10.2 Floquet Theory
341
In Chap. 5, we showed that classical nonintegrable systems, with three or more
degrees of freedom, have a dense set of resonances (the Arnol’d web) embedded in
their phase space. For conservative systems, trajectories are confined to the energy
surface, but can diffuse throughout the energy surface along the resonance lines
(or across them for large enough deviation of integrability). This diffusion is called
Arnol’d diffusion and is a universal property of nonintegrable systems with three or
more degrees of freedom. If the system is driven by a time-periodic field then the
system can show energy instability. In Sect. 10.8, we show that Arnol’d diffusion is
also a universal feature of quantum systems with three or more degrees of freedom,
with similar consequences.
In Sect. 10.9, we show that it is possible to use Floquet theory to analyze
a widely used method of quantum control of atomic and molecular transitions,
called STIRAP (stimulated Raman adiabatic passage), and we find interesting
consequences due to the chaos induced in the underlying phase space of the driven
system.
Finally, in Sect. 10.10, we make some concluding remarks.
10.2 Floquet Theory
Systems governed by time-periodic Hamiltonians do not have continuous time
translational invariance, so energy is not a conserved quantity. However, they do
have discrete time translational invariance so there is a conserved quasienergy.The
dynamics of such systems is governed by a Hermitian operator, the Floquet Hamiltonian, whose eigenvalues, the quasienergies, are conserved quantities (Shirley
1965; Zeldovich 1967; Sambe 1973). One can perform a spectral decomposition
of solutions of the Schrödinger equation in terms of eigenvalues and eigenfunctions
of the Floquet Hamiltonian.
There are two methods for treating the dynamics of such systems. One method
analyses the dynamics in terms of a unitary Floquet matrix. The other method
reformulates the system in terms of a time-independent Floquet Hamiltonian. We
discuss both methods below.
10.2.1 Floquet Matrix
Let us consider the Schrödinger equation
i ¯
h
∂
∂t
|(t) = ˆ
H (t)|
(10.1)
where |(t) is the state of the system at time t, ¯
h is Planck’s constant, and ˆ
H (t) is
a time-periodic Hamiltonian of the form
341
In Chap. 5, we showed that classical nonintegrable systems, with three or more
degrees of freedom, have a dense set of resonances (the Arnol’d web) embedded in
their phase space. For conservative systems, trajectories are confined to the energy
surface, but can diffuse throughout the energy surface along the resonance lines
(or across them for large enough deviation of integrability). This diffusion is called
Arnol’d diffusion and is a universal property of nonintegrable systems with three or
more degrees of freedom. If the system is driven by a time-periodic field then the
system can show energy instability. In Sect. 10.8, we show that Arnol’d diffusion is
also a universal feature of quantum systems with three or more degrees of freedom,
with similar consequences.
In Sect. 10.9, we show that it is possible to use Floquet theory to analyze
a widely used method of quantum control of atomic and molecular transitions,
called STIRAP (stimulated Raman adiabatic passage), and we find interesting
consequences due to the chaos induced in the underlying phase space of the driven
system.
Finally, in Sect. 10.10, we make some concluding remarks.
10.2 Floquet Theory
Systems governed by time-periodic Hamiltonians do not have continuous time
translational invariance, so energy is not a conserved quantity. However, they do
have discrete time translational invariance so there is a conserved quasienergy.The
dynamics of such systems is governed by a Hermitian operator, the Floquet Hamiltonian, whose eigenvalues, the quasienergies, are conserved quantities (Shirley
1965; Zeldovich 1967; Sambe 1973). One can perform a spectral decomposition
of solutions of the Schrödinger equation in terms of eigenvalues and eigenfunctions
of the Floquet Hamiltonian.
There are two methods for treating the dynamics of such systems. One method
analyses the dynamics in terms of a unitary Floquet matrix. The other method
reformulates the system in terms of a time-independent Floquet Hamiltonian. We
discuss both methods below.
10.2.1 Floquet Matrix
Let us consider the Schrödinger equation
i ¯
h
∂
∂t
|(t) = ˆ
H (t)|
(10.1)
where |(t) is the state of the system at time t, ¯
h is Planck’s constant, and ˆ
H (t) is
a time-periodic Hamiltonian of the form