1195 lines
58 KiB
TeX
1195 lines
58 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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\section{Introduction}
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In this chapter, we present the derivation of a general Hamiltonian
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that describes the coupling of atoms with far-detuned optical beams
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originally presented in Ref. \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. As this model extends the
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Bose-Hubbard Hamiltonian to include the effects of interactions with
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quantised light we also present a brief overview of the properties of
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the Bose-Hubbard model itself and its quantum phase transition. This is
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followed by a description of the behaviour of the scattered light with
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a particular focus on how to couple the optical fields to phase
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observables as opposed to density observables as is typically the
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case. Finally, we conclude with an overview of possible experimental
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realisability.
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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, from
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free particles to strongly correlated systems, due to the inherent
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tunability of such lattices. Furthermore, this model is capable of
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describing a range of different experimental setups ranging from a
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small number of sites with a large filling factor (e.g.~BECs trapped
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in a double-well potential) to an extended multi-site lattice with a
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low filling factor (e.g.~a system with one atom per site which will
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exhibit the Mott insulator to superfluid quantum phase transition).
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{LatticeDiagram}
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\caption[Experimental Setup in Free Space]{Atoms (green) trapped in
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an optical lattice are illuminated by a coherent probe beam (red),
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$a_0$, with a mode function $u_0(\b{r})$ which is at an angle
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$\theta_0$ to the normal to the lattice. The light scatters (blue)
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into the mode $\a_1$ in free space or into a cavity and is
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measured by a detector. Its mode function is given by $u_1(\b{r})$
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and it is at an angle $\theta_1$ relative to the normal to the
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lattice. 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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An optical lattice can be formed with classical light beams that form
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standing waves. Depending on the detuning with respect to the atomic
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resonance, the nodes or antinodes form the lattice sites in which
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atoms accumulate. As shown in Fig. \ref{fig:LatticeDiagram} the
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trapped bosons (green) are illuminated with a coherent probe beam
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(red) and scatter light into a different mode (blue) which is then
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measured with a detector. The most straightforward measurement is to
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simply count the number of photons with a photodetector, but it is
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also possible to perform a quadrature measurement by using a homodyne
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detection scheme \cite{carmichael, atoms2015}. The experiment can be
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performed in free space where light can scatter in any direction. The
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atoms can also be placed inside a cavity which has the advantage of
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being able to enhance light scattering in a particular direction
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\cite{bux2013, kessler2014, landig2015}. Furthermore, cavities allow
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for the formation of a fully quantum potential in contrast to the
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classical lattice trap.
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For simplicity we will be considering one-dimensional lattices most of
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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 two- and
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three-dimensional systems. This simplification allows us to present a
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much more intuitive picture of the physical setup where we only need
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to concern ourselves with a single angle for each optical mode. As
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shown in Fig. \ref{fig:LatticeDiagram} the angle between the normal to
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the lattice and the probe and detected beam are denoted by $\theta_0$
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and $\theta_1$ respectively. We will consider these angles to be
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tunable although the same effect can be achieved by varying the
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wavelength of the light modes. However, it is much more intuitive to
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consider variable angles in our model as this lends itself to a
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simpler geometrical representation.
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\section{Derivation of the Hamiltonian}
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\label{sec:derivation}
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A general many-body Hamiltonian coupled to a quantised light field in
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second quantised 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 $\a_l$ ($\ad_l$) are the annihilation (creation)
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operators of light modes with frequencies $\omega_l$, wave vectors
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$\b{k}_l$, and mode functions $u_l(\b{r})$, which can be pumped by
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coherent fields with amplitudes $\eta_l$. The second part of the
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Hamiltonian, $\H_a$, 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 quantised 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
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$V_\mathrm{cl}(\b{r})$. Therefore, we assume that the trapping is
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entirely due to the classical potential and the field operators
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$\Psi(\b{r})$ can be expanded using localised Wannier functions of
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$V_\mathrm{cl}(\b{r})$. By keeping only the lowest vibrational state
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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_{m,n}^M J^\mathrm{cl}_{m,n} \bd_m b_n +
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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}_j),
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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 separation between all its nearest neighbours (e.g. cubic or
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square lattice) we can define $J_{i,j}^\mathrm{cl} = - J$ (negative
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sign, because this way $J > 0$). For the inter-atomic interactions
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this simplifies to simply considering on-site collisions where
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$i=j=k=l$ and we define $U_{iiii} = U$. Finally, we end up with the
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canonical form for the Bose-Hubbard Hamiltonian
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\begin{equation}
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\H_a = -J \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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\label{eq:F}
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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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\label{eq:D}
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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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\label{eq:B}
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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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where $K$ denotes a sum over the illuminated sites and we neglect
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couplings beyond nearest neighbours for the same reason as before when
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deriving the matter Hamiltonian. Thus the interaction part of the
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Hamiltonian is 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 \hat{F}_{l,m}
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\end{equation}
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These equations encapsulate the simplicity and flexibility of the
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measurement scheme that we are proposing. The operators given above
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are entirely determined by the values of the $J^{l,m}_{i,j}$
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coefficients and despite its simplicity, this is sufficient to give
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rise to a host of interesting phenomena via measurement backaction
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such as the generation of multipartite entangled spatial modes in an
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optical lattice \cite{mekhov2009pra, elliott2015, atoms2015}, the
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appearance of long-range correlated tunnelling capable of entangling
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distant lattice sites, and in the case of fermions, the break-up and
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protection of strongly interacting pairs \cite{mazzucchi2016,
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kozlowski2016zeno}. Additionally, these coefficients are easy to
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manipulate experimentally by adjusting the optical geometry via the
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light mode functions $u_l(\b{r})$.
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It is important to note that we are considering a situation where the
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contribution of quantised 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 the Wannier functions in a self-consistent way
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which takes into account the depth of the quantum potential generated
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by the quantised 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 \cite{mekhov2008}.
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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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\label{eq:fullH}
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\H = \H_f -J \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 phenomenologically 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 interaction term, the
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light 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 beyond the Bose-Hubbard
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Hamiltonian paradigm.
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|
Furthermore, it is also vital to note that light couples to the matter
|
|
via an operator, namely $\hat{F}_{l,m}$, which makes it sensitive to
|
|
the quantum state of matter. This is a key element of our treatment of
|
|
the ultimate quantum regime of light-matter interaction that goes
|
|
beyond previous treatments.
|
|
|
|
\section{The Bose-Hubbard Hamiltonian}
|
|
|
|
\subsection{The Model}
|
|
|
|
The original Hubbard model was developed as a model for the motion of
|
|
electrons within transition metals \cite{hubbard1963}. The key
|
|
elements were the confinement of the motion to a lattice and the
|
|
approximation of the Coulomb screening interaction as a simple on-site
|
|
interaction. Despite these enormous simplifications the model was very
|
|
successful \cite{leggett}. The Bose-Hubbard model is an even simpler
|
|
variation where instead of fermions we consider spinless bosons. It
|
|
was originally devised as a toy model and applied to liquid helium
|
|
\cite{fisher1989}, but in 1998 it was shown by Jaksch \emph{et.~al.}
|
|
that it can be realised with ultracold atoms in an optical lattice
|
|
\cite{jaksch1998}. Shortly afterwards it was obtained in a
|
|
ground-breaking experiment \cite{greiner2002}. The model has been the
|
|
subject of intense research since then, because despite its simplicity
|
|
it possesses highly nontrivial properties such as the superfluid to
|
|
Mott insulator quantum phase transition. Furthermore, it is one of the
|
|
most controllable quantum many-body systems thus providing a solid
|
|
basis for new experiments and technologies.
|
|
|
|
The model we have derived is essentially an extension of the
|
|
well-known Bose-Hubbard model that also includes interactions with
|
|
quantised light. Therefore, it should come as no surprise that if we
|
|
eliminate all the quantised fields from the Hamiltonian we obtain
|
|
exactly the Bose-Hubbard model
|
|
\begin{equation}
|
|
\H_a = -J \sum_{\langle i,j \rangle}^M \bd_i b_j +
|
|
\frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1).
|
|
\end{equation}
|
|
The first term is the kinetic energy term and denotes the hopping of
|
|
atoms between neighbouring sites at a hopping rate given by $J$. The
|
|
second term describes the on-site repulsion which acts mainly to
|
|
suppress multiple occupancies of any site, but at the same time it
|
|
lies at the heart of strongly correlated phenomena in the Bose-Hubbard
|
|
model. Unfortunately, despite its simplicity, this system is not
|
|
solvable for any finite value $U/J$ using known techniques in finite
|
|
dimensions \cite{krauth1991, kolovsky2004, calzetta2006}. However,
|
|
exact solutions exist for the ground state in the two limits of
|
|
$U = 0$ and $J = 0$ which already provide some insight as they
|
|
correspond to the two different phases of the quantum phase transition
|
|
that is present in the model. In higher dimensions (e.g.~two or three)
|
|
the system is reasonably well described by a mean-field approximation
|
|
which provides us with a good understanding of the full quantum phase
|
|
diagram.
|
|
|
|
\subsection{The Superfluid and Mott Insulator Ground States}
|
|
|
|
Before trying to understand the quantum phase transition itself we
|
|
will look at the ground states of the two extremal cases first and for
|
|
simplicity we will consider only homogeneous systems with periodic
|
|
boundary conditions here. We will begin with the $U = 0$ limit where
|
|
the Hamiltonian's kinetic term can be diagonalised by transforming to
|
|
momentum space defined by the annihilation operator
|
|
\begin{equation}
|
|
b_\b{k} = \frac{1} {\sqrt{M}} \sum_m b_m e^{i \b{k} \cdot \b{r}_m},
|
|
\end{equation}
|
|
where $\b{k}$ denotes the wave vector running over the first Brillouin
|
|
zone. The Hamiltonian then is given by
|
|
\begin{equation}
|
|
\H_a = \sum_\b{k} \epsilon_\b{k} \bd_\b{k} b_\b{k},
|
|
\end{equation}
|
|
where the free particle dispersion is
|
|
\begin{equation}
|
|
\epsilon_\b{k} = -2 J \sum_{d = 1}^D \cos(k_d a),
|
|
\end{equation}
|
|
and $d$ denotes the dimension out of a total of $D$ dimensions. It
|
|
should now be obvious that the ground state is simply the state with
|
|
all the atoms in the $\b{k} = 0$ state and thus it is given by
|
|
\begin{equation}
|
|
\label{eq:GSSF}
|
|
| \Psi_\mathrm{SF} \rangle = \frac{ ( \bd_{\b{k} = 0} )^N } {\sqrt{N!}} | 0
|
|
\rangle = \frac{1} {\sqrt{N!}} \left( \frac{1} {\sqrt{M}} \sum_m^M
|
|
\bd_m \right)^N | 0 \rangle,
|
|
\end{equation}
|
|
where $| 0 \rangle$ denotes the vacuum state. This state is called a
|
|
superfluid (SF) due to its viscous-free flow properties
|
|
\cite{leggett1999}. The macroscopic occupancy of the single-particle
|
|
ground state is the signature and defining property of a
|
|
Bose-Einstein condensate. Note that the zero momentum states consist of a
|
|
superposition of an atom at all the sites of the optical lattice
|
|
highlighting the fact that kinetic energy is minimised by delocalising
|
|
the particles. This in turn implies that a superfluid state will have
|
|
infinite range correlations. It is also important to note that in the
|
|
thermodynamic limit the superfluid state is gapless, i.e.~it takes
|
|
zero energy to excite the system.
|
|
|
|
At the other end of the spectrum, i.e.~$J = 0$, the kinetic term
|
|
vanishes and the system is dominated by the on-site interactions. This
|
|
has the effect of decoupling the Hamiltonian into a sum of identical
|
|
single-site Hamiltonians which are already diagonal in the atom number
|
|
basis. Therefore, the ground state for $N = gM$, where $g$ is an
|
|
integer, is given by
|
|
\begin{equation}
|
|
\label{eq:GSMI}
|
|
| \Psi_\mathrm{MI} \rangle = \prod_m^M \frac{1} {\sqrt{g!}}
|
|
(\bd_m)^g | 0 \rangle.
|
|
\end{equation}
|
|
This state has precisely the same number of bosons, $g$, at each site
|
|
and is commonly known as the Mott insulator (MI). Unlike in the
|
|
superfluid state, the atoms are all localised to a specific lattice
|
|
site and thus not only are there no long-range correlations, there are
|
|
actually no two-site correlations in this state at all. Additionally,
|
|
this state is gapped, i.e.~it costs a finite amount of energy to
|
|
excite the system into its lowest excited state even in the
|
|
thermodynamic limit. In fact, the existence of the gap is the defining
|
|
property that distinguishes the Mott insulator from the superfluid in
|
|
the Bose-Hubbard model. Note that we have only considered a
|
|
commensurate case (number of atoms an integer multiple of the number
|
|
of sites). If we were to insert another atom on top of the Mott
|
|
insulator, the energy of the system would not depend on the lattice
|
|
site in which it is inserted. This implies that as soon as $J$ becomes
|
|
finite, the ground state would prefer this atom be delocalised and
|
|
effectively becomes gapless which means the system is a superfluid and
|
|
it will not exhibit a quantum phase transition.
|
|
|
|
\subsection{Mean-Field Theory}
|
|
|
|
Now that we have a basic understanding of the two limiting cases we
|
|
can now consider the model in between these two extremes. We have
|
|
already mentioned that an exact solution is not known, but fortunately
|
|
a very good mean-field approximation exists \cite{fisher1989}. In this
|
|
approach the interaction term is treated exactly, but the kinetic
|
|
energy term is decoupled as
|
|
\begin{equation}
|
|
\bd_m b_n = \langle \bd_m \rangle b_n + \bd_m \langle b_n \rangle -
|
|
\langle \bd_m \rangle \langle b_n \rangle = \Phi^* b_n + \Phi \bd_m
|
|
- | \Phi |^2,
|
|
\end{equation}
|
|
where the expectation values are for the $T = 0$ ground state. Note
|
|
that we have assumed that $\Phi = \langle b_m \rangle $ is non-zero
|
|
and homogeneous. $\Phi$ describes the influence of hopping between
|
|
neighbouring sites and is the mean -field order parameter. This
|
|
decoupling effect means that we can write the Hamiltonian as $\hat{H}_a
|
|
= \sum_m^M \hat{h}_a$, where
|
|
\begin{equation}
|
|
\hat{h}_a = -z J ( \Phi \bd + \Phi^* b) + z J | \Phi |^2 + \frac{U}{2}
|
|
\n (\n - 1) - \mu \n,
|
|
\end{equation}
|
|
and $z$ is the coordination number, i.e. the number of nearest
|
|
neighbours of a single site. We have also introduced the chemical
|
|
potential $\mu$ since we now consider the grand canonical ensemble as
|
|
the Hamiltonian no longer conserves the total atom number. The ground
|
|
state of the $| \Psi_0 \rangle$ of the overall system will be a
|
|
site-wise product of the individual ground states of
|
|
$\hat{h}_a$. These can be found very easily using standard
|
|
diagonalisation techniques where $\Phi$ is self-consistently
|
|
determined by minimising the energy of the ground state with respect
|
|
to $\Phi$.
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=0.8\linewidth]{BHPhase}
|
|
\caption[Mean-Field Bose-Hubbard Phase Diagram]{Mean-field phase
|
|
diagram of the Bose-Hubbard model in 1D, i.e.~ $z = 2$, from
|
|
Ref. \cite{StephenThesis}. The shaded regions are the Mott
|
|
insulator lobes and each lobe corresponds to a different on-site
|
|
filling labelled by $n$. The rest of the space corresponds to the
|
|
superfluid phase. The dashed lines are the phase boundaries
|
|
obtained from first-order perturbation theory. The solid lines are
|
|
lines of constant density in the superfluid
|
|
phase. \label{fig:BHPhase}}
|
|
\end{figure}
|
|
|
|
The main advantage of the mean-field treatment is that it lets us
|
|
study the quantum phase transition between the superfluid and Mott
|
|
insulator phases discussed in the previous sections. The phase
|
|
boundaries can be obtained from second-order perturbation theory as
|
|
\begin{equation}
|
|
\left( \frac{\mu} {U} \right)_\pm = \frac{1}{2} \left[ 2g - 1 -
|
|
\left( \frac{zJ}{U} \right) \pm \sqrt{ 1 - 2 (2g + 1) \left( \frac{zJ}{U}
|
|
\right) + \left( \frac{zJ}{U} \right)^2} \right].
|
|
\end{equation}
|
|
This yields a phase diagram with multiple Mott insulating lobes as
|
|
seen in Fig. \ref{fig:BHPhase} where each lobe represents a Mott
|
|
insulator with a different number of atoms per site. The tip of the
|
|
lobe which corresponds to the quantum phase transition along a line of
|
|
fixed density is obtained by equating the two branches of the above
|
|
equation to get
|
|
\begin{equation}
|
|
\left( \frac{J} {U} \right) = \frac{1} {z (2g + 1 + \sqrt{4g^2 + 4g} )}.
|
|
\end{equation}
|
|
It is important to note that the fact that this formalism predicts a
|
|
phase transition is in contrast to other mean-field theories such as
|
|
the Bogoliubov approximation \cite{PitaevskiiStringari}. This
|
|
highlights the fact that the interactions, which are treated exactly
|
|
here, are the dominant driver leading to this phase transition and
|
|
other strongly correlated effects in the Bose-Hubbard model.
|
|
|
|
\subsection{The Bose-Hubbard Model in One Dimension}
|
|
\label{sec:BHM1D}
|
|
|
|
The mean-field theory in the previous section is very useful tool for
|
|
studying the quantum phase transition in the Bose-Hubbard
|
|
model. However, it is effectively an infinite-dimensional theory and
|
|
in practice it only works in two dimensions or more. The phase
|
|
transition in 1D is poorly described, because it actually belongs to a
|
|
different universality class \cite{cazalilla2011, ejima2011,
|
|
kuhner2000, pino2012, pino2013}. This is clearly seen from the one
|
|
dimensional phase diagram shown in Fig. \ref{fig:1DPhase}.
|
|
|
|
Some general conclusions can be obtained by looking at Haldane's
|
|
prescription for Luttinger liquids \cite{haldane1981,
|
|
giamarchi}. Without a periodic potential the low-energy physics of
|
|
the system is described by the Hamiltonian
|
|
\begin{equation}
|
|
\hat{H}_a = \frac{1}{2 \pi} \int \mathrm{d} x \left\{ v K [ \hat{\Pi}(x)
|
|
]^2 + \frac{v} {K} [\partial_x \hat{\Phi}(x) ]^2 \right\},
|
|
\end{equation}
|
|
where we have expressed the bosonic field operators in terms of a
|
|
density operator $\hat{\rho}(x)$ and a phase operator $\hat{\Phi}(x)$
|
|
as $\hat{\Psi}(x) = \sqrt{\hat{\rho}(x)} e^{i \hat{\Phi}(x)}$
|
|
and $\hat{\Pi}(x)$ is the density fluctuation operator. Provided the
|
|
parameters $v$ and $K$ can be correctly determined this Hamiltonian
|
|
gives the correct description of the gapless superfluid phase of the
|
|
Bose-Hubbard model. Most importantly it gives an expression for the
|
|
spatial correlation functions such as
|
|
\begin{equation}
|
|
\langle \bd_i b_j \rangle = A \left( \frac{\alpha} {|i - j|} \right)^{K/2},
|
|
\end{equation}
|
|
where $A$ is some amplitude and $\alpha$ is a necessary cutoff to
|
|
regularise the theory at short distances. Unlike the superfluid ground
|
|
state in Eq. \eqref{eq:GSSF} this state does not have infinite range
|
|
correlations. They decay according to a power-law. However, for
|
|
non-interacting systems $K = 0$ and long-range order is re-established
|
|
as before though it is important to note that in higher dimensions
|
|
this long-range order persists in the whole superfluid phase even with
|
|
interactions present.
|
|
|
|
In order to describe the phase transition and the Mott insulating
|
|
phase it is necessary to introduce a periodic lattice potential. It
|
|
can be shown that this system exhibits at $T = 0$ a
|
|
Berezinskii-Kosterlitz-Thouless phase transition as the parameter $K$
|
|
is varied with a critical point at $K_c = \frac{1}{2}$ where
|
|
$K < \frac{1}{2}$ is a superfluid. Above $K = \frac{1}{2}$ the value
|
|
of $K$ jumps discontinuously to $K \rightarrow \infty$ producing the
|
|
Mott insulator phase. Unlike the gapless superfluid phase the spatial
|
|
correlations decay exponentially as
|
|
\begin{equation}
|
|
\langle \bd_i b_j \rangle = B e^{ - |i - j|/\xi},
|
|
\end{equation}
|
|
where $B$ is some constant and the correlation length is given by
|
|
$\xi = v / \Delta$ where $\Delta$ is the energy gap.
|
|
|
|
Using advanced numerical methods such as density matrix
|
|
renormalisation group (DMRG) calculations it is possible to identify
|
|
the critical point by fitting the power-law decay correlations in
|
|
order to obtain $K$. The resulting phase transition is shown in
|
|
Fig. \ref{fig:1DPhase} and the critical point was shown to be
|
|
$(U/zJ) = 1.68$. Note that unusually the phase diagram exhibits a
|
|
reentrant phase transition for a fixed $\mu$.
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=0.8\linewidth]{1DPhase}
|
|
\caption[Exact 1D Bose-Hubbard Phase Diagram]{Exact phase diagram
|
|
of the Bose-Hubbard model in 1D, i.e.~ $z = 2$, from
|
|
Ref. \cite{StephenThesis}. The shaded region corresponds to the
|
|
$n = 1$ Mott insulator lobe. The sharp tip distinguishes it from
|
|
the mean-field phase diagram. The red dotted line denotes the
|
|
reentrant phase transition which occurs for a fixed $\mu$.The
|
|
critical point for the fixed density $n = 1$ transition is denoted
|
|
by an 'x'. \label{fig:1DPhase}}
|
|
\end{figure}
|
|
|
|
\section{Scattered light behaviour}
|
|
\label{sec:a}
|
|
|
|
Having derived the full quantum light-matter Hamiltonian we will now
|
|
look at the behaviour of the scattered light. We begin by looking at
|
|
the equations of motion in the Heisenberg picture
|
|
\begin{equation}
|
|
\dot{\a_l} = \frac{i}{\hbar} [\hat{H}, \a_l] =
|
|
-i \left(\omega_l + U_{l,l}\hat{F}_{1,1} \right) \a_l
|
|
- i \sum_{m \ne l} U_{l,m} \hat{F}_{l,m} \a_m
|
|
- \kappa_l \a_l + \eta_l,
|
|
\end{equation}
|
|
where we have defined $U_{l,m} = g_l g_m / \Delta_a$. By considering
|
|
the steady state solution in the frame rotating with the probe
|
|
frequency $\omega_p$ we can show that the stationary solution is given
|
|
by
|
|
\begin{equation}
|
|
\a_l = \frac{\sum_{m \ne l} U_{l,m} \a_m \hat{F}_{l,m} + i \eta_l}
|
|
{(\Delta_{lp} - U_{l,l} \hat{F}_{l,l}) + i \kappa_l},
|
|
\end{equation}
|
|
where $\Delta_{lp} = \omega_p - \omega_l$ is the detuning between the
|
|
probe beam and the light mode $a_l$. This equation gives us a direct
|
|
relationship between the light operators and the atomic operators. In
|
|
the stationary limit, the quantum state of the light field
|
|
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 resonance 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, $a_0$ with mode
|
|
function $u_0(\b{r})$, and thus we neglect any cavity pumping
|
|
$\eta_l$. We also limit ourselves to only a single scattered mode,
|
|
$a_1$ with a mode function $u_1(\b{r})$. This leads to a simple linear
|
|
relationship between the light mode and the atomic operator
|
|
$\hat{F}_{1,0}$
|
|
\begin{equation}
|
|
\label{eq:a}
|
|
\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
|
|
$\hat{D}_{1,0} \equiv \hat{D}$, $\hat{B}_{1,0} \equiv \hat{B}$, and
|
|
the coefficients $J^{1,0}_{i,j} \equiv J_{i,j}$. We will adhere to
|
|
this convention from now on.
|
|
|
|
The operator $\a_1$ itself is not an observable. However, it is
|
|
possible to combine the outgoing light field with a stronger local
|
|
oscillator beam in order to measure the light quadrature
|
|
\begin{equation}
|
|
\hat{X}_\phi = \frac{1}{2} \left( \a_1 e^{-i \phi} + \ad_1 e^{i \phi} \right),
|
|
\end{equation}
|
|
which in turn can expressed via the quadrature of $\hat{F}$,
|
|
$\hat{X}^F_\beta$, as
|
|
\begin{equation}
|
|
\hat{X}_\phi = |C| \hat{X}_\beta^F = \frac{|C|}{2} \left( \hat{F}
|
|
e^{-i \beta} + \hat{F}^\dagger e^{i \beta} \right),
|
|
\end{equation}
|
|
where $\beta = \phi - \phi_C$, $C = |C| \exp(i \phi_C)$, and $\phi$ is
|
|
the local oscillator phase.
|
|
|
|
Whilst the light amplitude and the quadratures are only linear in
|
|
atomic operators, we can easily have access to higher moments via
|
|
related quantities such as the photon number
|
|
$\ad_1 \a_1 = |C|^2 \hat{F}^\dagger \hat{F}$ or the quadrature
|
|
variance
|
|
\begin{equation}
|
|
\label{eq:Xvar}
|
|
( \Delta X_\phi )^2 = \langle \hat{X}_\phi^2 \rangle - \langle
|
|
\hat{X}_\phi \rangle^2 = \frac{1}{4} + |C|^2 (\Delta X^F_\beta)^2,
|
|
\end{equation}
|
|
which reflect atomic correlations and fluctuations.
|
|
|
|
Finally, 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.
|
|
|
|
\section[Density and Phase Observables]
|
|
{Density and Phase Observables\footnote{The results of this
|
|
section were first published in Ref. \cite{kozlowski2015}}}
|
|
\label{sec:B}
|
|
|
|
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 existing 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 a lattice
|
|
geometry, one would ideally measure between two sites similarly to
|
|
single-site addressing \cite{greiner2009, bloch2011}, which would
|
|
measure a single term $\langle \bd_i b_{i+1}+b_i
|
|
\bd_{i+1}\rangle$. This could be achieved, for example, 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. However, a single-shot success rate of atom detection will be
|
|
small and as single-site addressing is challenging, we proceed with
|
|
our global scattering scheme.
|
|
|
|
In our model light couples to the operator $\hat{F}$ which consists of
|
|
a density component, $\hat{D} = \sum_i J_{i,i} \hat{n}_i$, and a phase
|
|
component, $\hat{B} = \sum_{\langle i, j \rangle} J_{i,j} \bd_i
|
|
b_j$. In general, the density component dominates, $\hat{D} \gg
|
|
\hat{B}$, and thus $\hat{F} \approx \hat{D}$
|
|
\cite{mekhov2012}. Physically, this is a consequence of the fact that
|
|
there are more atoms to scatter light at the lattice sites than in
|
|
between them. 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. Furthermore, it is not limited
|
|
to a double-well setup and naturally extends to a lattice structure
|
|
which is a key advantage. Such a counter-intuitive configuration may
|
|
affect works on quantum gases trapped in quantum potentials
|
|
\cite{mekhov2012, caballero2015, mekhov2008, larson2008, chen2009,
|
|
habibian2013, ivanov2014} and quantum measurement-induced
|
|
preparation of many-body atomic states \cite{mekhov2009prl,
|
|
elliott2015, mazzucchi2016, pedersen2014}.
|
|
|
|
For clarity we will consider a 1D lattice as shown in
|
|
Fig. \ref{fig:LatticeDiagram} 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:WannierProducts}a. For
|
|
the $\hat{B}$ operator we calculate the convolution with the nearest
|
|
neighbour overlap, $W_1(x) \equiv w(x - d/2) w(x + d/2)$, and for the
|
|
$\hat{D}$ operator we calculate the convolution with the square of the
|
|
Wannier function at a single site, $W_0(x) \equiv w^2(x)$. 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.
|
|
|
|
\begin{figure}[hbtp!]
|
|
\centering
|
|
\includegraphics[width=0.8\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
|
|
$J_{i,i+1} = J_1$. (b) Arrangement for spatially varying pattern
|
|
$J_{i,i+1}=(-1)^i J_2$; 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}
|
|
|
|
In order to calculate the $J_{i,j}$ coefficients we perform numerical
|
|
calculations using realistic Wannier functions
|
|
\cite{walters2013}. 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:WannierProducts}b. This is because for plane and standing
|
|
wave light modes the 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{align}
|
|
\label{eq:FTs}
|
|
\hat{D} = & \frac{1}{2}[\mathcal{F}[W_0](k_-)\sum_i\hat{n}_i\cos(k_-
|
|
x_i +\varphi_-) \nonumber\\
|
|
& + \mathcal{F}[W_0](k_+)\sum_i\hat{n}_i\cos(k_+ x_i +\varphi_+)],
|
|
\nonumber\\
|
|
\hat{B} = & \frac{1}{2}[\mathcal{F}[W_1](k_-)\sum_i\hat{B}_i\cos(k_- x_i
|
|
+\frac{k_-d}{2}+\varphi_-) \nonumber\\
|
|
& +\mathcal{F}[W_1](k_+)\sum_i\hat{B}_i\cos(k_+
|
|
x_i +\frac{k_+d}{2}+\varphi_+)],
|
|
\end{align}
|
|
where $k_\pm = k_{0x} \pm k_{1x}$,
|
|
$k_{(0,1)x} = k_{0,1} \sin(\theta_{0,1}$),
|
|
$\hat{B}_i=\bd_ib_{i+1}+b_i\bd_{i+1}$, 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.
|
|
|
|
The simplest case is to find a diffraction maximum where
|
|
$J_{i,i+1} = J^B_\mathrm{max}$, where $J^B_\mathrm{max}$ is a
|
|
constant. This results in a diffraction maximum where each bond
|
|
(inter-site term) scatters light in phase and the operator is given by
|
|
\begin{equation}
|
|
\hat{B}_\mathrm{max} = J^B_\mathrm{max} \sum_i^K \hat{B}_i .
|
|
\end{equation}
|
|
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:BDiagram}a
|
|
shows the resulting light mode functions and their product along the
|
|
lattice and Fig. \ref{fig:WannierProducts}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
|
|
\begin{equation}
|
|
\xi \equiv \varphi_0 = -\varphi_1 =
|
|
\frac{1}{2}\arccos\left[\frac{-\mathcal{F}[W_0](2\pi/d)}{\mathcal{F}[W_0](0)}\right].
|
|
\end{equation}
|
|
Under these conditions, the coefficient $J^B_\mathrm{max}$ is simply
|
|
given by $J^B_\mathrm{max} = \mathcal{F}[W_1](2 \pi / d)$. 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.
|
|
|
|
\begin{figure}
|
|
\centering
|
|
\includegraphics[width=\linewidth]{WF_S}
|
|
\caption[Wannier Function Products]{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:WannierProducts}
|
|
\end{figure}
|
|
|
|
Another possibility is to obtain an alternating pattern similar
|
|
corresponding to a diffraction minimum where each bond scatters light
|
|
in anti-phase with its neighbours giving
|
|
\begin{equation}
|
|
\hat{B}_\mathrm{min} = J^B_\mathrm{min} \sum_i^K (-1)^i \hat{B}_i,
|
|
\end{equation}
|
|
where $J^B_\mathrm{min}$ is a constant. 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{align}
|
|
\label{eq:DMin}
|
|
\hat{D} = & \mathcal{F}[W_0]\left(\frac{\pi}{d}\right) \sum_i (-1)^i \hat{n}_i
|
|
\cos(\varphi_0) \cos(\varphi_1) \nonumber \\
|
|
\hat{B} = & -\mathcal{F}[W_1]\left(\frac{\pi}{d}\right) \sum_i (-1)^i \hat{B}_i
|
|
\cos(\varphi_0) \sin(\varphi_1).
|
|
\end{align}
|
|
For $\varphi_0 = 0$ the corresponding $J_{i,j}$ coefficients are given
|
|
by $J_{i,i+1} = (-1)^i J^B_\mathrm{min}$, where
|
|
$J^B_\mathrm{min} = -\mathcal{F}[W_1](\pi / d)$, and are shown in
|
|
Fig. \ref{fig:WannierProducts}d. The light mode coupling along the
|
|
lattice is shown in Fig. \ref{fig:BDiagram}b. 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.
|
|
|
|
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.
|
|
|
|
\section[Electric Field Strength]
|
|
{Electric Field Strength\footnote{The derivation has not been
|
|
published before, but the final numerical results were
|
|
included in Ref. \cite{kozlowski2015}}}
|
|
\label{sec:Efield}
|
|
|
|
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 quantisation
|
|
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_A \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 $d_A$ 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 definition 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 tunnelling. 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{\hat{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
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|
\begin{equation}
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|
n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K.
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|
\end{equation}
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|
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|
\begin{table}
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|
\centering
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|
\begin{tabular}{l c c}
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|
\toprule
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|
Value & Miyake \emph{et al.} & Weitenberg \emph{et al.} \\ \midrule
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|
$\omega_a$ & \multicolumn{2}{ c }{$2 \pi \cdot 384$ THz}\\
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|
$\Gamma$ & \multicolumn{2}{ c }{$2 \pi \cdot 6.07$ MHz} \\
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|
$\Delta_a$ & $2\pi \cdot 30$ MHz & $2 \pi \cdot 85$ MHz \\
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|
$I$ & $4250$ Wm$^{-2}$ & N/A \\
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|
$\Omega_0$ & 293$\times 10^6$ s$^{-1}$ & 42.5$\times 10^6$ s$^{-1}$ \\
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|
$N_K$ & 10$^5$ & 147 \\ \midrule
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|
$n_{\Phi}$ & $6 \times 10^{11}$ s$^{-1}$ & $2 \times 10^6$ s$^{-1}$ \\
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|
\bottomrule
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|
\end{tabular}
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|
\caption[Photon Scattering Rates]{Experimental parameters used in
|
|
estimating the photon scattering rates.}
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|
\label{tab:photons}
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|
\end{table}
|
|
|
|
Estimates of the scattering rate using real experimental parameters
|
|
are given in Table \ref{tab:photons}. Rubidium atom data has been
|
|
taken from Ref. \cite{steck}. The two experiments were chosen as state
|
|
of the art setups that collected light scattered from ultracold atoms
|
|
in free space. Miyake \emph{et al.} experimental parameters are from
|
|
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 effects of saturation should be taken
|
|
into account in a more complete analysis. However, it is included for
|
|
discussion. The detuning was specified as ``a few linewidths'' and
|
|
note that it is much smaller than $\Omega_0$. The Rabi frequency 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.} experimental parameters are 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, but
|
|
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 $s_\mathrm{tot} = 2.5$.
|
|
|
|
\section{Possible Experimental Issues}
|
|
|
|
There are many possible experimental issues that we have neglected so
|
|
far in our theoretical treatment which need to be answered in order to
|
|
consider the experimental feasibility of our proposal. The two main
|
|
concerns are photodetector efficiency and heating losses.
|
|
|
|
In our theoretical models we treat the detectors as if they were
|
|
capable of detecting every scattered photon, but real photodetectors
|
|
can have efficiencies as low as 5\%. For the case of nondestructive
|
|
measurement as covered in Chapter \ref{chap:qnd} this is not an issue
|
|
provided a sufficient number of photons can be collected to calculate
|
|
reliable expectation values. The case of scattering into a cavity and
|
|
the effect of efficiency on the conditioned state was addressed in
|
|
Ref. \cite{mazzucchi2016njp} where it was shown that detector
|
|
efficiencies are not a problem provided that the photon scattering
|
|
pattern is periodic in some way, e.g.~oscillatory as was the case in
|
|
Ref. \cite{mazzucchi2016njp} or constant. This way it is only
|
|
necessary to detect a sufficient number of photons to deduce the
|
|
correct phase of the oscillations or the rate for the case of a
|
|
constant scattering rate. In this thesis we deal predominantly with
|
|
these two cases so photodetector efficiency is not an issue.
|
|
|
|
The other issue is the heating of the trapped gas which will limit the
|
|
lifetime of the experiment. For free space scattering appropriate
|
|
conditions have been achieved by for example using molasses beams that
|
|
simultaneously cool and trap the atoms \cite{weitenberg2011,
|
|
weitenbergThesis}. Similar feats have been achieved with atoms
|
|
coupled to a leaky cavity in Ref. \cite{brennecke2013} where
|
|
self-organisation effects were well observable. Crucially, the cavity
|
|
in said experiment has a decay rate of the order of MHz which is
|
|
necessary to observe measurement backaction which we will consider in
|
|
the subsequent chapters.
|