diff --git a/Chapter5/Figs/Zeno.pdf b/Chapter5/Figs/Zeno.pdf new file mode 100644 index 0000000..011cf53 Binary files /dev/null and b/Chapter5/Figs/Zeno.pdf differ diff --git a/Chapter5/chapter5.tex b/Chapter5/chapter5.tex index c24765e..413bad6 100644 --- a/Chapter5/chapter5.tex +++ b/Chapter5/chapter5.tex @@ -791,6 +791,287 @@ the $\gamma/U \gg 1$ regime is reached. \subsection{Emergent Long-Range Correlated Tunnelling} +When $\gamma \rightarrow \infty$ the measurement becomes +projective. This means that as soon as the probing begins, the system +collapses into one of the observable's eigenstates. Since this +measurement is continuous and doesn't stop after the projection the +system will be frozen in this state. This effect is called the quantum +Zeno effect from Zeno's classical paradox in which a ``watched arrow +never moves'' that stated since an arrow in flight is not seen to move +during any single instant, it cannot possibly be moving at +all. Classically the paradox was resolved with a better understanding +of infinity and infintesimal changes, but in the quantum world a +watched quantum arrow will in fact never move. The system is being +continuously projected into its initial state before it has any chance +to evolve away. If degenerate eigenspaces exist then we can observe +quantum Zeno dynamics where unitary evolution is uninhibited within +such a degenerate subspace, called the Zeno subspace. + +These effects can be easily seen in our model when +$\gamma \rightarrow \infty$. The system will be projected into one or +more degenerate eignstates of $\cd \c$, $| \psi_i \rangle$, for which +we define the projector +$P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$ +denotes a single degenerate subspace. The Zeno subspace is determined +randomly as per the Copenhagen postulates and thus it depends on the +initial state. If the projection is into the subspace +$\varphi^\prime$, the subsequent evolution is described by the +projected Hamiltonian +$P_{\varphi^\prime} \hat{H}_0 P_{\varphi^\prime}$. We have used the +original Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or +the quantum jumps as their combined effect is now described by the +projectors. Physically, in our model of ultracold bosons trapped in a +lattice this means that tunnelling between different spatial modes is +completely supressed since this process couples eigenstates belonging +to different Zeno subspaces. If a small connected part of the lattice +was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then +tunnelling would only be prohibited between the illuminated and +unilluminated areas, but dynamics proceeds normally within each zone +separately. Therefore, the goemetric patterns we have in which the +modes are sptailly delocalised in such a way that neighbouring sites +never belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$, +would lead to a complete suppression of tunnelling across the whole +lattice as there is no way for an atom to tunnel within this Zeno +subspace. + +This is an interesting example of the quantum Zeno effect and dynamics +and it can be used to prohibit parts of the dynamics of the +Bose-Hubbard Hamiltonian in order to engineer desired Hamiltonians for +quantum simulations or other applications. However, the infinite +projective limit is uninteresting in the context of a global +measurement scheme. The same effects and Hamiltonians can be achieved +using multiple independent measurements which address a few sites +each. The only advantage of the global scheme is that it might be +simpler to achieve as it requires a less complicated optical setup. In +order to take advantage of the nonlocal nature of the measurement it +turns out that we need to consider a finite limit for $\gamma \gg +J$. By considering a non-infinite $\gamma$ we observe additional +dynamics while the usual atomic tunnelling is still heavily +Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}. + +\begin{figure}[hbtp!] + \includegraphics[width=\textwidth]{Zeno.pdf} + \caption[Emergent Long-Range Correlated Tunnelling]{ Long-range + correlated tunneling and entanglement, dynamically induced by + strong global measurement in a single quantum + trajectory. (a),(b),(c) show different measurement geometries, + implying different constraints. Panels (1): schematic + representation of the long-range tunneling processes. Panels (2): + evolution of on-site densities. Panels (3): entanglement entropy + growth between illuminated and non-illuminated regions. Panels + (4): correlations between different modes (orange) and within the + same mode (green); $N_I$ ($N_{NI}$) is the atom number in the + illuminated (non-illuminated) mode. (a) (a.1) Atom number in the + central region is frozen: system is divided into three + regions. (a.2) Standard dynamics happens within each region, but + not between them. (a.3) Entanglement build up. (a.4) Negative + correlations between non-illuminated regions (green) and zero + correlations between the $N_I$ and $N_{NI}$ modes + (orange). Initial state: $|1,1,1,1,1,1,1 \rangle$, $\gamma/J=100$, + $J_{jj}=[0,0,1,1,1,0,0]$. (b) (b.1) Even sites are illuminated, + freezing $N_\text{even}$ and $N_\text{odd}$. Long-range tunneling + is represented by any pair of one blue and one red arrow. (b.2) + Correlated tunneling occurs between non-neighbouring sites without + changing mode populations. (b.3) Entanglement build up. (b.4) + Negative correlations between edge sites (green) and zero + correlations between the modes defined by $N_\text{even}$ and + $N_\text{odd}$ (orange). Initial state: $|0,1,2,1,0 \rangle$, + $\gamma/J=100$, $J_{jj}=[0,1,0,1,0]$. (c) (c.1,2) Atom number + difference between two central sites is frozen. (c.3) Entanglement + build up. (c.4) In contrast to previous examples, sites in the + same zones are positively correlated (green), while atoms in + different zones are negatively correlated (orange). Initial state: + $|0,2,2,0 \rangle$, $\gamma/J=100$, $J_{jj}=[0,-1,1,0]$. 1D + lattice, $U/J=0$.} + \label{fig:zeno} +\end{figure} + +There are two crucial features of the resulting dynamics that are of +note. First, just like in the infinite quantum Zeno limit the +evolution between nearest neighbours within the same mode is +unperturbed whilst tunnelling between different modes is heavily +suppressed by the measurement. Therefore, we see the usual quantum +Zeno dynamics within a single Zeno subspace and just like before, it +is also possible to use the global probing scheme to engineer these +eigenspaces and select which tunnelling processes should be +uninhibited and which should be suppressed. However, there is a second +effect that was not present before. In Fig. \ref{fig:zeno} we can +observe tunnelling that violates the boundaries established by the +spatial modes. When $\gamma$ is finite, second-order processes can now +occur, i.e.~two correlated tunnelling events, via an intermediate +(virtual) state outside of the Zeno subspace as long as the Zeno +subspace of the final state remains the same. Crucially, these +tunnelling events are only correlated in time, but not in space. This +means that the two events do not have to occur for the same atom or +even at the same site in the lattice. As long as the Zeno subspace is +preserved, these processes can occur anywhere in the system, that is a +pair of atoms separated by many sites is able to tunnel in a +correlated manner. This is only possible due to the possibility of +creating extensive and spatially nonlocal modes as described in +section \ref{sec:modes} which in turn is enabled by the global nature +of the measurement. This would not be possible to achieve with local +measurements as this would lead to Zeno subspaces described entirely +by local variables which cannot be preserved by such delocalised +tunnelling events. + +In the subsequent sections we will rigorously derive the following +Hamiltonian for the non-interacting dynamics within a single Zeno +subspace, $\varphi = 0$, for a lattice where the measurement defines +$Z = 2$ distinct modes, e.g. $\hat{D} = \hat{N}_K$ or +$\hat{D} = \hat{N}_\mathrm{odd}$ +\begin{equation} + \label{eq:hz} + \hat{H}_\varphi = P_0 \left[ -J \sum_{\langle i, j \rangle} + b^\dagger_i b_j - i \frac{J^2} {A \gamma} \sum_{\varphi} + \sum_{\substack{\langle i \in \varphi, j \in \varphi^\prime + \rangle \\ \langle k \in \varphi^\prime, l \in \varphi + \rangle}} b^\dagger_i b_j b^\dagger_k b_l \right] P_0, +\end{equation} +where +$A = (J_{\varphi,\varphi} - J_{\varphi^\prime,\varphi^\prime})^2$ is a +constant that depends on the measurement scheme, $\varphi$ denotes a +set of site belonging to a single mode and $\varphi^\prime$ is the +set's complement (e.g.~odd and even or illuminated and non-illuminated +sites). We see that this Hamiltonian consists of two parts. The first +term corresponds to the standard quantum Zeno first-order dynamics +that occurs within a Zeno subspace, i.e.~tunnelling between +neighbouring sites that belong to the same mode. Otherwise, +$P_0 \bd_i b_j P_0 = 0$. If $\gamma \rightarrow \infty$ we would +recover the quantum Zeno Hamiltonian where this would be the only +remaining term. It is the second term that shows the second-order +corelated tunnelling terms. This is evident from the inner sum which +requires that pairs of sites ($i$, $j$) and ($k$, $l$) between which +atoms tunnel must be nearest neighbours, but these pairs can be as far +apart from each other as possible within the mode structure. This is +in particular explicitly shown in Figs. \ref{fig:zeno}(a,b). The +imaginary coefficient means that the tinelling behaves like an +exponential decay (overdamped oscillations). This also implies that +the norm will decay, but this does not mean that there are physical +losses in the system. Instead, the norm itself represents the +probability of the system remaining in the $\varphi = 0$ Zeno +subspace. Since $\gamma$ is not infinite there is now a finite +probability that the stochastic nature of the measurement will lead to +a discontinous change in the system where the Zeno subspace rapidly +changes. However, later in this chapter we will see that steady states +of this Hamiltonian exist which will no longer change Zeno subspaces. + +Crucially, what sets this effect apart from usual many-body dynamics +with short-range interactions is that first order processes are +selectively suppressed by the global conservation of the measured +observable and not by the prohibitive energy costs of doubly-occupied +sites, as is the case in the $t$-$J$ model \cite{auerbach}. This has +profound consequences as this is the physical origin of the long-range +correlated tunneling events represented in \eqref{eq:hz} by the fact +that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. This is +because the projection $\hat{P}_0$ is not sensitive to individual site +occupancies, but instead enforces a fixed value of the observable, +i.e.~a single Zeno subspace. + +In Fig.~\ref{fig:zeno}a we consider illuminating only the central +region of the optical lattice and detecting light in the diffraction +maximum, thus we freeze the atom number in the $K$ illuminated sites +$\hat{N}_\text{K}$~\cite{mekhov2009prl,mekhov2009pra}. The measurement +scheme defines two different spatial modes: the non-illuminated zones +$1$ and $3$ and the illuminated one $2$. Figure~\ref{fig:zeno}(a.2) +illustrates the evolution of the mean density at each lattice site: +typical dynamics occurs within each region but the standard tunnelling +between different modes is suppressed. Importantly, second-order +processes that do not change $N_\text{K}$ are still possible since an +atom from $1$ can tunnel to $2$, if simultaneously one atom tunnels +from $2$ to $3$. Therfore, effective long-range tunneling between two +spatially disconnected zones $1$ and $3$ happens due to the two-step +processes $1 \rightarrow 2 \rightarrow 3$ or +$3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for +the negative (anti-)correlations +$\langle \delta N_1 \delta N_3 \rangle = \langle N_1 N_3 \rangle - +\langle N_1 \rangle \langle N_3 \rangle$ showing that an atom +disappearing from zone $1$ appears in zone $3$, while there are no +number correlations between illuminated and non-illuminated regions, +$\langle( \delta N_1 + \delta N_3 ) \delta N_2 \rangle = 0$ as shown +in Fig.~\ref{fig:zeno}(a.4). In contrast to fully-projective +measurement, the existence of an intermediate (virtual) step in the +correlated tunnelling process builds long-range entanglement between +illuminated and non-illuminated regions as shown in +Fig.~\ref{fig:zeno}(a.3). + +To make correlated tunneling visible even in the mean atom number, we +suppress the standard Bose-Hubbard dynamics by illuminating only the +even sites of the lattice in Fig.~\ref{fig:zeno}(b). Even though this +measurement scheme freezes both $N_\text{even}$ and $N_\text{odd}$, +atoms can slowly tunnel between the odd sites of the lattice, despite +them being spatially disconnected. This atom exchange spreads +correlations between non-neighbouring lattice sites on a time scale +$\sim \gamma/J^2$. The schematic explanation of long-range correlated +tunneling is presented in Fig.~\ref{fig:zeno}(b.1): the atoms can +tunnel only in pairs to assure the globally conserved values of +$N_\text{even}$ and $N_\text{odd}$, such that one correlated tunneling +event is represented by a pair of one red and one blue +arrow. Importantly, this scheme is fully applicable for a lattice with +large atom and site numbers, well beyond the numerical example in +Fig.~\ref{fig:zeno}(b.1), because as we can see in \eqref{eq:hz} +it is the geometry of quantum measurement that assures this mode +structure (in this example, two modes at odd and even sites) and thus +the underlying pairwise global tunnelling. + +This global pair tunneling may play a role of a building block for +more complicated many-body effects. For example, a pair tunneling +between the neighbouring sites has been recently shown to play +important role in the formation of new quantum phases, e.g., pair +superfluid \cite{sowinski2012} and lead to formulation of extended +Bose-Hubbard models \cite{omjyoti2015}. The search for novel +mechanisms providing long-range interactions is crucial in many-body +physics. One of the standard candidates is the dipole-dipole +interaction in, e.g., dipolar molecules, where the mentioned pair +tunneling between even neighboring sites is already considered to be +long-range \cite{sowinski2012,omjyoti2015}. In this context, our work +suggests a fundamentally different mechanism originating from quantum +optics: the backaction of global and spatially structured measurement, +which as we prove can successfully compete with other short-range +processes in many-body systems. This opens promising opportunities for +future research. + +The scheme in Fig.~\ref{fig:zeno}(b.1) can help to design a nonlocal +reservoir for the tunneling (or ``decay'') of atoms from one region to +another. For example, if the atoms are placed only at odd sites, +according to \eqref{eq:hz} their tunnelling is suppressed since the +multi-tunneling event must be successive, i.e.~an atom tunnelling into +a different mode, $\varphi^\prime$, must then also tunnel back into +its original mode, $\varphi$. If, however, one adds some atoms to even +sites (even if they are far from the initial atoms), the correlated +tunneling events become allowed and their rate can be tuned by the +number of added atoms. This resembles the repulsively bound pairs +created by local interactions \cite{winkler2006, folling2007}. In +contrast, here the atom pairs are nonlocally correlated due to the +global measurement. Additionally, these long-range correlations are a +consequence of the dynamics being constrained to a Zeno subspace: the +virtual processes allowed by the measurement entangle the spatial +modes nonlocally. Since the measurement only reveals the total number +of atoms in the illuminated sites, but not their exact distribution, +these multi-tunelling events cause the build-up of long range +entanglement. This is in striking contrast to the entanglement caused +by local processes which can be very confined, especially in 1D where +it is typically short range. This makes numerical calculations of our +system for large atom numbers really difficult, since well-known +methods such as DMRG and MPS \cite{schollwock2005} (which are successful +for short-range interactions) rely on the limited extent of +entanglement. + +The negative number correlations are typical for systems with +constraints (superselection rules) such as fixed atom number. The +effective dynamics due to our global, but spatially structured, +measurement introduces more general constraints to the evolution of +the system. For example, in Fig.~\ref{fig:zeno}(c) we show the +generation of positive number correlations shown in +Fig.~\ref{fig:zeno}(c.4) by freezing the atom number difference +between the sites ($N_\text{odd}-N_\text{even}$). Thus, atoms can only +enter or leave this region in pairs, which again is possible due to +correlated tunneling as seen in Figs.~\ref{fig:zeno}(c.1,c.2) and +manifests positive correlations. As in the previous example, two edge +modes in Fig.~\ref{fig:zeno}(c) can be considered as a nonlocal +reservoir for two central sites, where a constraint is applied. Note +that, using more modes, the design of higher-order multi-tunneling +events is possible. + \subsection{Non-Hermitian Dynamics in the Quantum Zeno Limit} % Contrast with t-J model here how U localises events, but measurement diff --git a/References/references.bib b/References/references.bib index 06be2f1..792c0b3 100644 --- a/References/references.bib +++ b/References/references.bib @@ -52,6 +52,12 @@ year = {2010} year={2004}, publisher={Springer Science \& Business Media} } +@book{auerbach, + title={Interacting electrons and quantum magnetism}, + author={Auerbach, Assa}, + year={2012}, + publisher={Springer Science \& Business Media} +} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Igor's original papers @@ -911,4 +917,64 @@ doi = {10.1103/PhysRevA.87.043613}, pages={012116}, year={2012}, publisher={APS} -} \ No newline at end of file +} +@article{sowinski2012, + title = {Dipolar Molecules in Optical Lattices}, + author = {Sowi\ifmmode \acute{n}\else \'{n}\fi{}ski, Tomasz and + Dutta, Omjyoti and Hauke, Philipp and Tagliacozzo, + Luca and Lewenstein, Maciej}, + journal = {Phys. Rev. Lett.}, + volume = {108}, + issue = {11}, + pages = {115301}, + numpages = {5}, + year = {2012}, + month = {Mar}, + publisher = {American Physical Society}, + doi = {10.1103/PhysRevLett.108.115301}, + url = {http://link.aps.org/doi/10.1103/PhysRevLett.108.115301} +} +@article{omjyoti2015, + author={Omjyoti Dutta and Mariusz Gajda and Philipp Hauke and Maciej + Lewenstein and Dirk-Soren Luhmann and Boris A + Malomed and Tomasz Sowinski and Jakub Zakrzewski}, + title={Non-standard Hubbard models in optical lattices: a review}, + journal={Reports on Progress in Physics}, + volume={78}, + number={6}, + pages={066001}, + url={http://stacks.iop.org/0034-4885/78/i=6/a=066001}, + year={2015} +} +@article{winkler2006, + title={{Repulsively bound atom pairs in an optical lattice}}, + author={Winkler, K. and Thalhammer, G. and Lang, F. and Grimm, + R. and Hecker Denschlag, J. and Daley, A. and + Kantian, A. and B\"{u}chler, H. P. and Zoller, P.}, + journal={Nature}, + volume={441}, + pages = {853-856}, + year={2006}, + publisher={Nature Publishing Group} +} +@article{folling2007, + title={{Direct observation of second-order atom tunnelling}}, + author={F\"{o}lling, S. and Trotzky, S. and Cheinet, P. and Feld, + M. and Saers, R. and Widera, A. and M\"{u}ller, + T. and Bloch, I.}, + journal={Nature}, + volume={448}, + pages={1029}, + year={2007}, + publisher={Nature Publishing Group} +} +@article{schollwock2005, + title = {The density-matrix renormalization group}, + author = {Schollw\"ock, U.}, + journal = {Rev. Mod. Phys.}, + volume = {77}, + issue = {1}, + pages = {259--315}, + year = {2005}, + publisher = {American Physical Society}, +} diff --git a/thesis-info.tex b/thesis-info.tex index 3095aa9..028866c 100644 --- a/thesis-info.tex +++ b/thesis-info.tex @@ -1,6 +1,7 @@ % ************************ Thesis Information & Meta-data ********************** %% The title of the thesis -\title{Interaction of Quantized Light with Ultracold Bosons} +\title{Competition between Weak Quantum Measurement and Many-Body +Dynamics in Ultracold Bosonic Gases} %\texorpdfstring is used for PDF metadata. Usage: %\texorpdfstring{LaTeX_Version}{PDF Version (non-latex)} eg., %\texorpdfstring{$sigma$}{sigma}