H_{fa} = \int \mathrm{d}^3 \b{r} \Psi^\dagger(\b{r}) \H_{1,fa}
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\Psi(\b{r}),
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\end{equation}
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where $\H_{1,fa}$ is the interaction part of the single-particle
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Hamiltonian, $\H_1$.
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The single-particle Hamiltonian in the rotating-wave and dipole
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approximation is given by
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\begin{equation}
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\H_1 = \H_f + \H_{1,a} + \H_{1,fa},
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\end{equation}
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\begin{equation}
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\H_{1,a} = \frac{\b{p}^2} {2 m_a} + \frac{\hbar \omega_a}{2} \sigma_z,
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\end{equation}
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\begin{equation}
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\H_{1,fa} = - i \hbar \sum_l \left[ \sigma^+ g_l \a_l u_l(\b{r}) - \sigma^- g^*_l
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\ad_l u^*_l(\b{r}) \right].
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\end{equation}
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In the equations above, $\b{p}$ and $\b{r}$ are the momentum and
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position operators of an atom of mass $m_a$ and resonance frequency
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$\omega_a$. The operators $\sigma^+ = |g \rangle \langle e|$,
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$\sigma^- = |e \rangle \langle g|$, and
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$\sigma_z = |e \rangle \langle e| - |g \rangle \langle g|$ are the
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atomic raising, lowering and population difference operators, where
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$|g \rangle$ and $| e \rangle$ denote the ground and excited states of
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the two-level atom respectively. $g_l$ are the atom-light coupling
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constants for each mode. It is the inclusion of the interaction of the
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boson with quantized light that distinguishes our work from the
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typical approach to ultracold atoms where all the optical fields,
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including the trapping potentials, are treated classically.
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We will now simplify the single-particle Hamiltonian by adiabatically
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eliminating the upper excited level of the atom. The equations of
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motion for the time evolution of operator $\hat{A}$ in the Heisenberg
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picture are given by
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\begin{equation}
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\dot{\hat{A}} = \frac{i}{\hbar} \left[\H, \hat{A} \right].
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\end{equation}
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Therefore, the Heisenberg equation for the lowering operator of a
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single particle is
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\begin{equation}
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\dot{\sigma}^- = \frac{i}{\hbar} \left[\H_1, \hat{\sigma}^- \right]
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= \hbar \omega_a \sigma^- + i \hbar \sum_l \sigma_z g_l \a_l u_l(\b{r}).
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\end{equation}
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We will consider nonresonant interactions between light and atoms
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where the detunings between the light fields and the atomic resonance,
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$\Delta_{la} = \omega_l - \omega_a$, are much larger than the
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spontaneous emission rate and Rabi frequencies $g_l \a_l$. Therefore,
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the atom will be predominantly found in the ground state and we can
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set $\sigma_z = -1$ which is also known as the linear dipole
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approximation as the dipoles respond linearly to the light amplitude
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when the excited state has negligible population. Moreover, we can
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adiabatically eliminate the polarization $\sigma^-$. Firstly we will
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re-write its equation of motion in a frame rotating at $\omega_p$, the
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external probe frequency, such that
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$\sigma^- = \tilde{\sigma}^- \exp(i \omega_p t)$, and similarly for
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$\tilde{\a}_l$. The resulting equation is given by
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\begin{equation}
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\dot{\tilde{\sigma}}^- = - \hbar \Delta_a \tilde{\sigma}^- - i \hbar
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\sum_l g_l \tilde{\a}_l u_l(\b{r}),
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\end{equation}
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where $\Delta_a = \omega_p - \omega_a$ is the atom-probe
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detuning. Within this rotating frame we will take
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$\dot{\tilde{\sigma}}^- \approx 0$ and thus obtain the following
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equation for the lowering operator
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\begin{equation}
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\sigma^- = - \frac{i}{\Delta_a} \sum_l g_l \a_l u_l(\b{r}).
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\end{equation}
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Therefore, by inserting this expression into the Heisenberg equation
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for the light mode $m$ given by
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\begin{equation}
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\dot{\a}_m = - \sigma^- g^*_m u^*_m(\b{r})
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\end{equation}
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we get the following equation of motion
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\begin{equation}
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\dot{\a}_m = \frac{i}{\Delta_a} \sum_l g_l g^*_m u_l(\b{r})
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u^*_m(\b{r}) \a_l.
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\end{equation}
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An effective Hamiltonian which results in the same optical equations
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of motion can be written as
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$\H^\mathrm{eff}_1 = \H_f + \H^\mathrm{eff}_{1,a} +
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\H^\mathrm{eff}_{1,fa}$. The effective atomic and interaction
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Hamiltonians are
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\begin{equation}
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\label{eq:aeff}
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\H^\mathrm{eff}_{1,a} = \frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}),
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\end{equation}
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\begin{equation}
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\label{eq:faeff}
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\H^\mathrm{eff}_{1,fa} = \frac{\hbar}{\Delta_a} \sum_{l,m}
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u_l^*(\b{r}) u_m(\b{r}) g_l g_m \ad_l \a_m,
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\end{equation}
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where we have explicitly extracted
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$V_\mathrm{cl}(\b{r}) = \hbar g_\mathrm{cl}^2 |a_\mathrm{cl}
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u_\mathrm{cl}(\b{r})|^2 / \Delta_{\mathrm{cl},a}$, the classical
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trapping potential, from the interaction terms. However, we consider
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the trapping beam to be sufficiently detuned from the other light
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modes that we can neglect any scattering between them. A later
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inclusion of this scattered light would not be difficult due to the
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linearity of the dipoles we assumed.
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Normally we will consider scattering of modes $a_l$ much weaker than
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the field forming the lattice potential $V_\mathrm{cl}(\b{r})$. We now
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proceed in the same way as when deriving the conventional Bose-Hubbard
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Hamiltonian in the zero temperature limit. The field operators
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$\Psi(\b{r})$ can be expanded using localised Wannier functions of
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$V_\mathrm{cl}(\b{r})$ and by keeping only the lowest vibrational
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state we get
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\begin{equation}
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\Psi(\b{r}) = \sum_i^M b_i w(\b{r} - \b{r}_i),
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\end{equation}
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where $b_i$ ($\bd_i$) is the annihilation (creation) operator of an
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atom at site $i$ with coordinate $\b{r}_i$, $w(\b{r})$ is the lowest
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band Wannier function, and $M$ is the number of lattice
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sites. Substituting this expression in Eq. \eqref{eq:FullH} with
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$\H_{1,a} = \H_{1,a}^\mathrm{eff}$ and
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$\H_{1,fa} = \H_{1,fa}^\mathrm{eff}$ given by Eq. \eqref{eq:aeff} and
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\eqref{eq:faeff} respectively yields the following generalised
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Bose-Hubbard Hamiltonian, $\H = \H_f + \H_a + \H_{fa}$,
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\begin{equation}
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\H = \H_f + \sum_{i,j}^M J^\mathrm{cl}_{i,j} \bd_i b_j +
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\sum_{i,j,k,l}^M \frac{U_{ijkl}}{2} \bd_i \bd_j b_k b_l +
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\frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m
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\left( \sum_{i,j}^K J^{l,m}_{i,j} \bd_i b_j \right).
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\end{equation}
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The optical field part of the Hamiltonian, $\H_f$, has remained
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unaffected by all our approximations and is given by
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Eq. \eqref{eq:Hf}. The matter-field Hamiltonian is now given by the
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well known Bose-Hubbard Hamiltonian
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\begin{equation}
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\H_a = \sum_{i,j}^M J^\mathrm{cl}_{i,j} \bd_i b_j +
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\sum_{i,j,k,l}^M \frac{U_{ijkl}}{2} \bd_i \bd_j b_k b_l,
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\end{equation}
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where the first term represents atoms tunnelling between sites with a
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hopping rate given by
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\begin{equation}
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J^\mathrm{cl}_{i,j} = \int \mathrm{d}^3 \b{r} w (\b{r} - \b{r}_i )
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\left( -\frac{\b{p}^2}{2 m_a} + V_\mathrm{cl}(\b{r}) \right) w(\b{r}
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- \b{r}_i),
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\end{equation}
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and $U_{ijkl}$ is the atomic interaction term given by
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\begin{equation}
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U_{ijkl} = \frac{4 \pi a_s \hbar^2}{m_a} \int \mathrm{d}^3 \b{r}
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w(\b{r} - \b{r}_i) w(\b{r} - \b{r}_j) w(\b{r} - \b{r}_k) w(\b{r} - \b{r}_l).
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\end{equation}
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Both integrals above depend on the overlap between Wannier functions
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corresponding to different lattice sites. This overlap decreases
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rapidly as the distance between sites increases. Therefore, in both
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cases we can simply neglect all terms but the one that corresponds to
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the most significant overlap. Thus, for $J_{i,j}^\mathrm{cl}$ we will
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only consider $i$ and $j$ that correspond to nearest neighbours.
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Furthermore, since we will only be looking at lattices that have the
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same separtion between all its nearest neighbours (e.g. cubic or
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square lattice) we can define $J_{i,j}^\mathrm{cl} = - J^\mathrm{cl}$
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(negative sign, because this way $J^\mathrm{cl} > 0$). For the
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inter-atomic interactions this simplifies to simply considering
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on-site collisions where $i=j=k=l$ and we define $U_{iiii} =
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U$. Finally, we end up with the canonical form for the Bose-Hubbard
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Hamiltonian
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\begin{equation}
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\H_a = -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j +
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\frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1),
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\end{equation}
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where $\langle i,j \rangle$ denotes a summation over nearest
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neighbours and $\hat{n}_i = \bd_i b_i$ is the atom number operator at
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site $i$.
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Finally, we have the light-matter interaction part of the Hamiltonian
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given by
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\begin{equation}
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\H_{fa} = \frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m
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\left( \sum_{i,j}^K J^{l,m}_{i,j} \bd_i b_j \right),
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\end{equation}
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where the coupling between the matter and optical fields is determined
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by the coefficients
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\begin{equation}
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J^{l,m}_{i,j} = \int \mathrm{d}^3 \b{r} w(\b{r} - \b{r}_i)
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u_l^*(\b{r}) u_m(\b{r}) w(\b{r} - \b{r}_j).
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\end{equation}
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This contribution can be separated into two parts, one which couples
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directly to the on-site atomic density and one that couples to the
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tunnelling operators. We will define the operator
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\begin{equation}
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\hat{F}_{l,m} = \hat{D}_{l,m} + \hat{B}_{l,m},
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\end{equation}
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where $\hat{D}_{l,m}$ is the direct coupling to atomic density
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\begin{equation}
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\hat{D}_{l,m} = \sum_{i}^K J^{l,m}_{i,i} \hat{n}_i,
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\end{equation}
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and $\hat{B}_{l,m}$ couples to the matter-field via the
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nearest-neighbour tunnelling operators
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\begin{equation}
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\hat{B}_{l,m} = \sum_{\langle i, j \rangle}^K J^{l,m}_{i,j} \bd_i b_j,
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\end{equation}
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and we neglect couplings beyond nearest neighbours for the same reason
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as before when deriving the matter Hamiltonian.
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It is important to note that we are considering a situation where the
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contribution of quantized light is much weaker than that of the
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classical trapping potential. If that was not the case, it would be
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necessary to determine thw Wannier functions in a self-consistent way
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which takes into account the depth of the quantum poterntial generated
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by the quantized light modes. This significantly complicates the
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treatment, but can lead to interesting physics. Amongst other things,
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the atomic tunnelling and interaction coefficients will now depend on
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the quantum state of light.
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\mynote{cite Santiago's papers and Maschler/Igor EPJD}
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Therefore, combining these final simplifications we finally arrive at
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our quantum light-matter Hamiltonian
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\begin{equation}
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\H = \H_f -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j +
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\frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) +
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\frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m} -
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i \sum_l \kappa_l \ad_l \a_l,
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\end{equation}
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where we have phenomologically included the cavity decay rates
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$\kappa_l$ of the modes $a_l$. A crucial observation about the
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structure of this Hamiltonian is that in the last term, the light
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modes $a_l$ couple to the matter in a global way. Instead of
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considering individual coupling to each site, the optical field
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couples to the global state of the atoms within the illuminated region
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via the operator $\hat{F}_{l,m}$. This will have important
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implications for the system and is one of the leading factors
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responsible for many-body behaviour beyong the Bose-Hubbard
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Hamiltonian paradigm.
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\subsection{Scattered light behaviour}
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Having derived the full quantum light-matter Hamiltonian we will no
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look at the behaviour of the scattered light. We begin by looking at
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the equations of motion in the Heisenberg picture
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\begin{equation}
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\dot{\a_l} = \frac{i}{\hbar} [\hat{H}, \a_l] =
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-i \left(\omega_l + U_{l,l}\hat{F}_{1,1} \right) \a_l
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- i \sum_{m \ne l} U_{l,m} \hat{F}_{l,m} \a_m
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- \kappa_l \a_l + \eta_l,
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\end{equation}
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where we have defined $U_{l,m} = g_l g_m / \Delta_a$. By considering
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the steady state solution in the frame rotating with the probe
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frequency $\omega_p$ we can show that the stationary solution is given
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by
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\begin{equation}
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\a_l = \frac{\sum_{m \ne l} U_{l,m} \a_m \hat{F}_{l,m} + i \eta_l}
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{(\Delta_{lp} - U_{l,l} \hat{F}_{l,l}) + i \kappa_l},
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\end{equation}
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|
where $\Delta_{lp} = \omega_p - \omega_l$ is the detuning between the
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probe beam and the light mode $a_l$. This equation gives us a direct
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relationship between the light operators and the atomic operators. In
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the stationary limit, the quantum state of the light field
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adiabatically follows the quantum state of matter.
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The above equation is quite general as it includes an arbitrary number
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of light modes which can be pumped directly into the cavity or
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produced via scattering from other modes. To simplify the equation
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slightly we will neglect the cavity resonancy shift,
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$U_{l,l} \hat{F}_{l,l}$ which is possible provided the cavity decay
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|
rate and/or probe detuning are large enough. We will also only
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|
consider probing with an external coherent beam and thus we neglect
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|
any cavity pumping $\eta_l$. This leads to a linear relationship
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|
between the light mode and the atomic operator $\hat{F}_{l,m}$
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|
|
\begin{equation}
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|
|
\a_l = \frac{\sum_{m \ne l} U_{l,m} \a_m}
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|
{\Delta_{lp} + i \kappa_l} \hat{F}_{l,m} = C \hat{F}_{l,m},
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|
\end{equation}
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|
|
where we have defined
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|
$C = \sum_{m \ne l} U_{l,m} \a_m / (\Delta_{lp} + i \kappa_l)$ which
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|
is essentially the Rayleigh scattering coefficient into the
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|
cavity. Whilst the light amplitude itself is only linear in atomic
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|
operators, we can easily have access to higher moments by simply
|
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|
simply considering higher moments of the $\a_l$ such as the photon
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|
number $\ad_l \a_l$.
|