= Release-expansion thermometry
{title2=$\sigma_x^2(t)=\sigma_{x0}^2+2C_{xv}t+\sigma_v^2t^2$}
A spatial image after a chosen dark interval is a different readout from continuously recording arrivals at a fixed probe. Here is an original idealised <fluorescence> imaging model: release a dilute cloud, let it move freely, then apply a short uniform probe pulse. Neglect <collisions>, residual <forces>, attenuation and motion during the pulse. Use each image's cloud centre to remove mean drift. Uniform <gravity> shifts that centre without changing the centred <variance>.
For each atom let $x(t)=x_0+v_0t$ along a horizontal imaging axis. Expanding the square about the <expected values> gives exactly
$$
\sigma_x^2(t)=\sigma_{x0}^2+2C_{xv}t+\sigma_v^2t^2,\qquad
C_{xv}=\operatorname{Cov}(x_0,v_0).
$$
No <Gaussian distribution> is needed for this <variance> identity. A jointly <Gaussian distribution> of initial positions and <velocities> does make the later density <Gaussian>. For a separable three-dimensional cloud viewed along $z$, integration along the viewing direction gives the column density
$$
n_{\rm col}(x,y;t)=\int n(x,y,z;t)\,dz
=\frac{N}{2\pi\sigma_x(t)\sigma_y(t)}
\exp\left[-\frac{(x-\bar x(t))^2}{2\sigma_x^2(t)}-\frac{(y-\bar y(t))^2}{2\sigma_y^2(t)}\right].
$$
Here $N$ is atom number. Integrating this image along $y$ leaves a one-dimensional <Gaussian distribution> with <standard deviation> $\sigma_x(t)$. This is a projected density measurement, not a three-dimensional radius or a direct <velocity> histogram. Under the uniform, optically thin probe assumptions, expected <fluorescence> is proportional to this column density. A known, independent <point spread function> with finite <variance> adds its own <variance> to the measured width; probe duration and intensity gradients need separate treatment.
For a <Maxwell-Boltzmann distribution>, $T_x=m\sigma_v^2/k_B$, where $m$ is atomic <mass> and $k_B$ the <Boltzmann constant>. For a nonthermal distribution this defines a directional second-moment <temperature>, which need not describe every feature of the distribution. A <covariance> between position and <velocity> can initially narrow the cloud even though $\sigma_v^2$ is positive. The physical <covariance matrix> requires $C_{xv}^2\le\sigma_{x0}^2\sigma_v^2$.
Choose $\sigma_{x0}=0.60\ \mathrm{mm}$, $\sigma_v=0.060\ \mathrm{m\,s^{-1}}$ and $C_{xv}=-0.018\ \mathrm{mm^2\,ms^{-1}}$, giving <correlation coefficient> $-1/2$. With $t$ in milliseconds, the numerical <variance> in square millimetres is $0.36-0.036t+0.0036t^2$. Its minimum occurs at $5\ \mathrm{ms}$, with <standard deviation> $\sqrt{0.27}\ \mathrm{mm}$. These chosen values satisfy the <covariance matrix> constraint; early contraction is compatible with free motion.
Generate thirteen synthetic <variance> observations at equally spaced times from $0$ to $30\ \mathrm{ms}$, adding independent <Gaussian noise> of <standard deviation> $0.012\ \mathrm{mm^2}$. <Linear least squares> fits all three coefficients in $a+bt+ct^2$. The example yields $a\simeq0.37026\ \mathrm{mm^2}$, $b\simeq-0.036413\ \mathrm{mm^2\,ms^{-1}}$ and $c\simeq0.0035820\ \mathrm{mm^2\,ms^{-2}}$. With the chosen approximate <sodium> <mass> $m=3.82\times10^{-26}\ \mathrm{kg}$, the true $T_x$ is $9.96\ \mathrm{\mu K}$ and the fitted value is $9.91\ \mathrm{\mu K}$. This is a fit to generated width data, not historical observations.
\Image[/release-expansion-covariance-thermometry.png]
{title=Cloud expansion with an initial position–velocity covariance}
{description=Original synthetic variance observations and ideal column-density marginals. All parameters and noise are chosen here. The three-dimensional Gaussian is integrated along the viewing axis and then along the image's other axis. The profiles omit imaging blur.}
If the same data are fitted to $a+ct^2$, forcing $C_{xv}=0$, the inferred $T_x$ falls to $6.79\ \mathrm{\mu K}$. The omitted term is partly absorbed into both fitted coefficients. In noiseless <linear least squares> with a free intercept, its precise effect on the quadratic coefficient is
$$
c_{\rm forced}=\sigma_v^2+2C_{xv}\frac{\operatorname{Cov}_{t}(t^2,t)}{\operatorname{Var}_{t}(t^2)},
$$
where the subscript denotes equal-weight moments over the chosen observation times, using one consistent unit system. Since the ratio is positive for distinct nonnegative times, negative $C_{xv}$ biases this estimate downwards. An incorrect release-time origin can also generate a linear term, so fitting it does not by itself identify a physical initial <covariance>. Timing calibration, spatial resolution, residual <magnetic fields> and probe-induced <recoil energy> remain separate controls. Neither a cold-looking image nor a single long expansion time identifies all three coefficients.
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