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In the quantum optics literature this simple Hamiltonian is called the Gardiner-Collett Hamiltonian owing to its appearance in the two famous papers [109,225], where their input-output approach to damped quantum systems was developed. 3 From (9.1), one can deduce (see Problem 9.1) the Lindblad master equation for high-Q cavity damping (6.68) introduced in 6. This is fine as a first approximation that holds well for high-Q cavities, where as we have seen in Section 6.2, the photon round trip-time in the cavity is much shorter than the average time it takes to escape to the outside (lifetime). But as soon as the cavity Q decreases a little, this approximation starts to break down [37, 43, 228, 390, 391, 408, 603]. An exact solution can be obtained by expanding the fields in terms of the normal modes of the entire closed system formed by the cavity and the outside, as we did in Section 2.5.2. This is called the modes-of-the-universe approach. Unfortunately, these modes can only be calculated for the simplest of structures. Moreover, as they are modes of the entire system, a photon in one of these modes of the continuum is completely delocalized. What one would like is an exact treatment where cavity modes were singled out explicitly. Then a deeper conceptual question also arises: What is a mode of an open cavity This is the main theme of this chapter.
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2That is. annihilating a photon in one of the outside continuum of modes and creating a photon in one of the discrete cavity modes. 3To be more precise. the Gardiner-ColIett Hamiltonian has a generic coupling constant that can be different for each pair of coupled modes.
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In quantum electronics, this question came to the forefront with the invention of the laser. The first masers used closed4 microwave cavities whose modes are calculated as in Appendix A. Any absorption at the cavity walls or transmissivity was modeled by assuming that there was an absorptive medium distributed homogeneously inside the cavity. The delocalization of absorption inside the cavity allows use of the Gardiner-Collett approach. 5 These closed cavities had to have dimensions comparable to the wavelength of the maser (centimeter) for the various cavity modes to be much farther apart from each other in frequency than the bandwidth of the molecular beam resonance. Otherwise, more than one frequency (cavity resonance) would be amplified and the maser would not be monochromatic. As the wavelength was reduced from microwaves (centimeter, millimeter) to the optical regime (nanometer), it was no longer possible at the time (late 1950s) to make a closed cavity of dimensions of such minute wavelength. The great idea that was put forward independently by Schawlow and Townes, Gordon, and Prokhorov was to effectively reduce the number of modes by selectively increasing the losses of all but one or just a few of them. This selective increase of losses would be done by removing the sidewalls of the closed cavity, transforming it into a Fabry-Perot. So the Fabry-Perot, which started life in 1899 [187] as an interferometer (see Figure 9.1), was now used as a resonator for the first time. 6 At that time, as Bennett [47] wrote: "It was not entirely obvious that normal modes would even exist in these open-walled structures .... " But as the citation in the prologue to this chapter suggests, open cavities have modes, too. Anyone who has experienced echo in a valley knows that intuitively. In quantum electronics this was clarified by the work of Fox and Li [220,221], Valnshteln [605-607], and others. This, however, is yet another example of an important early result being forgotten by the scientific community, only to be rediscovered much later. Long before Fox, Li, and Valnshteln tackled the problem of modes of open cavities, Joseph John Thomson, discoverer of the first subatomic particle, the electron, had already shown how to define and calculate such modes [594,595]. We discuss Thomson's ideas in Section 9.2. In quantum optics, the problem of open cavities is deeper than in the usual semiclassical approach of laser physics. Any modes used to expand the quantized electromagnetic field must form a complete set. Unfortunately, the usual Fox-Li or Valnshteln modes do not form a complete set (see, e.g., Sec. 63 p. 330, and Probs. 10.4 and 10.5 of [607]), due to adoption of the Sommerfeld radiation condition (discussed in Section 9.2), which excludes incoming radiation from infinity. In a semiclassical treatment, this is a problem only when excitation of the resonator by an external field is considered (see, e.g., Chap. 10 of [607]). With the quantized radiation field,
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4They had a small hole for the molecular beam to enter. But as Sommerfeld showed, as long as the dimensions of the hole are small in comparison with the wavelength of the maser, leakage of radiation through the hole is negligible. It is not so for optical frequencies. 5As mentioned in Section 6.2, when the photon round-trip time is much shorter than its lifetime in the cavity, the losses appear to be distributed evenly throughout the cavity rather than being concentrated at the semitransparent walls. 6 As far as the author knows.
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