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11 Perfect here is used just to indicate that these are the modes obtained when the semitransparent mirror is regarded as if it were a perfectly reflecting mirror.
using (9.38). However, we must not forget that the perfect intracavity and free-space modes do not describe correctly, beyond first order in E, the field configuration on the boundary formed by the semitransparent mirror. Because of this single point, the perfect mode projections that define an, a~, b(k), and bt(k) will miss some of the global a(k) and at(k). In other words, the an, a~, b(k), and bt(k) are not enough to cover the entire continuum of Fock spaces spanned by the global operators a(k) and at (k). Something is missing. Thus, (9.39) does not hold in general, only up to first order in E. As the breakdown of (9.39) beyond first order in E is a key point in our firstprinciples derivation of the Gardiner-Collett Hamiltonian (9.1), let us examine it a bit more closely. Let us call
the continuum part of a(k) according to (9.39), and
an(k) ==
'E {a nl(k)an +an2(k)a~}
[aD(k), ab(k')]
the discrete part. Then if (9.39) holds,
[a(k), at(k')]
+ [ac(k) , a~(k')].
Using (9.42) and (9.43), we find that
[ac(k), a~(k')] = 8(k - k')
1 -t -k . 1 + - \1 + r \ ' e-''kL .c*(k) coskL 0-+0+ k' - k + t'8 hm 7r 1 -t1 + - 11 + r I -k e ik'L , (k') cos k, . k - k' - t.8 L 0-+0+ hm 7r
~ 11 ~ r 12 ..Jkkiei(k'-k)L.c*(k) '(k') cos kL cosk' L
III. I} x { PkPF+2Pk'o~W+k'-k+i8'
Using (9.40) and (9.41) yields
[aD(k), ab(k')]
.c*(k) '(k')P k _ k' sinkLcosk' L .c*(k) '(k') k
x { _ e
sink' L cos kL } sin kL sin k'L. (9.48)
Now then. in general. (9.46) is not verified. But for E 1. up to first order in E. we find that (k) 1 ~ -E/2VL (9.49) ~ VL ~ k - k n + iE2 / 4L and
anl(k)~ ,fir
( -1)n
-iE2 /4L'
Under this approximation. [ac(k) and (9.52) From (9.51) and (9.52), we find that (9.46) does hold for In an analogous way, we can show that [a(k), a(k')J
at (k')] ~ 8(k -
k') _
.!. ~
~ k - kn -
iE2 /4L
k' - k n + iE2 /4L
1 up to first order.
[aD(k), aD(k')]
+ [ac(k), ac(k')]
also holds up to first order. Then (9.39) is valid up to first order and our expansion into perfect modes makes sense. but only up to first order (see also Problem 9.8). To go beyond the high-Q regime and be able to deal with larger losses. we have to use modes that satisfy the exact boundary conditions at the semitransparent mirror. These are the natural modes of the cavity and. unlike perfect modes, they are non orthogonal. We discuss natural modes in the rest of this chapter. But now let us determine the Hamiltonian and the coupling within this high-Q approximation. As mentioned before. (9.39) does suggest a parallel with the dressed operator version of Fano diagonaIization [38]. In the Fano diagonaIization method, however. the global operators a( k) and at (k) that diagonalize the Hamiltonian are the unknowns that we wish to determine, whereas the an, a~. b(k), bt(k) together with the nondiagonal Hamiltonian that couples then are known. In contrast, here we have the opposite. a(k) and at (k) and the diagonal Hamiltonian are known. while the an. a~. b(k). bt(k) and their nondiagonal Hamiltonian are not. So here we wish to do the reverse of the Fano diagonalization that was done in [37]. Using (9.39) in (9.17), we can write the Hamiltonian in terms of the operators an, a~, b(k). and bt(k) instead of a(k) and at(k). As we have seen above, this will be correct only up to first order in E. SO we should use the approximate expressions (9.50) for anl (k) and a n2 (k). as well as (9.54)