Collimated quasi-classical wavepackets and photodetection
In recent developments within quantum optics, the study of quasi-classical states presents fascinating insights into the behavior of wavepackets under specific conditions. In my latest research, I’ve focused on a particular scenario where a beam propagates along the x-axis, characterized by a sufficiently large transverse surface S such that diffraction effects over the length L are negligible. This setup allows us to treat the modes of the wavepacket as plane waves that are linearly polarized in the y-direction.
Theoretical Foundation
The modes of my beam can be described using periodic boundary conditions, defined mathematically by:
\mathbf{k}_\lambda = 2\pi \frac{n_\lambda}{L} \mathbf{e}_x = \frac{\omega_\lambda}{c}\mathbf{e}_x \quad n_\lambda \in \mathbb{N}
Here, n_\lambda represents the mode number, a critical factor in determining the mode’s contribution to the overall wavepacket. This wavepacket, in turn, is constructed as a tensor product of modes centered around a frequency \omega_\lambda:
|\Psi_{\text{qcwp}} \rangle = \prod_\lambda |\alpha_\lambda e^{-i\omega_\lambda t} \rangle
The coefficients \alpha_\lambda are crucial for understanding the distribution of the photons within the wavepacket. They are given by:
\alpha_\lambda = \frac{Ke^{i\omega_\lambda t_0}}{(\omega_\lambda - \omega_0) - i\frac{\Gamma}{2}}
where K is determined by the normalization condition that ensures the total number of photons follows a Lorentzian distribution, represented as:
\sum_\lambda |\alpha_\lambda|^2 = \frac{K^2 L}{c \Gamma} = \langle \mathbf N \rangle
Practical Implications and Calculations
Moving from theoretical constructs to measurable quantities, the classical form of the electric field related to this wavepacket can be calculated, leading to the following expression:
\langle \mathbf E \rangle (\mathbf r,t) = \sqrt{\frac{\hbar \omega_0\Gamma}{2\varepsilon_0 cS} \langle \mathbf N \rangle} H( au)e^{-\frac{\Gamma}{2}\tau}e^{i\omega_0\tau}
Here, au = t - t_0 - \frac{x}{c} represents the time variable adjusted for the propagation delay. The classical field’s magnitude directly contributes to the photodetection probability, which can be experimentally verified. This probability is calculated as:
w^{(1)}(\mathbf{r}, t) = \eta \frac{\Gamma}{S}H(\tau)e^{-\Gamma au}
where \eta symbolizes the efficiency of the photodetector.
Experimental Considerations and Quantum Mechanics
When exploring the implications of these theoretical insights, I conducted multiple experiments to validate the theoretical predictions. A notable result was the verification that the probability of double photodetection, contrary to classical expectations, is not null even for low average photon numbers:
w^{(2)}(\mathbf{r}_1, t_1, \mathbf{r}_2, t_2) = w^{(1)}(\mathbf{r}_1, t_1) w^{(1)}(\mathbf{r}_2, t_2) \neq 0
This result not only confirms the correctness of the quantum optic computations but also reinforces the quantum nature of light under conditions where classical physics would predict no correlation.
Conclusion
Through this research, I’ve demonstrated how classical and quantum perspectives can converge in the study of light, providing profound insights into the nature of photodetection and wavepacket behavior. The application of mathematical rigor paired with experimental verification forms the cornerstone of my approach to uncovering the nuanced dynamics of quasi-classical wavepackets.
For more insights into this topic, you can find the details here.