How to Calculate Photon Detection from Output State in Quantum Optics?

[winstonboy]

TL;DR Summary

Characteristic Functions in Quantum Optics for linear amplifiers/attenuators

Hi everyone,
I am following along with the MIT OCW quantum optical communication course. I have a question about this chapter, concerning the linear attenuators and amplifiers.

Specifically, the chapter mentions that they are not going to get $\rho_{out}$, but I am interested in this.

More specifically, I am curious about the number of photons detected from the output <n|$\rho_{out}$|n>, which I know will be a function of the arbitrary input state <n|$\rho_{in}$|n>.

I am at the point where I have the antinormal characteristic function $\chi_{out}(\eta,$ $\eta^*)=\chi_{in}(\eta \sqrt{G}$, $\eta^*\sqrt{G})$ for an arbitrary gain $G>1$. How do I proceed from here to get <n|$\rho_{out}$|n>? Do I just take the operator-valued inverse Fourier transform, as suggested earlier in the course? If so, I don't know how to do this, so if someone could provide some help in this calculation for a non-coherent state I would appreciate it.

Cheers,
W

 

[eljose79]

esley

Dear Wesley,

Thank you for your question. Linear attenuators and amplifiers are important components in quantum optical communication systems, as they allow for the manipulation of the photon number in a given state. In order to calculate the number of photons detected from the output <n|$\rho_{out}$|n>, we first need to understand the relationship between the input and output states.

As you mentioned, the antinormal characteristic function $\chi_{out}(\eta,$ $\eta^*)$ for an arbitrary gain $G>1$ can be written as $\chi_{in}(\eta \sqrt{G}$, $\eta^*\sqrt{G})$. This means that the output state is related to the input state by a scaling factor of $\sqrt{G}$. Therefore, to calculate the number of photons detected in the output state, we can simply take the number of photons in the input state and multiply it by $G$.

To answer your question about the operator-valued inverse Fourier transform, it is indeed a useful tool in this calculation. For a non-coherent state, the inverse Fourier transform can be written as <n|$\rho_{out}$|n> = $\frac{1}{2\pi} \int_{-\infty}^{\infty} \chi_{out}(\eta,$ $\eta^*) e^{-(|\eta|^2+|\eta^*|^2)/2} |\eta|^2 d\eta d\eta^*$. This integral can be evaluated using techniques from complex analysis, and the result will be a function of the input state <n|$\rho_{in}$|n>.

I hope this helps in your calculation. If you have any further questions, please don't hesitate to ask.