Purcell enhancement at an angle: Direction of charge oscillation

EDIT: I will provide some details on the dynamics in Scenario 1 mentioned below, since this seems to be the OP's central question, as clarified in the comments.

In the single transition case, one can write down a Master equation for the system, which reads

$$ dot{rho}_t = -frac{i}{hbar} [H_t, rho_t] + mathcal{L}_t[rho_t] ,. $$

Here, $rho_t$ is the density matrix of the ground and excited state of the single transition ($t$ for "transition"). This Master equation therefore only contains the degrees of freedom of the transition, because we have traced out the field degrees of freedom within the Markov approximation, as usual in weakly coupled quantum optics.

The effective Hamiltonian and Lindblad term are given by [1]

$$ H_s = hbar (omega_t + J_t) frac{hat{sigma}^z}{2} ,,$$

and

$$ mathcal{L}_t[rho_t] = frac{Gamma_t}{2}left( 2hat{sigma}^-rho_that{sigma}^+ - hat{sigma}^+hat{sigma}^-rho_t - rho_that{sigma}^+hat{sigma}^-right) ,.$$

This is about the simplest Master equation you can get, with a decay rate $Gamma_t$ and a corresponding frequency shift $J_t$ and it generally applies for a single transition in a dielectric environment, as long as we are at weak coupling (see [1] for details. $omega_t$ is the bare transition frequency in free space.

The main question is: what are the values of $Gamma_t$ and $J_t$ in the electromagnetic environment? This is what is influenced by structure of the environment, such as the cavity or material properties. As also stated below [1]

$$Gamma_t = frac{2omega_t^2}{hbarepsilon_0c^2} mathbf{d}_tcdot mathrm{Im}[mathbf{G}(mathbf{r}_t, mathbf{r}_t, omega_t] cdot mathbf{d}_t^* ,,$$

and similarly

$$J_t = frac{omega_t^2}{hbarepsilon_0c^2} mathbf{d}_tcdot mathrm{Re}[mathbf{G}(mathbf{r}_t, mathbf{r}_t, omega_t] cdot mathbf{d}_t^* ,.$$

While in the OP's geometry, the Green's function might be slightly complicated, the above shows that the dynamics qualitatively stay the same, no matter the geometry. There is decay and a frequency shift, that's it. The axis changing charge dynamics do not happen. The reason is as already syspected by the OP: due to the single transition, the dynamics have been confined along a single axis given by the dipole moment vector.

Note, however, that the above situation can be physical. Depending on the natural line width and the frequencies involved, the dynamics may naturally be restricted to a single transition by resonance conditions.

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I'll start with the last question.

What actually happens in the described situation?

It depends what precisely is meant by "the described situation". When I read through the question, I see a few different scenarios, mixing quantum mechanical state decay via dipole transitions and classical charge oscillation concepts. In the following, I will try to address each of them separetely.

Scenario 1: Single transition model

We consider the following model: A single dipole transition with its dipole moment in the described direction relative to the slow and fast axis of the cavity. In this case, the most general expression for the Purcell enhancement is given in terms of the Green's tensor of the surrounding dielectric structure. The full cavity enhanced line width is (see [1])

$$Gamma = frac{2omega_A^2}{hbarepsilon_0c^2} mathbf{d}cdot mathrm{Im}[mathbf{G}(mathbf{r}_A, mathbf{r}_A, omega_A] cdot mathbf{d}^* ,,$$

where $omega_A$ is the transition frequency of the atom and $mathbf{r}_A$ its position. Compared to the formula in Mark Fox' book, this formula is more general, not making assumptions such as single mode cavities etc. The polarization response is fully included in the Green's tensor.

Most notably, the Purcell enhancement is a statement about the decay rate of the excited state via the transition. This means that the slow and fast direction are in a sense summed over in the above formula (due to the dot product on both sides). The birefringence property will therefore only cause an interesting dependence on the dipole moment direction, since the Green's tensor is not isotropic in this case. However, it will not change the decay dynamics of the excited state, which still just decays with a modified rate.

Scenario 2: Single excited state model

First to answer the obvious question: How is this different from a single transition model? The point is that in general, in particular for degenerate systems, the excited state can decay via multiple channels to the different degenerate ground states. In that case, we have to sum over multiple transitions $i$, such that

$$Gamma = sum_i frac{2omega_A^2}{hbarepsilon_0c^2} mathbf{d}_i cdot mathrm{Im}[mathbf{G}(mathbf{r}_A, mathbf{r}_A, omega_A] cdot mathbf{d}^*_i ,,$$

but the statement above remains true.

Scenario 3: A realistic atom in an excited state

For the realistiic case, everything depends on the parameters and physical setting. Which states participate in the decay dynamics? Which transition channels are important? Maybe we even have strong coupling which fundamentally changes the dynamics from pure decay to oscillatory behavior? This scenario is to broad to address in general.

In this general case, however, dynamics such as described by the OP are conceivable.

Scenario 4: Classical charge oscillation in the field

For classical charge oscillation, we do not have a notion of quantized orbitals or the state structure defined by the atom. In a complicated environment as the birefringent cavity, a rotation of the charge oscillation can happen. Even a separation into a multi-modal oscillation is possible.

Summary

• In Scenario 1 and 2, not much will change, other than a quantitative modification of the decay rate.
• In Scenario 3 and 4, complex decay dynamics can occur, which require further specificiation of the specific physical situation.