Stimulated emission and photon interaction in quantized electromagnetic fields
After my post on absorption (here) and spontaneous emission (here), I conclude my series with an exploration of the intricate process of stimulated emission, focusing on the interaction between an electron in an excited state and a photon.
In my recent analysis, I start with an initial state where an electron is in state 2, and a photon is in mode 1. This state can be represented mathematically as:
| \hat \psi \rangle = \mathbf b_2^\dag \mathbf a_1^\dag | \mathbf 0 \rangle
Where the total energy of the system is given by:
E_q = E_b + E_f = E_2 + \hbar\omega_1
When considering the perturbing Hamiltonian \mathbf H_p, which drives the transitions within this quantum system, the expression becomes:
\mathbf H_p | \hat \psi \rangle = \sum_{\lambda,j,k} \mathbf {\hat H}_{\lambda,jk} \mathbf b^\dag_j \mathbf b_k \mathbf b_2^\dag \left(\mathbf a_\lambda \mathbf a_1^\dag - \mathbf a_\lambda^\dag\mathbf a_1^\dag\right) | \mathbf 0 \rangle
I explore the implications of the stimulated emission process, considering only those states that contribute to non-zero perturbation theory coefficients. This rigorous approach helps clarify the specific conditions under which the photon stimulates the emission from the electron.
Through mathematical rigor, I evaluate the transition rates using Fermi’s Golden Rule:
w_r = \frac{2\pi}{\hbar} \sum_\lambda \left| \mathbf {\hat H}_{\lambda,j2} \right|^2 \delta\left(E_j - E_2 + \hbar\omega_\lambda\right)
This formula is pivotal as it highlights the probabilities of transition from one state to another within the quantum system, which are influenced by the energy states of the electrons and the photon modes.
My study also extends to the normalization of states where photons are identical, an important aspect in understanding quantum mechanical processes in greater depth. This is reflected when considering the case where \lambda = 1, leading to a state normalization and a modification in the coefficient calculation:
\dot a_r^{(1)} = \frac{1}{i\hbar} e^{i\omega_{rq}t} \sqrt 2 \mathbf {\hat H}_{1,j2}
From this exploration, I draw significant conclusions about the energy configurations and the resulting probabilities of photon-induced transitions in quantum systems. This not only reinforces our understanding of fundamental quantum mechanics but also provides a framework that could be pivotal in technologies such as lasers and quantum computing, where controlled emission is crucial.
In summary, my investigation into stimulated emission sheds light on the complex interactions between photons and electrons under quantum mechanical rules. It underscores the precision required in manipulating quantum states, which is essential for the advancement of quantum technologies.
For more insights into this topic, you can find the details here.