One photon multimode interference in a Mach-Zehnder interferometer
In exploring the quantum mechanics of light, the intriguing phenomenon of one photon multimode interference within a Mach-Zehnder interferometer presents a fascinating case study. This post examines the subtle interplay of wave packet propagation and interference in such a setup, using Heisenberg formalism for the electric field to quantify the outcomes.
Introduction
When analyzing quantum optics experiments, particularly those involving interference, the Mach-Zehnder interferometer stands out for its clarity and versatility. Typically, this experiment involves splitting and then recombining a photon’s path, observing the resulting interference pattern to draw conclusions about the fundamental aspects of quantum mechanics.
Theoretical framework
The initial state of the system, when a single photon enters the interferometer, can be described as:
| \psi(t_0) \rangle = | \mathbf 1 \rangle = \sum_\lambda c_\lambda | \mathbf 1_\lambda \rangle
Here, c_\lambda represents the amplitudes of various modes of the photon, modeled as:
c_\lambda = \frac{Ke^{i\omega_\lambda t_0}}{(\omega_\lambda - \omega_0) - i\frac{\Gamma}{2}}, \quad k=\sqrt{\frac{c\Gamma}{L}}
where K is a normalization constant, \omega_\lambda and \omega_0 are the frequencies, \Gamma is the linewidth, and L the interaction length.
Experiment Setup
In the setup, I model the propagation and interaction of the photon across the beamsplitters and mirrors of the Mach-Zehnder interferometer. The complex amplitudes at various points in the setup can be related through the transformation matrices of the beamsplitters, which incorporate reflection (r) and transmission (t) coefficients.
Mathematical Analysis
The transformations at the first and second beamsplitters can be expressed as:
\begin{aligned} E_3^{(+)} & = r\,E_1^{(+)}(O) \\ E_4^{(+)} & = t\,E_1^{(+)}(O) \end{aligned}
and
\begin{aligned} E_5^{(+)} & = -r\,E_3^{(+)}(O^{'}) + t\,E_4^{(+)}(O^{'}) \\ E_6^{(+)} & = t\,E_3^{(+)}(O^{'}) + r\,E_4^{(+)}(O^{'}) \end{aligned}
By considering these transformations and the propagation effects (time delays due to different path lengths), the final expressions for the electric fields at the detectors are derived, leading to the calculation of photodetection probabilities.
Results and discussion
The interference pattern, characterized by variations in photodetection probability with changes in path length difference \delta L, provides deep insights into the wave-particle duality of light. The calculated probabilities exhibit the expected sinusoidal variation, reinforcing the quantum mechanical predictions.
Conclusion
Through this detailed analysis of one photon multimode interference in a Mach-Zehnder interferometer, I have demonstrated the complex yet beautiful nature of quantum mechanics. The precision in controlling and measuring quantum states in such experiments underscores the potential of quantum optics in advancing our understanding of the universe at the most fundamental levels.
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