Non-dispersive wave packets in periodically driven quantum systems

436 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547
“Floquet eigenstates” of the system—multiplied by oscillatory functions:
| (t)〉 =
cj exp −i Ejt
|Ej(t)〉 ; (73)
˝
with
j
|Ej(t + T )〉 = |Ej(t)〉 : (74)
The Ej are the “quasi-energies” of the system. Floquet states and quasi-energies are eigenstates and
eigenenergies of the Floquet Hamiltonian
H = H0 + V cos !t − i˝ 9
: (75)
9t
Note that, because of the time-periodicity with period T =2 =!, the quasi-energies are de ned
modulo ˝! [92].
The Floquet Hamiltonian (75) is nothing but the quantum analog of the classical Hamiltonian
(58) in extended phase space. Indeed, −i˝9=9t is the quantum version of the canonical momentum
Pt conjugate to time t. In strict analogy with the classical discussion of the previous section, it is
the Floquet Hamiltonian in extended phase space which will be the central object of our discussion.
It contains all the relevant information on the system, encoded in its eigenstates.
In a quantum optics or atomic physics context—with the external perturbation given by quantized
modes of the electromagnetic eld—the concept of “dressed atom” is widely used [18]. There, a
given eld mode and the atom are treated on an equal footing, as a composite quantum system,
leading to a time-independent Hamiltonian (energy is conserved for the entire system comprising
atom and eld). This picture is indeed very close to the Floquet picture. If the eld mode is in a
coherent state [18] with a large average number of photons, the electromagnetic eld can be treated
(semi)classically—i.e., replaced by a cos time dependence and a xed amplitude F—and the energy
spectrum of the dressed atom exactly coincides with the spectrum of the Floquet Hamiltonian [92].
By its mere de nition, Eq. (74), each Floquet eigenstate is associated with a strictly time-periodic
probability density in con guration space. Due to this periodicity with the period of the driving eld,
the probability density of a Floquet eigenstate in general changes its shape as time evolves, but
recovers its initial shape after each period. Hence, the Floquet picture provides clearly the simplest
approach to non-dispersive wave packets. Given the ability to build a Floquet state which is well
localized at a given phase of the driving eld, it will automatically represent a non-dispersive wave
packet. In our opinion, this is a much simpler approach than the attempt to build an a priori localized
wave packet and try to minimize its spreading during the subsequent evolution [44,45,34,47].
Note that also the reverse property holds true. Any state with T-periodic probability density (and,
in particular, any localized wave packet propagating along a T-periodic classical orbit) has to be a
single Floquet eigenstate: Such a state can be expanded into the Floquet eigenbasis, and during one
period, the various components of the expansion accumulate phase factors exp(−iEiT=˝). Hence,
the only solution which allows for a T -periodic density is a one-component expansion, i.e., a single
Floquet eigenstate. 10
10 One might argue that Floquet states di ering in energy by an integer multiple of ˝! could be used. However, the
Floquet spectrum is ˝!-periodic by construction [92], and two such states represent the same physical state.