qspec.normal_chi2_convolved_f_xi_pdf  (  f ,  f0 ,  xi ,  sigma_f ,  col = True  )[source]

The probability density at the frequency $f$ in the rest frame of an atom with kinetic energy $E_x$, determined through the parameter $\xi$, and distributed according to a convolution of a normal and a $\chi^2_1$ distribution $$\begin{aligned} \rho(f) &= \frac{1}{2\pi\sigma_\mathrm{f}\sqrt{\xi}}\int\limits_0^\infty \frac{1}{\sqrt{\eta}} \exp\left[-\frac{1}{2}\left(\frac{f - f_0 - \eta}{\sigma_\mathrm{f}}\right)^2 - \frac{\eta}{2\xi}\right]\mathrm{d}\eta\\[1ex] &= N\times\begin{cases} \sqrt{-\frac{\mu}{2\pi}}\exp[-(x + \mu)] K_{1/4}(x), & \text{if } \mu < 0 \\[1.5ex] \sqrt{\frac{\varpi}{\sqrt{\pi}}\sigma}, & \text{if } \mu = 0 \\[1.5ex] \sqrt{\pi\mu}\exp[-(x + \mu)] \frac{1}{2}\left[ I_{1/4}(x) + I_{-1/4}(x) \right], & \text{if } \mu > 0 \end{cases}\\[2ex] x &= \left(\!\frac{\mu}{2\sigma}\!\right)^{\!2},\quad\mu = \frac{f - f_0}{2\xi} - \sigma^2, \quad\sigma = \frac{\sigma_\mathrm{f}}{2\xi}\\[2ex] N &= \frac{1}{\sqrt{2\pi}\sigma_\mathrm{f}}\exp\left(-\frac{\sigma^2}{2}\right), \end{aligned}$$ where $I_\alpha$, and $K_\alpha$ are the modified Bessel functions of first and second kind, and $\varpi \approx 2.6220575543$ is the lemniscate constant.

Parameters:

farray_like

The frequency quantiles $f$ (arb. units).

f0array_like

A frequency offset $f_0$ ([f]).

xiarray_like

The asymmetry parameter $\xi$ ([f]).

sigma_farray_like

The standard deviation $\sigma_\mathrm{f}$ of the normal distribution ([f]).

colbool

The laser can be aligned collinearly (True) or anticollinearly (False) to the velocity of the atom.

Returns:

rho_fndarray

The probability density in thermal equilibrium at the frequency f (1/[f]).