228 lines
10 KiB
TeX
228 lines
10 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[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:TwoH}
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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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\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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\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. However, a
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later inclusion of this scattered light would not be difficult due to
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the linearity of the dipoles we assumed.
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