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Some more typo fixes

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Wojciech Kozlowski 2016-12-15 00:49:22 +00:00
parent 59f462d791
commit 1a62ccd830
2 changed files with 26 additions and 27 deletions
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35
Chapter5/chapter5.tex
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@ -819,24 +819,23 @@ unitary evolution is uninhibited within such a degenerate subspace,
called the Zeno subspace \cite{facchi2008, raimond2010, raimond2012, called the Zeno subspace \cite{facchi2008, raimond2010, raimond2012,
signoles2014}. signoles2014}.
These effects can be easily seen in our model when These effects can be easily seen in our model when $\gamma \rightarrow
$\gamma \rightarrow \infty$. The system will be projected into one or \infty$. The system will be projected into one or more degenerate
more degenerate eigenstates of $\cd \c$, $| \psi_i \rangle$, for which eigenstates of $\cd \c$, $| \psi_i \rangle$, for which we define the
we define the projector projector $P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle \langle
$P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle$ where $\varphi$ \psi_i |$ where $\varphi$ denotes a single degenerate subspace. The
denotes a single degenerate subspace. The Zeno subspace is determined Zeno subspace is determined randomly as per the Copenhagen postulates
randomly as per the Copenhagen postulates and thus it depends on the and thus it depends on the initial state. If the projection is into
initial state. If the projection is into the subspace $\varphi$, the the subspace $\varphi$, the subsequent evolution is described by the
subsequent evolution is described by the projected Hamiltonian projected Hamiltonian $P_{\varphi} \hat{H}_0 P_{\varphi}$. We have
$P_{\varphi} \hat{H}_0 P_{\varphi}$. We have used the original used the original Hamiltonian, $\hat{H}_0$, without the non-Hermitian
Hamiltonian, $\hat{H}_0$, without the non-Hermitian term or the term or the quantum jumps as their combined effect is now described by
quantum jumps as their combined effect is now described by the the projectors. Physically, in our model of ultracold bosons trapped
projectors. Physically, in our model of ultracold bosons trapped in a in a lattice this means that tunnelling between different spatial
lattice this means that tunnelling between different spatial modes is modes is completely suppressed since this process couples eigenstates
completely suppressed since this process couples eigenstates belonging belonging to different Zeno subspaces. If a small connected part of
to different Zeno subspaces. If a small connected part of the lattice the lattice was illuminated uniformly such that $\hat{D} = \hat{N}_K$
was illuminated uniformly such that $\hat{D} = \hat{N}_K$ then then tunnelling would only be prohibited between the illuminated and
tunnelling would only be prohibited between the illuminated and
unilluminated areas, but dynamics proceeds normally within each zone unilluminated areas, but dynamics proceeds normally within each zone
separately. However, the geometric patterns we have in which the modes separately. However, the geometric patterns we have in which the modes
are spatially delocalised in such a way that neighbouring sites never are spatially delocalised in such a way that neighbouring sites never

18
Chapter6/chapter6.tex
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@ -287,15 +287,15 @@ open systems described by the density matrix, $\hat{\rho}$,
\left[ \c \hat{\rho} \cd - \frac{1}{2} \left( \cd \c \hat{\rho} + \hat{\rho} \left[ \c \hat{\rho} \cd - \frac{1}{2} \left( \cd \c \hat{\rho} + \hat{\rho}
\cd \c \right) \right]. \cd \c \right) \right].
\end{equation} \end{equation}
The following results will not depend on the nature nor exact form of The following results will not depend on the nature nor the exact form
the jump operator $\c$. However, whenever we refer to our simulations of the jump operator $\c$. However, whenever we refer to our
or our model we will be considering $\c = \sqrt{2 \kappa} C(\D + \B)$ simulations or our model we will be considering $\c = \sqrt{2 \kappa}
as before, but the results are more general and can be applied to their C(\D + \B)$ as before, but the results are more general and can be
setups. This equation describes the state of the system if we discard applied to their setups. This equation describes the state of the
all knowledge of the outcome. The commutator describes coherent system if we discard all knowledge of the outcome. The commutator
dynamics due to the isolated Hamiltonian and the remaining terms are describes coherent dynamics due to the isolated Hamiltonian and the
due to measurement. This is a convenient way to find features of the remaining terms are due to measurement. This is a convenient way to
dynamics common to every measurement trajectory. find features of the dynamics common to every measurement trajectory.
Just like in the preceding chapter we define the projectors of the Just like in the preceding chapter we define the projectors of the
measurement eigenspaces, $P_m$, which have no effect on any of the measurement eigenspaces, $P_m$, which have no effect on any of the