Dynamical Origin of Decoherence in Clasically Chaotic Systems

The decay of overlap between a wave packet evolved with a HamiltonianH and the same state evolved withH+Σ serves as a measure of the decoherence timeτφ. Recent experimental and analytical evidence on classically chaotic systems suggest that, under certain conditions, τφ depends on H but not on Σ. By solving numerically a Hamiltonian model we find evidence of that property provided that the system shows a Wigner-Dyson spectrum (which defines quantum chaos) and the perturbation exceeds a crytical value defined by the parametric correlations of the spectra. The existence of chaos in classical mechanics is manifested in the evolution of a state as an extreme sensitivity to the initial conditions. Quantum mechanics, on the opposite, does not show this sensitivity.. This has raised several problems in a dynamical definition of quantum chaos. In particular, numerical [1] and experimental[2] studies show that time reversal can be achieved with great accuracy. Therefore, the search for a quantum definition of chaos, lead to investigate the spectral properties[3] of quantum systems whose classical equivalent is chaotic. Quantum chaos appears as the regime in which the properties of the eigenstates follow the predictions of the Random Matrix Theory (RMT). In particular the normalized spacing between energy levels s = (εi+1−εi)/∆ε, with ∆ε the mean level spacing, should have a probability distribution given by the Wigner Dyson distribution P WD(s) = (πs/2) exp(−πs /4) for an orthogonal ensemble and P WD(s) = (32s /π) exp(−4s/π) for the unitary ensemble. An infinite set of interacting spins is an example of a many-body system which is chaotic in its classical version (lattice gas) and hence it is expected to present the quantum signatures of chaos in the spectrum. The dynamics of this particular system can be studied by Nuclear Magnetic Resonance (NMR). Surprisingly, the “diffusive” dynamics of a local excitation exp[−iHtR]|0 > can Preprint submitted to Elsevier Preprint 1 February 2008 be reversed [4], generating a Polarization Echo M at time 2tR. In this case H is the many-body Hamiltonian of a network of spins with dipolar interaction. To accomplish this, the transformation H → −(H + Σ) at time tR is performed with standard NMR techniques [4]. This transformation is possible due to the anisotropic nature of the dipolar interaction. The perturbation Σ is a non-invertible component of the Hamiltonian. In some systems, the only contribution to Σ is proportional to the inverse of the radio frequency power and hence can be made arbitrarily small. In one body systems the Polarization Echo (i.e. magnitude of the excitation recovered at time 2tR) can then be written exactly as: M(t) = | 〈0| exp[i(H + Σ)t/h̄] exp[−iHt/h̄] |0〉 |, (1) where |0〉 is the initial wave function, H the unperturbed Hamiltonian and Σ the perturbation which can be associated with an environmental disturbance. Then the magnitude of experimental interest is the overlap between the same initial wave function evolved with the two different Hamiltonians, H and −(H + Σ). We should note that the second evolution can be seen as an “imperfect time reversal” of the wave function!. Consistently, more than 10 years ago Peres [5] proposed that dynamical signatures of quantum chaos should be searched on the sensitivity to perturbations in the Hamiltonian. Actually, for a classically chaotic system a perturbation in the initial conditions is equivalent to a perturbation in the Hamiltonian. The experiments show that M decays rapidly with tR with Gaussian law [6] indicating a progressive failure in rebuilding the original state. We can define a decoherence time τφ from this failure, as the width of the Gaussian. This is found[7] to be roughly independent of Σ and it extrapolates to a finite value when Σ → 0. Using a semiclassical one-body analytical approach in classically chaotic systems characterized by a Lyapunov exponent λ, Pastawski and Jalabert[8] have shown that there is a regime where the attenuation of M it is independent of the perturbation Σ and becoming 1/τφ = λ. This non-perturbative result is valid for long times and as long as Σ does not change the Hamiltonian nature. Our general aim is to find numerical evidence of this regime where M(t) is independent of Σ considering the simplest Hamiltonian that could model spin diffusion. In this work, we study one-body Hamiltonian in quasi-1D systems with N states which we called the Star’s necklace model. More specifically, we use a tight binding model of a ring-shaped lattice with on-site disorder, hopping matrix elements V , and a magnetic flux Φ perpendicular to the plane of the ring (see inset in Fig.1). Let us discuss the general features through one representative class, each star has 20 sites and there are L = 35 beads in the necklace which makes N = 700. In our case, perturbation acts only between two star beads: Σ(δΦ) = V exp[i2πΦ/Φo](exp[i2πδΦ/Φo]− 1) |1〉 〈L|+ c.c. Bra