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Physical Review A

Linear polarizabilities of two- and three-level atoms

Peter W. Milonni, R. Loudon, P. R. Berman, Stephen Mark Barnett

Published Apr 29, 2008• 34 citations
Publisher

Abstract

Different expressions for the linear polarizability of a two-level atom with radiative corrections have been derived recently. We show that an expression said to differ from that obtained by the present authors is in fact consistent with it. The same-sign and opposite-sign prescriptions for linewidths are revisited with respect to the polarizability, the scattering amplitude, and the optical theorem. Both prescriptions represent approximations to more general expressions in the two-level case, and neither is correct for transitions between excited atomic states, as we demonstrate by calculating the linear polarizability of a three-level atom.

DOI
10.1103/physreva.77.043835

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•Jan 1, 1974•0 citations
Different expressions for the linear polarizability of a two-level atom with radiative corrections have been derived recently. We show that an expression said to differ from that obtained by the present authors is in fact consistent with it. The same-sign and opposite-sign prescriptions for linewidths are revisited with respect to the polarizability, the scattering amplitude, and the optical theorem. Both prescriptions represent approximations to more general expressions in the two-level case, and neither is correct for transitions between excited atomic states, as we demonstrate by calculating the linear polarizability of a three-level atom.
The polarizability of a two-level atom in the presence of radiative damping alone is derived by a Green function calculation. The rotating-wave approximation is not made, either in the interaction of the atom with the incident light or in its interaction with the scattered optical field. In contrast to previous work, the derivation takes account of the modification of the system ground state brought about by the non-rotating-wave part of the latter interaction. The final expression satisfies the optical theorem in addition to the other general requirements on the polarizability.
The effect of spontaneous decay on the linear polarizability of an atom is typically included by adding imaginary parts to the frequency denominators that appear in the Kramers-Heisenberg formula. It has been shown for a two-level atom with radiative broadening that these (frequency-dependent) imaginary parts must be included in both the resonant and antiresonant frequency denominators [P. W. Milonni and R. W. Boyd, Phys. Rev. A 69, 023814 (2004)]; however, the expression obtained by Milonni and Boyd for the polarizability does not satisfy the optical theorem, if contributions from non-rotating-wave terms are included. In this paper, we derive a more accurate expression for the polarizability. The calculations are rather complicated and require that we go beyond the standard Weisskopf-Wigner approximation. We present calculations carried out in both the Heisenberg and Schr\"odinger pictures, since they offer complementary methods for understanding the dynamics of the Rayleigh scattering associated with the atomic polarizability. Moreover, it is shown that the shifts associated with the excited state are not the Lamb shifts of an isolated atom, but depend on the dynamics of the atom-field interaction. Our results for the polarizability are consistent with those obtained recently by Loudon and Barnett using a Green's-function approach.
The effect of the counterrotating terms on the linear polarizability is investigated, which is responsible for the validity of the optical theorem in all frequency regions. A unitary transformation method [H. Zheng, S. -Y. Zhu, and M.S. Zubairy, Rev. Lett. 101, 200404 (2008)] is adopted to overcome the difficulty brought in by the counterrotating terms, which yields a rotating-wave-approximation-like Hamiltonian with modified coupling constant due to the counterrotating terms. A simple expression for the polarizability is obtained, which is a sum of resonant (minus sign) and antiresonant (plus sign) parts, and from which the role of the counterrotating terms and quantum interference between the counterrotating terms and rotating terms at far off-resonance are discussed.
The excitation spectrum is calculated for a three-level atom interacting with two strong electromagnetic fields. The two fields are in resonance with the atomic transition frequencies from the ground state to the two excited states. The excitation spectrum consists of a central peak and two pairs of side bands for each of the two transitions. If the decay rates of the two excited states are equal a relatively simple closed form expression is derived for the excitation spectrum. For unequal decay rates numerical methods are used to determine the excitation spectrum for selected special cases.
Several calculations have been carried out using the transition operator technique. The radiative decay in a two-level quantum system with excited state coupled by an external perturbation has been investigated and an expression obtained for P(t), the probability of finding the atom in its excited state at time t. This is seen to possess a steady value for yt >> 1 and a third term which decays at nearly half the rate of its second. Next radiative decay of an atom with two close-lying excited states is considered and |bq| 2 , the spectral density for spontaneous emission from the uppermost level, calculated. This shows that the proximity of the upper two levels makes the line-shape non-Lorentxian. Heisenberg equations of motion were then derived for the transitions operators of an ℓ-level atom undergoing radiative decay, the possibility of overlapping pairs of levels being ignored. These master equations were then used to obtain the spectral profiles of certain atomic lines, employing the so-far avoided Markoff approximation. First the spectral density of the scattered radiation was found for transitions between levels 5P3/2 and 4S (ground state), and 6S and 5P 3/2, in the potassium atom when driving fields of arbitrary strength coupled levels 6S and 4P3/2, and 4P 3/2 and 4S. Triple-peaked profiles were obtained. Spectral profiles for emission were then derived for transitions between levels 3P and (ground state), and 3P and in the hydrogen atom when a driving field of arbitrary strength coupled levels The former profile was double the latter quadruple-peaked. Lastly master equations were derived for an atom, with two excited levels undergoing radiative decay, when allowance was made for thepossibility of their overlapping. A driving field of arbitrary magnitude coupled the uppermost and ground states. The spectrum for the scattered light resulting from transitions between the uppermost and ground state was found and graphs computed for various field strengths and separations of the excited states.