It doesn't seem like energy is conserved for "input-output theory" of a cavity

Question

For a long time I have struggled to make sense of "input-output theory." And unfortunately, I still cannot make any intuitive sense out of it. The key equation (found in this famous resource) is this boundary condition

$$ \alpha_\text{out}(t) = \alpha_\text{in}(t) - \sqrt{\kappa}\alpha(t) \, , $$

but immediately, this is already unclear. If for example I have a one-sided cavity that sends light into a cavity, then this suggests that what I get is a linear combination of the light that has originally entered, and destructive interference of the light in the mode of the cavity.

What this actually physically represents is not clear to me. What I would expect would be linear interference of light that was rejected from the cavity, interfering with the light that is leaking from the cavity.

$$a_{out} = R(a_{in}) a_{in}(t) + \sqrt{\kappa} a_{cavity}(t).$$

Something like this would be more sensible. $R(a_{in})$ is the reflection coefficienct of transmission function of the cavity, which is dependent on the input frequency of the light. And this interferes with some leackage light, which has some amplitude and phase.

So how exactly do I make physical sense of this weird form for the boundary condition. It has been for a long time really unclear to me what this actually represents.

Answer 1

OK, let's consider a one-sided cavity, which is given by a semi-transparent mirror and a fully reflective mirror at some distance behind the first. Such a system can be modeled with beamsplitter representing the two input ports as the opposite sides of the first mirror and the one output port of the beamsplitter directed back to one of the input ports with some phase delay. (If I have time, I draw some figures to add here).

What complicates the calculation is the implied feedback. If the initial state is parameterized by a single parameter function (such as a single photon state or a coherent state), one can perform the calculation of the parameter function for the output state in the same way one would calculate it for a classical field in this scenario. So, let's consider the classical case. Here one would solve it by setting up self-consistent equations where one of the outputs is equal to one of the inputs. It can be represented in matrix form as $$ \left[\begin{array}{c}f_{\text{out}} \\ f_{\text{cav}}\end{array}\right] = \left[\begin{array}{cc}1 & 0 \\ 0 & \Phi \end{array}\right] \left[\begin{array}{cc} C & -iS \\ -iS & C \end{array}\right] \left[\begin{array}{c}f_{\text{in}} \\ f_{\text{cav}} \end{array}\right] . $$ All the functions are spectral functions. The first matrix one the right introduces the propagation delay with transfer function $\Phi$ and the second matrix is the beamsplitter (or mirror) with $C=S=1/\sqrt{2}$ for a 50:50 beamsplitter. After multiplying out the matrices, one gets two equations, leading to a relationship between the input and output fields after eliminating the cavity field: $$ f_{\text{out}} = \frac{C-\Phi}{1-\Phi C} f_{\text{in}} . $$

If the initial state cannot be represented by a single parameter function (as for example with squeezed states and thermal states), one needs a more general approach. I've seen some people using ladder operators in the way one would use the parameter functions to come up with some way to address such situations. What it basically comes down to is a phase space-like approach where the ladder operators double for phase space variables.

Let's then do this on phase space in a formally justifiable way. (There is a formal one-to-one relationship between operators on Hilbert space and Wigner functions on phase space.) If we are only interested in the particle number degrees of freedom, the state can be represented as a Wigner function $W(\alpha)$ on a two-dimensional phase space with $\alpha$ as complex variable. However, for the current system that would be trivial because particles are not created or destroyed here. So, we are also interest in the other degrees of freedom (the spatiotemporal degrees of freedom). Then the state is represented by a Wigner functional $W[\alpha]$ and the phase space becomes a functional phase space with $\alpha(\mathbf{k})$ being a field variable. Now the effect of the beamsplitter and the phase delay gives a transformation on the field variables: $$ \alpha \rightarrow C\mu +i S\nu $$ and $$ \nu \rightarrow \Phi C\nu +i \Phi S\mu $$ where $\nu$ is the field variable for the cavity and $\mu$ is the field variable for the output state. Note that the signs are different because field variables transform with the opposite signs compared to parameter functions. Or wth the Hermitian adjoint of the unitary operations. When we again eliminate the cavity field variables, we get the output state in terms of the input state with the field variable transformation: $$ \mu = \frac{C-\Phi^*}{1-\Phi^* C} \alpha . $$ Hope this helps.

Answer 2

The reflection properties of the cavity are contained in the input-output formula via the mode parameters. Or rather, the input-output approach models the reflection coefficient as a sum of Lorentzians.

For the single mode case in the OP, the evolution of the mode comes out as

$$\dot{a}(t) = (i\omega_c-\kappa) a(t) + \sqrt{\kappa} a_{in}(t)$$

(for the case where the input/output channel is also the only leakage channel). The first term here is the free evolution of the mode according to its corresponding Master equation and the second term the driving from the external input mode.

Fourier transforming this relation gives in the frequency domain

$$a(\omega) = \frac{\sqrt{\kappa} a_{in}(\omega)}{i(\omega-\omega_0)+\kappa}$$

and similarly for the input-output equation

$$a_{out}(\omega) = a_{in}(\omega) - \sqrt{\kappa} a(\omega) \,.$$

Together, that is substituting the Master equation solution into the input-output relation, we get

$$a_{out}(\omega) = \left[1 - \frac{\sqrt{\kappa}}{i(\omega-\omega_0)+\kappa} \right] a_{in}(\omega) = R(\omega) a_{in}(\omega) \,.$$

This is something of the form the OP is looking for. The reflection coefficient therefore comes out of solving the Master equation and input-output relation as coupled equations. It does not have to be put in as an extra ingredient.

What one is effectively doing here is that one is decomposing the cavity response into modes (above a single mode). One can prove in general that input-output theory exactly reproduces scattering theory if certain modifications are included, see this paper.

(Disclaimer: there are probably some wrong signs and $i$ factors here...)

Answer 3

In a related practical experiment one can place a mirror at the output of a laser ... effectively the energy of the laser has no where to go/dissipate. We can say there is destructive interference of the field BUT not of the energy! In practice the laser will stop lasing, energy consumption is reduced to near zero. How far one can place this mirror from the source would depend on beam quality/coherence etc. The laser can also get damaged in such an experiment, in multimode laser diodes a few modes may be able to escape ... all lasing energy will tend to dump there, which can damage the crystal or exit mirror. I would think this effect would be true even for an optical cavity ... an optical cavity after a laser beam can improve beam quality. In all these cases it is the EM field that extends from the laser into the cavity that is responsible (not the energy).