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\@writefile{lof}{\contentsline {figure}{\numberline {4.1}{\ignorespaces Time of flight images (equation \ref {tof_eq2}) obtained from QMC-DWA simulations mimicking optical lattice experiments as realistically as possible, with interaction strength $U/t=8.11$ (left) , $27.5$ (right) at temperature $T/t=1$. The anisotropic optical lattice, with lattice strength $\mathaccentV {vec}17E{V}_0 = (8.8 E_{rx}, 8 E_{ry} , 8 E_{rz})$ (left), $(12.64 E_{rx}, 11.75 E_{ry} , 11.75 E_{rz})$ (right), and laser wavelength $\mathaccentV {vec}17E{\lambda } = (765,843,843)\mathrm {nm}$, confines $2.8 \times 10^5$ (left), $9.4 \times 10^4$ (right) bosons in a parabolic trap $\mathaccentV {vec}17E{V}_T = (17.1, 10.9, 11.3)\mathrm {Hz}$ (left), $(19.9, 13.0, 13.4)\mathrm {Hz}$ (right). The horizontal axes are $k_x$ and $k_y$ in units of $2\pi $, and the vertical axis is the time-of-flight distribution $\delimiter "426830A n_f (k_x, k_y)\delimiter "526930B $ in unit of inverse momentum area resolution $(\Delta k_x \Delta k_y)^{-1}$, taking experimental value $(\Delta k_x \Delta k_y) \approx 0.1^2$ in units of $(2\pi )^2$. See appendix \ref {chapter_tof_image} for further details. }}{47}{figure.4.1}}
\newlabel{fig_tof_image}{{4.1}{47}{Time of flight images (equation \ref {tof_eq2}) obtained from QMC-DWA simulations mimicking optical lattice experiments as realistically as possible, with interaction strength $U/t=8.11$ (left) , $27.5$ (right) at temperature $T/t=1$. The anisotropic optical lattice, with lattice strength $\vec {V}_0 = (8.8 E_{rx}, 8 E_{ry} , 8 E_{rz})$ (left), $(12.64 E_{rx}, 11.75 E_{ry} , 11.75 E_{rz})$ (right), and laser wavelength $\vec {\lambda } = (765,843,843)\mathrm {nm}$, confines $2.8 \times 10^5$ (left), $9.4 \times 10^4$ (right) bosons in a parabolic trap $\vec {V}_T = (17.1, 10.9, 11.3)\mathrm {Hz}$ (left), $(19.9, 13.0, 13.4)\mathrm {Hz}$ (right). The horizontal axes are $k_x$ and $k_y$ in units of $2\pi $, and the vertical axis is the time-of-flight distribution $\langle n_f (k_x, k_y)\rangle $ in unit of inverse momentum area resolution $(\Delta k_x \Delta k_y)^{-1}$, taking experimental value $(\Delta k_x \Delta k_y) \approx 0.1^2$ in units of $(2\pi )^2$. See appendix \ref {chapter_tof_image} for further details}{figure.4.1}{}}
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\newlabel{fig_density_profile}{{4.2}{48}{Cross-sectional (top) and column-integrated (bottom) density profiles obtained from QMC-DWA simulations mimicking optical lattice experiments as realistically as possible, with interaction strength $U/t=8.11$ (left) , $27.5$ (right) at temperature $T/t=1$. The anisotropic optical lattice experiments take the same setup parameters as in figure \ref {fig_tof_image}. See appendix \ref {chapter_density_profile} for further details}{figure.4.2}{}}
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\@writefile{lot}{\contentsline {table}{\numberline {4.1}{\ignorespaces Quantifying error budgets in energy $\delimiter "426830A E \delimiter "526930B $, density at trap centre $\delimiter "426830A n(0,0,0) \delimiter "526930B $, column-integrated density at trap centre $\delimiter "426830A n(0,0)\delimiter "526930B $, condensate fraction $f_c$, and visibility $\mathcal {V}$, for a $\pm 5\%$ fluctuation in total particle number $\delimiter "426830A N \delimiter "526930B $, s-wave scattering length $a$, lattice strength $V_{0x}$, and laser wavelength $\lambda _x$. The anisotropic optical lattice experiments take the same setup parameters as in figure \ref {fig_tof_image}. The estimated fluctuation is computed via QMC-DWA simulations, where the simulations indicated by (*) are performed such that the total particle number is fixed at $\delimiter "426830A N \delimiter "526930B = 2.8 \times 10^5$. See appendix \ref {chapter_tuning_chemical_potential} for further details. }}{49}{table.4.1}}
\newlabel{table:errorbudgets}{{4.1}{49}{Quantifying error budgets in energy $\langle E \rangle $, density at trap centre $\langle n(0,0,0) \rangle $, column-integrated density at trap centre $\langle n(0,0)\rangle $, condensate fraction $f_c$, and visibility $\mathcal {V}$, for a $\pm 5\%$ fluctuation in total particle number $\langle N \rangle $, s-wave scattering length $a$, lattice strength $V_{0x}$, and laser wavelength $\lambda _x$. The anisotropic optical lattice experiments take the same setup parameters as in figure \ref {fig_tof_image}. The estimated fluctuation is computed via QMC-DWA simulations, where the simulations indicated by (*) are performed such that the total particle number is fixed at $\langle N \rangle = 2.8 \times 10^5$. See appendix \ref {chapter_tuning_chemical_potential} for further details}{table.4.1}{}}
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\newlabel{fig_waist}{{4.3}{50}{Cross-sectional density profiles obtained from QMC-DWA simulations for $2.8 \times 10^5$ bosons trapped in an isotropic optical lattice with interaction strength $U/t=8.1$ at temperature $T/t=1$. The lattice strength is taken to be 8.35 $E_r$, the laser wavelength 843 nm, and the s-wave scattering length 101 $a_B$. Blue: The trapping potential is assumed to be parabolic, with trapping frequency 10.5 Hz. Red: The trapping potential is corrected according to equation \ref {trap_potential_correction} with $w_0 = 150 \mu m$. Waist correction reduces the density at the trap center $\langle n(0,0,0) \rangle $ by approximately 1\%, therefore only spreading out slightly in the wings of the bosonic cloud which is however statistically irrelevant}{figure.4.3}{}}
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