# Detailed derivation of the Input-Output theory

#### Basic definitions

Think of a system consisted of:

• A long waveguide with length accommodating a photon field .
• A small resonator at accommodating an electric oscillation .

that has the RWA Hamiltonian (5.60):

--- (5.60)

where we use the unit system with . So, the momentum and the wavenumber are the same . (In general .)

The wavelength of the photon field is numbered as . So, the wavenumber is counted as:

Hence, assuming that , we can replace .

Applying it to the last term of , we have:

So the operator has a scale dependency as . From the second term of , the coupling constant has the same scale dependency

Now we define as a scale independent spectral function. Then, we have the relationship (5.62):

--- (5.62)

Also, in the following discussion, we use the Fourier transformation of the Heaviside step function :

--- (0)

Note that (0) is a relationship as a hyperfunction. So it's valid only when they are combined with the operation .

#### Formal solution of the photon field

From (5.60), we have the following Heisenberg equations (B.22):

--- (1)

--- (2)

Define as with the initial condition . Then, from (2), we have:

--- (B.23)

#### A Langevin equation for the resonator

By substituting (B.23) into (1), we have:

--- (5.61)

where .

#### Markovian approximation

We assume that the resonator resonates with the photon field with some frequency with a slow modulation .

and the contribution to the integration in (5.61) is dominated by the oscillation . So,

In the limit , we have:

( Contribution from around is dominant.)

--- (B.29)

where,

--- (B.30)

By combining these results, we have:

--- (B.26)

And from (5.61), we have:

From this result, you can see that oscillates as . So,

.

This means that is (implicitly) decided from the consistency condition with (B.30):

#### Solution of the resonator

Now we have the Langevin equation for the resonator:

--- (3)

--- (4)

Define as . Then from (3), we have:

Hence, the solution is:

--- (B.32)

The constant is determined with the initial condition.

From this result, you can see that the contribution from coupling is dominant around the resonant frequency . So changing the coupling strength outside doesn't change the behavior of the system, and we can safely assume that is almost constant. In terms of , this means that

Then, compared to the free plane wave solution , is rescaled with .

So, we define to have a renormalized operator, and (3) becomes:

--- (5.64)

Note that in the discussions above, we used the initial condition at . As a result, corresponds to the input photon generated at . So we named to indicate the input photon.

#### Input-Output relations

It's also possible to consider the boundary condition at in the Markovian approximation. In this case, the calculation to get (B.29) becomes:

--- (B.29)'

Note that we used the fact that is an even function and is an odd function.

Compared to (B.29), (B.29)' has an opposite sign for , and (5.64) becomes:

--- (5.64)'

where corresponds to the photon field pulling back to the current time from the final status at .

By equating (5.64) and (5.64)', we have:

--- (B.38)