350
10 Time-Periodic Quantum Systems
time t = 50T for initial condition m = 130. On the scale shown in the figures, the
resonance zones remain separated in the Hilbert space.
The condition for resonance overlap for this system is ¯
n n 0 − ¯
n m 0 = 2
√
U 0 +
2
U 0
9 . This predicts that the resonances will overlap for U 0 ≥ 1406. In Fig. 10.3c,
we show the solution at time t = 18.75T for the double-resonance Schrödinger
equation for m 0 = 1, n 0 = 3, ω = 300, U 0 = 1478, V 0 = U 0 /9 = 164, and initial
state m = 130. We see that overlap has occurred between the quantum resonance
zones. In fact, overlap was actually observed to occur at U 0 ≈ 1100, at least on the
scale shown. In Fig. 10.3d, for the same parameters as in Fig. 10.3c, we show the
spread of probability at time τ = 26.5T 0 for an initial state n 0 = 700 well away
from the double-resonance region. There is very little spread in the probability.
10.3.4 Floquet Eigenvalue Nearest Neighbor Spacing
The distribution of spacings between Floquet eigenvalues in nonlinear resonance
regions as the resonances overlap has been studied in Lin and Reichl (1987, 1988,
1989); Berman and Kolovsky (1987). They find a change in the spectral statistics
from Poisson-like to Wigner-like behavior, indicating that a quantum number is
destroyed locally during the process of quantum resonance overlap.
The Floquet eigenvalue nearest neighbor spacing distribution can be found in
several ways. We can construct the Floquet matrix (see Sect. 10.2.1) numerically
and find its eigenvalues, or we can directly take the Fourier transform of the time
series ψ(θ, t). From Eq. (10.8), the peaks in the Fourier transform occur at the
Floquet eigenvalues. In Lin and Reichl (1987, 1989), the Fourier transform of the
function ψ(θ = 0, t) =
∞
n=0 ψ n (t), for V 0 = U 0 /9, was computed, and the nearest
neighbor spacing distribution for the peaks in the Fourier spectrum was obtained.
The solution to the Schrödinger equation was evolved for 1240 periods, T =
2π
ω ,
and then the Fourier transform of the time series for the next 1240 periods was taken.
In Fig. 10.4a, we show the nearest neighbor spacing histogram and the best fit Brody
distribution for U 0 = 657. In Fig. 10.4b, we show them for U 0 = 2465. The hatch
marks in these figures indicate spacings that could not be resolved in the Fourier
transform. A truncated Brody distribution was used to fit the histograms. There
is a clear transition from a Poisson-like to a Wigner-like distribution. The singleresonance case was also studied for U 0 = 2465 and V 0 = 0. The nearest neighbor
spacing was clearly Poisson-like, indicating that the single-resonance system is
integrable.
The nearest neighbor spacing distribution has also been studied in Berman and
Kolovsky (1987) and Lin and Reichl (1989) using the Floquet map. Similar results
are found.
10 Time-Periodic Quantum Systems
time t = 50T for initial condition m = 130. On the scale shown in the figures, the
resonance zones remain separated in the Hilbert space.
The condition for resonance overlap for this system is ¯
n n 0 − ¯
n m 0 = 2
√
U 0 +
2
U 0
9 . This predicts that the resonances will overlap for U 0 ≥ 1406. In Fig. 10.3c,
we show the solution at time t = 18.75T for the double-resonance Schrödinger
equation for m 0 = 1, n 0 = 3, ω = 300, U 0 = 1478, V 0 = U 0 /9 = 164, and initial
state m = 130. We see that overlap has occurred between the quantum resonance
zones. In fact, overlap was actually observed to occur at U 0 ≈ 1100, at least on the
scale shown. In Fig. 10.3d, for the same parameters as in Fig. 10.3c, we show the
spread of probability at time τ = 26.5T 0 for an initial state n 0 = 700 well away
from the double-resonance region. There is very little spread in the probability.
10.3.4 Floquet Eigenvalue Nearest Neighbor Spacing
The distribution of spacings between Floquet eigenvalues in nonlinear resonance
regions as the resonances overlap has been studied in Lin and Reichl (1987, 1988,
1989); Berman and Kolovsky (1987). They find a change in the spectral statistics
from Poisson-like to Wigner-like behavior, indicating that a quantum number is
destroyed locally during the process of quantum resonance overlap.
The Floquet eigenvalue nearest neighbor spacing distribution can be found in
several ways. We can construct the Floquet matrix (see Sect. 10.2.1) numerically
and find its eigenvalues, or we can directly take the Fourier transform of the time
series ψ(θ, t). From Eq. (10.8), the peaks in the Fourier transform occur at the
Floquet eigenvalues. In Lin and Reichl (1987, 1989), the Fourier transform of the
function ψ(θ = 0, t) =
∞
n=0 ψ n (t), for V 0 = U 0 /9, was computed, and the nearest
neighbor spacing distribution for the peaks in the Fourier spectrum was obtained.
The solution to the Schrödinger equation was evolved for 1240 periods, T =
2π
ω ,
and then the Fourier transform of the time series for the next 1240 periods was taken.
In Fig. 10.4a, we show the nearest neighbor spacing histogram and the best fit Brody
distribution for U 0 = 657. In Fig. 10.4b, we show them for U 0 = 2465. The hatch
marks in these figures indicate spacings that could not be resolved in the Fourier
transform. A truncated Brody distribution was used to fit the histograms. There
is a clear transition from a Poisson-like to a Wigner-like distribution. The singleresonance case was also studied for U 0 = 2465 and V 0 = 0. The nearest neighbor
spacing was clearly Poisson-like, indicating that the single-resonance system is
integrable.
The nearest neighbor spacing distribution has also been studied in Berman and
Kolovsky (1987) and Lin and Reichl (1989) using the Floquet map. Similar results
are found.