# Effective Hamiltonian for Transmission Coefficient

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 $$kappa^{prime}$$?