Multiatom effects in cavity QED with atomic beams

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 60, Issue 3, 1999, Pages 2497-2504

  Carmichael, H J ^a   Sanders, B C ^b  

^a UNIVERSITY OF OREGON   (United States)

^b MACQUARIE UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0001191310     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.60.2497     Document Type: Article

References (12)

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6. H.J. Carmichael, P. Kochan, and B.C. Sanders, Phys. Rev. Lett. 77, 631 (1996).
7. B.C. Sanders, H.J. Carmichael, and B.F. Wielinga, Phys. Rev. A 55, 1358 (1997).
8. R.J. Thompson, G. Rempe, and H.J. Kimble, Phys. Rev. Lett. 68, 1132 (1992).
9. The assumption of a Poisson distribution for (Formula presented) leads cleanly to Eq. (2.17), for all F, but is not essential. More generally we might write the right-hand side of Eq. (2.16) as (Formula presented)without specifying the probabilities (Formula presented). Then, on taking the limit (Formula presented) (Formula presented), we require that (Formula presented) be peaked about (Formula presented), and note that (Formula presented) [Eq. (2.18)]. Equation (2.17) follows, in its limiting forms (2.24) and (2.25), from the result (Formula presented), for (Formula presented) constant.
10. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, (Dover, New York, 1965), p. 231.
11. H. J. Bremermann, Distributions, Complex Variables, and Fourier Transforms (Addison-Wesley, Reading, MA, 1965).
12. The existence of problematic behavior near (Formula presented) can be appreciated by expanding the characteristic function (2.17) before taking the limit (Formula presented). Making the expansion to first order in (Formula presented) and taking the Fourier transform (2.23), for a ring cavity, (Formula presented)with (Formula presented)where (Formula presented) is the unit step function. The normalization is preserved by the approximation, since with increasing (Formula presented) the weight of the (Formula presented) function decreases to compensate for the probability carried by the one-atom tail of the distribution. But when (Formula presented), the tail has a nonintegrable singularity at (Formula presented). Hence, for any (Formula presented) no matter how small, the weight of the (Formula presented) function inevitably becomes negative as (Formula presented). It vanishes for (Formula presented). The truncation certainly fails, therefore, below (Formula presented). For a standing-wave cavity, the first-order expansion gives (Formula presented)which behaves in the same way.