Spontaneous emission in ultracold spin-polarized anisotropic Fermi seas

We examine and explain the spatial emission patterns of ultracold excited fermions in highly anisotropic trapping potentials in the presence of a spin-polarized Fermi sea of ground-state atoms. Due to the Pauli principle, the Fermi sea modifies the available phase space for the recoiling atom. This leads to the well-known modification of the atomic decay rate, but also to a spatial modulation in the probability of the emitted photon’s direction. In this work we carry out the first detailed investigation into these spatial anisotropies and show that they are due to an intricate interplay between Fermi energies and degeneracy values of specific energy levels. We identify different regimes and show that the emission can be engineered to become highly directional. As the latter is usually only possible in cavity settings, our results describe an alternative idea for a directional photon source.

Figure 1

(Color online) Top row: emission probability, $P_e$, into individual shells in a pancake-shaped trap for ${\eta }^2=36$ and $\lambda =10,23,46$. Arrow indicates the $n=20$ energy level of the harmonic trap, which is referred to in the text. Bottom row: decay rate of the excited particle, $M_f$, for the same parameters as above.

Figure 2

(Color online) Top row: emission probability, $P_e$, into individual shells in a cigar-shaped trap for ${\eta }^2=36$ and $\lambda =10,23,46$. Arrow indicates the $n=20$ energy level of the harmonic trap, which is referred to in the text. Bottom row: decay rate of the excited particle, $M_f$, for the same parameters as above.

Figure 3

(Color online) (a) Emission probability into individual states within the shell $n=5$ in an isotropic trap. The triplets represent $\left(n_x,n_y,n_z\right)$ and all permutations of each triplet have the same probability. The Lamb-Dicke factor is $\eta =5$, however changing this value only scales all values. (b) Emission probability into individual states within the shell $n=5$ in a pancake trap with $\lambda =5$. The values for the three states on the left are not visible on this scale.

Figure 4

(Color online) $M_f$ along the tight axis at $T=0$. ${\eta }^2=49$ and $n_F=60$. Note that we use a continuous distribution of $\lambda$ for this graph.

Figure 5

(Color online) (a) $M_f\left(\theta \right)$ in a pancake-shaped trap at $T=0$. ${\eta }^2=25$ and $\lambda =4$ with $n_F=31$ (outermost) to $n_F=36$ (innermost). (b) $M_f\left(\theta \right)$ in a pancake-shaped trap with $\lambda =11$, ${\eta }^2=25$, and $n_F=23$.

Figure 6

(Color online) $M_f\left(\theta ,n_z\right)$ for a pancake-shaped trap, with $\lambda =11$, ${\eta }^2=25$, and $n_F=23$. In the four graphs the decay is only allowed into quantum states of the harmonic trap with $n_z=0$, 1, 2, and $n_z\ge 3$, respectively.

Figure 7

(Color online) (a) $M_f\left(\theta \right)$ in a cigar-shaped trap at $T=0$, with $n_F=45$ and ${\eta }^2=49$. $\lambda =46$ (outermost), $\lambda =96$ (center plot), and $\lambda =\infty$ (innermost). The plot is symmetric through a $2\pi$ rotation about the $\left(0,\pi \right)$ axis. (b) A three-dimensional illustration of the excited particles decay rate in a cigar trap in the large anisotropy limit.