Feedback control of waiting times

Feedback loops are known as a versatile tool for controlling transport in small systems, which usually have large intrinsic fluctuations. Here we investigate the control of a temporal correlation function, the waiting-time distribution, under active and passive feedback conditions. We develop a general formalism and then specify to the simple unidirectional transport model, where we compare costs of open-loop and feedback control and use methods from optimal control theory to optimize waiting-time distributions.

Figure 1

Left: Scheme of active (measurement-based) feedback control for the example of a unidirectional stochastic process, Eq. (22). The feedback protocol is realized via the single time-dependent rate $\gamma (t-t_n)$, starting again and again at the time $t_n$ of the previous jump, cf. Eq. (30). Right: Passive feedback scheme (“hardwiring”) with series of $N+1$ inelastic transitions (with phonon emissions from quantum dot with $N$ levels) at identical rates $(N+1){\gamma }_0$, and electrons leaving from the lowest dot level only, to simulate an equivalent waiting-time distribution without feedback from measurement devices, cf. Sec. 8c.

Figure 2

Waiting-time distributions $w\left(\tau \right)$ [Eq. (23)] (left) of single jump process with rates $\gamma \left(t\right)$ (right) with a time dependence according to Eq. (47) that starts again after each jump. Feedback strength $g=0$ (black, dashed), $g=5$ (magenta, dotted), and $g=15$ (blue, solid).

Figure 3

Results from the optimizsation of the cost functional $J$ with $a=0$, in which case we only impose a cost with the gradient of $\gamma \left(t\right)$. The target waiting-time distribution is that in Eq. (46) with $g=5$. Left: The optimized rate function $\gamma \left(t\right)$. Right: The corresponding waiting-time distribution. The thin black lines show the target $\gamma \left(t\right)$ and $w\left(\tau \right)$, and optimized results are given for $b=0.01,0.1,1.0$ (dashed lines; black, blue, magenta).

Figure 4

As in Fig. 3, but here we set the rate parameter $b=0$ and investigate behavior with cost parameter $a=0.05,0.25,2.0$ (dashed lines; black, magenta, blue), i.e., we impose a cost associated with high rates.

Figure 5

Left: Comparison of control schemes for piecewise constant rates, Eq. (59). Upper: Feedback (closed-loop) control starting after each jump at time $t_n$ with delay ${\tau }_0$. Lower: Open-loop control with rates Eq. (60) switched on during time intervals $\mathrm{\Delta }T$ with period $T$. Red arrows show a possible charging and decharging sequence in the two-barrier model explained in the text. Right: Comparision of costs for feedback control (dotted curves), $J_{\mathrm{fb}}$ [Eq. (64)], with costs for open-loop control (solid curve), $J_{\mathrm{ol}}$ [Eq. (63)], as a function of scaled Fano factor $F$ [Eq. (58)]. Parameters for the observation cost $c$ in $J_{\mathrm{fb}}$ increase from bottom to top curve with $c=0$ (black), $c=0.5$ (blue), and $c=1$ (magenta). Switch cost parameter $b=0.1$.

Figure 6

Comparison of time-versus-number feedback (blue solid line) [Eq. (27)] and feedback conditioned on previous jump (dashed black line) [Eq. (46)] with dimensionless control parameter $g$. Left: Fano factors $F\equiv \frac{\langle {\tau }^2\rangle -{\langle \tau \rangle }^2}{{\langle \tau \rangle }^2}$. Right: Fisher information $\mathcal{F}\left(g\right)$ for waiting times [Eq. (71)], and ${\mathcal{F}}_p\left(g\right)$ (dotted lines) for $p(n,t)$ in Gaussian approximation (see text).