Cooling the Collective Motion of Trapped Ions to Initialize a Quantum Register

Abstract

We report preparation in the ground state of collective modes of motion of two trapped 9 Be + ions. This is a crucial step towards realizing quantum logic gates which can entangle the ions' internal electronic states. We find that heating of the modes of relative ion motion is substantially suppressed relative to that of the center-of-mass modes, suggesting the importance of these modes in future experiments.03.67. Lx,32.80.Pj In physics, quantum computation [1] provides a general framework for fundamental investigations into subjects such as entanglement, quantum measurement, and quantum information theory. Since quantum computation relies on entanglement between qubits, any implementation of a quantum computer must offer isolation from the effects of decoherence, but also allow controllable and coherent interaction between the qubits. Cirac and Zoller [2] have proposed an attractive scheme for realizing a quantum computer, which is scalable to an arbitrary number of qubits. Their scheme is based on a collection of trapped atomic ions, where each qubit (one per ion) is comprised of a pair of the ions' internal states, while quantum information is transferred between different ions using a particular quantized mode of the ions' collective motion. This "quantum data bus" must first be initialized in a pure quantum state [2]: for example, its ground state [3]. The basics of this scheme have been demonstrated experimentally in a fundamental logic gate (a Controlled-NOT) operating between a motional mode of a single trapped ion and two of the ion's internal states [4]. In that work, the motional state was initialized in the ground state by laser cooling [5]. The next step towards implementing the Cirac-Zoller scheme is to cool at least one mode of collective motion of multiple ions to the ground state. In this Letter, we describe the first experiments to realize this goal. We also report significant difference between the decoherence rates of the center-of-mass and non-center-of-mass modes of motion.We confine 9 Be + ions in a coaxial-resonator-based rf (Paul) trap, similar to that described in Ref. [6]. The electrodes in this trap are made from 125 µm-thick sheets of Be metal, as shown in Fig. 1. We apply a potential φ(t) = V 0 cos(Ω T t) + U 0 to the (elliptical) ring electrode relative to the endcap electrodes. If several ions are trapped and cooled, they will naturally align themselves along the major axis of the ring electrode. The electrode's elliptical shape, in combination with U 0 > 0, allows a linear crystal to be maintained while suppressing rf-micromotion of the ions along this direction [7]. With V 0 ≈ 520 V, Ω T /2π ≈ 238 MHz, and U 0 =0 V, the pseudopotential oscillation frequencies are (ω x , ω y , ω z )/2π ≈ (4.6,12.7,17.0) MHz. With U 0 = 18.2 V, the frequencies become (8.6,17.6,9.3) MHz. Fig. 1 shows two ions confined in the trap and imaged with an f /3 lens system onto a position-sensitive photomultiplier tube.The ions are cooled and probed with laser beams whose geometry is indicated in Fig. ...

An efficient single-step scheme for manipulating quantum information of two trapped ions beyond the Lamb–Dicke limit

“…Jefferts et al [27] point out that applying a static electric field to push the ions along x may control the amplitude of ξ j and thus the Ω j of the ion micro-motion. Finally, we note that the duration of the two applied simultaneous pulses for realizing the above quantum controlled operation is not much longer than that for other schemes (see, e.g., [5,9,12,17]) operating in the LD regime. The shortest duration of the applied synchronous pulses for realizing the above manipulations of two trapped ions is about 10 −4 seconds, of the same order of the gate speed operating in the LD regime [8], for Ω 1 /2π ≈ 225 kHz [10].…”

“…the usual collective CM vibrational mode, must first be initialized in a pure quantum state, e.g., its ground state. Recently, the collective motion of two and four 9 Be + ions has been successfully cooled to its ground state [5,9]. This is a further step towards realizing the ion trap quantum computer.…”

“…Thus, hereafter we neglect all off-resonant transitions and the excitations of the higher vibrational modes [23,24,25], i.e., let F j =Î, and only label the CM mode excitations. We stress the following important fact: the effective Hamiltonian (3) reduces to that in previous works (e.g., [5,9,12,17] under the usual LD approximation: (m+ 1)η 2 j ≪ 1), to the lowest order of the LD parameter η j . Here m is the occupation number of the Fock state of the CM vibrational quanta.…”

“…The collective normal modes of oscillation shared by all of the ions form the information bus, through which all gate operations can be performed. In the past few years, several key features of the proposal in [6], including the production of entangled states and the implementation of quantum controlled operations between a pair of trapped ions, have already been experimentally demonstrated * lfwei@riken.jp † fnori@riken.jp ‡ Permanent address [5,8,9,10]. Also, several alternative theoretical schemes (see, e.g., [11,12,13,14,15,16,17]) have been developed for overcoming various difficulties in realizing a practical iontrap quantum information processor.…”

“…The Lamb-Dicke (LD) approximation is made in almost all of these schemes (see, e.g., [5,9,12,16,17]), in order to simplify the treatment of the laser-ion interaction. This approximation requires that the coupling between the external and internal degrees of freedom of the ion is very weak, i.e., the spatial dimension of the motion of the ground state of the trapped ion should be much smaller than the effective wavelength of the applied laser field (see, e.g., [18]).…”