10.3 Quantum Nonlinear Resonances
345
10.3 Quantum Nonlinear Resonances
In classical systems, nonlinear resonance overlap provides the fundamental mechanism by which KAM tori are destroyed and chaos emerges. Nonlinear resonances
also exist in quantum dynamical systems (Berman and Zaslavsky 1977; Berman
et al. 1982; Berman and Kolovsky 1983a,b, 1987; Reichl and Lin 1986).
The behavior of nonlinear resonances, in quantum systems, is particularly easy
to study with the help of Floquet theory. Floquet eigenstates become localized on
nonlinear resonance structures as long as the resonance structures occupy phase
space areas in the underlying classical phase space that are of order Planck’s
constant or larger. Also the statistical distribution of spacings between Floquet
eigenvalues changes from Poisson-like to Wigner-like behavior, indicating that a
quantum number is destroyed locally during the process of quantum resonance
overlap. We will show this in some detail in the subsections below.
10.3.1 Two Primary Resonance Model
One of the simplest models with which to study nonlinear resonance overlap
classically is that of a one-dimensional rotor driven by two traveling cosine waves.
The classical Hamiltonian is
H =
J 2
2I
+ U cos(m 0 θ − ˜
ω˜ t) + V cos(n 0 θ − ˜
ω˜ t),
(10.20)
where J is the angular momentum, I is the moment of inertia, U and V are
amplitudes of the cosine traveling waves, ˜
ω is the radial frequency, ˜
t is the time,
and m 0 and n 0 are integers.
It is useful to write the Hamiltonian Eq. (10.20) in terms of dimensionless
variables. If we let J = ˜
n ¯
h, U = ¯
h 2
2I U 0 , H = ¯
h 2
2I H, ˜
ω = ¯
h
2I ω, and ˜
t =
2I
¯
h t,
then the dimensionless Hamiltonian takes the form
H = ˜
n
2
+ U 0 cos(m 0 θ − ωt) + V 0 cos(n 0 θ − ωt).
(10.21)
In Eq. (10.21), the dimensionless classical action variable, ˜
n, can take on a continuum of values. The speeds of the cosine waves with amplitudes U 0 and V 0 are
˙
θ =
ω
m 0
and ˙
θ =
ω
n 0
, respectively. The system governed by Eq. (10.21) will have
primary nonlinear resonances at angular momenta n = ¯
n i (i = m 0 , n 0 ) that are
given by the resonance condition, ˙
θ =
∂H
∂ ˜
n . This yields ¯
n m 0 =
ω
2m 0
and ¯
n n 0 =
ω
2n 0
for the system in Eq. (10.20). The widths of the classical resonance zones are given
by ˜
n U = 2
√
2U 0 and ˜
n V = 2
√
2V 0 , respectively.
The Chirikov condition for overlap of the resonances in Eq. (10.21) is that the
spacing between the resonances be equal to the sum of the half-widths, or | ¯
n m 0 −
345
10.3 Quantum Nonlinear Resonances
In classical systems, nonlinear resonance overlap provides the fundamental mechanism by which KAM tori are destroyed and chaos emerges. Nonlinear resonances
also exist in quantum dynamical systems (Berman and Zaslavsky 1977; Berman
et al. 1982; Berman and Kolovsky 1983a,b, 1987; Reichl and Lin 1986).
The behavior of nonlinear resonances, in quantum systems, is particularly easy
to study with the help of Floquet theory. Floquet eigenstates become localized on
nonlinear resonance structures as long as the resonance structures occupy phase
space areas in the underlying classical phase space that are of order Planck’s
constant or larger. Also the statistical distribution of spacings between Floquet
eigenvalues changes from Poisson-like to Wigner-like behavior, indicating that a
quantum number is destroyed locally during the process of quantum resonance
overlap. We will show this in some detail in the subsections below.
10.3.1 Two Primary Resonance Model
One of the simplest models with which to study nonlinear resonance overlap
classically is that of a one-dimensional rotor driven by two traveling cosine waves.
The classical Hamiltonian is
H =
J 2
2I
+ U cos(m 0 θ − ˜
ω˜ t) + V cos(n 0 θ − ˜
ω˜ t),
(10.20)
where J is the angular momentum, I is the moment of inertia, U and V are
amplitudes of the cosine traveling waves, ˜
ω is the radial frequency, ˜
t is the time,
and m 0 and n 0 are integers.
It is useful to write the Hamiltonian Eq. (10.20) in terms of dimensionless
variables. If we let J = ˜
n ¯
h, U = ¯
h 2
2I U 0 , H = ¯
h 2
2I H, ˜
ω = ¯
h
2I ω, and ˜
t =
2I
¯
h t,
then the dimensionless Hamiltonian takes the form
H = ˜
n
2
+ U 0 cos(m 0 θ − ωt) + V 0 cos(n 0 θ − ωt).
(10.21)
In Eq. (10.21), the dimensionless classical action variable, ˜
n, can take on a continuum of values. The speeds of the cosine waves with amplitudes U 0 and V 0 are
˙
θ =
ω
m 0
and ˙
θ =
ω
n 0
, respectively. The system governed by Eq. (10.21) will have
primary nonlinear resonances at angular momenta n = ¯
n i (i = m 0 , n 0 ) that are
given by the resonance condition, ˙
θ =
∂H
∂ ˜
n . This yields ¯
n m 0 =
ω
2m 0
and ¯
n n 0 =
ω
2n 0
for the system in Eq. (10.20). The widths of the classical resonance zones are given
by ˜
n U = 2
√
2U 0 and ˜
n V = 2
√
2V 0 , respectively.
The Chirikov condition for overlap of the resonances in Eq. (10.21) is that the
spacing between the resonances be equal to the sum of the half-widths, or | ¯
n m 0 −