10.6 Quantum Delta-Kicked Rotor
363
The experimental data (Fig. 10.9b) were able to resolve the dominant frequencies,
f exp < 3 kHz, in the interval α ≈ 8.7 and α ≈ 10.3. The theoretical analysis
(Fig. 10.10b) reproduces those experimental results. In the amplitude range α ≈ 7.6
to α ≈ 11.6, two frequencies dominate and give rise to the beats seen in Figs. 10.9a
and 10.10a. A fundamental change occurs for α > 14, where a different set of
Floquet states begins to dominate the dynamics.
Approximately eleven Floquet states (the number of momentum states) have
probability distributions that lie in the region of phase space between n = −5 and
n = 5.Only three Floquet states dominate the dynamics. The eigenphase differences
of these three Floquet states, (( b − a )/2π = 2.8 kHz and (( a − c )/2π =
2.4 kHz, correspond to the two dominant oscillation frequencies observed in the
experiment at α = 9.7. The “dynamical tunneling” between positive and negative
momentum states observed in the experiment appears to be due to interference
among these three Floquet states. The regions of phase space where the dominant
Floquet states have their support is determined by the distribution of chaotic regions
of the underlying classical phase space.
A similar dynamical tunneling experiment was performed by Hensinger et al.
(2001) using a dilute Bose-Einstein condensate of sodium atoms. The effective
Hamiltonian governing the dynamics of the sodium atoms was slightly different
from the experiment described here but also exhibited a localized chaotic region
in the phase space. The sodium atoms underwent a similar type of dynamical
tunneling. A Floquet analysis of that experiment is also given in Luter and Reichl
(2002).
10.6 Quantum Delta-Kicked Rotor
The delta-kicked rotor has been one of the most intensely studied quantum systems
because its Floquet matrix can be obtained analytically and because the classical
version has played such an important role in conservative chaos theory. Many of the
concepts used in quantum chaos theory were developed first for the quantum deltakicked rotor. However, caution also must be used in generalizing properties of the
driven rotor to other systems because some of its features are nongeneric.
10.6.1 The Schrödinger Equation for the Delta-Kicked Rotor
As we have shown in Chap. 3, the classical delta-kicked rotor has a Hamiltonian of
the form
H =
J 2
2I
+ K cos(θ )δ T (t),
(10.42)
363
The experimental data (Fig. 10.9b) were able to resolve the dominant frequencies,
f exp < 3 kHz, in the interval α ≈ 8.7 and α ≈ 10.3. The theoretical analysis
(Fig. 10.10b) reproduces those experimental results. In the amplitude range α ≈ 7.6
to α ≈ 11.6, two frequencies dominate and give rise to the beats seen in Figs. 10.9a
and 10.10a. A fundamental change occurs for α > 14, where a different set of
Floquet states begins to dominate the dynamics.
Approximately eleven Floquet states (the number of momentum states) have
probability distributions that lie in the region of phase space between n = −5 and
n = 5.Only three Floquet states dominate the dynamics. The eigenphase differences
of these three Floquet states, (( b − a )/2π = 2.8 kHz and (( a − c )/2π =
2.4 kHz, correspond to the two dominant oscillation frequencies observed in the
experiment at α = 9.7. The “dynamical tunneling” between positive and negative
momentum states observed in the experiment appears to be due to interference
among these three Floquet states. The regions of phase space where the dominant
Floquet states have their support is determined by the distribution of chaotic regions
of the underlying classical phase space.
A similar dynamical tunneling experiment was performed by Hensinger et al.
(2001) using a dilute Bose-Einstein condensate of sodium atoms. The effective
Hamiltonian governing the dynamics of the sodium atoms was slightly different
from the experiment described here but also exhibited a localized chaotic region
in the phase space. The sodium atoms underwent a similar type of dynamical
tunneling. A Floquet analysis of that experiment is also given in Luter and Reichl
(2002).
10.6 Quantum Delta-Kicked Rotor
The delta-kicked rotor has been one of the most intensely studied quantum systems
because its Floquet matrix can be obtained analytically and because the classical
version has played such an important role in conservative chaos theory. Many of the
concepts used in quantum chaos theory were developed first for the quantum deltakicked rotor. However, caution also must be used in generalizing properties of the
driven rotor to other systems because some of its features are nongeneric.
10.6.1 The Schrödinger Equation for the Delta-Kicked Rotor
As we have shown in Chap. 3, the classical delta-kicked rotor has a Hamiltonian of
the form
H =
J 2
2I
+ K cos(θ )δ T (t),
(10.42)