Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
Non-dispersive wave packets in periodically driven quantum systems
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
436 A. Buchleitner et al. / Physics Reports 368 (2002) 409–547<br />
“Floquet eigenstates” of the system—multiplied by oscillatory functions:<br />
| (t)〉 = <br />
<br />
cj exp −i Ejt<br />
<br />
|Ej(t)〉 ; (73)<br />
˝<br />
with<br />
j<br />
|Ej(t + T )〉 = |Ej(t)〉 : (74)<br />
The Ej are the “quasi-energies” of the system. Floquet states and quasi-energies are eigenstates and<br />
eigenenergies of the Floquet Hamiltonian<br />
H = H0 + V cos !t − i˝ 9<br />
: (75)<br />
9t<br />
Note that, because of the time-periodicity with period T =2 =!, the quasi-energies are de ned<br />
modulo ˝! [92].<br />
The Floquet Hamiltonian (75) is noth<strong>in</strong>g but the <strong>quantum</strong> analog of the classical Hamiltonian<br />
(58) <strong>in</strong> extended phase space. Indeed, −i˝9=9t is the <strong>quantum</strong> version of the canonical momentum<br />
Pt conjugate to time t. In strict analogy with the classical discussion of the previous section, it is<br />
the Floquet Hamiltonian <strong>in</strong> extended phase space which will be the central object of our discussion.<br />
It conta<strong>in</strong>s all the relevant <strong>in</strong>formation on the system, encoded <strong>in</strong> its eigenstates.<br />
In a <strong>quantum</strong> optics or atomic physics context—with the external perturbation given by quantized<br />
modes of the electromagnetic eld—the concept of “dressed atom” is widely used [18]. There, a<br />
given eld mode and the atom are treated on an equal foot<strong>in</strong>g, as a composite <strong>quantum</strong> system,<br />
lead<strong>in</strong>g to a time-<strong>in</strong>dependent Hamiltonian (energy is conserved for the entire system compris<strong>in</strong>g<br />
atom and eld). This picture is <strong>in</strong>deed very close to the Floquet picture. If the eld mode is <strong>in</strong> a<br />
coherent state [18] with a large average number of photons, the electromagnetic eld can be treated<br />
(semi)classically—i.e., replaced by a cos time dependence and a xed amplitude F—and the energy<br />
spectrum of the dressed atom exactly co<strong>in</strong>cides with the spectrum of the Floquet Hamiltonian [92].<br />
By its mere de nition, Eq. (74), each Floquet eigenstate is associated with a strictly time-periodic<br />
probability density <strong>in</strong> con guration space. Due to this periodicity with the period of the driv<strong>in</strong>g eld,<br />
the probability density of a Floquet eigenstate <strong>in</strong> general changes its shape as time evolves, but<br />
recovers its <strong>in</strong>itial shape after each period. Hence, the Floquet picture provides clearly the simplest<br />
approach to non-<strong>dispersive</strong> <strong>wave</strong> <strong>packets</strong>. Given the ability to build a Floquet state which is well<br />
localized at a given phase of the driv<strong>in</strong>g eld, it will automatically represent a non-<strong>dispersive</strong> <strong>wave</strong><br />
packet. In our op<strong>in</strong>ion, this is a much simpler approach than the attempt to build an a priori localized<br />
<strong>wave</strong> packet and try to m<strong>in</strong>imize its spread<strong>in</strong>g dur<strong>in</strong>g the subsequent evolution [44,45,34,47].<br />
Note that also the reverse property holds true. Any state with T-periodic probability density (and,<br />
<strong>in</strong> particular, any localized <strong>wave</strong> packet propagat<strong>in</strong>g along a T-periodic classical orbit) has to be a<br />
s<strong>in</strong>gle Floquet eigenstate: Such a state can be expanded <strong>in</strong>to the Floquet eigenbasis, and dur<strong>in</strong>g one<br />
period, the various components of the expansion accumulate phase factors exp(−iEiT=˝). Hence,<br />
the only solution which allows for a T -periodic density is a one-component expansion, i.e., a s<strong>in</strong>gle<br />
Floquet eigenstate. 10<br />
10 One might argue that Floquet states di er<strong>in</strong>g <strong>in</strong> energy by an <strong>in</strong>teger multiple of ˝! could be used. However, the<br />
Floquet spectrum is ˝!-periodic by construction [92], and two such states represent the same physical state.