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# Entanglement of Atomic Qubits using an Optical Frequency Comb 

D. Hayes, ∗ D. N. Matsukevich, P. Maunz, D. Hucul, Q. Quraishi, S. Olmschenk, W. Campbell, J. Mizrahi, C. Senko, and C. Monroe Joint Quantum Institute and Department of Physics, University of Maryland, College Park, MD 20742, USA 

(Dated: October 23, 2018) 

We demonstrate the use of an optical frequency comb to coherently control and entangle atomic qubits. A train of off-resonant ultrafast laser pulses is used to efficiently and coherently transfer population between electronic and vibrational states of trapped atomic ions and implement an entangling quantum logic gate with high fidelity. This technique can be extended to the high field regime where operations can be performed faster than the trap frequency. This general approach can be applied to more complex quantum systems, such as large collections of interacting atoms or molecules. 

The optical frequency comb generated from an ultra- fast laser pulse train has revolutionized optical frequency metrology [1–4] and is now playing an important role in high resolution spectroscopy [5]. The spectral purity yet large bandwidth of optical frequency combs also provides a means for the precise control of generic quantum sys- tems, with examples such as the quantum control of mul- tilevel atomic systems [6, 7], laser cooling of molecules or exotic atomic species [8, 9], and quantum state engineer- ing of particular rovibronic states in molecules [10, 11]. 

The optical frequency comb may become a crucial com- ponent in the field of quantum information science, where complex multilevel quantum systems must be controlled with great precision [12]. 

In this Letter, we report the use of an optical frequency comb generated from an ultrafast mode-locked laser to efficiently control and faithfully entangle two trapped atomic ion qubits. The optical pulse train drives stimu- lated Raman transitions between hyperfine levels [13, 14], accompanied by qubit state dependent momentum kicks [15]. The coherent accumulation of these pulses generates particular quantum gate operations that are controlled through the phase relationship between successive pulses. 

This precise spectral control of the process along with the large optical bandwidth required for bridging the qubit frequency splitting forms a simple method for control- ling both the internal electronic and external motional states of trapped ion qubits, and may be extended to most atomic species. This same approach can be applied to control larger trapped ion crystals with more advanced pulse-shaping techniques, and can also be extended to a strong pulse regime where only a few high power pulses are needed for fast quantum gate operations in trapped ions [15–17]. 

High fidelity qubit operations through Raman transi- tions are typically achieved by phase-locking frequency components separated by the energy difference of the qubit states. This is traditionally accomplished in a bottom-up type of approach where either two monochro- 

PACS numbers: 03.67.-a, 32.80.Qk, 37.10.Vz, 37.10.Rs 

matic lasers are phase-locked or a single cw laser is modu- lated by an acousto-optic (AO) or an electro-optic (EO) modulator. However, the technical demands of phase- locked lasers and the limited bandwidths of the modu- lators hinder their application to experiments. Here we exploit the large bandwidth of ultrafast laser pulses in a simple top-down approach toward bridging large fre- quency gaps and controlling complex atomic systems. By starting with the broad bandwidth of an ultrafast laser pulse, a spectral landscape can be sculpted by interfer- ence from sequential pulses, pulse shaping and frequency shifting. In this paper, we start with a picosecond pulse and, through the application of many pulses, generate a frequency comb that drives Raman transitions by stim- ulating absorption from one comb tooth and stimulating emission into another comb tooth as depicted in Fig. 1. 

Because this process only relies on the frequency differ- ence between comb teeth, their absolute position is ir- relevant and the carrier-envelope phase does not need to be locked [2]. As an example of how this new technique promises to ease experimental complexities, the control of metastable-state qubits separated by a terahertz was recently achieved using cw lasers that are phase-locked through a frequency comb [18], but might be controlled directly with a 100 fs Ti:sapph pulsed laser. 

At a fixed point in space, an idealized train of laser pulses has a time-dependent electric field that can be written as 

where f ( t ) is the pulse envelope, T is the time between successive pulses (repetition rate ν$_{R}$ = 1 /T ), N is the number of pulses in the train and ω$_{c}$ is the carrier fre- quency of the pulse. For simplicity, any pulse-to-pulse optical phase shift is ignored since the offset frequency in the comb is unimportant. The Fourier transform of Eq. (1) defines a frequency comb characterized by an envelope ˜ f ( ω ) ≡ F [ f ( t )] centered around the opti- 

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