# Derivation of Input-Output relations for waveguide-QED

#### Target system

The system is similar to the following one.

Here we restrict the states with only one excitation (either qubit or a propagating photon).

--- (7.13)

The Heisenberg equations are:

--- (7.14)

Using the same argument to derive (B.23) in the above article, we have the formal solution for as:

--- (1)

#### Position space

Wavefunction of the photon field is defined as:

We included so that the wavefunction is scale-independent.

We split it into right-moving part and left-moving part as:

By substituting (1), we have:

--- (7.17)

where represents a group of plane waves corresponds to input () or output () photon beam. Note that in some locations, they are just formal expressions. For example, corresponds to a right-moving output beam (before reaching the resonator) that doesn't exist in reality. It's a formal extension of the real output beam in to .

#### Approximate calculation

We will apply approximation to the last term in (7.17) with the following assumptions:

The last one is the same as in this. Now,

--- (2)

As the integration with will be dominant around ,

• If the integral interval of contains , we replace with , and extend the integral interval of to to .
• If not, we approximate as .

The followings are the cases that result in . They corresponds to the following trivial cases.

• and : In this case, is an input beam before reaching to the resonator. So it's naturally the same as .
• and : In this case, is an output beam leaving the resonator. So it's naturally the same as .
• and : In this case, is an input beam before reaching to the resonator. So it's naturally the same as .
• and : In this case, is an output beam leaving the resonator. So it's naturally the same as .

Note: Strictly speaking, is different from the asymptotic plane waves . In this rough approximation, we simply drop this information. Instead, we extract that information from . In other words, we shouldn't use the trivial cases above to extract any information about the photon-resonator interaction.

In other cases, (2) represents the difference of the beam before and after interacting with the resonator. In the following calculation, we implicitly exclude the trivial cases above.

For the case ,

For the case ,

So we have the following non-trivial cases:

where

#### Input-output relations

From the equation:

we have:

Using , this can be written as:

--- (7.20)