The Loschmidt Echo Homepage

Josef Loschmidt

The study of the Loschmidt Echo is an ongoing international collaboration with people from Latin-America and Europe. Up to now, the people involved are the following: Horacio M. Pastawski (FAMAF, Córdoba, Argentina) Patricia R. Levstein (FAMAF, Córdoba, Argentina) Rodolfo Jalabert (IPCMS, Strasbourg, France) Fernando M. Cucchietti (FAMAF, Córdoba, Argentina) Gonzalo Usaj (Duke University, Duke, USA) Diego A. Wisniacki (CNEA, Buenos Aires, Argentina) Eduardo Vergini (CNEA, Buenos Aires, Argentina) Eduardo R. Mucciolo (PUC, Rio de Janeiro, Brazil) Caio H. Lewenkopf (UERJ, Rio de Janeiro, Brazil) Raúl O. Vallejos (CBPF, Rio de Janeiro, Brazil)

The study of the Loscmidt Echo started in 1995 with NMR experiments in many-body spin systems [1]. There, a hipersensitivity of time reversal to perturbations was found, along with a characteristic decoherence time roughly independent of the perturbation. In these experiments, quantum irreversibility is monitored through the attenuation of the amount of revival of a local density excitation upon time reversal of its quantum evolution, i.e. the Loschmidt Echo. Its mathematical expression is quite simple,

M(t) = |<0| exp(it(H+S)) exp(-itH) |0>|^2

Where |0> is an initial state of the system, H its Hamiltonian, and S the perturbation. The first attempt to explain the experimental results used that such a complex many-body system could be assumed to be chaotic. This was reinforced by the fact that the decoherence time was found to be proportional to the only dynamical scale of the system. Using analytical semiclassical theory, Jalabert and Pastawski [2] predicted that if time reversal were attempted in a classically chaotic system, when a quenched disorder perturbation is present, the Loschmid Echo would attenuate exponentially. Notably, its decay rate is only determined by the Lyapunov exponent of the classical system.

A bunch of numerical has come out since then. Using different systems, methods and perturbations, we are exploring the limits of the theory (parameter ranges, soft chaos, etc). The main results have been obtained in the Lorentz Gas, the Bunimovich Stadium and the Smooth Billiard, all standard examples of classical chaos. Some nice pictures are available in this page, but for the serious work see [2] and [3].

[1] P. R. Levstein, G. Usaj and H. M. Pastawski, J. Chem. Phys. 108, 2718 (1998); G. Usaj, H. M. Pastawski P. R. Levstein, Mol. Phys.95, 1229 (1998); H. M. Pastawski, P. R. Levstein, G. Usaj, J. Raya and J. Hirschinger, Physica A 283, 166 (2000).

[2] R. Jalabert and H. M. Pastawski, Phys. Rev. Lett. 86, 2490 (2001)

[3] F. M. Cucchietti, H. M. Pastawski and D. A. Wisniacki, Phys. Rev. E 65, 045206(R) (2002)

The Lorentz Gas

We consider a particle in a square billiard of area L^2 where we place an irregular array of N circular scatterers of radius R.

(a)		(b)
(c)		(d)

(a) A particular realization of such system, where the length, in a minimal unit a, is L=200a. In the center of the figure is the density of the initial Gaussian wave packet.

(b) The density resulting from the evolution of the initial state in (a), where the initial momentum points to the left. The propagation time is t=30 \hbar/V.

(c) An evolution for an equivalent time, but with the perturbed Hamiltonian H+S

(d) Evolution for time t=30 \hbar/V with Hamiltonian (-H+S) of the state depicted in panel (b)

While the densities in panels (b) and (c) look similar to the eye, the corresponding square of the overlap, M(t), between both evolutions is about 0.09 indicating the relevant role of the quantum phase.

Here are the animated sequences of the system shown above, with the Forward (H) Backward (-H+S) evolution (zoomed for convenience)

(The file has 118Kb, so be patient)

The Smooth Billiard

As stated above, another system where the Loschmidt Echo is being investigated is the Smooth Billiard, which is defined by the potential

The exponent n sets the steepness of the confining potential. For n=1 the smooth stadium is integrable. As the value of n is increased, the borders become steeper. In the limit of n->infinity, the stadium gains hard walls, becoming the well-know Bunimovich billiard. Thus, by varying n, we can tune the system dynamics from integrable to progressively more chaotic and study the behavior of the Loschmidt Echo in this situations.

A little animation showing the evolution of a wave packet in this system (266 KB) is below. Later on I will add some time reversal animations...

The Bunimovich Stadium

This section is under construction

For more information contact Fernando M. Cucchietti