Photodetection rates in quasi-classical states
In the realm of quantum optics, understanding the interaction between light and matter is crucial, particularly when dealing with the detection of photons. Utilizing the Heisenberg formalism, I explore the computation of photodetection rates in quasi-classical states. This approach allows the state of radiation to remain constant, represented by the initial state \alpha_\lambda, while the operators, specifically the electric field operator, evolve with time.
The Heisenberg picture of electric field evolution
The electric field operator \mathbf{E}(\mathbf{r}, t) in the Heisenberg picture is given by:
\mathbf{E}(\mathbf{r}, t) = i\mathbf{e}_\lambda \mathscr E^{(1)}_\lambda \left(\mathbf a_\lambda e^{i(\mathbf{k}_\lambda \cdot \mathbf{r} - \omega_\lambda t)} + \mathbf a_\lambda^\dag e^{-(i\mathbf{k}_\lambda \cdot \mathbf{r} - \omega_\lambda t)} \right) = \mathbf{E}^{(+)}(\mathbf{r}, t) + \mathbf{E}^{(-)}(\mathbf{r}, t)
This represents the classical field with the complex amplitude replaced by the quantum operators \mathbf a_\lambda and \mathbf a_\lambda^\dag.
Positive electric field component
Considering only the positive component \mathbf{E}^{(+)}(\mathbf{r}, t), which involves the destruction operator, we have:
\mathbf{E}^{(+)}(\mathbf{r}, t) = i\mathbf{e}_\lambda \mathscr E^{(1)}_\lambda \mathbf a_\lambda e^{i(\mathbf{k}_\lambda \cdot \mathbf{r} - \omega_\lambda t)}
For a quasi-classical state \alpha_\lambda, the action on this state by \mathbf{E}^{(+)}(\mathbf{r}, t) yields an eigenvalue equation, with the eigenvalue being the mean of the electric field.
Photodetection signal calculation
To calculate the photodetection signal w^{(1)}(\mathbf{r}, t), we consider:
w^{(1)}(\mathbf{r}, t) = s \left|\mathbf{E}^{(+)}(\mathbf{r}, t)\right|^2
This expression evaluates the magnitude squared of the electric field, reflecting the classical analogy where the field’s strength directly influences the detection probability.
Double detection signals
Furthermore, for double detection events at points (\mathbf{r}_1, t_1) and (\mathbf{r}_2, t_2), the detection rate w^{(2)}(\mathbf{r}_1, t_1, \mathbf{r}_2, t_2) is determined by:
w^{(2)}(\mathbf{r}_1, t_1, \mathbf{r}_2, t_2) = s^2 \left|\mathbf{E}^{(+)}(\mathbf{r}_1, t_1)\right|^2 \left|\mathbf{E}^{(+)}(\mathbf{r}_2, t_2)\right|^2
This rate is a product of the individual detection probabilities at each point, showcasing that even in weak quasi-classical states, where the average photon number is less than one, non-trivial detection patterns emerge.
Conclusion
Through this analysis, I have shown that the Heisenberg formalism provides a robust framework for calculating photodetection rates in quasi-classical states. The method not only simplifies the evaluation by treating the electric field classically but also reveals subtle quantum mechanical effects in photon detection processes. For more insights into this topic, you can find the details here.