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

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 ComoV. Sokolov Como-NovosibirskT. 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)

Unitaryevolution

1- The quantum “Loschmidt Echo”(fidelity)

Joseph Loschmidt

“His work forms a mightycornerstone that will be visible as long as scienceexists”

Loschmidt paradox

Jalabert, PastawskyBeenakker, Jacquod, SilvestrovProzen, Znidarich, SeligmanTomsovich, CeruttiHeller, VanicekZurek, et al.Cucchietti et al.Wisniacki, CohenEmerson, 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) 066205

3- Lyapounov regime:

-The decay is perturbation independentand 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 055202 (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.

BenentiG.C.MontangeroShepelyanskyprl (2001)

The degree of stability of quantum algorithms doesnot 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 056216 (2004)

Rossini, Benenti , G. C. PRE 056216 (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/0504141

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, 032111 (2005)

Snapshots at time half Heisenberg time

G. Benenti ComoV. Sokolov ComoC. Monasteiro TorinoS. Montangero PisaD. Rossini PisaLi Baowen SingaporeWeng ge Wang SingaporeG. Veble LublianaT. Prosen Lubliana