Highlights

## Semi linear response theory (2005-2006)()

The absorption of energy by driven system can be calculated by linear response (Kubo) only if the driven transitions are much slower compared with the relaxation rate. In the mesoscopic regime we assume that the relaxation effect is weak, and consequently a theory that goes beyond Kubo is required.  Our semi linear response theory (SLRT) uses a resistor network analogy (see illustration below) in order to calculate the rate of energy absorption. Connected sequences of transitions are essential in order to have non-vanishing result, as in the theory of percolation. The original motivation for introducing this theory [1] was the study of mesosopic conductance.  The same idea was later used [2] to develop a theory for the rate of energy absorption by metallic grains.

[1] D. Cohen, T. Kottos and H. Schanz, J. Phys. A 39, 11755 (2006) [arXiv] [pdf]
[2] M. Wilkinson, B. Mehlig, D. Cohen, Europhysics Letters 75, 709 (2006) [arXiv] [pdf], Editorial Board highlight of 2006

## Quantum Chaos perspective of SLRT

Semi-linear response theory (SLRT) applies to the calculation of energy absorption of driven weak-quantum-chaos systems. Such systems features a "sparse" Hamiltonian matrix:

$\mathcal{H}=\{E_n\}+f(t)\{W_{nm}\}$

The theory uses a resistor network picture of transitions, and gives different results compared with linear response theory (Kubo) whenever the perturbation matrix in the Hamiltonian is "textured" or "sparse" [notably if its elements have log-normal or log-box rather than Gaussian distribution].

The theory has been applied to obtain novel results for the mesoscopic conductance of rings with very strong or very weak disorder, and also in order to resolve long standing questions regarding the applicability of the variable range hopping picture and the absorption of low frequency radiation by small metallic grains.

The following figure illustrates model systems of interest: a Billiard with a moving wall, and a Ring with a time dependent magnetic flux. The "sparsity" of the perturbation matrix reflects the slight deviation of the Billiard from intelligibility, due to the deformation of the boundary (as in the figure) or due to the presence of disorder.

Review: Energy absorption by "sparse" systems [arXiv][pdf] (Prague 2012).

## Heating of atoms in vibrating traps (2009-2011)

An application of semi linear response theory (SLRT) concerns the rate of heating of cold atoms in weakly chaotic optical billiards. This work extends older works where we have improved the "wall formula" of nuclear physics taking into account the type of wall deformation involved [notably we have argued that shaking or dilating a box does not lead to heating in the DC limit, unlike any other generic deformation]. We emphasize that SLRT rather than liner-response theory is essential in experimentally relevant circumstance if the system is quasi-isolated from its environment. SLRT is also the appropriate framework for the incorporation of weak interaction effects.

## The mesoscopic conductance of closed rings (2006-2008)

Consider a multichannel closed ring with disorder (W). In the semiclassical treatment its conductance (G) is given by the Drude formula. Our theory challenges this result both in the limit of strong disorder (eigenstates are not quantum-ergodic in real space) and in the limit of weak disorder (eigenstates are not quantum-ergodic in momentum space). The analysis of the conductance is based on our "semi linear response theory".
• Our theory implies that the correct quantum result for the conductance of a ballistic ring is neither Landauer-like nor Drude.
• In the strong disorder limit our theory provides a firm unified framework from which the "hopping" phenomenology of Mott can be derived.

[1] S. Bandopadhyay, Y. Etzioni and D. Cohen, EPL (2006). [arXiv] [pdf]
[2] D. Cohen, Phys. Rev. B 75, 125316 (2007). [arXiv] [pdf]
[3] T. Peer, R. Budoyo, A. Stotland, T. Kottos and D. Cohen, JPA (FTC, 2008). [arXiv] [pdf], Editor's choice of 2007-2008