Modulated two-level system: Exact work statistics

We consider an open two-level system driven by a piecewise constant periodic field and described by a rate equation with Fermi, Bose, and Arrhenius rates, respectively. We derive an analytical expression for the generating function and large deviation function of the work performed by the field and show that a work fluctuation theorem holds.

Figure 1

External field following a piecewise constant protocol oscillating between the values $h_0-a$ to $h_0+a$ over a period of duration $\tau$.

Figure 2

(Top) Cumulant generating function ${\varphi }_{\mu }$ of work per period as a function of the Laplace parameter $\mu$. (Bottom) Large deviation function $I_w$ versus work per period. Various values of the field are plotted: high field, $h_0=5$ and $a=0.5$ (light green dotted dashed line); intermediate field, $h_0=1$ and $a=0.5$ (blue dashed line); and low amplitude of the driving, $h_0=1$ and $a=0.1$ (orange solid line). Symbols encode the types of rates: Arrhenius (squares), Bose (triangles), and Fermi (circle). The other parameters are $\tau =1$ and $\alpha =0.3$.

Figure 3

(Top) Cumulant generating function ${\varphi }_{\mu }$ for the work per period versus the Laplace parameter $\mu$. (Bottom) Large deviation function $I_w$ versus work per period for various characteristic time scales of the driving: $\tau =100$ (light green dotted dashed line), $\tau =1$ (blue dashed line), and $\tau =0.01$ (orange solid line). Same symbols code as described in the legend of Fig. 2. Here, $a=0.5$, $h_0=1$, and $\alpha =0.3$.

Figure 4

Average and variance of work per period as a function of the modulation amplitude $a$ for (a) and (b) or as a function of the period $\tau$ for (c). Parameters are $\alpha =0.3$, $h_0=1$, with $\tau =1$ for (a), $\tau =100$ for inset (b), and $a=0.5$ for (c). The symbol code is the same as described in the legend of Fig. 2.