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\@writefile{toc}{\contentsline {section}{\numberline {3.1}Introduction}{27}{section.3.1}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3.1}{\ignorespaces Cross sectional density $n(r)$ of a 3D bosonic $^{87}$Rb optical lattice system with $N=125 000$, averaged over 1000 independent measurements obtained from a QMC simulation. Here, the parameters are $U/t = 10$ (left), $50$ (right), $T/t=1$, and $V_T/t = 0.0277$.}}{29}{figure.3.1}}
\newlabel{thermometry_figure_1}{{3.1}{29}{Cross sectional density $n(r)$ of a 3D bosonic $^{87}$Rb optical lattice system with $N=125 000$, averaged over 1000 independent measurements obtained from a QMC simulation. Here, the parameters are $U/t = 10$ (left), $50$ (right), $T/t=1$, and $V_T/t = 0.0277$}{figure.3.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1.3}Single site addressability}{29}{subsection.3.1.3}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.2}{\ignorespaces A single measurement of atom distribution for an ultracold quantum gas held in a two-dimensional optical lattice. The bosons, indicated by bright spots, are confined towards the parabolic trap centre. The interaction strength U/t increases from left to right, and thus transiting the system from being a superfluid into being a Mott insulator. Recent experimental advancement enables good visualisation of the Mott plateau (middle) around the trap centre up to single-site resolution. At extremely large interaction strength $U/t$, there exists a high probability to locate 2 bosons per site in the vicinity around the trap centre. However, current fluorescence experiments could not distinguish a doublon from a hole, therefore indicating dark spots around the centre (right). {\it {This figure is replicated from Nature 467, 68 (2010).}} \cite {bloch1} }}{30}{figure.3.2}}
\newlabel{single_site}{{3.2}{30}{A single measurement of atom distribution for an ultracold quantum gas held in a two-dimensional optical lattice. The bosons, indicated by bright spots, are confined towards the parabolic trap centre. The interaction strength U/t increases from left to right, and thus transiting the system from being a superfluid into being a Mott insulator. Recent experimental advancement enables good visualisation of the Mott plateau (middle) around the trap centre up to single-site resolution. At extremely large interaction strength $U/t$, there exists a high probability to locate 2 bosons per site in the vicinity around the trap centre. However, current fluorescence experiments could not distinguish a doublon from a hole, therefore indicating dark spots around the centre (right). {\it {This figure is replicated from Nature 467, 68 (2010).}} \cite {bloch1}\relax }{figure.3.2}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3.2}Fluctuation-dissipation thermometry}{30}{section.3.2}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3.3}{\ignorespaces Illustration of the quantities entering the FD thermometry formula. Shown from top to bottom are: (1) cross-sectional density $n(r)$, (2) column-integrated density $n(\rho )$, (3) dissipation term $L(\rho )$, (4) fluctuation term ($\xi = 3$) $R_3(\rho )$, and (5) fluctuation term ($\xi = \infty $) $R_\infty (\rho )$. We take a 3D bosonic $^{87}$Rb optical lattice system with $N=125 000$, and we average over 1000 independent measurements obtained from a QMC simulation. The parameters in the left column are $U/t=10$, $T/t=1$, and the parameters in the right column are $U/t=50$, $T/t=1$. The trapping frequency is $V_T/t=0.0091$ (left), $0.0277$ (right). }}{31}{figure.3.3}}
\newlabel{FD_1}{{3.3}{31}{Illustration of the quantities entering the FD thermometry formula. Shown from top to bottom are: (1) cross-sectional density $n(r)$, (2) column-integrated density $n(\rho )$, (3) dissipation term $L(\rho )$, (4) fluctuation term ($\xi = 3$) $R_3(\rho )$, and (5) fluctuation term ($\xi = \infty $) $R_\infty (\rho )$. We take a 3D bosonic $^{87}$Rb optical lattice system with $N=125 000$, and we average over 1000 independent measurements obtained from a QMC simulation. The parameters in the left column are $U/t=10$, $T/t=1$, and the parameters in the right column are $U/t=50$, $T/t=1$. The trapping frequency is $V_T/t=0.0091$ (left), $0.0277$ (right)}{figure.3.3}{}}
\citation{zhou1}
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\@writefile{lot}{\contentsline {table}{\numberline {3.1}{\ignorespaces Number of uncorrelated measurements needed to determine the temperature accurately within 5\% error for 3D ${}^{87}$Rb optical lattice experiments trapping 125,000 bosons. The variance reduction through window-sizing leads to orders of magnitudes improvement, and thus making FD thermometry scheme as a feasible tool for accurate temperature determination. The parameters are the same as in figure \ref {FD_1}. }}{33}{table.3.1}}
\newlabel{table:nr_shot1}{{3.1}{33}{Number of uncorrelated measurements needed to determine the temperature accurately within 5\% error for 3D ${}^{87}$Rb optical lattice experiments trapping 125,000 bosons. The variance reduction through window-sizing leads to orders of magnitudes improvement, and thus making FD thermometry scheme as a feasible tool for accurate temperature determination. The parameters are the same as in figure \ref {FD_1}}{table.3.1}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.4}{\ignorespaces Illustration of the FD thermometry scheme by showing different $L(\rho )- R_\xi (\rho )$ plots at various window sizes $\xi =0,1,2,3,5,\infty $. This approach illustrates how the density-density correlation length can be found in an experimental system. When the window size is smaller than density-density correlation length, systematic errors set in, and this results in nonlinear $L(\rho )-R(\rho )$ behaviour, while for $\xi $ larger than the denisty-density correlation length, the behaviour of $L(\rho )-R(\rho )$ is linear. However, statistical noise also increases with increasing window size. The parameters are the same as in figure \ref {FD_1}. }}{34}{figure.3.4}}
\newlabel{FD_2}{{3.4}{34}{Illustration of the FD thermometry scheme by showing different $L(\rho )- R_\xi (\rho )$ plots at various window sizes $\xi =0,1,2,3,5,\infty $. This approach illustrates how the density-density correlation length can be found in an experimental system. When the window size is smaller than density-density correlation length, systematic errors set in, and this results in nonlinear $L(\rho )-R(\rho )$ behaviour, while for $\xi $ larger than the denisty-density correlation length, the behaviour of $L(\rho )-R(\rho )$ is linear. However, statistical noise also increases with increasing window size. The parameters are the same as in figure \ref {FD_1}}{figure.3.4}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.5}{\ignorespaces The FD thermometry scheme remains valid over 20\% spread in temperature T (left), and 1\% spread in the particle number N (right). The parameters for the 3D bosonic $^{87}$Rb system are $N=125 000$, $U/t=10$, $T/t=1$, $V_T/t=0.0091$. Here, the optimal window size is $\xi =3$, and 20 independent measurements are obtained from a QMC simulation for accurate temperature determination within 5\% error.}}{35}{figure.3.5}}
\newlabel{FD_4}{{3.5}{35}{The FD thermometry scheme remains valid over 20\% spread in temperature T (left), and 1\% spread in the particle number N (right). The parameters for the 3D bosonic $^{87}$Rb system are $N=125 000$, $U/t=10$, $T/t=1$, $V_T/t=0.0091$. Here, the optimal window size is $\xi =3$, and 20 independent measurements are obtained from a QMC simulation for accurate temperature determination within 5\% error}{figure.3.5}{}}
\@writefile{lof}{\contentsline {figure}{\numberline {3.6}{\ignorespaces FD thermometry scheme at slightly lower temperature, for a 3D bosonic $^{87}$-Rb optical lattice with parameters $N= 125 000$, $U/t=10$, $T/t=0.5$, $V_T/t = 0.0091$. Here, the optimal window size is $\xi =5$, and the data is obtained by averaging over 100 independent measurements from a QMC simulation.}}{35}{figure.3.6}}
\newlabel{FD_3}{{3.6}{35}{FD thermometry scheme at slightly lower temperature, for a 3D bosonic $^{87}$-Rb optical lattice with parameters $N= 125 000$, $U/t=10$, $T/t=0.5$, $V_T/t = 0.0091$. Here, the optimal window size is $\xi =5$, and the data is obtained by averaging over 100 independent measurements from a QMC simulation}{figure.3.6}{}}
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\citation{bloch1}
\@writefile{lof}{\contentsline {figure}{\numberline {3.7}{\ignorespaces The FD thermometry scheme remains applicable to in-situ density experiments, which have a resolution of a few sites. The system consists of a 3D optical lattice with $N=125 000$ bosonic $^{87}$Rb atoms. Increasing the bin width increases the systematic error in the dissipation term $L(\rho )$, but the temperature estimate remains reliable. The parameters are the same as in figure \ref {FD_1}. Here, the optimal window size is $\xi =3$, and 1000 independent measurements are averaged out from a QMC simulation.}}{37}{figure.3.7}}
\newlabel{FD_6}{{3.7}{37}{The FD thermometry scheme remains applicable to in-situ density experiments, which have a resolution of a few sites. The system consists of a 3D optical lattice with $N=125 000$ bosonic $^{87}$Rb atoms. Increasing the bin width increases the systematic error in the dissipation term $L(\rho )$, but the temperature estimate remains reliable. The parameters are the same as in figure \ref {FD_1}. Here, the optimal window size is $\xi =3$, and 1000 independent measurements are averaged out from a QMC simulation}{figure.3.7}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3.8}{\ignorespaces FD thermometry scheme in the presence of double-hole indistinguishability. The parameters are the same as in figure \ref {FD_1}. Blue circles (green squares) show the curve where doublons can (cannot) be distinguished from holes.}}{38}{figure.3.8}}
\newlabel{FD_7}{{3.8}{38}{FD thermometry scheme in the presence of double-hole indistinguishability. The parameters are the same as in figure \ref {FD_1}. Blue circles (green squares) show the curve where doublons can (cannot) be distinguished from holes}{figure.3.8}{}}
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\citation{zhou2}
\@writefile{lof}{\contentsline {figure}{\numberline {3.9}{\ignorespaces Illustrating the concept of wing thermometry for a 3D bosonic ${}^{87}$Rb optical lattice system, {\it ie.,} describing the normal state by high temperature series expansions. Blue circles: In-situ density profile obtained from 100 uncorrelated measurements obtained by a QMC simulation with parameters $U/t = 10$ , $T/t = 3$ , $N = 125,000$. The superfluid-normal phase boundary occurs at the density $\delimiter "426830A n \delimiter "526930B =0.42$ or chemical potential $\mu / t = -2.75$. The second order series captures all the physics in the normal regime, whereas the zeroth order has a very small validity range.}}{43}{figure.3.9}}
\newlabel{HTE2_1}{{3.9}{43}{Illustrating the concept of wing thermometry for a 3D bosonic ${}^{87}$Rb optical lattice system, {\it ie.,} describing the normal state by high temperature series expansions. Blue circles: In-situ density profile obtained from 100 uncorrelated measurements obtained by a QMC simulation with parameters $U/t = 10$ , $T/t = 3$ , $N = 125,000$. The superfluid-normal phase boundary occurs at the density $\langle n \rangle =0.42$ or chemical potential $\mu / t = -2.75$. The second order series captures all the physics in the normal regime, whereas the zeroth order has a very small validity range}{figure.3.9}{}}
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\@writefile{lof}{\contentsline {figure}{\numberline {3.10}{\ignorespaces Second order high temperature series expansion thermometry scheme for a bosonic system that is entirely in the normal phase. No more than a single shot of cross-sectional density is needed to estimate the temperature and chemical potential within 10\% accuracy. We take a 3D optical lattice system with bosonic ${}^{87}$Rb atoms and parameters $N = 125,000$, $\mu /t = 25.97$, $U/t = 50$, $T/t = 3$. The blue circles are obtained from a single measurement in a QMC simulation; and the red line is a least-square fit over the entire normal region where $\mu _{\rm fit}/t = 25.92$ and $T_{\rm fit}/t = 2.824$ nK.}}{44}{figure.3.10}}
\newlabel{HTE2_2}{{3.10}{44}{Second order high temperature series expansion thermometry scheme for a bosonic system that is entirely in the normal phase. No more than a single shot of cross-sectional density is needed to estimate the temperature and chemical potential within 10\% accuracy. We take a 3D optical lattice system with bosonic ${}^{87}$Rb atoms and parameters $N = 125,000$, $\mu /t = 25.97$, $U/t = 50$, $T/t = 3$. The blue circles are obtained from a single measurement in a QMC simulation; and the red line is a least-square fit over the entire normal region where $\mu _{\rm fit}/t = 25.92$ and $T_{\rm fit}/t = 2.824$ nK}{figure.3.10}{}}
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