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We can see from this equation that as the time derivative of 15 is proportional to 15 with a negative constant of proportionality (because a is positive). it is always of opposite sign to 15; the tendency is to restore the oscillator to its Vs = va/1J' steady state.
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Thus we may conclude that given a small perturbation, such as the random voltage fluctuations that are present in any electronic circuit, (8.3) will predict the buildup of oscillations from V = 0, settling at the stable steady state Vs = J a /1]' after an initial transient period. As the frequency increases, some problems begin to appear with the basic idea of the electronic circuit in Figure 8.1. First, we can no longer find a pure inductor, capacitor, and so on. There are not really any truly pure electronic components anyway. A capacitor is always a bit of an inductor and of a resistor, but at low frequencies (such as radio), most of the capacity in the circuit will be lumped at the capacitors, most of the inductance at the inductors, and so on. As the frequency gets higher and higher, however, this approximation breaks down and inductors acquire as much capacity as capacitors, and there is inductance, capacity, and resistance (impedance) all over the place instead of only in tidy little lumped elements (see, e.g., 23 of [206]). So what can we use at higher frequencies instead of our old electronic LG circuit If you have read the previous chapters of this book, you should have the answer at the tip of your tongue: a cavity. The second problem is what to use instead of the vacuum tube to amplify these higher-frequency electromagnetic fields. Charles Townes's great idea was to use atomic or molecular electronic transitions to do that. He imagined sending excited atoms or molecules through a small hole across the cavity, each of which could then decay to their ground state emitting a photon into the cavity. Although a cavity mode is equivalent to .a harmonic oscillator as an LG circuit is, a cavity has many modes. How can we make sure that the photons will go all (or at least mostly) to a single mode Conservation of energy will provide selectivity by allowing emission only in the mode that lies within the linewidth of the atomic or molecular transition. But how can this radiation generated by many independent atomic/molecular emissions be as coherent as that of a radio transmitter, which is generated by a single-dipole antenna It was Einstein who in 1905 discovered that excited atoms or molecules can emit light in two qualitatively different ways: spontaneous and stimulated emission. In spontaneous emission, the photon is emitted randomly without particular preference for any mode (see 6). In stimulated emission, however, the photon is preferentially emitted into a mode that is already occupied by other photon(s), preserving the coherence of the radiation field. Stimulated emission is a consequence of photons being bosons, as we shall see briefly next. Emission of photons into a given mode k is represented by the term proportional to al in the coupling part of the Hamiltonian, as discussed in 5. A simple Fermi golden rule argument then shows that the probability of emission into a given mode . k is proportional to 10,11'1/1)1 2 We can rewrite lall'l/l)12 as Nlakall'l/l). Using the commutator between ak and al, [ak, all = 1, which is a consequence of the photon being a boson, we find that lall'l/lW = 1 + nb where nk = ('I/Ilnkl'l/l) is the average number of photons in mode k. Thus, the probability of emission into a mode that is
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