%******************************************************************************* %*********************************** Sixth Chapter ***************************** %******************************************************************************* \chapter[Phase Measurement Induced Dynamics] {Phase Measurement Induced Dynamics\footnote{The results of this chapter were first published in Ref. \cite{kozlowski2016phase}}} % Title of the Sixth Chapter \ifpdf \graphicspath{{Chapter6/Figs/Raster/}{Chapter6/Figs/PDF/}{Chapter6/Figs/}} \else \graphicspath{{Chapter6/Figs/Vector/}{Chapter6/Figs/}} \fi \section{Introduction} Light scatters due to its interaction with the dipole moment of the atoms which for off-resonant light results in an effective coupling with atomic density, not the matter-wave amplitude. Therefore, it is challenging to couple light to the phase of the matter-field, as is typical in quantum optics for optical fields. In the previous chapter we only considered measurement that couples directly to atomic density operators just like most of the existing work \cite{mekhov2012, LP2009, rogers2014, ashida2015, ashida2015a}. However, we have shown in section \ref{sec:B} that it is possible to couple to the relative phase differences between sites in an optical lattice by illuminating the bonds between them. Furthermore, we have also shown how it can be applied to probe the Bose-Hubbard order parameter or even matter-field quadratures in Chapter \ref{chap:qnd}. This concept has also been applied to the study of quantum optical potentials formed in a cavity and shown to lead to a host of interesting quantum phase diagrams \cite{caballero2015, caballero2015njp, caballero2016, caballero2016a}. This is a multi-site generalisation of previous double-well schemes \cite{cirac1996, castin1997, ruostekoski1997, ruostekoski1998, rist2012}, although the physical mechanism is fundamentally different as it involves direct coupling to the interference terms caused by atoms tunnelling rather than combining light scattered from different sources. We have already mentioned that there are three primary avenues in which the field of quantum optics with ultracold gases can be taken: nondestructive measurement, quantum measurement backaction, and quantum optical potentials. All three have been covered in the context of density-based measurement either here or in other works. However, coupling to phase observables in lattices has only been proposed and considered in the context of nondestructive measurements (see Chapter \ref{chap:qnd}) and quantum optical potentials \cite{caballero2015, caballero2015njp, caballero2016, caballero2016a}. In this chapter, we go in a new direction by considering the effect of measurement backaction on the atomic gas that results from such coupling. We investigate this mechanism using light scattered from these phase-related observables. The novel combination of measurement backaction as the physical mechanism driving the dynamics and phase coherence as the observable to which the optical fields couple to provides a completely new opportunity to affect and manipulate the quantum state. We first show how this scheme enables us to prepare energy eigenstates of the lattice Hamiltonian. Furthermore, in the second part we also demonstrate a novel type of a projection due to measurement which occurs even when there is significant competition with the Hamiltonian dynamics. This projection is fundamentally different to the standard formulation of the Copenhagen postulate projection or the quantum Zeno effect \cite{misra1977, facchi2008, raimond2010, raimond2012, signoles2014} thus providing an extension of the measurement postulate to dynamical systems subject to weak measurement. \section{Diffraction Maximum and Energy Eigenstates} In this chapter we will only consider measurement backaction when $\a_1 = C \hat{B}$ and $\c = \sqrt{2 \kappa} \a_1$ as introduced in section \ref{sec:B}. We have seen that there are two ways of engineering the spatial profile of the measurement such that the density contribution from $\hat{D}$ is suppressed. We will first consider the case when the profile is uniform, i.e.~the diffraction maximum of scattered light, when our measurement operator is given by \begin{equation} \B = \Bmax = J^B_\mathrm{max} \sum_j^K \left( \bd_j b_{j+1} + b_j \bd_{j+1} \right) = 2 J^B_\mathrm{max} \sum_k \bd_k b_k \cos(ka), \end{equation} where the second equality follows from converting to momentum space, denoted by index $k$, via $b_m = \frac{1}{\sqrt{M}} \sum_k e^{-ikma} b_k$ and $b_k$ annihilates an atom in the Brillouin zone. Note that this operator is diagonal in momentum space which means that its eigenstates are simply momentum Fock states. We have also seen in Chapter \ref{chap:backaction} how the global nature of the jump operators introduces a nonlocal quadratic term to the Hamiltonian, $\hat{H} = \hat{H}_0 - i \cd \c / 2$. In order to focus on the competition between tunnelling and measurement backaction we again consider non-interacting atoms, $U = 0$. Therefore, $\B$ is proportional to the Hamiltonian and both operators have the same eigenstates. The combined Hamiltonian for the non-interacting gas subject to measurement of the from $\a_1 = C \B$ is thus \begin{equation} \hat{H} = -J \sum_j \left( \bd_j b_{j+1} + b_j \bd_{j+1} \right) - i \gamma \Bmax^\dagger \Bmax. \end{equation} Furthermore, whenever a photon is detected, the operator $\c = \sqrt{2 \kappa} C \Bmax$ is applied to the wave function. We can rewrite the above equation as \begin{equation} \hat{H} = -\frac{J}{J^B_\mathrm{max}} \Bmax - i \gamma \Bmax^\dagger \Bmax. \end{equation} The eigenstates of this operator will be exactly the same as of the isolated Hamiltonian which we will label as $| h_l \rangle$, where $h_l$ denotes the corresponding eigenvalue. Therefore, for an initial state \begin{equation} | \Psi \rangle = \sum_l z_l^0 | h_l \rangle \end{equation} it is easy to show that the state after $m$ photocounts and time $t$ is given by \begin{equation} | \Psi (m,t) \rangle = \frac{1}{\sqrt{F(t)}} \sum_l h_l^m z_l^0 e^{[i h_l - \gamma(J^B_\mathrm{max}/J)^2 h_l^2]t} | h_l \rangle, \end{equation} where $\sqrt{F(t)}$ is the normalisation factor. Therefore, the probability of being in state $| h_l \rangle$ at time $t$ after $m$ photocounts is given by \begin{equation} p(h_l, m, t) = \frac{h_l^{2m}} {F(t)} \exp\left[ - 2 \gamma \left( \frac{J^B_\mathrm{max}} {J} \right)^2 h_l^2 t \right] |z_l^0|^2. \end{equation} As a lot of eigenstates are degenerate, we are actually more interested in the probability of being in eigenspace with the eigenvalue $h_l = (J/J^B_\mathrm{max})B_\mathrm{max}$. This probability is given by \begin{equation} \label{eq:bmax} p(B_\mathrm{max}, m, t) = \frac{B_\mathrm{max}^{2m}} {F(t)} \exp\left[ - 2 \gamma B_\mathrm{max}^2 t \right] p_0 (B_\mathrm{max}), \end{equation} where $p_0(B_\mathrm{max}) = \sum_{J_\mathrm{max} h_l = J B_\mathrm{max}} |z_l^0|^2$. This distribution will have two distinct peaks at $B_\mathrm{max} = \pm \sqrt{m/2\kappa |C|^2 t}$ and an initially broad distribution will narrow down around these two peaks with successive photocounts. The final state is in a superposition, because we measure the photon number, $\ad_1 \a_1$ and not field amplitude. Therefore, the measurement is insensitive to the phase of $\a_1 = C \B$ and we get a superposition of $\pm B_\mathrm{max}$. This is exactly the same situation that we saw for the macroscopic oscillations of two distinct components when the atom number difference between two modes is measured as seen in Fig. \ref{fig:oscillations}(b). However, this means that the matter is still entangled with the light as the two states scatter light with different phase which the photocount detector cannot distinguish. Fortunately, this is easily mitigated at the end of the experiment by switching off the probe beam and allowing the cavity to empty out or by measuring the light phase (quadrature) to isolate one of the components \cite{mekhov2012, mekhov2009pra, atoms2015}. Unusually, we do not have to worry about the timing of the quantum jumps, because the measurement operator commutes with the Hamiltonian. This highlights an important feature of this measurement - it does not compete with atomic tunnelling, and represents a quantum non-demolition (QND) measurement of the phase-related observable \cite{brune1992}. Eq. \eqref{eq:bmax} shows that regardless of the initial state or the photocount trajectory the system will project onto a superposition of eigenstates of the $\Bmax$ operator. In fact, the final state probability distribution would be exactly the same if we were to simply employ a projective measurement of the same operator. What is unusual in our case is that this has been achieved with weak measurement in the presence of significant atomic dynamics. The conditions for such a measurement have only recently been rigorously derived in Ref. \cite{weinberg2016} and here we provide a practical realisation using in ultracold bosonic gases. This projective behaviour is in contrast to conventional density based measurements which squeeze the atom number in the illuminated region and thus are in direct competition with the atom dynamics (which spreads the atoms), effectively requiring strong couplings for a projection as seen in the previous chapter. Here a projection is achieved at any measurement strength which allows for a weaker probe and thus effectively less heating and a longer experimental lifetime. This is in contrast to the quantum Zeno effect which requires a very strong probe to compete effectively with the Hamiltonian \cite{misra1977, facchi2008, raimond2010, raimond2012, signoles2014}. Interestingly, the $\Bmax$ measurement will even establish phase coherence across the lattice, $\langle \bd_j b_j \rangle \ne 0$, in contrast to density based measurements where the opposite is true: Fock states with no coherences are favoured. \section{General Model for Weak Measurement Projection} \subsection{Projections for Incompatible Dynamics and Measurement} In section \ref{sec:B} we have also shown that it is possible to achieve a more complicated spatial profile of the $\B$-measurement. The optical geometry can be adjusted such that each bond scatters light in anti-phase with its neighbours leading to a diffraction minimum where the expectation value of the amplitude is zero. In this case the $\B$ operator is given by \begin{equation} \B = \Bmin = J^B_\mathrm{min} \sum_m^K (-1)^m \hat{B}_m = 2 i J^B_\mathrm{min} \sum_k \bd_k b_{k - \pi/a} \sin(ka). \end{equation} Note how the measurement operator now couples the momentum mode $k$ with mode $k - \pi/a$. This measurement operator no longer commutes with the Hamiltonian so it is no longer QND and we do not expect there to be a steady state as before. In order to understand the measurement it will be easier to work in a basis in which it is diagonal. We perform the transformation $\beta_k = \frac{1}{\sqrt{2}} \left( b_k + i b_{k - \pi/a} \right)$, $\tilde{\beta}_k = \frac{1}{\sqrt{2}} \left( b_k - i b_{k - \pi/a} \right)$, which yields the following forms of the measurement operator and the Hamiltonian: \begin{equation} \label{eq:BminBeta} \Bmin = 2 J^B_\mathrm{min} \sum_{\mathrm{RBZ}} \sin(ka) \left( \beta^\dagger_k \beta_k - \tilde{\beta}_k^\dagger \tilde{\beta}_k \right), \end{equation} \begin{equation} \label{eq:H0Beta} \hat{H}_0 = - 2 J \sum_{\mathrm{RBZ}} \cos(ka) \left( \beta_k^\dagger \tilde{\beta}_k + \tilde{\beta}^\dagger_k \beta_k \right), \end{equation} where the summations are performed over the reduced Brillouin Zone (RBZ), $0 < k \le \pi/a$, to ensure the transformation is canonical. We see that the measurement operator now consists of two types of modes, $\beta_k$ and $\tilde{\beta_k}$, which are superpositions of two momentum states, $k$ and $k - \pi/a$. Note how a spatial pattern with a period of two sites leads to a basis with two modes whilst a uniform pattern had only one mode, $b_k$. Furthermore, note the similarities to $\D = \Delta \hat{N} = \hat{N}_\mathrm{even} - \hat{N}_\mathrm{odd}$ which is the density measurement operator obtained by illuminated the lattice such that neighbouring sites scatter light in anti-phase. This further highlights the importance of geometry for global measurement. Trajectory simulations confirm that there is no steady state. However, unexpectedly, for each trajectory we observe that the dynamics always ends up confined to some subspace as seen in Fig. \ref{fig:projections}. In general, this subspace is neither an eigenspace of the measurement operator or the Hamiltonian. In Fig. \ref{fig:projections}(b) it in fact clearly consists of multiple measurement eigenspaces. This clearly distinguishes it from the fundamental projections predicted by the Copenhagen postulates. It is also not the quantum Zeno effect which predicts that strong measurement can confine the evolution of a system as this subspace must be an eigenspace of the measurement operator \cite{misra1977, facchi2008, raimond2010, raimond2012, signoles2014}. Furthermore, the projection we see in Fig. \ref{fig:projections} occurs for even weak measurement strengths compared to the Hamiltonian's own evolution, a regime in which the quantum Zeno effect does not happen. It is also possible to dissipatively prepare quantum states in an eigenstate of a Hamiltonian provided it is also a dark state of the jump operator, $\c | \Psi \rangle = 0$~\cite{diehl2008}. However, this is also clearly not the case as the final state in Fig. \ref{fig:projections}(c) is not only not confined to a single measurement operator eigenspace, it also spans multiple Hamiltonian eigenspaces. Therefore, the dynamics induced by $\a_1 = C\Bmin$ project the system into some subspace, but since this does not happen via any of the mechanisms described above it is not immediately obvious what this subspace is. \begin{figure}[hbtp!] \includegraphics[width=\linewidth]{Projections} \caption[Projections for Non-Commuting Observable and Hamiltonian] {Subspace projections. Projection to a $\mathcal{P}_M$ space for four atoms on eight sites with periodic boundary conditions. The parameters used are $J=1$, $U=0$, $\gamma = 0.1$, and the initial state was $| 0,0,1,1,1,1,0,0 \rangle$. (a) The $\langle \hat{O}_k \rangle = \langle \n_k + \n_{k - \pi/a} \rangle$ distribution reaches a steady state at $Jt \approx 8$ indicating the system has been projected. (b) Populations of the $\Bmin$ eigenspaces. (c) Population of the $\hat{H}_0$ eigenspaces. Once the projection is achieved at $Jt\approx8$ we can see from (b-c) that the system is not in an eigenspace of either $\Bmin$ or $\hat{H}_0$, but it becomes confined to some subspace. The system has been projected onto a subspace, but it is neither that of the measurement operator or the Hamiltonian. \label{fig:projections}} \end{figure} To understand this dynamics we will look at the master equation for open systems described by the density matrix, $\hat{\rho}$, \begin{equation} \dot{\hat{\rho}} = -i \left[\hat{H}_0, \hat{\rho} \right] + \left[ \c \hat{\rho} \cd - \frac{1}{2} \left( \cd \c \hat{\rho} + \hat{\rho} \cd \c \right) \right]. \end{equation} The following results will not depend on the nature nor the exact form of the jump operator $\c$. However, whenever we refer to our simulations or our model we will be considering $\c = \sqrt{2 \kappa} C(\D + \B)$ as before, but the results are more general and can be applied to their setups. This equation describes the state of the system if we discard all knowledge of the outcome. The commutator describes coherent dynamics due to the isolated Hamiltonian and the remaining terms are due to measurement. This is a convenient way to find features of the dynamics common to every measurement trajectory. Just like in the preceding chapter we define the projectors of the measurement eigenspaces, $P_m$, which have no effect on any of the (possibly degenerate) eigenstates of $\c$ with eigenvalue $c_m$, but annihilate everything else, thus $P_m = \sum_{c_n = c_m} | c_n \rangle \langle c_n |,$ where $| c_n \rangle$ is an eigenstate of $\c$ with eigenvalue $c_n$. Note that in the specific case of our quantum gas model $\c = \sqrt{2 \kappa} C(\D + \B)$ so these projectors act on the matter state. Recall from the previous chapter that this allows us to decompose the master equation in terms of the measurement basis as a series of equations $P_m \dot{\hat{\rho}} P_n$. We have seen that for $m = n$ we obtain decoherence free subspaces, $P_m \dot{\hat{\rho}} P_m = -i P_m \left[\hat{H}_0, \hat{\rho} \right] P_m$, where the measurement terms disappear which shows that a state in a single eigenspace is unaffected by observation. On the other hand, for $m \ne n$ the Hamiltonian evolution actively competes against measurement. In general, if $\c$ does not commute with the Hamiltonian then a projection to a single eigenspace $P_m$ is impossible unless the measurement is strong enough for the quantum Zeno effect to occur. We now go beyond what we previously did and define a new type of projector \begin{equation} \mathcal{P}_M = \sum_{m \in M} P_m, \end{equation} such that \begin{equation} \mathcal{P}_M \mathcal{P}_N = \delta_{M,N} \mathcal{P}_M \end{equation} \begin{equation} \sum_M \mathcal{P}_M = \hat{1} \end{equation} where $M$ denotes some arbitrary subspace. The first equation implies that the subspaces can be built from $P_m$ whilst the second and third equation are properties of projectors and specify that these projectors do not overlap and that they cover the whole Hilbert space. Furthermore, we will also require that \begin{equation} [\mathcal{P}_M, \hat{H}_0 ] = 0, \end{equation} \begin{equation} [\mathcal{P}_M, \c] = 0. \end{equation} The second commutator simply follows from the definition $\mathcal{P}_M = \sum_{m \in M} P_m$, but the first one is non-trivial. However, if we can show that $\mathcal{P}_M = \sum_{m \in M} | h_m \rangle \langle h_m |$, where $| h_m \rangle$ is an eigenstate of $\hat{H}_0$ then the commutator is guaranteed to be zero. This is a complex set of requirements and it is unclear if it is possible to satisfy all of them at once. However, we note that there exists a trivial case where all these conditions are satisfied and that is when there is only one such projector $\mathcal{P}_M = \hat{1}$. Assuming that it is possible to have non-trivial cases where $\mathcal{P}_M \ne \hat{1}$ we can write the master equation as \begin{equation} \label{eq:bigP} \mathcal{P}_M \dot{\hat{\rho}} \mathcal{P}_N = -i \left[\hat{H}_0, \mathcal{P}_M \hat{\rho} \mathcal{P}_N \right] + \left[ \c \mathcal{P}_M \hat{\rho} \mathcal{P}_N \cd - \frac{1}{2} \left( \cd \c \mathcal{P}_M \hat{\rho} \mathcal{P}_N + \mathcal{P}_M \hat{\rho} \mathcal{P}_N \cd \c \right) \right]. \end{equation} Crucially, thanks to the commutation relations we were able to divide the density matrix in such a way that each submatrix's time evolution depends only on itself. Every submatrix $\mathcal{P}_M \dot{\hat{\rho}} \mathcal{P}_N$ depends only on the current state of itself and its evolution is ignorant of anything else in the total density matrix. This is in contrast to the partitioning we achieved with $P_m$. Previously we only identified subspaces that were decoherence free, i.e.~unaffected by measurement. However, those submatrices could still couple with the rest of the density matrix via the coherent term $P_m [\hat{H}_0, \hat{\rho}] P_n$. We note that when $M = N$ the equations for $\mathcal{P}_M \hat{\rho} \mathcal{P}_M$ will include decoherence free subspaces, i.e.~$P_m \hat{\rho} P_m$. Therefore, parts of the $\mathcal{P}_M \hat{\rho} \mathcal{P}_M$ submatrices will also remain unaffected by measurement. However, the submatrices $\mathcal{P}_M \hat{\rho} \mathcal{P}_N$, for which $M \ne N$, are guaranteed to not contain any of these measurement free subspaces thanks to the orthogonality of $\mathcal{P}_M$. Therefore, for $M \ne N$ all elements of $\mathcal{P}_M \hat{\rho} \mathcal{P}_N$ will experience a non-zero measurement term whose effect is always dissipative/lossy. Furthermore, these coherence submatrices $\mathcal{P}_M \hat{\rho} \mathcal{P}_N$ are not coupled to any other part of the density matrix as seen from Eq. \eqref{eq:bigP} and so they can never increase in magnitude; the remaining coherent evolution is unable to counteract the dissipative term without an `external pump' from other parts of the density matrix. The combined effect is such that all $\mathcal{P}_M \hat{\rho} \mathcal{P}_N$ for which $M \ne N$ will always go to zero due to dissipation. When all these cross-terms vanish, we are left with a density matrix that is a mixed state of the form $\hat{\rho} = \sum_M \mathcal{P}_M \hat{\rho} \mathcal{P}_M$. Since there are no coherences, $\mathcal{P}_M \hat{\rho} \mathcal{P}_N$, this state contains only classical uncertainty about which subspace, $\mathcal{P}_M$, is occupied - there are no quantum superpositions between different $\mathcal{P}_M$ spaces. Therefore, in a single measurement run we are guaranteed to have a state that lies entirely within a subspace defined by $\mathcal{P}_M$. This is once again analogous to the qubit example from section \ref{sec:master}. Such a non-trivial case is indeed possible for our $\hat{H}_0$ and $\a_1 = C\Bmin$ and we can see the effect in Fig. \ref{fig:projections}. We can clearly see how a state that was initially a superposition of a large number of eigenstates of both operators becomes confined to some small subspace that is neither an eigenspace of $\a_1$ or $\hat{H}_0$. In this case the projective spaces, $\mathcal{P}_M$, are defined by the parities (odd or even) of the combined number of atoms in the $\beta_k$ and $\tilde{\beta}_k$ modes for different momenta $0 < k < \pi/a$ that are distinguishable to $\Bmin$. The explanation requires careful consideration of where the eigenstates of the two operators overlap. \subsection{Determining the Projection Subspace} To find $\mathcal{P}_M$ we need to identify the subspaces $M$ which satisfy the following relation \begin{equation} \mathcal{P}_M = \sum_{m \in M} P_m = \sum_{m \in M} | h_m \rangle \langle h_m |. \end{equation} This can be done iteratively by \begin{enumerate} \item selecting some $P_m$, \item identifying the $| h_m \rangle$ which overlap with this subspace, \item identifying any other $P_m$ which also overlap with these $| h_m \rangle$ from step (ii). \item Repeat 2-3 for all the $P_m$ found in 3 until we have identified all the subspaces $P_m$ linked in this way and they will form one of our $\mathcal{P}_M$ projectors. If $\mathcal{P}_M \ne 1$ then there will be other subspaces $P_m$ which we have not included so far and thus we repeat this procedure on the unused projectors until we identify all $\mathcal{P}_M$. \end{enumerate} Computationally this can be straightforwardly solved with some basic algorithm that can compute the connected components of a graph. The above procedure, whilst mathematically correct and always guarantees to generate the projectors $\mathcal{P}_M$, is very unintuitive and gives poor insight into the nature or physical meaning of $\mathcal{P}_M$. In order to get a better understanding of these subspaces we need to define a new operator $\hat{O}$, with eigenspace projectors $R_m$, which commutes with both $\hat{H}_0$ and $\c$, \begin{equation} [\hat{O}, \hat{H}_0 ] = 0, \end{equation} \begin{equation} [\hat{O}, \c] = 0. \end{equation} Physically this means that $\hat{O}$ is a compatible observable with $\c$ and corresponds to a quantity conserved by the Hamiltonian. The fact that $\hat{O}$ commutes with the Hamiltonian implies that the projectors can be written as a sum of Hamiltonian eigenstates \begin{equation} R_m = \sum_{h_i = h_m} | h_i \rangle \langle h_i | \end{equation} and thus a projector \begin{equation} \mathcal{P}_M = \sum_{m \in M} R_m \end{equation} is guaranteed to commute with the Hamiltonian and similarly since $[\hat{O}, \c] = 0$ $\mathcal{P}_M$ will also commute with $\c$ as required. Therefore, \begin{equation} \mathcal{P}_M = \sum_{m \in M} R_m = \sum_{m \in M} P_m \end{equation} will satisfy all the necessary prerequisites. This is illustrated in Fig. \ref{fig:spaces}. \begin{figure}[hbtp!] \includegraphics[width=\linewidth]{spaces} \caption[A Visual Representation of the Projection Spaces of the Measurement]{A visual representation of the projection spaces of the measurement. The light blue areas (bottom layer) are $R_m$, the eigenspaces of $\hat{O}$. The green areas are measurement eigenspaces, $P_m$, and they overlap non-trivially with the $R_m$ subspaces. The $\mathcal{P}_M$ projection space boundary (dashed line) runs through the Hilbert space, $\mathcal{H}$, where there is no overlap between $P_m$ and $R_m$. \label{fig:spaces}} \end{figure} In the simplest case the projectors $\mathcal{P}_M$ can consist of only single eigenspaces of $\hat{O}$, $\mathcal{P}_M = R_m$. The interpretation is straightforward - measurement projects the system onto a eigenspace of an observable $\hat{O}$ which is a compatible observable with $\a_1$ and corresponds to a quantity conserved by the coherent Hamiltonian evolution. However, this may not be possible and we have the more general case when $\mathcal{P}_M = \sum_{m \in M} R_m$. In this case, one can simply think of all $R_{m \in M}$ as degenerate just like eigenstates of the measurement operator, $\a_1$, that are degenerate, can form a single eigenspace $P_m$. However, these subspaces correspond to different eigenvalues of $\hat{O}$ distinguishing it from conventional projections. Instead, the degeneracies are identified by some other feature. In our case, it is apparent from the form of $\Bmin$ and $\hat{H}_0$ in Eqs. \eqref{eq:BminBeta} and \eqref{eq:H0Beta} that \begin{equation} \hat{O}_k = \beta_k^\dagger \beta_k + \tilde{\beta}_k^\dagger \tilde{\beta_k} = \n_k + \n_{k - \pi/a} \end{equation} commutes with both operators for all $k$. Thus, we can easily construct \begin{equation} \hat{O} = \sum_\mathrm{RBZ} g_k \hat{O}_k \end{equation} for any arbitrary $g_k$. Its eigenspaces, $R_m$, can then be easily constructed and their relationship with $P_m$ and $\mathcal{P}_M$ is illustrated in Fig. \ref{fig:spaces} whilst the time evolution of $\langle \hat{O}_k \rangle$ for a sample trajectory is shown in Fig. \ref{fig:projections}(a). Note that unlike the $\c$ or $\H_0$ we can actually see that this observable's distribution does indeed freeze. These eigenspaces are composed of Fock states in momentum space that have the same number of atoms within each pair of $k$ and $k - \pi/a$ modes. Having identified an appropriate $\hat{O}$ operator we proceed to identifying $\mathcal{P}_M$ subspaces for the operator $\Bmin$. Firstly, since $\Bmin$ contains $\sin(ka)$ coefficients atoms in different $k$ modes that have the same $\sin(ka)$ value are indistinguishable to the measurement and will lie in the same $P_m$ eigenspaces. This will happen for the pairs ($k$, $\pi/a - k$). Therefore, the $R_m$ spaces that have the same $\hat{O}_k + \hat{O}_{\pi/a - k}$ eigenvalues must belong to the same $\mathcal{P}_M$. Secondly, if we re-write $\hat{O}$ in terms of the $\beta_k$ and $\tilde{\beta}_k$ modes we get \begin{equation} \hat{O} = \sum_{\mathrm{RBZ}} g_k \left( \beta^\dagger_k \beta_k + \tilde{\beta}_k^\dagger \tilde{\beta}_k \right), \end{equation} and so it's not hard to see that $\B_{\mathrm{min},k} = (\beta^\dagger_k \beta_k - \tilde{\beta}_k^\dagger \tilde{\beta}_k)$ can have the same eigenvalues for different values of $\hat{O}_k = \beta^\dagger_k \beta_k + \tilde{\beta}_k^\dagger \tilde{\beta}_k$. Specifically, if a given subspace $R_m$ corresponds to the eigenvalue $O_k$ of $\hat{O}_k$ then the possible values of $B_{\mathrm{min},k}$ will be $\{-O_k, -O_k + 2, ..., O_k - 2, O_k\}$ just like $\Delta N = N_\mathrm{even} - N_\mathrm{odd}$ can have multiple possible values for a fixed $N = N_\mathrm{even} + N_\mathrm{odd}$. Therefore, we can see that all $R_m$ with even values of $O_k$ will share $B_{\mathrm{min},k}$ eigenvalues and thus they will overlap with the same $P_m$ subspaces. The same is true for odd values of $O_k$. However, $R_m$ with an even value of $O_k$ will never have the same value of $B_{\mathrm{min},k}$ as a subspace with an odd value of $O_k$. Therefore, a single $\mathcal{P}_M$ will contain all $R_m$ that have the same parities of $O_k$ for all $k$, e.g.~if it includes the $R_m$ with $O_k = 6$, it will also include the $R_m$ for which $O_k = 0, 2, 4, 6, ..., N$, but not any of $O_k = 1, 3, 5, 7, ..., N$, where $N$ is the total number of atoms. Finally, the $k = \pi/a$ mode is special, because $\sin(\pi) = 0$ which means that $B_{\mathrm{min},k=\pi/a} = 0$ always. This in turn implies that all possible values of $O_{\pi/a}$ are degenerate to the measurement. Therefore, we exclude this mode when matching the parities of the other modes. To illustrate the above let us consider a specific example. Let us consider two atoms, $N=2$, on eight sites $M=8$. This particular configuration has eight momentum modes $ka = \{-\frac{3\pi}{4}, -\frac{\pi}{2}, -\frac{\pi}{4}, 0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi\}$ and so the RBZ has only four modes, $\mathrm{RBZ} := \{\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}, \pi\}$. There are 10 different ways of splitting two atoms into these four modes and thus we have 10 different $R_m = \{O_{\pi/4a}, O_{\pi/2a} ,O_{3\pi/4a} ,O_{\pi/a}\}$ eigenspaces of $\hat{O}$ \begin{table}[!htbp] \centering \begin{tabular}{l c c} \toprule $m$ & $R_m$ & Possible values of $B_\mathrm{min}$ \\ \midrule 0 & $\{2,0,0,0\}$ & $ -\sqrt{2}, 0, \sqrt{2} $ \\ 1 & $\{1,1,0,0\}$ & $ -\frac{1 + \sqrt{2}}{\sqrt{2}}, -\frac{1 - \sqrt{2}}{\sqrt{2}}, \frac{1 - \sqrt{2}}{\sqrt{2}}, \frac{1 + \sqrt{2}}{\sqrt{2}} $ \\ 2 & $\{1,0,1,0\}$ & $ -\sqrt{2}, 0, \sqrt{2} $ \\ 3 & $\{1,0,0,1\}$ & $ -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} $ \\ 4 & $\{0,2,0,0\}$ & $ -2, 0, 2 $ \\ 5 & $\{0,1,1,0\}$ & $ -\frac{1 + \sqrt{2}}{\sqrt{2}}, -\frac{1 - \sqrt{2}}{\sqrt{2}}, \frac{1 - \sqrt{2}}{\sqrt{2}}, \frac{1 + \sqrt{2}}{\sqrt{2}} $ \\ 6 & $\{0,1,0,1\}$ & $ -1, 1 $ \\ 7 & $\{0,0,2,0\}$ & $ -\sqrt{2}, 0, \sqrt{2} $ \\ 8 & $\{0,0,1,1\}$ & $ -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} $ \\ 9 & $\{0,0,0,2\}$ & $0$ \\ \bottomrule \end{tabular} \caption[Eigenspace Overlaps]{A list of all $R_m$ eigenspaces for $N = 2$ atoms at $M = 8$ sites. The third column displays the eigenvalues of all the eigenstates of $\Bmin$ that lie in the given $R_m$.} \label{tab:Rm} \end{table} In the third column we have also listed the eigenvalues of the $\Bmin$ eigenstates that lie within the given $R_m$. We note that $ka = \pi/4$ will be degenerate with $ka = 3\pi/4$ since $\sin(ka)$ is the same for both. Therefore, we already know that we can combine $(R_0, R_2, R_7)$, $(R_1, R_5)$, and $(R_3, R_8)$, because those combinations have the same $O_{\pi/4a} + O_{3\pi/4a}$ values. This is very clear in the table as these subspaces span exactly the same values of $B_\mathrm{min}$, i.e.~they have exactly the same values in the third column. Now we have to match the parities. Subspaces that have the same parity combination for the pair $(O_{\pi/4a} + O_{3\pi/4a}, O_{\pi/2a})$ will be degenerate in $\mathcal{P}_M$. Note that we excluded $O_{\pi/a}$, because as we discussed earlier they are all degenerate due to $\sin(\pi) = 0$. Therefore, the (even,even) subspace will include $(R_0, R_2, R_4, R_7, R_9)$, the (odd,even) will contain $(R_3, R_8)$, the (even, odd) will contain $(R_6)$ only, and the (odd, odd) contains $(R_1, R_5)$. These overlaps should be evident from the table as we can see that these combinations combine all $R_m$ that share any eigenstates of $\Bmin$ with the same eigenvalues. Therefore, we have ended up with four distinct $\mathcal{P}_M$ subspaces \begin{align} \mathcal{P}_\mathrm{even,even} = & R_0 + R_2 + R_4 + R_7 + R_9 \nonumber \\ \mathcal{P}_\mathrm{odd,even} = & R_3 + R_8 \nonumber \\ \mathcal{P}_\mathrm{even,odd} = & R_6 \nonumber \\ \mathcal{P}_\mathrm{odd,odd} = & R_1 + R_5 \nonumber. \end{align} At this point it should be clear that these projectors satisfy all our requirement. The conditions $\sum_M \mathcal{P}_M = 1$ and $\mathcal{P}_M \mathcal{P}_N = \delta_{M,N} \mathcal{P}_M$ should be evident from the form above. The commutator requirements are also easily satisfied since the subspaces $R_m$ are of an operator that commutes with both the Hamiltonian and the measurement operator. And finally, one can also verify using the table that all of these projectors are built from complete subspaces of $\Bmin$ (i.e.~each subspace $P_m$ belongs to only one $\mathcal{P}_M$) and thus $\mathcal{P}_M = \sum_{m \in M} P_m$. \section{Conclusions} In summary we have investigated measurement backaction resulting from coupling light to an ultracold gas's phase-related observables. We demonstrated how this can be used to prepare the Hamiltonian eigenstates even if significant tunnelling is occurring as the measurement can be engineered to not compete with the system's dynamics. Furthermore, we have shown that when the observable of the phase-related quantities does not commute with the Hamiltonian we still project to a specific subspace of the system that is neither an eigenspace of the Hamiltonian or the measurement operator. This is in contrast to quantum Zeno dynamics \cite{misra1977, facchi2008, raimond2010, raimond2012, signoles2014} or dissipative state preparation \cite{diehl2008}. We showed how this projection is essentially an extension of the measurement postulate to weak measurement on dynamical systems where the competition between the two processes is significant.