%******************************************************************************* %*********************************** Fifth Chapter ***************************** %******************************************************************************* \chapter[Density Measurement Induced Dynamics] {Density Measurement Induced Dynamics\footnote{The results of this chapter were first published in Refs. \cite{mazzucchi2016, kozlowski2016zeno, mazzucchi2016njp}}} % Title of the Fifth Chapter \ifpdf \graphicspath{{Chapter5/Figs/Raster/}{Chapter5/Figs/PDF/}{Chapter5/Figs/}} \else \graphicspath{{Chapter5/Figs/Vector/}{Chapter5/Figs/}} \fi \section{Introduction} In the previous chapter we have introduced a theoretical framework which will allow us to study measurement backaction using discontinuous quantum jumps and non-Hermitian evolution due to null outcomes using quantum trajectories. We have also wrapped our quantum gas model in this formalism by considering ultracold bosons in an optical lattice coupled to a cavity which collects and enhances light scattered in one particular direction. One of the most important conclusions of the previous chapter was that the introduction of measurement introduces a new energy and time scale into the picture which competes with the intrinsic dynamics of the bosons. In this chapter, we investigate the effect of quantum measurement backaction on the many-body state and dynamics of atoms. In particular, we will focus on the competition between the backaction and the two standard short-range processes, tunnelling and on-site interactions, in optical lattices. We show that the possibility to spatially structure the measurement at a microscopic scale comparable to the lattice period without the need for single site resolution enables us to engineer efficient competition between the three processes in order to generate new nontrivial dynamics. However, unlike tunnelling and on-site interactions our measurement scheme is global in nature which makes it capable of creating long-range correlations which enable nonlocal dynamical processes. Furthermore, global light scattering from multiple lattice sites creates nontrivial spatially nonlocal coupling to the environment, as seen in section \ref{sec:modes}, which is impossible to obtain with local interactions \cite{diehl2008, syassen2008, daley2014}. These spatial modes of matter fields can be considered as designed systems and reservoirs opening the possibility of controlling dissipation in ultracold atomic systems without resorting to atom losses and collisions which are difficult to manipulate. Thus the continuous measurement of the light field introduces a controllable decoherence channel into the many-body dynamics. Such a quantum optical approach can broaden the field even further allowing quantum simulation models unobtainable using classical light and the design of novel systems beyond condensed matter analogues. In the weak measurement limit, where the quantum jumps do not occur frequently compared to the tunnelling rate, this can lead to global macroscopic oscillations of bosons between odd and even sites. These oscillations occur coherently across the whole lattice enabled by the fact that measurement is capable of generating nonlocal spatial modes. When on-site interactions are included we obtain a system with three competing energy scales of which two correspond to local processes and one is global. This complicates the picture immensely. We show how under certain circumstances interactions prevent measurement from generating globally coherent dynamics, but on the other hand when the measurement is strong both processes collaborate in squeezing the atomic distribution. On the other end of the spectrum, when measurement is strong we enter the regime of quantum Zeno dynamics. Frequent measurements can slow the evolution of a quantum system leading to the quantum Zeno effect where a quantum state is frozen in its initial configuration \cite{misra1977, facchi2008}. One can also devise measurements with multi-dimensional projections which lead to quantum Zeno dynamics where unitary evolution is uninhibited within this degenerate subspace, usually called the Zeno subspace \cite{facchi2008, raimond2010, raimond2012, signoles2014}. Our flexible setup where global light scattering can be engineered allows us to suppress or enhance specific dynamical processes thus realising spatially nonlocal quantum Zeno dynamics. This unconventional variation occurs when measurement is near, but not in, its projective limit. The system is still confined to Zeno subspaces, but intermediate transitions are allowed via virtual Raman-like processes. We show that this result can, in general (i.e.~beyond the ultracold gas model), be approximated by a non-Hermitian Hamiltonian thus extending the notion of quantum Zeno dynamics into the realm of non-Hermitian quantum mechanics joining the two paradigms. \section{Quantum Measurement Induced Dynamics} \subsection{Large-Scale Dynamics due to Weak Measurement} We start by considering the weak measurement limit when photon scattering does not occur frequently compared to the tunnelling rate of the atoms, i.e.~$\gamma \ll J$. When the system is probed in this way, the measurement is unable to project the quantum state of the bosons to an eigenspace as postulated by the Copenhagen interpretation of quantum mechanics. The backaction of the photodetections is simply not strong or frequent enough to confine the atoms. However, instead of confining the evolution of the quantum state, it has been shown in Refs. \cite{mazzucchi2016, mazzucchi2016njp} that the measurement leads to coherent global oscillations between the modes generated by the spatial profile of the light field which we have seen in section \ref{sec:modes}. Fig. \ref{fig:oscillations} illustrates the atom number distributions in the odd sites for $Z = 2$ and one of the three modes for $Z = 3$. These oscillations correspond to atoms flowing from one mode to another. We only observe a small number of well defined components which means that this flow happens in phase, all the atoms are tunnelling between the modes together in unison. Furthermore, this exchange of population is macroscopic in scale. The trajectories reach a state where the maximum displacement point corresponds to all the atoms being entirely within a single mode. Finally, we note that these oscillating distributions are squeezed by the measurement and the individual components have a width smaller than the initial state. By contrast, in the absence of the external influence of measurement these distributions would spread out significantly and the centre of the broad distribution would oscillate with an amplitude comparable to the initial imbalance, i.e.~small oscillations for a small initial imbalance. \begin{figure}[htbp!] \centering \includegraphics[width=\textwidth]{Oscillations} \caption[Macroscopic Oscillations due to Weak Measurement]{Large oscillations between the measurement-induced spatial modes resulting from the competition between tunnelling and weak measurement induced backaction. The plots show the atom number distributions $p(N_l)$ in one of the modes in individual quantum trajectories. These distributions show various numbers of well-squeezed components reflecting the creation of macroscopic superposition states depending on the measurement configuration. $U/J = 0$, $\gamma/J = 0.01$, $M=N$, initial states: bosonic superfluid. (a) Measurement of the atom number at odd sites $\hat{N}_\mathrm{odd}$ creates one strongly oscillating component in $p(N_\mathrm{odd})$ ($N = 100$ bosons, $J_{j,j} = 1$ if $j$ is odd and 0 otherwise). (b) Measurement of $(\hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even})^2$ introduces $Z = 2$ modes and preserves the superposition of positive and negative atom number differences in $p(N_\mathrm{odd})$ ($N = 100$ bosons, $J_{j,j} = (-1)^{j+1}$). (c) Measurement for $Z = 3$ modes preserves three components in $p(N_1)$ ($N = 108$ bosons, $J_{j,j} = e^{i 2 \pi j / 3}$).} \label{fig:oscillations} \end{figure} In Figs. \ref{fig:oscillations}(b,c) we also see that the system is composed of multiple components. This depends on the quantity that is being measured and it is a consequence of the fact that the detected light intensity $\ad_1 \a_1$ is not sensitive to the light phase. The measurement will not distinguish between permutations of mode occupations that scatter light with the same intensity, but with a different phase. For example, when measuring $\hat{D} = \hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even}$, the light intensity will be proportional to $\hat{D}^\dagger \hat{D} = (\hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even})^2$ and thus it cannot distinguish between a positive and negative imbalance leading to the two components seen in Fig. \ref{fig:oscillations}. More generally, the number of components of the atomic state, i.e.~the degeneracy of $\ad_1 \a_1$, can be computed from the eigenvalues of Eq. \eqref{eq:Zmodes}, \begin{equation} \hat{D} = \sum_l^Z \exp\left[-i 2 \pi l R / Z \right] \hat{N}_l. \end{equation} Each eigenvalue can be represented as the sum of the individual terms in the above sum which are vectors on the complex plane with phases that are integer multiples of $2 \pi / Z$: $N_1 e^{-i 2 \pi R / Z}$, $N_2 e^{-i 4 \pi R / Z}$, ..., $N_Z$. Since the set of possible sums of these vectors is invariant under rotations by $2 \pi l R / Z$, $l \in \mathbb{Z}$, and reflection in the real axis, the state of the system is 2-fold degenerate for $Z = 2$ (reflections leave $Z = 2$ unchanged) and $2Z$-fold degenerate for $Z > 2$. Fig. \ref{fig:oscillations} shows the three mode case, where there are in fact $6$ components ($2Z = 6$), but in this case they all occur in pairs resulting in only three visible components. We will now limit ourselves to a specific illumination pattern with $\hat{D} = \hat{N}_\mathrm{odd}$ as this leads to the simplest multimode dynamics with $Z = 2$ and only a single component as seen in Fig. \ref{fig:oscillations}(a), i.e.~no multiple peaks like in Figs. \ref{fig:oscillations}(b,c). This pattern can be obtained by crossing two beams such that their projections on the lattice are identical and the even sites are positioned at their nodes. However, even though this is the simplest possible case and we are only dealing with non-interacting atoms solving the full dynamics of the Bose-Hubbard Hamiltonian combined with measurement is nontrivial. The backaction introduces a highly nonlinear global term. However, it has been shown in Ref. \cite{mazzucchi2016njp} that the non-interacting dynamics with quantum measurement backaction for $Z$-modes reduce to an effective Bose-Hubbard Hamiltonian with $Z$-sites provided the initial state is a superfluid. In this simplified model the $N_j$ atoms in the $j$-th site correspond to a superfluid of $N_j$ atoms within a single spatial mode as defined in section \ref{sec:modes}. Therefore, we now proceed to study the dynamics for $\hat{D} = \hat{N}_\mathrm{odd}$ using this reduced effective double-well model. The atomic state can be written as \begin{equation} \label{eq:discretepsi} | \psi \rangle = \sum_l^N q_l |l, N - l \rangle, \end{equation} where the ket $| l, N - l \rangle$, represents a superfluid with $l$ atoms in the odd sites and $N-l$ atoms in the even sites. The non-Hermitian Hamiltonian describing the time evolution in between the jumps is given by \begin{equation} \label{eq:doublewell} \hat{H} = -J \left( \bd_o b_e + b_o \bd_e \right) - i \gamma \n_o^2 \end{equation} and the quantum jump operator which is applied at each photodetection is $\c = \sqrt{2 \kappa} C \n_o$. $b_o$ ($\bd_o$) is the annihilation (creation) operator in the left site of the effective double-well corresponding to the superfluid at odd sites of the physical lattice. $b_e$ ($\bd_e$) is defined similarly, but for the right site and the superfluid at even sites of the physical lattice. $\n_o = \bd_o b_o$ is the atom number operator in the left site. Even though Eq. \eqref{eq:doublewell} is relatively simple as it is only a non-interacting two-site model, the non-Hermitian term complicates the situation making the system difficult to solve. However, a semiclassical approach to boson dynamics in a double-well in the limit of many atoms $N \gg 1$ has been developed in Ref. \cite{juliadiaz2012}. It was originally formulated to treat squeezing in a weakly interacting bosonic gas, but it can easily be applied to our system as well. In the limit of large atom number, the wave function in Eq. \eqref{eq:discretepsi} can be described using continuous variables by defining $\psi (x = l / N) = \sqrt{N} q_l$. Note that this requires the coefficients $q_l$ to vary smoothly which is the case for a superfluid state. We now rescale the Hamiltonian in Eq. \eqref{eq:doublewell} to be dimensionless by dividing by $NJ$ and define the relative population imbalance between the two wells $z = 2x - 1$. Finally, by taking the expectation value of the Hamiltonian and looking for the stationary points of $\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi \rangle$ we obtain the semiclassical Schr\"{o}dinger equation \begin{equation} \label{eq:semicl} i h \partial_t \psi(z, t) = \mathcal{H} \psi(z, t), \end{equation} \begin{equation} \label{eq:semiH} \mathcal{H} \approx -2 h^2 \partial^2_z \psi(z, t) + \left[ \frac{\omega^2 z^2} {8} - \frac{i \Gamma} {4} \left( z + 1 \right)^2 \right] \psi(z, t), \end{equation} where $\Gamma = N \kappa |C|^2 / J$, $h = 1/N$, $\omega = 2 \sqrt{1 + \Lambda - h}$, and $\Lambda = NU / (2J)$. The full derivation is not straightforward, but the introduction of the non-Hermitian term requires only a minor modification to the original formalism presented in detail in Ref. \cite{juliadiaz2012} so we have omitted it here. We will also be considering $U = 0$ as the effective model is only valid in this limit, thus $\Lambda = 0$. However, this model is valid for an actual physical double-well setup in which case interacting bosons can also be considered. The equation is defined on the interval $z \in [-1, 1]$, but $z \ll 1$ has been assumed in order to simplify the kinetic term and approximate the potential as parabolic. This does mean that this approximation is not valid for the maximum amplitude oscillations seen in Fig. \ref{fig:oscillations}(a), but since they already appear early on in the trajectory we are able to obtain a valid analytic description of the oscillations and their growth. A superfluid state in our continuous variable approximation corresponds to a Gaussian wave function $\psi$. Furthermore, since the potential is parabolic, even with the inclusion of the non-Hermitian term, it will remain Gaussian during subsequent time evolution. Therefore, we will use a very general Gaussian wave function of the form \begin{equation} \label{eq:ansatz} \psi(z, t) = \frac{1}{\pi b^2}\exp\left[ i \epsilon - \frac{(z - z_0)^2} {2 b^2} + \frac{i \phi (z - z_\phi)^2} {2 b^2} \right] \end{equation} as our ansatz to Eq. \eqref{eq:semicl}. The parameters $b$, $\phi$, $z_0$, and $z_\phi$ are real-valued functions of time whereas $\epsilon$ is a complex-valued function of time. Physically, the value $b^2$ denotes the width, $z_0$ the position of the centre, $\phi$ and $z_\phi$ contain the local phase information, and $\epsilon$ only affects the global phase and norm of the Gaussian wave packet. The non-Hermitian Hamiltonian and an ansatz are not enough to describe the full dynamics due to measurement. We also need to know the effect of each quantum jump. Within the continuous variable approximation, our quantum jump become $\c \propto 1 + z$. We neglect the constant prefactors, because the wave function is normalised after a quantum jump. Expanding around the peak of the Gaussian ansatz we get \begin{equation} 1 + z \approx \exp \left[ \ln (1 + z_0) + \frac{z - z_0}{1 + z_0} - \frac{(z - z_0)^2}{2 (1 + z_0)^2} \right]. \end{equation} Multiplying the wave function in Eq. \eqref{eq:ansatz} with the jump operator above yields a Gaussian wave function as well, but the parameters change discontinuously according to \begingroup \allowdisplaybreaks \begin{align} \label{eq:jumpb2} b^2 & \rightarrow \frac{ b^2 (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\ \phi & \rightarrow \frac{ \phi (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\ \label{eq:jumpz0} z_0 & \rightarrow z_0 + \frac{ b^2 (1 + z_0) } { (1 + z_0)^2 + b^2}, \\ z_\phi & \rightarrow z_\phi, \\ \epsilon & \rightarrow \epsilon. \end{align} \endgroup The fact that the wave function remains Gaussian after a photodetection is a huge advantage, because it means that the combined time evolution of the system can be described with a single Gaussian ansatz in Eq. \eqref{eq:ansatz} subject to non-Hermitian time evolution according to Eq. \eqref{eq:semicl} with discontinuous changes to the parameter values at each quantum jump. Having identified an appropriate ansatz and the effect of quantum jumps we proceed with solving the dynamics of wave function in between the photodetections. The initial values of the parameters for a superfluid state of $N$ atoms across the whole lattice are $b^2 = 2h$, $\phi =0$, $a_0 = 0$, $a_\phi = 0$, $\epsilon = 0$. However, we use the most general initial conditions at time $t = t_0$ which we denote by $b(t_0) = b_0$, $\phi(t_0) = \phi_0$, $z_0(t_0) = a_0$, $z_\phi(t_0) = a_\phi$, and $\epsilon(t_0) = \epsilon_0$. The reason for keeping them as general as possible is that after every quantum jump the system changes discontinuously. The subsequent time evolution is obtained by solving the Schr\"{o}dinger equation with the post-jump parameter values as the new initial conditions. By plugging the ansatz in Eq. \eqref{eq:ansatz} into the Schr\"{o}dinger equation in Eq. \eqref{eq:semicl} we obtain three differential equations \begin{equation} \label{eq:p} -2 h^2 p^2 + \left( \frac{ \omega^2 } { 8 } - \frac{ i \Gamma } { 4 } \right) + \frac{ i h } { 2 } \frac{ \mathrm{d} p } { \mathrm{d} t } = 0, \end{equation} \begin{equation} \label{eq:pq} 4 h^2 p q - \frac{ i \Gamma } { 2 } - i h \frac{ \mathrm{d} q } { \mathrm{d} t } = 0 \end{equation} \begin{equation} \label{eq:pqr} -2 h^2 (q^2 - p) - \frac{ i \Gamma } { 4 } - i h \left( \frac{ 1 } { 4 x } \frac{ \mathrm{d} x } {\mathrm{d} t } + i \frac{ \mathrm{d} \epsilon } { \mathrm{d} t } - \frac{1}{2} \frac{ \mathrm{d} r } { \mathrm{d} t } \right) = 0, \end{equation} where $x = 1/b^2$, $p = (1 - i \phi)/b^2$, $q = (z_0 - i \phi z_\phi)/b^2$, and $r = (z_0^2 - \phi z_\phi^2)/b^2$. The corresponding initial conditions are $x(t_0) = x_0 = 1/b_0^2$, $p(t_0) = p_0 = (1 - i \phi_0)/b_0^2$, $q(t_0) = q_0 = (a_0 - \phi_0 a_\phi)/b_0^2$, and $r(t_0) = r_0 = (a_0^2 - \phi_0 a_\phi^2)/b_0^2$. The original parameters can be extracted from these auxiliary variables by $b^2 = 1 / \Re \{ p \}$, $\phi = - \Im \{ p \} / \Re \{ p \}$, $z_0 = \Re \{ q \} / \Re \{ p \}$, $z_\phi = \Im \{ q \} / \Im \{ p \}$, and $\epsilon$ appears explicitly in the equations above. First, it is worth noting that all parameters of interest can be extracted from $p(t)$ and $q(t)$ alone. We are not interested in $\epsilon(t)$ as it is only related to the global phase and the norm of the wave function and it contains little physical information. Furthermore, an interesting and incredibly convenient feature of these equations is that the Eq. \eqref{eq:p} is a function of $p(t)$ alone and Eq. \eqref{eq:pq} is a function of $p(t)$ and $q(t)$ only. Therefore, we only need to solve first two equations and we can neglect Eq. \eqref{eq:pqr}. However, in order to actually perform Monte-Carlo simulations of quantum trajectories Eq. \eqref{eq:pqr} would need to be solved in order to obtain correct jump statistics. We start with Eq. \eqref{eq:p} and we note it can be rearranged into the form \begin{equation} \frac{ \mathrm{d} p } { (\zeta \omega / 4 h)^2 - p^2 } = i 4 h \mathrm{d} t, \end{equation} where $\zeta^2 = (\alpha - i \beta)^2 = 1 - i 2 \Gamma / \omega^2$, and \begin{equation} \alpha = \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{1 + \frac{ 4\Gamma^2 }{ \omega^4 }}}, \end{equation} \begin{equation} \beta = -\sqrt{ -\frac{1}{2} + \frac{1}{2} \sqrt{1 + \frac{ 4\Gamma^2 }{ \omega^4 }}}. \end{equation} This is a standard integral\footnotemark and thus yields \begin{equation} \label{eq:psol} p(t) = \frac{ \zeta \omega } { 4 h } \frac{ ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} - ( \zeta \omega - 4 h p_0 ) e^{-i \zeta \omega t} } { ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} + ( \zeta \omega - 4 h p_0 ) e^{-i \zeta \omega t} }. \end{equation} \footnotetext{ \[ \int \frac{\mathrm{d} x}{a^2 - x^2} = \frac{1}{2a} \ln \left( \frac{a+x}{a-x} \right) + \mathrm{const.} \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\] } Having found an expression for $p(t)$ we can now solve Eq. \eqref{eq:pq} for $q(t)$. To do that we first define the integrating factor \begin{equation} I(t) = \exp \left[ i 4 h \int p \mathrm{d} t \right] = ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} + ( \zeta \omega - 4 h p_0 ) e^{-i \zeta \omega t}, \end{equation} which lets us rewrite Eq. \eqref{eq:pq} as \begin{equation} \label{eq:Iq} \frac{\mathrm{d}} {\mathrm{d} t}(Iq) = - \frac{\Gamma}{2 h} I. \end{equation} %Upon integrating the equation above we obtain %\begin{equation} % \label{eq:Iq} % Iq = - \frac{ \Gamma } {2 h} \int I \mathrm{d} t. %\end{equation} Upon integrating and the substitution of the explicit form of the integration factor into this equation we obtain the solution \begin{equation} \label{eq:qsol} q(t) = \frac{1}{2 h \zeta \omega} \frac{4 h \zeta^2 \omega^2 q_0 - i 8 h \Gamma p_0 + i \Gamma [( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} - ( \zeta \omega - 4 h p_0 )e^{-i \zeta \omega t}]} { ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} + ( \zeta \omega - 4 h p_0 )e^{-i \zeta \omega t}}. \end{equation} The solutions we have obtained to $p(t)$ in Eq. \eqref{eq:psol} and $q(t)$ in Eq. \eqref{eq:qsol} are sufficient to completely describe the physics of the system. Unfortunately, these expressions are fairly complex and it is difficult to extract the physically meaningful parameters in a form that is easy to analyse. Therefore, we instead consider the case when $\Gamma = 0$, but we do not neglect the effect of quantum jumps. It may seem counter-intuitive to neglect the term that appears due to measurement, but we are considering the weak measurement regime where $\gamma \ll J$ and thus the dynamics between the quantum jumps are actually dominated by the tunnelling of atoms rather than the null outcomes. Furthermore, the effect of the quantum jump is independent of the value of $\Gamma$ ($\Gamma$ only determined their frequency). However, this is only true at times shorter than the average time between two consecutive quantum jumps. Therefore, this approach will not yield valid answers on the time scale of a whole quantum trajectory, but it will give good insight into the dynamics immediately after a quantum jump. The solutions for $\Gamma = 0$ are \begin{equation} b^2(t) = \frac{b_0^2}{2} \left[ \left(1 + \frac{16 h^2 (1 + \phi_0^2)} {b_0^4 \omega^2} \right) + \left(1 - \frac{16 h^2 (1 + \phi_0^2)} {b_0^4 \omega^2} \right) \cos (2 \omega t) + \frac{8 h \phi_0}{b_0^2 \omega} \sin(2 \omega t) \right], \end{equation} \begin{equation} \phi(t) = \frac{b_0^2 \omega} {8 h} \left[ \left( \frac{16 h^2 (1 + \phi_0^2)} {b_0^4 \omega^2} - 1 \right) \sin (2 \omega t) + \frac{8 h \phi_0} {b_0^2 \omega} \cos (2 \omega t) \right], \end{equation} \begin{equation} z_0(t) = a_0 \cos(\omega t) + \frac{4 h \phi_0} {b_0^2 \omega} (a_0 - a_\phi) \sin (\omega t), \end{equation} \begin{equation} \phi(t) z_\phi(t) = \phi_0 a_\phi \cos (\omega t) + \frac{4 h} {b_0^2 \omega} (a_0 - \phi_0^2 a_\phi) \sin( \omega t). \end{equation} First, these equations show that all quantities oscillate with a frequency $\omega$ or $2 \omega$. We are in particular interested in the quantity $z_0(t)$ as it represents the position of the peak of the wave function and we see that it oscillates with an amplitude $\sqrt{a_0^2 + 16 h^2 \phi_0^2 (a_0 - a_\phi)^2 / (b_0^4 \omega^2)}$. Thus we have obtained a solution that clearly shows oscillations of a single Gaussian wave packet. The fact that this appears even when $\Gamma = 0$ shows that the oscillations are a property of the Bose-Hubbard model itself. However, they also depend on the initial conditions and for these oscillations to occur, $a_0$ and $a_\phi$ cannot be zero, but this is exactly the case for an initial superfluid state. We have seen in Eq. \eqref{eq:jumpz0} that the effect of a photodetection is to displace the wave packet by approximately $b^2$, i.e.~the width of the Gaussian, in the direction of the positive $z$-axis. Therefore, even though the atoms can oscillate in the absence of measurement it is the quantum jumps that are the driving force behind this phenomenon. Furthermore, these oscillations grow because the quantum jumps occur at an average instantaneous rate proportional to $\langle \cd \c \rangle (t)$ which itself is proportional to $(1+z)^2$. This means they are most likely to occur at the point of maximum displacement in the positive $z$ direction at which point a quantum jump provides positive feedback and further increases the amplitude of the wave function leading to the growth seen in Fig. \ref{fig:oscillations}(a). The oscillations themselves are essentially due to the natural dynamics of coherently displaced atoms in a lattice , but it is the measurement that causes the initial and more importantly coherent displacement and the positive feedback drive which causes the oscillations to continuously grow. Furthermore, it is by engineering the measurement, and through it the geometry of the modes, that we have control over the nature of the correlated dynamics of the oscillations. We have now seen the effect of the quantum jumps and how that leads to oscillations between odd and even sites in a lattice. However, we have neglected the effect of null outcomes on the dynamics. Even though it is small, it will not be negligible on the time scale of a quantum trajectory with multiple jumps. First, we note that all the oscillatory terms $p(t)$ and $q(t)$ actually appear as $\zeta \omega = (\alpha - i \beta) \omega$. Therefore, we can see that the null outcomes lead to two effects: an increase in the oscillation frequency by a factor of $\alpha$ to $\alpha \omega$ and a damping term with a time scale $1/(\beta \omega)$. For weak measurement, both $\alpha$ and $\beta$ will be close to $1$ so the effects are not visible on short time scales. Instead, we look at the long time limit. Unfortunately, since all the quantities are oscillatory a stationary long time limit does not exist especially since the quantum jumps provide a driving force. However, the width of the Gaussian, $b^2$, is unique in that it does not oscillate around $b^2 = 0$. Furthermore, from Eq. \eqref{eq:jumpb2} we see that even though it will decrease discontinuously at every jump, this effect is fairly small since $b^2 \ll 1$ generally. Therefore, we expect $b^2$ to oscillate, but with an amplitude that decreases approximately monotonically with time due to quantum jumps and the $1/(\beta \omega)$ decay terms, because unlike for $z_0$ the quantum jumps do not cause further displacement in this quantity. Thus, neglecting the effect of quantum jumps and taking the long time limit yields \begin{equation} \label{eq:b2} b^2(t \rightarrow \infty) = \frac{4 h} {\gamma \omega} \approx b^2_\mathrm{SF} \left( 1 - \frac{\Gamma^2}{32} \right), \end{equation} where the approximation on the right-hand side follows from the fact that $\omega \approx 2$ since we are considering the $N \gg 1$ limit, and because we are considering the weak measurement limit $\Gamma^2 / \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ denotes the width of the initial superfluid state. This result is interesting, because it shows that the width of the Gaussian distribution is squeezed as compared with its initial state which is exactly what we see in Fig. \ref{fig:oscillations}(a). However, if we substitute the parameter values used in that trajectory we only get a reduction in width by about $3\%$, but the maximum amplitude oscillations look like they have a significantly smaller width than the initial distribution. This discrepancy is due to the fact that the continuous variable approximation is only valid for $z \ll 1$ and thus it cannot explain the final behaviour of the system. Furthermore, it has been shown that the width of the distribution $b^2$ does not actually shrink to a constant value, but rather it keeps oscillating around the value given in Eq. \eqref{eq:b2} \cite{mazzucchi2016njp}. However, what we do see is that during the early stages of the trajectory, which are well described by this model, is that the width does in fact stay roughly constant. It is only in the later stages when the oscillations reach maximal amplitude that the width becomes visibly reduced. \subsection{Three-Way Competition} Now it is time to turn on the inter-atomic interactions, $U/J \ne 0$. As a result the atomic dynamics will change as the measurement now competes with both the tunnelling and the on-site interactions. A common approach to study such open systems is to map a dissipative phase diagram by finding the steady state of the master equation for a range of parameter values \cite{kessler2012}. However, here we adopt a quantum optical approach in which we focus on the conditional dynamics of a single quantum trajectory as this corresponds to a single realisation of an experiment. The resulting evolution does not necessarily reach a steady state and usually occurs far from the ground state of the system. A key feature of the quantum trajectory approach is that each trajectory evolves differently as it is conditioned on the photodetection times which are determined stochastically. Furthermore, even states in the same measurement subspace, i.e.~indistinguishable to the measurement , can have minimal overlap. This is in contrast to the unconditioned solutions obtained with the master equation which only yields a single outcome that is an average taken over all possible outcomes. However, this makes it difficult to study the three-way competition in some meaningful way across the whole parameter range. Ultimately, regardless of its strength measurement always tries to project the quantum state onto one of its eigenstates (or eigenspaces if there are degeneracies). If the probe is strong enough this will succeed, but we have seen in the previous section that when this is not the case, measurement leads to new dynamical phenomena. However, despite this vast difference in behaviour, there is a single quantity that lets us determine the degree of success of the projection, namely the fluctuations, $\sigma_D^2$ (or equivalently the standard deviation, $\sigma_D$), of the observable that is being measured, $\hat{D}$. For a perfect projection this value is exactly zero, because the system at that point is in the corresponding eigenstate. When the system is unable to project the state into such a state, the variance will be non-zero. However, the smaller its value is the closer it is to being in such an eigenstate and on the other hand a large variance means that the internal processes dominate the competition. Finally, this quantity is perfect to study quantum trajectories, because its value in the long-time limit it is only a function of $\gamma$, $J$, and $U$. It does not depend on the explicit history of photodetections. Fig. \ref{fig:squeezing} shows a plot of this quantity for $\hat{D} = \hat{N}_\mathrm{odd}$ averaged over multiple trajectories, $\langle \sigma^2_D \rangle_\mathrm{traj}$, as a function of $\gamma/J$ and $U/J$ for a lattice of six atoms on six sites (we cannot use the effective double-well model, because $U \ne 0$). We use a ground state for the corresponding $U$ and $J$ values as this provides a realistic starting point and a reference for comparing the measurement induced dynamics. We will also consider only $\hat{D} = \hat{N}_\mathrm{odd}$ unless stated otherwise. \begin{figure}[htbp!] \centering \includegraphics[width=\textwidth]{Squeezing} \caption[Squeezing in the presence of Interactions]{Atom number fluctuations at odd sites for $N = 6$ atoms at $M = 6$ sites subject to a $\hat{D} = \hat{N}_\mathrm{odd}$ measurement demonstrating the competition of global measurement with local interactions and tunnelling. Number variances are averaged over 100 trajectories. Error bars are too small to be shown ($\sim 1\%$) which emphasizes the universal nature of the squeezing. The initial state used was the ground state for the corresponding $U$ and $J$ value. The fluctuations in the ground state without measurement decrease as $U / J$ increases, reflecting the transition between the superfluid and Mott insulator phases. For weak measurement values $\langle \sigma^2_D \rangle_\mathrm{traj}$ is squeezed below the ground state value for $U = 0$, but it subsequently increases and reaches its maximum as the atom repulsion prevents the accumulation of atoms prohibiting coherent oscillations thus making the squeezing less effective. In the strongly interacting limit, the Mott insulator state is destroyed and the fluctuations are larger than in the ground state as weak measurement is not strong enough to project into a state with smaller fluctuations than the ground state.} \label{fig:squeezing} \end{figure} First, it is important to note that even though we are dealing with an average over many trajectories this information cannot be extracted from a master equation solution. This is because the variance of $\hat{D}$ as calculated from the density matrix would be dominated by the uncertainty of the final state. In other words, the fact that the final value of $\hat{D}$ is undetermined is included in this average and thus the fluctuations obtained this way are representative of the variance in the final outcome rather than the squeezing of an individual conditioned trajectory. This highlights the fact that interesting physics happens on a single trajectory level which would be lost if we studied an ensemble average. \begin{figure}[htbp!] \centering \includegraphics[width=\textwidth]{panel_U} \caption[Trajectories in the presence of Interactions]{Conditional dynamics of the atom-number distributions at odd sites illustrating competition of the global measurement with local interactions and tunnelling. The plots are for single quantum trajectories starting from the ground state for $N = 6$ atoms on $M = 6$ sites with $\hat{D} = \hat{N}_\mathrm{odd}$, $\gamma/J = 0.1$. (a) Weakly interacting bosons $U/J = 1$: the on-site repulsion prevents the formation of well-defined oscillation in the population of the mode. As states with different imbalance evolve with different frequencies, the squeezing is not as efficient for the non-interacting case. (b) Strongly interacting bosons $U/J = 10$: oscillations are completely suppressed and the number of atoms in the mode is rather well-defined although clearly worse than in a Mott insulator.} \label{fig:Utraj} \end{figure} Looking at Fig. \ref{fig:squeezing} we see many interesting things happening suggesting different regimes of behaviour. For the ground state (i.e.~no measurement) we see that the fluctuations decrease monotonically as $U$ increases reflecting the superfluid to Mott insulator quantum phase transition. The measured state on the other hand behaves very differently and $\langle \sigma^2_D \rangle_\mathrm{traj}$ varies non-monotonically. For weak interactions the fluctuations are strongly squeezed below those of the ground state followed by a rapid increase as $U$ is increased before peaking and eventually decreasing. We have already seen in the previous section and in particular Fig. \ref{fig:oscillations} that the macroscopic oscillations at $U = 0$ are well squeezed when compared to the initial state and this is the case over here as well. However, as $U$ is increased the interactions prevent the atoms from accumulating in one place thus preventing oscillations with a large amplitude which effectively makes the squeezing less effective as seen in Fig. \ref{fig:Utraj}(a). In fact, we have seen towards the end of the last section how for small amplitude oscillations that can be described by the effective double-well model the width of the number distribution does not change by much. Even though that model is not valid for $U \ne 0$ we should not be surprised that without macroscopic oscillations the fluctuations cannot be significantly reduced. On the other end of the spectrum, for weak measurement, but strong on-site interactions we note that the backaction leads to a significant increase in fluctuations compared to the ground state. This is simply due to the fact that the measurement destroys the Mott insulating state, which has small fluctuations due to strong local interactions, but then subsequently is not strong enough to squeeze the resulting dynamics as shown in Fig. \ref{fig:Utraj}(b). To see why this is so easy for the quantum jumps to do we look at the ground state in first-order perturbation theory given by \begin{equation} | \Psi_{J/U} \rangle = \left[ 1 + \frac{J}{U} \sum_{\langle i, j \rangle} \bd_i b_j \right] | \Psi_0 \rangle, \end{equation} where we have neglected the non-Hermitian term as we're in the weak measurement regime and $| \Psi_0 \rangle$ is the Mott insulator state and the second term in the brackets represents a uniform distribution of particle-hole excitation pairs across the lattice. In the $U \rightarrow \infty$ limit a quantum jump has no effect as $| \Psi_0 \rangle$ is already an eigenstate of $\hat{D}$. However, for finite $U$, each photocount will amplify the present excitations increasing the fluctuations in the system. In fact, consecutive detections lead to an exponential growth of these excitations. For $K \gg 1$ illuminated sites and unit filling of the lattice, the atomic state after $m$ consecutive quantum jumps becomes $\c^m | \Psi_{J/U} \rangle \propto | \Psi_{J/U} \rangle + | \Phi_m \rangle$ where \begin{equation} | \Phi_m \rangle = \frac{2^m J} {K U} \sum_{i \in \mathrm{odd}} \left( \bd_i b_{i-1} - \bd_{i-1} b_i - \bd_{i+1} b_i + \bd_i b_{i+1} \right) | \Psi_0 \rangle. \end{equation} In the weak measurement regime the effect of non-Hermitian decay is negligible compared to the local atomic dynamics combined with the quantum jumps so there is minimal dissipation occurring. Therefore, because of the exponential growth of the excitations, even a small number of photons arriving in succession can destroy the ground state. We have neglected all dynamics in between the jumps which would distribute the new excitations in a way which will affect and possibly reduce the effects of the subsequent quantum jumps. However, due to the lack of any serious decay channels they will remain in the system and subsequent jumps will still amplify them further destroying the ground state and thus quickly leading to a state with large fluctuations. In the strong measurement regime ($\gamma \gg J$) the measurement becomes more significant than the local dynamics and the system will freeze the state in the measurement operator eigenstates. In this case, the squeezing will always be better than in the ground state, because measurement and on-site interaction cooperate in suppressing fluctuations. This cooperation did not exist for weak measurement, because it tried to induce dynamics which produced squeezed states (either successfully as seen with the macroscopic oscillations or unsuccessfully as seen with the Mott insulator). This suffered heavily from the effects of interactions as they would prevent this dynamics by dephasing different components of the coherent excitations. Strong measurement, on the other hand, squeezes the quantum state by trying to project it onto an eigenstate of the observable \cite{mekhov2009prl, mekhov2009pra}. For weak interactions where the ground state is a highly delocalised superfluid it is obvious that projections onto $\hat{D} = \hat{N}_\mathrm{odd}$ will suppress fluctuations significantly. However, the strongly interacting regime is much less evident, especially since we have just demonstrated how sensitive the Mott insulating phase is to the quantum jumps when the measurement is weak. To understand the strongly interacting case we will again use first-order perturbation theory and consider a postselected $\langle \hat{D}^\dagger \hat{D} \rangle = 0$ trajectory. This corresponds to a state that scatters no photons and thus is fully described by the non-Hermitian Hamiltonian alone. Squeezing depends on the measurement and interaction strengths and is common to all the possible trajectories so we can gain insight into the general behaviour by considering a specific special case. However, we will instead consider $\hat{D} = \Delta \hat{N} = \hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even}$, because this measurement also has only $Z = 2$ modes, but its $\langle \hat{D}^\dagger \hat{D} \rangle = 0$ trajectory would be very close to the Mott insulating ground state, because $\hat{D}^\dagger \hat{D} | \Psi_0 \rangle = 0$ and we can expand around the Mott insulating state. Applying perturbation theory to obtain the modified ground state we get \begin{equation} | \Psi_{J,U, \gamma} \rangle = \left[ 1 + \frac{J}{U - i 4 \gamma} \sum_{\langle i, j \rangle} \bd_i b_j \right] | \Psi_0 \rangle. \end{equation} The variance of the measurement operator for this state is given by \begin{equation} \sigma^2_{\Delta N} = \frac{16 J^2 M} {U^2 + 16 \gamma^2}. \end{equation} From the form of the denominator we immediately see that both interaction and measurement squeeze with the same quadratic dependence and that the squeezing is always better than in the ground state ($\gamma = 0$) regardless of the value of $U$. Also, depending on the ratio of $\gamma/U$ the squeezing can be dominated by measurement ($\gamma/U \gg 1$) or by interactions ($\gamma/U \ll 1$) or both processes can contribute equally ($\gamma/U \approx 1$). The $\hat{D} = \hat{N}_\mathrm{odd}$ measurement will behave similarly since the geometry is exactly the same. Furthermore, the Mott insulator state is also an eigenstate of this operator, just not the zero eigenvalue vector and thus the final state would need to be described using a balance of quantum jumps and non-Hermitian evolution complicating the picture. However, the particle-hole excitation term would be proportional to $(U^2 + \gamma^2)^{-1}$ instead since the $\gamma$ coefficient in the perturbative expansion depends on $(J_{i,i} - J_{i\pm1,i\pm1})^2$. We can see the system transitioning into the strong measurement regime in Fig. \ref{fig:squeezing} as the $U$-dependence flattens out with increasing measurement strength as 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. Furthermore, since this measurement is continuous and does not stop after the projection the system will be frozen in this state. This effect is called the quantum Zeno effect \cite{misra1977, facchi2008} from Zeno's classical paradox in which a ``watched arrow never moves'' that stated that 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 infinitesimal 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. 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 \cite{facchi2008, raimond2010, raimond2012, signoles2014}. These effects can be easily seen in our model when $\gamma \rightarrow \infty$. The system will be projected into one or more degenerate eigenstates 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 $\varphi$, the subsequent evolution is described by the projected Hamiltonian $P_{\varphi} \hat{H}_0 P_{\varphi}$. 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 suppressed 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. However, the geometric patterns we have in which the modes are spatially 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 without first having to leave it. 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 Model 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. 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 tunnelling 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 tunnelling 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 tunnelling is represented by any pair of one blue and one red arrow. (b.2) Correlated tunnelling 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, i.e.~two correlated tunnelling events, can now occur 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 ability 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 the Zeno subspaces would be 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 sites 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, if $i$ and $j$ belong to different modes $P_0 \bd_i b_j P_0 = 0$. When $\gamma \rightarrow \infty$ we recover the quantum Zeno Hamiltonian where this would be the only remaining term. It is the second term that shows the second-order correlated 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 anywhere on the lattice within the constraints of the mode structure. This is in particular explicitly shown in Figs. \ref{fig:zeno}(a,b). The imaginary coefficient means that the tunnelling 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 discontinuous change in the system where the Zeno subspace rapidly changes which can be seen in Fig. \ref{fig:zeno}(a). 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 tunnelling events represented in Eq. \eqref{eq:hz} by the fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The projection $P_0$ is not sensitive to individual site occupancies, but instead enforces a fixed value of the observable, i.e.~a single Zeno subspace. This is a striking difference with the $t$-$J$ and other strongly interacting models. The strong interaction also leads to correlated events in which atoms can tunnel over each other by creating an unstable doubly occupied site during the intermediate step. However, these correlated events are by their nature localised. Due to interactions the doubly-occupied site cannot be present in the final state which means that any tunnelling event that created this unstable configuration must be followed by another tunnelling event which takes an atom away. In the case of global measurement this process is delocalised, because since the modes consist of many sites the stable configuration can be restored by a tunnelling event from a completely different lattice site that belongs to the same mode. 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 $3$. Therefore, effective long-range tunnelling 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 tunnelling 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 $\sim \gamma/J^2$ as seen in Eq. \eqref{eq:hz}. The schematic explanation of long-range correlated tunnelling 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 tunnelling 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 Eq. \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. The scheme in Fig.~\ref{fig:zeno}(b.1) can help design a nonlocal reservoir for the tunnelling (or ``decay'') of atoms from one region to another. For example, if the atoms are placed only at odd sites, according to Eq. \eqref{eq:hz} their tunnelling is suppressed since the multi-tunnelling 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 tunnelling 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-tunnelling 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 tunnelling 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-tunnelling events is possible. This global pair tunnelling may play a role of a building block for more complicated many-body effects. For example, a pair tunnelling 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 tunnelling between even neighbouring 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. \section{Non-Hermitian Dynamics in the Quantum Zeno Limit} In the previous section we provided a rather high-level analysis of the strong measurement limit in our quantum gas model. We showed that global measurement in the strong, but not projective, limit leads to correlated tunnelling events which can be highly delocalised. Multiple examples for different optical geometries and measurement operators demonstrated the incredible flexibility and potential in engineering dynamics for ultracold gases in an optical lattice. We also claimed that the behaviour of the system is described by the Hamiltonian given in Eq. \eqref{eq:hz}. Having developed a physical and intuitive understanding of the dynamics in the quantum Zeno limit we will now provide a more rigorous, low-level and fundamental understanding of the process. \subsection{Suppression of Coherences in the Density Matrix} At this point we deviate from the quantum trajectory approach and we resort to a master equation as introduced in section \ref{sec:master}. We do this, because we have seen that the emergent long-range correlated tunnelling is a feature of all trajectories and mostly depends on the geometry of the measurement. Therefore, a general approach starting from an unconditioned state should be able to reveal these features. However, we will later make use of the fact that we are in possession of a measurement record and obtain a conditioned state. Furthermore, we first consider the most general case of an open system subject to a quantum measurement and only limit ourselves to the quantum gas model later on. This demonstrates that the dynamics we observed in the previous section are a feature of measurement rather than our specific model. As introduced in section \ref{sec:master} we consider a state described by the density matrix $\hat{\rho}$ whose isolated behaviour is described by the Hamiltonian $\H_0$ and when measured the jump operator $\c$ is applied to the state at each detection \cite{MeasurementControl}. The master equation describing its time evolution when we ignore the measurement outcomes is given by \begin{equation} \dot{\hat{\rho}} = -i [ \H_0 , \hat{\rho} ] + \c \hat{\rho} \cd - \frac{1}{2}( \cd\c \hat{\rho} + \hat{\rho} \cd\c ). \end{equation} We also define $\c = \lambda \op$ and $\H_0 = \nu \h$. The exact definition of $\lambda$ and $\nu$ is not so important as long as these coefficients can be considered to be some measure of the relative size of these operators. They would have to be determined on a case-by-case basis, because the operators $\c$ and $\H_0$ may be unbounded. If these operators are bounded, one can simply define them such that $||\op|| \sim O(1)$ and $||\h|| \sim O(1)$. If they are unbounded, one possible approach would be to identify the relevant subspace of which dynamics we are interested in and scale the operators such that the eigenvalues of $\op$ and $\h$ in this subspace are $\sim O(1)$. We will once again use projectors $P_m$ which have no effect on states within a degenerate subspace of $\c$ ($\op$) with eigenvalue $c_m$ ($o_m$), but annihilate everything else. For convenience we will also use the following definition $\hat{\rho}_{mn} = P_m \hat{\rho} P_n$. Note that these are submatrices of the density matrix, which in general are not single matrix elements. Therefore, we can write the master equation that describes this open system as a set of equations \begin{equation} \label{eq:master} \dot{\hat{\rho}}_{mn} = -i \nu P_m \left[ \h \sum_r \hat{\rho}_{rn} - \sum_r \hat{\rho}_{mr} \h \right] P_n + \lambda^2 \left[ o_m o_n^* - \frac{1}{2} \left( |o_m|^2 + |o_n|^2 \right) \right] \hat{\rho}_{mn}, \end{equation} where the first term describes coherent evolution whereas the second term causes dissipation. First, note that for the density submatrices for which $m = n$, $\hat{\rho}_{mm}$, the dissipative term vanishes. This means that these submatrices are subject to coherent evolution only and do not experience losses and they are thus decoherence free subspaces. It is crucial to note that these submatrices are simply the density matrices of the individual degenerate Zeno subspaces. Interestingly, any state that consists only of these decoherence free subspaces, i.e.~ $\hat{\rho} = \sum_m \hat{\rho}_{mm}$, and that commutes with the Hamiltonian, $[\hat{\rho}, \hat{H}_0] = 0$, will be a steady state. This can be seen by substituting this ansatz into Eq. \eqref{eq:master} which yields $\dot{\hat{\rho}}_{mn} = 0$ for all $m$ and $n$. These states can be prepared dissipatively using known techniques \cite{diehl2008}, but it is not required that the state be a dark state of the dissipative operator as is usually the case. Second, we consider a large detection rate, $\lambda^2 \gg \nu$, for which the coherences, i.e.~ the density submatrices $\hat{\rho}_{mn}$ for which $m \ne n$, will be heavily suppressed by dissipation. We can adiabatically eliminate these cross-terms by setting $\dot{\hat{\rho}}_{mn} = 0$, to get \begin{equation} \label{eq:intermediate} \hat{\rho}_{mn} = \frac{\nu}{\lambda^2} \frac{i P_m \left[ \h \sum_r \hat{\rho}_{rn} - \sum_r \hat{\rho}_{mr} \h \right] P_n } {o_m o_n^* - \frac{1}{2} \left( |o_m|^2 + |o_n|^2 \right)} \end{equation} which tells us that they are of order $\nu/\lambda^2 \ll 1$. Therefore, the resulting density matrix will be given by $\hat{\rho} \approx \sum_m \hat{\rho}_{mm}$ which consists solely of the individual Zeno subspace density matrices. One can easily recover the projective Zeno limit by considering $\lambda \rightarrow \infty$ when all the subspaces completely decouple. This is exactly the $\gamma \rightarrow \infty$ limit discussed in the previous section. However, we have seen that it is crucial we only consider $\lambda^2 \gg \nu$, but not infinite. If the subspaces do not decouple completely, then transitions within a single subspace can occur via other subspaces in a manner similar to Raman transitions. In Raman transitions population is transferred between two states via a third, virtual, state that remains empty throughout the process. By avoiding the infinitely projective Zeno limit we open the option for such processes to happen in our system where transitions within a single Zeno subspace occur via a second, different, Zeno subspace even though the occupation of the intermediate states will remain negligible at all times. A single quantum trajectory results in a pure state as opposed to the density matrix and in general, there are many density matrices that have non-zero and non-negligible $m = n$ submatrices, $\hat{\rho}_{mm}$, even when the coherences are small. They correspond to a mixed states containing many Zeno subspaces and it is not clear what the pure states that make up these density matrices are. However, we note that for a single pure state the density matrix can consist of only a single diagonal submatrix $\hat{\rho}_{mm}$. To understand this, consider the state $| \Phi \rangle$ and take it to span exactly two distinct subspaces $P_a$ and $P_b$ ($a \ne b$). This wave function can thus be written as $| \Phi \rangle = P_a | \Phi \rangle + P_b | \Phi \rangle$. The corresponding density matrix is given by \begin{equation} \hat{\rho}_\Psi = P_a | \Phi \rangle \langle \Phi | P_a + P_a | \Phi \rangle \langle \Phi | P_b + P_b | \Phi \rangle \langle \Phi | P_a + P_b | \Phi \rangle \langle \Phi | P_b. \end{equation} If the wave function has significant components in both subspaces then in general the density matrix will not have negligible coherences, $\hat{\rho}_{ab} = P_a | \Phi \rangle \langle \Phi | P_b$. A density matrix with just diagonal components must be in either subspace $a$, $| \Phi \rangle = P_a | \Phi \rangle$, or in subspace $b$, $| \Phi \rangle = P_b | \Phi \rangle$. Therefore, a density matrix of the form $\hat{\rho} = \sum_m \hat{\rho}_{mm}$ without any cross-terms between different Zeno subspaces can only be composed of pure states that each lie predominantly within a single subspace. However, because we will not be dealing with the projective limit, the wave function will in general not be entirely confined to a single Zeno subspace. We have seen that the coherences are of order $\nu/\lambda^2$. This would require the wave function components to satisfy $P_a | \Phi \rangle \approx O(1)$ and $P_b | \Phi \rangle \approx O(\nu/\lambda^2)$ (or vice-versa). This in turn implies that the population of the states outside of the dominant subspace (and thus the submatrix $\hat{\rho}_{bb}$) will be of order $\langle \Phi | P_b^2 | \Phi \rangle \approx O(\nu^2/\lambda^4)$. Therefore, these pure states, even though they span multiple Zeno subspaces, cannot exist in a meaningful coherent superposition in this limit. This means that a density matrix that spans multiple Zeno subspaces has only classical uncertainty about which subspace is currently occupied as opposed to the uncertainty due to a quantum superposition. This is analogous to the simple qubit example we considered in section \ref{sec:master}. \subsection{Quantum Measurement vs. Dissipation} This is where quantum measurement deviates from dissipation. If we have access to a measurement record we can infer which Zeno subspace is occupied, because we know that only one of them can be occupied at any time. We have seen that since the density matrix cross-terms are small we know \emph{a priori} that the individual wave functions comprising the density matrix mixture will not be coherent superpositions of different Zeno subspaces and thus we only have classical uncertainty which means we can resort to classical probability methods. Each individual experiment will at any time be predominantly in a single Zeno subspace with small cross-terms and negligible occupations in the other subspaces. With no measurement record our density matrix would be a mixture of all these possibilities. We can try and determine the Zeno subspace around which the state evolves in a single experiment from the number of detections, $m$, in time $t$. The detection distribution on time-scales shorter than dissipation (so we can approximate as if we were in a fully Zeno regime) can be obtained by integrating over the detection times \cite{mekhov2009pra} to get \begin{equation} P(m,t) = \sum_n \frac{[|c_n|^2 t]^m} {m!} e^{-|c_n|^2 t} \mathrm{Tr} (\rho_{nn}). \end{equation} For a state that is predominantly in one Zeno subspace, the distribution will be approximately Poissonian (up to $O(\nu^2 / \lambda^4)$, the population of the other subspaces). Therefore, in a single experiment we will measure $m = |c_0|^2t \pm \sqrt{|c_0|^2t}$ detections (note, we have assumed $|c_0|^2 t$ is large enough to approximate the distribution as normal. This is not necessary, we simply use it here to not have to worry about the asymmetry in the deviation around the mean value). The uncertainty does not come from the fact that $\lambda$ is not infinite. The jumps are random events with a Poisson distribution. Therefore, even in the full projective limit we will not observe the same detection trajectory in each experiment even though the system evolves in exactly the same way and remains in a perfectly pure state. If the basis of $\c$ is continuous (e.g. free particle position or momentum) then the deviation around the mean will be our upper bound on the deviation of the system from a pure state evolving around a single Zeno subspace. However, continuous systems are beyond the scope of this work and we will confine ourselves to discrete systems. Though it is important to remember that continuous systems can be treated this way, but the error estimate (and thus the mixedness of the state) will be different. For a discrete system it is easier to exclude all possibilities except for one. The error in our estimate of $|c_0|^2$ in a single experiment decreases as $1/\sqrt{t}$ and thus it can take a long time to confidently determine $|c_0|^2$ to a sufficient precision this way. However, since we know that it can only take one of the possible values from the set $\{|c_n|^2 \}$ it is much easier to instead exclude all the other values. In an experiment we can use Bayes' theorem to infer the state of our system as follows \begin{equation} p(c_n = c_0 | m) = \frac{ p(m | c_n = c_0) p(c_n = c_0) }{ p(m) }, \end{equation} where $p(x)$ denotes the probability of the discrete event $x$ and $p(x|y)$ the conditional probability of $x$ given $y$. We know that $p(m | c_n = c_0)$ is simply given by a Poisson distribution with mean $|c_0|^2 t$. $p(m)$ is just a normalising factor and $p(c_n = c_0)$ is our \emph{a priori} knowledge of the state. Therefore, one can get the probability of being in the right Zeno subspace from \begin{align} p(c_n & = c_0 | m) = \frac{ p_0(c_n = c_0) \frac{ \left( |c_0|^2 t \right)^{2m} } {m!} e^{-|c_0|^2 t}} {\sum_n p_0(c_n) \frac{ \left( |c_n|^2 t \right)^{2m} } {m!} e^{-|c_n|^2 t}} \nonumber \\ & = p_0(c_n = c_0) \left[ \sum_n p_0(c_n) \left( \frac{ |c_n|^2 } { |c_0|^2 } \right)^{2m} e^{\left( |c_0|^2 - |c_n|^2 \right) t} \right]^{-1}, \end{align} where $p_0$ denotes probabilities at $t = 0$. In a real experiment one could prepare the initial state to be close to the Zeno subspace of interest and thus it would be easier to deduce the state. Furthermore, in the middle of an experiment if we have already established the Zeno subspace this will be reflected in these \emph{a priori} probabilities again making it easier to infer the correct subspace. However, we will consider the worst case scenario which might be useful if we don't know the initial state or if the Zeno subspace changes during the experiment, a uniform $p_0(c_n)$. This probability is a rather complicated function as $m$ is a stochastic quantity that also increases with $t$. We want it to be as close to $1$ as possible. In order to devise an appropriate condition for this we note that in the first line all terms in the denominator are Poisson distributions of $m$. Therefore, if the mean values $|c_n|^2 t$ are sufficiently spaced out, only one of the terms in the sum will be significant for a given $m$ and if this happens to be the one that corresponds to $c_0$ we get a probability close to unity. Therefore, we set the condition such that it is highly unlikely that our measured $m$ could be produced by two different distributions \begin{align} \sqrt{|c_0|^2 t} \ll ||c_0|^2 - |c_n|^2| t, \forall n \ne 0 \\ \sqrt{|c_n|^2 t} \ll ||c_0|^2 - |c_n|^2| t, \forall n \ne 0 \end{align} The left-hand side is the standard deviation of $m$ if the system was in subspace $P_0$ or $P_n$. The right-hand side is the difference in the mean detections between the subspace $n$ and the one we are interested in. The condition becomes more strict if the subspaces become less distinguishable as it becomes harder to confidently determine the correct state. Once again, using $\c = \lambda \hat{o}$ where $\hat{o} \sim O(1)$ we get \begin{equation} t \gg \frac{1}{\lambda^2} \frac{|o_{0,n}|^2} {(|o_0|^2 - |o_n|^2|)^2}. \end{equation} Since detections happen on average at an average rate of order $\lambda^2$ we only need to wait for a few detections to satisfy this condition. Therefore, we see that even in the worst case scenario of complete ignorance of the state of the system we can very easily determine the correct subspace. Once it is established for the first time, the \emph{a priori} information can be updated and it will become even easier to monitor the system. However, it is important to note that physically once the quantum jumps deviate too much from the mean value the system is more likely to change the Zeno subspace (due to measurement backaction) and the detection rate will visibly change. Therefore, if we observe a consistent detection rate it is extremely unlikely that it can be produced by two different Zeno subspaces so in fact it is even easier to determine the correct state, but the above estimate serves as a good lower bound on the necessary detection time. Having derived the necessary conditions to confidently determine which Zeno subspace is being observed in the experiment we can make another approximation thanks to measurement which would be impossible in a purely dissipative open system. If we observe a number of detections consistent with the subspace $P_m = P_0$ we can set $\hat{\rho}_{mn} \approx 0$ for all cases when both $m \ne 0$ and $n \ne 0$ leaving our density matrix in the form \begin{equation} \label{eq:approxrho} \hat{\rho} = \hat{\rho}_{00} + \sum_{r\ne0} (\hat{\rho}_{0r} + \hat{\rho}_{r0}). \end{equation} We can do this, because the other states are inconsistent with the measurement record. We know from the previous section that the system must lie predominantly in only one of the Zeno subspaces and when that is the case, $\hat{\rho}_{0r} \approx O(\nu/\lambda^2)$ and for $m \ne 0$ and $n \ne 0$ we have $\hat{\rho}_{mn} \approx O(\nu^2/\lambda^4)$. Therefore, this amounts to keeping first order terms in $\nu/\lambda^2$ in our approximation. This is a crucial step as all $\hat{\rho}_{mm}$ matrices are decoherence free subspaces and thus they can all coexist in a mixed state decreasing the purity of the system without measurement. Physically, this means we exclude trajectories in which the Zeno subspace has changed (measurement is not fully projective). By substituting Eq. \eqref{eq:intermediate} into Eq. \eqref{eq:master} we see that this happens at a rate of $\nu^2 / \lambda^2$. However, since the two measurement outcomes cannot coexist any transition between them happens in discrete transitions (which we know about from the change in the detection rate as each Zeno subspace will correspond to a different rate) and not as continuous coherent evolution. Therefore, we can postselect in a manner similar to Refs. \cite{otterbach2014, lee2014prx, lee2014prl}, but our requirements are significantly more relaxed - we do not require a specific single trajectory, only that it remains within a Zeno subspace. Furthermore, upon reaching a steady state, these transitions become impossible as the coherences vanish. This approximation is analogous to optical Raman transitions where the population of the excited state is neglected. Here, we can make a similar approximation and neglect all but one Zeno subspace thanks to the additional knowledge we gain from knowing the measurement outcomes. \subsection{The Non-Hermitian Hamiltonian} Rewriting the master equation using $\c = c_0 + \delta \c$, where $c_0$ is the eigenvalue corresponding to the eigenspace defined by the projector $P_0$ which we used to obtain the density matrix in Eq. \eqref{eq:approxrho}, we get \begin{equation} \label{eq:finalrho} \dot{\hat{\rho}} = -i \left( \H_\mathrm{eff} \hat{\rho} - \hat{\rho} \H_\mathrm{eff}^\dagger \right) + \delta \c \hat{\rho} \delta \cd, \end{equation} \begin{equation} \label{eq:Ham} \H_\mathrm{eff} = \H_0 + i \left( c_0^*\c - \frac{|c_0|^2}{2} - \frac{\cd\c}{2} \right). \end{equation} The first term in Eq. \eqref{eq:finalrho} describes coherent evolution due to the non-Hermitian Hamiltonian $\H_\mathrm{eff}$ and the second term is decoherence due to our ignorance of measurement outcomes. When we substitute our approximation of the density matrix $\hat{\rho} = \hat{\rho}_{00} + \sum_{r\ne0} (\hat{\rho}_{0r} + \hat{\rho}_{r0})$ into Eq. \eqref{eq:finalrho}, the last term vanishes, $\delta \c \hat{\rho} \delta \cd = 0$. This happens, because $\delta \c P_0 \hat{\rho} = \hat{\rho} P_0 \delta \c^\dagger = 0$. The projector annihilates all states except for those with eigenvalue $c_0$ and so the operator $\delta \c = \c - c_0$ will always evaluate to $c_0 - c_0 = 0$. Recall that we defined $\hat{\rho}_{mn} = P_m \hat{\rho} P_n$ which means that every term in our approximate density matrix contains the projector $P_0$. However, it is important to note that this argument does not apply to other second order terms in the master equation, because some terms only have the projector $P_0$ applied from one side, e.g.~$\hat{\rho}_{0m}$. The term $\delta \c \hat{\rho} \delta \cd$ applies the fluctuation operator from both sides so it does not matter in this case, but it becomes relevant for terms such as $ \hat{\rho} \delta \cd \delta \c$. It is important to note that this term does not automatically vanish, but when the explicit form of our approximate density matrix is inserted, it is in fact zero. Therefore, we can omit this term using the information we gained from measurement, but keep other second order terms, such as $\delta \cd \delta \c \rho$ in the Hamiltonian which are the origin of other second-order dynamics. This could not be the case in a dissipative system. Ultimately we find that a system under continuous measurement for which $\lambda^2 \gg \nu$ in the Zeno subspace $P_0$ is described by the deterministic non-Hermitian Hamiltonian $\H_\mathrm{eff}$ in Eq. \eqref{eq:Ham} and thus obeys the following Schr\"{o}dinger equation \begin{equation} i \frac{\mathrm{d} | \Psi \rangle}{\mathrm{d}t} = \left[\H_0 + i \left( c_0^*\c - \frac{|c_0|^2}{2} - \frac{\cd\c}{2} \right) \right] | \Psi \rangle. \end{equation} Of the three terms in the parentheses the first two represent the effects of quantum jumps due to detections (which one can think of as `reference frame' shifts between different degenerate eigenspaces) and the last term is the non-Hermitian decay due to information gain from no detections. It is important to emphasize that even though we obtained a deterministic equation, we have not neglected the stochastic nature of the detection events. The detection trajectory seen in an experiment will have fluctuations around the mean determined by the Zeno subspace, but there simply are many possible measurement records with the same outcome. This is just like the fully projective Zeno limit where the system remains perfectly pure in one of the possible projections, but the detections remain randomly distributed in time. One might then be concerned that purity is preserved even though we might be averaging over many trajectories within this Zeno subspace. We have neglected the small terms $\hat{\rho}_{m,n}$ ($m,n \ne 0$) which are $O(\nu^2/\lambda^4)$ and thus they are not correctly accounted for by our approximation. This means that we have an $O(\nu^2/\lambda^4)$ error in our density matrix. The purity given by \begin{equation} \mathrm{Tr}(\hat{\rho}^2) = \mathrm{Tr}(\hat{\rho}^2_{00} + \sum_{m \ne 0} \hat{\rho}_{0m}\hat{\rho}_{m0}) + \mathrm{Tr}(\sum_{m,n\ne0} \hat{\rho}_{mn} \hat{\rho}_{nm}) \end{equation} where the second term contains the terms not accounted for by our approximation thus introduces an $O(\nu^4/\lambda^8)$ error. Therefore, this discrepancy is negligible in our approximation. The pure state predicted by $\H_\mathrm{eff}$ is only an approximation, albeit a good one, and the real state will be mixed to a small extent. Whilst perfect purity within the Zeno subspace $\hat{\rho}_{00}$ is expected due to the measurement's strong decoupling effect, the nearly perfect purity when transitions outside the Zeno subspace are included is a nontrivial result. Similarly, in Raman transitions the population of the neglected excited state is also non-zero, but negligible. Furthermore, this equation does not actually require the adiabatic elimination used in Eq. \eqref{eq:intermediate} (we only used it to convince ourselves that the coherences are small) and such situations may be considered provided all approximations remain valid. In a similar way the limit of linear optics is derived from the physics of a two-level nonlinear medium, when the population of the upper state is neglected and the adiabatic elimination of coherences is not required. \subsection{Non-Hermitian Dynamics in Ultracold Gases} We finally return to our quantum gas model inside of a cavity. We start by considering the simplest case of a global multi-site measurement of the form $\hat{D} = \hat{N}_K = \sum_i^K \n_i$, where the sum is over $K$ illuminated sites. The effective Hamiltonian becomes \begin{equation} \label{eq:nHH2} \hat{H}_\mathrm{eff} = \hat{H}_0 - i \gamma \left( \delta \hat{N}_K \right)^2, \end{equation} where $ \delta \hat{N}_K = \hat{N}_K - N^0_K$ and $N^0_K$ is the Zeno subspace eigenvalue. It is now obvious that continuous measurement squeezes the fluctuations in the measured quantity, as expected, and that the only competing process is the system's own dynamics. In this case, if we adiabatically eliminate the density matrix cross-terms and substitute Eq. \eqref{eq:intermediate} into Eq. \eqref{eq:master} for this system we obtain an effective Hamiltonian within the Zeno subspace defined by $N_K$ \begin{equation} \H_\varphi = P_0 \left[ \H_0 - i \frac{J^2}{\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 $\varphi$ denotes a set of sites belonging to a single mode and $\varphi^\prime$ is the set's complement (e.g. odd and even or illuminated and non-illuminated sites) and $P_0$ is the projector onto the eigenspace with $N_K^0$ atoms in the illuminated area. We focus on the case when the second term is not only significant, but also leads to dynamics within a Zeno subspace that are not allowed by conventional quantum Zeno dynamics accounted for by the first term. The second term represents second-order transitions via other subspaces which act as intermediate states much like virtual states in optical Raman transitions. This is in contrast to the conventional understanding of the Zeno dynamics for infinitely frequent projective measurements (corresponding to $\gamma \rightarrow \infty$) where such processes are forbidden \cite{facchi2008}. Thus, it is the weak quantum measurement that effectively couples the states. Note that this is a special case of the equation in Eq. \eqref{eq:hz} which can be obtained by considering a more general two mode setup. \subsection{Small System Example} To get clear physical insight, we initially consider three atoms in three sites and choose our measurement operator such that $\hat{D} = \n_2$, i.e.~only the middle site is subject to measurement, and the Zeno subspace defined by $n_2 = 1$. Such an illumination pattern can be achieved with global addressing by crossing two beams and placing the nodes at the odd sites and the antinodes at even sites. This means that $P_0 \H_0 P_0 = 0$. However, the first and third sites are connected via the second term. Diagonalising the Hamiltonian reveals that out of its ten eigenvalues all but three have a significant negative imaginary component of the order $\gamma$ which means that the corresponding eigenstates decay on a time scale of a single quantum jump and thus quickly become negligible. The three remaining eigenvectors are dominated by the linear superpositions of the three Fock states $|2,1,0 \rangle$, $|1, 1, 1 \rangle$, and $|0,1,2 \rangle$. Whilst it is not surprising that these components are the only ones that remain as they are the only ones that actually lie in the Zeno subspace $n_2 = 1$, it is impossible to solve the full dynamics by just considering these Fock states alone as they are not coupled to each other in $\hat{H}_0$. The components lying outside of the Zeno subspace have to be included to allow intermediate steps to occur via states that do not belong in this subspace, much like virtual states in optical Raman transitions. An approximate solution for $U=0$ can be written for the $\{|2,1,0 \rangle, |1,1,1 \rangle, |0,1,2 \rangle\}$ subspace by multiplying each eigenvector with its corresponding time evolution \begin{equation} | \Psi(t) \rangle \propto \left( \begin{array}{c} z_1 + \sqrt{2} z_2 e^{-6 J^2 t / \gamma} + z_3 e^{-12 J^2 t / \gamma} \\ -\sqrt{2} \left(z_1 - z_3 e^{-12 J^2 t / \gamma} \right) \\ z_1 - \sqrt{2} z_2 e^{-6 J^2 t / \gamma} + z_3 e^{-12 J^2 t / \gamma} \\ \end{array} \right), \end{equation} where $z_i$ denote the overlap between the eigenvectors and the initial state, $z_i = \langle v_i | \Psi (0) \rangle$, with $| v_1 \rangle = (1, -\sqrt{2}, 1)/2$, $| v_2 \rangle = (1, 0, -1)/\sqrt{2}$, and $| v_3 \rangle = (1, \sqrt{2}, 1)/2$. The steady state as $t \rightarrow \infty$ is given by $| v_1 \rangle = (1, -\sqrt{2}, 1)/2$. This solution is illustrated in Fig. \ref{fig:comp} which clearly demonstrates dynamics beyond the canonical understanding of quantum Zeno dynamics as tunnelling occurs between states coupled via a different Zeno subspace. \begin{figure}[hbtp!] \includegraphics[width=\linewidth]{comp} \caption[Fock State Populations in a Zeno Subspace]{Populations of the Fock states in the Zeno subspace for $\gamma/J = 100$ and initial state $| 2,1,0 \rangle$. It is clear that quantum Zeno dynamics occurs via Raman-like processes even though none of these states are connected in $\hat{H}_0$. The dynamics occurs via virtual intermediate states outside the Zeno subspace. The system also tends to a steady state which minimises tunnelling effectively suppressing fluctuations. The lines are solutions to the non-Hermitian Hamiltonian, and the dots are points from a stochastic trajectory calculation.\label{fig:comp}} \end{figure} \subsection{Steady State of non-Hermitian Dynamics} A distinctive difference between Bose-Hubbard model ground states and the final steady state, $| \Psi \rangle = [|2,1,0 \rangle - \sqrt{2} |1,1,1\rangle + |0,1,2\rangle]/2$, is that its components are not in phase. Squeezing due to measurement naturally competes with inter-site tunnelling which tends to spread the atoms. However, from Eq. \eqref{eq:nHH2} we see the final state will always be the eigenvector with the smallest fluctuations as it will have an eigenvalue with the largest imaginary component. This naturally corresponds to the state where tunnelling between Zeno subspaces (here between every site) is minimised by destructive matter-wave interference, i.e.~the tunnelling dark state defined by $\hat{T} |\Psi \rangle = 0$, where $\hat{T} = \sum_{\langle i, j \rangle} \bd_i b_j$. This is simply the physical interpretation of the steady states we predicted for Eq. \eqref{eq:master}. Crucially, this state can only be reached if the dynamics are not fully suppressed by measurement and thus, counter-intuitively, the atomic dynamics cooperate with measurement to suppress itself by destructive interference. Therefore, this effect is beyond the scope of traditional quantum Zeno dynamics and presents a new perspective on the competition between a system's short-range dynamics and global measurement backaction. We now consider a one-dimensional lattice with $M$ sites so we extend the measurement to $\hat{D} = \N_\text{even}$ where every even site is illuminated. The wave function in a Zeno subspace must be an eigenstate of $\c$ and we combine this with the requirement for it to be in the dark state of the tunnelling operator (eigenstate of $\H_0$ for $U = 0$) to derive the steady state. These two conditions in momentum space are \begin{align} \hat{T} | \Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_k - \bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0, \\ \Delta \N |\Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_{-q} + \bd_{-q} b_k \right] | \Psi \rangle= \Delta N |\Psi \rangle, \end{align} where $b_k = \frac{1}{\sqrt{M}} \sum_j e^{i k j a} b_j$, $\Delta \hat{N} = \hat{D} - N/2$, $q = \pi/a - k$, $a$ is the lattice spacing, $N$ the total atom number, and we perform summations over the reduced Brillouin zone (RBZ), $-\pi/2a < k \le \pi/2a$, as the symmetries of the system are clearer this way. Now we define \begin{equation} \hat{\alpha}_k^\dagger = \bd_k \bd_q - \bd_{-k} \bd_{-q}, \end{equation} \begin{equation} \hat{\beta}_\varphi^\dagger = \bd_{\pi/2a} + \varphi \bd_{-\pi/2a}, \end{equation} where $\varphi = \Delta N / | \Delta N |$, which create the smallest possible states that satisfy the two equations for $\Delta N = 0$ and $\Delta N \ne 0$ respectively. Therefore, by noting that \begin{align} \left[ \hat{T}, \hat{\alpha}_k^\dagger \right] & = 0, \\ \left[ \hat{T}, \hat{\beta}_\varphi^\dagger \right] & = 0, \\ \left[ \Delta \N, \hat{\alpha}_k^\dagger \right] & = 0, \\ \left[ \Delta \N, \hat{\beta}_\varphi^\dagger \right] & = \varphi \hat{\beta}_\varphi^\dagger, \end{align} we can now write the equation for the $N$-particle steady state \begin{equation} \label{eq:ss} | \Psi \rangle \propto \left[ \prod_{i=1}^{(N - |\Delta N|)/2} \left( \sum_{k = 0}^{\pi/2a} \phi_{i,k} \hat{\alpha}_k^\dagger \right) \right] \left( \hat{\beta}_\varphi^\dagger \right)^{| \Delta N |} | 0 \rangle, \end{equation} where $\phi_{i,k}$ are coefficients that depend on the trajectory taken to reach this state and $|0 \rangle$ is the vacuum state defined by $b_k |0 \rangle = 0$. Since this a dark state (an eigenstate of $\H_0$) of the atomic dynamics, this state will remain stationary even with measurement switched-off. Interestingly, this state is very different from the ground states of the Bose-Hubbard Hamiltonian, it is even orthogonal to the superfluid state, and thus it cannot be obtained by cooling or projecting from an initial ground state. The combination of tunnelling with measurement is necessary. \begin{figure}[hbtp!] \includegraphics[width=\linewidth]{steady} \caption[Non-Hermitian Steady State]{A trajectory simulation for eight atoms in eight sites, initially in $|1,1,1,1,1,1,1,1 \rangle$, with periodic boundary conditions and $\gamma/J = 100$. (a), The fluctuations in $\c$ where the stochastic nature of the process is clearly visible on a single trajectory level. However, the general trend is captured by the non-Hermitian Hamiltonian. (b), The local density variance. Whilst the fluctuations in the global measurement operator decrease, the fluctuations in local density increase due to tunnelling via states outside the Zeno subspace. (c), The momentum distribution. The initial Fock state has a flat distribution which with time approaches the steady state distribution of two identical and symmetric distributions centred at $k = \pi/2a$ and $k = -\pi/2a$.\label{fig:steady}} \end{figure} In order to prepare the steady state one has to run the experiment and wait until the photocount rate remains constant for a sufficiently long time. Such a trajectory is illustrated in Fig. \ref{fig:steady} and compared to a deterministic trajectory calculated using the non-Hermitian Hamiltonian. It is easy to see from Fig. \ref{fig:steady}(a) how the stochastic fluctuations around the mean value of the observable have no effect on the general behaviour of the system in the strong measurement regime. By discarding these fluctuations we no longer describe a pure state, but we showed how this only leads to a negligible error. Fig. \ref{fig:steady}(b) shows the local density variance in the lattice. Not only does it grow showing evidence of tunnelling between illuminated and non-illuminated sites, but it grows to significant values. This is in contrast to conventional quantum Zeno dynamics where no tunnelling would be allowed at all. Finally, Fig. \ref{fig:steady}(c) shows the momentum distribution of the trajectory. We can clearly see that it deviates significantly from the initial flat distribution of the Fock state. Furthermore, the steady state does not have any atoms in the $k=0$ state and thus is orthogonal to the superfluid state as discussed. To obtain a state with a specific value of $\Delta N$ postselection may be necessary, but otherwise it is not needed. The process can be optimised by feedback control since the state is monitored at all times \cite{ivanov2014, mazzucchi2016feedback, ivanov2016}. Furthermore, the form of the measurement operator is very flexible and it can easily be engineered by the geometry of the optical setup \cite{elliott2015, mazzucchi2016} which can be used to design a state with desired properties. \section{Conclusions} In this chapter we have demonstrated that global quantum measurement backaction can efficiently compete with standard local processes in many-body systems. This introduces a completely new energy and time scale into quantum many-body research. This is made possible by the ability to structure the spatial profile of the measurement on a microscopic scale comparable to the lattice period without the need for single site addressing. The extreme flexibility of the setup considered allowed us to effectively tailor long-range entanglement and correlations present in the system. We showed that the competition between the global backaction and usual atomic dynamics leads to the production of spatially multimode macroscopic superpositions which exhibit large-scale oscillatory dynamics which could be used for quantum information and metrology. We subsequently demonstrated that when on-site atomic interactions are introduced the dynamics become much more complicated with different regimes of behaviour where measurement and interactions can either compete or cooperate. In the strong measurement regime we showed that conventional quantum Zeno dynamics can be realised, but more interestingly, by considering a strong, but not projective, limit of measurement we observe a new type of nonlocal dynamics. It turns out that a global measurement scheme leads to correlations between spatially separated tunnelling events which conserve the Zeno subspace via Raman-like processes which would be forbidden in the canonical fully projective limit. We subsequently presented a rigorous analysis of the underlying process of this new type of quantum Zeno dynamics in which we showed that in this limit quantum trajectories can be described by a deterministic non-Hermitian Hamiltonian. In contrast to previous works, it is independent of the underlying system and there is no need to postselect a particular exotic trajectory \cite{lee2014prx, lee2014prl}. Finally, we have shown that the system will always tend towards the eigenstate of the Hamiltonian with the best squeezing of the observable and the atomic dynamics, which normally tend to spread the distribution, cooperates with measurement to produce a state in which tunnelling is suppressed by destructive matter-wave interference. A dark state of the tunnelling operator will have zero fluctuations and we provided an expression for the steady state which is significantly different from the ground state of the Hamiltonian. This is in contrast to previous works on dissipative state preparation where the steady state had to be a dark state of the measurement operator \cite{diehl2008}. Such globally paired tunnelling due to a fundamentally new phenomenon, global quantum measurement backaction, can enrich the physics of long-range correlated systems beyond relatively short-range interactions expected from standard dipole-dipole interactions \cite{sowinski2012, omjyoti2015}. These nonlocal high-order processes entangle regions of the optical lattice that are disconnected by the measurement. Using different detection schemes, we showed how to tailor density-density correlations between distant lattice sites. Quantum optical engineering of nonlocal coupling to environment, combined with quantum measurement, can allow the design of nontrivial system-bath interactions, enabling new links to quantum simulations~\cite{stannigel2013} and thermodynamics~\cite{erez2008} and extend these directions to the field of non-Hermitian quantum mechanics, where quantum optical setups are particularly promising~\cite{lee2014prl}. Importantly, both systems and baths, designed by our method, can be strongly correlated systems with internal long-range entanglement.