G k ==

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e- rw {COSh (w~ + ~ sinh (wJ1'2 - n) },

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1'_ '1. =

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(8.146)

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with the dimensionless parameters rand w given by an d

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w == 2t into 9

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(8.147)

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Now suppose that l' is large enough for 1'2 always to be much larger than n. In other words, we assume that whatever steady-state field is established inside the cavity, only numbers of photons n 1'2 have a nonnegligible probability. As the field in the cavity is only being generated by the incoming excited atoms, this criterion is satisfied when N ex /2 1'2. With this assumption, we can approximate (8.146) by (8.148) G k ~ exp (

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-n; ),

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w/1', as w can be comparable to 1'. If, however, N ex nw /21' 1, and approximate (8.148) further by

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Gk~1-n2r'

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where we are neglecting terms on the order of n /1'2 and higher, but not terms of order 41' /w, we can safely take

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(8.149)

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In that case, substituting (8.149) in (8.145), we get

Pn = (

NexW)n --::tr po

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(8.150)

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THE THRESHOLDLESS LASER

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This means that for N ex <t:: 4r Iw, the photon statistics is thermal. From the requirement that the probabilities have to add up to 1, we can determine Po and then find the following expression for the average number of photons in the cavity, (8.151) This is the same average photon number that a thermal field would have if its temperature were nw 1 (8.l52) T=

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kB In (Nexwl [4r])'

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On the other hand, for N ex and then

4r I w, we can approximate 29 (8.148) by G k

Nex)n 1 2 ~po.

~ 0,

Pn= (

(8.153)

This photon probability distribution is Poissonian. So N ex = 4r Iw is the threshold of this laser. So far, our hypothetical device works like a normal laser with a well-defined threshold. But what if w 12r is so large that even for n = 1, exp( -nw 12r) is already almost zero Then the photon statistics will be Poissonian for any pump power (i.e., even for arbitrarily small values of N ex ). In other words, this device will become thresholdless. What does wl2r 1 mean Rewriting it in terms of the interaction time, the vacuum Rabi frequency, and the decay rate of the polarization, we find that this condition means that

tint

(~2)-1

(8.l54)

As 4g 2 I 11. is the spontaneous emission rate into the cavity mode derived earlier, this condition says that the interaction time must be longer than the average time required for an atom to emit a photon spontaneously into the cavity. So now we can get a physical picture of what is going on. When an atom enters the cavity, the strong coupling conditions g t;;-a~, III will ensure that all emission will be in the cavity mode. But as 11. > g, the atom can go through the cavity without emitting. Only when the transit time tint is much larger than the spontaneous emission time will the atom emit before leaving the cavity. Notice that this gives us a way of controlling {3, the fraction of spontaneous emission into the lasing mode, just by varying the transit time tint. Condition (8.154) takes us to the {3 --+ 1 regime, which, together with 4g 2 Ill. t;;-a~' makes the laser become an ideal thresholdless laser. This transition from the small {3 regime to the {3 --+ 1 regime is illustrated in Figure 8.8.

29The key assumption here is that the steady-state photon distribution will be peaked around N ex /2, so that even though the approximation G k l:::: 0 breaks down for small k, it does not matter because these low photon numbers have a very low probability anyway. The only photon numbers that have a nonnegligible probability are those near N ex /2, and for those the approximation Gk l:::: 0 holds well.

THE MASER, THE LASER, AND THEIR CAVITY QED COUSINS

8 6 v 4 '" 2 b 0

s:::

0 16 32 10-3 Nex

.......

'" 8

25 20 15

.- ....... _-.........

_- --

.......

5 0 0

10 20 30 40 50