= Synthetic bimodal profile of a Bose gas
{title2=$P(u)=(1-f_0)G_{\sigma_t}(u)+f_0G_{\sigma_c}(u)$}
A broad component plus a narrow central component can have one local maximum while still displaying two distinct width scales. This is what “bimodal” means in this imaging context; it does not require two separated peaks. To isolate the population effect, construct an original normalized line profile
$$
G_\sigma(u)=\frac{e^{-u^2/(2\sigma^2)}}{\sqrt{2\pi}\sigma},\qquad
P(u)=(1-f_0)G_1(u)+f_0G_{0.20}(u).
$$
Both terms are <normal distributions>; their integrals are one, so the component areas are $1-f_0$ and $f_0$. With $f_0=0.488$, the narrow component has central height $f_0/(\sqrt{2\pi}\,0.20)\simeq0.9734$, while the broad component contributes $(1-f_0)/\sqrt{2\pi}\simeq0.2043$. A tall central feature therefore need not contain most of the particles. Holding these widths fixed deliberately isolates changing weights; it does not model how <temperature>, interactions or camera resolution determine actual widths.
\Image[/bose-harmonic-trap-fraction-and-profiles.png]
{title=Ideal harmonic-trap condensate fraction and synthetic profiles}
{description=Original calculated <condensate fraction> at left; normalized two-width <normal distribution> mixtures at right, with weights from the cubic law. Chosen widths and parameters, no measured data. A profile is a line marginal, not an unintegrated camera pixel cut.}
For <absorption imaging> in the weak-probe regime with negligible saturation, the <optical depth> is $\mathrm{OD}(x,z)=-\ln[I(x,z)/I_0(x,z)]=\sigma_{\rm abs}n_{\rm col}(x,z)$, where $I_0$ is the reference <light intensity> without atoms, $I$ the transmitted <light intensity>, $\sigma_{\rm abs}$ an effective absorption <cross section>, and $n_{\rm col}$ the <column density>. Integrating $n_{\rm col}$ over both image coordinates gives the particle number; a central pixel height alone does not. Finite resolution can obscure a narrow component, and saturation invalidates this simple logarithmic calibration.
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