From 7f3f3e59c14719c597f518e814478b309ca81d4c Mon Sep 17 00:00:00 2001 From: Wojciech Kozlowski Date: Fri, 15 Jul 2016 19:16:19 +0100 Subject: [PATCH] End for the day - rewriting chapter 3 content to make more sense within the scope of the thesis --- Chapter2/chapter2.tex | 11 +- Chapter3/chapter3.tex | 476 ++++++++++++++++++++++++------------------ thesis.tex | 2 +- 3 files changed, 284 insertions(+), 205 deletions(-) diff --git a/Chapter2/chapter2.tex b/Chapter2/chapter2.tex index 976fe73..2ceff37 100644 --- a/Chapter2/chapter2.tex +++ b/Chapter2/chapter2.tex @@ -95,6 +95,7 @@ variable angles in our model as this lends itself to a simpler geometrical representation. \subsection{Derivation of the Hamiltonian} +\label{sec:derivation} A general many-body Hamiltonian coupled to a quantized light field in second quantized can be separated into three parts, @@ -358,6 +359,7 @@ the quantum state of light. \mynote{cite Santiago's papers and Therefore, combining these final simplifications we finally arrive at our quantum light-matter Hamiltonian \begin{equation} + \label{eq:fullH} \H = \H_f -J^\mathrm{cl} \sum_{\langle i,j \rangle}^M \bd_i b_j + \frac{U}{2} \sum_{i}^M \hat{n}_i (\hat{n}_i - 1) + \frac{\hbar}{\Delta_a} \sum_{l,m} g_l g_m \ad_l \a_m \hat{F}_{l,m} - @@ -492,7 +494,14 @@ 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. +sources. Such a counter-intuitive configuration may affect works on +quantum gases trapped in quantum potentials \cite{mekhov2012, + mekhov2008, larson2008, chen2009, habibian2013, ivanov2014, + caballero2015} and quantum measurement-induced preparation of +many-body atomic states \cite{mazzucchi2016, mekhov2009prl, + pedersen2014, elliott2015}. + +\mynote{add citiations above if necessary} For clarity we will consider a 1D lattice as shown in Fig. \ref{fig:LatticeDiagram} with lattice spacing $d$ along the diff --git a/Chapter3/chapter3.tex b/Chapter3/chapter3.tex index f65d7ee..14f68a9 100644 --- a/Chapter3/chapter3.tex +++ b/Chapter3/chapter3.tex @@ -19,53 +19,69 @@ Nondestructive Addressing} %Title of the Third Chapter Having developed the basic theoretical framework within which we can treat the fully quantum regime of light-matter interactions we now consider possible applications. There are three prominent directions -in which we can apply our model: nondestructive probing, quantum -measurement backaction and quantum optical lattices. Here, we deal -with the first of the three options. +in which we can proceed: nondestructive probing of the quantum state +of matter, quantum measurement backaction induced dynamics and quantum +optical lattices. Here, we deal with the first of the three options. -\mynote{adjust for the fact that the derivation of B operator has been moved} -\mynote{update some outdated mentions to previous experiments} In this chapter we develop a method to measure properties of ultracold -gases in optical lattices by light scattering. We show that such -measurements can reveal not only density correlations, but also -matter-field interference. Recent quantum non-demolition (QND) -schemes \cite{rogers2014, mekhov2007prl, eckert2008} probe density -fluctuations and thus inevitably destroy information about phase, -i.e.~the conjugate variable, and as a consequence destroy matter-field -coherence. In contrast, we focus on probing the atom interference -between lattice sites. Our scheme is nondestructive in contrast to -totally destructive methods such as time-of-flight measurements. It +gases in optical lattices by light scattering. In the previous chapter +we have shown that quantum light field couples to the bosons via the +operator $\hat{F}$. This is the key element of the scheme we propose +as this makes it sensitive to the quantum state of the matter and all +of its possible superpositions which will be reflected in the quantum +state of the light itself. We have also shown in section +\ref{sec:derivation} that this coupling consists of two parts, a +density component $\hat{D}$ given by Eq. \eqref{eq:D}, and a phase +component $\hat{B}$ given by Eq. \eqref{eq:B}. Therefore, when probing +the quantum state of the ultracold gas we can have access to not only +density correlations, but also matter-field interference at its +shortest possible distance in an optical lattice, i.e.~the lattice +period. Previous work on quantum non-demolition (QND) schemes +\cite{rogers2014, mekhov2007prl, eckert2008} probe only the density +component as it is generally challenging to couple to the matter-field +observables directly. Here, we will consider nondestructive probing of +both density and interference operators. + +Firstly, we will consider the simpler and more typical case of +coupling to the atom number operators via $\hat{F} = +\hat{D}$. However, we show that light diffraction in this regime has +several nontrivial characteristics due to the fully quantum nature of +the interaction. Firstly, we show that the angular distribution has +multiple interesting features even when classical diffraction is +forbidden facilitating their experimental observation. We derive new +generalised Bragg diffraction conditions which are different to their +classical counterpart. Furthermore, due to the fully quantum nature of +the interaction our proposal is capable of probing the quantum state +beyond mean-field prediction. We demonstrate this by showing that this +scheme is capable of distinguishing all three phases in the Mott +insulator - superfluid - Bose glass phase transition in a 1D +disordered optical lattice which is not very well described by a +mean-field treatment. We underline that transitions in 1D are much +more visible when changing an atomic density rather than for +fixed-density scattering. It was only recently that an experiment +distinguished a Mott insulator from a Bose glass \cite{derrico2014} +via a series of destructive measurements. Our proposal, on the other +hand, is nondestructive and is capable of extracting all the relevant +information in a single experiment. + +Having shown the possibilities created by this nondestructive +measurement scheme we move on to considering light scattering from the +phase related observables via the operator $\hat{F} = \hat{B}$. This enables in-situ probing of the matter-field coherence at its shortest possible distance in an optical lattice, i.e. the lattice period, which defines key processes such as tunnelling, currents, phase -gradients, etc. This is achieved by concentrating light between the -sites. 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}. Such a counter-intuitive configuration -may affect works on quantum gases trapped in quantum potentials -\cite{mekhov2012, mekhov2008, larson2008, chen2009, habibian2013, - ivanov2014, caballero2015} and quantum measurement-induced -preparation of many-body atomic states \cite{mazzucchi2016, - mekhov2009prl, pedersen2014, elliott2015}. Within the mean-field -treatment, this enables measurements of the order parameter, -matter-field quadratures and squeezing. This can have an impact on -atom-wave metrology and information processing in areas where quantum -optics already made progress, e.g., quantum imaging with pixellized -sources of non-classical light \cite{golubev2010, kolobov1999}, as an -optical lattice is a natural source of multimode nonclassical matter -waves. +gradients, etc. This is in contrast to standard destructive +time-of-flight measurements which deal with far-field interference +although a relatively near-field scheme was use in +Ref. \cite{miyake2011}. We show how within the mean-field treatment, +this enables measurements of the order parameter, matter-field +quadratures and squeezing. This can have an impact on atom-wave +metrology and information processing in areas where quantum optics +already made progress, e.g., quantum imaging with pixellized sources +of non-classical light \cite{golubev2010, kolobov1999}, as an optical +lattice is a natural source of multimode nonclassical matter waves. -Furthermore, the scattering angular distribution is nontrivial, even -when classical diffraction is forbidden and we derive generalized -Bragg conditions for this situation. The method works beyond -mean-field, which we demonstrate by distinguishing all three phases in -the Mott insulator - superfluid - Bose glass phase transition in a 1D -disordered optical lattice. We underline that transitions in 1D are -much more visible when changing an atomic density rather than for -fixed-density scattering. It was only recently that an experiment -distinguished a Mott insulator from a Bose glass \cite{derrico2014}. - -\section{Global Nondestructive Measurement} +\section{Coupling to the Quantum State of Matter} As we have seen in section \ref{sec:a} under certain approximations the scattered light mode, $\a_1$, is linked to the quantum state of @@ -77,9 +93,10 @@ matter via where the atomic operators $\hat{D}$ and $\hat{B}$, given by Eq. \eqref{eq:D} and Eq. \eqref{eq:B}, are responsible for the coupling to on-site density and inter-site interference -respectively. It crucial to note that light couples to the bosons via -an operator as this makes it sensitive to the quantum state of the -matter. +respectively. It is crucial to note that light couples to the bosons +via an operator as this makes it sensitive to the quantum state of the +matter as this will imprint the fluctuations in the quantum state of +the scattered light. Here, we will use this fact that the light is sensitive to the atomic quantum state due to the coupling of the optical and matter fields via @@ -87,97 +104,149 @@ operators in order to develop a method to probe the properties of an ultracold gas. Therefore, we neglect the measurement back-action and we will only consider expectation values of light observables. Since the scheme is nondestructive (in some cases, it even satisfies the -stricter requirements for a QND measurement \cite{mekhov2007pra, - mekhov2012}) and the measurement only weakly perturbs the system, +stricter requirements for a QND measurement \cite{mekhov2012, + mekhov2007pra}) and the measurement only weakly perturbs the system, many consecutive measurements can be carried out with the same atoms -without preparing a new sample. Again, we will show how the extreme +without preparing a new sample. We will show how the extreme flexibility of the the measurement operator $\hat{F}$ allows us to probe a variety of different atomic properties in-situ ranging from density correlations to matter-field interference. -\subsection{On-site density measurements} +\subsection{On-site Density Measurements} -Typically, the dominant term in $\hat{F}$ is the density term -$\hat{D}$, rather than inter-site matter-field interference $\hat{B}$ -\cite{mekhov2007pra, rist2010, lakomy2009, ruostekoski2009, - LP2009}. However, before we move onto probing the interference -terms, $\hat{B}$, we will first discuss typical light scattering. We -start with a simpler case when scattering is faster than tunneling and -$\hat{F} = \hat{D}$. This corresponds to a QND scheme -\cite{mekhov2007prl, mekhov2007pra, eckert2008, rogers2014}. The -density-related measurement destroys some matter-phase coherence in -the conjugate variable \cite{mekhov2009pra, LP2010, LP2011} -$\bd_i b_{i+1}$, but this term is neglected. For this purpose we will -define an auxiliary quantity, +We have seen in section \ref{sec:B} that typically the dominant term +in $\hat{F}$ is the density term $\hat{D}$ \cite{LP2009, + mekhov2007pra, rist2010, lakomy2009, ruostekoski2009}. This is +simply due to the fact that atoms are localised with lattice sites +leading to an effective coupling with atom number operators instead of +inter-site interference terms. Therefore, we will first consider +nondestructive probing of the density related observables of the +quantum gas. However, we will focus on the novel nontrivial aspects +that go beyond the work in Ref. \cite{mekhov2012, mekhov2007prl, + mekhov2007pra} which only considered a few extremal cases. + +As we are only interested in the quantum information imprinted in the +state of the optical field we will simplify our analysis by +considering the light scattering to be much faster than the atomic +tunnelling. Therefore, our scheme is actually a QND scheme +\cite{rogers2014, mekhov2007prl, mekhov2007pra, eckert2008} as +normally density-related measurements destroy the matter-phase +coherence since it is its conjugate variable, but here we neglect the +$\bd_i b_j$ terms. Furthermore, we will consider a deep +lattice. Therefore, the Wannier functions will be well localised +within their corresponding lattice sites and thus the coefficients +$J_{i,i}$ reduce to $u_1^*(\b{r}_i) u_0(\b{r}_i)$ leading to +\begin{equation} + \label{eq:D-3} + \hat{D}=\sum_i^K u_1^*(\b{r}_i) u_0(\b{r}_i) \hat{n}_i, +\end{equation} +which for travelling +[$u_l(\b{r})=\exp(i \b{k}_l \cdot \b{r}+i\varphi_l)$] or standing +[$u_l(\b{r})=\cos(\b{k}_l \cdot \b{r}+\varphi_l)$] waves is just a +density Fourier transform at one or several wave vectors +$\pm(\b{k}_1 \pm \b{k}_0)$. + +We will now define a new auxiliary quantity to aid our analysis, \begin{equation} \label{eq:R} R = \langle \ad_1 \a_1 \rangle - | \langle \a_1 \rangle |^2, \end{equation} -which we will call the ``quantum addition'' to light scattering. $R$ -is simply the full light intensity minus the classical field intensity -and thus it faithfully represents the new contribution from the -quantum light-matter interaction to the diffraction pattern. +which we will call the ``quantum addition'' to light scattering. By +construction $R$ is simply the full light intensity minus the +classical field diffraction. In order to justify its name we will show +that this quantity depends purely quantum mechanical properties of the +ultracold gase. We will substitute $\a_1 = C \hat{D}$ using +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} + R = |C|^2 \sum_{i,j}^K A^*_i A_j \langle \delta \hat{n}_i \delta + \hat{n}_j \rangle, +\end{equation} +where $\delta \hat{n}_i = \hat{n}_i - \langle \hat{n}_i +\rangle$. Thus, we can clearly see that $R$ is a result of light +scattering from fluctuations in the atom number which is a purely +quantum mechanical property of a system. Therefore, $R$, the ``quantum +addition'' faithfully represents the new contribution from the quantum +light-matter interaction to the diffraction pattern. + +If instead we are interested in quantities linear in $\hat{D}$, we can +measure the quadrature of the light fields which in section +\ref{sec:a} we saw that $\hat{X}_\phi = |C| \hat{X}^F_\beta$. For the +case when both the scattered mode and probe are travelling waves the +quadrature +\begin{equation} + \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]. +\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. + +The ``quantum addition'', $R$, and the quadrature variance, +$(\Delta X^F_\beta)^2$, are both quadratic in $\a_1$. Therefore, they +will havea 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 +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 +fluctuations entirely leading to absolutely no ``quantum addition''. \begin{figure}[htbp!] \centering \includegraphics[width=\linewidth]{Ep1} \caption[Light Scattering Angular Distribution]{Light intensity scattered into a standing wave mode from a superfluid in a 3D - lattice (units of $R/N_K$). Arrows denote incoming travelling wave - probes. The Bragg condition, $\Delta \b{k} = \b{G}$, is not - fulfilled, so there is no classical diffraction, but intensity - still shows multiple peaks, whose heights are tunable by simple - phase shifts of the optical beams: (a) $\varphi_1=0$; (b) - $\varphi_1=\pi/2$. Interestingly, there is also a significant - uniform background level of scattering which does not occur in its - classical counterpart. } - \label{fig:Scattering} + lattice (units of $R/(|C|^2N_K)$). Arrows denote incoming + travelling wave probes. The classical Bragg condition, + $\Delta \b{k} = \b{G}$, is not fulfilled, so there is no classical + diffraction, but intensity still shows multiple peaks, whose + heights are tunable by simple phase shifts of the optical beams: + (a) $\varphi_1=0$; (b) $\varphi_1=\pi/2$. Interestingly, there is + also a significant uniform background level of scattering which + does not occur in its classical counterpart. } + \label{fig:scattering} \end{figure} -In a deep lattice, -\begin{equation} - \hat{D}=\sum_i^K u_1^*({\bf r}_i) u_0({\bf r}_i) \hat{n}_i, -\end{equation} -which for travelling -[$u_l(\b{r})=\exp(i \b{k}_l \cdot \b{r}+i\varphi_l)$] or standing -[$u_l(\b{r})=\cos(\b{k}_l \cdot \b{r}+\varphi_l)$] waves is just a -density Fourier transform at one or several wave vectors -$\pm(\b{k}_1 \pm \b{k}_0)$. The quadrature, as defined in section -\ref{sec:a}, for two travelling waves is reduced to -\begin{equation} - \hat{X}^F_\beta = \sum_i^K \hat{n}_i\cos[(\b{k}_1 - \b{k}_2) \cdot - \b{r}_i - \beta]. -\end{equation} -Note that different light quadratures are differently coupled to the -atom distribution, hence varying local oscillator phase and detection -angle, one scans the coupling from maximal to zero. An identical -expression exists for $\hat{D}$ for a standing wave, where $\beta$ is -replaced by $\varphi_l$, and scanning is achieved by varying the -position of the wave with respect to atoms. Thus, variance -$(\Delta X^F_\beta)^2$ and quantum addition $R$, have a non-trivial -angular dependence, showing more peaks than classical diffraction and -the peaks can be tuned by the light-atom coupling. +We can also observe maxima at several different angles in +Fig. \ref{fig:scattering}. Interestingly, they occur at different +angles than predicted by the classical Bragg condition. Moreover, the +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}. -Fig. \ref{fig:Scattering} shows the angular dependence of $R$ for -standing and travelling waves scattering from bosons in a 3D optical -lattice. The isotropic background gives the density fluctuations -[$R = K( \langle \hat{n}^2 \rangle - \langle \hat{n} \rangle^2 )/2$ in -mean-field with inter-site correlations neglected]. The radius of the -sphere changes from zero, when it is a Mott insulator with suppressed -fluctuations, to half the atom number at $K$ sites, $N_K/2$, in the -deep superfluid. There exist peaks at angles different than the -classical Bragg ones and thus, can be observed without being masked by -classical diffraction. Interestingly, even if 3D diffraction -\cite{miyake2011} is forbidden as seen in Fig. \ref{fig:Scattering}, -the peaks are still present. 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 +% Derive and show these Bragg conditions + +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. +\ref{fig:scattering}b. In section \ref{sec:Efield} we have estimated the mean photon scattering rates integrated over the solid angle for the only two @@ -187,97 +256,15 @@ where the measurement object was light n_{\Phi}= \left(\frac{\Omega_0}{\Delta_a}\right)^2 \frac{\Gamma K}{8} (\langle\hat{n}^2\rangle-\langle\hat{n}\rangle^2). \end{equation} -The background signal should reach $n_\Phi \approx 10^6$ s$^{-1}$ in -Ref. \cite{weitenberg2011} (150 atoms in 2D), and -$n_\Phi \approx 10^{11}$ s$^{-1}$ in Ref. \cite{miyake2011} ($10^5$ -atoms in 3D). These numbers show that the diffraction patterns we have -seen due to the ``quantum addition'' should be visible using currently -available technology, especially since the most prominent features, -such as Bragg diffraction peaks, do not coincide at all with the -classical diffraction pattern. - -\subsection{Matter-field interference measurements} - -We now focus on enhancing the interference term $\hat{B}$ in the -operator $\hat{F}$. - -Firstly, we will use this result to show how one can probe -$\langle \hat{B} \rangle$ which in MF gives information about the -matter-field amplitude, $\Phi = \langle b \rangle$. - -Hence, by measuring the light quadrature we probe the kinetic energy -and, in MF, the matter-field amplitude (order parameter) $\Phi$: -$\langle \hat{X}^F_{\beta=0} \rangle = | \Phi |^2 -\mathcal{F}[W_1](2\pi/d) (K-1)$. - -Secondly, we show that it is also possible to access the fluctuations -of matter-field quadratures $\hat{X}^b_\alpha = (b e^{-i\alpha} + \bd -e^{i\alpha})/2$, which in MF can be probed by measuring the variance -of $\hat{B}$. Across the phase transition, the matter field changes -its state from Fock (in MI) to coherent (deep SF) through an -amplitude-squeezed state as shown in Fig. \ref{Quads}(a,b). - -Assuming $\Phi$ is real in MF: -\begin{equation} - \label{intensity} - \langle \ad_1 \a_1 \rangle = 2 |C|^2(K-1)\mathcal{F}^2[W_1](\frac{\pi}{d}) - \times [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ] -\end{equation} -and it is shown as a function of $U/(zJ^\text{cl})$ in -Fig. \ref{Quads}. Thus, since measurement in the diffraction maximum -yields $\Phi^2$ we can deduce $\langle b^2 \rangle - \Phi^2$ from the -intensity. This quantity is of great interest as it gives us access to -the quadrature variances of the matter-field -\begin{equation} - (\Delta X^b_{0,\pi/2})^2 = 1/4 + [(n - \Phi^2) \pm - (\langle b^2 \rangle - \Phi^2)]/2, -\end{equation} -where $n=\langle\hat{n}\rangle$ is the mean on-site atomic density. - -\begin{figure}[htbp!] - \centering - \includegraphics[width=\linewidth]{Quads} - \captionsetup{justification=centerlast,font=small} - \caption[Mean-Field Matter Quadratures]{Photon number scattered in a - diffraction minimum, given by Eq. (\ref{intensity}), where - $\tilde{C} = 2 |C|^2 (K-1) \mathcal{F}^2 [W_1](\pi/d)$. More - light is scattered from a MI than a SF due to the large - uncertainty in phase in the insulator. (a) The variances of - quadratures $\Delta X^b_0$ (solid) and $\Delta X^b_{\pi/2}$ - (dashed) of the matter field across the phase transition. Level - 1/4 is the minimal (Heisenberg) uncertainty. There are three - important points along the phase transition: the coherent state - (SF) at A, the amplitude-squeezed state at B, and the Fock state - (MI) at C. (b) The uncertainties plotted in phase space.} - \label{Quads} -\end{figure} - -Probing $\hat{B}^2$ gives us access to kinetic energy fluctuations -with 4-point correlations ($\bd_i b_j$ combined in pairs). Measuring -the photon number variance, which is standard in quantum optics, will -lead up to 8-point correlations similar to 4-point density -correlations \cite{mekhov2007pra}. These are of significant interest, -because it has been shown that there are quantum entangled states that -manifest themselves only in high-order correlations -\cite{kaszlikowski2008}. - -Surprisingly, inter-site terms scatter more light from a Mott -insulator than a superfluid Eq. \eqref{intensity}, as shown in -Fig. \eqref{Quads}, although the mean inter-site density -$\langle \hat{n}(\b{r})\rangle $ is tiny in a MI. This reflects a -fundamental effect of the boson interference in Fock states. It indeed -happens between two sites, but as the phase is uncertain, it results -in the large variance of $\hat{n}(\b{r})$ captured by light as shown -in Eq. \eqref{intensity}. The interference between two macroscopic -BECs has been observed and studied theoretically -\cite{horak1999}. When two BECs in Fock states interfere a phase -difference is established between them and an interference pattern is -observed which disappears when the results are averaged over a large -number of experimental realizations. This reflects the large -shot-to-shot phase fluctuations corresponding to a large inter-site -variance of $\hat{n}(\b{r})$. By contrast, our method enables the -observation of such phase uncertainty in a Fock state directly between -lattice sites on the microscopic scale in-situ. +Therefore, applying these results to the scattering patters in +Fig. \ref{fig:scattering} the background signal should reach +$n_\Phi \approx 10^6$ s$^{-1}$ in Ref. \cite{weitenberg2011} (150 +atoms in 2D), and $n_\Phi \approx 10^{11}$ s$^{-1}$ in +Ref. \cite{miyake2011} ($10^5$ atoms in 3D). These numbers show that +the diffraction patterns we have seen due to the ``quantum addition'' +should be visible using currently available technology, especially +since the most prominent features, such as Bragg diffraction peaks, do +not coincide at all with the classical diffraction pattern. \subsection{Mapping the quantum phase diagram} @@ -461,6 +448,89 @@ all of which are destructive techniques. Our method is simpler as it only requires measurement of the quantity $R$ and additionally, it is nondestructive. +\subsection{Matter-field interference measurements} + +We now focus on enhancing the interference term $\hat{B}$ in the +operator $\hat{F}$. + +Firstly, we will use this result to show how one can probe +$\langle \hat{B} \rangle$ which in MF gives information about the +matter-field amplitude, $\Phi = \langle b \rangle$. + +Hence, by measuring the light quadrature we probe the kinetic energy +and, in MF, the matter-field amplitude (order parameter) $\Phi$: +$\langle \hat{X}^F_{\beta=0} \rangle = | \Phi |^2 +\mathcal{F}[W_1](2\pi/d) (K-1)$. + +Secondly, we show that it is also possible to access the fluctuations +of matter-field quadratures $\hat{X}^b_\alpha = (b e^{-i\alpha} + \bd +e^{i\alpha})/2$, which in MF can be probed by measuring the variance +of $\hat{B}$. Across the phase transition, the matter field changes +its state from Fock (in MI) to coherent (deep SF) through an +amplitude-squeezed state as shown in Fig. \ref{Quads}(a,b). + +Assuming $\Phi$ is real in MF: +\begin{equation} + \label{intensity} + \langle \ad_1 \a_1 \rangle = 2 |C|^2(K-1)\mathcal{F}^2[W_1](\frac{\pi}{d}) + \times [ ( \langle b^2 \rangle - \Phi^2 )^2 + ( n - \Phi^2 ) ( 1 +n - \Phi^2 ) ] +\end{equation} +and it is shown as a function of $U/(zJ^\text{cl})$ in +Fig. \ref{Quads}. Thus, since measurement in the diffraction maximum +yields $\Phi^2$ we can deduce $\langle b^2 \rangle - \Phi^2$ from the +intensity. This quantity is of great interest as it gives us access to +the quadrature variances of the matter-field +\begin{equation} + (\Delta X^b_{0,\pi/2})^2 = 1/4 + [(n - \Phi^2) \pm + (\langle b^2 \rangle - \Phi^2)]/2, +\end{equation} +where $n=\langle\hat{n}\rangle$ is the mean on-site atomic density. + +\begin{figure}[htbp!] + \centering + \includegraphics[width=\linewidth]{Quads} + \captionsetup{justification=centerlast,font=small} + \caption[Mean-Field Matter Quadratures]{Photon number scattered in a + diffraction minimum, given by Eq. (\ref{intensity}), where + $\tilde{C} = 2 |C|^2 (K-1) \mathcal{F}^2 [W_1](\pi/d)$. More + light is scattered from a MI than a SF due to the large + uncertainty in phase in the insulator. (a) The variances of + quadratures $\Delta X^b_0$ (solid) and $\Delta X^b_{\pi/2}$ + (dashed) of the matter field across the phase transition. Level + 1/4 is the minimal (Heisenberg) uncertainty. There are three + important points along the phase transition: the coherent state + (SF) at A, the amplitude-squeezed state at B, and the Fock state + (MI) at C. (b) The uncertainties plotted in phase space.} + \label{Quads} +\end{figure} + +Probing $\hat{B}^2$ gives us access to kinetic energy fluctuations +with 4-point correlations ($\bd_i b_j$ combined in pairs). Measuring +the photon number variance, which is standard in quantum optics, will +lead up to 8-point correlations similar to 4-point density +correlations \cite{mekhov2007pra}. These are of significant interest, +because it has been shown that there are quantum entangled states that +manifest themselves only in high-order correlations +\cite{kaszlikowski2008}. + +Surprisingly, inter-site terms scatter more light from a Mott +insulator than a superfluid Eq. \eqref{intensity}, as shown in +Fig. \eqref{Quads}, although the mean inter-site density +$\langle \hat{n}(\b{r})\rangle $ is tiny in a MI. This reflects a +fundamental effect of the boson interference in Fock states. It indeed +happens between two sites, but as the phase is uncertain, it results +in the large variance of $\hat{n}(\b{r})$ captured by light as shown +in Eq. \eqref{intensity}. The interference between two macroscopic +BECs has been observed and studied theoretically +\cite{horak1999}. When two BECs in Fock states interfere a phase +difference is established between them and an interference pattern is +observed which disappears when the results are averaged over a large +number of experimental realizations. This reflects the large +shot-to-shot phase fluctuations corresponding to a large inter-site +variance of $\hat{n}(\b{r})$. By contrast, our method enables the +observation of such phase uncertainty in a Fock state directly between +lattice sites on the microscopic scale in-situ. + \section{Conclusions} In summary, we proposed a nondestructive method to probe quantum gases diff --git a/thesis.tex b/thesis.tex index 0e6eb48..24525c8 100644 --- a/thesis.tex +++ b/thesis.tex @@ -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]{Classes/PhDThesisPSnPDF} +\documentclass[a4paper,12pt,times,numbered,print,index,chapter]{Classes/PhDThesisPSnPDF} % ****************************************************************************** % ******************************* Class Options ********************************