Time-dependent theory of nonlinear response and current fluctuations

A general nonlinear response theory is derived for an arbitrary time-dependent Hamiltonian, not necessarily obeying time-reversal symmetry. We consider the application of this theory to a multiterminal mesoscopic system with arbitrary interactions and time-dependent voltages. This allows us to obtain a generalized Kubo-type formula. We derive a microscopic expression for the finite frequency differential conductance matrix, which preserves current conservation and gauge invariance. We exploit this result to show that the asymmetric part of the current fluctuation matrix at finite frequency obeys a generalized time-dependent fluctuation-dissipation theorem. In the stationary regime, this theorem provides a common explanation for the asymmetry of the excess noise with respect to positive and negative frequencies that has been obtained in several systems as a consequence of nonlinearity. It also explains the origin of the unexpected negative sign of the excess noise. Finally, we apply these general results to the case of a tunnel junction and obtain a nonperturbative out-of-equilibrium link between conductance and current fluctuations. We also derive a universal property of the finite frequency noise in the perturbative regime.

Figure 1

A mesoscopic system coupled to many terminals, including a gate, with arbitrary time dependence of their electrochemical potentials. The time-dependent Hamiltonian $\mathcal{H}$ includes arbitrary interactions or disorder and can depend on other parameters $X\left(t^{\prime }\right)$ in a nonlocal and nonlinear way. The differential of the current average $\langle I_n\left(t\right)\rangle$ at terminal $n$ either with respect to $V_{n^{\prime }}\left(t^{\prime }\right)$ or to $X\left(t^{\prime }\right)$ can be expressed through a generalized response formula keeping all $V_n$ and $X$ finite.

Figure 2

Tunnel junction between two conductors 1 and 2 for arbitrary dimension, interactions or disorder, subject to time-dependent potential $V_{1,2}\left(t\right)$, and with time-dependent tunneling (not necessarily weak) of charges $q$. These conductors can be either similar or different, such as fractional edge states with equal or different filling factors, leads that are superconducting, insulating, or normal, etc. One can also view one system as a probe, such as the tip of a scanning tunneling microscope or a Fermi liquid coupled to edge states or to a quantum dot. Capacitive coupling is possible, and we require only charges $Q_1$ and $Q_2$ to be well defined, commuting with the total Hamiltonian. Coupling to an electromagnetic environment can be incorporated as well.