From 86e3b2644c126fb512daf3c300b58a1f7ca2dd1d Mon Sep 17 00:00:00 2001 From: Wojciech Kozlowski Date: Sat, 16 Jul 2016 18:56:16 +0100 Subject: [PATCH] Finished for the day. Chapter 3 still needs an update to the B operator section and possibly another look at the conclusion --- Chapter3/chapter3.tex | 456 ++++++++++++++++++++++++++---------------- thesis.tex | 2 +- 2 files changed, 284 insertions(+), 174 deletions(-) diff --git a/Chapter3/chapter3.tex b/Chapter3/chapter3.tex index 14f68a9..5bd4a02 100644 --- a/Chapter3/chapter3.tex +++ b/Chapter3/chapter3.tex @@ -188,23 +188,23 @@ 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 +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 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 @@ -236,7 +236,7 @@ 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}. -% Derive and show these Bragg conditions +\mynote{Derive and show these Bragg conditions below} As $(\Delta X^F_\beta)^2$ and $R$ are quadratic variables, the generalized Bragg conditions for the peaks are @@ -248,18 +248,22 @@ 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. -In section \ref{sec:Efield} we have estimated the mean photon -scattering rates integrated over the solid angle for the only two -experiments so far on light diffraction from truly ultracold bosons -where the measurement object was light +A quantum signal that isn't masked by classical diffraction is very +useful for future experimental realisability. However, it is still +unclear whether this signal would be strong enough to be +visible. After all, a classical signal scales as $N_K^2$ whereas here +we have only seen a scaling of $N_K$. In section \ref{sec:Efield} we +have estimated the mean photon scattering rates integrated over the +solid angle for the only two experiments so far on light diffraction +from truly ultracold bosons where the measurement object was light \begin{equation} 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} -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 +These results can be applied directly to the scattering patters in +Fig. \ref{fig:scattering}. Therefore, 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 @@ -268,6 +272,123 @@ not coincide at all with the classical diffraction pattern. \subsection{Mapping the quantum phase diagram} +We have shown that scattering from atom number operators leads to a +purely quantum diffraction pattern which depends on the density +fluctuations and their correlations. We have also seen that this +signal should be strong enough to be visible using currently available +technology. However, so far we have not looked at what this can tell +us about the quantum state of matter. We have briefly mentioned that a +deep superfluid will scatter a lot of light due to its infinite range +correlations and a Mott insulator will not contriute any ``quantum +addition'' at all, but we have not look at the quantum phase +transition between these two phases. In two or higher dimensions this +has a rather simple answer as the Bose-Hubbard phase transition is +described well by mean-field theories and it has a sharp transition at +the critical point. This means that the ``quantum addition'' signal +would drop rapidly at the critical point and go to zero as soon as it +was crossed. However, $R$ given by Eq. \eqref{eq:R} clearly contains +much more information. + +There are many situations where the mean-field approximation is not a +valid description of the physics. A prominent example is the +Bose-Hubbard model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, + pino2012, pino2013}. Observing the transition in 1D by light at +fixed density was considered to be difficult \cite{rogers2014} or even +impossible \cite{roth2003}. This is, because the one-dimensional +quantum phase transition is in a different universality class than its +higher dimensional counterparts. The energy gap, which is the order +parameter, decays exponentially slowly across the phase transition +making it difficult to identify the phase transition even in numerical +simulations. Here, we will show the avaialable tools provided by the +``quantum addition'' that allows one to nondestructively map this +phase transition and distinguish the superfluid and Mott insulator +phases. + +The 1D phase transition is best understood in terms of two-point +correlations as a function of their separation \cite{giamarchi}. In +the Mott insulating phase, the two-point correlations $\langle \bd_i +b_j \rangle$ and $\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$ +($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay +exponentially with $|i-j|$. This is a characteristic of insulators. On +the other hand the superfluid will exhibit long-range order which in +dimensions higher than one, manifests itself with an infinite +correlation length. However, in 1D only pseudo long-range order +happens and both the matter-field and density fluctuation correlations +decay algebraically \cite{giamarchi}. + +The method we propose gives us direct access to the structure factor, +which is a function of the two-point correlation $\langle \delta +\hat{n}_i \delta \hat{n}_j \rangle$. This quantity can be extracted +from the measured light intensity bu considering the ``quantum +addition''. We will consider the case when both the probe and +scattered modes are plane waves which can be easily achieved in free +space. We will again consider the case of light being maximally +coupled to the density ($\hat{F} = \hat{D}$). Therefore, the quantum +addition is given by +\begin{equation} + R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0) + (\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta + \hat{n}_j \rangle. +\end{equation} + +\mynote{can put in more detail here with equations} + +This alone allows us to analyse the phase transition quantitatively +using our method. Unlike in higher dimensions where an order parameter +can be easily defined within the mean-field approximation as a simple +expectation value of some quantity, the situation in 1D is more +complex as it is difficult to directly access the excitation energy +gap which defines this phase transition. However, a valid description +of the relevant 1D low energy physics is provided by Luttinger liquid +theory \cite{giamarchi}. In this model correlations in the supefluid +phase as well as the superfluid density itself are characterised by +the Tomonaga-Luttinger parameter, $K_b$. This parameter also +identifies the critical point in the thermodynamic limit at $K_b = +1/2$. This quantity can be extracted from various correlation +functions and in our case it can be extracted directly from $R$ +\cite{ejima2011}. This quantity was used in numerical calculations +that used highly efficient density matrix renormalisation group (DMRG) +methods to calculate the ground state to subsequently fit the +Luttinger theoru to extract this parameter $K_b$. These calculations +yield a theoretical estimate of the critical point in the +thermodynamic limit for commensurate filling in 1D to be at +$U/2J^\text{cl} \approx 1.64$ \cite{ejima2011}. Our proposal provides +a method to directly measure $R$ nondestructively in a lab which can +then be used to experimentally determine the location of the critical +point in 1D. + +However, whilst such an approach will yield valuable quantitative +results we will instead focus on its qualitative features which give a +more intuitive understanding of what information can be extracted from +$R$. This is because the superfluid to Mott insulator phase transition +is well understood, so there is no reason to dwell on its quantitative +aspects. However, our method is much more general than the +Bose-Hubbard model as it can be easily applied to many other systems +such as fermions, photonic circuits, optical lattices qith quantum +potentials, etc. Therefore, by providing a better physical picture of +what information is carried by the ``quantum addition'' it should be +easier to see its usefuleness in a broader context. + +We calculate the phase diagram of the Bose-Hubbard Hamiltonian given +by +\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) - \mu + \sum_i \hat{n}_i, +\end{equation} +where the $\mu$ is the chemical potential. We have introduced the last +term as we are interested in grand canonical ensemble calculations as +we want to see how the system's behaviour changes as density is +varied. We perform numerical calculations of the ground state using +DMRG methods \cite{tnt} from which we can compute all the necessary +atomic observables. Experiments typically use an additional harmonic +confining potential on top of the optical lattice to keep the atoms in +place which means that the chemical potential will vary in +space. However, with careful consideration of the full +($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in +Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the +system \cite{batrouni2002}. + \begin{figure}[htbp!] \centering \includegraphics[width=\linewidth]{oph11} @@ -288,121 +409,106 @@ not coincide at all with the classical diffraction pattern. \label{fig:SFMI} \end{figure} -We have shown how in mean-field, we can track the order parameter, -$\Phi$, by probing the matter-field interference using the coupling of -light to the $\hat{B}$ operator. In this case, it is very easy to -follow the superfluid to Mott insulator quantum phase transition since -we have direct access to the order parameter which goes to zero in the -insulating phase. In fact, if we're only interested in the critical -point, we only need access to any quantity that yields information -about density fluctuations which also go to zero in the MI phase and -this can be obtained by measuring -$\langle \hat{D}^\dagger \hat{D} \rangle$. However, there are many -situations where the mean-field approximation is not a valid -description of the physics. A prominent example is the Bose-Hubbard -model in 1D \cite{cazalilla2011, ejima2011, kuhner2000, pino2012, - pino2013}. Observing the transition in 1D by light at fixed density -was considered to be difficult \cite{rogers2014} or even impossible -\cite{roth2003}. By contrast, here we propose varying the density or -chemical potential, which sharply identifies the transition. We -perform these calculations numerically by calculating the ground state -using DMRG methods \cite{tnt} from which we can compute all the -necessary atomic observables. Experiments typically use an additional -harmonic confining potential on top of the optical lattice to keep the -atoms in place which means that the chemical potential will vary in -space. However, with careful consideration of the full -($\mu/2J^\text{cl}$, $U/2J^\text{cl}$) phase diagrams in -Fig. \ref{fig:SFMI}(d,e) our analysis can still be applied to the -system \cite{batrouni2002}. - -The 1D phase transition is best understood in terms of two-point -correlations as a function of their separation \cite{giamarchi}. In -the Mott insulating phase, the two-point correlations -$\langle \bd_i b_j \rangle$ and -$\langle \delta \hat{n}_i \delta \hat{n}_j \rangle$ -($\delta \hat{n}_i =\hat{n}_i-\langle \hat{n}_i\rangle$) decay -exponentially with $|i-j|$. On the other hand the superfluid will -exhibit long-range order which in dimensions higher than one, -manifests itself with an infinite correlation length. However, in 1D -only pseudo long-range order happens and both the matter-field and -density fluctuation correlations decay algebraically \cite{giamarchi}. - -The method we propose gives us direct access to the structure factor, -which is a function of the two-point correlation $\langle \delta -\hat{n}_i \delta \hat{n}_j \rangle$, by measuring the light -intensity. For two travelling waves maximally coupled to the density -(atoms are at light intensity maxima so $\hat{F} = \hat{D}$), the -quantum addition is given by -\begin{equation} - R =\sum_{i, j} \exp[i (\mathbf{k}_1 - \mathbf{k}_0) - (\mathbf{r}_i - \mathbf{r}_j)] \langle \delta \hat{n}_i \delta - \hat{n}_j \rangle, -\end{equation} - -The angular dependence of $R$ for a Mott insulator and a superfluid is -shown in Fig. \ref{fig:SFMI}a, and there are two variables -distinguishing the states. Firstly, maximal $R$, -$R_\text{max} \propto \sum_i \langle \delta \hat{n}_i^2 \rangle$, -probes the fluctuations and compressibility $\kappa'$ -($\langle \delta \hat{n}^2_i \rangle \propto \kappa' \langle \hat{n}_i -\rangle$). The Mott insulator is incompressible and thus will have -very small on-site fluctuations and it will scatter little light -leading to a small $R_\text{max}$. The deeper the system is in the MI -phase (i.e. that larger the $U/2J^\text{cl}$ ratio is), the smaller -these values will be until ultimately it will scatter no light at all -in the $U \rightarrow \infty$ limit. In Fig. \ref{fig:SFMI}a this can -be seen in the value of the peak in $R$. The value $R_\text{max}$ in -the SF phase ($U/2J^\text{cl} = 0$) is larger than its value in the MI +We then consider probing these ground states using our optical scheme +and we calculate the ``quantum addition'', $R$. The angular dependence +of $R$ for a Mott insulator and a superfluid is shown in +Fig. \ref{fig:SFMI}a, and we note that there are two variables +distinguishing the states. Firstly, maximal $R$, $R_\text{max} \propto +\sum_i \langle \delta \hat{n}_i^2 \rangle$, probes the fluctuations +and compressibility $\kappa'$ ($\langle \delta \hat{n}^2_i \rangle +\propto \kappa' \langle \hat{n}_i \rangle$). The Mott insulator is +incompressible and thus will have very small on-site fluctuations and +it will scatter little light leading to a small $R_\text{max}$. The +deeper the system is in the insulating phase (i.e. that larger the +$U/2J^\text{cl}$ ratio is), the smaller these values will be until +ultimately it will scatter no light at all in the $U \rightarrow +\infty$ limit. In Fig. \ref{fig:SFMI}a this can be seen in the value +of the peak in $R$. The value $R_\text{max}$ in the superfluid phase +($U/2J^\text{cl} = 0$) is larger than its value in the Mott insulating phase ($U/2J^\text{cl} = 10$) by a factor of $\sim$25. Figs. \ref{fig:SFMI}(b,d) show how the value of -$R_\text{max}$ changes across the phase transition. We see that the -transition shows up very sharply as $\mu$ is varied. +$R_\text{max}$ changes across the phase transition. There are a few +things to note at this point. Firstly, if we follow the transition +along the line corresponding to commensurate filling (i.e.~any line +that is in between the two white lines in Fig. \ref{fig:SFMI}d) we see +that the transition is very smooth and it is hard to see a definite +critical point. This is due to the energy gap closing exponentially +slowly which makes precise identification of the critical point +extremely difficult. The best option at this point would be to fit +Tomonage-Luttinger theory to the results in order to find this +critical point. However, we note that there is a drastic change in +signal as the chemical potential (and thus the density) is +varied. This is highlighted in Fig. \ref{fig:SFMI}b which shows how +the Mott insulator can be easily identified by a dip in the quantity +$R_\text{max}$. Secondly, being a Fourier transform, the width $W_R$ of the dip in $R$ is a direct measure of the correlation length $l$, $W_R \propto 1/l$. The Mott insulator being an insulating phase is characterised by exponentially decaying correlations and as such it will have a very -large $W_R$. However, the superfluid in 1D exhibits pseudo long-range -order which manifests itself in algebraically decaying two-point -correlations \cite{giamarchi} which significantly reduces the dip in -the $R$. This can be seen in Fig. \ref{fig:SFMI}a and we can also see -that this identifies the phase transition very sharply as $\mu$ is -varied in Figs. \ref{fig:SFMI}(c,e). One 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 $1/M$. - -It is also possible to analyse the phase transition quantitatively -using our method. Unlike in higher dimensions where an order parameter -can be easily defined within the MF approximation there is no such -quantity in 1D. However, a valid description of the relevant 1D low -energy physics is provided by Luttinger liquid theory -\cite{giamarchi}. In this model correlations in the supefluid phase as -well as the superfluid density itself are characterised by the -Tomonaga-Luttinger parameter, $K_b$. This parameter also identifies -the phase transition in the thermodynamic limit at $K_b = 1/2$. This -quantity can be extracted from various correlation functions and in -our case it can be extracted directly from $R$ \cite{ejima2011}. By -extracting this parameter from $R$ for various lattice lengths from -numerical DMRG calculations it was even possible to give a theoretical -estimate of the critical point for commensurate filling, $N = M$, in -the thermodynamic limit to occur at $U/2J^\text{cl} \approx 1.64$ -\cite{ejima2011}. Our proposal provides a method to directly measure -$R$ in a lab which can then be used to experimentally determine the -location of the critical point in 1D. +large $W_R$. On the other hand, the superfluid in 1D exhibits pseudo +long-range order which manifests itself in algebraically decaying +two-point correlations \cite{giamarchi} which significantly reduces +the dip in the $R$. This can be seen in +Fig. \ref{fig:SFMI}a. Furthermore, just like for $R_\text{max}$ we see +that the transition is much sharper as $\mu$ is varied. This is shown +in Figs. \ref{fig:SFMI}(c,e). Notably, the difference in angle between +a superfluid and an insulating state is fairly significant $\sim +20^\circ$ 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}. + diff --git a/thesis.tex b/thesis.tex index 24525c8..0e6eb48 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,chapter]{Classes/PhDThesisPSnPDF} +\documentclass[a4paper,12pt,times,numbered,print,index]{Classes/PhDThesisPSnPDF} % ****************************************************************************** % ******************************* Class Options ********************************