2 FIG. 2: Schematic of the experimental setup showing the paths of the pulse trains emitted by a mode-locked Ti:Sapphire (Ti:Sapph) laser, where the optical pulses are fre- quency shifted by AOs. Single qubit rotations only require a single pulse train, but to address the motional modes the pulse train is split into two and sent through AOs to tune the rela- tive offset of the two combs. We lock the repetition rate ( ν$_{R}$ ) by first detecting ν$_{R}$ with a photodetector (PD). The output of the PD is an RF frequency comb spaced by ν$_{R}$ . We band- pass filter (BP) the RF comb at 12.685 GHz and then mix the signal with a local oscillator (LO). The output of the mixer is sent into a feedback loop (PID) which stabilizes ν$_{R}$ by means of a piezo mounted on one of the laser cavity mirrors. When locked, ν$_{R}$ is stable to within 1 Hz for more than an hour. As an alternative, instead of locking the repetition rate of the pulsed laser, an error signal could be sent to one of the AOs to use the relative offset of the two combs to compensate for a change in the comb spacing. FIG. 1: The Stokes Raman process driven by frequency combs is shown here schematically. An atom starting in the |↓〉 state can be excited to a virtual level by absorbing a photon from the blue comb and then driven to the |↑〉 state by emitting a photon into the red comb. Although drawn here as two dif- ferent combs, if the pulsed laser’s repetition rate or one of its harmonics is in resonance with the hyperfine frequency, the absorption and emission can both be stimulated by the same frequency comb. Because of the even spacing of the frequency comb, all of the comb teeth contribute through different vir- tual states which result in indistinguishable paths and add constructively. of q . As shown in Fig. 3, when n = 2 ( q = 313 and ν$_{R}$ = 40 . 39 MHz), application of the pulse train drives oscillations between the qubit states of a single ion. How- ever, when n = 3 ( q = 469 . 5 and ν$_{R}$ = 26 . 93 MHz), the qubit does not evolve. The Rabi frequency of these oscillations can be esti- mated by considering the Hamiltonian resulting from an infinite train of pulses. After adiabatically eliminating cal frequency ω$_{c}$ and teeth separated by ν$_{R}$ whose in- dividual widths scale like ∼ ν$_{R}$/N . The Raman res- onance condition will be satisfied when a harmonic of the repetition rate is equal to the hyperfine qubit split- ting ω$_{0}$ , implying that the parameter q ≡ ω$_{0}$/ 2 πν$_{R}$ , is an integer. To demonstrate coherent control with a pulse train, $^{171}$Yb + ions confined in a linear Paul trap are used to encode qubits in the $^{2}$S$_{1}$$_{/}$$_{2}$ hyperfine clock states | F = 0 , m$_{F}$ = 0 〉 ≡ |↓〉 and | F = 1 , m$_{F}$ = 0 〉 ≡ |↑〉 , having hyperfine splitting ω$_{0}$/ 2 π = 12 . 6428 GHz. For state preparation and detection we use standard Doppler cooling, optical pumping, and state-dependent fluores- cence methods on the 811 THz $^{2}$S$_{1}$$_{/}$$_{2}$ ↔ $^{2}$P$_{1}$$_{/}$$_{2}$ electronic transition [19]. The frequency comb is produced by a frequency-doubled mode-locked Ti:Sapphire laser at a carrier frequency of 802 THz, detuned by Δ / 2 π = 9 THz from the electronic transition. The repetition rate of the laser is ν$_{R}$ = 80 . 78 MHz, with each pulse having a du- ration of τ ≈ 1 psec. The repetition rate is phase-locked to a stable microwave oscillator as shown in Fig. 2, pro- viding a ratio of hyperfine splitting to comb spacing of q = 156 . 5. An EO pulse picker is used to allow the pas- sage of one out of every n pulses, decreasing the comb spacing by a factor of n and permitting integral values FIG. 3: After Doppler cooling and optical pumping to the |↓〉 state, a single pulse train is directed onto the ion. When the ratio of qubit splitting to pulse repetition rate, q , is an integer, pairs of comb teeth can drive Raman transitions as shown by the blue circular data points. However, if the q parameter is a half integer, the qubit remains in the initial state as shown by the red square data points.