 Synthetic bimodal profile of a Bose gas

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 profileBoth 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 $f_0/(\sqrt{2\pi }0.20)\simeq 0.9734$, while the broad component contributes $(1-f_0)/\sqrt{2\pi }\simeq 0.2043$. 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.

For absorption imaging in the weak-probe regime with negligible saturation, the optical depth is $\mathrm{OD}(x,z)=-\ln [I(x,z)/I_0(x,z)]={\sigma }_{\mathrm{abs}}{n}_{\mathrm{col}}(x,z)$, where $I_0$ is the reference light intensity without atoms, $I$ the transmitted light intensity, ${\sigma }_{\mathrm{abs}}$ an effective absorption cross section, and $n_{\mathrm{col}}$ the column density. Integrating $n_{\mathrm{col}}$ 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.