Coherent dynamics in stochastic systems revealed by full counting statistics

Stochastic systems feature, in general, both coherent dynamics and incoherent transitions between different states. We propose a method to identify the coherent part in the full counting statistics for the transitions. The proposal is illustrated for electron transfer through a quantum-dot spin valve, which combines quantum-coherent spin precession with electron tunneling. We show that by counting the number of transferred electrons as a function of time, it is possible to distill out the coherent dynamics from the counting statistics even in transport regimes, in which other tools such as the frequency-dependent current noise and the waiting-time distribution fail.

Figure 1

Electron transfer through a quantum-dot spin valve: an electron tunnels in from the left lead, precesses about the exchange field $\mathbf{B}$, and then tunnels out to the right lead. A coupled quantum-point contact (QPC) or single-electron transistor (SET) measures the electron occupation of the quantum dot.

Figure 2

${|\langle N\rangle }_s\left(t\right)|$ as a function of time for $s=1,-0.7,$ and $-0.9$. A regular peak pattern with peak-to-peak distance $\pi /\mathrm{\Omega }$ occurs for negative $s$, where $\mathrm{\Omega }$ is approximatively given by the Larmor frequency $\left|\mathbf{B}\right|$ of the exchange field. Parameters are $p=0.3$, $\varphi =\pi /2$, $\epsilon =U/3$, $eV=13U/6$, $k_{\text{B}}T=U/30$, and $a=0.8$.

Figure 3

(a) Position of the zeros $z_j\left(t\right)$ of ${\mathcal{M}}_1(z,t)$ in the complex plane for $\mathrm{\Gamma }t=61$. Complex-valued zeros (blue dots) witness the presence of Coulomb interaction in the system. Real-valued zeros (black dots) indicate the presence of Larmor precession. ${\langle N\rangle }_{-0.7}$ and ${\langle N\rangle }_{-0.9}$ diverge when a real-valued zero crosses the points marked by a red and green cross, respectively. (b) Position of the real-valued zeros (black dots) as a function of time. Other parameters are as in Fig. 2.

Figure 4

(a) Waiting-time distribution and (b) finite-frequency Fano factor for two different choices for $p$ and asymmetry $a$, depicted in Fig. 5 by a triangle (blue curves) and a dot (yellow curves). Other parameters are as in Fig. 2.

Figure 5

Comparison of different detection tools for spin precession as a function of the leads' polarizations $p$ and the asymmetry $a$ of the tunnel couplings to the leads. Other parameters are as in Fig. 2. Above the blue, red, and green lines, waiting times, Fano factors, and ${\langle N\rangle }_s\left(t\right)$ are useful tools, respectively. Obviously, ${\langle N\rangle }_s\left(t\right)$ provides the largest area of application. Below the dashed black line, ${\langle N\rangle }_s\left(t\right)$ shows divergencies with an $s$-dependent peak-to-peak distance, which are not connected to spin precession. The black dot, square, and triangle marks the values for $p$ and $a$ used in Figs. 4, 6, and 7.

Figure 6

${|\langle N\rangle }_s\left(t\right)|$ as a function of time for $s=1$ (blue curve), $-0.7$ (red curve), and $-0.9$ (green curve). From (a) to (c), the precision $\text{d}{P}_N$ of $P_N\left(t\right)$ increases and more peaks are resolved. The parameters are as in Fig. 2, with polarization $p$ and asymmetry $a$ as for the dot in Fig. 5.

Figure 7

${|\langle N\rangle }_s\left(t\right)|$ as a function of time for parameters $a=-0.8$ and $p=0.3$ (corresponding to the square in Fig. 5). The other parameters are as in Fig. 2 of the main text.