All animations show the time evolution of the echoed Wigner function of a quantized top, i.e. the Wigner function of a state after a forward time evolution with Hamiltonian H followed by a backward time evolution with a perturbed Hamiltonian H'. The initial state is the same coherent state in all animations. See full article for details.

The fidelity F(t) at time t is just an overlap between the echoed Wigner function at time t and the Wigner function of the initial state, which is a Gaussian. The same property holds for a classical phase-space density and the classical fidelity.

Animations for two different systems are shown (see full article), one with ω'' = 0, having strong resonances, and another for ω'' = 4 where resonances are supressed. For each case we show three movies in MPEG format, one for short times of correspondence with the classical fidelity (also shown in parallel), one for intermediate times when quantum fidelity has a plateau, and one for long times when fidelity exhibits balistic decay.

The unitary one step propagator in the case of ω'' = 0 is: U₀ = exp(-iαS_z²/2S), while in the case where we suppress the resonances by a nonvanishing second derivative of the frequency ω''=4, the propagator reads U₀ = exp(-iS[α(S_z/S)²/2 + 4(S_z/S - j*)³/6]), where j* is the action coordinate of the centre of the coherent state.

The perturbed evolution is in both cases generated by the x-component of the spin: U_δ = U₀ exp(-iS_xδ).

For all animations we use S = 200, δ = 0.0016, and show only the upper hemisphere of the phase space, 0 <  < 2π, 0 < cos θ < 1.

System with ω'' = 0

Short time quantum (red curve) and classical (green curve) fidelity decay. Note pointwise agreement up to the 'regular Ehrenfest time' which is in this case ~14.

short.mpg (758 KB MPEG) Short time animation from t = 0 to t = 240. The lower part shows the classical density producing the classical fidelity. At time t₁ = 14 the correspondence between the classical and quantum evolution stops. The classical fidelity decays in a power law fashion while the quantum fidelity freezes on a plateau.

middle.mpg (906 KB MPEG) Intermediate time animation from t = 80 to t = 2000, in steps of 10. This is the regime of a plateau in the quantum fidelity. The plateau can here be seen to originate from a part of a packet which stays localized at the initial position, giving a constant overlap with the initial Wigner function. The mechanism producing the quantum resonances is also nicely illustrated.

long.mpg (736 KB MPEG) Long time animation from t = 2000 to t = 140000, in steps of 1000. Here the Gaussian decay of fidelity due to a balistic movement of an echo begins.

System with ω'' = 4

Long time quantum fidelity decay for ω'' = 0 (green) and ω'' = 4 (red):

g4short.mpg (658 KB MPEG) Short time animation from t = 0 to t = 240.

g4middle.mpg (8106 KB MPEG) Intermediate time animation from t = 80 to t = 2000. The regime of the plateau.

g4long.mpg (796 KB MPEG) Long time animation from t = 2000 to t = 150000, in steps of 1000. The regime of ballistic decay.