Deterministic entanglement and tomography of ion–spin qubits

J P Home, M J McDonnell, D M Lucas, G Imreh, B C Keitch, D J Szwer, N R Thomas, S C Webster, D N Stacey and A M Steane^,1

Centre for Quantum Computation, Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK

¹ Author to whom any correspondence should be addressed.

Email: a.steane1@physics.ox.ac.uk

Received 7 June 2006
Published 12 September 2006

Contents

• Acknowledgments
• References

In recent years, entanglement has been detected and used in a variety of physical systems, but the number in which it can be created under deterministic control with good fidelity remains small. We present the implementation of a universal two-qubit quantum logic gate in a trapped ion experiment, demonstrating deterministic entanglement with 83% fidelity. Tomography was performed using a method robust against noise sources of practical relevance in the experiments. Our results illustrate the relative insensitivity of the gate to the initial temperature of the ions.

The logic gate exploits the oscillating force method [1]–[5]. Our study differs from previous trapped-ion experiments in the physical nature of the qubit, the trapping parameters, the light fields used to implement logic at the ions, and the laser sources, as follows.

First, each qubit is stored in a pure spin-half system, the spin of a ⁴⁰Ca ion in its ground state. Previous studies used hyperfine structure [1], [6]–[8] or a pair of electron orbitals [9, 10].^Note2  The spin-half system is convenient in its simplicity. The absence of `spectator levels', i.e. other states nearby in energy, reduces the movement of population out of the controlled Hilbert space. This process, sometimes called leakage error, is not directly correctable by quantum error correction [11, 12]. Spectator levels also exacerbate decoherence by photon scattering during the gate operation.

Next, our trap is comparatively large [13], so electric field noise from fluctuating patch potentials and Johnson noise in the electrodes is very small [14, 15]. We have measured very low heating rates, of order a few phonons per second, in the trap.

We use a single laser field to apply all the operations (single ion rotations and the two-qubit gate), changing only the amplitude and frequency to switch from one operation to another. This reduces the experimental complexity, and opens the possibility of future work in which the different parts of a pulse sequence could be brought together in a single chirped and shaped pulse.

Finally, the laser sources in our apparatus are all small semiconductor diodes, which do not require frequency doubling and which could in principle be packaged on a small optical chip. We thus begin to address the optical part of the challenge to scale this type of system up to a powerful quantum computer [16]–[18].

Let the spin-up/down states |↑, |↓ be the computational basis states of a qubit. We implement the two-qubit controlled-phase gate Z₁(₁) Z₂(₁) G(Ψ) where Z_k(₁) is a rotation of qubit k  =  1,2 by ₁ about the quantization axis, and G(Ψ) is a two-qubit operator represented in the computational basis |↑↑,|↑↓,|↓↑,|↓↓ by the diagonal matrix diag(1,exp(i Ψ),exp(i Ψ),1). For Ψ  =  π/2, or an odd multiple of π/2, this is equivalent to controlled-not up to single qubit rotations.

The gate mechanism is described in [1]. A spin-dependent force oscillating at frequency ω is applied to the pair of ions. Let f_k(m) be the complex amplitude of the force on ion k when it is in spin state m  =  ↑,↓. The COM (stretch) mode is excited by the sum (difference) force f₁(m₁) ± f₂(m₂) respectively. The effect on the quantum harmonic motion is simply to displace the state in its z–p phase space (as long as the position-dependence of the force is negligible during the displacement [19]). By choosing ω close to one of the normal mode frequencies, one can enhance the displacement and also simplify the dynamics, because then the excited mode simply describes a circle in phase space, which closes after time 2π/δ, and the excitation of the other mode can be neglected. This motion causes the system to acquire a phase _L proportional to the area of the loop in phase space.

Many arrangements are possible. We adopt f₁(m)  =  f₂(m), then the stretch mode is excited only for anti-aligned spin states. By tuning close to the stretch mode frequency ω_s, i.e. |δ|    ω_s – ω_c where δ ≡ ω – ω_s and ω_c is the COM mode frequency, we then have Ψ  =  _L  =  (π/2)(Ω_f/δ)² where Ω_f  =  |f(↑)–f(↓)| z_0s/, z_0s  =  (/(4 M ω_s))^1/2 and M is the mass of one ion.^Note3 

To measure the spin state of the ion pairs, we developed the following quantum state tomography method. Tomography in general requires the accumulation of a sufficient set {M_i} of measurements of a given state ρ (many copies of which are needed), in order to allow ρ to be reconstructed to some desired accuracy from {M_i}. Usually rotations are applied to the system which is then measured in a fixed basis.

This problem has been studied in various settings and, for example, sets of {M_i} of minimal size, or designed for certain experimental situations, have been discovered [20]. Our method is designed to be robust against noise and drift problems, and to be convenient in a fairly broad range of experimental settings. We achieve this by a judicious choice of rotation angles and by accumulating data sets in a form that should be well fitted by sums of orthogonal sinusoidal functions of known period. The least squares fit is robust because the functions in the fit are orthogonal, it averages the data over many timescales, and many noise processes affect the residuals not the fitted parameter values.

Let R(θ,) be a rotation of a single qubit through θ on the Bloch sphere about an axis of azimuthal angle . To analyse a given density matrix ρ, the qubits are first rotated and then measured in the |↑,|↓ basis, then ρ is re-prepared and one repeats with either the same or new rotation angles, until a sufficient set { M_i} has been acquired. Experimentally θ depends on a pulse area, on the phase of an r.f. oscillator. The latter is more easily adjusted than the former to a variety of precisely known values. Therefore we only use two values of θ, but many values of , see figure 3. For clarity, we give details for the case of two-qubits both undergoing the same rotation, but the principles apply more generally. In this restricted case, a general ρ can be almost, but not fully reconstructed.

Write where are the Pauli spin operators (starting with the identity). Let P(m₁, m₂) be the populations of ρ and P_R(m₁,m₂) be the populations after the rotation R(θ,) has been applied to both qubits. Then

and similarly for ↑↓,↓↑, where the coefficients a–e are simple sinusoidal functions of θ. By choosing different values of θ and , 12 independent real numbers can be extracted from the population measurement outcomes. Therefore three of the 15 independent real numbers which fully characterize ρ are not available when both qubits undergo the same rotation.

The tomography method consists of accumulating data at given θ as a function of , then fitting equation (1) and its partners for P_R(↑↓), P_R(↓↑) to the data, with the a–e coefficients as fitted parameters, cf figure 3. After using two θ values, the density matrix coefficients c_ij can be found from θ and a–e (with some redundancy). The inferred matrix ρ^M is not guaranteed to be positive definite. One can handle this by any suitable approach, for example by searching among physical density matrices ρ^P for the one which most closely matches ρ^M by some measure. We adopt this `maximum likelihood' method using the cost function where ρ^P is positive definite [21].

The apparatus consists of a linear r.f. Paul trap and laser system largely as described in [13]. We load single or pairs of ⁴⁰Ca ions by photoionization. A 6.3 MHz r.f. field provides radial confinement, with radial COM frequency ≊ 1 MHz. Axial confinement is provided by dc endcaps, tuned as described below. 397 and 866 nm lasers are used for Doppler cooling, fluorescence detection and spin state preparation (with 99% fidelity) by optical pumping. A further pair of lasers performs spin-state-selective shelving to D_5/2, which allows the fluorescence to be used to read out the spin state (with ~ 90% fidelity), see [22]. The ions are cooled in three dimensions by Doppler cooling, and along z by Raman sideband cooling to , measured as in [23].

The light field used for pulsed sideband cooling and all the coherent operations is a walking wave created by laser beams A, B (shown in figure 1) propagating at θ_L  =  58.9° to each other, with difference wavevector Δ k  =  4π sin (θ_L/2)/λ directed along the trap axis z. A quantization axis is set by a weak magnetic field directed at 3° to beam B (and θ_A  =  62° to beam A). All these axes are horizontal. To reduce sensitivity to the pulse area, we balance the ac Stark shifts on |↑, |↓ from either beam acting alone, using Ramsey interferometry. This results in polarization very close to linear,^Note4  which we make at angle β to the vertical for A, and horizontal for B. Each beam illuminates both ions equally. They are derived from the same laser, at a frequency detuned Δ_L/2π  =  30 GHz above the S_1/2–P_1/2 transition. A precise frequency difference ω  =  ω_A – ω_B is introduced between them by acousto-optic modulation.

The Zeeman splitting of |↑ from |↓ is ω₀  =  2π × 4800 kHz. At ω  =  ω₀ spin-flip transitions are resonantly driven by the π component of A combined with the σ^– component of B. Let Ω_c be the Rabi frequency for this `carrier' Raman process. The oscillating force is created by the periodic light shift from the σ^± components of the walking wave. The coupling strength Ω_s is given by , where ε ₊ , ε_π are polarization amplitudes for beam A. For our geometry, and Ω_f  =  2 η Ω_ssin(Δ/2), where η  =  Δ kz_0s is the stretch mode Lamb–Dicke parameter, and Δ is the phase angle between the forces, given by tan(Δ/2)  =  (cos θ_A tan β)^–1.

During the force pulse at ω ≊ ω_s, the carrier process is driven off-resonantly at detuning ω₀ ± ω. For ω₀ ± ω    Ω_c, the qubit states are pushed apart by a light shift

This is of the order 4 kHz in our experiments, and it gives rise to the single-qubit rotations Z_k(₁) in the gate, with ₁  =  Δ_c τ.

To achieve f₁(m)  =  f₂(m), we adjust ω_c so that the ions' separation d ≊ 9 μm is an integer number p of standing wave periods, Δ kd  =  2 π p. We used p  =  22,  ω_c/2π  =  500 kHz (η  =  0.133) in one experiment, and p  =  21, ω_c/2π  =  536.5 kHz (η  =  0.128) in another.

A given experimental sequence consists of cooling, spin preparation in |↓↓, then a spin echo sequence with the force pulse of duration τ in one or both of the gaps, followed by an analysis pulse and then spin state measurement; see inset to figure 2(b). The analysis pulse is the rotation R(θ,) in the tomography scheme. A perfect implementation would produce the maximally entangled state

before the analysis pulse, with r  =  2 ₁ – π/2. To assess the state ρ obtained in practice we use the fidelity F ≡ max_r E(r)|ρ|E(r). This compares ρ with the most closely matching member of the class |E(r).^Note5 

Figure 2. (a) Measured populations P(↑ ↑), P(↑ ↓) + P(↓ ↑) (•, ) versus τ after a spin-echo sequence with in the first gap, at δ/2π  =  12.6 kHz. Each point is the average of 500 repeats of the experimental sequence. The lines are fitted curves, using a model which assumes each qubit decoheres independently at rate Γ. (b) Parity signal vs. after a θ  =  0.46 π analysis pulse, for τ  =  77 μs. Each point is the average of 1000 repeats of the experimental sequence. The inset shows the pulse sequence.

We use `Schrödinger cat' experiments with a single ion [19, 24] to analyse the forced motion and characterize light shifts. We then use the two-ion behaviour as a function of τ with no analysis pulse to diagnose the setup. We model the results by assuming each qubit decoheres independently at rate Γ. For the case of a pulse only in the first gap, we then expect

where A  =  (1/4)  +  e^–2Γτ(cos(2Δ_c τ)  +  e^{–2|α(τ)|2})/8 and α(τ), Ψ(τ) are the motional displacement and phase as described in [1]. Since the 866 nm laser is on throughout the sequence of coherent manipulations, there is negligible population in the D_3/2 level, and thus P(↓↓)  =  1 – P(↑↑) – P(↑↓) – P(↓↑). Data is shown in figure 2(a), fitted with floated parameters Γ,δ,Ω_f and Δ . We obtained Γ  =  5.4 ms^–1, δ/2π  =  13 kHz, Ω_f/2π  =  23 kHz and Δ  =  1.6 from the fit; these values were consistent with our other information on δ, Ω_c and Δ. Note that so in this experiment Ψ  =  3π/2. When we use the model and fitted values to infer ρ at τ  =  2π/δ, we obtain fidelity F  =  0.7(1).

We next measured a lower bound more directly, F ≥ 2 |C|, by using an analysis pulse at θ  =  π/2 to deduce the coherence C  =  ρ_{↑↑,↓↓}  =  c₁₁–c₂₂ +  i(c₁₂ + c₂₁). This is obtained from the component at frequency 2 in the parity signal 1 – 2(P_R(↑↓) + P_R(↓↑)) versus (data shown in figure 2(b)). We observed F ≥ 0.74 (3). This result is more precise than the previous one because it does not rely on assumptions about the decoherence.

Finally, we performed tomography in another experiment where the pulse was inserted in both gaps. This has the effect of cancelling the single-bit rotations Z_k(₁). Both pulses had the duration τ  =  2π/δ, so the motion completes two loops in phase space, and the gate requires Ψ  =  π/4 for each loop. We set δ/2π  =  22.7 kHz and observed the signal with pulses of double length in order to adjust the laser intensity. We then used the carrier flopping rate to infer Ω_f, obtaining 2 π × 16.3 (9) kHz, giving the consistency check (Ω_f/δ)²  =  0.52(5).

The data and fitted curves for the tomography are shown in figure 3, and the inferred (maximum likelihood) density matrix is shown in figure 4. Owing to the absence of single ion addressing in the rotations and the measurements, the tomography only gives partial information on the internal elements of ρ (i.e. away from the corners in figure 4). However, we find that there is negligible population in |↑↓ and |↓↑ and therefore for the state in question, the tomography is complete. The inferred ρ had fidelity F  =  0.83 (2) and entanglement of formation [12] 0.54. The improvement is owing to better precision in pulse areas and a reduction in the total area by , which reduces decoherence by photon scattering.

Figure 3. Tomography signals for spin state after the spin echo ( in both gaps, δ/2π  =  22.7 kHz). The data points show PR(↑↑), PR(↑↓) + PR(↓↑) (•, ) versus for θ  =  0.66π (top panel) and θ  =  0.54π (bottom panel), as deduced from 500 repeats of the experimental sequence. The lines are the fitted curves (1). θ values were deduced from carrier flopping signals, taking into account a 0.1 μs dead time in the AOM; is accurately known from the r.f. signal generators. Inset: locus of θ, values on the Bloch sphere.

We estimate that the primary source of infidelity is photon scattering (12%), with additional contributions from imbalance in light intensity at the ions, pulse area imprecision, and motional decoherence. Had we used a gate method which is more sensitive to the prepared motional state, such as [25], the thermal effect alone would contribute a further 6% infidelity through imprecise sideband pulses. We therefore obtained a useful improvement by using a temperature-insensitive method.

To conclude, we have implemented a two-qubit quantum logic gate between a pair of trapped ⁴⁰Ca ions with 83% fidelity. One of the notable features of our study is that the qubit is physically represented by a very good approximation to a strictly two-state system: leakage out of the computational Hilbert space is completely negligible after repumping by the 866 nm laser. We have also demonstrated the technical simplification of working with a single pair of laser beams for all coherent manipulations. This introduces an ac Stark shift whose effects are O(π) and can be taken into account.

Acknowledgments

This study was supported by the EPSRC through QIP IRC, the Royal Society, the European Union through the FET project `QGATES' and research training network `CONQUEST', the National Security Agency (NSA) and Advanced Research and Development Activity (ARDA) (P-43513-PH-QCO-02107-1)

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Notes

Note2  Ion–spin qubits are entangled in H Häffner et al (2005 Appl. Phys. B 81 151), by transfer from an S_1/2/D_5/2 qubit.

Note3  The motion leads to a two-qubit effect, Ψ ≠ 0, only if ω_s ≠ ω_c. Since this difference is owing to the ions' Coulomb repulsion, the two-qubit gate relies, as expected, on a two-body interaction.

Note4  Beam A drives the carrier by a Raman process off-resonant by the Zeeman splitting of |↑ and |↓. This creates a light shift whose cancellation results in ε_–/ε ₊  ≊ 1.02 for this beam.

Note5  This choice is reasonable because it is equivalent to using the data to extract the value of r.