Spin-Dependent Forces on Trapped Ions for Phase-Stable Quantum Gates and Entangled States of Spin and Motion

Abstract

Favored schemes for trapped-ion quantum logic gates use bichromatic laser fields to couple internal qubit states with external motion through a "spin-dependent force." We introduce a new degree of freedom in this coupling that reduces its sensitivity to phase decoherence. We demonstrate bichromatic spin-dependent forces on a single trapped 111 Cd + ion, and show that phase coherence of the resulting "Schrödinger-cat" states of motion depends critically upon the spectral arrangement of the optical fields. This applies directly to the operation of entangling gates on multiple ions.PACS numbers: 42.50.Vk,03.67.Mn,03.67.Lx Trapped atomic ions have a number of desirable features that make them well suited for quantum information applications [1]. Pairs of hyperfine ground states provide an ideal host for quantum bits (qubits) that can be manipulated and entangled via optical Raman transitions [2]. Since the first proposal for entangling trapped ions [3], significant theoretical advances have relaxed the constraints on implementation to realize more robust entanglement schemes [4,5,6,7]. These schemes rely on a spin-dependent force acting on each ion where "spin" refers to the effective Pauli spin associated with the ionqubit's two-level system. Acting on a single ion, the force can entangle the ion's spin degree of freedom with its motion; following disentanglement, the ion acquires a net geometric phase that is spin dependent [6]. When the force is applied to two ions, the geometric phase depends nonlinearly on their spins through their mutual Coulomb interaction, generating entanglement.Optical Raman fields are a convenient way to create strong spin-dependent forces for hyperfine ion-qubits [8,9]. For example, an entangling gate based on aσ zdependent force has been realized with a state-dependent AC Stark shift from Raman fields [10], whereσ x,y,z are the Pauli operators. Unfortunately, suchσ z gates are not compatible with magnetic field insensitive (or "clock") qubit states [2,11], and are therefore open to qubit phase decoherence from fluctuating magnetic fields. An alternative solution is the Mølmer-Sørensen (MS) gate [4,12,13], which uses a more complicated arrangement of bichromatic fields to realize aσ φ -type force whereσ φ is a linear combination ofσ x andσ y operators. While the gate does work on clock states, it can have a significant phase instability due to the hyperfine coherences required to generate the spin dependence. All gates, whatever the spin dependence, are susceptible to fast phase fluctuations of the Raman fields during the course of gate evolution. However, the MS gate can also be susceptible to slow phase drifts, which cause changes in the spin dependence of the force and results in a phase instability between gates.In this Letter we show how to overcome phase drifts in the MS gate, allowing clock-state benefits to be realized for ion-based qubits. Such ideas may be useful in other experimental settings since the MS scheme is a general paradigm for entanglement [14]. A bichromatic for...

Implementation of Grover’s quantum search algorithm in a scalable system

“…The implementation of Grover's algorithm reported here complements the repertoire of multi-qubit quantum algorithms recently demonstrated in the scalable system of trapped atomic ions [15,16,17,18]. Unlike these earlier ion trap demonstrations, we use magnetically-insensitive "clock state" qubits and particular entangling gates that are uniquely suited to such qubits while remaining insensitive to external phase drifts between gates [19,20,21].…”

“…Finally, each qubit is detected with greater than 97% efficiency by uniformly illuminating the ions with resonant laser radiation and observing the ion fluorescence on an intensified CCD camera. The Mølmer-Sørensen gate directly entangles the clock state qubits and is insensitive to the relative optical phase of the Raman laser beams between gates [19,20,26]. This is an important consideration when multiple entangling gates are implemented because it suppresses decoherence from magnetic fields and optical phase noise that may fluctuate from gate to gate.…”

“…There are additional phases not included in the above equation pertaining to the ion-ion spacing, the phase of the oscillator that defines the Raman beam beatnote, and Stark shifts from the applied Raman beams [19,21]. These phases are set to zero for the present case since they are fixed before the experiment is run by first synchronizing the phase of the entangling gate with the phase of microwave π/2 pulses [19].…”

“…The experiment is performed with two 111 Cd + ions confined in a three-layer linear ion trap with axial frequency ω z /2π=2.0MHz [19,20,25]. The S 1/2 ground state hyperfine levels |F = 0, m f = 0 (denoted by |0 ) and |F = 1, m f = 0 (denoted by |1 ), separated in frequency by ω 0 /2π= 14.5 GHz, serve as qubit levels.…”

Entanglement of trapped-ion clock states

“…Robust schemes for gates based on spin-dependent forces have been proposed [9,10,11,12] and experimentally implemented [13,14] that, for example, relax the purity requirement on the initial motional state of the ions. Here we report the realization of one such entangling gate for pairs of trapped 111 Cd + ions that uses an advantageous implementation [15,16] of the Mølmer-Sørensen (MS) scheme [9,13]. The implementation reduces sensitivity to optical phase drifts through an appropriate Raman beam setup and reduces sensitivity to magnetic field fluctuations through the use of magnetic-field insensitive clock states [17].…”