Figure 1

Two upper graphs show the spectrum for a Student’s t-distribution of the detunings [see Eq. (13)]. On the left, the spectrum is plotted for $r=0.5$ for which the first moment of the distribution diverges, whereas on the right, the spectrum is plotted for $r=1.5$ for which only the second moment diverges, but the first moment exists. For both spectra, the spectrum is plotted without fluctuations (solid line) and with fluctuations at $\Gamma =10{\delta }_0$ (dashed line). In the lower graph, we plot the FWHM normalized to the FWHM without fluctuations, as a function of $\Gamma$.Reuse & Permissions

Figure 2

Coherence without fluctuations (dotted lines) and with fluctuations separated by $\Delta t_j=0.25{\delta }_0^{-1}$ (solid lines). The detuning distribution is assumed to be the Student’s t-distribution with $r=0.5$ and $1.5$. Since the coherence is given by $R(T)=R_0(\Delta t_1)R_0(\Delta t_2)\dots R_0(\Delta t_n)$, a Zeno/anti-Zeno-like behavior explains the transition from motional narrowing to broadening for a diverging first moment of $P_0$, at which point $\partial {}_T{R}_0|{}_{T=0}$ changes from $0$ to $-\infty$.Reuse & Permissions

Figure 3

Numerical simulation of the logarithm of the coherence using an ensemble of 10 000 particles following the Poisson discrete fluctuations model with three different rates $\Gamma$ and detunings following a Student’s t-distribution with $r=0.5$. The dotted (black) lines are linear fits, validating the prediction of Eq. (16) that as $\Gamma$ increases, the decay becomes exponential. We extract the decay rate $\gamma$ by fitting the simulated coherence for $T>10{\Gamma }^{-1}$ to an exponentially decaying function $A{e}^{-\gamma T}$. The inset shows $\gamma$ as a function of $\Gamma$ for different values of the distribution parameter $r$. The dotted lines are fits to the function $\gamma =a{\Gamma }^{1-\mathrm{\alpha ̄}}+b$, which has the functional form derived in Eq. (16). The extracted exponents agree well with the expected values $\mathrm{\alpha ̄}=r$.Reuse & Permissions

Figure 4

Effect of a cutoff in the detuning distribution on the motional broadening phenomenon. The figure shows numerical simulation of the normalized coherence $R(T)/R_0(T)$ at time $T=0.5{\delta }_0^{-1}$ versus the fluctuations rate $\Gamma$. The detuning distribution is taken to be the Student’s t-distribution [see Eq. (13)] with $r=0.5$, and the fluctuations are Poissonian. Each curve is calculated for $P_0$ truncated at a cutoff detuning ${\delta }_c$. As expected in motional broadenings, the coherence is decreasing for small $\Gamma$, and only for higher values, the behavior is changing to motional narrowing.Reuse & Permissions