New stuff in section 2

This commit is contained in:
Wojciech Kozlowski 2016-07-01 17:37:26 +01:00
parent 00c09dda60
commit 91bb08ff1a
2 changed files with 196 additions and 0 deletions

View File

@ -193,6 +193,7 @@ detuning. Within this rotating frame we will take
$\dot{\tilde{\sigma}}^- \approx 0$ and thus obtain the following $\dot{\tilde{\sigma}}^- \approx 0$ and thus obtain the following
equation for the lowering operator equation for the lowering operator
\begin{equation} \begin{equation}
\label{eq:sigmam}
\sigma^- = - \frac{i}{\Delta_a} \sum_l g_l \a_l u_l(\b{r}). \sigma^- = - \frac{i}{\Delta_a} \sum_l g_l \a_l u_l(\b{r}).
\end{equation} \end{equation}
Therefore, by inserting this expression into the Heisenberg equation Therefore, by inserting this expression into the Heisenberg equation
@ -399,3 +400,170 @@ cavity. Whilst the light amplitude itself is only linear in atomic
operators, we can easily have access to higher moments by simply operators, we can easily have access to higher moments by simply
simply considering higher moments of the $\a_l$ such as the photon simply considering higher moments of the $\a_l$ such as the photon
number $\ad_l \a_l$. number $\ad_l \a_l$.
\subsection{Electric Field Stength}
The Electric field operator at position $\b{r}$ and at time $t$ is
usually written in terms of its positive and negative components:
\begin{equation}
\b{\hat{E}}(\b{r},t) = \b{\hat{E}}^{(+)}(\b{r},t) + \b{\hat{E}}^{(-)}(\b{r},t),
\end{equation}
where
\begin{subequations}
\begin{equation}
\b{\hat{E}}^{(+)}(\b{r},t) = \sum_{\b{k}} \hat{\epsilon}_{\b{k}}
\mathcal{E}_{\b{k}} \a_{\b{k}} e^{-i \omega_{\b{k}} t + i \b{k} \cdot \b{r}},
\end{equation}
\begin{equation}
\b{\hat{E}}^{(-)}(\b{r},t) = \sum_{\b{k}} \hat{\epsilon}_{\b{k}}
\mathcal{E}_{\b{k}} \a^\dagger_{\b{k}} e^{i \omega_{\b{k}} t - i \b{k} \cdot \b{r}},
\end{equation}
\end{subequations}
$\hat{\epsilon}_{\b{k}}$ is a unit polarization vector, $\b{k}$ is the
wave vector,
$\mathcal{E}_{\b{k}} = \sqrt{\hbar \omega_{\b{k}} / 2 \epsilon_0 V}$,
$\epsilon_0$ is the free space permittivity, $V$ is the quantization
volume and $\a_\b{k}$ and $\a_\b{k}^\dagger$ are the annihilation and
creation operators respectively of a photon in mode $\b{k}$, and
$\omega_\b{k}$ is the angular frequency of mode $\b{k}$.
In Ref. \cite{Scully} it was shown that the operator
$\b{\hat{E}}^{(+)}(\b{r},t)$ due to light scattering from a single
atom located at $\b{r}^\prime$ at the observation point $\b{r}$ is
given by
\begin{equation}
\label{eq:Ep}
\b{\hat{E}}^{(+)}(\b{r},\b{r}^\prime,t) = \frac{\omega_a^2 d \sin \eta}{4 \pi
\epsilon_0 c^2 |\b{r} - \b{r}^\prime|} \hat{\epsilon} \sigma^-
\left( \b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right),
\end{equation}
with a similar expression for
$\b{E}^{(-)}(\b{r},\b{r}^\prime,t)$. Eq. \eqref{eq:Ep} is valid only
in the far field, $\eta$ is the angle the dipole makes with
$\b{r} - \b{r}^\prime$, $\hat{\epsilon}$ is the polarization vector
which is perpendicular to $\b{r} - \b{r}^\prime$ and lies in the plane
defined by $\b{r} - \b{r}^\prime$ and the dipole, $\omega_a$ is the
atomic transition frequency, and $d$ is the dipole matrix element
between the two levels, and $c$ is the speed of light in vacuum.
We have already derived an expression for the atomic lowering
operator, $\sigma^-$, in Eq. \eqref{eq:sigmam} and it is given by
\begin{equation}
\sigma^- = - \frac{i} {\Delta_a} \sum_\b{k} g_\b{k} a_\b{k} 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.
We now use the definititon of the atom-light coupling constant
\cite{Scully} to make the substitution
$g_0 a_0 = -i \Omega_0 e^{-i \omega_0 t} / 2$, where
$\Omega_0$ is the Rabi frequency for the probe-atom system. We now
evaluate the polarisation operator at the retarded time
\begin{equation}
\sigma^- \left(\b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right) =
-\frac{\Omega_0}{2 \Delta_a} u_0 (\b{r}^\prime) \exp \left[-i \omega_0
\left( t - \frac{|\b{r} - \b{r}^\prime|}{c} \right)\right].
\end{equation}
We are considering observation in the far field regime so
$|\b{r} - \b{r}^\prime| \approx r - \hat{\b{r}} \cdot
\b{r}^\prime$. We also consider the incoming, $\b{k}_0$, and the
outgoing, $\mathbf{k}_1$, waves to be of the same wavelength,
$|\b{k}_0| = |\b{k}_1| = k = \omega_0 / c$, hence
$(\omega_0 / c) |\b{r} - \b{r}^\prime| \approx k (r - \hat{\b{r}}
\cdot \b{r}^\prime) = \b{k}_1 \cdot (\b{r} - \b{r}^\prime )$ and
\begin{equation}
\sigma^- \left(\b{r}^\prime, t - \frac{|\b{r} - \b{r}^\prime|}{c} \right) \approx
-\frac{\Omega_0}{2 \Delta_a} u_0 (\b{r}^\prime) \exp \left[i
\b{k}_1 \cdot (\b{r} - \b{r}^\prime ) - i \omega_0 t \right].
\end{equation}
The many-body version of the electric field operator is given by
\begin{equation}
\b{\hat{E}}^{(+)}_N(\b{r},t) = \int \mathrm{d}^3 \b{r}^\prime
\Psi^\dagger (\b{r}^\prime) \b{\hat{E}}^{(+)}(\b{r},\b{r}^\prime,t) \Psi(\b{r}^\prime),
\end{equation}
where $\Psi(\b{r})$ is the atomic matter-field operator. As before, we
expand this field operator using localized Wannier functions
corresponding to the lattice potential and keeping only the lowest
vibrational state at each site:
$\Psi(\b{r}) = \sum_i b_i w(\b{r} - \b{r}_i)$, where $b_i$ is the
annihilation operator of an atom at site $i$ with the coordinate
$\b{r}_i$. Thus, the relevant many-body operator is
\begin{subequations}
\begin{equation}
\b{\hat{E}}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E
\sum_{i,j}^K \bd_i b_j \int \mathrm{d}^3 \b{r}^\prime
w^* (\b{r}^\prime - \b{r}_i) \frac{u_0
(\b{r}^\prime)}{|\b{r} - \b{r}^\prime|} e^ {i \b{k}_1
\cdot (\b{r} - \b{r}^\prime ) - i \omega_0 t }
w(\b{r}^\prime - \b{r}_j),
\end{equation}
\begin{equation}
C_E = -\frac{\omega_a^2 \Omega_0 d \sin \eta}{8 \pi \epsilon_0 c^2
\Delta_a}
\end{equation}
\end{subequations}
where $K$ is the number of illuminated sites. We now assume that the
lattice is deep and that the light scattering occurs on time scales
much shorter than atom tunneling. Therefore, we ignore all terms for
which $i \ne j$ and the remaining integrals become $\int \mathrm{d}^3
\b{r}_0 w^*(\b{r}_0 - \b{r}_i) f (\b{r})
w(\b{r}_0 - \b{r}_i) = f(\b{r}_i)$. The final form of
the many body operator is then
\begin{equation}
\b{E}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E
\sum_{j = 1}^K \hat{n}_j \frac{u_0 (\b{r}_j)}{|\b{r} -
\b{r}_j|} e^ {i \b{k}_1 \cdot (\b{r} - \b{r}_j
) - i \omega_0 t },
\end{equation}
where $\hat{n}_i = b^\dagger_ib_i$ is the atom number operator at site
$i$. We will now consider the incoming probe to be a plane wave,
$u_0(\b{r}) = e^{i \b{k}_0 \cdot \b{r}}$, and we
approximate $|\b{r} - \b{r}_j| \approx r$ in the
denominator. Thus,
\begin{equation}
\b{E}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E \frac{e^ {ikr - i\omega_0t}}{r}
\sum_{j = 1}^K \hat{n}_j e^ {i (\b{k}_0 - \b{k}_1) \cdot \b{r}_j}.
\end{equation}
We note that this is exactly the same form of the optical field as
derived from a generalised Bose-Hubbard model which considers a fully
quantum light-matter interaction Hamiltonian \cite{mekhov2007prl,
mekhov2007pra, mekhov2012}. We can even express the result in terms of
the $\hat{D} = \sum_j^K u_1^*(\b{r}_j) u_0(\b{r}_j)
\hat{n}_j$ formalism used in those works
\begin{equation}
\b{E}^{(+)}_N(\b{r},t) = \hat{\epsilon} C_E \frac{e^
{ikr - i\omega_0t}}{r} \hat{D}.
\end{equation}
The average light intensity at point $\b{r}$ at time $t$ is given
by the formula $I = c \epsilon_0 |E|^2/2$ and yields
\begin{equation}
\langle I (\b{r}, t) \rangle = c \epsilon_0 \langle
E^{(-)} (\b{r}, t) E^{(+)} (\b{r}, t) \rangle =
\frac{c \epsilon_0}{r^2} C_E^2 \langle \hat{D}^* \hat{D} \rangle.
\end{equation}
The scattering is dominated by a uniform background for which $\langle
\hat{D}^* \hat{D} \rangle \approx N_K$ for a superfluid
\cite{mekhov2012}, where $N_K$ is the mean number of atoms in the
illuminated volume. Thus, the approximate number of photons scattered
per second can be obtained by integrating over a sphere
\begin{equation}
n_{\Phi} = \frac{4 \pi c \epsilon_0}{3 \hbar \omega_a} \left(\frac{\omega_a^2
\Omega_0 d}{8 \pi \epsilon_0 c^2 \Delta_a}\right)^2 N_K.
\end{equation}
In the Weisskopf-Wigner approximation it can be shown \cite{Scully}
that for a two level system the decay constant, $\Gamma$, is
\begin{equation}
\Gamma = \frac{\omega_a^3 d^2}{3 \pi \epsilon_0 \hbar c^3}.
\end{equation}
Therefore, we can now express the quantity $n_{\Phi}$ as
\begin{equation}
n_{\Phi} = \frac{1}{8} \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma}{2} N_K.
\end{equation}

View File

@ -35,4 +35,32 @@
author={Mazzucchi, Gabriel and Caballero-Benitez, Santiago F and Mekhov, Igor B}, author={Mazzucchi, Gabriel and Caballero-Benitez, Santiago F and Mekhov, Igor B},
journal={arXiv preprint arXiv:1510.04883}, journal={arXiv preprint arXiv:1510.04883},
year={2015} year={2015}
}
@inbook{Scully,
author = {Scully, M. and Zubairy, S.},
title = {{Quantum Optics}},
publisher = {Cambridge University Press},
chapter = {10.1},
pages = {293},
year = {1997}
}
@article{mekhov2007prl,
title={Cavity-enhanced light scattering in optical lattices to probe atomic quantum statistics},
author={Mekhov, Igor B and Maschler, Christoph and Ritsch, Helmut},
journal={Physical review letters},
volume={98},
number={10},
pages={100402},
year={2007},
publisher={APS}
}
@article{mekhov2007pra,
title={Light scattering from ultracold atoms in optical lattices as an optical probe of quantum statistics},
author={Mekhov, Igor B and Maschler, Christoph and Ritsch, Helmut},
journal={Physical Review A},
volume={76},
number={5},
pages={053618},
year={2007},
publisher={APS}
} }