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Physical Review A

Renormalized Frequency Shift of Coherent Radiation

Y. C. Lee, D. L. Lin

Published Jul 1, 1972• 8 citations
Publisher

Abstract

The radiative correction to the frequency of the spontaneous radiation from a system of two-level atoms in correlated states is calculated in the resonance approximation. A general prescription for renormalization without explicit reference to plane-wave states is given and used to obtain the renormalized frequency shifts. It is shown that the coherence frequency shift is not affected by the mass renormalization.

DOI
10.1103/physreva.6.388

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The radiative correction to the frequency of the spontaneous radiation from a system of two-level atoms in correlated states is calculated in the resonance approximation. A general prescription for renormalization without explicit reference to plane-wave states is given and used to obtain the renormalized frequency shifts. It is shown that the coherence frequency shift is not affected by the mass renormalization.
Density-matrix equations of motion derived from Heisenberg picture source-field quantum electrodynamics contain damping constants and frequency shifts which are not associated with a single radiative transition but which instead couple two separate transitions together. In the absence of degeneracies in the atomic levels, these coupling terms are normally neglected when the rotating-wave approximation is invoked. These terms cannot be neglected when there are degeneracies or near degeneracies. In this paper, we interpret the generalized frequency shifts, show how they can be renormalized and included in shifts of atomic energy levels and altered damping constants, and discuss their experimental significance.
A model radiating system consisting of two atoms in close proximity is analyzed. This system demonstrates the influence of spatial coherence on the spectrum of the radiation field. Explicit expressions for the degree of coherence, the source spectrum, and the spectrum of the radiation field are derived. The results are discussed in terms of Wolf's work [Phys. Rev. Lett. 56, 1370 (1986)] on this effect, which can be considered in terms of a multiple-atom analog of the effects of radiation reaction on a single atom, i.e., spontaneous decay and the Lamb shift.
The velocity-dependent spontaneous emission of a two-level atom in a Fabry-P\'erot cavity in the strong-coupling regime and the deflection of a beam of two-level atoms in a classical standing wave inside a cavity are discussed using a renormalization group approach. In this way we are able to renormalize the leading-order solutions for both problems through calculations of the corrections at first order. In fact, the first-order terms are not bounded for large times and no sense can be attached to this higher-order correction unless small times are considered. These are like the divergences of quantum field theory. To make them harmless, the condition for the Raman-Nath regime is recovered. The renormalization group methods permit one to eliminate those divergences generating a renormalized leading-order wave function without any condition of applicability. For the spontaneous emission of a two-level atom in a Fabry-P\'erot cavity in the strong regime, using a Hamiltonian without losses, it is shown that the unperturbed levels are shifted by a term proportional to the zeroth-order Bessel function with an argument yielded by the ratio of the Rabi frequency and the Doppler-shifted frequency of the mode of the cavity. When the detuning is zero, the correction to the leading-order wave function is not present and known results are recovered. For the beam of two-level atoms in a classical standing wave, when the detuning is much larger than the Rabi frequency, it is shown that the renormalization group equation, which gives the correction for the renormalized leading-order wave function, is a time-dependent Schr\"odinger equation for a free particle that induces a spreading of the initial Gaussian wave packet.
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.
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.