History corner
Milestones concerning quantum Floquet theory
Quantum systems under the influence of an external time-periodic force,
such as atoms interacting with a classical laser field, admit a particular
set of solutions to the time-dependent Schr�dinger equation known as
Floquet states. These states constitute an analog to the Bloch waves
of solid state physics: Exactly as the discrete
spatial translational
symmetry of a crystalline lattice leads to Bloch states characterized by some
quasimomentum, discrete
temporal translational symmetry gives rise to
Floquet states characterized by a
quasienergy. Floquet states are of
great value for a non-perturbative understanding of the processes induced by
forcing. Here are some "historical" examples:
Of particular importance is the fact that Floquet states
respond adiabatically to slowly changing parameters, such as the envelope of
a laser pulse. Thus, under appropriate conditions the wave function simply
evolves adiabatically on its quasienergy surface when, e.g., the driving
amplitude rises during a laser pulse, with multiphoton-like nonadiabatic
transitions at avoided quasienergy crossings. The theoretical description of
this adiabatic-diabatic scenario rests on a Schr�dinger-like
evolution
equation in an extended Hilbert space which makes explicit use of two
different time variables: One variable for the "slow" time-dependence
of the parameters, another variable for the "fast" periodic oscillation.
This equation, formulated in the paper linked above, allows one to first apply
standard adiabatic techniques in the extended Hilbert space, and then to obtain
the true wave function by projecting back to the system's actual Hilbert space.
It has been applied, for instance, for describing the entrance of highly excited
atoms into a microwave cavity, for designing generalized pi-pulses, and for
investigating adiabatic following of forced Bose-Einstein condensates.
Floquet states, although explicitly time-dependent, are
in many ways analogous to the usual energy eigenstates. This fact is underlined
by the observation that they are tied, in the semiclassical limit, to invariant
manifolds in an extended classical phase space spanned by position, momentum,
and time in the same manner as energy eigenstates are semiclassically linked to
invariant manifolds in the usual, even-dimensional phase space. In particular,
there exist
semiclassical quantization rules for Floquet states which
constitute a direct generalization of the well-known Bohr-Sommerfeld conditions.
The above link leads to a pictorial which contains some illustrative examples,
and gives further references.
An interesting situation occurs when a spatially periodic
system undergoes time-periodic forcing, as happens with far-infrared irradiated
semiconductor superlattices, or ultracold atoms in modulated optical lattices.
Then the energy bands of the unforced system turn into
quasienergy bands,
the width of which depends on the parameters of the force. Since the extension of
the Floquet states in lattices with disorder depends on the ratio of disorder
strength and quasienergy band width, this fact allows one to control their size,
and hence several phenomena connected to quantum localization, by adjusting the
amplitude or the frequency of the external drive. Some aspects of that effect have
been explored in the references compiled in the above review.
Disclaimer
Druckversion
Martin Holthaus � Last modified: October 07 2006