Beam splitter for quasi-classical states
In this blog post, I will explore the fascinating realm of beam splitters and quasi-classical states, an essential part of quantum optics. Beam splitters are devices where two input modes interact and are transformed into two output modes, governed by a unitary transformation. The challenge here lies in linking the input state, | \Psi_{12} \rangle, with the output state |\Psi_{34} \rangle, through the transformation \mathbf{U}. In classical optics, this can be achieved using a 2 imes 2 symmetric matrix to describe the electric field, greatly simplifying the calculations.
The electric field transformation
The transformation can be represented as:
\begin{bmatrix} \mathbf E_3^{(+)} \\ \mathbf E_4^{(+)} \end{bmatrix} = \begin{bmatrix} r & t \\ t & -r \end{bmatrix} \begin{bmatrix} \mathbf E_1^{(+)} \\ \mathbf E_2^{(+)} \end{bmatrix}
where r and t are the reflection and transmission coefficients, respectively, obeying the condition r^2 + t^2 = 1. This transformation applies not only to the electric field but also to the destruction operators:
\begin{bmatrix} \mathbf a_3 \\ \mathbf a_4\end{bmatrix} = \begin{bmatrix} r & t \\ t & -r \end{bmatrix} \begin{bmatrix} \mathbf a_1 \\ \mathbf a_2 \end{bmatrix}
This formulation simplifies the quantum states’ description, reducing the complexity typically associated with larger matrices and state combinations.
Single photon photodetection probability
I explore the single photon detection probability at one of the output modes, say (4), which can be expressed as:
w^{(1)}(\mathbf r_4,t) = s t^2 \left[ \mathscr E^{(1)}_\lambda\right]^2 | \alpha |^2
This result holds for a semi-classical state \alpha in the input channel (1), with the vacuum state in channel (2). This expression represents the detection probability of the electric field observable and showcases the elegance of semi-classical treatments.
Field observables in output modes
The average electric field in one of the output modes can be calculated from the creation and annihilation operators. For example, for the mode (3), the field is:
\mathbf E_3(\mathbf r) = i\mathbf{e}_3 \mathscr E^{(1)}_\omega \left( r \mathbf a_1 e^{i\mathbf{k}_3 \cdot \mathbf{r}} -r \mathbf a_1^\dag e^{-i\mathbf{k}_3 \cdot \mathbf{r}} \right)
Taking the expectation value with respect to the quasi-classical input state | \alpha_1 e^{-i\omega t} \rangle, I obtain:
\langle \Psi_{12} (t) | \mathbf E_3(\mathbf r) | \Psi_{12} (t) \rangle = i\mathbf{e}_1 \mathscr E^{(1)}_\omega \left(r \alpha_1 e^{i(\mathbf{k}_1 \cdot \mathbf{r} - \omega t)} -r \bar \alpha_1 e^{-i(\mathbf{k}_1 \cdot \mathbf{r} - \omega t)} \right)
This shows that the output field behaves quasi-classically, proportional to r \alpha_1.
Quasi-classical state transformation
In a broader context, a quasi-classical state in one of the input modes results in two semi-classical states in the output channels. For example, for a coherent state | \alpha \rangle_1, the output state becomes:
| \Psi \rangle = | r \alpha \rangle_3 | t \alpha \rangle_4
This transformation illustrates the direct correspondence between the classical field and the quantum field in the semi-classical regime. The reflection and transmission coefficients r and t effectively determine the amplitude of the quasi-classical states in the output channels, providing an intuitive and clean description.
Conclusion
This method of treating beam splitters simplifies quantum optics by drawing parallels to classical fields. The output states transform in a way that mirrors classical optics, but with the added quantum mechanical operators. For those working with semi-classical and coherent states, this approach is particularly useful.
For more insights into this topic, you can find the details here.