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arXiv:quant-ph/0509135 · Physical Review A

Algebraic solution for a two-level atom in radiation fields and the Freeman resonances

D. Guo, Yong Wu, Linn Van Woerkom

Published Feb 17, 2006• 8 citations
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Abstract

Using techniques of complex analysis in an algebraic approach, we solve the wave equation for a two-level atom interacting with a monochromatic light field exactly. A closed-form expression for the quasienergies is obtained, which shows that the Bloch-Siegert shift is always finite, regardless of whether the original or the shifted level spacing is an integral multiple of the driving frequency $\ensuremath{\omega}$. We also find that the wave functions, though finite when the original level spacing is an integral multiple of $\ensuremath{\omega}$, become divergent when the intensity-dependent shifted energy spacing is an integral multiple of the photon energy. This result provides an ab initio theoretical explanation for the occurrence of the Freeman resonances observed in above-threshold ionization experiments.

DOI
10.1103/physreva.73.023419

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Using techniques of complex analysis in an algebraic approach, we solve the wave equation for a two-level atom interacting with a monochromatic light field exactly. A closed-form expression for the quasienergies is obtained, which shows that the Bloch-Siegert shift is always finite, regardless of whether the original or the shifted level spacing is an integral multiple of the driving frequency \ensuremathω\ensuremath{\omega}\ensuremathω. We also find that the wave functions, though finite when the original level spacing is an integral multiple of \ensuremathω\ensuremath{\omega}\ensuremathω, become divergent when the intensity-dependent shifted energy spacing is an integral multiple of the photon energy. This result provides an ab initio theoretical explanation for the occurrence of the Freeman resonances observed in above-threshold ionization experiments.
The time evolution of a two-level system interacting with a classical monochromatic radiation field is calculated by Fourier transforming the equations of motion. This can be done exactly if one uses the convolution of the product of two Fourier transformations. The method produces recurrence relationships for the transforms. The time-dependent amplitudes of the states and the transition probability are evaluated by using two such recurrence relationships. The time-averaged probability shows characteristic resonances at odd multiples of the ratio of the natural frequency of the system to the frequency of the radiation field. The peaks of the resonances are displaced by the Bloch-Siegert shift. At these resonances the system is completely saturated with the flipping frequency dependent on the strength of the radiation field. The time dependence and the frequency spectrum of the dipole moment is determined at the main resonance and at the three-quantum resonance. The spectrum contains the zero-frequency terms which are responsible for the coherence resonances.
A computation with unprecedented precision is presented for quasienergies of a two-level atom in a monochromatic 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.
Based on the vanishing-damping continued-fraction method which I developed previously for the analysis of the resonance behavior of a two-level atom in a two-frequency field with equal intensities in the two field modes [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], I analyze here the related problem in which the two field intensities are unequal. Several analytical results are derived, namely, expressions for the lowest-order Bloch-Siegert shifts of all resonances, and an expression for the positions of the resonances in the limit of one strong and one weak field. It is shown that, in this limit, the resonances exhibit level crossings of the field-atom system.
Expressions for energy-level shifts and transition amplitudes of atoms interacting with an incoming radiation field are developed in a completely quantum theoretic way and the results are compared with others. The continued-fraction expressions reproduce both those of Swain and of Gontier et al., and the numerical results for the particular case of Bloch-Siegert shifts in mercury vapor agree with Stenholm's but go further in giving the shape of the resonances. These results are also obtained by a method of direct diagonalization of the Hamiltonian of the coupled system.
This work presents an analytical calculation of the time-dependent radiation forces on a two-level atom interacting with a single-mode laser field. Such a closed and compact expression of the radiation forces is derived by solving the optical Bloch equations analytically. It is confirmed, in particular, that the radiation force consists of reactive as well as dissipative components, whose explicit analytical forms of the temporal solutions can be explicitly obtained. The succinct analytical solutions of the radiation forces may be helpful for a convenient and intuitive description of the complex atomic dynamics such as interaction with various laser fields.
The Schrodinger equation for the interaction of a decaying two-level atom with an external oscillating field is separated via a series of matrix transformations into a time independent and a time dependent part; the former yields the radiofrequency Stark shift for decaying states, the latter is normally negligible for atomic beam resonance experiments. In the limit of stable states, or for the case of two levels of equal lifetime, the shift reduces to that of the Bloch-Siegert effect. Exact analytical expressions for the state amplitudes are obtained for the case when the time dependent matrix can be neglected; the resulting transition probabilities are investigated as functions of frequency, time, and interaction strength and are compared with the rotating field theory and the exact numerical solutions.