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10 Time-Periodic Quantum Systems
like eigenvalues of the Mathieu equation, which is the equation for the quantum
pendulum. The Floquet eigenstates associated with these pendulum-like Floquet
eigenvalues are localized in the nonlinear resonance zones, which have pendulumlike structure at these low values of .
10.4.2 Avoided Crossings and High Harmonic Radiation
It is also of interest to look at the spectrum of radiation emitted by a driven particle
in a square-well potential (Chism et al. 1998). We consider the case where the
amplitude of the time-periodic field varies in time and is turned on at time t = 0.
The Schrödinger equation for the system we now consider can be written
i
∂
∂t
x|ψ(t) =
−
∂ 2
∂x 2 + ξ(t)xcos(ω 0 t)
x|ψ(t),
(10.36)
where ξ(t) is a “turn-on" function and is defined as ξ(t) = sin
2 (ω 0 t/(4n c )) for
t < 2πn c /ω 0 and ξ(t) = 1 for t > 2πn c /ω 0 . Here n c is the number of external
field periods required for the driving field to be fully turned on.
The radiation is proportional to the Fourier transform of the average acceleration,
a(t) and is given by
a(t) = =ψ(t)| ¨
x|ψ(t) =
α
β
e
−i(( α − β )t
β (t)| ¨
x| α (t)
××ψ(0)| β α |ψ(0).
(10.37)
The Fourier transform of a(t) is ˜
a(ω). The power spectrum is given by χ(ω) =
|| ˜
a(ω)| 2 .
In Fig. 10.7, we show the radiation spectrum, as a function of radiated frequency,
for external field strength = 320 and three different initial conditions. The turnon takes n c = 12 cycles of the external driving field. The spectrum was computed
from a time series for the acceleration that ran for a time interval t = 128T 0 (128
external field cycles), starting after the turn-on was complete. At time t = 0, we
start the system in an energy eigenstate, |φ n , for each of the following three cases:
n = 35 in the region dominated by KAM tori; n = 16 in the region of the large
nonlinear resonance; and n = 3 in the chaotic region. We see that in the region
dominated by KAM tori (Fig. 10.7a), the radiation spectrum is typical of what might
be found using perturbation theory. There is a small amount of radiation at the lower
harmonics of the driving field frequency, ω 0 , but nothing at higher harmonics.
For cases where the initial condition lies in the chaotic region, in Figs. 10.7b
and c, there is significant high harmonic radiation. In both these cases, the frequency
at which the radiation cuts off is determined by the spread in energy of the chaotic
sea of the underlying classical phase space (Chism et al. 1998). The power spectrum
10 Time-Periodic Quantum Systems
like eigenvalues of the Mathieu equation, which is the equation for the quantum
pendulum. The Floquet eigenstates associated with these pendulum-like Floquet
eigenvalues are localized in the nonlinear resonance zones, which have pendulumlike structure at these low values of .
10.4.2 Avoided Crossings and High Harmonic Radiation
It is also of interest to look at the spectrum of radiation emitted by a driven particle
in a square-well potential (Chism et al. 1998). We consider the case where the
amplitude of the time-periodic field varies in time and is turned on at time t = 0.
The Schrödinger equation for the system we now consider can be written
i
∂
∂t
x|ψ(t) =
−
∂ 2
∂x 2 + ξ(t)xcos(ω 0 t)
x|ψ(t),
(10.36)
where ξ(t) is a “turn-on" function and is defined as ξ(t) = sin
2 (ω 0 t/(4n c )) for
t < 2πn c /ω 0 and ξ(t) = 1 for t > 2πn c /ω 0 . Here n c is the number of external
field periods required for the driving field to be fully turned on.
The radiation is proportional to the Fourier transform of the average acceleration,
a(t) and is given by
a(t) = =ψ(t)| ¨
x|ψ(t) =
α
β
e
−i(( α − β )t
β (t)| ¨
x| α (t)
××ψ(0)| β α |ψ(0).
(10.37)
The Fourier transform of a(t) is ˜
a(ω). The power spectrum is given by χ(ω) =
|| ˜
a(ω)| 2 .
In Fig. 10.7, we show the radiation spectrum, as a function of radiated frequency,
for external field strength = 320 and three different initial conditions. The turnon takes n c = 12 cycles of the external driving field. The spectrum was computed
from a time series for the acceleration that ran for a time interval t = 128T 0 (128
external field cycles), starting after the turn-on was complete. At time t = 0, we
start the system in an energy eigenstate, |φ n , for each of the following three cases:
n = 35 in the region dominated by KAM tori; n = 16 in the region of the large
nonlinear resonance; and n = 3 in the chaotic region. We see that in the region
dominated by KAM tori (Fig. 10.7a), the radiation spectrum is typical of what might
be found using perturbation theory. There is a small amount of radiation at the lower
harmonics of the driving field frequency, ω 0 , but nothing at higher harmonics.
For cases where the initial condition lies in the chaotic region, in Figs. 10.7b
and c, there is significant high harmonic radiation. In both these cases, the frequency
at which the radiation cuts off is determined by the spread in energy of the chaotic
sea of the underlying classical phase space (Chism et al. 1998). The power spectrum