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Figure 8.3 shows how the steady-state photon number and its fluctuations behave as we vary the rate at which energy is fed into the laser (Le., the pump parameter A). The plot on the left shows the typical behavior of a laser (class A). For A < K/ /3, the laser does not oscillate. In this regime, spontaneous emission is dominant and the laser behaves like an ordinary thermal source of light. For A > K/ /3, the laser oscillates and the steady-state photon number increases linearly with the pump. The region around A = K/ /3, which separates the nonoscillatory from the oscillatory regime, is

called the laser threshold. As the laser crosses its threshold, the fluctuations in the photon number peale. This is the signature of a phase transition, as when water boils. In fact, there is an analogy [147] between the one-photon laser phase threshold and a second-order phase transition. 16 The laser is in this sense a macroscopic device, operating with a large number (thermodynamic limit) of atoms (or molecules) and photons. As /3 becomes larger, however, this analogy begins to break because the very notion of a laser threshold starts losing its meaning [518]. Large /3 can tum a laser into a microscopic device operating with only a few photons, for which the thermodynamic limit of a large number of particles no longer applies. The plot on the left in Figure 8.3 illustrates this point. You can see the kink at M = 1 in the photon number curve become smoother and start disappearing. The peak in the Fano parameter curve also becomes wider, and this curve starts becoming flat for large M. But this is all that this simple rate equation analysis can reveal. We cannot go to the limit of /3 --t 1 here, because the rate equations break down well before this limit. We will go back to this problem later using a fully quantum theory. 1.2

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To create a Subtotal Item, follow these steps: 1. Go to Lists Items. Click the Item button and choose New. 2. Enter the Item name.

The Pretoria metro region is host to over 2 million people, and Johannesburg is home to another 24 million, according to the United Nations A few biotech companies are also located in Cape Town, with a population of over 3 million, located on the south east tip of the country Aside from an established chemical industry and an allegedly exten sive biological weapons development program, the South African biotech industry can best be characterized as early stage For example, the Nature Biotechnology Directly lists only three biotech companies in South Africa, and as of 2003, there was only one biotech venture fund in the country Furthermore, the level of venture capital investment is on the order of $1 million per deal The government is reportedly interested in establishing biotech incubators, such as the Cape Regional Biotech nology Innovation Center, which received about $14 million from the government in 2003.

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Figure B.3 The solid line represents the steady-state photon number and the dashed line the Fano parameter f == (8n 2 ) / n s , a good measure of the fl uctuations in the photon number. Both ns and f are plot against the normalized pump parameter M == {JA/K. On the left plot we used a typical spontaneous emission factor of {J rv 10- 5 On the right plot we used {J rv 0.05. On both plots, we assumed that K 'YII' so that No/no 1.

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