Physics Asked by Aubrey Sharansky on January 4, 2021

Suppose we have a ring resonator. For the m-th resonant mode, its wave vector along the waveguide $beta_{m}$ satisfies $left(beta_{m}-beta_{0}right) L=2 pi m$. where L is the circumference for the ring. Assuming zero

group-velocity dispersion in the waveguide, the resonant frequency is $

omega_{m}=omega_{0}+m Omega

$,

where $Omega$ is the free spectral range of the ring.

Suppose we place in the ring a phase modulator, which has a time-dependent transmission coefficient

$$

T=e^{i 2 kappa cos (Omega t+phi)}

$$

where we choose the modulation frequency to be equal to $Omega$ and the modulation strength $kappa . phi$ is the modulation phase. In the presence of an incident wave $e^{i omega t},$ the transmitted wave has the form $T e^{i omega t},$ which to the lowest order of $kappa$ can be approximated as $T e^{i omega t} approx e^{i omega t}+$ $i kappaleft[e^{i(omega+Omega) t+i phi}+e^{i(omega-Omega) t-i phi}right] .$ I have to prove that this system can be

described by the Hamiltonian

$$

H=sum_{m} omega_{m} a_{m}^{dagger} a_{m}+sum_{m}left[2 kappa cos (Omega t+phi) a_{m}^{dagger} a_{m+1}+mathrm{H.c.}right]

$$

where $a_{m}left(a_{m}^{dagger}right)$ is the annihilation (creation) operator for the $m$ th mode.

How can one intuitively show that the transmission co-efficient couples two neighboring modes with coupling strength $kappa$? Extending this, if there were to be another term in T such that $T=e^{ileft[2 kappa cos (Omega t)+2 kappa^{prime} cos (N Omega t)right]}$, can I claim that now the m and m+N th mode will be coupled with modulation strength $kappa^{prime}$?

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