FLOQUET Theory

The natural quantum analogue of the classical map (1.21) is
obtained by considering the evolution of quantum states during one period
of the excitation.
Therefore I define, in close analogy with (1.19), the
state
immediately before the -th kick as

(2.6) |

is a special case of the general time evolution operator :

where is defined to take the quantum state from time to time ,

(2.9) |

(2.10) |

(2.11) |

For a system with a time-periodic Hamiltonian

Therefore the propagator (2.8) is the same for all ,

and the quantum map (2.7) simplifies to

for all iterations.

The time--propagator is also known as the FLOQUET operator of the quantum system. This naming convention is due to the fact that, using a time-independent orthonormal basis of HILBERT space for expanding into , the time-dependent SCHRÖDINGER equation (2.1) can be transformed into a system of ordinary linear differential equations, the coefficients of which are -periodic because exhibits the same periodicity. For a finite basis this is the setting of the FLOQUET theorem which asserts existence and uniqueness of the solutions and explicitly states their functional dependence on [Flo83,YS75]. In the present case the FLOQUET theorem does not apply as the HILBERT space is infinite-dimensional, but nevertheless the typical form of the FLOQUET solution and several other properties do carry over [Sal74]. Therefore, by analogy, the quantum theory of systems with time-periodic Hamiltonians is often also called FLOQUET theory.

Using the
time ordering operator
,

(2.16) |

(2.17) |

Consider the (normalized) eigenstates
of the FLOQUET operator
with respect to the eigenvalues :

(2.18) |

(2.19) |

of the SCHRÖDINGER equation (2.1) with respect to the initial condition can be constructed for all times . I now discuss some properties of these solutions for general , although later on mainly solutions for the stroboscopic times are needed.

Since
is an eigenstate of
with respect to the eigenvalue ,
is an eigenstate of
with respect to the
same eigenvalue :

where in the last step the periodicity (2.13) of has been used. Therefore the can all be labelled by the same index , indicating the same eigenvalue for all .

Because of the unitarity of
its
eigenvalues are of unit modulus and can be written as

for the

This property provides the basis for some of the considerations in chapter 5.

The definition (2.23) is tailored to make the description
of the time dependence of the
as similar to the dynamics
of an autonomous system as possible.
What is more, for stroboscopic times these two types of dynamics coincide,

Since the parameter plays a similar role as the energy eigenvalue of a
time-independent system,
is
called
a
*quasienergy* of the time-periodic Hamiltonian, and the
FLOQUET states
are referred to as its *quasienergy states*
[Zel67]. For brevity, often the states
are
called (reduced) quasienergy states, too.
The
quasienergy is defined modulo only, since
it originates from the exponential in equation
(2.22).
Due to this nonuniqueness of the quasienergy it cannot be identified
with any physical observable
in a straightforward way,
but note that, for an unscaled system, the quasienergy has the dimension
of an energy.
A discussion of the problems potentially arising from identifying the
quasienergy with the conventional energy
-- and thereby linking the quasienergy spectrum directly with the
resonance (emission/absorption) spectrum
of the respective system --
may be found in [DM98].
Normally, one restricts to the interval
.
By equation (2.21), the case of , i.e. ,
corresponds to the quantum map's stationary states,
for which not only the reduced
, but also the full
FLOQUET states
are periodic with period .

The quasienergy states
are characterized by
several useful properties.
It is easy to show that for the quasienergy states
,
are
orthogonal: their scalar product is

(2.26) |

with

Summarizing, with respect to a time-periodic Hamiltonian the quasienergies and the quasienergy states play much the same role as the energy eigenvalues and the stationary energy eigenstates do with respect to a time-independent Hamiltonian [Sam73]. This analogy includes the observation that in the same way as any solution of the time-independent SCHRÖDINGER equation can be expanded in terms of energy eigenstates with constant coefficients, the same can be accomplished using quasienergy states in the FLOQUET case.