Advances in Laser Spectroscopy by Christian J. Bordé (auth.), F. T. Arecchi, F. Strumia, H.

By Christian J. Bordé (auth.), F. T. Arecchi, F. Strumia, H. Walther (eds.)

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5. a a a' a a a a' a' a a a 13' Figure 5 a (3' c. J. BORDE 34 With these new diagrams we have the choice between two possibilities : I) Use second-quantization for the electromagnetic field as well as for molecules. It is then necessary to introduce a propagator for the radiation density matrix elements. 2) Use the semiclassical approximation in which the classical electromagnetic field enters at each vertex with the corresponding space-time argument. ' s ,whereas the exponential factor exp i0 oscillates rapidly ex~ept when the phase is stationary,and the greatest contribution to the velocity and.

2 )1/2 If Yb is negligible,we obtain a Gaussian line shape for each transv~rse velocity,with a width where the usual Doppler broadening and the transit broadening are combined quadratically llv =~ 2v'1U2 [1 + 222 2]1/2 k wo e: . (115) In this limit of negligible relaxation,a very general expression for the line shape is obtained with the convolution theorem : +00 +00 2Re f dT exp [ i (YO>- o =.!. v where f +00 "'b. I:lx 2 1- J dx G(x) G*(x-vx T) _00 dx G(x) e V (116) -00 II = yoo - CIl. ba - 6 - kv z That is, the line shape is the square of the modulus of the Fourier transform of the transverse field distribution.

J. BORDE 24 From the integral equation for p we may write + 3 similar terms. This equation is obviously covariant and displays the usual Feynman propagators. as a sum of density matrix elements between energy-~omentum states : (86) performed over small intervals around p + m hk and p + mt:). hk. o a. 1 fJ,mj3 0 + ~ a' g ~ , i (x 2)! 1 fJ,mj3 (x 2)"! 0> t dt' _00 (87) + 3 similar terms. If we write (*) -+ -+ YVl: -+, rl-r l ,t-t' = Yl: (88) and use (64),we recover the same invariant propagator as before, apart from the rank-two bispinors : and (89) which reduce to unity only in the rest frame when basis spinors are eigenvectors of the z-component of the spin in that frame.

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