Finished section on D-op scattering in chapter 3
This commit is contained in:
Wojciech Kozlowski
parent
86e3b2644c
commit
ec18bd353e
@ -160,6 +160,7 @@ Eq. \eqref{eq:D-3} into our expression for $R$ in Eq. \eqref{eq:R} and
|
||||
we will make use of the shorthand notation
|
||||
$A_i = u_1^*(\b{r}_i) u_0(\b{r}_i)$. The result is
|
||||
\begin{equation}
|
||||
\label{eq:Rfluc}
|
||||
R = |C|^2 \sum_{i,j}^K A^*_i A_j \langle \delta \hat{n}_i \delta
|
||||
\hat{n}_j \rangle,
|
||||
\end{equation}
|
||||
@ -176,35 +177,46 @@ measure the quadrature of the light fields which in section
|
||||
case when both the scattered mode and probe are travelling waves the
|
||||
quadrature
|
||||
\begin{equation}
|
||||
\label{eq:Xtrav}
|
||||
\hat{X}^F_\beta = \frac{1}{2} \left( \hat{F} e^{-i \beta} +
|
||||
\hat{F}^\dagger e^{i \beta} \right) = \sum_i^K \hat{n}_i\cos[(\b{k}_1 - \b{k}_2) \cdot
|
||||
\b{r}_i - \beta].
|
||||
\hat{F}^\dagger e^{i \beta} \right) = \sum_i^K \hat{n}_i\cos[(\b{k}_0 - \b{k}_1) \cdot
|
||||
\b{r}_i + (\phi_0 - \phi_1) - \beta].
|
||||
\end{equation}
|
||||
Note that different light quadratures are differently coupled to the
|
||||
atom distribution, hence by varying the local oscillator phase, and
|
||||
thus effectively $\beta$, and/or the detection angle one can scan the
|
||||
whole range of couplings. A similar expression exists for $\hat{D}$
|
||||
for a standing wave probe, where $\beta$ is replaced by $\varphi_0$,
|
||||
and scanning is achieved by varying the position of the wave with
|
||||
respect to atoms.
|
||||
for a standing wave probe, where instead of varying $\beta$ scanning
|
||||
is achieved by varying the position of the wave with respect to
|
||||
atoms. Additionally, the quadrature variance, $(\Delta X^F_\beta)^2$,
|
||||
will have a similar form to $R$ given in Eq. \eqref{eq:Rfluc},
|
||||
\begin{equation}
|
||||
(\Delta X^F_\beta)^2 = |C|^2 \sum_{i.j}^K A_i^\beta A_j^\beta
|
||||
\langle \dn_i \dn_j \rangle,
|
||||
\end{equation}
|
||||
where $A_i^\beta = (A_i e^{-i\beta} + A_i^* e^{i \beta})/2$. However,
|
||||
this has the advantage that unlike in the case of $R$ there is no need
|
||||
to subtract a spatially varying classical signal to obtain this
|
||||
quantity.
|
||||
|
||||
The ``quantum addition'', $R$, and the quadrature variance, $(\Delta
|
||||
X^F_\beta)^2$, are both quadratic in $\a_1$ and both rely heavily on
|
||||
the quantum state of the matter. Therefore, they will have a
|
||||
nontrivial angular dependence, showing more peaks than classical
|
||||
The ``quantum addition'', $R$, and the quadrature variance,
|
||||
$(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$ and both rely
|
||||
heavily on the quantum state of the matter. Therefore, they will have
|
||||
a nontrivial angular dependence, showing more peaks than classical
|
||||
diffraction. Furthermore, these peaks can be tuned very easily with
|
||||
$\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular
|
||||
dependence of $R$ for the case when the scattered mode is a standing
|
||||
wave and the probe is a travelling wave scattering from bosons in a 3D
|
||||
optical lattice. The first noticeable feature is the isotropic
|
||||
background which does not exist in classical diffraction. This
|
||||
background yields information about density fluctuations which,
|
||||
according to mean-field estimates (i.e.~inter-site correlations are
|
||||
ignored), are related by $R = K( \langle \hat{n}^2 \rangle - \langle
|
||||
\hat{n} \rangle^2 )/2$. In Fig. \ref{fig:scattering} we can see a
|
||||
significant signal of $R = |C|^2 N_K/2$, because it shows scattering
|
||||
from an ideal superfluid which has significant density fluctuations
|
||||
with correlations of infinte range. However, as the parameters of the
|
||||
wave and the probe is a travelling wave scattering from an ideal
|
||||
superfluid in a 3D optical lattice. The first noticeable feature is
|
||||
the isotropic background which does not exist in classical
|
||||
diffraction. This background yields information about density
|
||||
fluctuations which, according to mean-field estimates (i.e.~inter-site
|
||||
correlations are ignored), are related by
|
||||
$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. In
|
||||
Fig. \ref{fig:scattering} we can see a significant signal of
|
||||
$R = |C|^2 N_K/2$, because it shows scattering from an ideal
|
||||
superfluid which has significant density fluctuations with
|
||||
correlations of infinte range. However, as the parameters of the
|
||||
lattice are tuned across the phase transition into a Mott insulator
|
||||
the signal goes to zero. This is because the Mott insulating phase has
|
||||
well localised atoms at each site which suppresses density
|
||||
@ -233,20 +245,125 @@ classical Bragg condition is actually not satisfied which means there
|
||||
actually is no classical diffraction on top of the ``quantum
|
||||
addition'' shown here. Therefore, these features would be easy to see
|
||||
in an experiment as they wouldn't be masked by a stronger classical
|
||||
signal. We can even derive the generalised Bragg conditions for the
|
||||
peaks that we can see in Fig. \ref{fig:scattering}.
|
||||
signal. This difference in behaviour is due to the fact that
|
||||
classical diffraction is ignorant of any quantum correlations. This
|
||||
signal is given by the square of the light field amplitude squared
|
||||
\begin{equation}
|
||||
|\langle \a_1 \rangle|^2 = |C|^2 \sum_{i,j} A_i^*
|
||||
A_j \langle \n_i \rangle \langle \n_j \rangle,
|
||||
\end{equation}
|
||||
which is idependent of any two-point correlations unlike $R$. On the
|
||||
other hand the full light intensity (classical signal plus ``quantum
|
||||
addition'') of the quantum light does include higher-order
|
||||
correlations
|
||||
\begin{equation}
|
||||
\langle \ad_1 \a_1 \rangle = |C|^2 \sum_{i,j} A_i^*
|
||||
A_j \langle \n_i \n_j \rangle.
|
||||
\end{equation}
|
||||
Therefore, we see that in the fully quantum picture light scattering
|
||||
not only depends on the diffraction structure due to the distribution
|
||||
of atoms in the lattice, but also on the quantum correlations between
|
||||
different lattice sites which will be dependent on the quantum state
|
||||
of the matter. These correlations are imprinted in $R$ as shown in
|
||||
Eq. \eqref{eq:Rfluc} and it highlights the key feature of our model,
|
||||
i.e.~the light couples to the quantum state directly via operators.
|
||||
|
||||
\mynote{Derive and show these Bragg conditions below}
|
||||
We can even derive the generalised Bragg conditions for the peaks that
|
||||
we can see in Fig. \ref{fig:scattering}. The exact conditions under
|
||||
which diffraction peaks emerge for the ``quantum addition'' will
|
||||
depend on the optical setup as well as on the quantum state of the
|
||||
matter as it is the density fluctuation correlations,
|
||||
$\langle \dn_i \dn_j \rangle$, that provide the structure for $R$ and
|
||||
not the lattice itself as seen in Eq. \eqref{eq:Rfluc}. For classical
|
||||
light it is straightforward to develop an intuitive physical picture
|
||||
to find the Bragg condition by considering angles at which the
|
||||
distance travelled by light scattered from different points in the
|
||||
lattice is equal to an integer multiple of wavelength. The ``quantum
|
||||
addition'' is more complicated and less intuitive as we now have to
|
||||
consider quantum correlations which are not only nonlocal, but can
|
||||
also be nagative.
|
||||
|
||||
As $(\Delta X^F_\beta)^2$ and $R$ are quadratic variables,
|
||||
the generalized Bragg conditions for the peaks are
|
||||
$2 \Delta \b{k} = \b{G}$ for quadratures of travelling waves, where
|
||||
$\Delta \b{k} = \b{k}_0 - \b{k}_1$ and $\b{G}$ is the reciprocal
|
||||
lattice vector, and $2 \b{k}_1 = \b{G}$ for standing wave $\a_1$ and
|
||||
travelling $\a_0$, which is clearly different from the classical Bragg
|
||||
condition $\Delta \b{k} = \b{G}$. The peak height is tunable by the
|
||||
local oscillator phase or standing wave shift as seen in Fig.
|
||||
\ref{fig:scattering}b.
|
||||
We will consider scattering from a superfluid, because the Mott
|
||||
insulator has no ``quantum addition'' due to a lack of density
|
||||
fluctuations. The wavefunction of a superfluid on a lattice is given
|
||||
by
|
||||
\begin{equation}
|
||||
\frac{1}{\sqrt{M^N N!}} \left( \sum_i^M \bd_i \right)^N |0 \rangle,
|
||||
\end{equation}
|
||||
where $| 0 \rangle$ denotes the vacuum state. This state has infinte
|
||||
range correlations and thus has the convenient property that all
|
||||
two-point density fluctuation correlations are equal regardless of
|
||||
their separation,
|
||||
i.e.~$\langle \dn_i \dn_j \rangle \equiv \langle \dn_a \dn_b \rangle$
|
||||
for all $(i \ne j)$, where the right hand side is a constant
|
||||
value. This allows us to extract all correlations from the sum in
|
||||
Eq. \eqref{eq:Rfluc} to obtain
|
||||
\begin{equation}
|
||||
\label{eq:RSF}
|
||||
\frac{R}{|C|^2} = (\langle \dn^2 \rangle - \langle \dn_a \dn_b \rangle) \sum_i^K
|
||||
|A_i|^2 + \langle \dn_a \dn_b \rangle |A|^2 = \frac{N}{M} \sum_i^K
|
||||
|A_i|^2 - \frac{N}{M^2} |A|^2,
|
||||
\end{equation}
|
||||
where $A = \sum_i^K A_i$, and we have dropped the index from
|
||||
$\langle \dn^2 \rangle$ as it is equal at every site. The second
|
||||
equality follows from the fact that for a superfluid state
|
||||
$\langle \dn^2 \rangle = N/M - N/M^2$ and
|
||||
$\langle \dn_a \dn_b \rangle = -N/M^2$. Naturally, we get an identical
|
||||
expression for the quadrature variance $(\Delta X^F_\beta)^2$ with the
|
||||
coefficients $A_i$ replaced by the real $A_i^\beta$.
|
||||
|
||||
The ``quantum addition'' for the case when both the scattered and
|
||||
probe modes are travelling waves is actually trivial. It has no peaks
|
||||
and thus it has no generalised Bragg condition and it only consists of
|
||||
a uniform background. This is a consequence of the fact that
|
||||
travelling waves couple equally strongly with every site as only the
|
||||
phase differs. Therefore, since superfluid correlations lack structure
|
||||
as they're uniform we do no get a strong coherent peak. The
|
||||
contribution from the lattice structure is included in the classical
|
||||
Bragg peaks which we have subtracted in order to obtain the quantity
|
||||
$R$. However, if we consider the case where the scattered mode is
|
||||
collected as a standing wave using a pair of mirrors we get the
|
||||
diffraction pattern that we saw in Fig. \ref{fig:scattering}. This
|
||||
time we get strong visible peaks, because at certain angles the
|
||||
standing wave couples to the atoms maximally at all lattice sites and
|
||||
thus it uses the structure of the lattice to amplify the signal from
|
||||
the quantum fluctuations. This becomes clear when we look at
|
||||
Eq. \eqref{eq:RSF}. We can neglect the second term as it is always
|
||||
negative and it has the same angular distribution as the classical
|
||||
diffraction pattern and thus it is mostly zero except when the
|
||||
classical Bragg condition is satisfied. Since in
|
||||
Fig. \ref{fig:scattering} we have chosen an angle such that the Bragg
|
||||
is not satisfied this term is essentially zero. Therefore, we are left
|
||||
with the first term $\sum_i^K |A_i|^2$ which for a travelling wave
|
||||
probe and a standing wave scattered mode is
|
||||
\begin{equation}
|
||||
\sum_i^K |A_i|^2 = \sum_i^K \cos^2(\b{k}_0 \cdot \b{r}_i + \phi_0) =
|
||||
\frac{1}{2} \sum_i^K \left[1 + \cos(2 \b{k}_0 \cdot \b{r}_i + 2
|
||||
\phi_0) \right].
|
||||
\end{equation}
|
||||
Therefore, it is straightforward to see that unless
|
||||
$2 \b{k}_0 = \b{G}$, where $\b{G}$ is a reciprocal lattice vector,
|
||||
there will be no coherent signal and we end up with the mean uniform
|
||||
signal of strength $N_k/2$. When this condition is satisifed all the
|
||||
cosine terms will be equal and they will add up constructively instead
|
||||
of cancelling each other out. Note that this new Bragg condition is
|
||||
different from the classical one $\b{k}_0 - \b{k}_1 = \b{G}$. This
|
||||
result makes it clear that the uniform background signal is not due to
|
||||
any coherent scattering, but rather due to the lack of structure in
|
||||
the quantum correlations. Furthermore, we see that the peak height is
|
||||
actually tunable via the phase, $\phi_0$, which is illustrated in
|
||||
Fig. \ref{fig:scattering}b.
|
||||
|
||||
For light field quadratures the situation is different, because as we
|
||||
have seen in Eq. \eqref{eq:Xtrav} even for travelling waves we can
|
||||
tune the contributions to the signal from different sites by changing
|
||||
the angle of measurement or tuning the local oscillator signal
|
||||
$\beta$. The rest is similar to the case we discussed for $R$ with a
|
||||
standing wave mode and we can show that the new Bragg condition in
|
||||
this case is $2 (\b{k}_0 - \b{k}_1) = \b{G}$ which is different from
|
||||
the condition we had for $R$ and is still different from the classical
|
||||
condition $2 (\b{k}_0 - \b{k}_1) = \b{G}$. Furthermore, just like in
|
||||
Fig. \ref{fig:scattering}b the peak height can be tuned using $\beta$.
|
||||
|
||||
A quantum signal that isn't masked by classical diffraction is very
|
||||
useful for future experimental realisability. However, it is still
|
||||
|
||||
@ -205,6 +205,7 @@
|
||||
|
||||
\renewcommand{\H}{\hat{H}}
|
||||
\newcommand{\n}{\hat{n}}
|
||||
\newcommand{\dn}{\delta \hat{n}}
|
||||
\newcommand{\ad}{a^\dagger}
|
||||
\newcommand{\bd}{b^\dagger}
|
||||
\renewcommand{\a}{a} % in case we decide to put hats on
|
||||
|
||||
Reference in New Issue
Block a user