J. Phys. Chem. 1994,98, 9777-9780
9777
Stable and Unstable Localization in Two-Level Systems Driven by Two Time-Dependent Fields Yuri Dakhnovskii* and Raanan Bavli Department of Chemistry, University of California, Santa Barbara, Califomia 93106 Received: July 7, 1994@
The effect of localization of a charged particle in two-level systems dnven by two laser fields has been analyzed. An analytical solution verified by direct numerical integration for a time-dependent population has been found for arbitrary field amplitudes and frequencies in the limit of small tunneling matrix element. The topology of the localization regions in the parameter space has been studied. Stable and unstable localization regions have been found. The unstable regions originate from the nonanalytical behavior of the population with respect to a small incommensurability. This phenomenon mathematically resembles phase transition instabilities in a solid state. For the stable regions in the frequency domain threshold values have been found.
I. Introduction The laser-induced localization of a charge in double-well quantum systems has been intensively studied during recent years.'-" Such systems can be experimentally fabricated as superlattices with periods up to a few hundred angstroms,I2 a charge transfer chemical reaction driven by a laser,13 nuclear spin flips in strong magnetic fields,14 etc. The initial explanations of this effect were based on the Floquet theory According to this, the wave function of a particle can be described as a superposition of symmetric and antisymmetric field-dependent Floquet states. The full localization is obtained when this superposition consists of only two degenerate states with equal coefficients and phase difference of nn (n = 0, +1, 1 2 , ...). According to Grossmann, Dittrich, Jung, and Hanggi,' the Floquet states originate from the bare states which are initially separated by energy A. The field renormalizes the splitting 6, and for particular values of the field amplitude 6 vanishes. The localization phenomenon is closely connected with generation of even harmonics in symmetric potential^.^ The second harmonic susceptibility is proportional to the product of three dipole matrix elements. In the systems with parity eigenstates this product always vanishes unless two of the states involved in this process are degenerate. This is the important condition for localization. When laser parameters are tuned slightly off degeneracy (localization) conditions, even harmonic lines split to doublets with the separation of 26. The odd harmonics are unchanged. The mechanism of localization and even harmonic generation has been explained in the framework of Floquet theory.'-4 Although this approach gives an excellent qualitative analysis, it has to be supplemented by numerical or analytical solution of the time-dependent Schrodinger equation. Floquet theory is valid only for periodic fields, and therefore, an analysis of interaction with nonperiodic fields requires a different approach. In this paper we use a method of a time-dependent perturbative expansion in the small parameter A2 for a two-level system driven by an arbitrary time-dependent field. A two-level system gives a good description of the dynamics in a double-well
potential when two levels with splitting A are well separated in energy from other ones. Using such a method, localization has been found in the first order of perturbative e x p a n s i ~ n . ~ . ~ The analysis is restricted to finite times because otherwise the perturbative expansion is invalid. It becomes unclear whether localization is a short time effect or it takes place for long times. This question was clarified in refs 6-9 by summing all orders of the perturbative series. It has been shown that localization takes place for any time. This method also allows to calculate the time-dependent tunneling probability in two electric fields where one is a constant and the other is a CW optical field7 or two optical fields with slightly incommensurate frequencies.* In the former case new localization conditions have been found. The latter case describes the dynamics in nonperiodic fields. In this paper we study in detail a localization phenomenon in two fields with different frequencies: (w, w 4-Aw), (w, 2w Am), and ( w , 3w Aw) where Aw is a small frequency shift. The first frequency combinations can be created using one laser and an ac field source and the others using stimulated Raman, two- and three-photon stimulated Raman techniques, second and third harmonic generation for Aw = 0, and amplitude modulation for the case (w, w Am).* The generation of the second field from the first is preferred because a phase between the fields can easily be controlled.
+
+
+
11. Theory
We start from the two-level Hamiltonian
H = Au,
+ V(t)u,
(1)
where V(t)is a driving force (h = 1) and ai (i = x,y, z ) are the Pauli matrices. According to ref 15, the time-dependent population is defined as
P(t) = [l
+ x(t)]/2
(2)
where x ( t ) = (u,)ll, with (11) the matrix element of a,. From the equations of motions for a,,ay,and a, the following master equation for the population has been obtained6 dxldt = -A2hfdtl x ( t , ) cos[F(t) - F(tl)l
(3a)
~~
* T o whom correspondence should be addressed: Department of Chemistry, Camegie Mellon University, 4400 Fifth Ave., Pittsburgh, PA 15213. Abstract published in Advance ACS Abstracts, September 1, 1994. @
with the initial condition x(t=O) = 1
0022-3654/94/2098-9777$04.50/0 0 1994 American Chemical Society
(3b)
Dakhnovskii and Bavli
9778 J. Phys. Chem., Vol. 98, No. 39, 1994
a
where
F(t) = 2Jtdt, V(tl)
(4)
The derivation of eqs 3 and 4 has been done in ref 6. The driving force consists of two fields of frequencies w1, 02 and amplitudes VI, V2. We do not restrict ourselves to cases of commensurate frequencies so that the driving force is in general nonperiodic V(t) = v, cos(w,t)
+ v, cos(w,t)
(5)
Then
F(t) = 2V,/o, sin(w,t)
+ 2V2/02sin(o,t)
2v1I w1
(6)
b
exp[ -W2)1X(t2) ( 7 )
\
Equation 3 may be rewritten in a convenient form x ( t ) = 1 - (NoJ2 R e h t d t l exp[iF(tl)l~r'dt,x 3"
Here we have introduced the following notation
where p = 1, 2, 3, .... In the region where ( A / w ~
