10.8 Arnol’d Diffusion in Quantum Systems
377
Fig. 10.17 A strobe plot showing some fractional and primary resonances for classical onedimensional hydrogen with Hamiltonian H =
1
2 p 2 −
1
z + λz cos(ω 0 t). The field strength is
λ = 1.9 × 10 −10 a.u. and the frequency is ω 0 = 1.5 × 10 −6 . The fractions in the figure indicate
values of m = n 3 ω 0 (Burns and Reichl 1992)
thermalization of quantum systems. Until now there has not been much work
analyzing the mechanisms leading to Arnol’d diffusion in quantum systems. One
of the earliest works (von Milczewski et al. 1996) involved the computation of the
Arnol’d web for a hydrogen atom in crossed electric and magnetic fields. They
related the Arnol’d web in the underlying classical system to the diffusion of the
electron wavefunction in the quantum system. Arnol’d diffusion has also been
reported to increase the conductance of open billiards with 3 DoF (Nakamura and
Harayama 2004). More recently, Malyshev and Chizhova (2010) have studied the
classical and quantum dynamics of two weakly coupled oscillators placed in a timeperiodic external field. For the system they considered, which has 2.5 DoF, they
found diffusion in the quantum system that parallels that in the classical system, but
a diffusion rate that was an order of magnitude slower in some parameter regimes.
In classical systems, primary resonances and “daughter” resonances that occur
due to interaction between primary resonances, form a self-similar network in the
phase space, and the diffusion process can occur at all length scales. Quantum
systems are primarily influenced by classical structures that occupy a volume in
the phase space of order Planck’s constant or larger. In resonance regions that are
larger than Planck’s constant, quantum states can spread throughout resonances
and chaotic regions (Reichl 1989; Morrow and Reichl 1994). Therefore, it is not
surprising that the analog of Arnol’d diffusion exists in quantum systems, although
perhaps on a more subdued scale. The effect of Arnol’d diffusion in a quantum
system can be seen clearly in the driven optical lattice, as we shall now show.
377
Fig. 10.17 A strobe plot showing some fractional and primary resonances for classical onedimensional hydrogen with Hamiltonian H =
1
2 p 2 −
1
z + λz cos(ω 0 t). The field strength is
λ = 1.9 × 10 −10 a.u. and the frequency is ω 0 = 1.5 × 10 −6 . The fractions in the figure indicate
values of m = n 3 ω 0 (Burns and Reichl 1992)
thermalization of quantum systems. Until now there has not been much work
analyzing the mechanisms leading to Arnol’d diffusion in quantum systems. One
of the earliest works (von Milczewski et al. 1996) involved the computation of the
Arnol’d web for a hydrogen atom in crossed electric and magnetic fields. They
related the Arnol’d web in the underlying classical system to the diffusion of the
electron wavefunction in the quantum system. Arnol’d diffusion has also been
reported to increase the conductance of open billiards with 3 DoF (Nakamura and
Harayama 2004). More recently, Malyshev and Chizhova (2010) have studied the
classical and quantum dynamics of two weakly coupled oscillators placed in a timeperiodic external field. For the system they considered, which has 2.5 DoF, they
found diffusion in the quantum system that parallels that in the classical system, but
a diffusion rate that was an order of magnitude slower in some parameter regimes.
In classical systems, primary resonances and “daughter” resonances that occur
due to interaction between primary resonances, form a self-similar network in the
phase space, and the diffusion process can occur at all length scales. Quantum
systems are primarily influenced by classical structures that occupy a volume in
the phase space of order Planck’s constant or larger. In resonance regions that are
larger than Planck’s constant, quantum states can spread throughout resonances
and chaotic regions (Reichl 1989; Morrow and Reichl 1994). Therefore, it is not
surprising that the analog of Arnol’d diffusion exists in quantum systems, although
perhaps on a more subdued scale. The effect of Arnol’d diffusion in a quantum
system can be seen clearly in the driven optical lattice, as we shall now show.