From 1a62ccd8302b17ba086dba561f4414a5c1b109da Mon Sep 17 00:00:00 2001 From: Wojciech Kozlowski Date: Thu, 15 Dec 2016 00:49:22 +0000 Subject: [PATCH] Some more typo fixes --- Chapter5/chapter5.tex | 35 +++++++++++++++++------------------ Chapter6/chapter6.tex | 18 +++++++++--------- 2 files changed, 26 insertions(+), 27 deletions(-) diff --git a/Chapter5/chapter5.tex b/Chapter5/chapter5.tex index 04c1da4..6374975 100644 --- a/Chapter5/chapter5.tex +++ b/Chapter5/chapter5.tex @@ -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 diff --git a/Chapter6/chapter6.tex b/Chapter6/chapter6.tex index e416000..0b422b8 100644 --- a/Chapter6/chapter6.tex +++ b/Chapter6/chapter6.tex @@ -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