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(6.49)
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where we have dropped the primes and expressed the displacement field in terms of the creation and annihilation operators defined by equation (5.117). The coupling constant ge (k) is obtained from equation (6.46) when the atomic dipole is expressed in terms of the two-level atom operators [i.e.,d = -iliJ-L (O"t - 0") x). We have shifted the zero of energy to the average of the energies of the two atomic levels as in 5. Now we want to obtain a nonperturbative expression for the expectation value of the atomic energy 0" zliwo/2 as a function of time for the case where the field is initially in the vacuum state. We can solve for the field operators as we have done before:
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= ae(k,O)e- iwt + ige(k) lot dt l [O"t(t/) - O"(t')] e-iw(t-t').
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(6.50)
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oAt) = -i2 [O"t(t) - O"(t)]
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(6.51)
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Because we are interested in the total rate of change of atomic energy, we can choose normal ordering6 when we substitute equation (6.50) in (6.51). This yields
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~9;(k) [O"tW) -
O"(t')] e-iw(t-t' )
+ c.c.
(6.52) We notice that the time evolution of the atomic creation and annihilation operators O"t and 0" consists of a fast oscillation of angular frequency Wo and a slowly varying envelope [13] O"(t) = o-(t)e- iwot . (6.53)
6The choice of normal ordering simplifies the calculation of the expectation value of (6.51) for the case where the field is initially in the vacuum state because the first term on the right-hand side of equation (6.50). the free field. will not contribute.
SPONTANEOUS EMISSION
Using (6.53) in (6.52), we obtain
(o-z(t)) = -21tdt,/d3k
~9~(k)
a(t')e-i(W-Wo)(t-t')])
(at (t) [at (t')eiwo(tH')-iw(t-t') -
+ c.c.
~-(O't(t)O'(t)47r
d3 kL9:(k)o(w-wo)
(6.54)
When we integrate equation (6.54), we find that the atomic energy decays exponentially with the rate -I: (6.55) The exponential decay in (6.55) is typical of irreversible processes. The atom has an infinite continuum of modes to emit the photon, and once emitted, it is extremely unlikely that the photon will find its way back to the atom and be reabsorbed. In the next section we will see that a cavity can change this decay rate, and in the strong coupling regime, even make spontaneous emission a reversible process.
6.2 SPONTANEOUS EMISSION IN A CAVITY
Let us assume that the atomic transition is resonant with the mode oflowest frequency supported by the cavity and very far off resonance from any other modes. In this case, we can neglect all cavity modes except for the lowest one, and the Hamiltonian of the entire system becomes (6.56) where
(a,a t ] = 1.
(6.57)
The terms O'a and 0' tat in the Hamiltonian (6.56) do not conserve energy and can only contribute to the dynamics through virtual transitions. This contribution is much smaller than that of the real energy-conserving transitions associated with the terms O'a t and O't a. So we will make the rotating-wave approximation and keep only the energy-conserving terms in (6.56):
Ii~O'z + Iiw (ata + ~) + lig (O't a + at O') .
(6.58)
Real cavities are not made of perfectly reflective mirrors. In a real cavity, radiation eventually decays because of losses. The usual approach is to introduce losses by
EMISSION INA CAVITY
coupling the cavity to a large reservoir. When the reservoir variables are traced over, the evolution of the atom + lossy cavity system is given by a master equation for the reduced density matrix7 of such a system, p. As this is a well-known procedure (see, e.g., [533]), we will not repeat such calculations here. We will, however, derive the same result from a physical argument for the case where the environment is at zero temperature. Let us consider for a moment a hypothetical one-dimensional cavity composed of two mirrors a distance l apart. After a time t = 2Nl/c corresponding to N round trips in the cavity,8 the electric field of a classical electromagnetic wave at position x, E(x), will decrease from its initial value at t = 0, due to the partial reflections at the mirrors, according to E(x,t) = RNE(x,O), (6.59) where R is the reflectivity of the mirrors; assumed the same for both, with no change of phase on reflection. Substituting N = tc/2l in (6.59), we can rewrite (6.59) as
E(x, t) = E(x, 0)e- ltt / 2 ,
(6.60)
where K = -c In(R)/4l. Equation (6.60) is purely formal, because the field does not decay continuously in time but only at those discrete times when it hits one of the mirrors. If, however, the mirrors are sufficiently good reflectors so that electromagnetic radiation can complete many round trips before it loses a very small fraction of its energy,9 we can neglect the fact that the losses are localized at the mirrors and assume them to be distributed homogeneously in the cavity and occur continuously in time. lO In this case, equation (6.60) is a good approximation for any time t > 0, even for times that are not integral multiples of a round-trip time. In our crude argument above, we have omitted an important detail. When the mean energy of the field in the cavity becomes comparable to the mean energy of blackbody radiation, equation (6.60) breaks down. This is because the energy leaking out of the cavity will be balanced by energy being fed into the cavity by blackbody radiation. At zero temperature, however, there is no blackbody radiation and we would expect (6.60) to be valid all the way down to the quantum zero-point energy. The quantum states that recover classical radiation in the limit of high intensity are coherent states. So if (6.60) can be carried out to the quantum limit, a coherent state la) at time t decays to (6.61) We will use this result to derive the master equation describing the decay of any cavity field.
7 An alternative approach keeps the state vector description but replaces SchrOdinger's equation by a stochastic equation [88, 131,590]. SHere c is the speed of light in the cavity. 9That is. if the decay time is much longer than the time of flight of a light signal through the cavity. IOThis implies coarse graining at a time scale on the order of the time of flight of a light signal through the cavity (see [362]).