Loschmidt echo and the local density of states.

Loschmidt echo (LE) is a measure of reversibility and sensitivity to perturbations of quantum evolutions. For weak perturbations its decay rate is given by the width of the local density of states (LDOS). When the perturbation is strong enough, it has been shown in chaotic systems that its decay is dictated by the classical Lyapunov exponent. However, several recent studies have shown an unexpected nonuniform decay rate as a function of the perturbation strength instead of that Lyapunov decay. Here we study the systematic behavior of this regime in perturbed cat maps. We show that some perturbations produce coherent oscillations in the width of LDOS that imprint clear signals of the perturbation in LE decay. We also show that if the perturbation acts in a small region of phase space (local perturbation) the effect is magnified and the decay is given by the width of the LDOS.

[1]  Doron Cohen,et al.  Quantum irreversibility, perturbation independent decay, and the parametric theory of the local density of states. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  D. Shepelyansky,et al.  Stable quantum computation of unstable classical chaos. , 2001, Physical review letters.

[3]  J. Raimond,et al.  Manipulating quantum entanglement with atoms and photons in a cavity , 2001 .

[4]  D. Wisniacki,et al.  Stadium billiard with moving walls. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  L. Sohn,et al.  Mesoscopic electron transport , 1997 .

[6]  Quantum computers in phase space , 2002, quant-ph/0204149.

[7]  S. Datta.  Quantum Transport: Atom to Transistor , 2004 .

[8]  Thomas H. Seligman,et al.  Dynamics of Loschmidt echoes and fidelity decay , 2006, quant-ph/0607050.

[9]  Klaus Richter,et al.  Loschmidt echo for local perturbations: non-monotonic cross-over from the Fermi-golden-rule to the escape-rate regime , 2008, 0804.0571.

[10]  Jirí Vanícek.  Dephasing representation: Employing the shadowing theorem to calculate quantum correlation functions. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  R A Jalabert,et al.  Environment-independent decoherence rate in classically chaotic systems. , 2001, Physical review letters.

[12]  Ezequiel N Pozzo,et al.  Fidelity and quantum chaos in the mesoscopic device for the josephson flux qubit. , 2007, Physical review letters.

[13]  Stuart A. Rice,et al.  Optical Control of Molecular Dynamics , 2000 .

[14]  P G Silvestrov,et al.  Golden rule decay versus Lyapunov decay of the quantum Loschmidt echo. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  N Davidson,et al.  Decay of quantum correlations in atom optics billiards with chaotic and mixed dynamics. , 2006, Physical review letters.

[16]  G. Casati,et al.  Stability of quantum motion: beyond Fermi-golden-rule and Lyapunov decay. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  E. Heller,et al.  Parametric evolution for a deformed cavity. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Asher Peres,et al.  Stability of quantum motion in chaotic and regular systems , 1984 .

[19]  Arseni Goussev,et al.  Long-time saturation of the Loschmidt echo in quantum chaotic billiards. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  M. Berry,et al.  Quantization of linear maps on a torus-fresnel diffraction by a periodic grating , 1980 .

[21]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[22]  Joseph Ford,et al.  The Arnol'd cat: failure of the correspondence principle , 1991 .

[23]  U. Kuhl,et al.  Algebraic fidelity decay for local perturbations. , 2007, Physical review letters.

[24]  M. Dematos,et al.  Quantization of Anosov Maps , 1995 .

[25]  Diego A Wisniacki.  Short-time decay of the Loschmidt echo. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  M. Esposti,et al.  The quantum perturbed cat map and symmetry , 2005 .