Deriving the total spontaneous emission rate
In my research into quantum mechanics and particularly in studying transitions within atomic systems, I recently focused on deriving the total spontaneous emission rate for two-state electron systems. This derivation not only enhances our understanding of quantum processes but also provides crucial data for various applications in quantum computing and spectroscopy.
Spontaneous emission is a fundamental phenomenon where an electron in an excited state in an atom or molecule transitions to a lower energy state, emitting a photon in the process. The rate at which this occurs is essential for understanding light-matter interactions in fields ranging from quantum optics to communication technologies.
Theoretical foundation
The spontaneous emission rate is calculated by starting with the basic interaction Hamiltonian between a quantum system and the electromagnetic field. For a two-state system transitioning from state |2\rangle to |1\rangle, the transition rate can be expressed as:
W = \frac{2\pi}{\hbar}\sum_\lambda\left| \mathbf{\hat{H}}_{\lambda,12} \right|^2 \delta\left(E_1 - E_2 + \hbar\omega_\lambda\right)
In the equation above, \mathbf{\hat{H}}_{\lambda,12} represents the interaction Hamiltonian component responsible for the transition, influenced by the matrix element that includes the photon’s electric field and the dipole moment of the electron transition.
Simplifying the interaction hamiltonian
Considering the electric dipole approximation, where the spatial variation of the electromagnetic field across the atom can be neglected, the Hamiltonian simplifies considerably. This approximation allows the use of the dipole matrix element \mu_{21}, defined as:
\mu_{21} = \int \phi_2(\mathbf{r}) \mathbf{r} \phi_1(\mathbf{r}) , \mathrm{d}^3 \mathbf{r}
The simplification leads to the following form of the squared matrix element:
\left| \mathbf{\hat{H}}_{\lambda,12} \right|^2 = e^2 \frac{\hbar \omega_\lambda}{2 \varepsilon_0} \left| \mathbf{u}_\lambda \cdot \mu_{21} \right|^2
Integration over photon states
The transition rate integral over the electromagnetic field modes (or photon states) involves converting the sum over states into an integral over the photon density of states, \rho(\omega). Integrating over this density, considering the energy conservation delta function, leads us to:
W = \frac{e^2 \omega^3 \left| \mu_{21} \right|^2}{3 \pi \varepsilon_0 \hbar c^3}
This formula encapsulates the spontaneous emission rate in terms of the energy difference between the two states (expressed as \omega), the dipole matrix element, and fundamental constants.
Practical implications
Understanding this rate is crucial for predicting the behavior of quantum systems and designing devices that operate on principles of quantum mechanics. The spontaneous emission rate not only influences the design of lasers and optical amplifiers but also plays a critical role in the development of quantum computing elements, where controlling such emissions is paramount.
For more insights into this topic, you can find the details here.