1811 lines
100 KiB
TeX
1811 lines
100 KiB
TeX
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%*********************************** Fifth Chapter *****************************
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%*******************************************************************************
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\chapter[Density Measurement Induced Dynamics]
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{Density Measurement Induced Dynamics\footnote{The results of
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this chapter were first published in
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Refs. \cite{mazzucchi2016, kozlowski2016zeno,
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mazzucchi2016njp}}}
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% Title of the Fifth Chapter
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\ifpdf
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\graphicspath{{Chapter5/Figs/Raster/}{Chapter5/Figs/PDF/}{Chapter5/Figs/}}
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\else
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\graphicspath{{Chapter5/Figs/Vector/}{Chapter5/Figs/}}
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\fi
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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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outcomes using quantum trajectories. We have also wrapped our quantum
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gas model in this formalism by considering ultracold bosons in an
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optical 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, as seen in section
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\ref{sec:modes}, which is impossible to obtain with local interactions
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\cite{daley2014, diehl2008, syassen2008}. These spatial modes of
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matter fields can be considered as designed systems and reservoirs
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opening the possibility of controlling dissipations in ultracold
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atomic systems without resorting to atom losses and collisions which
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are difficult to manipulate. Thus the continuous measurement of the
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light field introduces a controllable decoherence channel into the
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many-body dynamics. Such a quantum optical approach can broaden the
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field even further allowing quantum simulation models unobtainable
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using classical light and the design of novel systems beyond condensed
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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{Quantum Measurement Induced Dynamics}
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\subsection{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 the 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 \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 $NJ$ and define the relative population imbalance between
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the two wells $z = 2x - 1$. Finally, by taking the expectation value
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of the Hamiltonian and looking for the stationary 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 $\Lambda = NU / (2J)$. The
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full derivation is not straightforward, but the introduction of the
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non-Hermitian term requires only a minor modification to the original
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formalism presented in detail in Ref. \cite{juliadiaz2012} so we have
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omitted it here. We will also be considering $U = 0$ as the effective
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model is only valid in this limit, thus $\Lambda = 0$. However, this
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model is valid for an actual physical double-well setup in which case
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interacting bosons can also be considered. The equation is defined on
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the interval $z \in [-1, 1]$, but $z \ll 1$ has been assumed in order
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to simplify the kinetic term and approximate the potential as
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parabolic. This does mean that this approximation is not valid for the
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maximum amplitude oscillations seen in Fig. \ref{fig:oscillations}(a),
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but since they already appear early on in the trajectory we are able
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to obtain a valid analytic description of the oscillations and their
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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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\begingroup
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\allowdisplaybreaks
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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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\endgroup
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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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$\epsilon(t)$ as it is only related to the global phase and the norm
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of 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}
|
|
\frac{ \mathrm{d} p } { (\zeta \omega / 4 h)^2 - p^2 } = i 4 h
|
|
\mathrm{d} t,
|
|
\end{equation}
|
|
where $\zeta^2 = (\alpha - i \beta)^2 = 1 - i 2 \Gamma / \omega^2$, and
|
|
\begin{equation}
|
|
\alpha = \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{1 + \frac{ 4\Gamma^2
|
|
}{ \omega^4 }}},
|
|
\end{equation}
|
|
\begin{equation}
|
|
\beta = -\sqrt{ -\frac{1}{2} + \frac{1}{2} \sqrt{1 + \frac{ 4\Gamma^2
|
|
}{ \omega^4 }}}.
|
|
\end{equation}
|
|
This is a standard integral\footnotemark and thus yields
|
|
\begin{equation}
|
|
\label{eq:psol}
|
|
p(t) = \frac{ \zeta \omega } { 4 h }
|
|
\frac{ ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} - ( \zeta
|
|
\omega - 4 h p_0 ) e^{-i \zeta \omega t} }
|
|
{ ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} + ( \zeta \omega
|
|
- 4 h p_0 ) e^{-i \zeta \omega t} }.
|
|
\end{equation}
|
|
|
|
\footnotetext{ \[ \int \frac{\mathrm{d} x}{a^2 - x^2} = \frac{1}{2a}
|
|
\ln \left( \frac{a+x}{a-x} \right) + \mathrm{const.}
|
|
\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
|
|
\quad\quad\quad\quad\quad\] }
|
|
|
|
Having found an expression for $p(t)$ we can now solve
|
|
Eq. \eqref{eq:pq} for $q(t)$. To do that we first define the
|
|
integrating factor
|
|
\begin{equation}
|
|
I(t) = \exp \left[ i 4 h \int p \mathrm{d} t \right] = ( \zeta
|
|
\omega + 4 h p_0 )e^{i \zeta \omega t} + ( \zeta \omega - 4 h p_0 )
|
|
e^{-i \zeta \omega t},
|
|
\end{equation}
|
|
which lets us rewrite Eq. \eqref{eq:pq} as
|
|
\begin{equation}
|
|
\label{eq:Iq}
|
|
\frac{\mathrm{d}} {\mathrm{d} t}(Iq) = - \frac{\Gamma}{2 h} I.
|
|
\end{equation}
|
|
%Upon integrating the equation above we obtain
|
|
%\begin{equation}
|
|
% \label{eq:Iq}
|
|
% Iq = - \frac{ \Gamma } {2 h} \int I \mathrm{d} t.
|
|
%\end{equation}
|
|
Upon integrating and the substitution of the explicit form of the
|
|
integration factor into this equation we obtain the solution
|
|
\begin{equation}
|
|
\label{eq:qsol}
|
|
q(t) = \frac{1}{2 h \zeta \omega}
|
|
\frac{4 h \zeta^2 \omega^2 q_0 - i 8 h \Gamma p_0
|
|
+ i \Gamma [( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} -
|
|
( \zeta \omega - 4 h p_0 )e^{-i \zeta \omega t}]}
|
|
{ ( \zeta \omega + 4 h p_0 )e^{i \zeta \omega t} +
|
|
( \zeta \omega - 4 h p_0 )e^{-i \zeta \omega t}}.
|
|
\end{equation}
|
|
|
|
The solutions we have obtained to $p(t)$ in Eq. \eqref{eq:psol} and
|
|
$q(t)$ in Eq. \eqref{eq:qsol} are sufficient to completely describe
|
|
the physics of the system. Unfortunately, these expressions are fairly
|
|
complex and it is difficult to extract the physically meaningful
|
|
parameters in a form that is easy to analyse. Therefore, we instead
|
|
consider the case when $\Gamma = 0$, but we do not neglect the effect
|
|
of quantum jumps. It may seem counter-intuitive to neglect the term
|
|
that appears due to measurement, but we are considering the weak
|
|
measurement regime where $\gamma \ll J$ and thus the
|
|
dynamics between the quantum jumps are actually dominated by the
|
|
tunnelling of atoms rather than the null outcomes. Furthermore, the
|
|
effect of the quantum jump is independent of the value of $\Gamma$
|
|
($\Gamma$ only determined their frequency). However, this is only true
|
|
at times shorter than the average time between two consecutive quantum
|
|
jumps. Therefore, this approach will not yield valid answers on the
|
|
time scale of a whole quantum trajectory, but it will give good
|
|
insight into the dynamics immediately after a quantum jump. The
|
|
solutions for $\Gamma = 0$ are
|
|
\begin{equation}
|
|
b^2(t) = \frac{b_0^2}{2} \left[ \left(1 + \frac{16 h^2 (1 + \phi_0^2)}
|
|
{b_0^4 \omega^2} \right) + \left(1 - \frac{16 h^2 (1 + \phi_0^2)}
|
|
{b_0^4 \omega^2} \right) \cos (2 \omega t) + \frac{8 h \phi_0}{b_0^2
|
|
\omega} \sin(2 \omega t) \right],
|
|
\end{equation}
|
|
\begin{equation}
|
|
\phi(t) = \frac{b_0^2 \omega} {8 h} \left[ \left( \frac{16 h^2 (1 + \phi_0^2)}
|
|
{b_0^4 \omega^2} - 1 \right) \sin (2 \omega t) + \frac{8 h
|
|
\phi_0} {b_0^2 \omega} \cos (2 \omega t) \right],
|
|
\end{equation}
|
|
\begin{equation}
|
|
z_0(t) = a_0 \cos(\omega t) + \frac{4 h \phi_0} {b_0^2 \omega} (a_0 -
|
|
a_\phi) \sin (\omega t),
|
|
\end{equation}
|
|
\begin{equation}
|
|
\phi(t) z_\phi(t) = \phi_0 a_\phi \cos (\omega t) + \frac{4 h}
|
|
{b_0^2 \omega} (a_0 - \phi_0^2 a_\phi) \sin( \omega t).
|
|
\end{equation}
|
|
First, these equations show that all quantities oscillate with a
|
|
frequency $\omega$ or $2 \omega$. We are in particular interested in
|
|
the quantity $z_0(t)$ as it represents the position of the peak of the
|
|
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 atoms can
|
|
oscillate in the absence of measurement it is the quantum jumps that
|
|
are the driving force behind this phenomenon. Furthermore, these
|
|
oscillations grow because the quantum jumps occur at an average
|
|
instantaneous rate proportional to $\langle \cd \c \rangle (t)$ which
|
|
itself is proportional to $(1+z)^2$. This means they are most likely
|
|
to occur at the point of maximum displacement in the positive $z$
|
|
direction at which point a quantum jump provides positive feedback and
|
|
further increases the amplitude of the 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. Furthermore, it is by engineering the measurement, and through
|
|
it the geometry of the modes, that we have control over the nature of
|
|
the correlated dynamics of the oscillations.
|
|
|
|
We have now seen the effect of the quantum jumps and how that leads to
|
|
oscillations between odd and even sites in a lattice. However, we have
|
|
neglected the effect of null outcomes on the dynamics. Even though it
|
|
is small, it will not be negligible on the time scale of a quantum
|
|
trajectory with multiple jumps. First, we note that all the
|
|
oscillatory terms $p(t)$ and $q(t)$ actually appear as
|
|
$\zeta \omega = (\alpha - i \beta) \omega$. Therefore, we can see that
|
|
the null outcomes lead to two effects: an increase in the oscillation
|
|
frequency by a factor of $\alpha$ to $\alpha \omega$ and a damping
|
|
term with a time scale $1/(\beta \omega)$. For weak measurement, both
|
|
$\alpha$ and $\beta$ will be close to $1$ so the effects are not
|
|
visible on short time scales. Instead, we look at the long time
|
|
limit. Unfortunately, since all the quantities are oscillatory a
|
|
stationary long time limit does not exist especially since the quantum
|
|
jumps provide a driving force. However, the width of the Gaussian,
|
|
$b^2$, is unique in that it 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 look like
|
|
they have a significantly smaller width than the initial
|
|
distribution. This discrepancy is due to the fact that the continuous
|
|
variable approximation is only valid for $z \ll 1$ and thus it cannot
|
|
explain the final behaviour of the system. Furthermore, it has been
|
|
shown that the width of the distribution $b^2$ does not actually
|
|
shrink to a constant value, but rather it keeps oscillating around the
|
|
value given in Eq. \eqref{eq:b2} \cite{mazzucchi2016njp}. However,
|
|
what we do see is that during the early stages of the trajectory,
|
|
which are well described by this model, is that the width does in fact
|
|
stay roughly constant. It is only in the later stages when the
|
|
oscillations reach maximal amplitude that the width becomes visibly
|
|
reduced.
|
|
|
|
\subsection{Three-Way Competition}
|
|
|
|
Now it is time to turn on the inter-atomic interactions, $U/J \ne
|
|
0$. As a result the atomic dynamics will change as the measurement now
|
|
competes with both the tunnelling and the on-site interactions. A
|
|
common approach to study such open systems is to map a dissipative
|
|
phase diagram by finding the steady state of the master equation for a
|
|
range of parameter values \cite{kessler2012}. However, here we adopt a
|
|
quantum optical approach in which we focus on the conditional dynamics
|
|
of a single quantum trajectory as this corresponds to a single
|
|
realisation of an experiment. The resulting evolution does not
|
|
necessarily reach a steady state and usually occurs far from the
|
|
ground state of the system.
|
|
|
|
A key feature of the quantum trajectory approach is that each
|
|
trajectory evolves differently as it is conditioned on the
|
|
photodetection times which are determined stochastically. Furthermore,
|
|
even states in the same measurement subspace, i.e.~indistinguishable
|
|
to the measurement , can have minimal overlap. This is in contrast to
|
|
the unconditioned solutions obtained with the master equation which
|
|
only yields a single outcome that is an average taken over all
|
|
possible outcomes. However, this makes it difficult to study the
|
|
three-way competition in some meaningful way across the whole
|
|
parameter range.
|
|
|
|
Ultimately, regardless of its strength measurement always tries to
|
|
project the quantum state onto one of its eigenstates (or eigenspaces
|
|
if there are degeneracies). If the probe is strong enough this will
|
|
succeed, but we have seen in the previous section that when this is
|
|
not the case, measurement leads to new dynamical phenomena. However,
|
|
despite this vast difference in behaviour, there is a single quantity
|
|
that lets us determine the degree of success of the projection, namely
|
|
the fluctuations, $\sigma_D^2$ (or equivalently the standard
|
|
deviation, $\sigma_D$), of the observable that is being measured,
|
|
$\hat{D}$. For a perfect projection this value is exactly zero,
|
|
because the system at that point is in the corresponding
|
|
eigenstate. When the system is unable to project the state into such a
|
|
state, the variance will be non-zero. However, the smaller its value
|
|
is the closer it is to being in such an eigenstate and on the other
|
|
hand a large variance means that the internal processes dominate the
|
|
competition. Finally, this quantity is perfect to study quantum
|
|
trajectories, because its value in the long-time limit it is only a
|
|
function of $\gamma$, $J$, and $U$. It does not depend on the explicit
|
|
history of photodetections. Fig. \ref{fig:squeezing} shows a plot of
|
|
this quantity for $\hat{D} = \hat{N}_\mathrm{odd}$ averaged over
|
|
multiple trajectories, $\langle \sigma^2_D \rangle_\mathrm{traj}$, as
|
|
a function of $\gamma/J$ and $U/J$ for a lattice of six atoms on six
|
|
sites (we cannot use the effective double-well model, because
|
|
$U \ne 0$). We use a ground state 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 serious decay channels they will remain in the system
|
|
and subsequent jumps will still amplify them further destroying the
|
|
ground state and thus quickly leading to a state with large
|
|
fluctuations.
|
|
|
|
In the strong measurement regime ($\gamma \gg J$) the measurement
|
|
becomes more significant than the local dynamics and the system will
|
|
freeze the state in the measurement operator eigenstates. In this
|
|
case, the squeezing will always be better than in the ground state,
|
|
because measurement and on-site interaction cooperate in suppressing
|
|
fluctuations. This cooperation did not exist for weak measurement,
|
|
because it tried to induce dynamics which produced squeezed states
|
|
(either 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.
|
|
|
|
\subsection{Emergent Long-Range Correlated Tunnelling}
|
|
|
|
When $\gamma \rightarrow \infty$ the measurement becomes
|
|
projective. This means that as soon as the probing begins, the system
|
|
collapses into one of the observable's eigenstates. Furthermore, since
|
|
this measurement is continuous and doesn't stop after the projection
|
|
the system will be frozen in this state. This effect is called the
|
|
quantum Zeno effect \cite{misra1977, facchi2008} from Zeno's classical
|
|
paradox in which a ``watched arrow never moves'' that stated that
|
|
since an arrow in flight is not seen to move during any single
|
|
instant, it cannot possibly be moving at all. Classically the paradox
|
|
was resolved with a better understanding of infinity and infintesimal
|
|
changes, but in the quantum world a watched quantum arrow will in fact
|
|
never move. The system is being continuously projected into its
|
|
initial state before it has any chance to evolve. If degenerate
|
|
eigenspaces exist then we can observe quantum Zeno dynamics where
|
|
unitary evolution is uninhibited within such a degenerate subspace,
|
|
called the Zeno subspace \cite{facchi2008, raimond2010, raimond2012,
|
|
signoles2014}.
|
|
|
|
These effects can be easily seen in our model when
|
|
$\gamma \rightarrow \infty$. The system will be projected into one or
|
|
more degenerate eignstates of $\cd \c$, $| \psi_i \rangle$, for which
|
|
we define the projector
|
|
$P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$
|
|
denotes a single degenerate subspace. The Zeno subspace is determined
|
|
randomly as per the Copenhagen postulates and thus it depends on the
|
|
initial state. If the projection is into the subspace $\varphi$, the
|
|
subsequent evolution is described by the projected Hamiltonian
|
|
$P_{\varphi} \hat{H}_0 P_{\varphi}$. We have used the original
|
|
Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or the
|
|
quantum jumps as their combined effect is now described by the
|
|
projectors. Physically, in our model of ultracold bosons trapped in a
|
|
lattice this means that tunnelling between different spatial modes is
|
|
completely supressed since this process couples eigenstates belonging
|
|
to different Zeno subspaces. If a small connected part of the lattice
|
|
was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then
|
|
tunnelling would only be prohibited between the illuminated and
|
|
unilluminated areas, but dynamics proceeds normally within each zone
|
|
separately. However, the goemetric patterns we have in which the modes
|
|
are spatially delocalised in such a way that neighbouring sites never
|
|
belong to the same mode, e.g. $\hat{D} = \hat{N}_\mathrm{odd}$, would
|
|
lead to a complete suppression of tunnelling across the whole lattice
|
|
as there is no way for an atom to tunnel within this Zeno subspace
|
|
without first having to leave it.
|
|
|
|
This is an interesting example of the quantum Zeno effect and dynamics
|
|
and it can be used to prohibit parts of the dynamics of the
|
|
Bose-Hubbard Model in order to engineer desired Hamiltonians for
|
|
quantum simulations or other applications. However, the infinite
|
|
projective limit is uninteresting in the context of a global
|
|
measurement scheme. The same effects and Hamiltonians can be achieved
|
|
using multiple independent measurements which address a few sites
|
|
each. In order to take advantage of the nonlocal nature of the
|
|
measurement it turns out that we need to consider a finite limit for
|
|
$\gamma \gg J$. By considering a non-infinite $\gamma$ we observe
|
|
additional dynamics while the usual atomic tunnelling is still heavily
|
|
Zeno-suppressed. These new effects are shown in Fig. \ref{fig:zeno}.
|
|
|
|
\begin{figure}[hbtp!]
|
|
\includegraphics[width=\textwidth]{Zeno.pdf}
|
|
\caption[Emergent Long-Range Correlated Tunnelling]{ Long-range
|
|
correlated tunneling and entanglement, dynamically induced by
|
|
strong global measurement in a single quantum
|
|
trajectory. (a),(b),(c) show different measurement geometries,
|
|
implying different constraints. Panels (1): schematic
|
|
representation of the long-range tunneling processes. Panels (2):
|
|
evolution of on-site densities. Panels (3): entanglement entropy
|
|
growth between illuminated and non-illuminated regions. Panels
|
|
(4): correlations between different modes (orange) and within the
|
|
same mode (green); $N_I$ ($N_{NI}$) is the atom number in the
|
|
illuminated (non-illuminated) mode. (a) (a.1) Atom number in the
|
|
central region is frozen: system is divided into three
|
|
regions. (a.2) Standard dynamics happens within each region, but
|
|
not between them. (a.3) Entanglement build up. (a.4) Negative
|
|
correlations between non-illuminated regions (green) and zero
|
|
correlations between the $N_I$ and $N_{NI}$ modes
|
|
(orange). Initial state: $|1,1,1,1,1,1,1 \rangle$, $\gamma/J=100$,
|
|
$J_{jj}=[0,0,1,1,1,0,0]$. (b) (b.1) Even sites are illuminated,
|
|
freezing $N_\text{even}$ and $N_\text{odd}$. Long-range tunneling
|
|
is represented by any pair of one blue and one red arrow. (b.2)
|
|
Correlated tunneling occurs between non-neighbouring sites without
|
|
changing mode populations. (b.3) Entanglement build up. (b.4)
|
|
Negative correlations between edge sites (green) and zero
|
|
correlations between the modes defined by $N_\text{even}$ and
|
|
$N_\text{odd}$ (orange). Initial state: $|0,1,2,1,0 \rangle$,
|
|
$\gamma/J=100$, $J_{jj}=[0,1,0,1,0]$. (c) (c.1,2) Atom number
|
|
difference between two central sites is frozen. (c.3) Entanglement
|
|
build up. (c.4) In contrast to previous examples, sites in the
|
|
same zones are positively correlated (green), while atoms in
|
|
different zones are negatively correlated (orange). Initial state:
|
|
$|0,2,2,0 \rangle$, $\gamma/J=100$, $J_{jj}=[0,-1,1,0]$. 1D
|
|
lattice, $U/J=0$.}
|
|
\label{fig:zeno}
|
|
\end{figure}
|
|
|
|
There are two crucial features of the resulting dynamics that are of
|
|
note. First, just like in the infinite quantum Zeno limit the
|
|
evolution between nearest neighbours within the same mode is
|
|
unperturbed whilst tunnelling between different modes is heavily
|
|
suppressed by the measurement. Therefore, we see the usual quantum
|
|
Zeno dynamics within a single Zeno subspace and just like before, it
|
|
is also possible to use the global probing scheme to engineer these
|
|
eigenspaces and select which tunnelling processes should be
|
|
uninhibited and which should be suppressed. However, there is a second
|
|
effect that was not present before. In Fig. \ref{fig:zeno} we can
|
|
observe tunnelling that violates the boundaries established by the
|
|
spatial modes. When $\gamma$ is finite, second-order processes,
|
|
i.e.~two correlated tunnelling events, can now occur via an
|
|
intermediate (virtual) state outside of the Zeno subspace as long as
|
|
the Zeno subspace of the final state remains the same. Crucially,
|
|
these tunnelling events are only correlated in time, but not in
|
|
space. This means that the two events do not have to occur for the
|
|
same atom or even at the same site in the lattice. As long as the Zeno
|
|
subspace is preserved, these processes can occur anywhere in the
|
|
system. That is, a pair of atoms separated by many sites is able to
|
|
tunnel in a correlated manner. This is only possible due to the
|
|
ability of creating extensive and spatially nonlocal modes as
|
|
described in section \ref{sec:modes} which in turn is enabled by the
|
|
global nature of the measurement. This would not be possible to
|
|
achieve with local measurements as the Zeno subspaces would be
|
|
described entirely by local variables which cannot be preserved by
|
|
such delocalised tunnelling events.
|
|
|
|
In the subsequent sections we will rigorously derive the following
|
|
Hamiltonian for the non-interacting dynamics within a single Zeno
|
|
subspace, $\varphi = 0$, for a lattice where the measurement defines
|
|
$Z = 2$ distinct modes, e.g. $\hat{D} = \hat{N}_K$ or
|
|
$\hat{D} = \hat{N}_\mathrm{odd}$
|
|
\begin{equation}
|
|
\label{eq:hz}
|
|
\hat{H}_\varphi = P_0 \left[ -J \sum_{\langle i, j \rangle}
|
|
b^\dagger_i b_j - i \frac{J^2} {A \gamma} \sum_{\varphi}
|
|
\sum_{\substack{\langle i \in \varphi, j \in \varphi^\prime
|
|
\rangle \\ \langle k \in \varphi^\prime, l \in \varphi
|
|
\rangle}} b^\dagger_i b_j b^\dagger_k b_l \right] P_0,
|
|
\end{equation}
|
|
where
|
|
$A = (J_{\varphi,\varphi} - J_{\varphi^\prime,\varphi^\prime})^2$ is a
|
|
constant that depends on the measurement scheme, $\varphi$ denotes a
|
|
set of sites belonging to a single mode and $\varphi^\prime$ is the
|
|
set's complement (e.g.~odd and even or illuminated and non-illuminated
|
|
sites). We see that this Hamiltonian consists of two parts. The first
|
|
term corresponds to the standard quantum Zeno first-order dynamics
|
|
that occurs within a Zeno subspace, i.e.~tunnelling between
|
|
neighbouring sites that belong to the same mode. Otherwise, if $i$ and
|
|
$j$ belong to different modes $P_0 \bd_i b_j P_0 = 0$. When
|
|
$\gamma \rightarrow \infty$ we recover the quantum Zeno Hamiltonian
|
|
where this would be the only remaining term. It is the second term
|
|
that shows the second-order corelated tunnelling terms. This is
|
|
evident from the inner sum which requires that pairs of sites ($i$,
|
|
$j$) and ($k$, $l$) between which atoms tunnel must be nearest
|
|
neighbours, but these pairs can be anywhere on the lattice within the
|
|
constraints of the mode structure. This is in particular explicitly
|
|
shown in Figs. \ref{fig:zeno}(a,b). The imaginary coefficient means
|
|
that the tinelling behaves like an exponential decay (overdamped
|
|
oscillations). This also implies that the norm will decay, but this
|
|
does not mean that there are physical losses in the system. Instead,
|
|
the norm itself represents the probability of the system remaining in
|
|
the $\varphi = 0$ Zeno subspace. Since $\gamma$ is not infinite there
|
|
is now a finite probability that the stochastic nature of the
|
|
measurement will lead to a discontinous change in the system where the
|
|
Zeno subspace rapidly changes which can be seen in
|
|
Fig. \ref{fig:zeno}(a). However, later in this chapter we will see
|
|
that steady states of this Hamiltonian exist which will no longer
|
|
change Zeno subspaces.
|
|
|
|
Crucially, what sets this effect apart from usual many-body dynamics
|
|
with short-range interactions is that first order processes are
|
|
selectively suppressed by the global conservation of the measured
|
|
observable and not by the prohibitive energy costs of doubly-occupied
|
|
sites, as is the case in the $t$-$J$ model \cite{auerbach}. This has
|
|
profound consequences as this is the physical origin of the long-range
|
|
correlated tunneling events represented in Eq. \eqref{eq:hz} by the
|
|
fact that the pairs ($i$, $j$) and ($k$, $l$) can be very distant. The
|
|
projection $P_0$ is not sensitive to individual site
|
|
occupancies, but instead enforces a fixed value of the observable,
|
|
i.e.~a single Zeno subspace. This is a striking difference with the
|
|
$t$-$J$ and other strongly interacting models. The strong interaction
|
|
also leads to correlated events in which atoms can tunnel over each
|
|
other by creating an unstable doubly occupied site during the
|
|
intermediate step. However, these correlated events are by their
|
|
nature localised. Due to interactions the doubly-occupied site cannot
|
|
be present in the final state which means that any tunnelling event
|
|
that created this unstable configuration must be followed by another
|
|
tunnelling event which takes an atom away. In the case of global
|
|
measurement this process is delocalised, because since the modes
|
|
consist of many sites the stable configuration can be restored by a
|
|
tunnelling event from a completely different lattice site that belongs
|
|
to the same mode.
|
|
|
|
In Fig.~\ref{fig:zeno}(a) we consider illuminating only the central
|
|
region of the optical lattice and detecting light in the diffraction
|
|
maximum, thus we freeze the atom number in the $K$ illuminated sites
|
|
$\hat{N}_\text{K}$~\cite{mekhov2009prl,mekhov2009pra}. The measurement
|
|
scheme defines two different spatial modes: the non-illuminated zones
|
|
$1$ and $3$ and the illuminated one $2$. Figure~\ref{fig:zeno}(a.2)
|
|
illustrates the evolution of the mean density at each lattice site:
|
|
typical dynamics occurs within each region but the standard tunnelling
|
|
between different modes is suppressed. Importantly, second-order
|
|
processes that do not change $N_\text{K}$ are still possible since an
|
|
atom from $1$ can tunnel to $2$, if simultaneously one atom tunnels
|
|
from $2$ to $3$. Therefore, effective long-range tunneling between two
|
|
spatially disconnected zones $1$ and $3$ happens due to the two-step
|
|
processes $1 \rightarrow 2 \rightarrow 3$ or
|
|
$3 \rightarrow 2 \rightarrow 1$. These transitions are responsible for
|
|
the negative (anti-)correlations
|
|
$\langle \delta N_1 \delta N_3 \rangle = \langle N_1 N_3 \rangle -
|
|
\langle N_1 \rangle \langle N_3 \rangle$ showing that an atom
|
|
disappearing from zone $1$ appears in zone $3$, while there are no
|
|
number correlations between illuminated and non-illuminated regions,
|
|
$\langle( \delta N_1 + \delta N_3 ) \delta N_2 \rangle = 0$ as shown
|
|
in Fig.~\ref{fig:zeno}(a.4). In contrast to fully-projective
|
|
measurement, the existence of an intermediate (virtual) step in the
|
|
correlated tunnelling process builds long-range entanglement between
|
|
illuminated and non-illuminated regions as shown in
|
|
Fig.~\ref{fig:zeno}(a.3).
|
|
|
|
To make correlated tunneling visible even in the mean atom number, we
|
|
suppress the standard Bose-Hubbard dynamics by illuminating only the
|
|
even sites of the lattice in Fig.~\ref{fig:zeno}(b). Even though this
|
|
measurement scheme freezes both $N_\text{even}$ and $N_\text{odd}$,
|
|
atoms can slowly tunnel between the odd sites of the lattice, despite
|
|
them being spatially disconnected. This atom exchange spreads
|
|
correlations between non-neighbouring lattice sites on a time scale
|
|
$\sim \gamma/J^2$ as seen in Eq. \eqref{eq:hz}. The schematic
|
|
explanation of long-range correlated tunneling is presented in
|
|
Fig.~\ref{fig:zeno}(b.1): the atoms can tunnel only in pairs to assure
|
|
the globally conserved values of $N_\text{even}$ and $N_\text{odd}$,
|
|
such that one correlated tunneling event is represented by a pair of
|
|
one red and one blue arrow. Importantly, this scheme is fully
|
|
applicable for a lattice with large atom and site numbers, well beyond
|
|
the numerical example in Fig.~\ref{fig:zeno}(b.1), because as we can
|
|
see in Eq. \eqref{eq:hz} it is the geometry of quantum measurement that
|
|
assures this mode structure (in this example, two modes at odd and
|
|
even sites) and thus the underlying pairwise global tunnelling.
|
|
|
|
The scheme in Fig.~\ref{fig:zeno}(b.1) can help design a nonlocal
|
|
reservoir for the tunneling (or ``decay'') of atoms from one region to
|
|
another. For example, if the atoms are placed only at odd sites,
|
|
according to Eq. \eqref{eq:hz} their tunnelling is suppressed since the
|
|
multi-tunneling event must be successive, i.e.~an atom tunnelling into
|
|
a different mode, $\varphi^\prime$, must then also tunnel back into
|
|
its original mode, $\varphi$. If, however, one adds some atoms to even
|
|
sites (even if they are far from the initial atoms), the correlated
|
|
tunneling events become allowed and their rate can be tuned by the
|
|
number of added atoms. This resembles the repulsively bound pairs
|
|
created by local interactions \cite{winkler2006, folling2007}. In
|
|
contrast, here the atom pairs are nonlocally correlated due to the
|
|
global measurement. Additionally, these long-range correlations are a
|
|
consequence of the dynamics being constrained to a Zeno subspace: the
|
|
virtual processes allowed by the measurement entangle the spatial
|
|
modes nonlocally. Since the measurement only reveals the total number
|
|
of atoms in the illuminated sites, but not their exact distribution,
|
|
these multi-tunelling events cause the build-up of long-range
|
|
entanglement. This is in striking contrast to the entanglement caused
|
|
by local processes which can be very confined, especially in 1D where
|
|
it is typically short range. This makes numerical calculations of our
|
|
system for large atom numbers really difficult, since well-known
|
|
methods such as DMRG and MPS \cite{schollwock2005} (which are
|
|
successful for short-range interactions) rely on the limited extent of
|
|
entanglement.
|
|
|
|
The negative number correlations are typical for systems with
|
|
constraints (superselection rules) such as fixed atom number. The
|
|
effective dynamics due to our global, but spatially structured,
|
|
measurement introduces more general constraints to the evolution of
|
|
the system. For example, in Fig.~\ref{fig:zeno}(c) we show the
|
|
generation of positive number correlations shown in
|
|
Fig.~\ref{fig:zeno}(c.4) by freezing the atom number difference
|
|
between the sites ($N_\text{odd}-N_\text{even}$). Thus, atoms can only
|
|
enter or leave this region in pairs, which again is possible due to
|
|
correlated tunneling as seen in Figs.~\ref{fig:zeno}(c.1,c.2) and
|
|
manifests positive correlations. As in the previous example, two edge
|
|
modes in Fig.~\ref{fig:zeno}(c) can be considered as a nonlocal
|
|
reservoir for two central sites, where a constraint is applied. Note
|
|
that, using more modes, the design of higher-order multi-tunneling
|
|
events is possible.
|
|
|
|
This global pair tunneling may play a role of a building block for
|
|
more complicated many-body effects. For example, a pair tunneling
|
|
between the neighbouring sites has been recently shown to play
|
|
important role in the formation of new quantum phases, e.g., pair
|
|
superfluid \cite{sowinski2012} and lead to formulation of extended
|
|
Bose-Hubbard models \cite{omjyoti2015}. The search for novel
|
|
mechanisms providing long-range interactions is crucial in many-body
|
|
physics. One of the standard candidates is the dipole-dipole
|
|
interaction in, e.g., dipolar molecules, where the mentioned pair
|
|
tunneling between even neighboring sites is already considered to be
|
|
long-range \cite{sowinski2012,omjyoti2015}. In this context, our work
|
|
suggests a fundamentally different mechanism originating from quantum
|
|
optics: the backaction of global and spatially structured measurement,
|
|
which as we prove can successfully compete with other short-range
|
|
processes in many-body systems. This opens promising opportunities for
|
|
future research.
|
|
|
|
\section{Non-Hermitian Dynamics in the Quantum Zeno Limit}
|
|
|
|
In the previous section we provided a rather high-level analysis of
|
|
the strong measurement limit in our quantum gas model. We showed that
|
|
global measurement in the strong, but not projective, limit leads to
|
|
correlated tunnelling events which can be highly delocalised. Multiple
|
|
examples for different optical geometries and measurement operators
|
|
demonstrated the incredible felixbility and potential in engineering
|
|
dynamics for ultracold gases in an optical lattice. We also claimed
|
|
that the behaviour of the system is described by the Hamiltonian given
|
|
in Eq. \eqref{eq:hz}. Having developed a physical and intuitive
|
|
understanding of the dynamics in the quantum Zeno limit we will now
|
|
provide a more rigorous, low-level and fundamental understanding of
|
|
the process.
|
|
|
|
\subsection{Suppression of Coherences in the Density Matrix}
|
|
|
|
At this point we deviate from the quantum trajectory approach and we
|
|
resort to a master equation as introduced in section
|
|
\ref{sec:master}. We do this, because we have seen that the emergent
|
|
long-range correlated tunnelling is a feature of all trajectories and
|
|
mostly depends on the geometry of the measurement. Therefore, a
|
|
general approach starting from an unconditioned state should be able
|
|
to reveal these features. However, we will later make use of the fact
|
|
that we are in possession of a measurement record and obtain a
|
|
conditioned state. Furthermore, we first consider the most general
|
|
case of an open system subject to a quantum measurement and only limit
|
|
ourselves to the quantum gas model later on. This demonstrates that
|
|
the dynamics we observed in the previous section are a feature of
|
|
measurement rather than our specific model.
|
|
|
|
As introduced in section \ref{sec:master} we consider a state
|
|
described by the density matrix $\hat{\rho}$ whose isolated behaviour
|
|
is described by the Hamiltonian $\H_0$ and when measured the jump
|
|
operator $\c$ is applied to the state at each detection
|
|
\cite{MeasurementControl}. The master equation describing its time
|
|
evolution when we ignore the measurement outcomes is given by
|
|
\begin{equation}
|
|
\dot{\hat{\rho}} = -i [ \H_0 , \hat{\rho} ] + \c \hat{\rho} \cd - \frac{1}{2}(
|
|
\cd\c \hat{\rho} + \hat{\rho} \cd\c ).
|
|
\end{equation}
|
|
We also define $\c = \lambda \op$ and $\H_0 = \nu \h$. The exact
|
|
definition of $\lambda$ and $\nu$ is not so important as long as these
|
|
coefficients can be considered to be some measure of the relative size
|
|
of these operators. They would have to be determined on a case-by-case
|
|
basis, because the operators $\c$ and $\H_0$ may be unbounded. If
|
|
these operators are bounded, one can simply define them such that
|
|
$||\op|| \sim O(1)$ and $||\h|| \sim O(1)$. If they are unbounded, one
|
|
possible approach would be to identify the relevant subspace of which
|
|
dynamics we are interested in and scale the operators such that the
|
|
eigenvalues of $\op$ and $\h$ in this subspace are $\sim O(1)$.
|
|
|
|
We will once again use projectors $P_m$ which have no effect on states
|
|
within a degenerate subspace of $\c$ ($\op$) with eigenvalue $c_m$
|
|
($o_m$), but annihilate everything else. For convenience we will also
|
|
use the following definition $\hat{\rho}_{mn} = P_m \hat{\rho} P_n$.
|
|
Note that these are submatrices of the density matrix, which in
|
|
general are not single matrix elements. Therefore, we can write the
|
|
master equation that describes this open system as a set of equations
|
|
\begin{equation}
|
|
\label{eq:master}
|
|
\dot{\hat{\rho}}_{mn} = -i K P_m \left[ \h \sum_r \hat{\rho}_{rn}
|
|
- \sum_r \hat{\rho}_{mr} \h \right] P_n + \lambda^2 \left[ o_m
|
|
o_n^* - \frac{1}{2} \left( |o_m|^2 + |o_n|^2 \right) \right] \hat{\rho}_{mn},
|
|
\end{equation}
|
|
where the first term describes coherent evolution whereas the second
|
|
term causes dissipation.
|
|
|
|
First, note that for the density submatrices for which $m = n$,
|
|
$\hat{\rho}_{mm}$, the dissipative term vanishes. This means that
|
|
these submatrices are subject to coherent evolution only and do not
|
|
experience losses and they are thus decoherence free subspaces. It is
|
|
crucial to note that these submatrices are simply the density matrices
|
|
of the individual degenerate Zeno subspaces. Interestingly, any state
|
|
that consists only of these decoherence free subspaces, i.e.~
|
|
$\hat{\rho} = \sum_m \hat{\rho}_{mm}$, and that commutes with the
|
|
Hamiltonian, $[\hat{\rho}, \hat{H}_0] = 0$, will be a steady state.
|
|
This can be seen by substituting this ansatz into
|
|
Eq. \eqref{eq:master} which yields $\dot{\hat{\rho}}_{mn} = 0$ for all
|
|
$m$ and $n$. These states can be prepared dissipatively using known
|
|
techniques \cite{diehl2008}, but it is not required that the state be
|
|
a dark state of the dissipative operator as is usually the case.
|
|
|
|
Second, we consider a large detection rate, $\lambda^2 \gg \nu$, for
|
|
which the coherences, i.e.~ the density submatrices $\hat{\rho}_{mn}$
|
|
for which $m \ne n$, will be heavily suppressed by dissipation. We can
|
|
adiabatically eliminate these cross-terms by setting
|
|
$\dot{\hat{\rho}}_{mn} = 0$, to get
|
|
\begin{equation}
|
|
\label{eq:intermediate}
|
|
\hat{\rho}_{mn} = \frac{\nu}{\lambda^2} \frac{i P_m \left[ \h \sum_r \hat{\rho}_{rn} - \sum_r \hat{\rho}_{mr} \h \right] P_n } {o_m o_n^* - \frac{1}{2} \left( |o_m|^2 + |o_n|^2 \right)}
|
|
\end{equation}
|
|
which tells us that they are of order $\nu/\lambda^2 \ll
|
|
1$. Therefore, the resulting density matrix will be given by
|
|
$\hat{\rho} \approx \sum_m \hat{\rho}_{mm}$ which consists solely of
|
|
the individual Zeno subspace density matrices. One can easily recover
|
|
the projective Zeno limit by considering $\lambda \rightarrow \infty$
|
|
when all the subspaces completely decouple. This is exactly the
|
|
$\gamma \rightarrow \infty$ limit discussed in the previous
|
|
section. However, we have seen that it is crucial we only consider
|
|
$\lambda^2 \gg \nu$, but not infinite. If the subspaces do not
|
|
decouple completely, then transitions within a single subspace can
|
|
occur via other subspaces in a manner similar to Raman transitions. In
|
|
Raman transitions population is transferred between two states via a
|
|
third, virtual, state that remains empty throughout the process. By
|
|
avoiding the infinitely projective Zeno limit we open the option for
|
|
such processes to happen in our system where transitions within a
|
|
single Zeno subspace occur via a second, different, Zeno subspace even
|
|
though the occupation of the intermediate states will remian
|
|
negligible at all times.
|
|
|
|
A single quantum trajectory results in a pure state as opposed to the
|
|
density matrix and in general, there are many density matrices that
|
|
have non-zero and non-negligible $m = n$ submatrices,
|
|
$\hat{\rho}_{mm}$, even when the coherences are small. They correspond
|
|
to a mixed states containing many Zeno subspaces and it is not clear
|
|
what the pure states that make up these density matrices are. However,
|
|
we note that for a single pure state the density matrix can consist of
|
|
only a single diagonal submatrix $\hat{\rho}_{mm}$. To understand
|
|
this, consider the state $| \Phi \rangle$ and take it to span exactly
|
|
two distinct subspaces $P_a$ and $P_b$ ($a \ne b$). This wavefunction
|
|
can thus be written as
|
|
$| \Phi \rangle = P_a | \Phi \rangle + P_b | \Phi \rangle$. The
|
|
corresponding density matrix is given by
|
|
\begin{equation}
|
|
\hat{\rho}_\Psi = P_a | \Phi \rangle \langle \Phi | P_a + P_a | \Phi
|
|
\rangle \langle \Phi | P_b + P_b | \Phi \rangle \langle \Phi | P_a +
|
|
P_b | \Phi \rangle \langle \Phi | P_b.
|
|
\end{equation}
|
|
If the wavefunction has significant components in both subspaces then
|
|
in general the density matrix will not have negligible coherences,
|
|
$\hat{\rho}_{ab} = P_a | \Phi \rangle \langle \Phi | P_b$. A density
|
|
matrix with just diagonal components must be in either subspace $a$,
|
|
$| \Phi \rangle = P_a | \Phi \rangle$, or in subspace $b$,
|
|
$| \Phi \rangle = P_b | \Phi \rangle$. Therefore, a density matrix of
|
|
the form $\hat{\rho} = \sum_m \hat{\rho}_{mm}$ without any cross-terms
|
|
between different Zeno subspaces can only be composed of pure states
|
|
that each lie predominantly within a single subspace. However, because
|
|
we will not be dealing with the projective limit, the wavefunction
|
|
will in general not be entirely confined to a single Zeno subspace. We
|
|
have seen that the coherences are of order $\nu/\lambda^2$. This would
|
|
require the wavefunction components to satisfy
|
|
$P_a | \Phi \rangle \approx O(1)$ and
|
|
$P_b | \Phi \rangle \approx O(\nu/\lambda^2)$ (or vice-versa). This in
|
|
turn implies that the population of the states outside of the dominant
|
|
subspace (and thus the submatrix $\hat{\rho}_{bb}$) will be of order
|
|
$\langle \Phi | P_b^2 | \Phi \rangle \approx
|
|
O(\nu^2/\lambda^4)$. Therefore, these pure states, even though they
|
|
span multiple Zeno subspaces, cannot exist in a meaningful coherent
|
|
superposition in this limit. This means that a density matrix that
|
|
spans multiple Zeno subspaces has only classical uncertainty about
|
|
which subspace is currently occupied as opposed to the uncertainty due
|
|
to a quantum superposition. This is anlogous to the simple qubit
|
|
example we considered in section \ref{sec:master}.
|
|
|
|
\subsection{Quantum Measurement vs. Dissipation}
|
|
|
|
This is where quantum measurement deviates from dissipation. If we
|
|
have access to a measurement record we can infer which Zeno subspace
|
|
is occupied, because we know that only one of them can be occupied at
|
|
any time. We have seen that since the density matrix cross-terms are
|
|
small we know \emph{a priori} that the individual wavefunctions
|
|
comprising the density matrix mixture will not be coherent
|
|
superpositions of different Zeno subspaces and thus we only have
|
|
classical uncertainty which means we can resort to clasical
|
|
probability methods. Each individual experiment will at any time be
|
|
predominantly in a single Zeno subspace with small cross-terms and
|
|
negligible occupations in the other subspaces. With no measurement
|
|
record our density matrix would be a mixture of all these
|
|
possibilities. We can try and determine the Zeno subspace around which
|
|
the state evolves in a single experiment from the number of
|
|
detections, $m$, in time $t$.
|
|
|
|
The detection distribution on time-scales shorter than dissipation (so
|
|
we can approximate as if we were in a fully Zeno regime) can be
|
|
obtained by integrating over the detection times \cite{mekhov2009pra}
|
|
to get
|
|
\begin{equation}
|
|
P(m,t) = \sum_n \frac{[|c_n|^2 t]^m} {m!} e^{-|c_n|^2 t} \mathrm{Tr} (\rho_{nn}).
|
|
\end{equation}
|
|
For a state that is predominantly in one Zeno subspace, the
|
|
distribution will be approximately Poissonian (up to
|
|
$O(\nu^2 / \lambda^4)$, the population of the other
|
|
subspaces). Therefore, in a single experiment we will measure
|
|
$m = |c_0|^2t \pm \sqrt{|c_0|^2t}$ detections (note, we have assumed
|
|
$|c_0|^2 t$ is large enough to approximate the distribution as
|
|
normal. This is not necessary, we simply use it here to not have to
|
|
worry about the asymmetry in the deviation around the mean value). The
|
|
uncertainty does not come from the fact that $\lambda$ is not
|
|
infinite. The jumps are random events with a Poisson
|
|
distribution. Therefore, even in the full projective limit we will not
|
|
observe the same detection trajectory in each experiment even though
|
|
the system evolves in exactly the same way and remains in a perfectly
|
|
pure state.
|
|
|
|
If the basis of $\c$ is continuous (e.g. free particle position or
|
|
momentum) then the deviation around the mean will be our upper bound
|
|
on the deviation of the system from a pure state evolving around a
|
|
single Zeno subspace. However, continuous systems are beyond the scope
|
|
of this work and we will confine ourselves to discrete systems. Though
|
|
it is important to remember that continuous systems can be treated
|
|
this way, but the error estimate (and thus the mixedness of the state)
|
|
will be different.
|
|
|
|
For a discrete system it is easier to exclude all possibilities except
|
|
for one. The error in our estimate of $|c_0|^2$ in a single experiment
|
|
decreases as $1/\sqrt{t}$ and thus it can take a long time to
|
|
confidently determine $|c_0|^2$ to a sufficient precision this
|
|
way. However, since we know that it can only take one of the possible
|
|
values from the set $\{|c_n|^2 \}$ it is much easier to instead
|
|
exclude all the other values.
|
|
|
|
In an experiment we can use Bayes' theorem to infer the state of our
|
|
system as follows
|
|
\begin{equation}
|
|
p(c_n = c_0 | m) = \frac{ p(m | c_n = c_0) p(c_n = c_0) }{ p(m) },
|
|
\end{equation}
|
|
where $p(x)$ denotes the probability of the discrete event $x$ and
|
|
$p(x|y)$ the conditional probability of $x$ given $y$. We know that
|
|
$p(m | c_n = c_0)$ is simply given by a Poisson distribution with mean
|
|
$|c_0|^2 t$. $p(m)$ is just a normalising factor and $p(c_n = c_0)$ is
|
|
our \emph{a priori} knowledge of the state. Therefore, one can get the
|
|
probability of being in the right Zeno subspace from
|
|
\begin{align}
|
|
p(c_n & = c_0 | m) = \frac{ p_0(c_n = c_0) \frac{ \left( |c_0|^2 t
|
|
\right)^{2m} } {m!} e^{-|c_0|^2 t}} {\sum_n p_0(c_n) \frac{
|
|
\left( |c_n|^2 t \right)^{2m} } {m!} e^{-|c_n|^2 t}} \nonumber \\
|
|
& = p_0(c_n = c_0) \left[ \sum_n p_0(c_n) \left( \frac{
|
|
|c_n|^2 } { |c_0|^2 } \right)^{2m} e^{\left( |c_0|^2 -
|
|
|c_n|^2 \right) t} \right]^{-1},
|
|
\end{align}
|
|
where $p_0$ denotes probabilities at $t = 0$. In a real experiment one
|
|
could prepare the initial state to be close to the Zeno subspace of
|
|
interest and thus it would be easier to deduce the state. Furthermore,
|
|
in the middle of an experiment if we have already established the Zeno
|
|
subspace this will be reflected in these \emph{a priori} probabilities
|
|
again making it easier to infer the correct subspace. However, we will
|
|
consider the worst case scenario which might be useful if we don't
|
|
know the initial state or if the Zeno subspace changes during the
|
|
experiment, a uniform $p_0(c_n)$.
|
|
|
|
This probability is a rather complicated function as $m$ is a
|
|
stochastic quantity that also increases with $t$. We want it to be as
|
|
close to $1$ as possible. In order to devise an appropriate condition
|
|
for this we note that in the first line all terms in the denominator
|
|
are Poisson distributions of $m$. Therefore, if the mean values
|
|
$|c_n|^2 t$ are sufficiently spaced out, only one of the terms in the
|
|
sum will be significant for a given $m$ and if this happens to be the
|
|
one that corresponds to $c_0$ we get a probability close to
|
|
unity. Therefore, we set the condition such that it is highly unlikely
|
|
that our measured $m$ could be produced by two different distributions
|
|
\begin{align}
|
|
\sqrt{|c_0|^2 t} \ll ||c_0|^2 - |c_n|^2| t, \forall n \ne 0 \\
|
|
\sqrt{|c_n|^2 t} \ll ||c_0|^2 - |c_n|^2| t, \forall n \ne 0
|
|
\end{align}
|
|
The left-hand side is the standard deviation of $m$ if the system was
|
|
in subspace $P_0$ or $P_n$. The right-hand side is the difference in
|
|
the mean detections between the subspace $n$ and the one we are
|
|
interested in. The condition becomes more strict if the subspaces
|
|
become less distinguishable as it becomes harder to confidently
|
|
determine the correct state. Once again, using $\c = \lambda \hat{o}$
|
|
where $\hat{o} \sim O(1)$ we get
|
|
\begin{equation}
|
|
t \gg \frac{1}{\lambda^2} \frac{|o_{0,n}|^2} {(|o_0|^2 - |o_n|^2|)^2}.
|
|
\end{equation}
|
|
Since detections happen on average at an average rate of order
|
|
$\lambda^2$ we only need to wait for a few detections to satisfy this
|
|
condition. Therefore, we see that even in the worst case scenario of
|
|
complete ignorance of the state of the system we can very easily
|
|
determine the correct subspace. Once it is established for the first
|
|
time, the \emph{a priori} information can be updated and it will
|
|
become even easier to monitor the system.
|
|
|
|
However, it is important to note that physically once the quantum
|
|
jumps deviate too much from the mean value the system is more likely
|
|
to change the Zeno subspace (due to measurement backaction) and the
|
|
detection rate will visibly change. Therefore, if we observe a
|
|
consistent detection rate it is extremely unlikely that it can be
|
|
produced by two different Zeno subspaces so in fact it is even easier
|
|
to determine the correct state, but the above estimate serves as a
|
|
good lower bound on the necessary detection time.
|
|
|
|
Having derived the necessary conditions to confidently determine which
|
|
Zeno subspace is being observed in the experiment we can make another
|
|
approximation thanks to measurement which would be impossible in a
|
|
purely dissipative open system. If we observe a number of detections
|
|
consistent with the subspace $P_m = P_0$ we can set
|
|
$\hat{\rho}_{mn} \approx 0$ for all cases when both $m \ne 0$ and
|
|
$n \ne 0$ leaving our density matrix in the form
|
|
\begin{equation}
|
|
\label{eq:approxrho}
|
|
\hat{\rho} = \hat{\rho}_{00} + \sum_{r\ne0} (\hat{\rho}_{0r} +
|
|
\hat{\rho}_{r0}).
|
|
\end{equation}
|
|
We can do this, because the other states are inconsistent with the
|
|
measurement record. We know from the previous section that the system
|
|
must lie predominantly in only one of the Zeno subspaces and when that
|
|
is the case, $\hat{\rho}_{0r} \approx O(\nu/\lambda^2)$ and for
|
|
$m \ne 0$ and $n \ne 0$ we have
|
|
$\hat{\rho}_{mn} \approx O(\nu^2/\lambda^4)$. Therefore, this amounts
|
|
to keeping first order terms in $\nu/\lambda^2$ in our approximation.
|
|
|
|
This is a crucial step as all $\hat{\rho}_{mm}$ matrices are
|
|
decoherence free subspaces and thus they can all coexist in a mixed
|
|
state decreasing the purity of the system without
|
|
measurement. Physically, this means we exclude trajectories in which
|
|
the Zeno subspace has changed (measurement isn't fully projective). By
|
|
substituting Eq. \eqref{eq:intermediate} into Eq. \eqref{eq:master} we
|
|
see that this happens at a rate of $\nu^2 / \lambda^2$. However, since
|
|
the two measurement outcomes cannot coexist any transition between
|
|
them happens in discrete transitions (which we know about from the
|
|
change in the detection rate as each Zeno subspace will correspond to
|
|
a different rate) and not as continuous coherent evolution. Therefore,
|
|
we can postselect in a manner similar to Refs. \cite{otterbach2014,
|
|
lee2014prx, lee2014prl}, but our requirements are significantly more
|
|
relaxed - we do not require a specific single trajectory, only that it
|
|
remains within a Zeno subspace. Furthermore, upon reaching a steady
|
|
state, these transitions become impossible as the coherences
|
|
vanish. This approximation is analogous to optical Raman transitions
|
|
where the population of the excited state is neglected. Here, we can
|
|
make a similar approximation and neglect all but one Zeno subspace
|
|
thanks to the additional knowledge we gain from knowing the
|
|
measurement outcomes.
|
|
|
|
\subsection{The Non-Hermitian Hamiltonian}
|
|
|
|
Rewriting the master equation using $\c = c_0 + \delta \c$, where
|
|
$c_0$ is the eigenvalue corresponding to the eigenspace defined by the
|
|
projector $P_0$ which we used to obtain the density matrix in
|
|
Eq. \eqref{eq:approxrho}, we get
|
|
\begin{equation}
|
|
\label{eq:finalrho}
|
|
\dot{\hat{\rho}} = -i \left( \H_\mathrm{eff} \hat{\rho} - \hat{\rho}
|
|
\H_\mathrm{eff}^\dagger \right) + \delta \c \hat{\rho} \delta \cd,
|
|
\end{equation}
|
|
\begin{equation}
|
|
\label{eq:Ham}
|
|
\H_\mathrm{eff} = \H_0 + i \left( c_0^*\c - \frac{|c_0|^2}{2} - \frac{\cd\c}{2} \right).
|
|
\end{equation}
|
|
The first term in Eq. \eqref{eq:finalrho} describes coherent evolution
|
|
due to the non-Hermitian Hamiltonian $\H_\mathrm{eff}$ and the second
|
|
term is decoherence due to our ignorance of measurement outcomes. When
|
|
we substitute our approximation of the density matrix
|
|
$\hat{\rho} = \hat{\rho}_{00} + \sum_{r\ne0} (\hat{\rho}_{0r} +
|
|
\hat{\rho}_{r0})$ into Eq. \eqref{eq:finalrho}, the last term
|
|
vanishes, $\delta \c \hat{\rho} \delta \cd = 0$. This happens, because
|
|
$\delta \c P_0 \hat{\rho} = \hat{\rho} P_0 \delta \c^\dagger = 0$. The
|
|
projector annihilates all states except for those with eigenvalue
|
|
$c_0$ and so the operator $\delta \c = \c - c_0$ will always evaluate
|
|
to $c_0 - c_0 = 0$. Recall that we defined
|
|
$\hat{\rho}_{mn} = P_m \hat{\rho} P_n$ which means that every term in
|
|
our approximate density matrix contains the projector $P_0$. However,
|
|
it is important to note that this argument does not apply to other
|
|
second order terms in the master equation, because some terms only
|
|
have the projector $P_0$ applied from one side,
|
|
e.g.~$\hat{\rho}_{0m}$. The term $\delta \c \hat{\rho} \delta \cd$
|
|
applies the fluctuation operator from both sides so it does not matter
|
|
in this case, but it becomes relevant for terms such as
|
|
$ \hat{\rho} \delta \cd \delta \c$. It is important to note that this
|
|
term does not automatically vanish, but when the explicit form of our
|
|
approximate density matrix is inserted, it is in fact zero. Therefore,
|
|
we can omit this term using the information we gained from
|
|
measurement, but keep other second order terms, such as
|
|
$\delta \cd \delta \c \rho$ in the Hamiltonian which are the origin of
|
|
other second-order dynamics. This could not be the case in a
|
|
dissipative system.
|
|
|
|
Ultimately we find that a system under continuous measurement for
|
|
which $\lambda^2 \gg \nu$ in the Zeno subspace $P_0$ is described by
|
|
the deterministic non-Hermitian Hamiltonian $\H_\mathrm{eff}$ in
|
|
Eq. \eqref{eq:Ham} and thus obeys the following Schr\"{o}dinger
|
|
equation
|
|
\begin{equation}
|
|
i \frac{\mathrm{d} | \Psi \rangle}{\mathrm{d}t} = \left[\H_0 + i \left(
|
|
c_0^*\c - \frac{|c_0|^2}{2} - \frac{\cd\c}{2} \right) \right] |
|
|
\Psi \rangle.
|
|
\end{equation}
|
|
Of the three terms in the parentheses the first two represent the
|
|
effects of quantum jumps due to detections (which one can think of as
|
|
`reference frame' shifts between different degenerate eigenspaces) and
|
|
the last term is the non-Hermitian decay due to information gain from
|
|
no detections. It is important to emphasize that even though we
|
|
obtained a deterministic equation, we have not neglected the
|
|
stochastic nature of the detection events. The detection trajectory
|
|
seen in an experiment will have fluctuations around the mean
|
|
determined by the Zeno subspace, but there simply are many possible
|
|
measurement records with the same outcome. This is just like the fully
|
|
projective Zeno limit where the system remains perfectly pure in one
|
|
of the possible projections, but the detections remain randomly
|
|
distributed in time.
|
|
|
|
One might then be concerned that purity is preserved even though we
|
|
might be averaging over many trajectories within this Zeno
|
|
subspace. We have neglected the small terms $\hat{\rho}_{m,n}$
|
|
($m,n \ne 0$) which are $O(\nu^2/\lambda^4)$ and thus they are not
|
|
correctly accounted for by our approximation. This means that we have
|
|
an $O(\nu^2/\lambda^4)$ error in our density matrix. The purity
|
|
given by
|
|
\begin{equation}
|
|
\mathrm{Tr}(\hat{\rho}^2) = \mathrm{Tr}(\hat{\rho}^2_{00} + \sum_{m \ne
|
|
0} \hat{\rho}_{0m}\hat{\rho}_{m0}) + \mathrm{Tr}(\sum_{m,n\ne0}
|
|
\hat{\rho}_{mn} \hat{\rho}_{nm})
|
|
\end{equation}
|
|
where the second term contains the terms not accounted for by our
|
|
approximation thus introduces an $O(\nu^4/\lambda^8)$
|
|
error. Therefore, this discrepancy is negligible in our
|
|
approximation. The pure state predicted by $\H_\mathrm{eff}$ is only
|
|
an approximation, albeit a good one, and the real state will be mixed
|
|
to a small extent. Whilst perfect purity within the Zeno subspace
|
|
$\hat{\rho}_{00}$ is expected due to the measurement's strong
|
|
decoupling effect, the nearly perfect purity when transitions outside
|
|
the Zeno subspace are included is a nontrivial result. Similarly, in
|
|
Raman transitions the population of the neglected excited state is
|
|
also non-zero, but negligible. Furthermore, this equation does not
|
|
actually require the adiabatic elimination used in
|
|
Eq. \eqref{eq:intermediate} (we only used it to convince ourselves
|
|
that the coherences are small) and such situations may be considered
|
|
provided all approximations remain valid. In a similar way the limit
|
|
of linear optics is derived from the physics of a two-level nonlinear
|
|
medium, when the population of the upper state is neglected and the
|
|
adiabatic elimination of coherences is not required.
|
|
|
|
\subsection{Non-Hermitian Dynamics in Ultracold Gases}
|
|
|
|
We finally return to our quantum gas model inside of a cavity. We
|
|
start by considering the simplest case of a global multi-site
|
|
measurement of the form $\hat{D} = \hat{N}_K = \sum_i^K \n_i$, where
|
|
the sum is over $K$ illuminated sites. The effective Hamiltonian
|
|
becomes
|
|
\begin{equation}
|
|
\label{eq:nHH2}
|
|
\hat{H}_\mathrm{eff} = \hat{H}_0 - i \gamma \left( \delta \hat{N}_K \right)^2,
|
|
\end{equation}
|
|
where $ \delta \hat{N}_K = \hat{N}_K - N^0_K$ and $N^0_K$ is the Zeno
|
|
subspace eigenvalue. It is now obvious that continuous measurement
|
|
squeezes the fluctuations in the measured quantity, as expected, and
|
|
that the only competing process is the system's own dynamics.
|
|
|
|
In this case, if we adiabatically eliminate the density matrix
|
|
cross-terms and substitute Eq. \eqref{eq:intermediate} into
|
|
Eq. \eqref{eq:master} for this system we obtain an effective
|
|
Hamiltonian within the Zeno subspace defined by $N_K$
|
|
\begin{equation}
|
|
\H_\varphi = P_0 \left[ \H_0 - i \frac{J^2}{\gamma}
|
|
\sum_\varphi \sum_{\substack{\langle i \in \varphi, j \in \varphi^\prime
|
|
\rangle \\ \langle k \in \varphi^\prime, l \in \varphi
|
|
\rangle}} b^\dagger_i b_j b^\dagger_k b_l \right] P_0,
|
|
\end{equation}
|
|
where $\varphi$ denotes a set of sites belonging to a single mode and
|
|
$\varphi^\prime$ is the set's complement (e.g. odd and even or
|
|
illuminated and non-illuminated sites) and $P_0$ is the projector onto
|
|
the eigenspace with $N_K^0$ atoms in the illuminated area. We focus on
|
|
the case when the second term is not only significant, but also leads
|
|
to dynamics within a Zeno subspace that are not allowed by
|
|
conventional quantum Zeno dynamics accounted for by the first
|
|
term. The second term represents second-order transitions via other
|
|
subspaces which act as intermediate states much like virtual states in
|
|
optical Raman transitions. This is in contrast to the conventional
|
|
understanding of the Zeno dynamics for infinitely frequent projective
|
|
measurements (corresponding to $\gamma \rightarrow \infty$) where such
|
|
processes are forbidden \cite{facchi2008}. Thus, it is the weak
|
|
quantum measurement that effectively couples the states. Note that
|
|
this is a special case of the equation in Eq. \eqref{eq:hz} which can
|
|
be obtained by considering a more general two mode setup.
|
|
|
|
\subsection{Small System Example}
|
|
|
|
To get clear physical insight, we initially consider three atoms in
|
|
three sites and choose our measurement operator such that
|
|
$\hat{D} = \n_2$, i.e.~only the middle site is subject to measurement,
|
|
and the Zeno subspace defined by $n_2 = 1$. Such an illumination
|
|
pattern can be achieved with global addressing by crossing two beams
|
|
and placing the nodes at the odd sites and the antinodes at even
|
|
sites. This means that $P_0 \H_0 P_0 = 0$. However, the
|
|
first and third sites are connected via the second term. Diagonalising
|
|
the Hamiltonian reveals that out of its ten eigenvalues all but three
|
|
have a significant negative imaginary component of the order $\gamma$
|
|
which means that the corresponding eigenstates decay on a time scale
|
|
of a single quantum jump and thus quickly become negligible. The three
|
|
remaining eigenvectors are dominated by the linear superpositions of
|
|
the three Fock states $|2,1,0 \rangle$, $|1, 1, 1 \rangle$, and
|
|
$|0,1,2 \rangle$. Whilst it is not surprising that these components
|
|
are the only ones that remain as they are the only ones that actually
|
|
lie in the Zeno subspace $n_2 = 1$, it is impossible to solve the full
|
|
dynamics by just considering these Fock states alone as they are not
|
|
coupled to each other in $\hat{H}_0$. The components lying outside of
|
|
the Zeno subspace have to be included to allow intermediate steps to
|
|
occur via states that do not belong in this subspace, much like
|
|
virtual states in optical Raman transitions.
|
|
|
|
An approximate solution for $U=0$ can be written for the
|
|
$\{|2,1,0 \rangle, |1,1,1 \rangle, |0,1,2 \rangle\}$ subspace by
|
|
multiplying each eigenvector with its corresponding time evolution
|
|
\begin{equation}
|
|
| \Psi(t) \rangle \propto \left( \begin{array}{c}
|
|
z_1 + \sqrt{2} z_2 e^{-6 J^2 t / \gamma} + z_3 e^{-12 J^2 t / \gamma} \\
|
|
-\sqrt{2} \left(z_1 - z_3 e^{-12 J^2 t / \gamma} \right) \\
|
|
z_1 - \sqrt{2} z_2 e^{-6 J^2 t / \gamma} + z_3 e^{-12 J^2 t /
|
|
\gamma} \\
|
|
\end{array}
|
|
\right),
|
|
\end{equation}
|
|
where $z_i$ denote the overlap between the eigenvectors and the
|
|
initial state, $z_i = \langle v_i | \Psi (0) \rangle$, with
|
|
$| v_1 \rangle = (1, -\sqrt{2}, 1)/2$,
|
|
$| v_2 \rangle = (1, 0, -1)/\sqrt{2}$, and
|
|
$| v_3 \rangle = (1, \sqrt{2}, 1)/2$. The steady state as
|
|
$t \rightarrow \infty$ is given by
|
|
$| v_1 \rangle = (1, -\sqrt{2}, 1)/2$. This solution is illustrated in
|
|
Fig. \ref{fig:comp} which clearly demonstrates dynamics beyond the
|
|
canonical understanding of quantum Zeno dynamics as tunnelling occurs
|
|
between states coupled via a different Zeno subspace.
|
|
|
|
\begin{figure}[hbtp!]
|
|
\includegraphics[width=\linewidth]{comp}
|
|
\caption[Fock State Populations in a Zeno
|
|
Subspace]{Populations of the Fock states in the Zeno subspace
|
|
for $\gamma/J = 100$ and initial state $| 2,1,0 \rangle$. It
|
|
is clear that quantum Zeno dynamics occurs via Raman-like
|
|
processes even though none of these states are connected in
|
|
$\hat{H}_0$. The dynamics occurs via virtual intermediate
|
|
states outside the Zeno subspace. The system also tends to a
|
|
steady state which minimises tunnelling effectively
|
|
suppressing fluctuations. The lines are solutions to the
|
|
non-Hermitian Hamiltonian, and the dots are points from a
|
|
stochastic trajectory calculation.\label{fig:comp}}
|
|
\end{figure}
|
|
|
|
\subsection{Steady State of non-Hermitian Dynamics}
|
|
|
|
A distinctive difference between Bose-Hubbard model ground states and
|
|
the final steady state,
|
|
$| \Psi \rangle = [|2,1,0 \rangle - \sqrt{2} |1,1,1\rangle +
|
|
|0,1,2\rangle]/2$, is that its components are not in phase. Squeezing
|
|
due to measurement naturally competes with inter-site tunnelling which
|
|
tends to spread the atoms. However, from Eq. \eqref{eq:nHH2} we see
|
|
the final state will always be the eigenvector with the smallest
|
|
fluctuations as it will have an eigenvalue with the largest imaginary
|
|
component. This naturally corresponds to the state where tunnelling
|
|
between Zeno subspaces (here between every site) is minimised by
|
|
destructive matter-wave interference, i.e.~the tunnelling dark state
|
|
defined by $\hat{T} |\Psi \rangle = 0$, where
|
|
$\hat{T} = \sum_{\langle i, j \rangle} \bd_i b_j$. This is simply the
|
|
physical interpretation of the steady states we predicted for
|
|
Eq. \eqref{eq:master}. Crucially, this state can only be reached if
|
|
the dynamics aren't fully suppressed by measurement and thus,
|
|
counter-intuitively, the atomic dynamics cooperate with measurement to
|
|
suppress itself by destructive interference. Therefore, this effect is
|
|
beyond the scope of traditional quantum Zeno dynamics and presents a
|
|
new perspective on the competition between a system's short-range
|
|
dynamics and global measurement backaction.
|
|
|
|
We now consider a one-dimensional lattice with $M$ sites so we extend
|
|
the measurement to $\hat{D} = \N_\text{even}$ where every even site is
|
|
illuminated. The wavefunction in a Zeno subspace must be an
|
|
eigenstate of $\c$ and we combine this with the requirement for it to
|
|
be in the dark state of the tunnelling operator (eigenstate of $\H_0$
|
|
for $U = 0$) to derive the steady state. These two conditions in
|
|
momentum space are
|
|
\begin{align}
|
|
\hat{T} | \Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_k -
|
|
\bd_{q} b_{q} \right] \cos(ka) |\Psi \rangle = 0, \\
|
|
\Delta \N |\Psi \rangle = \sum_{\text{RBZ}} \left[ \bd_k b_{-q} +
|
|
\bd_{-q} b_k \right] | \Psi \rangle= \Delta N |\Psi \rangle,
|
|
\end{align}
|
|
where $b_k = \frac{1}{\sqrt{M}} \sum_j e^{i k j a} b_j$,
|
|
$\Delta \hat{N} = \hat{D} - N/2$, $q = \pi/a - k$, $a$ is the lattice
|
|
spacing, $N$ the total atom number, and we perform summations over the
|
|
reduced Brillouin zone (RBZ), $-\pi/2a < k \le \pi/2a$, as the
|
|
symmetries of the system are clearer this way. Now we define
|
|
\begin{equation}
|
|
\hat{\alpha}_k^\dagger = \bd_k \bd_q - \bd_{-k} \bd_{-q},
|
|
\end{equation}
|
|
\begin{equation}
|
|
\hat{\beta}_\varphi^\dagger = \bd_{\pi/2a} + \varphi \bd_{-\pi/2a},
|
|
\end{equation}
|
|
where $\varphi = \Delta N / | \Delta N |$, which create the smallest
|
|
possible states that satisfy the two equations for $\Delta N = 0$ and
|
|
$\Delta N \ne 0$ respectively. Therefore, by noting that
|
|
\begin{align}
|
|
\left[ \hat{T}, \hat{\alpha}_k^\dagger \right] & = 0, \\
|
|
\left[ \hat{T}, \hat{\beta}_\varphi^\dagger \right] & = 0, \\
|
|
\left[ \Delta \N, \hat{\alpha}_k^\dagger \right] & = 0, \\
|
|
\left[ \Delta \N, \hat{\beta}_\varphi^\dagger \right] & = \varphi
|
|
\hat{\beta}_\varphi^\dagger,
|
|
\end{align}
|
|
we can now write the equation for the $N$-particle steady state
|
|
\begin{equation}
|
|
\label{eq:ss}
|
|
| \Psi \rangle \propto \left[ \prod_{i=1}^{(N - |\Delta N|)/2}
|
|
\left( \sum_{k = 0}^{\pi/2a} \phi_{i,k} \hat{\alpha}_k^\dagger
|
|
\right) \right] \left( \hat{\beta}_\varphi^\dagger \right)^{|
|
|
\Delta N |} | 0 \rangle,
|
|
\end{equation}
|
|
where $\phi_{i,k}$ are coefficients that depend on the trajectory
|
|
taken to reach this state and $|0 \rangle$ is the vacuum state defined
|
|
by $b_k |0 \rangle = 0$. Since this a dark state (an eigenstate of
|
|
$\H_0$) of the atomic dynamics, this state will remain stationary even
|
|
with measurement switched-off. Interestingly, this state is very
|
|
different from the ground states of the Bose-Hubbard Hamiltonian, it
|
|
is even orthogonal to the superfluid state, and thus it cannot be
|
|
obtained by cooling or projecting from an initial ground state. The
|
|
combination of tunnelling with measurement is necessary.
|
|
|
|
\begin{figure}[hbtp!]
|
|
\includegraphics[width=\linewidth]{steady}
|
|
\caption[Non-Hermitian Steady State]{A trajectory simulation
|
|
for eight atoms in eight sites, initially in
|
|
$|1,1,1,1,1,1,1,1 \rangle$, with periodic boundary
|
|
conditions and $\gamma/J = 100$. (a), The fluctuations in
|
|
$\c$ where the stochastic nature of the process is clearly
|
|
visible on a single trajectory level. However, the general
|
|
trend is captured by the non-Hermitian Hamiltonian. (b), The
|
|
local density variance. Whilst the fluctuations in the
|
|
global measurement operator decrease, the fluctuations in
|
|
local density increase due to tunnelling via states outside
|
|
the Zeno subspace. (c), The momentum distribution. The
|
|
initial Fock state has a flat distribution which with time
|
|
approaches the steady state distribution of two identical
|
|
and symmetric distributions centred at $k = \pi/2a$ and
|
|
$k = -\pi/2a$.\label{fig:steady}}
|
|
\end{figure}
|
|
|
|
In order to prepare the steady state one has to run the experiment and
|
|
wait until the photocount rate remains constant for a sufficiently
|
|
long time. Such a trajectory is illustrated in Fig. \ref{fig:steady}
|
|
and compared to a deterministic trajectory calculated using the
|
|
non-Hermitian Hamiltonian. It is easy to see from
|
|
Fig. \ref{fig:steady}(a) how the stochastic fluctuations around the
|
|
mean value of the observable have no effect on the general behaviour
|
|
of the system in the strong measurement regime. By discarding these
|
|
fluctuations we no longer describe a pure state, but we showed how
|
|
this only leads to a negligible error. Fig. \ref{fig:steady}(b) shows
|
|
the local density variance in the lattice. Not only does it grow
|
|
showing evidence of tunnelling between illuminated and non-illuminated
|
|
sites, but it grows to significant values. This is in contrast to
|
|
conventional quantum Zeno dynamics where no tunnelling would be
|
|
allowed at all. Finally, Fig. \ref{fig:steady}(c) shows the momentum
|
|
distribution of the trajectory. We can clearly see that it deviates
|
|
significantly from the initial flat distribution of the Fock
|
|
state. Furthermore, the steady state does not have any atoms in the
|
|
$k=0$ state and thus is orthogonal to the superfluid state as
|
|
discussed.
|
|
|
|
To obtain a state with a specific value of $\Delta N$ postselection
|
|
may be necessary, but otherwise it is not needed. The process can be
|
|
optimised by feedback control since the state is monitored at all
|
|
times \cite{ivanov2014, mazzucchi2016feedback,
|
|
ivanonv2016}. Furthermore, the form of the measurement operator is
|
|
very flexible and it can easily be engineered by the geometry of the
|
|
optical setup \cite{elliott2015, mazzucchi2016} which can be used to
|
|
design a state with desired properties.
|
|
|
|
\section{Conclusions}
|
|
|
|
In this chapter we have demonstrated that global quantum measurement
|
|
backaction can efficiently compete with standard local processes in
|
|
many-body systems. This introduces a completely new energy and time
|
|
scale into quantum many-body research. This is made possbile by the
|
|
ability to structure the spatial profile of the measurement on a
|
|
microscopic scale comparable to the lattice period without the need
|
|
for single site addressing. The extreme flexibility of the setup
|
|
considered allowed us to effectively tailor long-range entanglement
|
|
and correlations present in the system. We showed that the competition
|
|
between the global backaction and usual atomic dynamics leads to the
|
|
production of spatially multimode macroscopic superpositions which
|
|
exhibit large-scale oscillatory dynamics which could be used for
|
|
quantum information and metrology. We subsequently demonstrated that
|
|
when on-site atomic interactions are introduced the dynamics become
|
|
much more complicated with different regimes of behaviour where
|
|
measurement and interactions can either compete or cooperate. In the
|
|
strong measurement regime we showed that conventional quantum Zeno
|
|
dynamics can be realised, but more interestingly, by considering a
|
|
strong, but not projective, limit of measurement we observe a new type
|
|
of nonlocal dynamics. It turns out that a global measurement scheme
|
|
leads to correlations between spatially separated tunnelling events
|
|
which conserve the Zeno subspace via Raman-like processes which would
|
|
be forbidden in the canonical fully projective limit. We subsequently
|
|
presented a rigorous analysis of the underlying process of this new
|
|
type of quantum Zeno dynamics in which we showed that in this limit
|
|
quantum trajectories can be described by a deterministic non-Hermitian
|
|
Hamiltonian. In contrast to previous works, it is independent of the
|
|
underlying system and there is no need to postselect a particular
|
|
exotic trajectory \cite{lee2014prx, lee2014prl}. Finally, we have
|
|
shown that the system will always tend towards the eigenstate of the
|
|
Hamiltonian with the best squeezing of the observable and the atomic
|
|
dynamics, which normally tend to spread the distribution, cooperates
|
|
with measurement to produce a state in which tunnelling is suppressed
|
|
by destructive matter-wave interference. A dark state of the
|
|
tunnelling operator will have zero fluctuations and we provided an
|
|
expression for the steady state which is significantly different from
|
|
the ground state of the Hamiltonian. This is in contrast to previous
|
|
works on dissipative state preparation where the steady state had to
|
|
be a dark state of the measurement operator \cite{diehl2008}.
|
|
|
|
Such globally paired tunnelling due to a fundamentally new phenomenon,
|
|
global quantum measurement backaction, can enrich the physics of
|
|
long-range correlated systems beyond relatively short-range
|
|
interactions expected from standard dipole-dipole interactions
|
|
\cite{sowinski2012, omjyoti2015}. These nonlocal high-order processes
|
|
entangle regions of the optical lattice that are disconnected by the
|
|
measurement. Using different detection schemes, we showed how to
|
|
tailor density-density correlations between distant lattice
|
|
sites. Quantum optical engineering of nonlocal coupling to
|
|
environment, combined with quantum measurement, can allow the design
|
|
of nontrivial system-bath interactions, enabling new links to quantum
|
|
simulations~\cite{stannigel2013} and thermodynamics~\cite{erez2008}
|
|
and extend these directions to the field of non-Hermitian quantum
|
|
mechanics, where quantum optical setups are particularly
|
|
promising~\cite{lee2014prl}. Importantly, both systems and baths,
|
|
designed by our method, can be strongly correlated systems with
|
|
internal long-range entanglement.
|