Spatially uniform single-qubit gate operations with near-field microwaves and composite pulse compensation

The trap conforms to a symmetric five-wire surface-electrode Paul trap geometry [23] fabricated on a 11 × 11 mm² silicon die (figure 1(a)) similar to the designs reported in [24–26]. Electrodes etched into three sputtered aluminum layers are separated by insulating silicon dioxide films. rf potentials applied to two parallel electrodes provide radial ion confinement (in the x–y plane, figure 1(b)) 59 μm above the Metal 2 layer (figure 1(c)). Quasi-static potentials applied to segmented dc electrodes confine and transport ions along z. The rf electrodes are 30 μm wide along x with an inner edge-to-edge separation of 92 μm. The segmented dc electrodes are 56 μm wide along z except for six 100 μm wide electrodes bordering the loading slot. Two additional dc electrodes traversing the entire length of the trapping region are used to apply uniform x–y fields and to rotate the radial principal axes. Each electrode is separated from neighboring conductors by 4 μm gaps. Each dc electrode incorporates a 60 pF plate capacitor (1 mm² area) to filter unwanted rf pickup [24]. A loading slot allows a thermal beam of neutral Yb to reach the trapping volume from an oven located below the trap.

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The trap includes a pair of conductor-backed coplanar waveguides that generate local microwave magnetic fields. Each waveguide includes a 40 μm wide electrode with 4 μm gaps to neighboring conductors, and 10 μm of SiO₂ separate the coplanar layer from the ground plane below (figure 1(b)). The waveguides support a ω_mw = 2π × 12.64 GHz quasi-transverse electromagnetic (TEM) guided mode resonant with the hyperfine splitting between the F = 0 and 1 manifolds in the ²S_1/2 ground state of ¹⁷¹Yb⁺ (see figure 3(a)). In the ideal case, currents in each waveguide generate a magnetic field along the trapping axis

$$\begin{equation} \vec{B}_k(z,t)=(\hat{x}\beta_{x,k} + \hat{y}\beta_{y,k})I_{k}(z)\cos{( \omega_{\mathrm{mw}}t + \phi_{k})}, \end{equation} \tag{ 1 }$$

where k = 1,2 is an index for the two waveguides, ϕ_k is the phase of the microwave current source, I_k(z) is the current in each waveguide, and β_x,1 = β_x,2 ≃ 0.08 mT A⁻¹ and β_y,1 = −β_y,2 ≃ 0.17 mT A⁻¹ are properties of the waveguide mode. β_y,1 = −β_y,2 due to the symmetric placement of the waveguides around the trapping axis. The waveguides terminate in an open circuit at a position that is approximately a quarter-wavelength from the trap center, which produces a standing wave field with maximum amplitude and uniformity in the gate region. A smaller traveling wave component also exists due to on-chip attenuation that generates an amplitude difference between forward and backward propagating waves. Although a traveling wave would produce more uniform fields, a standing wave produces a larger maximum field amplitude for a given input power. A standing wave design simplifies the trap by eliminating the need for on-chip resistive termination or for coupling power off the chip.

Far from the trap center, the waveguides meander to fit a complete wavelength on the chip and then terminate on wirebond pads at the edge of the chip. Extending the waveguides to a full wavelength in this way places a current node at the wirebond pads and reduces the potential for resistive power loss in the connections. A series of 25.4 μm diameter aluminum wirebonds connect the chip waveguides to two printed circuit board (PCB) waveguides that route microwaves from the edge of the trap package (figure 2). Connections between the PCB top level ground and the on-chip metal 1 and 2 ground planes are symmetric about each microwave electrode. Quarter-wave transformers match the 50 Ω impedance of the PCB waveguide to the 27 Ω on-chip characteristic impedance. The PCB is fabricated using a 254 μm thick Rogers 4350B substrate with two 18 μm thick copper foil conductive layers and a 3–6 μm electroless nickel immersion gold (ENIG) finish. The skin depth in the PCB at ω_mw is comparable to the thickness of the lossy nickel layer resulting in ≈3 dB of power loss between the microwave connector and wirebonds. Much of this power loss could be recovered by replacing the ENIG finish with an electroplated gold surface.

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Figure 3(a) shows the transitions in the ground state hyperfine manifold of ¹⁷¹Yb⁺ addressed by the microwave field. A static 0.74 mT field along y defines the quantization axis and lifts the degeneracy of the F = 1 triplet. We select the clock states |²S_1/2,F = 1,m_F = 0〉 ≡ |↑〉 and |²S_1/2,F = 0,m_F  = 0〉 ≡ |↓〉 as the qubit states. We observe Rabi oscillations at a frequency Ω = 2π × 0.49 MHz (figure 3(b)) by optically pumping into |↓〉, applying microwave power for a variable interval of time, and measuring the resulting population transfer into the F = 1 manifold through state-selective fluorescence of the 369.5 nm cycling transition (figure 3(a)) [27].

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Currents in each waveguide generate an oscillating magnetic field which contains both a π − polarized component that couples to the Δm_F  = 0 clock transition and also a transverse-polarized component that couples to the Δm_F  = ± 1 transitions. Figure 3(c) shows the hyperfine spectrum measured with microwave power applied to a single waveguide. From the relative Rabi frequencies of the transitions, we estimate the ratio of polarization components for each waveguide as |β_x,k|/|β_y,k| ≈ 0.46.

The polarization of the near-field microwaves may be controlled by adjusting the relative amplitude and phase of the microwave currents in the two waveguides. In particular, the polarization may be aligned along the quantization axis, thereby maximizing the qubit transition Rabi frequency while also suppressing off-resonant Δm_F  = ± 1 transitions. The active and passive microwave components supplying energy to these electrodes are not perfectly power-balanced and phase-matched. We calibrate the microwave sources by first driving each waveguide independently to map the relationship between source power and Rabi frequency (figure 4(a)). Once the field amplitudes from the waveguides have been equalized, the relative phase between microwave currents ϕ_r can be varied to produce an arbitrary linear polarization in the x–y plane. Figure 4(b) shows the resonant Rabi frequency for each of the three transitions as ϕ_r is varied, demonstrating the desired suppression of the Δm_F  = ± 1 transitions at ϕ_r = π. The minimum measured Rabi frequencies of the Δm_F  = ± 1 transitions are less than 8% of the Δm_F  = 0 frequency. We suspect that the suppression is limited by a residual z component of the quantization field, and the mismatch between the Δm_F  = ± 1 curves in figure 4(b) is caused by frequency dependence in the microwave delivery system.

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Polarization control is useful with any hyperfine qubit because it can suppress unwanted transitions as gate speeds are increased or Zeeman splittings are reduced. In the context of the experiments presented here, with a 10.4 MHz Zeeman splitting and Ω = 2π × 0.5 MHz, off-resonant coupling would limit the fidelity of a π/2 gate (square envelope) to $\mathcal {F} < 1 - 4 \times 10^{-4}$ if only a single microwave waveguide were used.

The standing wave current in the waveguides produces a microwave field with non-uniform amplitude along the trap axis. Figure 5 shows the measured Rabi frequency of the qubit transition at several positions along the trap axis. We observe a maximum Rabi frequency of 2π × 0.52 MHz, corresponding to a field amplitude of 0.037 mT, located z₀ = 957 μm from the loading slot center. Finite element calculations predict an antinode location at z = 895 μm in reasonable agreement with the experiment. These models indicate that the maximum field corresponds to a local current in each electrode of I_z(z₀) ≈ 0.1 A. The maximum Rabi frequency is presently limited by the output power of the 3 W amplifiers and losses in the microwave delivery system (see the appendix). Heating due to on-chip dissipation in the 1 μm thick aluminum electrodes would ultimately determine the maximum current density and associated Rabi frequency.

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The microwave field non-uniformity appears as an effective amplitude error when implementing global single-qubit rotations on multiple ions located at different positions in the trap. To improve single-qubit gate uniformity, we implement global gates using broadband compensating pulse sequences [28, 29]. This technique replaces a simple pulse with a sequence of pulses whose phases are chosen to nearly cancel the effective microwave amplitude error. The excitation profiles of such pulse sequences enable global rotations on many qubits, although the microwave amplitude may differ significantly between distant ions.

Here we demonstrate the error-canceling properties of composite X-gates constructed from the first-order SK1 [30, 31] and second-order BB1 [32] sequences. The experiments prepare the qubit in |↓〉, apply a logical X-gate, and then measure the population in the F = 1 manifold. To simulate the effect of systematic over/under rotations, we uniformly scale the pulse areas of every pulse in the sequence by adjusting the pulse duration. Figure 6 plots the measured excitation profiles produced by compensated gates, overlayed on the signal predicted by theory. Gates performed with the BB1 sequence are more robust to large variations in pulse area and require no additional time to execute compared to SK1, therefore BB1 is a superior choice for compensating the non-uniform field amplitude observed in figure 5. In contrast, the passband behavior of the SK1 sequence may make it a better choice for applications where localized gates are a primary objective. Assuming the field non-uniformity is the sole source of error, we calculate theoretical fidelities of X-gates. For the 6% amplitude deviation observed at z = 300 μm, a simple rotation performs a global X-gate with a minimum fidelity $\mathcal {F} \geqslant 0.995$, whereas SK1 and BB1 perform the same gate with minimum fidelities of $\mathcal {F} \geqslant 1 - 1.5 \times 10^{-4}$ and 1 − 2.2 × 10⁻⁷, respectively.

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As a demonstration of uniform global gates, we perform an experiment where n sequential logical X-gates are applied to a qubit initialized in |↓〉. We calibrate gate times so that an ion located at the microwave amplitude maximum (z₀ = 957 μm) experiences nearly perfect rotations. Qubits displaced from the field maximum rotate at lower Rabi frequencies, acquiring an under-rotation error that accumulates as n increases. We measure the F = 1 population as a function of ion axial position and number of sequential X-gates. For X-gates implemented by simple rotations we observe fringes (figure 7(a)) arising from the local qubit falling behind by an entire Rabi cycle relative to the maximal Rabi frequency. Instead, when implementing SK1 or BB1 pulses (figures 7(b) and (c)) the error accumulates so slowly that the excitation profile remains flat over the trapping region after n = 55 logical X-gates. Our ability to resolve fringe structure in these cases is currently limited by systematic state-preparation and measurement errors and by the number of simple pulse operations we can implement. In theory, neither the BB1 nor the SK1 sequence imposes a penalty on the maxiumum achievable fidelity when the primary sources of error can be compensated by the sequences, and we observe no reduction in F = 1 population due to the pulse sequences at the field calibration position, z₀, within our measurement error.

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We analytically calculate the fidelity scaling of the sequential logical X-gates as a function of the microwave field strength. For simple rotations, the fidelity drops as $\mathcal {F} = |\kern-2pt\cos [ \epsilon (z) \pi n / 2 ] |$, where (z) = [Ω(z) − Ω(z₀)]/Ω(z₀) is the fractional difference in Rabi frequencies between the ion location z and the field maximum. For SK1 pulses the fidelity scales as $\mathcal {F} = |1 - \frac {15}{128} \pi ^4 \epsilon (z)^4 n + O( \epsilon (z)^6)|$, and for BB1 the fidelity scales as $\mathcal {F} = |1 - \frac {5}{1024} \pi ^6 \epsilon (z)^6 n + O( \epsilon (z)^8)|$.

A schematic view of the microwave supply system is shown in figure A.1. An Agilent 83623B Swept-Signal Generator provides a stable microwave local oscillator (LO) signal that is approximately 300 MHz detuned from the qubit resonant frequency near 12.64 GHz. Delivery of this signal is controlled by an American Microwave SW-218 high-speed rf switch. The signal is split to supply the local oscillator ports of two frequency mixers. Two separate direct digital synthesis (DDS) boards with synchronized clocks supply the intermediate frequency (IF) signals near 300 MHz. The DDS outputs have independently controllable amplitude, phase and frequency and these signals are amplified prior to mixing with the LO signal. The mixers reject the carrier frequency and produce two rf sidebands at ω_mw = ω_lo ± ω_dds where one of these sidebands is tuned to the frequency of the desired hyperfine transition, while the other is far off resonance. Both signals are amplified in separate Mini-Circuits ZVE-3W-183 +  amplifiers and routed to a coaxial feedthrough port on the UHV chamber. A pair of circulators protect the amplifiers from accidental high power reflections and also provide a dc ground on the microwave lines. Inside the chamber the microwaves are carried by two coaxial lines with Kapton dielectric and braided conductor, selected for UHV compatibility. The combined power loss in the feedthrough and Kapton cables is ≈5.3 dB.

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We mount chips to a gold electroplated stainless steel plate that also serves as a structurally rigid platform for mounting other components, including microwave connectors, PCB, grounded screen and a 100-pin ceramic pin-grid array (CPGA). Vacuum compatible Au/Sn solder bonds the microwave connectors (Corning Gilbert GPO) to the top metal layer of the PCB, and a retention block secures these connectors to the steel plate to reduce stress in the solder joints. The chip is bonded to the interface plate with electrically conductive epoxy (Epoxy Technology H21D) which accommodates the differential thermal expansion between the stainless steel mounting plate and the silicon chip during vacuum bake-out. Wire bonds at the chip perimeter establish electrical connections to the CPGA to provide both dc and rf trapping potentials, while wire bonds to the PCB supply microwave energy to the chip (figure 2(b)). To minimize scattered laser light, wire bonds are excluded from regions where cross-chip laser access is required.