Mach-Zehnder interferometer: quantum optics approach
In my previous analysis of classical optics approaches, I examined the behaviors of fields and interference within various optical setups. This foundation allows for a seamless transition into the quantum realm, particularly when considering single photon inputs in a Mach-Zehnder interferometer. The full details of the classical approach can be found in my earlier work here.
Quantum approach
By considering the quantum state of a single photon in mode (1) and no photon in mode (2):
| \Psi_{12} \rangle = | \mathbf 1 \rangle_1 | \mathbf 0 \rangle_2
I adapt the semi-classical equations for field propagation through the interferometer, introducing field operators that replace classical fields:
\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}
The propagation between points O and O’ remains consistent with classical mechanics:
\begin{aligned} \mathbf E_3^{(+)}(O^{'}) & = e^{ikL_3}\mathbf E_3^{(+)} \\ \mathbf E_4^{(+)}(O^{'}) & = e^{ikL_4}\mathbf E_4^{(+)} \end{aligned}
Analysis of detection rates
At the second beam splitter, the transformations lead to the following expressions:
\begin{aligned} \mathbf E_5^{(+)} & = -r\,e^{ikL_3}\mathbf E_3^{(+)} + t\,e^{ikL_4}\mathbf E_4^{(+)} \\ \mathbf E_6^{(+)} & = t\,e^{ikL_3}\mathbf E_3^{(+)} + r\,e^{ikL_4}\mathbf E_4^{(+)} \end{aligned}
Applying these operators to the initial quantum state, only \mathbf E_1^{(+)} contributes, due to the absence of photons in mode (2):
\mathbf E_2^{(+)} | \Psi_{12} \rangle = 0
The remaining terms for detection at outputs 5 and 6 become significant only due to their dependence on \mathbf E_1^{(+)}. The calculation of the detection rate illustrates the role of quantum mechanical principles even in scenarios typically analyzed using classical optics.
Conclusion
My analysis reveals that the quantum description of light in an interferometer, while rooted in complex quantum mechanics, mirrors classical predictions. The observed interference patterns reaffirm the classical results, underscoring quantum mechanics’ predictive power even in single-photon experiments.
For more insights into this topic, you can find the details here.