Finished for the day. Chapter 3 still needs an update to the B operator section and possibly another look at the conclusion
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Wojciech Kozlowski
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@ -188,23 +188,23 @@ for a standing wave probe, where $\beta$ is replaced by $\varphi_0$,
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and scanning is achieved by varying the position of the wave with
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respect to atoms.
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The ``quantum addition'', $R$, and the quadrature variance,
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$(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$. Therefore, they
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will havea nontrivial angular dependence, showing more peaks than
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classical diffraction. Furthermore, these peaks can be tuned very
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easily with $\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows
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the angular dependence of $R$ for the case when the scattered mode is
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a standing wave and the probe is a travelling wave scattering from
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bosons in a 3D optical lattice. The first noticeable feature is the
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isotropic background which does not exist in classical
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diffraction. This background yields information about density
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fluctuations which, according to mean-field estimates (i.e.~inter-site
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correlations are ignored), are related by
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$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$. In
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Fig. \ref{fig:scattering} we can see a significant signal of
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$R = |C|^2 N_K/2$, because it shows scattering from an ideal
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superfluid which has significant density fluctuations with
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correlations of infinte range. However, as the parameters of the
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The ``quantum addition'', $R$, and the quadrature variance, $(\Delta
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X^F_\beta)^2$, are both quadratic in $\a_1$ and both rely heavily on
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the quantum state of the matter. Therefore, they will have a
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nontrivial angular dependence, showing more peaks than classical
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diffraction. Furthermore, these peaks can be tuned very easily with
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$\beta$ or $\varphi_l$. Fig. \ref{fig:scattering} shows the angular
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dependence of $R$ for the case when the scattered mode is a standing
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wave and the probe is a travelling wave scattering from bosons in a 3D
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optical lattice. The first noticeable feature is the isotropic
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background which does not exist in classical diffraction. This
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background yields information about density fluctuations which,
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according to mean-field estimates (i.e.~inter-site correlations are
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ignored), are related by $R = K( \langle \hat{n}^2 \rangle - \langle
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\hat{n} \rangle^2 )/2$. In Fig. \ref{fig:scattering} we can see a
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significant signal of $R = |C|^2 N_K/2$, because it shows scattering
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from an ideal superfluid which has significant density fluctuations
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with correlations of infinte range. However, as the parameters of the
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lattice are tuned across the phase transition into a Mott insulator
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the signal goes to zero. This is because the Mott insulating phase has
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well localised atoms at each site which suppresses density
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@ -236,7 +236,7 @@ in an experiment as they wouldn't be masked by a stronger classical
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signal. We can even derive the generalised Bragg conditions for the
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peaks that we can see in Fig. \ref{fig:scattering}.
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% Derive and show these Bragg conditions
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\mynote{Derive and show these Bragg conditions below}
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As $(\Delta X^F_\beta)^2$ and $R$ are quadratic variables,
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the generalized Bragg conditions for the peaks are
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@ -248,18 +248,22 @@ condition $\Delta \b{k} = \b{G}$. The peak height is tunable by the
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local oscillator phase or standing wave shift as seen in Fig.
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\ref{fig:scattering}b.
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In section \ref{sec:Efield} we have estimated the mean photon
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scattering rates integrated over the solid angle for the only two
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experiments so far on light diffraction from truly ultracold bosons
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where the measurement object was light
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A quantum signal that isn't masked by classical diffraction is very
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useful for future experimental realisability. However, it is still
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unclear whether this signal would be strong enough to be
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visible. After all, a classical signal scales as $N_K^2$ whereas here
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we have only seen a scaling of $N_K$. In section \ref{sec:Efield} we
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have estimated the mean photon scattering rates integrated over the
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solid angle for the only two experiments so far on light diffraction
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from truly ultracold bosons where the measurement object was light
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\begin{equation}
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n_{\Phi}= \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma K}{8}
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(\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2).
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\end{equation}
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Therefore, applying these results to the scattering patters in
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Fig. \ref{fig:scattering} the background signal should reach
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$n_\Phi \approx 10^6$ s$^{-1}$ in Ref. \cite{weitenberg2011} (150
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atoms in 2D), and $n_\Phi \approx 10^{11}$ s$^{-1}$ in
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These results can be applied directly to the scattering patters in
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Fig. \ref{fig:scattering}. Therefore, the background signal should
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reach $n_\Phi \approx 10^6$ s$^{-1}$ in Ref. \cite{weitenberg2011}
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(150 atoms in 2D), and $n_\Phi \approx 10^{11}$ s$^{-1}$ in
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Ref. \cite{miyake2011} ($10^5$ atoms in 3D). These numbers show that
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the diffraction patterns we have seen due to the ``quantum addition''
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should be visible using currently available technology, especially
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@ -268,6 +272,123 @@ not coincide at all with the classical diffraction pattern.
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\subsection{Mapping the quantum phase diagram}
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We have shown that scattering from atom number operators leads to a
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purely quantum diffraction pattern which depends on the density
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fluctuations and their correlations. We have also seen that this
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signal should be strong enough to be visible using currently available
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technology. However, so far we have not looked at what this can tell
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us about the quantum state of matter. We have briefly mentioned that a
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deep superfluid will scatter a lot of light due to its infinite range
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correlations and a Mott insulator will not contriute any ``quantum
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addition'' at all, but we have not look at the quantum phase
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transition between these two phases. In two or higher dimensions this
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has a rather simple answer as the Bose-Hubbard phase transition is
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described well by mean-field theories and it has a sharp transition at
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the critical point. This means that the ``quantum addition'' signal
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would drop rapidly at the critical point and go to zero as soon as it
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was crossed. However, $R$ given by Eq. \eqref{eq:R} clearly contains
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much more information.
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There are many situations where the mean-field approximation is not a
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valid description of the physics. A prominent example is the
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Bose-Hubbard model in 1D \cite{cazalilla2011, ejima2011, kuhner2000,
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pino2012, pino2013}. Observing the transition in 1D by light at
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fixed density was considered to be difficult \cite{rogers2014} or even
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impossible \cite{roth2003}. This is, because the one-dimensional
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quantum phase transition is in a different universality class than its
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higher dimensional counterparts. The energy gap, which is the order
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parameter, decays exponentially slowly across the phase transition
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making it difficult to identify the phase transition even in numerical
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simulations. Here, we will show the avaialable tools provided by the
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``quantum addition'' that allows one to nondestructively map this
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phase transition and distinguish the superfluid and Mott insulator
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phases.
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The 1D phase transition is best understood in terms of two-point
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correlations as a function of their separation \cite{giamarchi}. In
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the Mott insulating phase, the two-point correlations $\langle \bd_i
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b_j \rangle$ and $\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$
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($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay
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exponentially with $|i-j|$. This is a characteristic of insulators. On
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the other hand the superfluid will exhibit long-range order which in
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dimensions higher than one, manifests itself with an infinite
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correlation length. However, in 1D only pseudo long-range order
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happens and both the matter-field and density fluctuation correlations
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decay algebraically \cite{giamarchi}.
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The method we propose gives us direct access to the structure factor,
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which is a function of the two-point correlation $\langle \delta
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\hat{n}_i \delta \hat{n}_j \rangle$. This quantity can be extracted
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from the measured light intensity bu considering the ``quantum
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addition''. We will consider the case when both the probe and
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scattered modes are plane waves which can be easily achieved in free
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space. We will again consider the case of light being maximally
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coupled to the density ($\hat{F} = \hat{D}$). Therefore, the quantum
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addition is given by
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\begin{equation}
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R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0)
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(\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta
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\hat{n}_j \rangle.
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\end{equation}
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\mynote{can put in more detail here with equations}
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This alone allows us to analyse the phase transition quantitatively
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using our method. Unlike in higher dimensions where an order parameter
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can be easily defined within the mean-field approximation as a simple
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expectation value of some quantity, the situation in 1D is more
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complex as it is difficult to directly access the excitation energy
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gap which defines this phase transition. However, a valid description
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of the relevant 1D low energy physics is provided by Luttinger liquid
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theory \cite{giamarchi}. In this model correlations in the supefluid
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phase as well as the superfluid density itself are characterised by
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the Tomonaga-Luttinger parameter, $K_b$. This parameter also
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identifies the critical point in the thermodynamic limit at $K_b =
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1/2$. This quantity can be extracted from various correlation
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functions and in our case it can be extracted directly from $R$
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\cite{ejima2011}. This quantity was used in numerical calculations
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that used highly efficient density matrix renormalisation group (DMRG)
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methods to calculate the ground state to subsequently fit the
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Luttinger theoru to extract this parameter $K_b$. These calculations
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yield a theoretical estimate of the critical point in the
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thermodynamic limit for commensurate filling in 1D to be at
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$U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our proposal provides
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a method to directly measure $R$ nondestructively in a lab which can
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then be used to experimentally determine the location of the critical
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point in 1D.
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However, whilst such an approach will yield valuable quantitative
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results we will instead focus on its qualitative features which give a
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more intuitive understanding of what information can be extracted from
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$R$. This is because the superfluid to Mott insulator phase transition
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is well understood, so there is no reason to dwell on its quantitative
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aspects. However, our method is much more general than the
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Bose-Hubbard model as it can be easily applied to many other systems
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such as fermions, photonic circuits, optical lattices qith quantum
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potentials, etc. Therefore, by providing a better physical picture of
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what information is carried by the ``quantum addition'' it should be
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easier to see its usefuleness in a broader context.
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We calculate the phase diagram of the Bose-Hubbard Hamiltonian given
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by
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\begin{equation}
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\hat{H}_\mathrm{dis} = -J^\mathrm{cl} \sum_{\langle i, j \rangle}
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\bd_i b_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) - \mu
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\sum_i \hat{n}_i,
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\end{equation}
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where the $\mu$ is the chemical potential. We have introduced the last
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term as we are interested in grand canonical ensemble calculations as
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we want to see how the system's behaviour changes as density is
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varied. We perform numerical calculations of the ground state using
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DMRG methods \cite{tnt} from which we can compute all the necessary
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atomic observables. Experiments typically use an additional harmonic
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confining potential on top of the optical lattice to keep the atoms in
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place which means that the chemical potential will vary in
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space. However, with careful consideration of the full
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($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in
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Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the
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system \cite{batrouni2002}.
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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=\linewidth]{oph11}
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@ -288,121 +409,106 @@ not coincide at all with the classical diffraction pattern.
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\label{fig:SFMI}
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\end{figure}
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We have shown how in mean-field, we can track the order parameter,
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$\Phi$, by probing the matter-field interference using the coupling of
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light to the $\hat{B}$ operator. In this case, it is very easy to
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follow the superfluid to Mott insulator quantum phase transition since
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we have direct access to the order parameter which goes to zero in the
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insulating phase. In fact, if we're only interested in the critical
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point, we only need access to any quantity that yields information
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about density fluctuations which also go to zero in the MI phase and
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this can be obtained by measuring
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$\langle \hat{D}^\dagger \hat{D} \rangle$. However, there are many
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situations where the mean-field approximation is not a valid
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description of the physics. A prominent example is the Bose-Hubbard
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model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, pino2012,
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pino2013}. Observing the transition in 1D by light at fixed density
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was considered to be difficult \cite{rogers2014} or even impossible
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\cite{roth2003}. By contrast, here we propose varying the density or
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chemical potential, which sharply identifies the transition. We
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perform these calculations numerically by calculating the ground state
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using DMRG methods \cite{tnt} from which we can compute all the
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necessary atomic observables. Experiments typically use an additional
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harmonic confining potential on top of the optical lattice to keep the
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atoms in place which means that the chemical potential will vary in
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space. However, with careful consideration of the full
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($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in
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Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the
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system \cite{batrouni2002}.
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The 1D phase transition is best understood in terms of two-point
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correlations as a function of their separation \cite{giamarchi}. In
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the Mott insulating phase, the two-point correlations
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$\langle \bd_i b_j \rangle$ and
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$\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$
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($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay
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exponentially with $|i-j|$. On the other hand the superfluid will
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exhibit long-range order which in dimensions higher than one,
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manifests itself with an infinite correlation length. However, in 1D
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only pseudo long-range order happens and both the matter-field and
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density fluctuation correlations decay algebraically \cite{giamarchi}.
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The method we propose gives us direct access to the structure factor,
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which is a function of the two-point correlation $\langle \delta
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\hat{n}_i \delta \hat{n}_j \rangle$, by measuring the light
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intensity. For two travelling waves maximally coupled to the density
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(atoms are at light intensity maxima so $\hat{F} = \hat{D}$), the
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quantum addition is given by
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\begin{equation}
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R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0)
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(\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta
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\hat{n}_j \rangle,
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\end{equation}
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The angular dependence of $R$ for a Mott insulator and a superfluid is
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shown in Fig. \ref{fig:SFMI}a, and there are two variables
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distinguishing the states. Firstly, maximal $R$,
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$R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$,
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probes the fluctuations and compressibility $\kappa'$
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($\langle \delta \hat{n}^2_i \rangle \propto \kappa' \langle \hat{n}_i
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\rangle$). The Mott insulator is incompressible and thus will have
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very small on-site fluctuations and it will scatter little light
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leading to a small $R_\text{max}$. The deeper the system is in the MI
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phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the smaller
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these values will be until ultimately it will scatter no light at all
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in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a this can
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be seen in the value of the peak in $R$. The value $R_\text{max}$ in
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the SF phase ($U/2J^\text{cl} = 0$) is larger than its value in the MI
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We then consider probing these ground states using our optical scheme
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and we calculate the ``quantum addition'', $R$. The angular dependence
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of $R$ for a Mott insulator and a superfluid is shown in
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Fig. \ref{fig:SFMI}a, and we note that there are two variables
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distinguishing the states. Firstly, maximal $R$, $R_\text{max} \propto
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\sum_i \langle \delta \hat{n}_i^2 \rangle$, probes the fluctuations
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and compressibility $\kappa'$ ($\langle \delta \hat{n}^2_i \rangle
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\propto \kappa' \langle \hat{n}_i \rangle$). The Mott insulator is
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incompressible and thus will have very small on-site fluctuations and
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it will scatter little light leading to a small $R_\text{max}$. The
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deeper the system is in the insulating phase (i.e. that larger the
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$U/2J^\text{cl}$ ratio is), the smaller these values will be until
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ultimately it will scatter no light at all in the $U \rightarrow
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\infty$ limit. In Fig. \ref{fig:SFMI}a this can be seen in the value
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of the peak in $R$. The value $R_\text{max}$ in the superfluid phase
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($U/2J^\text{cl} = 0$) is larger than its value in the Mott insulating
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phase ($U/2J^\text{cl} = 10$) by a factor of
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$\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of
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$R_\text{max}$ changes across the phase transition. We see that the
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transition shows up very sharply as $\mu$ is varied.
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$R_\text{max}$ changes across the phase transition. There are a few
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things to note at this point. Firstly, if we follow the transition
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along the line corresponding to commensurate filling (i.e.~any line
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that is in between the two white lines in Fig. \ref{fig:SFMI}d) we see
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that the transition is very smooth and it is hard to see a definite
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critical point. This is due to the energy gap closing exponentially
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slowly which makes precise identification of the critical point
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extremely difficult. The best option at this point would be to fit
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Tomonage-Luttinger theory to the results in order to find this
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critical point. However, we note that there is a drastic change in
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signal as the chemical potential (and thus the density) is
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varied. This is highlighted in Fig. \ref{fig:SFMI}b which shows how
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the Mott insulator can be easily identified by a dip in the quantity
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$R_\text{max}$.
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Secondly, being a Fourier transform, the width $W_R$ of the dip in $R$
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is a direct measure of the correlation length $l$, $W_R \propto
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1/l$. The Mott insulator being an insulating phase is characterised by
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exponentially decaying correlations and as such it will have a very
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large $W_R$. However, the superfluid in 1D exhibits pseudo long-range
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order which manifests itself in algebraically decaying two-point
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correlations \cite{giamarchi} which significantly reduces the dip in
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the $R$. This can be seen in Fig. \ref{fig:SFMI}a and we can also see
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that this identifies the phase transition very sharply as $\mu$ is
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varied in Figs. \ref{fig:SFMI}(c,e). One possible concern with
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experimentally measuring $W_R$ is that it might be obstructed by the
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classical diffraction maxima which appear at angles corresponding to
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the minima in $R$. However, the width of such a peak is much smaller
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as its width is proportional to $1/M$.
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It is also possible to analyse the phase transition quantitatively
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using our method. Unlike in higher dimensions where an order parameter
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can be easily defined within the MF approximation there is no such
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quantity in 1D. However, a valid description of the relevant 1D low
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energy physics is provided by Luttinger liquid theory
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\cite{giamarchi}. In this model correlations in the supefluid phase as
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well as the superfluid density itself are characterised by the
|
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Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies
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the phase transition in the thermodynamic limit at $K_b = 1/2$. This
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quantity can be extracted from various correlation functions and in
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our case it can be extracted directly from $R$ \cite{ejima2011}. By
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extracting this parameter from $R$ for various lattice lengths from
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numerical DMRG calculations it was even possible to give a theoretical
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estimate of the critical point for commensurate filling, $N = M$, in
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the thermodynamic limit to occur at $U/2J^\text{cl} \approx 1.64$
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\cite{ejima2011}. Our proposal provides a method to directly measure
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$R$ in a lab which can then be used to experimentally determine the
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location of the critical point in 1D.
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large $W_R$. On the other hand, the superfluid in 1D exhibits pseudo
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long-range order which manifests itself in algebraically decaying
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two-point correlations \cite{giamarchi} which significantly reduces
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the dip in the $R$. This can be seen in
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Fig. \ref{fig:SFMI}a. Furthermore, just like for $R_\text{max}$ we see
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that the transition is much sharper as $\mu$ is varied. This is shown
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in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle between
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a superfluid and an insulating state is fairly significant $\sim
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20^\circ$ which should make the two phases easy to identify using this
|
||||
measure. In this particular case, measuring $W_R$ in the Mott phase is
|
||||
not very practical as the insulating phase does not scatter light
|
||||
(small $R_\mathrm{max}$). The phase transition information is easier
|
||||
extracted from $R_\mathrm{max}$. However, this is not always the case
|
||||
and we will shortly see how certain phases of matter scatter a lot of
|
||||
light and can be distinguished using measurements of $W_R$. Another
|
||||
possible concern with experimentally measuring $W_R$ is that it might
|
||||
be obstructed by the classical diffraction maxima which appear at
|
||||
angles corresponding to the minima in $R$. However, the width of such
|
||||
a peak is much smaller as its width is proportional to $1/M$.
|
||||
|
||||
So far both variables we considered, $R_\text{max}$ and $W_R$, provide
|
||||
similar information. Next, we present a case where it is very
|
||||
different. The Bose glass is a localized insulating phase with
|
||||
exponentially decaying correlations but large compressibility and
|
||||
on-site fluctuations in a disordered optical lattice. Therefore,
|
||||
measuring both $R_\text{max}$ and $W_R$ will distinguish all the
|
||||
phases. In a Bose glass we have finite compressibility, but
|
||||
exponentially decaying correlations. This gives a large $R_\text{max}$
|
||||
and a large $W_R$. A Mott insulator will also have exponentially
|
||||
decaying correlations since it is an insulator, but it will be
|
||||
incompressible. Thus, it will scatter light with a small
|
||||
similar information. They both take on values at one of its extremes
|
||||
in the Mott insulating phase and they change drastically across the
|
||||
phase transition into the superfluid phase. Next, we present a case
|
||||
where it is very different. We will again consider ultracold bosons in
|
||||
an optical lattice, but this time we introduce some disorder. We do
|
||||
this by adding an additional periodic potential on top of the exisitng
|
||||
setup that is incommensurate with the original lattice. The resulting
|
||||
Hamiltonian can be shown to be
|
||||
\begin{equation}
|
||||
\hat{H}_\mathrm{dis} = -J^\mathrm{cl} \sum_{\langle i, j \rangle}
|
||||
\bd_i b_j + \frac{U}{2} \sum_i \hat{n}_i (\hat{n}_i - 1) +
|
||||
\frac{V}{2} \sum_i \left[ 1 + \cos (2 r \pi m + 2 \phi) \right]
|
||||
\hat{n}_i,
|
||||
\end{equation}
|
||||
where $V$ is the strength of the superlattice potential, $r$ is the
|
||||
ratio of the superlattice and trapping wave vectors and $\phi$ is some
|
||||
phase shift between the two lattice potentials \cite{roux2008}. The
|
||||
first two terms are the standard Bose-Hubbard Hamiltonian. The only
|
||||
modification is an additionally spatially varying potential shift. We
|
||||
will only consider the phase diagram at fixed density as the
|
||||
introduction of disorder makes the usual interpretation of the phase
|
||||
diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane for a
|
||||
fixed ratio $V/U$ complicated due to the presence of multiple
|
||||
compressible and incompressible phases between successive MI lobes
|
||||
\cite{roux2008}. Therefore, the chemical potential no longer appears
|
||||
in the Hamiltonian as we are no longer considering the grand canonical
|
||||
ensemble.
|
||||
|
||||
The reason for considering such a system is that it introduces a
|
||||
third, competing phase, the Bose glass into our phase diagram. It is
|
||||
an insulating phase like the Mott insulator, but it has local
|
||||
superfluid susceptibility making it compressible. Therefore this
|
||||
localized insulating phase will have exponentially decaying
|
||||
correlations just like the Mott phase, but it will have large on-site
|
||||
fluctuations just like the compressible superfluid phase. As these are
|
||||
the two physical variables encoded in $R$ measuring both
|
||||
$R_\text{max}$ and $W_R$ will provide us with enough information to
|
||||
distinguish all three phases. In a Bose glass we have finite
|
||||
compressibility, but exponentially decaying correlations. This gives a
|
||||
large $R_\text{max}$ and a large $W_R$. A Mott insulator will also
|
||||
have exponentially decaying correlations since it is an insulator, but
|
||||
it will be incompressible. Thus, it will scatter light with a small
|
||||
$R_\text{max}$ and large $W_R$. Finally, a superfluid will have long
|
||||
range correlations and large compressibility which results in a large
|
||||
$R_\text{max}$ and a small $W_R$.
|
||||
@ -423,24 +529,26 @@ $R_\text{max}$ and a small $W_R$.
|
||||
|
||||
We confirm this in Fig. \ref{fig:BG} for simulations with the ratio of
|
||||
superlattice- to trapping lattice-period $r\approx 0.77$ for various
|
||||
disorder strengths $V$ \cite{roux2008}. Here, we only consider
|
||||
calculations for a fixed density, because the usual interpretation of
|
||||
the phase diagram in the ($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) plane
|
||||
for a fixed ratio $V/U$ becomes complicated due to the presence of
|
||||
multiple compressible and incompressible phases between successive MI
|
||||
lobes \cite{roux2008}. This way, we have limited our parameter space
|
||||
to the three phases we are interested in: superfluid, Mott insulator,
|
||||
and Bose glass. From Fig. \ref{fig:BG} we see that all three phases
|
||||
can indeed be distinguished. In the 1D BHM there is no sharp MI-SF
|
||||
phase transition in 1D at a fixed density \cite{cazalilla2011,
|
||||
ejima2011, kuhner2000, pino2012, pino2013} just like in
|
||||
Figs. \ref{fig:SFMI}(d,e) if we follow the transition through the tip
|
||||
of the lobe which corresponds to a line of unit density. However,
|
||||
despite the lack of an easily distinguishable critical point it is
|
||||
possible to quantitatively extract the location of the transition
|
||||
lines by extracting the Tomonaga-Luttinger parameter from the
|
||||
scattered light, $R$, in the same way it was done for an unperturbed
|
||||
BHM \cite{ejima2011}.
|
||||
disorder strengths $V$ \cite{roux2008}. From Fig. \ref{fig:BG} we see
|
||||
that all three phases can indeed be distinguished. From the
|
||||
$R_\mathrm{max}$ plot we can distinguish the two compressible phases,
|
||||
the superfluid and Bose glass, from the incompressible phase, the Mott
|
||||
insulator. We can now also distinguish the Bose glass phase from the
|
||||
superfluid, because only one of them is an insulator and thus the
|
||||
angular width of their scattering patterns will reveal this
|
||||
information. However, unlike the Mott insulator, the Bose glass
|
||||
scatters a lot of light (large $R_\mathrm{max}$) enabling such
|
||||
measurements. Since we performed these calculations at a fixed density
|
||||
the Mott to superfluid phase transition is not particularly sharp
|
||||
\cite{cazalilla2011, ejima2011, kuhner2000, pino2012, pino2013} just
|
||||
like we have seen in Figs. \ref{fig:SFMI}(d,e) if we follow the
|
||||
transition through the tip of the lobe which corresponds to a line of
|
||||
unit density. However, despite the lack of an easily distinguishable
|
||||
critical point, as we have already discussed, it is possible to
|
||||
quantitatively extract the location of the transition lines by
|
||||
extracting the Tomonaga-Luttinger parameter from the scattered light,
|
||||
$R$, in the same way it was done for an unperturbed BHM
|
||||
\cite{ejima2011}.
|
||||
|
||||
Only recently \cite{derrico2014} a Bose glass phase was studied by
|
||||
combined measurements of coherence, transport, and excitation spectra,
|
||||
@ -533,31 +641,33 @@ lattice sites on the microscopic scale in-situ.
|
||||
|
||||
\section{Conclusions}
|
||||
|
||||
In summary, we proposed a nondestructive method to probe quantum gases
|
||||
in an optical lattice. Firstly, we showed that the density-term in
|
||||
scattering has an angular distribution richer than classical
|
||||
diffraction, derived generalized Bragg conditions, and estimated
|
||||
parameters for the only two relevant experiments to date
|
||||
\cite{weitenberg2011, miyake2011}. Secondly, we proposed how to
|
||||
measure the matter-field interference by concentrating light between
|
||||
the sites. This corresponds to interference at the shortest possible
|
||||
distance in an optical lattice. By contrast, standard destructive
|
||||
time-of-flight measurements deal with far-field interference and a
|
||||
relatively near-field one was used in Ref. \cite{miyake2011}. This
|
||||
defines most processes in optical lattices. E.g. matter-field phase
|
||||
changes may happen not only due to external gradients, but also due to
|
||||
intriguing effects such quantum jumps leading to phase flips at
|
||||
neighbouring sites and sudden cancellation of tunneling
|
||||
\cite{vukics2007}, which should be accessible by our method. In
|
||||
mean-field, one can measure the matter-field amplitude (order
|
||||
parameter), quadratures and squeezing. This can link atom optics to
|
||||
areas where quantum optics has already made progress, e.g., quantum
|
||||
imaging \cite{golubev2010, kolobov1999}, using an optical lattice as
|
||||
an array of multimode nonclassical matter-field sources with a high
|
||||
degree of entanglement for quantum information processing. Thirdly, we
|
||||
In this chapter we explored the possibility of nondestructively
|
||||
probing a quantum gas trapped in an optical lattice using quantized
|
||||
light. Firstly, we showed that the density-term in scattering has an
|
||||
angular distribution richer than classical diffraction, derived
|
||||
generalized Bragg conditions, and estimated parameters for two
|
||||
relevant experiments \cite{weitenberg2011, miyake2011}. Secondly, we
|
||||
demonstrated how the method accesses effects beyond mean-field and
|
||||
distinguishes all the phases in the Mott-superfluid-glass transition,
|
||||
which is currently a challenge \cite{derrico2014}. Based on
|
||||
which is currently a challenge \cite{derrico2014}. Finally, we looked
|
||||
at measuring the matter-field interference via the operator $\hat{B}$
|
||||
by concentrating light between the sites. This corresponds to probing
|
||||
interference at the shortest possible distance in an optical
|
||||
lattice. This is in contrast to standard destructive time-of-flight
|
||||
measurements which deal with far-field interference and a relatively
|
||||
near-field one was used in Ref. \cite{miyake2011}. This defines most
|
||||
processes in optical lattices. E.g. matter-field phase changes may
|
||||
happen not only due to external gradients, but also due to intriguing
|
||||
effects such quantum jumps leading to phase flips at neighbouring
|
||||
sites and sudden cancellation of tunneling \cite{vukics2007}, which
|
||||
should be accessible by our method. We showed how in mean-field, one
|
||||
can measure the matter-field amplitude (order parameter), quadratures
|
||||
and squeezing. This can link atom optics to areas where quantum optics
|
||||
has already made progress, e.g., quantum imaging \cite{golubev2010,
|
||||
kolobov1999}, using an optical lattice as an array of multimode
|
||||
nonclassical matter-field sources with a high degree of entanglement
|
||||
for quantum information processing. Since our scheme is based on
|
||||
off-resonant scattering, and thus being insensitive to a detailed
|
||||
atomic level structure, the method can be extended to molecules
|
||||
\cite{LP2013}, spins, and fermions \cite{ruostekoski2009}.
|
||||
|
||||
|
||||
@ -1,7 +1,7 @@
|
||||
% ******************************* PhD Thesis Template **************************
|
||||
% Please have a look at the README.md file for info on how to use the template
|
||||
|
||||
\documentclass[a4paper,12pt,times,numbered,print,index,chapter]{Classes/PhDThesisPSnPDF}
|
||||
\documentclass[a4paper,12pt,times,numbered,print,index]{Classes/PhDThesisPSnPDF}
|
||||
|
||||
% ******************************************************************************
|
||||
% ******************************* Class Options ********************************
|
||||
|
||||
Reference in New Issue
Block a user