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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,
signoles2014}.
These effects can be easily seen in our model when
$\gamma \rightarrow \infty$. The system will be projected into one or
more degenerate eigenstates 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 suppressed 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
These effects can be easily seen in our model when $\gamma \rightarrow
\infty$. The system will be projected into one or more degenerate
eigenstates of $\cd \c$, $| \psi_i \rangle$, for which we define the
projector $P_\varphi = \sum_{i \in \varphi} | \psi_i \rangle \langle
\psi_i |$ 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 suppressed 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 geometric patterns we have in which the modes
are spatially delocalised in such a way that neighbouring sites never

18
Chapter6/chapter6.tex
View File

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