The theory of linear response (formalized by R. Kubo in 1956) allows to express the variations of physical observables (the "response") when a system is displaced slightly out-of-equilibrium by a "foreign perturbation". This linear response is analogous for physics to the first order Taylor expansion for a mathematical function with, in addition, dynamical aspects: the system response usually lags after the perturbation that caused it.

A well-known example of linear response function is the impedance of an electrical circuit, which gives the response of the current in an electrical circuit to a change in the voltage. The real part of impedance measures the dissipative response, which is fundamentally connected to voltage fluctuations in the unperturbed system by the "Fluctuation-Dissipation Theorem" (FDT). This theorem is a crucial consequence of the Kubo formula and is encountered in all areas of science.

Despite its universality and its huge success, the linear response theory cannot handle many physical systems that are intrinsically nonlinear. This is the case for instance of nanostructures where the Coulomb repulsion between electrons plays a crucial role. Several approaches were used to address the response of such systems, but, being specific to each problem, they have a limited scope.

However, we could show that there is a rigorous generalization of the Kubo formula giving the response of any physical system to an arbitrary excitation. This generalization is valid with virtually no restrictions on the system (it may be out of equilibrium, time-dependent, interacting...) and respects conservation laws by construction. This formalism is therefore suited to address the response of nonlinear systems. Consequently, and to illustrate the power of this generalization, we establish a new non-equilibrium FDT-type relation for correlations between the currents measured at two different times in an arbitrary conductor connected to multiple terminals.

Then, we use this new FDT relation to demonstrate a universal property of the current correlations in circuits. Under stationary conditions, their Fourier transform at negative and positive frequencies respectively gives the spectrum of absorption and emission of the system. We show that the deviations of this spectrum with respect to its value at equilibrium is frequency-asymmetric if the conductance is non-linear with respect to the DC voltage applied to the system. This allows, in particular, to explain such an asymmetry between emission and absorption observed in experiments carried out at LPS (R. Deblock and colleagues) with Josephson junctions that are intrinsically non-linear. This general result also allows to unify the understanding of such asymmetries predicted in various nonlinear systems with or without Coulomb repulsion.

**Figure:** A conductor described by a Hamiltonian (potentially time-dependent) connected to several terminals connected to time-dependent voltages sources V_{n}(t) where n is the index of the terminal. In a manner similar to the Kubo formula, we can express the conductance G_{nn’}(t,t’), which gives the variation of the average electric current I_{n’}(t’) in a terminal n’ in response to a variation of V_{n}(t) while keeping all voltages finite. The result is a new FDT-type theorem relating the correlations between the currents measured in n and n’: < I_{n}(t) I_{n’}(t’) >, and the conductance G_{nn’}(t,t’).

**References:**

I. Safi et P. Joyez, Phys. Rev. B **84**, 205129 (2011).

I. Safi, arXiv:0908.4382 (2009).

**Contact: **

**Inès Safi** (ines.safi@u-psud.fr)