Full quantum model for spontaneous emission?

Question

Often I hear it described that spontaneous emission is when vacuum fluctuations "kick" an atom and induce stimulated emission on its own, but I've never been able to find a convincing explanation using a quantum optics formalism.

My main question is: If I have an atom in an excited state, how do I describe the Hamiltonian or equations of motion for the effect of spontaneous emission?

Specifically I am interested in obtaining a result that includes the frequency modes $a_\omega^\dagger$ and their distribution $g(\omega)$.

If I were to guess I think that this system would have something of the form:

$\dot{\rho}_{excited} = -\gamma_{decay} \rho_{excited}\sum_\omega g(\omega)(a_\omega+a^\dagger_\omega)$

(That is over time there is decay from the excited state and production of photons with certain frequencies over some bandwidth, but I'm not sure this is correct.)

Answer 1

The minimal quantum model for spontaneous emission is given by the interaction Hamiltonian

$$ H = \int d\omega [g(\omega)a(\omega)\sigma^+ + g^*(\omega)a^\dagger(\omega)\sigma^-]$$

where $a(\omega), a^\dagger(\omega)$ are bosonic modes associated with the light field and $\sigma^\pm$ are the ladder operators of a two-level system. The coupling function $g(\omega)$ depends on the electromagnetic environment and can e.g. be evaluated for free space or for a cavity.

The sponteneous emission problem is then specifically given by the initial state where the atom is fully excited and the light field is in the vacuum state (or a thermal state at a given temperature, but I will only consider the vacuum case for simplicity).

With regards to solution methods, there are multiple levels of complexity that one can consider. The standard approach, which is probably covered in every modern quantum optics book, is Wigner-Weisskopf theory. The latter makes some approximations such as the Markov approximation, which make it applicable at weak light-matter coupling.

For strong light-matter coupling, the standard approach is to use an alternative starting Hamiltonian, namely the Jaynes-Cummings model as suggested by @NorbertSchuch in the comments. The reason is that strong coupling is often achieved by coupling the atom to a resonator where a single mode dominates.

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As an advanced topic, I would like to note that the spontaneous emission problem, specifically, can also be described at strong coupling using the above Hamiltonian. There are a bunch of resources on this topic, I think one of the Cohen-Tannoudji quantum optics textbooks also contains a chapter. My personal favourite is Krimer et al., PRA 89, 033820 (2014). It essentially presents a generalization of the Wigner-Weisskopf method, where the Markov approximation is eliminated by solving certain integrals exactly using Laplace transforms. With regards to the comments, the paper also shows the transition from the Wigner-Weisskopf type emission to the Jaynes-Cummings type dynamics as you turn up the coupling by making the electromagnetic environment more confined.

Note that the central reason why this model is so useful and can be solved so accurately is the rotating wave approximation. It results in the Hamiltonian being excitation number conserving, such that only the states $|1_\mathrm{atom}\rangle|0_\omega\rangle$ and $|0_\mathrm{atom}\rangle|1_\omega\rangle$ contribute.

Even this treatment breaks down eventually. In the ultra-strong coupling regime, the minimal model given above does not work anymore, since counter-rotating terms and stuff like gauge invariance start to matter. There is a lot of literature on this topic, see e.g. What are the "strong", "ultrastrong" and "deep strong" coupling regimes of the Rabi model? and references therein.

Answer 2

If I may help, there is an alternative to Wigner-Weiskopff theory to explain spontaneous emission, assuming that the electron oscillates between upper state $\vert b >$ to lower state $\vert a>$. This theory, as far as I know introduced by Roger Boudet in his book "Relativistic Transitions in the Hydrogenic Atoms", uses classical electromagnetism with vector potential $\boldsymbol{A}$ and probability current transition $\boldsymbol{j}$:

$$ \Box \boldsymbol{A} = \mu_0 \boldsymbol{j} $$

with the transition current provided (here non-relativistic) by:

$$ \boldsymbol{j} = \frac{\hbar e}{m} \text{Re} \lbrace i \left[ \Psi_b^* \boldsymbol{\nabla} \Psi_a - \Psi_a \boldsymbol{\nabla} \Psi_b^* \right] \rbrace $$

Computing $\boldsymbol{A}$ in far field, then $\boldsymbol{E}$ and $\boldsymbol{B}$ you can have the time-averaged Poynting vector then the radiated power: you divide by $E_b - E_a$ and you obtain the Einstein spontaneous coefficient. The full derivation is available here.