%******************************************************************************* %*********************************** Fifth Chapter ***************************** %******************************************************************************* \chapter{Density Measurement Induced Dynamics} % 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. We have also wrapped our quantum gas model in this quantum trajectory 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. In this chapter, we investigate the effect of quantum measurement backaction on the many-body state of atoms. In particular, we will focus on the competition between the backaction and the 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 micrscopic scalecomparable to the lattice period without the need for single site resolution enebales us to engineer efficient competition between the three processes in order to generate new nontrivial dynamics. Furthermore, the global nature of the measurement creates long-range correlations which enable nonlocal dynamical processes distinguishing it from the local processes. 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 in the picture 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. One can also devise measurements with multi-dimensional projections which lead to quantum Zeno dynamics where unitary evolution is uninhibited within this degenrate subspace, i.e.~the Zeno subspace. The 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 of quantum Zeno dynamics 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 considered here), be approimated 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. The measurement process generates spatial modes of matter fields that can be considered as designed systems and reservoirs opening the possibility of controlling dissipations in ultracold atomic systems without resorting to atom losses and collisions which are difficult to manipulate. The continuous measurement of the light field introduces a controllable decoherence channel into the many-body dynamics. Furthermore, global light scattering from multiple lattice sites creates nontrivial spatially nonlocal coupling to the environment which is impossible to obtain with local interactions. 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. \section{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 thus making it impossible to establish quantum Zeno Dynamics. 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. Fig. \ref{fig:oscillations} illustrates the atom number distributions in one of the modes for $Z = 2$ ($N_\mathrm{odd}$) and $Z = 3$ ($N_1$) \cite{mazzucchi2016}. In the absence of the external influence of measurement these distributions would spread out significantly and oscillate with an amplitude that is less than or equal to the initial imbalance, i.e.~small oscillations for a small initial imbalance. By contrast, here we observe a macroscopic exchange of atoms between the modes even in the absence of an initial imbalance, that the distributions consist of a small number of well defined components, and these components are squeezed by the weak measurement. \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 dstributions 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} Furthermore, depending on the quantity being addressed by the measurement, the state of the system has multiple components as seen in Figs. \ref{fig:oscillations}b and \ref{fig:oscillations}c. This is a consequence of the fact that the measured light intensity $\ad_1 \a_1$ is not sensitive to the light phase. The measurement will not distinguish between all permutations of mode occupations that scatter light with the same intensity, but 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} noting that they can be represented as the sum of 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 $Z = 2$ 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 three visible components. It has also been shown in Ref. \cite{mazzucchi2016njp} that the non-interacting dynamics with quantum measurement backaction for $R$-modes reduce to an effective Bose-Hubbard Hamiltonian with $R$-sites provided the initial state is a superfluid. In this simplified model the $N_j$ atoms in the $j$th site corresponds to a superfluid of $N_j$ atoms within a single spatial mode as defined by Eq. \eqref{eq:Zmodes}. Furthermore, the tunnelling term in the Bose-Hubbard model and the quantum jumps do not affect this correspondence. Therefore, we will now consider an illumination pattern with $\hat{D} = \hat{N}_\mathrm{odd}$. 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. Fig. \ref{fig:oscillations}a shows that this leads to macroscopic oscillations with a single peak. We will now attempt to get some physical insight into the process by using the 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^\mathrm{cl} \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-hand site in 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-hand 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-hand site. Even though Eq. \eqref{eq:doublewell} is relatively simple as it 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 be easily applied to our system as well. In the limit of large atom number, the wavefunction 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 $\Lambda = NU / (2J^\mathrm{cl})$. We will 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 we have assumed that $z \ll 1$ 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 wavefunction $\psi$. 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 wavefunction 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 center, and $\phi$ and $z_\phi$ contain the phase information 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 derive the effect of a single quantum jump. Within the continuous variable approximation, our quantum jump become $\c \propto 1 + z$. We neglect the constant prefactors, because the wavefunction 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 wavefunction in Eq. \eqref{eq:ansatz} with the jump operator above yields a Gaussian wavefunction as well, but the parameters change discontinuously according to \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. \end{align} The fact that the wavefunction 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 discontinous 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 wavefunction in between the photodetecions. The initial values of the parameters for a superfluid state of $N$ atoms across the whole lattice are $b^2 = 2h$, $\phi =0$, $a_0 = 0$, and $a_\phi = 0$. Howver, we use the most general initial conditions at time $t = t_0$ which we denote by $b(t_0) = b_0$, $\phi(t_0) = \phi_0$, $z_0(t_0) = a_0$, and $z_\phi(t_0) = a_\phi$. 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 paramater values as the new initial conditions. By plugging the ansatz in Eq. \eqref{eq:ansatz} into the 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 $x(0) = x_0 = 1/b_0^2$, $p(0) = p_0 = (1 - i \phi_0)/b_0^2$, $q(0) = q_0 = (a_0 - \phi_0 a_\phi)/b_0^2$, and $r(0) = r_0 = (a_0^2 - \phi_0 a_\phi^2)/b_0^2$. 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 $\epsilon$ is 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 $\epsilon$ as it is only related to the global phase and the norm of the wavefunction 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}. Eq. \eqref{eq:p} 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], \end{equation} which lets us rewrite Eq. \eqref{eq:pq} as \begin{equation} \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} The integrating factor can be evaluated and shown to be \begin{equation} I(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} which upon substitution into Eq. \eqref{eq:Iq} yields 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 $\Gamma = 0$. It may seem counter-intuitive to neglect the term that appears due to measurement, but we are considering the weak measurement regime where $\gamma \ll J^\mathrm{cl}$ and thus the dynamics between the quantum jumps are actually dominated by the tunnelling of atoms rather than the null outcomes. 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 wavefunction and we see that it oscillates with an amplitude $\sqrt{a_0^2 + 16 h^2 \phi_0^2 (a_0 - a_\phi)^2 / (b_0^4 \omega^2)}$. For these oscillations to occur, $a_0$ and $a_\phi$ cannot be zero, but this is exactly the case for an initial superfluid state. However, we have seen in Eq. \eqref{eq:jumpz0} that the effect of a photodetection is to displace the wavepacket by approximately $b^2$, i.e.~the width of the Gaussian, in the direction of the positive $z$-axis. Therefore, it is the quantum jumps that are the driving force behind this phenomenon. The oscillations themselves are essentially due to the natural dynamics of the atoms in a lattice, but it is the measurement that causes the initial displacement. Furthermore, since the quantum jumps occur at an average instantaneous rate proportional to $\langle \cd \c \rangle (t)$ which itself is proportional to $(1+z)^2$ they are most likely to occur at the point of maximum displacement in the positive $z$ direction at which point a quantum jump further increases the amplitude of the wavefunction leading to the growth seen in Fig. \ref{fig:oscillations}a. 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. Due to the complexity of the equations in the case of $\Gamma \ne 0$ our analysis will be less rigoruous and we will focus on the qualitative aspects of the dynamics. 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. Therefore, it would be worthwhile to look at the long time limit. Unfortunately, since all the quantities are oscillatory a long time limit is fairly meaningless especially since the quantum jumps provide a driving force leading to larger and larger oscillations. However, the width of the Gaussian, $b^2$, is unique in that it doesn't 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 monotonically with time, 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 and so $\Gamma^2 / \omega^4 \ll 1$. $b^2_\mathrm{SF} = 2h$ 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. However, if we substitute the parameter values from Fig. \ref{fig:oscillations}a we only get a reduction in width by about $3\%$. The maximum amplitude oscillations in Fig. \ref{fig:oscillations}a 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}. However, what we do see is that during the early stages of the trajectory, which should be well described by this model, is that the width does not in fact shrink by much. It is only in the later stages when the oscillations reach maximal amplitude that the width becomes visibly reduced. \section{Three-Way Competition} \section{Emergent Long-Range Correlated Tunnelling} \section{Non-Hermitian Dynamics in the Quantum Zeno Limit} % Contrast with t-J model here how U localises events, but measurement % does the opposite \section{Steady-State of the Non-Hermitian Hamiltonian} \section{Conclusions}