Beam splitter analysis for multimode quasi-classical states
In my exploration of beam splitters, I’ve examined how they interact with multimode quasi-classical states, with a particular focus on the dynamics of single and double detection probabilities in output channels. This analysis leverages the Heisenberg formalism and aims to provide an insightful discussion on the theoretical and practical implications of photon behavior in beam splitters.
Theoretical background
Beam splitters play a critical role in quantum optics, serving as fundamental components that manipulate the paths of photons. In my analysis, the setup involves two input modes (1) and (2) and two output modes (3) and (4), with a unitary transformation linking the input state |\Psi_{12}\rangle to the output state |\Psi_{34}\rangle. The input state I consider is:
|\Psi_{12}\rangle = |\Psi_{ ext{qcwp}}\rangle_1(0) |\mathbf{0}\rangle_2 = \prod_\lambda |\alpha_\lambda\rangle_1 |\mathbf{0}\rangle_2
This configuration includes a quasi-classical wavepacket in input mode (1) and the vacuum in input mode (2).
Analysis of detection probabilities
The transformation of the electric field operators from the input to the output can be expressed 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}
The probabilities of single photon detections in the output channels are derived by considering the intensity of the transformed electric fields. For example, the detection probability in output channel (3) is:
P_3 = \iiint w^{(1)}(\mathbf{r}_3, t) , dS , dt = |r|^2 \eta \langle \mathbf{N} \rangle
Similarly, for output channel (4):
P_4 = \iiint w^{(1)}(\mathbf{r}_4, t) , dS , dt = |t|^2 \eta \langle \mathbf{N} \rangle
These results elucidate how the characteristics of the beam splitter influence the distribution of photons between the two output modes.
Quantum mechanical considerations
Using quantum optics, I consider the state where two photons are injected into input (1) and the vacuum in input (2). The output state |\Psi_{34}\rangle is influenced by how the creation operator for channel (1) is expressed in terms of the output channels:
|\Psi_{34}\rangle = \frac{(r \mathbf{a}_3^\dagger + t \mathbf{a}_4^\dagger)^2}{\sqrt{2}} |\mathbf{0}\rangle
This reveals the probabilities of different configurations of photon distribution in the output, which include cases where both photons are in one channel, both are in another, or one is in each:
\begin{aligned} & P_{33} = |r|^4 \\ & P_{44} = |t|^4 \\ & P_{34} = 2|r|^2|t|^2 \end{aligned}
These insights help quantify the behavior of photons through beam splitters and underscore the nuances of quantum statistics that differ from classical expectations.
Conclusion
The analysis of beam splitters with multimode quasi-classical states and two-photon inputs provides a comprehensive understanding of photon interaction dynamics. It highlights the quantum mechanical principles governing photon distributions and their experimental validations. For more insights into this topic, you can find the details here.