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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

enter image description here

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 $$ \hat{U}_{QWP}(\alpha) = e^{i\Bigg[ \frac{1}{\sqrt{2}}\Big[ (1+i cos(2\alpha)) \hat{a}^{\dagger}_H + i sin(2\alpha)\hat{a}^{\dagger}_V\Big]\hat{a}_H + \frac{1}{\sqrt{2}}\Big[ i sin(2\alpha) \hat{a}_H^{\dagger} + (1-i cos(2\alpha))\hat{a}_V^{\dagger} \Big] \hat{a}_V\Bigg]} $$

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1 Answer 1

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The question isn't really sensible to me. You're trying to mix two things.

In one case, you have the unitary transformation, which is derived by knowing the properties of the object. These are "instantaneous" in the sense that you put the beam in with one state and you get the beam out in another state and don't bother about the exact time evolution. This is fine.

On the other hand you're trying to construct a Hamiltonian, which means you're considering the time evolution of the beams (or the field operators). This is a far more complicated process because it depends on the exact properties of the device, often including how it was made (e.g. different types of beamsplitters work on different basic physics). You almost never need to know how the field evolves as it propagates through the beamsplitter. If you do, you have to appeal to E&M to get the right results, and simply knowing/assuming what an "ideal black box" device does won't be adequate.

Sorry to post this as an answer, but it was too long for a comment.

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