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