10.10 Conclusions
393
avoided crossing in level |E 4 . As the laser pulses are turned on and off the system
transitions from the initial state |ψ(0) = |E 1 to the final state |ψ(+∞) = |E 4 .
Note that both Fig. 10.23b and c follow almost exactly the behavior of the Floquet
state A shown in Fig. 10.5c. This is an indication that we are in the adiabatic regime
in Fig. 10.23b and c.
The very large oscillations in the probability in Figs. 10.23b and c have been
explained by Berry (1990) in terms of a sequence of “super-adiabatic bases”. He
shows that the decrease in the amplitude of these oscillations as we increase t tot is
a sign that we are moving further into the adiabatic regime. The frequencies of the
oscillations in Fig. 10.7b and c appear to be determined by the difference in Floquet
eigenphases of the two Floquet states involved in the sharp avoided crossing. For
example, at t f ix = t f the period of the oscillation is T osc ≈ 400. The difference in
the Floquet eigenphases is || = | 1 − 4 | ≈ 0.016. Thus, T osc = 2π/|| =
393. Similarly, at t f ix = (τ I − t f )/2, T osc ≈ 600. The difference in the Floquet
eigenphases is || = | 1 − 4 | ≈ 0.011. Thus, T osc = 2π/|| = 571. The
observed oscillation periods are the same for both Fig. 10.23b and c.
As a result of the application of the two carefully chosen laser pulses, the system
has been excited from the ground state energy level to the third excited state with
a probability P ∼ 0.9, when the pulses have a duration t tot = 120. This is the
expected, and desired, result of STIRAP. However, for much longer pulse durations,
T tot = 21000 and T tot = 270000, the system transitions from the ground state to
the fourth excited state, with probability close to one. This unexpected transition is
due to an avoided crossing likely induced by the underlying chaos. The case with
pulse duration T tot = 270000 is a truly adiabatic process and the system transitions
to the fourth excited state with a probability close to one. Thus, the carefully applied
laser pulses have essentially allowed to us to control the transitions in the quantum
system with close to 100% accuracy.
10.10 Conclusions
Examination of the dynamical behavior of periodically driven quantum systems
has shown us that much of the nonlinear behavior found in classical mechanics
carries over to the quantum domain. Although the quantum systems we have studied
in this chapter do not become chaotic, they do exhibit a change in behavior as
nonlinear resonances overlap and destroy quantum numbers locally. The wave
function representing the state of the system becomes extended throughout the
region of nonlinear resonance overlap. However, we have the additional possibility
of dynamic Anderson localization, which can limit the extension of the wave
function. It is interesting that higher-order resonances also exist in quantum systems
and have been observed in microwave-driven hydrogen. This gives experimental
backing to the observation that nonlinear resonances form a self-similar structure in
quantum dynamical systems (at least down to sizes of order ¯
h) as they do in classical
systems.
393
avoided crossing in level |E 4 . As the laser pulses are turned on and off the system
transitions from the initial state |ψ(0) = |E 1 to the final state |ψ(+∞) = |E 4 .
Note that both Fig. 10.23b and c follow almost exactly the behavior of the Floquet
state A shown in Fig. 10.5c. This is an indication that we are in the adiabatic regime
in Fig. 10.23b and c.
The very large oscillations in the probability in Figs. 10.23b and c have been
explained by Berry (1990) in terms of a sequence of “super-adiabatic bases”. He
shows that the decrease in the amplitude of these oscillations as we increase t tot is
a sign that we are moving further into the adiabatic regime. The frequencies of the
oscillations in Fig. 10.7b and c appear to be determined by the difference in Floquet
eigenphases of the two Floquet states involved in the sharp avoided crossing. For
example, at t f ix = t f the period of the oscillation is T osc ≈ 400. The difference in
the Floquet eigenphases is || = | 1 − 4 | ≈ 0.016. Thus, T osc = 2π/|| =
393. Similarly, at t f ix = (τ I − t f )/2, T osc ≈ 600. The difference in the Floquet
eigenphases is || = | 1 − 4 | ≈ 0.011. Thus, T osc = 2π/|| = 571. The
observed oscillation periods are the same for both Fig. 10.23b and c.
As a result of the application of the two carefully chosen laser pulses, the system
has been excited from the ground state energy level to the third excited state with
a probability P ∼ 0.9, when the pulses have a duration t tot = 120. This is the
expected, and desired, result of STIRAP. However, for much longer pulse durations,
T tot = 21000 and T tot = 270000, the system transitions from the ground state to
the fourth excited state, with probability close to one. This unexpected transition is
due to an avoided crossing likely induced by the underlying chaos. The case with
pulse duration T tot = 270000 is a truly adiabatic process and the system transitions
to the fourth excited state with a probability close to one. Thus, the carefully applied
laser pulses have essentially allowed to us to control the transitions in the quantum
system with close to 100% accuracy.
10.10 Conclusions
Examination of the dynamical behavior of periodically driven quantum systems
has shown us that much of the nonlinear behavior found in classical mechanics
carries over to the quantum domain. Although the quantum systems we have studied
in this chapter do not become chaotic, they do exhibit a change in behavior as
nonlinear resonances overlap and destroy quantum numbers locally. The wave
function representing the state of the system becomes extended throughout the
region of nonlinear resonance overlap. However, we have the additional possibility
of dynamic Anderson localization, which can limit the extension of the wave
function. It is interesting that higher-order resonances also exist in quantum systems
and have been observed in microwave-driven hydrogen. This gives experimental
backing to the observation that nonlinear resonances form a self-similar structure in
quantum dynamical systems (at least down to sizes of order ¯
h) as they do in classical
systems.