10.9 Quantum Control
389
We can determine the behavior of Floquet eigenstates at fixed times t = t f ix
during which the pulses drive the system. The distribution of probability in the
Floquet eigenstates is sensitive to structures in the classical phase space which are
larger than Planck’s constant. The first pulse has carrier frequency ω f = (E 3 −
E 2 ) = 5π 2 /4 and the second pulse has carrier frequency ω s = (E 2 − E 1 ) = 3π 2 /4.
These frequencies are commensurate since ω f /ω s = 5/3. From Eq. (10.114), the
period of the Hamiltonian is T o = 8/π and the Floquet frequency is ω o = 2π/T o =
π 2 /4. Thus, ω f = 5ω o and ω s = 3ω o . We set U o = 3.0.
The classical dynamics of this system tells us that we only need to keep five
unperturbed energy eigenstates as a basis to form the Floquet matrix. This can be
seen from Fig. 10.20 where we show the underlying classical phase space at selected
values of t f ix during the time that the pulses are on. For J > 5, the classical
phase space is dominated by KAM tori with almost constant values of J and the
unperturbed energy states are very weakly coupled by the dynamics for n > 5.
Thus, it is sufficient to construct a 5 × 5 Floquet matrix with the five basis states
|E 1 , . . . , |E 5 . We find that only four of the five Floquet eigenstates of are actively
involved in the dynamics.
To keep track of the changes that occur in the Floquet eigenstates, we will give
each eigenstate a unique alphabetical label determined by its dominant dependence
on unperturbed energy states at time t f ix = 0. We find that at t f ix = 0 the Floquet
eigenstates have the following structure and we give them, and the corresponding
eigenphases, the following labels:
A = |φ 1 =|E 1 , D = |φ 4 =|E 4 and E = |φ 5 =|E 5 ,
BC
+
=|φ 2 =
1
√
2
(|E 2 +|E 3 ), BC
−
=|φ 3 =
1
√
2
(|E 2 −|E 3 ). (10.118)
The four relevant eigenphases, α , obtained from the 5 × 5 Floquet matrix are
plotted modulo ω o = π 2 /4 in Fig. 10.21a. Two of these Floquet eigenphases, A
and D, are almost degenerate over the time interval that the pulses act and are not
distinguishable on the scale shown in Fig. 10.21a. For t f ix = 0, the four Floquet
eigenphases are approximately degenerate modulo ω o . In Fig. 10.21b, we focus on
eigenphases A and D. They appear to undergo an avoided crossing at time t f ix =
τ avoid ≈ 10/23t tot just before t f ix = t c , and a crossing for t f ix > t c .
We can follow each Floquet eigenstate during the entire process by computing
the eigenstates for a sequence of values of t f ix over the interval 0 ≤ t f ix ≤ t tot .
For closely spaced values of t f ix , Floquet eigenstates at different times belonging
to different eigenphases will be orthogonal. This provides a means of following the
evolution of each eigenphase and eigenstate as a function of t f ix . As we will see,
the Floquet eigenstates can change structure when avoided crossings occur between
Floquet eigenphases.
389
We can determine the behavior of Floquet eigenstates at fixed times t = t f ix
during which the pulses drive the system. The distribution of probability in the
Floquet eigenstates is sensitive to structures in the classical phase space which are
larger than Planck’s constant. The first pulse has carrier frequency ω f = (E 3 −
E 2 ) = 5π 2 /4 and the second pulse has carrier frequency ω s = (E 2 − E 1 ) = 3π 2 /4.
These frequencies are commensurate since ω f /ω s = 5/3. From Eq. (10.114), the
period of the Hamiltonian is T o = 8/π and the Floquet frequency is ω o = 2π/T o =
π 2 /4. Thus, ω f = 5ω o and ω s = 3ω o . We set U o = 3.0.
The classical dynamics of this system tells us that we only need to keep five
unperturbed energy eigenstates as a basis to form the Floquet matrix. This can be
seen from Fig. 10.20 where we show the underlying classical phase space at selected
values of t f ix during the time that the pulses are on. For J > 5, the classical
phase space is dominated by KAM tori with almost constant values of J and the
unperturbed energy states are very weakly coupled by the dynamics for n > 5.
Thus, it is sufficient to construct a 5 × 5 Floquet matrix with the five basis states
|E 1 , . . . , |E 5 . We find that only four of the five Floquet eigenstates of are actively
involved in the dynamics.
To keep track of the changes that occur in the Floquet eigenstates, we will give
each eigenstate a unique alphabetical label determined by its dominant dependence
on unperturbed energy states at time t f ix = 0. We find that at t f ix = 0 the Floquet
eigenstates have the following structure and we give them, and the corresponding
eigenphases, the following labels:
A = |φ 1 =|E 1 , D = |φ 4 =|E 4 and E = |φ 5 =|E 5 ,
BC
+
=|φ 2 =
1
√
2
(|E 2 +|E 3 ), BC
−
=|φ 3 =
1
√
2
(|E 2 −|E 3 ). (10.118)
The four relevant eigenphases, α , obtained from the 5 × 5 Floquet matrix are
plotted modulo ω o = π 2 /4 in Fig. 10.21a. Two of these Floquet eigenphases, A
and D, are almost degenerate over the time interval that the pulses act and are not
distinguishable on the scale shown in Fig. 10.21a. For t f ix = 0, the four Floquet
eigenphases are approximately degenerate modulo ω o . In Fig. 10.21b, we focus on
eigenphases A and D. They appear to undergo an avoided crossing at time t f ix =
τ avoid ≈ 10/23t tot just before t f ix = t c , and a crossing for t f ix > t c .
We can follow each Floquet eigenstate during the entire process by computing
the eigenstates for a sequence of values of t f ix over the interval 0 ≤ t f ix ≤ t tot .
For closely spaced values of t f ix , Floquet eigenstates at different times belonging
to different eigenphases will be orthogonal. This provides a means of following the
evolution of each eigenphase and eigenstate as a function of t f ix . As we will see,
the Floquet eigenstates can change structure when avoided crossings occur between
Floquet eigenphases.