Single photon radiation emission and photodetection rates
In my recent exploration of quantum optics, I have focused on the intricacies of single photon detection in free space, an essential concept for understanding radiation emission and detection processes at the quantum level. This blog post presents the scenario where a photon emitter is strategically positioned at the focus of a deep parabolic mirror, enhancing the radiation directionality and efficiency.
Radiation emission and normalization
The first part of my analysis deals with the emission of radiation. By placing the emitter at the focus of a parabolic mirror, the radiation emitted can be ideally considered as expanding in plane waves with vectors predominantly along the z-axis. The theoretical framework requires setting discrete values for the wave vectors \mathbf{k}_\lambda, determined by periodic boundary conditions with a length L.
A crucial aspect of my study involves the normalization of the emitted radiation spectrum. To describe a one-photon packet resulting from spontaneous emission, I use the formulation:
|\psi(t_0)\rangle = |\mathbf{1}\rangle = \sum_\lambda c_\lambda |\mathbf{1}_\lambda \rangle
where c_\lambda is the probability amplitude given by:
c_\lambda = \frac{Ke^{i\omega_\lambda t_0}}{(\omega_\lambda - \omega_0) - i\frac{\Gamma}{2}}
Here, K is a normalization constant and \Gamma represents the decay rate associated with the lifetime of the excited state of the emitter. The spectrum of the radiation is a Lorentzian, characterized by its half-width at half-maximum, \frac{\Gamma}{2}. To normalize this spectrum, I transition from a discrete summation to an integral over the continuous variable \omega_\lambda, resulting in:
\int_{-\infty}^{\infty} \frac{K^2}{(\omega_\lambda - \omega_0)^2 + \frac{\Gamma^2}{4}} \frac{L}{2\pi c} d\omega_\lambda = 1
This transformation helps determine the precise value of K, ensuring that the total probability across the spectrum sums to unity.
Photodetection at a distance
The next section of my analysis addresses the rate of photodetection at a specific distance z from the emitter. By considering the Heisenberg representation of quantum mechanics, the photodetection rate can be expressed as:
w^{(1)}(\mathbf{r}, t) = s |\mathbf{E}^{(+)}(\mathbf{r}, t) | \psi(t_0) \rangle|^2
In practical terms, this represents the probability per unit time of detecting a photon at a distance z from the point of emission, considering that the emitter was excited at time t_0. The formalism integrates the effects of wave propagation and interference, leading to an expression for w^{(1)} as a function of the delay time t - t_0. The calculated rate includes the influences of the physical constants and the quantum efficiency of the detection process.
Conclusion
The theoretical exploration detailed here underscores the nuanced dynamics of photon emission and detection in quantum optics. By simulating the conditions under which these phenomena occur, I provide a foundation for experimental setups aimed at verifying the quantum mechanical properties of light. These insights are crucial for advancements in quantum communication and computational technologies.
For more insights into this topic, you can find the details here.