Condensed Matter Theory


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