820 lines
40 KiB
TeX
820 lines
40 KiB
TeX
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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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\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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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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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=1.0\textwidth]{LatticeDiagram}
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\caption[Experimental Setup]{Atoms (green) trapped in an optical
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lattice are illuminated by a coherent probe beam (red). The light
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scatters (blue) in free space or into a cavity and is measured by
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a detector. If the experiment is in free space light can scatter
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in any direction. A cavity on the other hand enhances scattering
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in 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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\label{eq:sigmam}
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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.
|
|
|
|
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 $\hat{F}_{1,0}$
|
|
\begin{equation}
|
|
\a_1 = \frac{U_{1,0} a_0} {\Delta_{p} + i \kappa} \hat{F} =
|
|
C \hat{F},
|
|
\end{equation}
|
|
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 $\hat{B}_{1,0} \equiv \hat{B}$.
|
|
|
|
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}
|
|
|
|
The Electric field operator at position $\b{r}$ and at time $t$ is
|
|
usually written in terms of its positive and negative components:
|
|
\begin{equation}
|
|
\b{\hat{E}}(\b{r},t) = \b{\hat{E}}^{(+)}(\b{r},t) + \b{\hat{E}}^{(-)}(\b{r},t),
|
|
\end{equation}
|
|
where
|
|
\begin{subequations}
|
|
\begin{equation}
|
|
\b{\hat{E}}^{(+)}(\b{r},t) = \sum_{\b{k}} \hat{\epsilon}_{\b{k}}
|
|
\mathcal{E}_{\b{k}} \a_{\b{k}} e^{-i \omega_{\b{k}} t + i \b{k} \cdot \b{r}},
|
|
\end{equation}
|
|
\begin{equation}
|
|
\b{\hat{E}}^{(-)}(\b{r},t) = \sum_{\b{k}} \hat{\epsilon}_{\b{k}}
|
|
\mathcal{E}_{\b{k}} \a^\dagger_{\b{k}} e^{i \omega_{\b{k}} t - i \b{k} \cdot \b{r}},
|
|
\end{equation}
|
|
\end{subequations}
|
|
$\hat{\epsilon}_{\b{k}}$ is a unit polarization vector, $\b{k}$ is the
|
|
wave vector,
|
|
$\mathcal{E}_{\b{k}} = \sqrt{\hbar \omega_{\b{k}} / 2 \epsilon_0 V}$,
|
|
$\epsilon_0$ is the free space permittivity, $V$ is the quantization
|
|
volume and $\a_\b{k}$ and $\a_\b{k}^\dagger$ are the annihilation and
|
|
creation operators respectively of a photon in mode $\b{k}$, and
|
|
$\omega_\b{k}$ is the angular frequency of mode $\b{k}$.
|
|
|
|
In Ref. \cite{Scully} it was shown that the operator
|
|
$\b{\hat{E}}^{(+)}(\b{r},t)$ due to light scattering from a single
|
|
atom located at $\b{r}^\prime$ at the observation point $\b{r}$ is
|
|
given by
|
|
\begin{equation}
|
|
\label{eq:Ep}
|
|
\b{\hat{E}}^{(+)}(\b{r},\b{r}^\prime,t) = \frac{\omega_a^2 d \sin \eta}{4 \pi
|
|
\epsilon_0 c^2 |\b{r} - \b{r}^\prime|} \hat{\epsilon} \sigma^-
|
|
\left( \b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right),
|
|
\end{equation}
|
|
with a similar expression for
|
|
$\b{E}^{(-)}(\b{r},\b{r}^\prime,t)$. Eq. \eqref{eq:Ep} is valid only
|
|
in the far field, $\eta$ is the angle the dipole makes with
|
|
$\b{r} - \b{r}^\prime$, $\hat{\epsilon}$ is the polarization vector
|
|
which is perpendicular to $\b{r} - \b{r}^\prime$ and lies in the plane
|
|
defined by $\b{r} - \b{r}^\prime$ and the dipole, $\omega_a$ is the
|
|
atomic transition frequency, and $d$ is the dipole matrix element
|
|
between the two levels, and $c$ is the speed of light in vacuum.
|
|
|
|
|
|
We have already derived an expression for the atomic lowering
|
|
operator, $\sigma^-$, in Eq. \eqref{eq:sigmam} and it is given by
|
|
\begin{equation}
|
|
\sigma^-(\b{r}, t) = - \frac{i} {\Delta_a} \sum_\b{k} g_\b{k}
|
|
a_\b{k}(t) u_\b{k}(\b{r}).
|
|
\end{equation}
|
|
We will want to substitute this expression into Eq. \eqref{eq:Ep}, but
|
|
first we note that in the setup we consider the coherent probe will be
|
|
much stronger than the scattered modes. This in turn implies that the
|
|
expression for $\sigma^-$ will be dominated by this external
|
|
probe. Therefore, we drop the sum and consider only a single wave
|
|
vector, $\b{k}_0$, of the dominant contribution from the external
|
|
beam corresponding to the mode $\a_0$.
|
|
|
|
We now use the definititon of the atom-light coupling constant
|
|
\cite{Scully} to make the substitution
|
|
$g_0 a_0 = -i \Omega_0 e^{-i \omega_0 t} / 2$, where
|
|
$\Omega_0$ is the Rabi frequency for the probe-atom system. We now
|
|
evaluate the polarisation operator at the retarded time
|
|
\begin{equation}
|
|
\sigma^- \left(\b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right) =
|
|
-\frac{\Omega_0}{2 \Delta_a} u_0 (\b{r}^\prime) \exp \left[-i \omega_0
|
|
\left( t - \frac{|\b{r} - \b{r}^\prime|}{c} \right)\right].
|
|
\end{equation}
|
|
We are considering observation in the far field regime so
|
|
$|\b{r} - \b{r}^\prime| \approx r - \hat{\b{r}} \cdot
|
|
\b{r}^\prime$. We also consider the incoming, $\b{k}_0$, and the
|
|
outgoing, $\mathbf{k}_1$, waves to be of the same wavelength,
|
|
$|\b{k}_0| = |\b{k}_1| = k = \omega_0 / c$, hence
|
|
$(\omega_0 / c) |\b{r} - \b{r}^\prime| \approx k (r - \hat{\b{r}}
|
|
\cdot \b{r}^\prime) = \b{k}_1 \cdot (\b{r} - \b{r}^\prime )$ and
|
|
\begin{equation}
|
|
\sigma^- \left(\b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right) \approx
|
|
-\frac{\Omega_0}{2 \Delta_a} u_0 (\b{r}^\prime) \exp \left[i
|
|
\b{k}_1 \cdot (\b{r} - \b{r}^\prime ) - i \omega_0 t \right].
|
|
\end{equation}
|
|
|
|
The many-body version of the electric field operator is given by
|
|
\begin{equation}
|
|
\b{\hat{E}}^{(+)}_N(\b{r},t) = \int \mathrm{d}^3 \b{r}^\prime
|
|
\Psi^\dagger (\b{r}^\prime) \b{\hat{E}}^{(+)}(\b{r},\b{r}^\prime,t) \Psi(\b{r}^\prime),
|
|
\end{equation}
|
|
where $\Psi(\b{r})$ is the atomic matter-field operator. As before, we
|
|
expand this field operator using localized Wannier functions
|
|
corresponding to the lattice potential and keeping only the lowest
|
|
vibrational state at each site:
|
|
$\Psi(\b{r}) = \sum_i b_i w(\b{r} - \b{r}_i)$, where $b_i$ is the
|
|
annihilation operator of an atom at site $i$ with the coordinate
|
|
$\b{r}_i$. Thus, the relevant many-body operator is
|
|
\begin{subequations}
|
|
\begin{equation}
|
|
\b{\hat{E}}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E
|
|
\sum_{i,j}^K \bd_i b_j \int \mathrm{d}^3 \b{r}^\prime
|
|
w^* (\b{r}^\prime - \b{r}_i) \frac{u_0
|
|
(\b{r}^\prime)}{|\b{r} - \b{r}^\prime|} e^ {i \b{k}_1
|
|
\cdot (\b{r} - \b{r}^\prime ) - i \omega_0 t }
|
|
w(\b{r}^\prime - \b{r}_j),
|
|
\end{equation}
|
|
\begin{equation}
|
|
C_E = -\frac{\omega_a^2 \Omega_0 d \sin \eta}{8 \pi \epsilon_0 c^2
|
|
\Delta_a}
|
|
\end{equation}
|
|
\end{subequations}
|
|
where $K$ is the number of illuminated sites. We now assume that the
|
|
lattice is deep and that the light scattering occurs on time scales
|
|
much shorter than atom tunneling. Therefore, we ignore all terms for
|
|
which $i \ne j$ and the remaining integrals become $\int \mathrm{d}^3
|
|
\b{r}_0 w^*(\b{r}_0 - \b{r}_i) f (\b{r})
|
|
w(\b{r}_0 - \b{r}_i) = f(\b{r}_i)$. The final form of
|
|
the many body operator is then
|
|
\begin{equation}
|
|
\b{E}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E
|
|
\sum_{j = 1}^K \hat{n}_j \frac{u_0 (\b{r}_j)}{|\b{r} -
|
|
\b{r}_j|} e^ {i \b{k}_1 \cdot (\b{r} - \b{r}_j
|
|
) - i \omega_0 t },
|
|
\end{equation}
|
|
where $\hat{n}_i = b^\dagger_ib_i$ is the atom number operator at site
|
|
$i$. We will now consider the incoming probe to be a plane wave,
|
|
$u_0(\b{r}) = e^{i \b{k}_0 \cdot \b{r}}$, and we
|
|
approximate $|\b{r} - \b{r}_j| \approx r$ in the
|
|
denominator. Thus,
|
|
\begin{equation}
|
|
\b{E}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E \frac{e^ {ikr - i\omega_0t}}{r}
|
|
\sum_{j = 1}^K \hat{n}_j e^ {i (\b{k}_0 - \b{k}_1) \cdot \b{r}_j}.
|
|
\end{equation}
|
|
We note that this is exactly the same form of the optical field as
|
|
derived from a generalised Bose-Hubbard model which considers a fully
|
|
quantum light-matter interaction Hamiltonian \cite{mekhov2007prl,
|
|
mekhov2007pra, mekhov2012}. We can even express the result in terms of
|
|
the $\hat{D} = \sum_j^K u_1^*(\b{r}_j) u_0(\b{r}_j)
|
|
\hat{n}_j$ formalism used in those works
|
|
\begin{equation}
|
|
\b{E}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E \frac{e^
|
|
{ikr - i\omega_0t}}{r} \hat{D}.
|
|
\end{equation}
|
|
|
|
The average light intensity at point $\b{r}$ at time $t$ is given
|
|
by the formula $I = c \epsilon_0 |E|^2/2$ and yields
|
|
\begin{equation}
|
|
\langle I (\b{r}, t) \rangle = c \epsilon_0 \langle
|
|
E^{(-)} (\b{r}, t) E^{(+)} (\b{r}, t) \rangle =
|
|
\frac{c \epsilon_0}{r^2} C_E^2 \langle \hat{D}^* \hat{D} \rangle.
|
|
\end{equation}
|
|
The scattering is dominated by a uniform background for which $\langle
|
|
\hat{D}^* \hat{D} \rangle \approx N_K$ for a superfluid
|
|
\cite{mekhov2012}, where $N_K$ is the mean number of atoms in the
|
|
illuminated volume. Thus, the approximate number of photons scattered
|
|
per second can be obtained by integrating over a sphere
|
|
\begin{equation}
|
|
n_{\Phi} = \frac{4 \pi c \epsilon_0}{3 \hbar \omega_a} \left(\frac{\omega_a^2
|
|
\Omega_0 d}{8 \pi \epsilon_0 c^2 \Delta_a}\right)^2 N_K.
|
|
\end{equation}
|
|
In the Weisskopf-Wigner approximation it can be shown \cite{Scully}
|
|
that for a two level system the decay constant, $\Gamma$, is
|
|
\begin{equation}
|
|
\Gamma = \frac{\omega_a^3 d^2}{3 \pi \epsilon_0 \hbar c^3}.
|
|
\end{equation}
|
|
Therefore, we can now express the quantity $n_{\Phi}$ as
|
|
\begin{equation}
|
|
n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K.
|
|
\end{equation}
|
|
Estimates of the scattering rate using real experimental parameters
|
|
are given in Table \ref{tab:photons}.
|
|
|
|
\begin{table}[!htbp]
|
|
\centering
|
|
\begin{tabular}{|c|c|c|}
|
|
\hline
|
|
Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \hline
|
|
$\omega_a$ & \multicolumn{2}{|c|}{$2 \pi \cdot 384$ THz}\\ \hline
|
|
$\Gamma$ & \multicolumn{2}{|c|}{$2 \pi \cdot 6.07$ MHz} \\ \hline
|
|
$\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\ \hline
|
|
$I$ & $4250$ Wm$^{-2}$ & N/A \\ \hline
|
|
$\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\ \hline
|
|
$N_K$ & 10$^5$ & 147 \\ \hline \hline
|
|
$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\ \hline
|
|
\end{tabular}
|
|
\caption[Photon Scattering Rates]{ Rubidium data taken from
|
|
Ref. \cite{steck}; Miyake \emph{et al.}: based on
|
|
Ref. \cite{miyake2011}. The $5^2S_{1/2}$, $F=2 \rightarrow
|
|
5^2P_{3/2}$, $F^\prime = 3$ transition of $^{87}$Rb is
|
|
considered. For this transition the Rabi frequency is actually
|
|
larger than the detuning and and effects of saturation should be
|
|
taken into account in a more complete analysis. However, it is
|
|
included for discussion. The detuning is said to be a few natural
|
|
linewidths. Note that it is much smaller than $\Omega_0$. The Rabi
|
|
fequency is $\Omega_0 = (d_\mathrm{eff}/\hbar)\sqrt{2 I / c
|
|
\epsilon_0}$ which is obtained from definition of Rabi
|
|
frequency, $\Omega = \mathbf{d} \cdot \mathbf{E} / \hbar$,
|
|
assuming the electric field is parallel to the dipole, and the
|
|
relation $I = c \epsilon_0 |E|^2 /2$. We used $d_\mathrm{eff} =
|
|
1.73 \times 10^{-29}$ Cm \cite{steck}; Weitenberg \emph{et al.}:
|
|
based on Ref. \cite{weitenberg2011, weitenbergThesis}. The
|
|
$5^2S_{1/2} \rightarrow 5^2P_{3/2}$ transition of $^{87}$Rb is
|
|
used. Ref. \cite{weitenberg2011} gives the free space detuning to
|
|
be $\Delta_\mathrm{free} = - 2 \pi \cdot 45$
|
|
MHz. Ref. \cite{weitenbergThesis} clarifies that the relevant
|
|
detuning is $\Delta = \Delta_\mathrm{free} + \Delta_\mathrm{lat}$,
|
|
where $\Delta_\mathrm{lat} = - 2 \pi \cdot 40$
|
|
MHz. Ref. \cite{weitenbergThesis} uses a saturation parameter
|
|
$s_\mathrm{tot}$ to quantify the intensity of the beams which is
|
|
related to the Rabi frequency, $s_\mathrm{tot} = 2 \Omega^2 /
|
|
\Gamma^2$ \cite{steck,foot}. We can extract $s_\mathrm{tot}$ for
|
|
the experiment from the total scattering rate by
|
|
$\Gamma_\mathrm{sc} = (\Gamma/2) (s_\mathrm{tot}) /
|
|
(1+s_\mathrm{tot}+(2 \Delta / \Gamma)^2)$. A scattering rate of 60
|
|
kHz per atom \cite{weitenberg2011} gives $s_\mathrm{tot} = 2.5$.}
|
|
\label{tab:photons}
|
|
\end{table}
|