Demonstration of the Jaynes-Cummings ladder with Rydberg-dressed atoms

We observe the nonlinearity of the Jaynes-Cummings (JC) ladder in the Autler-Townes spectroscopy of the hyperfine ground states for a Rydberg-dressed two-atom system. Here, the role of the two-level system in the JC model is played by the presence or absence of a collective Rydberg excitation, and the bosonic mode manifests as the number $n$ of single-atom spin flips, symmetrically distributed between the atoms. We measure the normal-mode splitting and $\sqrt{n}$ nonlinearity as a function of detuning and Rabi frequency, thereby experimentally establishing the isomorphism with the JC model.

Figure 1

(a) Energy-level diagram of the ${}^{133}\mathrm{Cs}$ atom. The states are $|0\rangle =|6S_{1/2},F=3,m_F=0\rangle$ and $|1\rangle =|6S_{1/2},F=4,m_F=0\rangle$, where ${\mathrm{\Delta }}_r$ is the detuning from $|1\rangle$ to $|r\rangle =|54P_{3/2},m_J=+3/2\rangle$ transition. (b) Energy-level diagram of an ensemble of $N$ bare state atoms, symmetrically coupled under the assumption of a perfect Rydberg blockade; $|g,n\rangle \equiv {\left\{{|0\rangle }^{\otimes N-n}{|1\rangle }^{\otimes n}\right\}}_{\mathrm{sym}}$ and $|e,n\rangle \equiv {\left\{{|0\rangle }^{\otimes N-n-1}{|1\rangle }^{\otimes n}|r\rangle \right\}}_{\mathrm{sym}}$. (c) Energy-level diagram of dressed ground-state atoms showing the Autler-Townes splitting and exhibiting the nonlinearity of the JC model. The states of $|\tilde{g},n\rangle$ and $|\tilde{e},n-1\rangle$ are the groundlike- and Rydberg-like-dressed states, respectively, $|\tilde{g},n\rangle \equiv cos({\theta }_n/2)|g,n\rangle +sin({\theta }_n/2)|e,n-1\rangle$ and $|\tilde{e},n-1\rangle \equiv cos({\theta }_n/2)|e,n-1\rangle -sin({\theta }_n/2)|g,n\rangle$, where $tan{\theta }_n=\sqrt{n}{\mathrm{\Omega }}_r/{\mathrm{\Delta }}_r$.

Figure 2

Experimental setup. The Rydberg laser and the Raman lasers are aligned along the $x$ axis. Two optical tweezers are formed by two lasers with an angular separation $\theta$. In this setup, eight electrodes null the electric fields near the trapped atoms. The bias magnetic field is applied along the $x$ axis.

Figure 3

Jaynes-Cummings ladder and its $\sqrt{n}$ nonlinearity for two Rydberg-dressed atoms. The $x$ axis is the normalized Rydberg detuning ${\mathrm{\Delta }}_r/{\mathrm{\Omega }}_r$, and the $y$ axis gives the normalized microwave detuning. Two-atom Rydberg-dressed states of $|\tilde{g},2\rangle$ ($|\tilde{e},1\rangle$) are located at the upper quadrant I and III (II and IV), and single-atom Rydberg-dressed states of $|\tilde{g},1\rangle$ ($|\tilde{e},0\rangle$) are positioned at the lower quadrant I and III (II and IV). The upper and lower quadrant I and III (II and IV) are related to the groundlike- (Rydberg-like-) dressed states. The red (blue) bands are theoretical predictions of the microwave frequencies for the Autler-Townes splitting, incorporating measured systematic drifts ($5\%$) in the experimental parameters (${\mathrm{\Delta }}_r$ and ${\mathrm{\Omega }}_r$). The green lines correspond to the energy of two atoms without the interaction. Two Rydberg transitions are considered for the theoretical plots, $|6S_{1/2},F=4,m_f=0\rangle \to |54P_{3/2},m_J=+3/2\rangle$ and $|54P_{3/2},m_J=+1/2\rangle$. The Rabi frequencies of $m_J=+3/2$ and $+1/2$ are ${\mathrm{\Omega }}_r$ and ${\mathrm{\Omega }}_r/\sqrt{3}$, respectively. States with $m_J=+3/2$ and $+1/2$ are separated by ${\mathrm{\Delta }}_{\mathrm{Zeeman}}\simeq 2.13{\mathrm{\Omega }}_r$ with a 4.6-G magnetic field.

Figure 4

Resonant (${\mathrm{\Delta }}_r\approx 0$) Autler-Townes splitting of two Rydberg-dressed atoms as a function of Rydberg-transition Rabi frequency. The $x$ axis is the Rabi frequency of a single-atom Rydberg excitation (${\mathrm{\Omega }}_r$), and the $y$ axis is the Autler-Townes splitting measured by microwave spectroscopy. The two upper and lower data trends are the splittings of two-atom Rydberg-dressed states between $|\tilde{g},2\rangle$ and $|\tilde{e},1\rangle$ and single-atom Rydberg-dressed states between $|\tilde{g},1\rangle$ and $|\tilde{e},0\rangle$. The two gray bands incorporate measured systematic drifts ($5\%$) in the experimental parameters (${\mathrm{\Omega }}_r$ and ${\mathrm{\Delta }}_r$). The green line corresponds to the splitting of two atoms without the interaction. Based on linear fits, the ratio of the splittings is $1.43\left(0.03\right)$, consistent with $\sqrt{2}$.