Multi-photon absorption: state transitions and energy calculations
In my latest study on quantum mechanics, I focus on the phenomenon of multi-photon absorption, an essential process in the field of quantum optics and a key component in understanding electron-photon interactions in quantum systems. This process involves an electron initially in a specific state interacting with multiple photons in a given mode, leading to a transition into a new state with one less photon.
Initial state and system setup
The system initially consists of an electron in state |1\rangle and n photons in the same mode. The state can be mathematically described as:
| \hat \psi \rangle = \frac{1}{\sqrt{n!}} \mathbf b_1^\dag \left(\mathbf a_1^\dag \right)^n | \mathbf 0 \rangle
Here, \mathbf{b}_1^\dag and \mathbf{a}_1^\dag are the creation operators for the electron and photons respectively, and | \mathbf 0 \rangle represents the vacuum state.
Total energy of the initial state
The total energy of this initial configuration is a summation of the electron’s energy and the energy contributed by the photons:
E_q = E_1 + n\hbar\omega_1
where E_1 is the energy of the electron in state |1\rangle, \hbar is the reduced Planck’s constant, and \omega_1 is the angular frequency of the photons.
Interaction Hamiltonian and state evolution
Upon introducing a perturbing Hamiltonian, the system evolves. The interaction is modeled by the Hamiltonian:
\mathbf H_p | \hat \psi \rangle = \frac{1}{\sqrt{n!}} \sum_{\lambda,j,k} \mathbf {\hat H}_{\lambda,jk} \mathbf b^\dag_j \left(\delta_{k1} - \mathbf b_1^\dag\mathbf b_k \right) \left(n \delta_{\lambda 1} \left(\mathbf{a}_1^\dag\right)^{n-1} + \left(\mathbf{a}_1^\dag\right)^n \mathbf{a}_\lambda - \mathbf a_\lambda^\dag\left(\mathbf a_1^\dag \right)^n\right) | \mathbf 0 \rangle
The result of this interaction leads to the state where an electron transitions to state |2\rangle and n-1 photons remain.
Transition coefficients and fermi golden rule
The transition rate can be derived using the coefficient:
\dot a_r^{(1)} = \frac{1}{i\hbar} e^{i\omega_{rq}t} \sqrt{n} \mathbf {\hat H}_{1,21}
and by applying Fermi’s Golden Rule, which gives:
w_r = \frac{2\pi}{\hbar}n \left| \mathbf {\hat H}_{1,12} \right|^2 \delta\left(E_2 - E_1 - \hbar\omega_1\right)
The delta function \delta\left(E_2 - E_1 - \hbar\omega_1\right) ensures that energy is conserved in the transition, meaning that the photon’s energy exactly compensates for the energy difference between the two electron states.
In conclusion, my analysis of multi-photon absorption in quantum systems illustrates the complex interplay between quantum states and the fundamental principles governing their transitions. This exploration not only enhances our understanding of quantum dynamics but also serves as an element in the development of quantum technologies.
For more insights into this topic, you can find the details here.