Trapped Rydberg ions are a promising novel approach to quantum computing and simulations [1][2][3]. They are envisaged to combine the exquisite control of trapped ion qubits [4] with the fast two-qubit Rydberg gates already demonstrated in neutral atom experiments [5][6][7]. Coherent Rydberg excitation is a key requirement for these gates. Here, we carry out the first coherent Rydberg excitation of an ion and perform a single-qubit Rydberg gate, thus demonstrating basic elements of a trapped Rydberg ion quantum computer.Systems of trapped ion qubits have set numerous benchmarks for single-qubit preparation, manipulation, and readout [8]. They can perform low error entanglement operations [9,10] with up to 14 ion qubits [11]. Still, a major limitation towards realizing a large-scale trapped ion quantum computer or simulator is the scalability of entangling quantum logic gates [12].Arrays of neutral atoms in dipole traps offer another promising approach to quantum computation and simulation. Here, qubits are stored in electronically low-lying states and multi-qubit gates may be realized by exciting atoms to Rydberg states [6,7,13,14]. Rydberg states are exotic states of matter in which the valence electron is excited to high principal quantum numbers. They can have extremely high dipole moments and may interact strongly with each other, which has allowed entanglement generation [15,16] and fast two-qubit Rydberg gates [5] in neutral atom systems.A system of trapped Rydberg ions may combine the advantages of both technologies. Electronically lowlying states may be used as qubit states and fast multiqubit gates are envisaged by coherently exciting ions to Rydberg states and employing dipolar interactions between them [1,17]. Multi-qubit gates commonly used in trapped ion systems suffer scalability restrictions due to spectral crowding of motional modes [12]. This issue does not affect multi-qubit Rydberg gates thus a trapped Rydberg ion quantum computer offers an alternate approach to a scalable system. An unanswered question was whether trapped ions can be excited to Rydberg states in a coherent fashion as is required for multi-qubit Rydberg gates. In our experiment we study a single 88 Sr + ion confined in a linear Paul trap. Three atomic levels in a ladder configuration are coupled using two UV lasers (Fig. 1). The qubit state |0 is coupled to the excited state |e by the pump laser at 243 nm with Rabi frequency Ω P . |e is coupled in turn to the Rydberg state |r (42S 1/2 , m J = −1/2) using the Stokes laser at 307 nm with Rabi frequency Ω S . The experimental setup is described in detail in the Methods section and in [3].We can use the two-photon coupling for coherent control of the Rydberg excitation. At two-photon resonance (|0 to |r ) the coupling Hamiltonian has a "dark" eigenstate |Φ dark ∼ Ω S e iφ |0 − Ω P |r (Methods), which is named so because it does not contain any component of
|0-|rRydberg excitation by STIRAP shown by comparing application of the single and the double STIRAP pulse sequences. The sin...