Exploring multi-photon emission: spontaneous and stimulated processes
In my ongoing exploration of quantum mechanics and its intricate phenomena, multi-photon emission stands out due to its significance in both theoretical research and practical applications. This blog post explores the mathematical descriptions of spontaneous and stimulated multi-photon emission processes, offering an analytical perspective.
Spontaneous Emission of multiple photons
Let’s start by considering an electron in an excited state, specifically state | 2 \rangle, in the presence of n photons in a mode |\lambda \rangle \ne | 1 \rangle. The quantum mechanical description of this scenario leads to an understanding of how the transition rate can be expressed. The transition rate for spontaneous emission is given by:
w_r = \frac{2\pi}{\hbar}\sum_\lambda \left| \mathbf {\hat H}_{1,12} \right|^2 \delta\left(E_1 - E_2 + \hbar\omega_\lambda\right)
This equation illustrates how the rate of emission is influenced by the overlap between the electron’s energy state changes and the photon modes.
Stimulated emission with multiple photons
Moving forward to stimulated emission, consider an electron in state | 2 \rangle and n photons in the mode | 1 \rangle. The state of the system can be described by:
| \hat \psi \rangle =\frac{1}{\sqrt{n!}} \mathbf b_2^\dag \left(\mathbf a_1^\dag \right)^n | \mathbf 0 \rangle
The perturbing Hamiltonian’s action on this initial state, and the interaction among different states and modes, leads to a more complex expression:
\mathbf H_p | \hat \psi \rangle = \frac{1}{\sqrt{n!}} \mathbf {\hat H}_{1,j2} \left(n \mathbf b^\dag_j \left(\mathbf{a}_1^\dag\right)^{n-1}| \mathbf 0 \rangle - \mathbf b^\dag_j \left(\mathbf a_1^\dag \right)^{n+1}| \mathbf 0 \rangle \right)
This detailed analysis aids in understanding how the photon’s count influences the system’s evolution and the resultant emission.
Transition rates and their implications
Through these analyses, I can derive transition rates that capture the dynamics of these quantum systems effectively. For example, the transition rate for stimulated emission where an electron moves from state | 2 \rangle to | 1 \rangle while interacting with photons can be detailed as follows:
w_r = \frac{2\pi}{\hbar} (n + 1) \left| \mathbf {\hat H}_{1,22} \right|^2 \delta\left(E_1 - E_2 + \hbar\omega_1\right)
These mathematical formulations not only elucidate the theoretical underpinnings but also set the stage for experimental validations.
Conclusion
Understanding the quantum mechanical processes of multi-photon emissions deepens our grasp of quantum optics and other related fields. By harnessing detailed mathematical models, I am able to convey complex phenomena in a manner that is accessible yet thorough. For more insights into this topic, you can find the details here.