Geometric effects in nonequilibrium electron transfer statistics in adiabatically driven quantum junctions

Cyclic Pancharatnam-Berry (PB) and adiabatic noncyclic geometric (ANG) effects are investigated in a single electron orbital system connected to two metal contacts with externally driven chemical potential and/or temperatures. The PB contribution doesn't affect the density matrix evolution, but has a quantitative effect on the statistics (fluctuations) of electron transfer. The ANG contribution, on the other hand, affects the net flux across the junction. Unlike the PB, the ANG contribution is nonzero when two parameters are identically driven. Closed analytical expressions are derived for the ANG contribution to the flux, and the PB contribution to the first two leading order fluctuations. Fluctuations can be modified by manipulating the relative phases of the drivings. Interestingly, we find that the fluctuations of the pumped charge do not satisfy the steady state fluctuation theorem in presence of nonzero geometric contribution, but can be recovered for a vanishing geometric contribution even in presence of the external driving.

Figure 1

Schematics of the driven single resonant level model. The leads are metallic and spinless charge transfer occurs between the single orbital and leads. The Fermi distributions of the metal leads in resonance with the orbital's energy, ${\epsilon }_s$, are driven periodically in time. ${\mathrm{\Gamma }}_{l\left(r\right)}^{}$ is the coupling of the orbital with the left (right) lead.

Figure 2

Fano factor $\left(F\right)$ contour plot for ${\mathrm{\Gamma }}_r$ and $\varphi$. The PB contribution is introduced by changing the Fermi functions as $f_l\left(t\right)=f_l(1-A^2{cos}^2\left(kt\right)),f_r\left(t\right)=f_r(1-A^2{cos}^2(kt+\varphi ))$. $k=\pi /t_p$, and $t_p$ is the driving time period, $\varphi$ is the phase difference between the two drivings, and $0<A<1$. Simulation parameters are $A=0.9,k=1,\nu =10,f_l=0.9,f_r=0.2.{\mathrm{\Gamma }}_l=0.9.$ Second curve from the top right represents the boundary between $F>1$ and $F<1$.

Figure 3

Net interpolated PDF in presence of PB contribution $(\varphi \ne 0)$, simulated at $f_l=0.75,f_r=0.25,k=1,\nu =20,A=0.9,\varphi =\frac{\pi }{4},{\mathrm{\Gamma }}_r=0.25$, and ${\mathrm{\Gamma }}_l=1$. Since the difference between $P(q,t)$ and $P_o(q,t)$ is very small, we only plot $P(q,t)$ in the main figure. The inset shows the difference between the PDFs in presence and absence of PB part.

Figure 4

Steady state driven fluctuation theorem, Eq. (44). All curves are simulated at ${\mathrm{\Gamma }}_l={\mathrm{\Gamma }}_r=0.25,f_l=0.75,f_r=0.25,k=1,\nu =20,A=0.9$. The three curves with different slopes $\left(R\right)$ correspond to $\varphi =0(R=1.38),\frac{\pi }{3}\left(1.61\right),\frac{\pi }{4}\left(1.53\right)$. The bottom-right inset shows results for $t\mathrm{\Delta }\mathfrak{L}$, Eq. (42), at $\varphi =0,{\mathrm{\Gamma }}_l=0.75,{\mathrm{\Gamma }}_r=0.25$ for a fixed $t=20\pi$. As $f_l$ increases the slope $\left(R\right)$ increases, $f_l=0.9(R=1.69),0.6\left(1.07\right),0.4\left(0.55\right)$. All other parameters are the same as in the main figure. The top-left inset represents the decay of Eq. (40), fitted with a power law: $t^{-0.49}$ for $q=9, \varphi =\pi /4,A=0.2,{\mathrm{\Gamma }}_l=0.75$, and ${\mathrm{\Gamma }}_r=0.25$. The time axis has been rescaled by a factor of ${10}^{-2}.$

Figure 5

Dynamic cumulant generating function, $S_d\left(\lambda \right)$ (solid), and linearly shifted, $S_d(-\lambda -R)$ (dotted). The curves are indistinguishable due to the GC symmetry, $S_d\left(\lambda \right)=S_d(-\lambda -R)$, where $R$ is the thermodynamic force given in Eq. (43). All curves are simulated at ${\mathrm{\Gamma }}_l=0.75,{\mathrm{\Gamma }}_r=0.25,f_l=0.75,k=0.1,\nu =20,\varphi =\pi /4$. The two curves, green (red), are evaluated for $A=0.2,f_r=0.1;R=3.12(A=0.5,f_r=0.25;R=2.29)$, respectively. The inset is evaluated at $\varphi =0$. The two curves, green (red), represent $A=0.2,f_r=0.1;R=2.15(A=0.5,f_r=0.25;R=2.11)$.

Figure 6

Asymmetric geometric cumulant generating function. ${\mathrm{\Gamma }}_l=1$ (higher peak); 0.75 (lower peak). All other parameters are the same as in Fig. 5. Because of the asymmetric behavior of $S_g\left(\lambda \right)$, the linear symmetry, $S\left(\lambda \right)=S(-\lambda -R)$, does not hold in the presence of the geometric contribution, resulting in violation of the FT, Eq. (44).

Figure 7

Average geometric flux [Eq. (67)] as a function of $n$. The upper (lower) curve is simulated for $t_p=10\pi \left(5\pi \right)$. It is zero at $n=0,1$. Simulation parameters used are $A=0.9,f_l=0.9,f_r=0.1,{\mathrm{\Gamma }}_l=1.25,{\mathrm{\Gamma }}_r=0.75,\varphi =\pi /4$.