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e-lal2lal2n L --,-- cos (2gtVn + 1)1. n.
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This is a superposition of Rabi oscillations of different frequencies weighted by a Poissonian distribution. The total sum will oscillate initially, but the oscillation will eventually collapse when the cosine terms become out of phase. We can verify that + 1 around lal 2 for lal 2 1 in equation (6.92): by expanding
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Pll(t) =
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~ [ ~ + e-~aI2 eigtlal exp (laI2eigt/lal) + c.c'] .
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. t 2t igt/lal '" 1 ~ _ ___ g e '" + lal 21a12' 2
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I, we can expand the exponential in (6.93): (6.94)
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Then equation (6.93) becomes [130] Pll (t)
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(1 + e-g2t2/2 cos 2gtlal) .
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So the collapse happens at a time on the order of II 9 [34, 178]. They occur because of the noise in the field, which leads to the finite variance of the photon
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12It will become evident later that for this to be true. the cavity decay constant K. has to be at least much smaller than 9 / v'fi. where n is the average number of photons in the cavity mode.
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number, causing the eventual dephasing of the various cosine terms in (6.92). This noise is of quantum origin here, but classical noise would be equally effective. Expression (6.95) is valid only for times much smaller than 10'1/ g. In fact, we notice by inspecting (6.93) that the destructive interference disappears when the oscillating complex exponential argument of the exponential function in (6.93) becomes unit again. So a revival of the oscillations should occur at (6.96) Although it is possible to recover the collapse of the oscillations in a calculation using a classical field with classical noise, the revival is a genuine quantum effect. Revivals can only take place because the field energy is discrete rather than continuous, so that the various cosine components of the oscillations can eventually come back in phase with each other. The recent observation of revivals [153, 517] provided more evidence of the quantum nature of the electromagnetic interaction. In this chapter we have applied QED to the problem of spontaneous emission by a single atom in the vacuum. We have seen that a cavity can make spontaneous emission become reversible, leading to Rabi oscillations. In 7 we study how QED can be modified to account for material media similar to the process in classical electrodynamics: through a dielectric permittivity.
RECOMMENDED READING For a discussion of atomic stability where coupling with the radiation field is neglected and only the Coulomb potential is considered, see [175, 177, 409. 575]. There are several approximations and assumptions in the Weisskopf-Wigner approach. For a detailed study of these and alternative improvements on the original Weisskopf- Wigner theory. see [7, 144, 360, 555]. For a discussion of renormalization in the context of atomic physics, I refer the reader to Weisskopf [630], Welton [632], Dalibard et al. [131], and Haroche [274]. For a thorough discussion of the two-level atom approximation, see Allen and Eberly [13].
Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box Sergio M. Dutra Copyright 2005 John Wiley & Sons, Inc.
Macroscopic QED: Quantum electrodynamics in material media
... zu jener Zeit ... elektrische bezw. magnetische "Feldstiirken" und "Verschiebungen" wurden als gleich element are Grossen behandelt, der leere Raum als Spezialfall eines dielektrischen Korpers. Die Materie erschien als Thliger des Feldes, nicht der Raum .... Es war das grosse Verdienst von H. A. Lorentz, dass er hier in iiberzeugender Weise Wandel schur. 1m Prinzip gibt es nach ihm ein Feld nur im leeren Raume. Die atomistisch gedachte Materie ist einziger Sitz der elektrischen Ladungenj zwischen den materiellen Teilchen ist leerer Raum, der Sitz des elektromagnetischen Feldes ... Dielektrizitat, Leitungsfahigkeit, etc. sind ausschliesslich durch die Art der mechanischen Bindung der Teilchen bedingt, aus welchen die Korper bestehen. 1 -Albert Einstein [184]
There are two main versions of classical electrodynamics. The first was originally created by Maxwell and is now referred to as macroscopic electrodynamics. It involves four vectorial field quantities in a material medium: the electric field E, the magnetic field M, the electric displacement D, and the magnetic induction B. The
1Translation: .. . at that time . . . electric or magnetic "field intensities" and "displacements" were treated as equally elementary variables, empty space as a special instance of a dielectric body. Matter appeared as the bearer of the field, not space . .. . It was the great merit of H. A. Lorentz that he brought about a change here in a convincing fashion. In principle a field exists, according to him, only in empty space. Matter-considered as atoms-is the only seat of electric charges; between the material particles there is empty space, the seat of the electromagnetic field .... Dielectricity, conductivity, etc. are determined exclusively by the type of mechanical tie connecting the particles, of which the bodies consist.