High-precision quasienergies for a driven two-level atom at the two-photon preresonance
Dong-Sheng Guo,
1
J. T. Wang,
1
and Yong-Shi Wu
2
1
Department of Physics, Southern University and A&M College, Baton Rouge, Louisiana 70813, USA
2
Department of Physics, University of Utah, Salt Lake City, Utah 84112, USA
Received 14 March 2007; published 4 February 2008
A computation with unprecedented precision is presented for quasienergies of a two-level atom in a mono-
chromatic radiation on the basis of a recently obtained exact expression D.-S. Guo et al., Phys. Rev. A 73,
023419 2006. We start with the proof of an expression theorem. With this theorem the quasienergies for any
two-level atom can be expressed in terms of the quasienergies for only those with the original energy spacing
per field photon energy being an integer preresonances. Then we carry out a numerical evaluation of the
quasienergies at the two-photon preresonance, which involves computing an infinite determinant, up to the
18th power of the coupling strength. The theoretical prediction presents an experimental challenge for high-
precision tests of quantum mechanics and could be exploited for precise calibration of high laser intensities.
DOI: 10.1103/PhysRevA.77.025401 PACS numbers: 32.80.Rm
In a recent paper 1, a closed expression for quasiener-
gies exhibiting explicit Floquet periodicity, together with the
corresponding wave functions, is derived for a two-level
atom in a monochromatic radiation field without any restric-
tion on the range of the parameters. The study of the two-
level atom 2 in a radiation field has occupied a central
position in quantum optics see, e.g., 3–6. The ultimate
goal of quantum optics is to achieve a complete understand-
ing of atoms interacting with radiation fields. To reach this
goal, the most important thing is to obtain accurate wave
functions with corresponding accurate quasienergies for an
arbitrary atom or ion interacting with a radiation field. A
real atom or ion has an infinite number of energy levels,
which may include a finite number of bound-state energy
levels occupied or nonoccupied ground states, an enumer-
able infinite number of excited states occupied or nonoccu-
pied Rydberg states, and a nonenumerable infinite number
of states in its continuous spectrum. With present computing
techniques, all continuous quantities are discretized to sets of
finite numbers. Thus the N-level atom model with an arbi-
trarily large N number is an appropriate way to describe a
real atom.
The importance of the model of a two-level atom in a
radiation field is in the following two aspects. i The model
can apply to many physical systems directly—e.g., atoms in
a laser field when the laser frequency is near a transition
frequency between two levels of a real atom and a two-level
system of magnetic resonances. ii The model is the first
step in the development of an accurate theory for the N-level
atom in radiation fields: Analytic or numerical techniques for
solving a driven two-level system may be generalized to the
driven N-level system; and as a necessary condition, the ana-
lytical and numerical solutions for a driven two-level atom
will provide strong checks for the correctness of the solu-
tions for a driven N-level atom. Now, obtaining accurate ana-
lytic and numerical solutions for an arbitrary N-level atom is
in our short-term plan. Thus an accurate numerical solution
for a two-level atom plays a key role in our current and
future researches.
There is a vast literature on the two-level atom in a mono-
chromatic classical field described by the Bloch equation 7.
Analytic methods with various approximations have been
used see, e.g., 3–6, while numerically accurate, direct in-
tegration of the Bloch equation in strong radiation fields still
remains elusive, because the pertinent time-dependent differ-
ential equations involve discrete quasienergies and discrete
boundary conditions to be determined.
The closed expression for quasienergies obtained in Ref.
1 involves infinite determinants. Numerical evaluations of
these infinite determinants involve some complicated alge-
braic manipulations, such as multiple infinite series and infi-
nite products, which cannot be found in any mathematics
textbook. One of the main purposes of the present paper is to
demonstrate, with physically interesting examples, how to
evaluate these infinite determinants in a numerically satisfac-
tory fashion. Moreover, the precise numerical values of
quasienergies provide theoretical predictions which can be
put to precise experimental tests and exploited to calibrate
laser intensities. The techniques developed here are useful
for evaluating quasienergies and wave functions at any pre-
resonance. The special case we are presenting here is when
the level spacing 2 is twice the photon energy of the
radiation. We call this the two-photon preresonance, which
has been proven 1 to be not a true resonance. The true
resonances occur only when the radiation-shifted level spac-
ing is an integral multiple of the laser-photon energy, such as
Freeman resonances in above-threshold photoelectron spec-
tra 8. However, as shown below, the preresonance cases
play a very important role in calculating quasienergies for a
generic value of 2 / n, since those quasienergies can be
expressed in terms of all quasienergies only at the prereso-
nances.
Theorem 1. The quasienergies of a driven two-level atom
whose original spacing before interacting with the driving
radiation field is a nonintegral multiple of the photon energy
of the driving field can be expressed in terms of the quasien-
ergies of all two-level atoms with an integral original spacing
interacting with the same radiation field.
Proof. In Ref. 1, the quasienergy when the original spac-
ing 2 integer we set =1, as in Ref. 1, the quasiener-
gies can be expressed as any one of the following two ex-
pressions:
PHYSICAL REVIEW A 77, 025401 2008
1050-2947/2008/772/0254014 ©2008 The American Physical Society 025401-1