Stability
of quantum motion

The problem of the
stability of quantum motion has attracted a great interest, also in
relation to the field of quantum computation.

A quantity of central importance which has been on the focus of many studies is the so-called fidelity f(t), which measures the accuracy to which a quantum state can be recovered by inverting, at time t, the dynamics with a perturbed Hamiltonian.

The analysis of this quantity has shown that, under appropriate conditions, the decay of f(t) is exponential with a rate given by the classical Lyapunov exponent (the Lyapunov exponent measures the rate of exponential instability of classical motion).

A quantity of central importance which has been on the focus of many studies is the so-called fidelity f(t), which measures the accuracy to which a quantum state can be recovered by inverting, at time t, the dynamics with a perturbed Hamiltonian.

The analysis of this quantity has shown that, under appropriate conditions, the decay of f(t) is exponential with a rate given by the classical Lyapunov exponent (the Lyapunov exponent measures the rate of exponential instability of classical motion).

Quantum-classical
correspondence in perturbed chaotic systems

We have discussed the behavior of fidelity for a classically chaotic quantum system. We have shown the existence of a critical value of the perturbation above which the quantum decay, exponential or power-law, follows the classical one [1].

The independence of the decay rate on the perturbation strength, discussed in the literature, is a consequence of the quantum-classical correspondence of the relaxation process.

Quantum-classical correspondence
for the fidelity decay in the diffusive regime until the Heisenberg
time scale.

References

[1] G. Benenti and G.
Casati, Quantum-classical correspondence in perturbed chaotic systems,
Phys. Rev. E 65, 066205 (2002).