Dephasing by a nonstationary classical intermittent noise

We consider a phenomenological model for a $1∕f^{\mu }$ classical intermittent noise and study its effects on the dephasing of a two-level system. Within this model, the evolution of the relative phase between the $\mid \pm ⟩$ states is described as a continuous time random walk (CTRW). Using renewal theory, we find exact expressions for the dephasing factor and identify the physically relevant various regimes in terms of the coupling to the noise. In particular, we point out the consequences of the nonstationarity and pronounced non-Gaussian features of this noise, including some anomalous and aging dephasing scenarios.

Figure 1

Representation of a low-frequency noise as a sum of contributions from telegraphic signals (a). In this first case, the switching rates for the up $\left({\gamma }_{+}\right)$ and down $\left({\gamma }_{-}\right)$ states are comparable, and a $1∕f$ spectrum is recovered for a distribution of switching rates $\sim 1∕\gamma$. The intermittent limit (b) corresponds to the limit where the noise stays in the down states most of the time $({\gamma }_{+}⪢{\gamma }_{-})$. In this paper, we will approximate this intermittent noise by a spike field. For this intermittent noise, a $1∕f$ spectrum implies a nonstationarity whose consequences on dephasing are studied in this paper.

Figure 2

Representation of the random spike field used to model the intermittent low-frequency noise in this work. This noise is described by the distributions of the phase pulses $x_i$ and of the waiting time intervals ${\tau }_i$ between the spikes.

Figure 3

A configuration of the noise $X(t_p+\tau )$ in our model (as a function of $\tau$) between $t_p={10}^5{\tau }_0$ and $t_p+\tau =1.1{10}^5{\tau }_0$. In this figure, $h=0$ and the waiting times are distributed with an algebraic distribution $P\left({\tau }_i\right)\simeq {\tau }_i^{-2.1}$. The bottom part of the figure shows the corresponding continuous time random walk of the accumulated phase of the TLS between $t_p$ and $t_p+\tau$.

Figure 4

Intermittent noise between $t_p$ and $t_p+t$.

Figure 5

Dephasing time ${\tau }_{\varphi }$ in the asymmetric model as a function of the coupling $g$ for a huge asymmetry $h⪢g^2$, and different values of $\mu <1$. At weak coupling, $h<h_c\left(t_p\right)$, the dephasing time scales as ${\tau }_{\varphi }\propto 1∕h^{1∕\mu }$, whereas it becomes proportional to $t_p$ for $h>h_c\left(t_p\right)$. Note the $t_p$ dependence of the critical coupling $h_c$, visualized for $\mu =0.4$.

Figure 6

Decay of $\mid D_{t_p}\left(t\right)\mid$ in the asymmetric model for $\mu =0.8,t_p={10}^3{\tau }_0$ and different values of the coupling strength $h$. Here $h_c\left({10}^3{\tau }_0\right)\simeq {10}^{-2.4}$.