# Research Activity,  Highlights

### Semi-linear response for the heating rate of cold atoms in vibrating traps

We are working on the absorption rate of particles that are confined by vibrating walls in a box. This subject was of interest in past linear response theory (LRT) studies of nuclear friction, where it leads to the damping of the wall motion. More recently, it has become of interest in the context of cold atoms physics. One wonders what happens if the "billiard" is fully chaotic but with nearly integrable shape.  We show that the analysis that is relevant to the typical experimental conditions (collaboration with Nir Davidson at Weizmann Institute, Rehovot, Israel) should go beyond LRT, and should involve a "resistor network" picture of transitions in energy space, somewhat similar to a percolation problem. Each unperturbed energy level of a particle in a vibrating trap, or of an electron in a mesoscopic ring, is regarded as a node (n) of the network; The transition rates (w_{mn}) between the nodes are regarded as the elements of a random matrix that describes the network. This theory was called the semi-linear response theory (SLRT). We show that if the perturbation matrix is "sparse'' or "textured'' (notably if its elements (x) have log-normal or log-box rather than Gaussian distribution), as determined by the geometry of the system, the results are different compared with LRT.  Consequently, we predicted that the rate of energy absorption would be suppressed by orders of magnitude and introduced an improved sparse random matrix model which leads to a generalized variable range hopping picture. Our analytical estimates were supported by a numerical calculation.

[1]
A. Stotland, D. Cohen and N. Davidson, Europhysics Letters 86, 10004 (2009) [arXiv] [pdf]

In the first row there are different possible types of perturbation matrices: "banded Gaussian", "textured", "textured and sparse".
In the second row the "resistor network" in energy space is presented, and next to it there is a histogram of  sparse matrix elements.

### The mesoscopic conductance of disordered rings

We have also applied the semi-linear response theory to calculate the mesoscopic conductance of small metallic rings driven by Aharonov-Bohm flux (in collaboration with the group of Tsampikos Kottos at Wesleyan University, Middletown, Connecticut, USA).  Consider a multichannel closed ring with disorder (W). In the semi-classical 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).
• 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.
Also in this problem the texture and the sparsity of the perturbation matrix dictate the value of the conductance, and we studied its dependence on the disorder strength, ranging from the ballistic to the Anderson localization regime. In particular we have obtained practical approximations for the conductance of mesoscopic rings.

[1] A. Stotland, R. Budoyo, T. Peer, T. Kottos and D. Cohen, J. Phys. A 41, 262001 (FTC) (2008) [arXiv] [pdf], Editor's choice of 2007-2008
[2]
A. Stotland, T. Kottos and D. Cohen,  (2009) [arXiv]

The SLRT, LRT and Drude conductance vs. the disorder strength.

### Diffractive energy spreading and its semiclassical limit

We consider driven systems where the driving induces jumps in energy space:
• particles pulsed by a step potential;
• particles in a box with a moving wall;
• particles in a ring driven by an electro-motive-force.
In all these cases the route towards quantum-classical correspondence is highly non-trivial. Some insight is gained by observing that the dynamics in energy space, where n is the level index, is essentially the same as that of Bloch electrons in a tight binding model, where n is the site index. The mean level spacing is like a constant electric field and the driving induces long range hopping 1/(n-m).

[1] A. Stotland and D. Cohen, J. Phys. A 39, 10703 (2006) [arXiv] [pdf]

In the above illustration the EMF is concentrated at one point along the ring.  Whenever a particle crosses the EMF region its kinetic energy is boosted. The energy jump is eV. From quantum mechanical point of view this constitutes a non-perturbative effect. It is neither "adiabatic" nor "diabatic", but rather a "semiclassical" transition. In the analagous tight binding model the semicalssical dynamics is regarded as uni-directional Bloch oscillations.

### The information entropy of quantum mechanical states

It is well known that a Shannon based definition of information entropy leads in the classical case to the Boltzmann entropy. It is tempting to regard the Von Neumann entropy as the corresponding quantum mechanical definition. But the latter is problematic from quantum information point of view. Consequently
• we introduce a new definition of entropy that reflects the inherent uncertainty of quantum mechanical states.
• we derive for it an explicit expression, and discuss some of its general properties.
• we distinguish between the minimum uncertainty entropy of pure states, and the excess statistical entropy of mixtures.
$S[\rho]=S_0(N) + F(p_1,p_2,...,p_N) \ne -\sum_r p_r \ln p_r$

[1]
A. Stotland, A.A. Pomeransky, E. Bachmat, D. Cohen, Europhysics Letters 67, 700 (2004) [arXiv] [pdf]