801 lines
42 KiB
TeX
801 lines
42 KiB
TeX
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%*********************************** Fifth Chapter *****************************
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\chapter{Density Measurement Induced Dynamics}
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% Title of the Fifth Chapter
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\section{Introduction}
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In the previous chapter we have introduced a theoretical framework
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which will allow us to study measurement backaction using
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discontinuous quantum jumps and non-Hermitian evolution due to null
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outcomesquantum trajectories. We have also wrapped our quantum gas
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model in this formalism by considering ultracold bosons in an optical
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lattice coupled to a cavity which collects and enhances light
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scattered in one particular direction. One of the most important
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conclusions of the previous chapter was that the introduction of
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measurement introduces a new energy and time scale into the picture
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which competes with the intrinsic dynamics of the bosons.
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In this chapter, we investigate the effect of quantum measurement
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backaction on the many-body state and dynamics of atoms. In
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particular, we will focus on the competition between the backaction
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and the the two standard short-range processes, tunnelling and on-site
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interactions, in optical lattices. We show that the possibility to
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spatially structure the measurement at a microscopic scale comparable
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to the lattice period without the need for single site resolution
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enables us to engineer efficient competition between the three
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processes in order to generate new nontrivial dynamics. However,
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unlike tunnelling and on-site interactions our measurement scheme is
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global in nature which makes it capable of creating long-range
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correlations which enable nonlocal dynamical processes. Furthermore,
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global light scattering from multiple lattice sites creates nontrivial
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spatially nonlocal coupling to the environment which is impossible to
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obtain with local interactions \cite{daley2014, diehl2008,
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syassen2008}. These spatial modes of matter fields can be considered
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as designed systems and reservoirs opening the possibility of
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controlling dissipations in ultracold atomic systems without resorting
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to atom losses and collisions which are difficult to manipulate. Thus
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the continuous measurement of the light field introduces a
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controllable decoherence channel into the many-body dynamics. Such a
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quantum optical approach can broaden the field even further allowing
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quantum simulation models unobtainable using classical light and the
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design of novel systems beyond condensed matter analogues.
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In the weak measurement limit, where the quantum jumps do not occur
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frequently compared to the tunnelling rate, this can lead to global
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macroscopic oscillations of bosons between odd and even sites. These
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oscillations occur coherently across the whole lattice enabled by the
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fact that measurement is capable of generating nonlocal spatial
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modes. When on-site interactions are included we obtain a system with
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three competing energy scales of which two correspond to local
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processes and one is global. This complicates the picture
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immensely. We show how under certain circumstances interactions
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prevent measurement from generating globally coherent dynamics, but on
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the other hand when the measurement is strong both processes
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collaborate in squeezing the atomic distribution.
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On the other end of the spectrum, when measurement is strong we enter
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the regime of quantum Zeno dynamics. Frequent measurements can slow
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the evolution of a quantum system leading to the quantum Zeno effect
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where a quantum state is frozen in its initial configuration
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\cite{misra1977, facchi2008}. One can also devise measurements with
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multi-dimensional projections which lead to quantum Zeno dynamics
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where unitary evolution is uninhibited within this degenerate
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subspace, usually called the Zeno subspace \cite{facchi2008,
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raimond2010, raimond2012, signoles2014}. Our flexible setup where global light
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scattering can be engineered allows us to suppress or enhance specific
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dynamical processes thus realising spatially nonlocal quantum Zeno
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dynamics. This unconventional variation occurs when measurement is
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near, but not in, its projective limit. The system is still confined
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to Zeno subspaces, but intermediate transitions are allowed via
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virtual Raman-like processes. We show that this result can, in general
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(i.e.~beyond the ultracold gas model), be approximated by a
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non-Hermitian Hamiltonian thus extending the notion of quantum Zeno
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dynamics into the realm of non-Hermitian quantum mechanics joining the
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two paradigms.
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\section{Large-Scale Dynamics due to Weak Measurement}
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We start by considering the weak measurement limit when photon
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scattering does not occur frequently compared to the tunnelling rate
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of the atoms, i.e.~$\gamma \ll J$. When the system is probed in this
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way, the measurement is unable to project the quantum state of the
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bosons to an eigenspace as postulated by the Copenhagen interpretation
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of quantum mechanics. The backaction of the photodetections is simply
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not strong or frequent enough to confine the atoms. However, instead
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of confining the evolution of the quantum state, it has been shown in
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Refs. \cite{mazzucchi2016, mazzucchi2016njp} that the measurement
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leads to coherent global oscillations between the modes generated by
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the spatial profile of the light field which we have seen in section
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\ref{sec:modes}. Fig. \ref{fig:oscillations} illustrates the atom
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number distributions in the odd sites for $Z = 2$ and one of the three
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modes for $Z = 3$. These oscillations correspond to atoms flowing from
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one mode to another. We only observe a small number of well defined
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components which means that this flow happens in phase, all the atoms
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are tunnelling between the modes together in unison. Furthermore, this
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exchange of population is macroscopic in scale. The trajectories reach
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a state where the maximum displacement point corresponds to all the
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atoms being entirely within a single mode. Finally, we note that these
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oscillating distributions are squeezed by the measurement and the
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individual components have a width smaller than the initial state. By
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contrast, in the absence of the external influence of measurement
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these distributions would spread out significantly and the center of
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the broad distribution would oscillate with an amplitude comparable to
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the initial imbalance, i.e.~small oscillations for a small initial
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imbalance.
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\begin{figure}[htbp!]
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\centering
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\includegraphics[width=\textwidth]{Oscillations}
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\caption[Macroscopic Oscillations due to Weak Measurement]{Large
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oscillations between the measurement-induced spatial modes
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resulting from the competition between tunnelling and weak
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measurement induced backaction. The plots show the atom number
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distributions $p(N_l)$ in one of the modes in individual quantum
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trajectories. These dstributions show various numbers of
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well-squeezed components reflecting the creation of macroscopic
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superposition states depending on the measurement
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configuration. $U/J = 0$, $\gamma/J = 0.01$, $M=N$, initial
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states: bosonic superfluid. (a) Measurement of the atom number at
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odd sites $\hat{N}_\mathrm{odd}$ creates one strongly oscillating
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component in $p(N_\mathrm{odd})$ ($N = 100$ bosons, $J_{j,j} = 1$
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if $j$ is odd and 0 otherwise). (b) Measurement of
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$(\hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even})^2$ introduces
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$Z = 2$ modes and preserves the superposition of positive and
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negative atom number differences in $p(N_\mathrm{odd})$ ($N = 100$
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bosons, $J_{j,j} = (-1)^{j+1}$). (c) Measurement for $Z = 3$ modes
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preserves three components in $p(N_1)$ ($N = 108$ bosons,
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$J_{j,j} = e^{i 2 \pi j / 3}$).}
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\label{fig:oscillations}
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\end{figure}
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In Figs. \ref{fig:oscillations}(b,c) we also see that the system is
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composed of multiple components. This depends on the quantity that is
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being measured and it is a consequence of the fact that the detected
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light intensity $\ad_1 \a_1$ is not sensitive to the light phase. The
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measurement will not distinguish between permutations of mode
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occupations that scatter light with the same intensity, but with a
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different phase. For example, when measuring
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$\hat{D} = \hat{N}_\mathrm{odd} - \hat{N}_\mathrm{even}$, the light
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intensity will be proportional to
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$\hat{D}^\dagger \hat{D} = (\hat{N}_\mathrm{odd} -
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\hat{N}_\mathrm{even})^2$ and thus it cannot distinguish between a
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positive and negative imbalance leading to the two components seen in
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Fig. \ref{fig:oscillations}. More generally, the number of components
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of the atomic state, i.e.~the degeneracy of $\ad_1 \a_1$, can be
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computed from the eigenvalues of Eq. \eqref{eq:Zmodes},
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\begin{equation}
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\hat{D} = \sum_l^Z \exp\left[-i 2 \pi l R / Z \right] \hat{N}_l.
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\end{equation}
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Each eigenvalue can be represented as the sum of the individual terms
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in teh above sum which are vectors on the complex plane with phases
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that are integer multiples of $2 \pi / Z$: $N_1 e^{-i 2 \pi R / Z}$,
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$N_2 e^{-i 4 \pi R / Z}$, ..., $N_Z$. Since the set of possible sums
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of these vectors is invariant under rotations by $2 \pi l R / Z$,
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$l \in \mathbb{Z}$, and reflection in the real axis, the state of the
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system is 2-fold degenerate for $Z = 2$ (reflections leave $Z = 2$
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unchanged) and $2Z$-fold degenerate for $Z >
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2$. Fig. \ref{fig:oscillations} shows the three mode case, where there
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are in fact $6$ components ($2Z = 6$), but in this case they all occur
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in pairs resulting in only three visible components.
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We will now limit ourselves to a specific illumination pattern with
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$\hat{D} = \hat{N}_\mathrm{odd}$ as this leads to the simplest
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multimode dynamics with $Z = 2$ and only a single component as seen in
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Fig. \ref{fig:oscillations}a, i.e.~no multiple peaks like in
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Figs. \ref{fig:oscillations}(b,c). This pattern can be obtained by
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crossing two beams such that their projections on the lattice are
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identical and the even sites are positioned at their nodes. However,
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even though this is the simplest possible case and we are only dealing
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with non-interacting atoms solving the full dynamics of the
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Bose-Hubbard Hamiltonian combined with measurement is nontrivial. The
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backaction introduces a highly nonlinear global term. However, it has
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been shown in Ref. \cite{mazzucchi2016njp} that the non-interacting
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dynamics with quantum measurement backaction for $Z$-modes reduce to
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an effective Bose-Hubbard Hamiltonian with $Z$-sites provided the
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initial state is a superfluid. In this simplified model the $N_j$
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atoms in the $j$-th site correspond to a superfluid of $N_j$ atoms
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within a single spatial mode as defined in section
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\ref{sec:modes}. Therefore, we now proceed to study the dynamics for
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$\hat{D} = \hat{N}_\mathrm{odd}$ using this reduced effective
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double-well model.
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The atomic state can be written as
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\begin{equation}
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\label{eq:discretepsi}
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| \psi \rangle = \sum_l^N q_l |l, N - l \rangle,
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\end{equation}
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where the ket $| l, N - l \rangle$, represents a superfluid with $l$
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atoms in the odd sites and $N-l$ atoms in the even sites. The
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non-Hermitian Hamiltonian describing the time evolution in between the
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jumps is given by
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\begin{equation}
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\label{eq:doublewell}
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\hat{H} = -J^\mathrm{cl} \left( \bd_o b_e + b_o \bd_e \right) - i
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\gamma \n_o^2
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\end{equation}
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and the quantum jump operator which is applied at each photodetection
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is $\c = \sqrt{2 \kappa} C \n_o$. $b_o$ ($\bd_o$) is the annihilation
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(creation) operator in the left site of the effective double-well
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corresponding to the superfluid at odd sites of the physical
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lattice. $b_e$ ($\bd_e$) is defined similarly, but for the right site
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and the superfluid at even sites of the physical lattice.
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$\n_o = \bd_o b_o$ is the atom number operator in the left site.
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Even though Eq. \eqref{eq:doublewell} is relatively simple as it it is
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only a non-interacting two-site model, the non-Hermitian term
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complicates the situation making the system difficult to
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solve. However, a semiclassical approach to boson dynamics in a
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double-well in the limit of many atoms $N \gg 1$ has been developed in
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Ref. \cite{juliadiaz2012}. It was originally formulated to treat
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squeezing in a weakly interacting bosonic gas, but it can easily be
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applied to our system as well. In the limit of large atom number, the
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wavefunction in Eq. \eqref{eq:discretepsi} can be described using
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continuous variables by defining $\psi (x = l / N) = \sqrt{N}
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q_l$. Note that this requires the coefficients $q_l$ to vary smoothly
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which is the case for a superfluid state. We now rescale the
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Hamiltonian in Eq. \eqref{eq:doublewell} to be dimensionless by
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dividing by $NJ^\mathrm{cl}$ and define the relative population
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imbalance between the two wells $z = 2x - 1$. Finally, by taking the
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expectation value of the Hamiltonian and looking for the stationary
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points of
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$\langle \psi | \hat{H} | \psi \rangle - E \langle \psi | \psi
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\rangle$ we obtain the semiclassical Schr\"{o}dinger equation
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\begin{equation}
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\label{eq:semicl}
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i h \partial_t \psi(z, t) = \mathcal{H} \psi(z, t),
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\end{equation}
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\begin{equation}
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\label{eq:semiH}
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\mathcal{H} \approx -2 h^2 \partial^2_z \psi(z, t) + \left[
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\frac{\omega^2 z^2} {8} - \frac{i \Gamma} {4} \left( z + 1
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\right)^2 \right] \psi(z, t),
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\end{equation}
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where $\Gamma = N \kappa |C|^2 / J$, $h = 1/N$,
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$\omega = 2 \sqrt{1 + \Lambda - h}$, and
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$\Lambda = NU / (2J^\mathrm{cl})$. The full derivation is not
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straightforward, but the introduction of the non-Hermitian term
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requires only a minor modification to the original formalism presented
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in detail in Ref. \cite{juliadiaz2012} so we have omitted it here. We
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will also be considering $U = 0$ as the effective model is only valid
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in this limit, thus $\Lambda = 0$. However, this model is valid for an
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actual physical double-well setup in which case interacting bosons can
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also be considered. The equation is defined on the interval
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$z \in [-1, 1]$, but $z \ll 1$ has been assumed in order to simplify
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the kinetic term and approximate the potential as parabolic. This does
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mean that this approximation is not valid for the maximum amplitude
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oscillations seen in Fig. \ref{fig:oscillations}a, but since they
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already appear early on in the trajectory we are able to obtain a
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valid analytic description of the oscillations and their growth.
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A superfluid state in our continuous variable approximation
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corresponds to a Gaussian wavefunction $\psi$. Furthermore, since the
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potential is parabolic, even with the inclusion of the non-Hermitian
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term, it will remain Gaussian during subsequent time
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evolution. Therefore, we will use a very general Gaussian wavefunction
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of the form
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\begin{equation}
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\label{eq:ansatz}
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\psi(z, t) = \frac{1}{\pi b^2}\exp\left[ i \epsilon
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- \frac{(z - z_0)^2} {2 b^2} + \frac{i \phi (z - z_\phi)^2} {2 b^2} \right]
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\end{equation}
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as our ansatz to Eq. \eqref{eq:semicl}. The parameters $b$, $\phi$,
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$z_0$, and $z_\phi$ are real-valued functions of time whereas
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$\epsilon$ is a complex-valued function of time. Physically, the value
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$b^2$ denotes the width, $z_0$ the position of the center, $\phi$ and
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$z_\phi$ contain the local phase information, and $\epsilon$ only
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affects the global phase and norm of the Gaussian wave packet.
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The non-Hermitian Hamiltonian and an ansatz are not enough to describe
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the full dynamics due to measurement. We also need to know the effect
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of each quantum jump. Within the continuous variable approximation,
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our quantum jump become $\c \propto 1 + z$. We neglect the constant
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prefactors, because the wavefunction is normalised after a quantum
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jump. Expanding around the peak of the Gaussian ansatz we get
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\begin{equation}
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1 + z \approx \exp \left[ \ln (1 + z_0) + \frac{z - z_0}{1 + z_0} -
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\frac{(z - z_0)^2}{2 (1 + z_0)^2} \right].
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\end{equation}
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Multiplying the wavefunction in Eq. \eqref{eq:ansatz} with the jump
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operator above yields a Gaussian wavefunction as well, but the
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parameters change discontinuously according to
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\begin{align}
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\label{eq:jumpb2}
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b^2 & \rightarrow \frac{ b^2 (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\
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\phi & \rightarrow \frac{ \phi (1 + z_0)^2 } { (1 + z_0)^2 + b^2 }, \\
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\label{eq:jumpz0}
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z_0 & \rightarrow z_0 + \frac{ b^2 (1 + z_0) } { (1 + z_0)^2 + b^2}, \\
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z_\phi & \rightarrow z_\phi, \\
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\epsilon & \rightarrow \epsilon.
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\end{align}
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The fact that the wavefunction remains Gaussian after a photodetection
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is a huge advantage, because it means that the combined time evolution
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of the system can be described with a single Gaussian ansatz in
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Eq. \eqref{eq:ansatz} subject to non-Hermitian time evolution
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according to Eq. \eqref{eq:semicl} with discontinous changes to the
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parameter values at each quantum jump.
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Having identified an appropriate ansatz and the effect of quantum
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jumps we proceed with solving the dynamics of wavefunction in between
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the photodetecions. The initial values of the parameters for a
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superfluid state of $N$ atoms across the whole lattice are $b^2 = 2h$,
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$\phi =0$, $a_0 = 0$, $a_\phi = 0$, $\epsilon = 0$. However, we use
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the most general initial conditions at time $t = t_0$ which we denote
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by $b(t_0) = b_0$, $\phi(t_0) = \phi_0$, $z_0(t_0) = a_0$,
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$z_\phi(t_0) = a_\phi$, and $\epsilon(t_0) = \epsilon_0$. The reason
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for keeping them as general as possible is that after every quantum
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jump the system changes discontinuously. The subsequent time evolution
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is obtained by solving the Schr\"{o}dinger equation with the post-jump
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paramater values as the new initial conditions.
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By plugging the ansatz in Eq. \eqref{eq:ansatz} into the
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Schr\"{o}dinger equation in Eq. \eqref{eq:semicl} we obtain three
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differential equations
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\begin{equation}
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\label{eq:p}
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-2 h^2 p^2 + \left( \frac{ \omega^2 } { 8 } - \frac{ i \Gamma } { 4
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} \right) + \frac{ i h } { 2 } \frac{ \mathrm{d} p } { \mathrm{d}
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t } = 0,
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\end{equation}
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\begin{equation}
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\label{eq:pq}
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4 h^2 p q - \frac{ i \Gamma } { 2 } - i h \frac{ \mathrm{d} q } {
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\mathrm{d} t } = 0
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\end{equation}
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\begin{equation}
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\label{eq:pqr}
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-2 h^2 (q^2 - p) - \frac{ i \Gamma } { 4 } - i h \left( \frac{ 1 } {
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4 x } \frac{ \mathrm{d} x } {\mathrm{d} t } + i \frac{
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\mathrm{d} \epsilon } { \mathrm{d} t } - \frac{1}{2} \frac{
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\mathrm{d} r } { \mathrm{d} t } \right) = 0,
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\end{equation}
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where $x = 1/b^2$, $p = (1 - i \phi)/b^2$,
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$q = (z_0 - i \phi z_\phi)/b^2$, and
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$r = (z_0^2 - \phi z_\phi^2)/b^2$. The corresponding initial
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conditions are $x(t_0) = x_0 = 1/b_0^2$,
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$p(t_0) = p_0 = (1 - i \phi_0)/b_0^2$,
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$q(t_0) = q_0 = (a_0 - \phi_0 a_\phi)/b_0^2$, and
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$r(t_0) = r_0 = (a_0^2 - \phi_0 a_\phi^2)/b_0^2$. The original
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parameters can be extracted from these auxiliary variables by
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$b^2 = 1 / \Re \{ p \}$, $\phi = - \Im \{ p \} / \Re \{ p \}$,
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$z_0 = \Re \{ q \} / \Re \{ p \}$,
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$z_\phi = \Im \{ q \} / \Im \{ p \}$, and $\epsilon$ appears
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explicitly in the equations above.
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First, it is worth noting that all parameters of interest can be
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extracted from $p(t)$ and $q(t)$ alone. We are not interested in
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$\epsilon$ as it is only related to the global phase and the norm of
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the wavefunction and it contains little physical
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information. Furthermore, an interesting and incredibly convenient
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feature of these equations is that the Eq. \eqref{eq:p} is a function
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of $p(t)$ alone and Eq. \eqref{eq:pq} is a function of $p(t)$ and
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$q(t)$ only. Therefore, we only need to solve first two equations and
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we can neglect Eq. \eqref{eq:pqr}. However, in order to actually
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perform Monte-Carlo simulations of quantum trajectories
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Eq. \eqref{eq:pqr} would need to be solved in order to obtain correct
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jump statistics.
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We start with Eq. \eqref{eq:p} and we note it can be rearranged into
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the form
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\begin{equation}
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\frac{ \mathrm{d} p } { (\zeta \omega / 4 h)^2 - p^2 } = i 4 h
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\mathrm{d} t,
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\end{equation}
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where $\zeta^2 = (\alpha - i \beta)^2 = 1 - i 2 \Gamma / \omega^2$, and
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\begin{equation}
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\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 $\gamma \ll J^\mathrm{cl}$ 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
|
|
wavefunction and we see that it oscillates with an amplitude
|
|
$\sqrt{a_0^2 + 16 h^2 \phi_0^2 (a_0 - a_\phi)^2 / (b_0^4
|
|
\omega^2)}$. 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 wavepacket by
|
|
approximately $b^2$, i.e.~the width of the Gaussian, in the direction
|
|
of the positive $z$-axis. Therefore, even though the 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 wavefunction 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.
|
|
|
|
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 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 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 in 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.
|
|
|
|
\section{Three-Way Competition}
|
|
|
|
Now it is time to turn on the inter-atomic interactions,
|
|
$U/J^\mathrm{cl} \ne 0$. 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 of 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 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 supefluid 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 isn't
|
|
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
|
|
trajectores 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 supressed 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 inital 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 occuring. 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 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 succesfully as seen with the macroscopic oscillations or
|
|
unsuccesfully 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, mekhov2009prl}. 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 supress
|
|
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.
|
|
|
|
\section{Quantum Zeno Dynamics}
|
|
|
|
\subsection{Emergent Long-Range Correlated Tunnelling}
|
|
|
|
\subsection{Non-Hermitian Dynamics in the Quantum Zeno Limit}
|
|
|
|
% Contrast with t-J model here how U localises events, but measurement
|
|
% does the opposite
|
|
|
|
\subsection{Steady-State of the Non-Hermitian Hamiltonian}
|
|
|
|
\section{Conclusions} |