390
10 Time-Periodic Quantum Systems
Fig. 10.21 Floquet
eigenphases, for the system
with maximum pulse strength
U o = 3.0 and frequencies
ω f = 5ω o and ω s = 3ω o , are
plotted over the entire interval
0≤t f ix ≤t tot . (a) Floquet
eigenphases for four Floquet
states A, BC + , BC − and D
plotted mod ω o = π 2 /4. The
wide avoided crossing at
t f ix = t c is clear. (b) Floquet
eigenphase curves for the
Floquet states A and D. The
sharp avoided crossing at
t f ix = τ I and the crossing at
t f ix = τ I I are clearly seen
(Na and Reichl 2004)
10.9.5 Avoided Crossings
The Floquet eigenphases for states A and D in Fig. 10.21b avoid crossing at time
t f ix = τ I . Before time t f ix = τ I Floquet state A is predominately composed of
level n = 1 and Floquet state D is predominately composed of level n = 4. After
the avoided crossing at time t f ix = τ I the states have changed their character and
Floquet state A is composed predominately of level n = 4 and Floquet state D
is composed predominately of level n = 1. Because of the avoided crossing at
t f ix = τ I , the entire population gets shifted from level n = 1 to level n = 4 before
the traditional STIRAP ladder process can take place. These transitions are clearly
seen in Fig. 10.22, where we show the level compositions of the four participating
Floquet states, A, BC + , BC − and D as a function of t f ix .
Now let us determine how behavior of the Floquet states affects the exact
behavior of the system, by solving the Schrödinger equation in Eq. (10.111). We
will assume that at time t = 0 the system is in state |ψ(0) = |E 1 . As we will
see, the actual dynamics of this system is determined by the length of time during
which the pulses are allowed to act. The pulse duration time necessary to achieve
adiabatic behavior of the system is determined largely by the avoided crossings in
the Floquet eigenphases.
Avoided crossings of Floquet eigenphases occur as the classical phase space
becomes chaotic, and a symmetry has been broken in that local region of the phase
10 Time-Periodic Quantum Systems
Fig. 10.21 Floquet
eigenphases, for the system
with maximum pulse strength
U o = 3.0 and frequencies
ω f = 5ω o and ω s = 3ω o , are
plotted over the entire interval
0≤t f ix ≤t tot . (a) Floquet
eigenphases for four Floquet
states A, BC + , BC − and D
plotted mod ω o = π 2 /4. The
wide avoided crossing at
t f ix = t c is clear. (b) Floquet
eigenphase curves for the
Floquet states A and D. The
sharp avoided crossing at
t f ix = τ I and the crossing at
t f ix = τ I I are clearly seen
(Na and Reichl 2004)
10.9.5 Avoided Crossings
The Floquet eigenphases for states A and D in Fig. 10.21b avoid crossing at time
t f ix = τ I . Before time t f ix = τ I Floquet state A is predominately composed of
level n = 1 and Floquet state D is predominately composed of level n = 4. After
the avoided crossing at time t f ix = τ I the states have changed their character and
Floquet state A is composed predominately of level n = 4 and Floquet state D
is composed predominately of level n = 1. Because of the avoided crossing at
t f ix = τ I , the entire population gets shifted from level n = 1 to level n = 4 before
the traditional STIRAP ladder process can take place. These transitions are clearly
seen in Fig. 10.22, where we show the level compositions of the four participating
Floquet states, A, BC + , BC − and D as a function of t f ix .
Now let us determine how behavior of the Floquet states affects the exact
behavior of the system, by solving the Schrödinger equation in Eq. (10.111). We
will assume that at time t = 0 the system is in state |ψ(0) = |E 1 . As we will
see, the actual dynamics of this system is determined by the length of time during
which the pulses are allowed to act. The pulse duration time necessary to achieve
adiabatic behavior of the system is determined largely by the avoided crossings in
the Floquet eigenphases.
Avoided crossings of Floquet eigenphases occur as the classical phase space
becomes chaotic, and a symmetry has been broken in that local region of the phase