Versatile laser-free trapped-ion entangling gates

Abstract

We present a general theory for laser-free entangling gates with trapped-ion hyperfine qubits, using either static or oscillating magnetic-field gradients combined with a pair of uniform microwave fields symmetrically detuned about the qubit frequency. By transforming into a 'bichromatic' interaction picture, we show that either ${\hat{\sigma }}_{\varphi }\otimes {\hat{\sigma }}_{\varphi }$ or ${\hat{\sigma }}_z\otimes {\hat{\sigma }}_z$ geometric phase gates can be performed. The gate basis is determined by selecting the microwave detuning. The driving parameters can be tuned to provide intrinsic dynamical decoupling from qubit frequency fluctuations. The ${\hat{\sigma }}_z\otimes {\hat{\sigma }}_z$ gates can be implemented in a novel manner which eases experimental constraints. We present numerical simulations of gate fidelities assuming realistic parameters.

Keywords: atomic physics; geometric phase gates; quantum computing; quantum gates; quantum logic; quantum physics; trapped-ions.

Figure 1.

Relative strengths of the gate Rabi frequencies versus 4Ω_μ/δ for the first three resonances in the bichromatic interaction picture when the microwave field term (∝Ŝ_i) does not commute with the gradient term (∝Ŝ_j) in the Hamiltonian. Note that at the point where intrinsic dynamical decoupling occurs (dotted line), the values of the J_1,2 Rabi frequencies are near their maximum values.

Figure 2.

Numerical simulation of the fidelity $\mathcal{F}$ of the maximally entangled Bell state of equation (30) versus time (normalized to t_f), for the ${\widehat{\sigma }}_{\varphi }\otimes {\widehat{\sigma }}_{\varphi }$ gate described in this section. In both panels, the high frequency blue line corresponds to equation (26), i.e. the ion frame Hamiltonian, and the the orange line corresponds to equation (28), i.e. the bichromatic interaction picture Hamiltonian. Panel (a) shows a gate with no pulse shaping, where large-amplitude oscillations at δ make the ion frame gate fidelity highly sensitive to the exact value of t_f. Panel (b) shows the same gate operation including a τ = 10 μs Blackman envelope at the beginning and the end of the gate sequence; the ion frame fidelity smoothly approaches the interaction picture fidelity at the end of the gate.

Figure 3.

(a) Fidelity $\mathcal{F}$ of the gate operation creating the maximally entangled Bell state of equation (30) versus static qubit frequency shift ε normalized to gradient strength Ω_g. Data in both panels are calculated by numerical integration of the full ion frame Hamiltonian given by equation (26). Here, Ŝ_i = Ŝ_x, Ŝ_j = Ŝ_z, Ω_g/2π = 1 kHz, and δ/2π = 1.5 MHz (chosen to be experimentally realistic), with varying values of Ω_μ. Fidelities are plotted for the intrinsically dynamically decoupled ${\widehat{\sigma }}_{\varphi }\otimes {\widehat{\sigma }}_{\varphi }$ gate (4Ω_μ/δ ≃ 2.405, black solid line), the fastest ${\widehat{\sigma }}_{\varphi }\otimes {\widehat{\sigma }}_{\varphi }$ gate (4Ω_μ/δ ≃ 1.841, red dashed line), the ${\widehat{\sigma }}_{\varphi }\otimes {\widehat{\sigma }}_{\varphi }$ gate shown in figure 2 (4Ω_μ/δ ≃ 1.333, orange solid line), as well as the fastest ${\widehat{\sigma }}_Z\otimes {\widehat{\sigma }}_Z$ gate with a spin-echo (4Ω_μ/δ ≃ 3.054, green dotted line) described in section 4.2. (b) Infidelity $1-\mathcal{F}$ of the ${\widehat{\sigma }}_{\varphi }\otimes {\widehat{\sigma }}_{\varphi }$ gate versus the frequency ω_ε at which the qubit shift ε oscillates, for the intrinsically dynamically decoupled gate (solid black) and the fastest ${\widehat{\sigma }}_{\varphi }\otimes {\widehat{\sigma }}_{\varphi }$ gate (red dashed), for a particular value of ε₀ = Ω_g. This value of ε₀ represents a significantly larger qubit shift than is typically seen experimentally, where ∣ε₀∣ ≪ Ω_g [16]. The gray lines show the effect of ±1% relative changes in the ratio 4Ω_μ/δ for the intrinsically dynamically decoupled gate.

Figure 4.

Infidelity $1-\mathcal{F}$ of the ${\widehat{\sigma }}_z\otimes {\widehat{\sigma }}_z$ gate versus the frequency ω_ε at which the qubit shift ε oscillates, for the intrinsically dynamically decoupled gate (solid black) and the fastest ${\widehat{\sigma }}_z\otimes {\widehat{\sigma }}_z$ gate (red dashed), for a particular value of ε₀ = Ω_g. This value of ε₀ represents a significantly larger qubit shift than is typically seen experimentally, where ∣ε₀∣ ≪ Ω_g [16]. The gray lines show the effect of ±1% relative changes in the ratio 4Ω_μ/δ for the intrinsically dynamically decoupled gate. Data are calculated by numerical integration of the full ion frame Hamiltonian given by equation (26).