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Equations (9.50) and (9.54)-(9.56) agree with Barnett and Radmore's expressions [37] for 'coupling strength 12:
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So we have derived from first principles the phenomenological Gardiner-Collett Hamiltonian (9.1), have demonstrated that it holds only for high-Q cavities, and have found the reason why it breaks down away from the high-Q regime. Moreover, we have derived an explicit expression for the coupling strength within our simple one-dimensional model. Let us look into what this expression tells us. As the cavity photons can only escape to the external continuum of modes, there is really just a single reservoir (this external continuum) for all cavity modes. In quantum optics, however, it is often tacitly assumed [261] that each mode is coupled to an independent reservoir. The explicit expression (9.57) for the coupling strength that we have found shows how this effective independent reservoir behavior emerges from a single reservoir in the high-Q regime. The sinc function in (9.57) splits the continuum of external modes into a collection of smaller reservoirs, each having a finite frequency width and coupled to a single-cavity mode. Their independency arises from the smallness of the overlap of a given cavity mode with neighboring cavity resonances in the high-Q regime (see Figure 9.3). Grangier and Poizat [247, 248] have shown that the breakdown of the independent reservoir assumption is essential for the appearance of excess quantum noise in lasers. This is a curious phenomenon first found in gain-guided semiconductor lasers by Petermann [491], where a laser has a linewidth larger than the usual SchawlowTownes Iinewidth, as if there were more than one spontaneous emission noise photon per mode. Later, Siegman developed a semiclassical theory [561, 562], where excess noise emerges as a consequence of mode nonorthogonality. Siegman showed that for those lasers the perfect cavity mode expansion of the radiation field breaks down and one must use instead the actual modes of the open laser cavity, which are nonorthogonaL Thus, the breakdown of the Gardiner-Collett Hamiltonian, the need to adopt the
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12Notice that unlike the constant coupling strength advocated by Barnett and Radmore [37] and Gardiner and Collett [225], among others, the frequency-dependent Vn(k) given by (9.57) does not make the phase shift integral in the Fano diagonalization method diverge. Moreover, (9.57) is practically flat (constant) within the cavity resonance peaks but falls off between peaks.
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THE RADIATION CONDITION
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natural cavity modes rather than perfect cavity modes, and the emergence of excess noise in lasers are all connected.
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Figure 9.3 Here we show the coupling strength Vn (k) in relation to the cavity resonances. The thin line is a plot of 1 (k)12; the peaks show the cavity resonances corresponding to k o, kl, and k2. The thick dotted line is a plot of sin2 ([k - kIJL)j(k - kI)2 properly scaled to appear in the figure. Notice how VI (k) is practically flat inside the n = 1-mode Iinewidth and falls off as it approaches the next modes. We have used an energy reflectivity Irl2 of 96% (i.e., = 0.2).
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Another interesting feature of this derivation is that even though we have not adopted the rotating-wave approximation, there are no counterrotating terms in (9.56). This absence of counterrotating terms is a consequence of a n 2(k) and i32(k,k') vanishing in the high-Q approximation. It guarantees that the master equation derived from (9.56) is of the Lindblad form [411], as (6.68), derived in 6.
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9.2 SOMMERFELD'S RADIATION CONDITION
Sommerfeld pointed out that for wave motion in an infinite medium, the usual boundary conditions at the finite surfaces involved in the problem are not enough to determine the solution uniquely. He was then led to propose restrictions on the behavior of the wave at infinity to guarantee a unique solution. These restrictions, known as Sommeifeld's radiation condition, amount to an asymptotic boundary condition at infinity. Sommerfeld's argument (see Sec. 28 of [571]) is this: Consider a spherical cavity of radius R and a scalar field ' whose boundary condition is that it vanishes on the surface of the sphere. Then for spherically symmetric radial oscillations, the eigenfunctions ' n of the Helmholtz equation (9.58)