Cavity QED with multiple hyperfine levels

We calculate the weak-driving transmission of a linearly polarized cavity mode strongly coupled to the $D2$ transition of a single cesium atom. Results are relevant to future experiments with microtoroid cavities, where the single-photon Rabi frequency $g$ exceeds the excited-state hyperfine splittings, and photonic band-gap resonators, where $g$ is greater than both the excited- and ground-state splitting.

Figure 1

(Color) Eigenfrequencies of excitations in the first manifold $\left\{{ϵ}_k^{\left(1\right)}\right\}$ vs cavity detuning. Color represents $⟨a^{†}a⟩$ for each corresponding eigenvector. Here ${\omega }_{4\to 4^{\prime }}-{\omega }_{4\to 5^{\prime }}=-251\mathrm{MHz}$, ${\omega }_{4\to 3^{\prime }}-{\omega }_{4\to 5^{\prime }}=-452\mathrm{MHz}$ [11], and $g=450\mathrm{MHz}$ [3]. Although there are 36 eigenvectors in the manifold, there are only 20 unique eigenvalues due to symmetry.

Figure 2

(Color) Transmission (intracavity photon number, normalized to the empty cavity on resonance) vs probe and cavity detunings in the weak-driving limit (calculated in a Fock basis of {0,1} photons) from Eq. (4). Red dotted lines indicate resonances for uncoupled atomic and cavity transitions. The blue dashed lines indicate the cavity detunings in Figs. 3, 4. Rates are $(g,\kappa ,\gamma )=(450,1.75,2.6)\mathrm{MHz}$ [3], $\epsilon =\kappa \gamma ∕g$, and ${\Omega }_r=\gamma$. The repump beam is resonant with the $F=3\to F^{\prime }=4^{\prime }$ transition, i.e., ${\Delta }_r={\Delta }_{4^{\prime }}$. Values were computed on a grid with a $50\mathrm{MHz}$ $\left(0.5\mathrm{MHz}\right)$ spacing in ${\omega }_c$ $\left({\omega }_p\right)$.

Figure 3

(Color) (a) Normalized transmission $T$ and (b) populations in various Zeeman ground states of the $F=4$ manifold as a function of probe detuning, with the cavity frequency fixed at ${\omega }_c={\omega }_{4\to 5^{\prime }}-100\mathrm{MHz}$. Parameters are the same as in Fig. 2. By symmetry, populations are the same in Zeeman level $m_F$ as in level $-m_F$. The sharp feature at ${\omega }_p={\omega }_{4\to 5^{\prime }}-251$ is caused by a coherent Raman effect between the probe and the repump light. Note that because of the weak cavity drive and the strong repump light, nearly all of the population is in the $F=4$ manifold.

Figure 4

(Color online) Normalized transmission $T$ of a cavity coupled to the $F=4\to F^{\prime }=5^{\prime }$ transitions (dashed) and to the $F=4\to F^{\prime }=\{3^{\prime },4^{\prime },5^{\prime }\}$ transitions (red). Parameters are the same as in Fig. 2, with the cavity frequency fixed at ${\omega }_c={\omega }_{4\to 5^{\prime }}$.

Figure 5

(Color) Eigenfrequencies of excitations in the first manifold $\left\{{\eta }_k^{\left(1\right)}\right\}$ vs cavity detuning. Color represents $⟨a^{†}a⟩$ for each corresponding eigenvector. The coupling $g=17\mathrm{GHz}$ [4]. Although there are 48 eigenvectors in the manifold, there are only 27 unique eigenvalues due to symmetry. These eigenvalues are clustered in five bands.

Figure 6

(Color) Difference frequencies for allowed single-quanta transitions between ground and first excited states, ${\eta }_k^{\left(1\right)}-{\eta }_j^{\left(0\right)}$, vs cavity detuning. Eigenfrequencies are the same as in Fig. 5. Black (red) lines indicate transitions from ground states with the atom in the $F=3$ $(F=4)$ manifold.

Figure 7

(Color) Transmission (intracavity photon number, normalized to the empty cavity on resonance) vs probe and cavity detunings in the weak-driving limit (calculated in a Fock basis of {0,1} photons) from Eq. (7). The red dotted line indicates resonance for the uncoupled cavity. The blue dashed lines indicate the cavity detunings in Figs. 8, 9, 10. Rates are $(g,\kappa ,\gamma )=(17,4.4,0.0026)\mathrm{GHz}$ [4], $\epsilon =100\kappa \gamma ∕g$. Values were computed on a grid with a $1\mathrm{GHz}$ $\left(0.05\mathrm{GHz}\right)$ spacing in ${\omega }_c$ $\left({\omega }_p\right)$.

Figure 8

(Color online) Normalized transmission $T$ (a) and atomic populations in the hyperfine ground states (b) as a function of probe detuning, with the cavity frequency fixed at ${\omega }_c={\omega }_{4\to 5^{\prime }}-13\mathrm{GHz}$. Parameters are the same as in Fig. 7. Note that because the cavity drive is weak, nearly all of the population is in the ground states.

Figure 9

(Color online) Normalized transmission $T$ (a) and atomic populations in the hyperfine ground states (b) as a function of probe detuning, with the cavity frequency fixed at ${\omega }_c={\omega }_{4\to 5^{\prime }}+20\mathrm{GHz}$. Parameters are the same as in Fig. 7. Note that because the cavity drive is weak, nearly all of the population is in the ground states.

Figure 10

(Color online) Normalized transmission $T$ (a) and atomic populations in the hyperfine ground states (b) as a function of probe detuning, with the cavity frequency fixed at ${\omega }_c={\omega }_{4\to 5^{\prime }}+4\mathrm{GHz}$. Parameters are the same as in Fig. 7. Note that because the cavity drive is weak, nearly all of the population is in the ground states.