# Interaction hamiltonians of quantum optics devices

## Question:

I am trying to write the interaction Hamiltonian of common quantum optic devices (beam-splitters, half-wave plates, ...) but I am not sure I am doing it correctly. Furthermore I am finding it nearly impossible to find any reference for this.

In particular I am assuming certain transformations for these optic devices and I am trying to handcraft an Hamiltonian in order to recover these transformations, therefore I have no theoretical basis for these Hamiltonians which are constructed ad hoc.

Can someone tell me if what I am writing is correct and what would be a formally correct way to obtain these operators?

## Hamiltonians

Mirror

The transformation I am trying to recover is the following

$$\begin{cases} |H> \longrightarrow -i|H>\\ |V> \longrightarrow i|V> \end{cases}$$

the corresponding Hamiltonian would be

$$\hat{H}_{mirror} = \hat{a}^{\dagger}_V\hat{a}_V - \hat{a}^{\dagger}_H\hat{a}_H$$ and the corresponding unitary operator $$\hat{U} = e^{i\frac{\pi}{2}\big(\hat{a}^{\dagger}_V\hat{a}_V - \hat{a}^{\dagger}_H\hat{a}_H\big)}$$

Beam-splitter

I am referring to the following notation

Transformation $$\begin{cases} |H>_1 \longrightarrow |H>_2-i|H>_1\\ |V>_1 \longrightarrow |V>_2+i|V>_1 \end{cases}$$ Hamiltonian $$\hat{H}_{BS} = \theta \big(\hat{a}^{\dagger}_{1V}\hat{a}_{2V} + \hat{a}_{1V}\hat{a}^{\dagger}_{2V}\big) - \theta\big(\hat{a}^{\dagger}_{1H}\hat{a}_{2H} + \hat{a}_{1H}\hat{a}^{\dagger}_{2H}\big)$$ Unitary operator $$\hat{U}_{BS} = e^{i\theta \Big[ \big(\hat{a}^{\dagger}_{1V}\hat{a}_{2V} + \hat{a}_{1V}\hat{a}^{\dagger}_{2V}\big) - \big(\hat{a}^{\dagger}_{1H}\hat{a}_{2H} + \hat{a}_{1H}\hat{a}^{\dagger}_{2H}\big) \Big]}$$

where if the beam-splitter is 50:50 then $$\theta = \frac{\pi}{4}$$

Polarizing beam-splitter Transformation $$\begin{cases} |H>_1 \longrightarrow |H>_2\\ |V>_1 \longrightarrow i|V>_1\\ |H>_2 \longrightarrow |H>_1\\ |V>_2 \longrightarrow i|V>_2 \end{cases}$$ Hamiltonian $$\hat{H}_{PBS} = \theta\big( \hat{a}^{\dagger}_{1V} \hat{a}_{2V} + \hat{a}_{1V} \hat{a}^{\dagger}_{2V} \big)-i \theta\big( \hat{a}^{\dagger}_{1H} \hat{a}_{1H} + \hat{a}_{2H} \hat{a}^{\dagger}_{2H}\big)$$ Unitary operator $$\hat{U}_{PBS}= e^{i\theta \hat{H}_{PBS}} = e^{i\theta \big[\big( \hat{a}^{\dagger}_{1V} \hat{a}_{2V} + \hat{a}_{1V} \hat{a}^{\dagger}_{2V} \big)-i \big( \hat{a}^{\dagger}_{1H} \hat{a}_{1H} + \hat{a}_{2H} \hat{a}^{\dagger}_{2H}\big)\big]}$$

Half-wave plate Transformation $$\begin{cases} |H> \longrightarrow icos(2\alpha)|H> + i sin(2\alpha)|V>\\ |V> \longrightarrow i sin(2\alpha)|H> -i cos(2\alpha)|V> \end{cases}$$ Hamiltonian $$\hat{H}_{HWP}(\alpha) = \big[cos(2\alpha)\hat{a}_H^{\dagger}+ sin(2\alpha)\hat{a}_V^{\dagger}\big]\hat{a}_H + \big[ sin(2\alpha)\hat{a}_H^{\dagger}-cos(2\alpha)\hat{a}_V^{\dagger} \big]\hat{a}_V$$ Unitary operator $$\hat{U}_{HWP}(\alpha) = e^{i\Big(\big[cos(2\alpha)\hat{a}_H^{\dagger}+ sin(2\alpha)\hat{a}_V^{\dagger}\big]\hat{a}_H + \big[ sin(2\alpha)\hat{a}_H^{\dagger}-cos(2\alpha)\hat{a}_V^{\dagger} \big]\hat{a}_V\Big)}$$

Quarter wave-plate Tranformation $$\begin{cases} |H> \longrightarrow \frac{1}{\sqrt{2}}\Big[ (1+icos(2\alpha))|H> +isin(2\alpha)|V> \Big]\\ |V> \longrightarrow \frac{1}{\sqrt{2}}\Big[i sin(2\alpha)|H> + (1-icos(2\alpha))|V>\Big] \end{cases}$$ Hamiltonian $$\hat{H}_{QWP}(\alpha) = \frac{1}{\sqrt{2}}\Big[ (1+i cos(2\alpha)) \hat{a}^{\dagger}_H + i sin(2\alpha)\hat{a}^{\dagger}_V\Big]\hat{a}_H + \frac{1}{\sqrt{2}}\Big[ i sin(2\alpha) \hat{a}_H^{\dagger} + (1-i cos(2\alpha))\hat{a}_V^{\dagger} \Big] \hat{a}_V$$ Unitary operator $$\hat{U}_{QWP}(\alpha) = e^{i\Bigg[ \frac{1}{\sqrt{2}}\Big[ (1+i cos(2\alpha)) \hat{a}^{\dagger}_H + i sin(2\alpha)\hat{a}^{\dagger}_V\Big]\hat{a}_H + \frac{1}{\sqrt{2}}\Big[ i sin(2\alpha) \hat{a}_H^{\dagger} + (1-i cos(2\alpha))\hat{a}_V^{\dagger} \Big] \hat{a}_V\Bigg]}$$