Semi-reflecting mirror: Quantum optics approach
I analyze this setup with a semiclassical approach here; quantum optics provides a rich framework for understanding the behavior of light at the quantum level, particularly in devices like beam-splitters which are pivotal in experiments involving light interference and entanglement. In this post, I explore the quantum mechanical treatment of a beam-splitter used for photo detection, detailing the transformation of states and the computation of detection probabilities through operator methods.
Beam-Splitter Model and State Transformation
A beam-splitter, or semi-reflecting mirror, can be modeled in quantum optics by considering how it transforms input light states into output states. The essence of this device’s operation can be encapsulated by the unitary transformations it imposes on the state vectors and operators associated with the light modes.
For two input modes labeled as (1) and (2) and two output modes as (3) and (4), the transformation can be expressed through the scattering matrix \mathbf{S}, which relates the input field operators to the output field operators:
\begin{aligned} \mathbf{E}_3^{(+)} & = r\,\mathbf{E}_1^{(+)} + t\,\mathbf{E}_2^{(+)} \\ \mathbf{E}_4^{(+)} & = t\,\mathbf{E}_1^{(+)} - r\,\mathbf{E}_2^{(+)} \end{aligned}
Here, r and t represent the reflectivity and transmissivity of the beam-splitter, respectively, and all the electric fields are operators.
Photo Detection Calculations
When considering the detection of photons, the output observable \mathbf{O}_o, related to the detection of photons at output mode (4), is given by the product of the negative-frequency and positive-frequency components of the electric field:
\mathbf{O}_o = E_4^{(-)}E_4^{(+)}
The transformation of this observable back to the input space using the transformation matrix \mathbf{S} allows us to compute the expected detection rates directly from the input state properties. For a state with one photon in mode (1) and none in mode (2):
| \Psi_{12} \rangle = | \mathbf{1} \rangle_1 | \mathbf{0} \rangle_2
The probability of detecting a photon at output (4) is calculated as:
w^{(1)}(\mathbf{r}_4,t) = s \left[ \mathscr{E}^{(1)}\right]^2 t^2
where s is a scaling factor related to the detector efficiency and \mathscr{E}^{(1)} represents the strength of the electric field associated with a single photon.
Theoretical implications and quantum coherence
These calculations underline the quantum nature of light and its coherence properties in a beam-splitting scenario. Notably, the process preserves the quantum commutation relations, ensuring that the transformation is physically valid. The derived expressions for the detection probabilities also allow comparisons between classical predictions and quantum mechanical results, highlighting significant differences especially in cases involving multiple photon detections.
For example, while a semiclassical model might predict a non-zero probability for double detection events (a photon being detected twice), quantum mechanics rigorously shows that such probabilities are zero due to the annihilation of the photon state upon measurement. This underlines a fundamental aspect of quantum mechanics where measurement affects the state being measured.
Conclusion
The study of quantum optics, particularly through the lens of operator transformations and photo detection in beam-splitters, offers profound insights into the fundamental nature of light and quantum mechanics. This analysis not only aids in conceptual understanding but also has practical implications in designing experiments and interpreting their outcomes in quantum information science and optical communication technologies.
For more insights into this topic, you can find the details here.