For I$_{sat}$ = 0 . 15 W/cm $^{2}$, the data shown in Fig. 3 is con- sistent with an average intensity ¯ I ≈ 500 W/cm $^{2}$. 3 the excited $^{2}$P$_{1}$$_{/}$$_{2}$ state and performing the rotating-wave approximation, the resonant Rabi frequency of Raman transitions between the qubit states is given by a sum over all spectral components of the comb teeth as indi- cated in Fig. 1 ($_{ℏ}$ = 1): In order to entangle multiple ions, we first address the motion of the ion by resolving motional sideband tran- sitions. As depicted in Fig. 2, the pulse train is split into two perpendicular beams with wavevector difference k along the x − direction of motion. Their polarizations are mutually orthogonal to each other and to a weak magnetic field that defines the quantization axis [20]. We control the spectral beatnotes between the combs by sending both beams through AO modulators (driven at frequencies ν$_{1}$ and ν$_{2}$ ), imparting a net offset frequency of Δ ω/ 2 π = ν$_{1}$ − ν$_{2}$ between the combs. For instance, in order to drive the first upper/lower sideband transi- tion we set | 2 πjν$_{R}$ + Δ ω | = ω$_{0}$ ± ω$_{t}$ , with j an integer and ω$_{t}$ the trap frequency. In order to see how the side- bands are spectrally resolved, we consider the following Hamiltonian of a single ion and single mode of harmonic motion interacting with the Raman pulse train: where µ is the dipole matrix element between the ground and excited electronic states, E$_{k}$ ≡ ν$_{R}$ ˜ f (2 πkν$_{R}$ ), and q is an integer. In the approximate expression above, the sum is replaced by an integral and each pulse is de- scribed by f ( t ) = $^{√}$π/ 2 E$_{0}$ sech( πt/τ ) with τ ≪ T , where Ω$_{0}$ = ( ν$_{R}$τ ) | µE$_{0}$ | $^{2}$/ Δ = sγ $^{2}$/ 2Δ is the time-averaged res- onant Rabi frequency of the pulse train and s = ¯ I/I$_{sat}$ is the average intensity ¯ I = ν$_{R}$cϵ$_{0}$/ 2 ∫ dt | f ( t ) | 2 scaled to the $^{2}$S$_{1}$$_{/}$$_{2}$ ↔ $^{2}$P$_{1}$$_{/}$$_{2}$ saturation intensity. Note the net transi- tion rate is suppressed unless the single-pulse bandwidth is large compared to the hyperfine frequency ( ω$_{0}$τ ≪ 1), in which case Ω ≈ Ω$_{0}$. In our experiments, ω$_{0}$τ ≈ 0 . 08. Ω = | µ | 2 ∑ l E$_{l}$E$_{l}$$_{−}$$_{q}$ Δ ≈ Ω$_{0}$$^{(}$ ω$_{0}$τ e ω$_{0}$τ/ 2 − e − ω$_{0}$τ/ 2 ) , (2) H$_{eff}$ = ω$_{t}$a $^{†}$a + ω$_{0}$ 2 σ$_{z}$ + θ$_{p}$ 2 ∑ n δ ( t − nT ) $^{(}$σ$_{+}$ e i ( k ˆ x +Δ ωt ) + σ$_{−}$e − i ( k ˆ x +Δ ωt ) $^{)}$, (3) N − 1 ∑ n =0 Q$_{n}$ ≈ iη sin Nϑ$_{r}$/ 2 sin ϑ$_{r}$/ 2 e iϑ$_{r}$ ( N − 1) / $^{2}$σ$_{+}$ a + h.c. (7) V = exp [ − iH$_{0}$T ] exp$^{[}$ − iθ$_{p}$ 2 $^{(}$σ$_{+}$ e ik ˆ x + σ$_{−}$e − ik ˆ x ) ] (4) to satisfy the resonance condition for the red sideband, ϑ$_{r}$ ≡ ( ω$_{0}$ + Δ ω − ω$_{t}$ ) T = 2 πj , where j is an integer, then the sum in Eq. (5) is approximately given by, where θ$_{p}$ = Ω T is the change in the Bloch angle due to a single pulse, σ$_{z}$ is the Pauli-z operator, σ$_{±}$ are raising and lowering operators, ˆ x is the x − position operator of the trapped ion, a † and a are the raising and lowering operators of the x − mode of harmonic motion and the q parameter has been assumed to not be an integer or half- integer. In the interaction picture, the evolution operator after N pulses is given by V $^{N}$, where The coefficient in Eq. (7) is the same as the field ampli- tude created by a diffraction grating of N slits, whose narrow peaks have an amplitude equal to N . In the limit ω$_{t}$T ≪ 1, the other terms in Eq. (5) that drive the carrier and other sideband transitions can be neglected when N ≫ ( ω$_{t}$T η ) − $^{1}$. This is analogous to the de- structive interference of amplitudes away from the bright peaks in a diffraction grating. For ω$_{t}$/ 2 π = 1 . 64 MHz, T = 12 . 4 ns and η = 0 . 1, the sidebands are well-resolved when N ≫ 80. and H$_{0}$ = ω$_{t}$a $^{†}$a + 1 / 2( ω$_{0}$ + Δ ω ) σ$_{z}$ . The time evolution operator is given by, where D ( α ) = exp[ αa † − α $^{∗}$a ] is the harmonic oscil- lator displacement operator in phase space, and η = k $^{√}$$_{ℏ}$/ 2 mω$_{t}$ is the Lamb-Dicke parameter. In the Lamb-Dicke regime, η $^{√}$〈 a $^{†}$a 〉 + 1 ≪ 1, we can write D ( iηe iω$_{t}$$^{nT}$) ≈ 1 + iη ( e iω$_{t}$$^{nT}$a † + e − iω$_{t}$$^{nT}$a ) turning the sum in Eq. (5) into a geometric series. If, for exam- ple, the offset frequency between the combs Δ ω is tuned For many applications in quantum information, the motional modes of the ion must be cooled and initial- ized to a nearly pure state. Fig. 4 shows that the pulsed laser can also be used to carry out the standard tech- niques of sideband cooling [20] to prepare the ion in the motional ground state with near unit fidelity. The set-up also easily lends itself to implementing a two-qubit entan- gling gate by applying two fields whose frequencies are symmetrically detuned from the red and blue sidebands [21, 22]. By simultaneously applying two modulation fre- Q$_{n}$ ≡ σ$_{+}$e i ( ω$_{0}$ +Δ ω ) $^{nT}$D ( iηe iω$_{t}$$^{nT}$) + h.c. , (6) V N = exp[ − iH$_{0}$NT ]( ˆ I − i θ$_{p}$ 2 N − 1 ∑ n =0 Q$_{n}$ + O ( θ 2 $_{p}$)) (5)