Multimode quasi-classical states and photon distribution
In exploring the dynamics of quantum systems, it becomes essential to understand the behavior of multimode quasi-classical states. These states are crucial in describing systems where multiple quantum states are entangled across different modes. The mathematical representation of a multimode quasi-classical state is a tensor product of quasi-classical states across these modes:
|\Psi_{qc} \rangle = | \alpha_1 \rangle \otimes | \alpha_2 \rangle \dots \otimes | \alpha_\lambda \rangle = \prod_\lambda | \alpha_\lambda \rangle
This expression allows us to analyze the system in a holistic manner where each mode evolves with its unique dynamics. Importantly, each mode evolves as e^{-i\omega_\lambda t}, which leads to a time-dependent state:
|\Psi_{qc} \rangle(t) = \prod_\lambda | \alpha_\lambda e^{-i\omega t} \rangle
In my analysis, I also explore the generalization of single-mode results to multimode scenarios. This includes examining the state’s evolution and its interaction with an electric field, which can be represented as an eigenstate of the positive frequency of the electric field observable \mathbf{E}^{(+)}_{\text{classical}}(\mathbf{r}, t):
| \Psi_{qc} \rangle = e^{-\frac{\sum_\lambda|\alpha_\lambda|^2}{2}}e^{\sum_\lambda \alpha_\lambda \mathbf a_\lambda^\dag} | \mathbf 0 \rangle
Here, \mathbf a_\lambda^\dag denotes the creation operator for the mode \lambda. The detection rates for single and double photon events mirror those expected from a classical field, which reinforces the classical-like behavior of these quantum states:
\begin{aligned} & w^{(1)}(\mathbf{r}, t) = s \left|\mathbf{E}^{(+)}_{\text{classical}}(\mathbf{r}, t)\right|^2 \\ & w^{(2)}(\mathbf{r}_1, t_1, \mathbf{r}_2, t_2) = s^2 \left|\mathbf{E}^{(+)}_{\text{classical}}(\mathbf{r}_1, t_1) \right|^2 \left|\mathbf{E}^{(+)}_{\text{classical}}(\mathbf{r}_2, t_2) \right|^2 \end{aligned}
Furthermore, I computed the variance and average of the photon number observable \mathbf N, which helped clarify the statistical nature of photon distribution in these states:
\begin{aligned} &\langle \mathbf N \rangle = \sum_\lambda \langle |\alpha_\lambda|^2 \rangle \\ & \left(\Delta \mathbf N\right)^2 = \langle \mathbf N \rangle \end{aligned}
The derivation of these quantities highlights the Poissonian nature of the photon number distribution in multimode quasi-classical states, which provides a significant insight into the quantum statistical mechanics involved in multimode interactions.
By defining a multimode creation operator, I further expanded the mathematical framework to simplify the description of interactions across modes, enhancing our understanding of their collective behavior:
\mathbf a_{\text{multi}}^\dag \equiv \frac{1}{\alpha_{\text{multi}}}\sum_\lambda \alpha_\lambda \mathbf a_\lambda^\dag = \frac{1}{\sum_\lambda |\alpha_\lambda|^2 }\sum_\lambda \alpha_\lambda \mathbf a_\lambda^\dag = \frac{1}{\langle \mathbf N \rangle }\sum_\lambda \alpha_\lambda \mathbf a_\lambda^\dag
In conclusion, the multimode quasi-classical states offer a robust platform for examining the intricate behaviors of quantum fields across multiple modes. The mathematical tools and methods I developed provide a clear pathway to not only predict but also manipulate the quantum states of complex systems in a more controlled manner.
For more insights into this topic, you can find the details here.