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(8.95)
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17This is a reasonable approximation as long as We is much larger than V2 in (8.91). See, for example, the discussion in Sec. II.A of [225].
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the ideal laser Hamiltonian becomes
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(8.99) In the interaction picture, the time evolution operator (h derived from this Hamiltonian is given by
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dw {eiwcta*(w)A(w)
+ e-iwcta(w)At(w)}.
(h(t)
where
= exp
(-~ lot dt' iII(t'))'
(8.100)
Integrating (8.101) over time and using (8.95), we find that
where
Ca(t)
Cb(7],t) =
-00 00
dw i(w dw.(
ei(wcw)t - 1 2 _ w) la(w)1 ,
(8.103) (8.104)
ei(wcw)t - 1
Z We -
) f3(w,7])a*(w).
Now, neglecting spontaneous emission is only a reasonable approximation to a real laser in the steady state, and even then only if this steady state is far above threshold. Taking the limit t ~ 00 of (8.103) and (8.104) to obtain the steady state, we find thatl8 (Problem 8.7) lim Ca(t) = and lim Cb(7], t)
t-+oo
t-+oo
= Z W 7] K . .( 2V) e
(8.105)
18Provided that V
i- O.
THE MASER, THE LASER, AND THEIR CAVITY QED COUSINS
Thus, in the steady state, the evolution operator is a product of Glauber displacement operators (the Glauber displacement operator was introduced in 4), lim UI(t)
t--+oo
= De. (-i2~) IIDb()d'1 (2~~), K '1 K W - TJ
{'1}
(8.106)
where the subscript in each displacement operator denotes the mode it corresponds to (e.g., De. corresponds to the cavity mode), and the productory is over the continuum of external modes. The steady-state quantum state of our closed system is obtained by applying (8.106) to the initial quantum state. Assuming that we started with the quantum vacuum before switching on our ideal laser, both the cavity and outside are in a product of coherent states in the steady state. 19 In particular, the cavity mode is in the coherent state (8.107) where Ii> is a number state with i photons. The main effect of spontaneous emission is to make the phase of this coherent state perform a random walk which leads to a very slow phase diffusion. For times short enough compared with this diffusion time, a coherent state is a good approximation to the quantum state of laser light.
8.5 THE SINGLE-ATOM MASER
The first ASER was the ammonia-beam maser, whose idea came to Charles Townes in 1951 on a park bench [602] and was realized three years later by James P. Gordon, Herbert J. Zeiger, and Townes at Columbia University [242]. It consisted of a microwave cavity through which a beam of excited ammonia molecules was sent (Figure 8.4). At any given time there would be a large number of ammonia molecules in the cavity; otherwise, it would not mase. Incoherent processes such as cavity damping, thermal blackbody radiation, decay from the molecular energy levels of the masing transition, and the velocity spread of the molecular beam played a major role in the dynamics of the device. It took 31 years to reduce most of these incoherent processes to a negligible level and to build a much simpler device20: a maser that mases in the quantum regime of no, No 1. The first such micromaser was realized by Dieter Meschede, Herbert Walther, and GUnter MUller at the Max-Planck Institute for Quantum Optics in Garching, Germany [440]. As in the first maser, the gain medium was also a material beam that flowed across the cavity rather than something that remained inside it (Figure 8.5). But the
19Notice that, in general, a dissipative system and its environment would become entangled rather than remain in a product state through time evolution. 2OFrom the point of view of theory.
THE SINGLE-ATOM MASER
OUTPUT GUIOE INPUT GUIOE
j1dJ
CAVITY FOCUSSER
ENO X-SECTION
FOCUSSER X-SECTION
Figure 8_4 Schematic diagram of the first maser. A beam of ammonia emerges from an oven and goes into the focuser, which separates out the molecules in the lower state. As the beam enters the microwave cavity, it consists basically of molecules in the upper state of the masing transition. (Reproduced from [242], 1955 The American Physical Society.)
similarities stop there, because a micromaser, as the name suggests, is a device where a field can be build inside a cavity due to interaction of the cavity mode with only a few atoms at a time-ideally, one atom at a time, as we assume in this chapter. For atomic decay times much larger than the transit time through the cavity, there are three parameters that determine whether such a regime of operation can be achieved: the interaction time tint, which is the time an atom spends in the cavity, the interval of time between successive atoms tat' and the cavity decay time t cav . In order to have at most one atom at a time in the cavity, we must require that
(8.108)
Now, the net gain will depend on the average number of atoms crossing the cavity during its decay time t cav . This number is given by
tv, _