Stability of quantum motion and correlation decay I & II

Tomaz Prosen

Department of Physics, Faculty of Mathematics and Physics
University of Ljubljana, Slovenia

In this two-hour lecture we will make a review of the theory and the (numerical) experiments on the behavior of quantum fidelity of classically chaotic and regular hamiltonian systems.

We will derive a simple and general relation between the fidelity of quantum motion, characterizing the stability of quantum dynamics with respect to arbitrary static perturbation of the unitary evolution propagator, and the integrated time auto-correlation function of the generator of perturbation. Quite surprisingly, this relation predicts the slower decay of fidelity the faster decay of correlations. In particular, for non-ergodic and non-mixing dynamics, where asymptotic decay of correlations is absent, a qualitatively different and faster decay of fidelity is predicted on a time scale $\propto 1/\delta$ as opposed to mixing dynamics where the fidelity is found to decay exponentially on a time-scale $\propto 1/\delta^2$, where $\delta$ is proportional to the strength of perturbation. A detailed discussion of a semi-classical regime of small effective values of Planck constant $\hbar$ is given where classical correlation functions can be used to predict quantum fidelity decay. Note that the correct and intuitively expected classical stability behavior is recovered in the classical limit $\hbar\to 0$, as the two limits $\delta\to 0$ and $\hbar\to 0$ do not commute. In addition we also discuss a non-trivial dependence on the number of degrees of freedom and the role of the thermodynamic limit.

The theoretical predictions will be demonstrated mainly in two families of models: (i) a quantized kicked top and a quantized pair of coupled kicked tops where the semiclassical regime is emphasized, and (ii) kicked Ising spin 1/2 chain where the thermodynamic regime is emphasized. We also need to stress that these results have important implications for the stability of quantum computation, and may be used in order to optimize the accuracy of quantum algorithms.

References
T. Prosen, ``General relation between quantum ergodicity and fidelity of quantum dynamics'', Phys. Rev. E 65, 036208 (2002).
T. Prosen and M. Znidaric, ``Can quantum chaos enhance the stability of quantum computation?'', J. Phys. A: Math. Gen. 34, L681 (2001).
T. Prosen and M. Znidaric, ``Stability of quantum motion and correlation decay'', J. Phys. A: Math. Gen. 35, 1455 (2002).
T. Prosen and T. H. Seligman, ``Decoherence of spin echoes'', J. Phys. A: Math. Gen. 35, 4707 (2002); nlin.CD/0201038.
T. Prosen, T. H. Seligman and M. Znidaric, ``Stability of quantum coherence and correlation decay'', preprint, quant-ph/0204043

Chaotic resonances in quantum many-body dynamics

Tomaz Prosen

Department of Physics, Faculty of Mathematics and Physics
University of Ljubljana, Slovenia

The questions about the mechanisms and the conditions for the relaxation to equilibrium in the thermodynamic limit of a generic isolated hamiltonian system constitute important open problems in statistical mechanics. In this talk we consider an established technique in describing relaxation of strongly chaotic single-particle classical systems, namely the concept of Perron-Frobenius-Ruelle resonance spectrum, and use it for the dynamical description of non-integrable quantum many-body systems in thermodynamic limit.

We define a quantum Perron-Frobenius master operator over a suitable normed space of translationally invariant states adjoint to the quasi-local $C^*$ algebra of quantum lattice gasses (e.g. spin chains), whose spectrum determines the exponents of decay of time correlation functions. The gap between the leading eigenvalue and the unit circle signals the exponential mixing (universal asymptotic exponential decay of arbitrary time correlation functions), whereas closing the gap typically corresponds to a transition to non-ergodic dynamics, which may as a consequence, lead to important anomalous transport properties. In particular, the conservation laws of completely integrable quantum lattices represent degenerate eigenvalue 1 eigenvectors of the Perron-Frobenius operator.

Theoretical ideas are applied and validated in a generic example of kicked Ising spin $1/2$ chains, namely a one dimensional spin $1/2$ lattice with nearest neighbor Ising interaction kicked with periodic pulses of a tilted homogeneous magnetic field. We show that the 'chaotic eigenmodes' corresponding to leading Perron-Frobenius-Ruelle eigenvalue resonances have fractal structure in the basis of local operators.

References
T. Prosen, ``Chaotic resonances in quantum many-body dynamics'', preprint.