CENTER FOR NONLINEAR AND COMPLEX SYSTEMS Giulio Casati - Istituto Nazionale di Fisica della Materia, and Universita’ dell’Insubria -National University of Singapore, Singapore. Como - Italy

1- The Loschmidt echo and the stability of classical and quantum motion. 2- Quantum dephasing and internal dynamical chaos. 3- A double slit experiment. This talk: In collaboration with: G. Benenti Como V. Sokolov Como-Novosibirsk T. Prosen, Liubliana

Main features of quantum dynamics: -discretness of phase space (finite h) -interference Main features of dynamical chaos: -Exponential local instability -Continuous spectrum of the motion

HYDROGEN ATOM IN EXTERNAL MICROWAVE FIELD classical quantum Time of reversal prl 56, 2437 (1986) classical quantum

HYDROGEN ATOM IN EXTERNAL MICROWAVE FIELD classical quantum Time of reversal prl 56, 2437 (1986) quantum

HYDROGEN ATOM IN EXTERNAL MICROWAVE FIELD classical quantum Time of reversal prl 56, 2437 (1986)

Unitary evolution

1- The quantum “Loschmidt Echo” (fidelity)

Joseph Loschmidt “His work forms a mighty cornerstone that will be visible as long as science exists” Loschmidt paradox

Jalabert, Pastawsky Beenakker, Jacquod, Silvestrov Prozen, Znidarich, Seligman Tomsovich, Cerutti Heller, Vanicek Zurek, et al. Cucchietti et al. Wisniacki, Cohen Emerson, Loyd + several others….

Fidelity decay for classically chaotic systems 1-Perturbative regime 2-Breit -Wigner regime (Fermi golden rule)

The time scale in which one regimes prevails over the other depends on which case the argument of the exponential takes on the lesser value. The crossover time is given by:

Benenti,g.c.PRE: 65 (2002) Lyapounov regime:

-The decay is perturbation independent and asymptotically same as correlations functions: - exponential with rate given by: i)short times: Lyapounov ii)asymptotic: the gap in the Perron- Frobenius operator - power law. - Noise leads to same decay as static perturbations. For chaotic classical systems : Benenti,G.C. Veble PRE (2003)

Quantum fidelity decay in regular systems Integrability is the exception. However: - quasi-integrable motion is typical - quantum computer should operate below the chaos border - some quantum algorithm (Grover) can be reduced to regular map The decay is perturbation dependent: Initially Gaussian followed by power law tail. W. Wenge, g.c. B. Li : preprint

Efficient quantum algorithms have been found to simulate quantum dynamics of complex systems Question: given a generic dynamical system is it possible to find its solution efficiently? When following a classical chaotic orbit one digit of accuracy is lost per suitable chosen unit of time: To follow an orbit up to time t we must input O(t) bits of information.

Benenti G.C. Montangero Shepelyansky prl (2001)

The degree of stability of quantum algorithms does not depends on the nature of the simulated dynamics Consider unitary errors modeled by noisy gates (unavoidable due to imperfections in the quantum hardware or interaction with environment). Quantum errors are non local in phase space Rossini, Benenti, G. C. PRE (2004)

2- Loschmidt echo and dephasing for a pure coherent state

For a mixed initial state: Not related to dephasing!

The first term is a sum of fidelity of individual pure states. If the number M of pure states in the initial mixed state is large M>>1, then

Consider a nonlinear oscillator driven by a periodic multimode external force g(t):

We analitically show that, due to dephasing induced by the underlying chaotic dynamics, the decay of can be directly connected to the decay of a Classical correlation funtion Contrary to decoherence produced by external noise, here dephasing is of purely dynamical nature. V. V. Sokolov, G. Benenti, G. C. quant-ph/

When the strength of the driving force exceeds a critical value the classical motion becomes chaotic and the function becomes random

Numerical results on the kicked rotator model

“….is impossible, absolutely impossible to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery.“ 3- The double slit experiment R. Feynmann

g.c., T. Prozen: Phys. Rev. A 72, (2005)

Snapshots at time half Heisenberg time

G. Benenti Como V. Sokolov Como C. Monasteiro Torino S. Montangero Pisa D. Rossini Pisa Li Baowen Singapore Weng ge Wang Singapore G. Veble Lubliana T. Prosen Lubliana