Characterization and active cancellation of power-line-induced motional-mode frequency noise in a trapped-ion system

Abstract

The stability of motional-mode frequency is essential for realizing high-fidelity quantum gates in trapped-ion quantum computing. While broadband Gaussian noise has been extensively studied and mitigated using pulse shaping techniques, the impact of coherent periodic noise has remained largely unexplored. Here we report a systematic investigation of 60-Hz power-line noise and its effect on the secular frequencies of a single ${}^{171}\mathrm{Yb}^{+}$ ion. Using spin-echo Ramsey spectroscopy, we characterize the amplitude and phase of the resulting secular-frequency modulation and validate this characterization via passive phase correction of the Ramsey sequence. Building on this, we implement active cancellation by injecting a compensation tone into the set-point of a PI controller that stabilizes the trap RF drive amplitude. A phasor-fitting procedure optimizes the amplitude and phase of the compensation signal, enabling near-complete suppression of the 60-Hz component. With active cancellation engaged, the coherence time of a radial motional mode is extended from approximately 10 ms to 35 ms, consistent with the limit set by motional heating. Our results provide both a clear characterization of periodic motional-mode noise and a practical framework for its suppression in trapped-ion quantum computing platforms.

Figures (4)

• Figure 1: Amplitude characterization of sinusoidal secular-frequency modulation in a radial motional mode using CP sequences with (a) $n = 1$ (echo Ramsey), (b) $n = 2$, and (c) $n = 3$ refocusing pulses. Blue circles show the measured Ramsey signal $\expval{\sigma_z}$ with error bars indicating shot noise ($N = 500$ shots per point). Solid red curves show fits to the analytic sinusoidal-modulation model [Eqs. (\ref{['eq:C1']})--(\ref{['eq:C3']})] including heating effects. Best-fit parameters are (a) $A / 2\pi = 53.9(11)~\mathrm{Hz}$, $\dot{\bar{n}} = 6.4(8)~\mathrm{s}^{-1}$; (b) $A / 2\pi = 40.4(12)~\mathrm{Hz}$, $\dot{\bar{n}} = 7.1(15)~\mathrm{s}^{-1}$; (c) $A / 2\pi = 45.5(17)~\mathrm{Hz}$, $\dot{\bar{n}} = 13.6(26)~\mathrm{s}^{-1}$. Dashed black curves show the independently simulated heating envelope alone; see Supplementary Information, Sec. 3 for simulation details. Revivals occur at evolution times set by the 60-Hz line cycle: for echo Ramsey at integer multiples of $1/30\,\mathrm{s}$, and for multi-pulse CP at characteristic fractions (e.g., $1/15\,\mathrm{s}$ and $1/10\,\mathrm{s}$), as predicted by the corresponding $C_n(\tau)$ expressions.
• Figure 2: Phase characterization of sinusoidal secular-frequency modulation in radial motional modes using delay-swept spin-echo Ramsey measurements. (a) Delay-swept spin-echo Ramsey measurement at a representative trigger delay of $2~\mathrm{ms}$. The data are plotted in the same format as in Fig. \ref{['fig:char_ampl']}: the experiment (blue circles with error bars indicating shot noise, $N = 500$ shots per point), a fit including sinusoidal noise and heating effects (solid red curve), and the heating effects alone (dashed black curve). Best-fit parameters are $A / 2\pi = 56.8(7)~\mathrm{Hz}$, $\phi_d = 0.913(3)\pi$, and $\dot{\bar{n}} = 15.5(8)~\mathrm{s^{-1}}$. Orange diamonds show the passive phase-correction verification with error bars indicating shot noise ($N = 150$ shots per point); see Verification via spin-echo Ramsey tracking. (b) Extracted modulation amplitude (dark-gray circle and bright-gray diamonds for the X and Y modes, respectively) and phase (green circle and purple diamonds for the X and Y modes, respectively) for the two radial modes as a function of the delay relative to the power-line trigger. The fitted phase-evolution slopes (dashed green and dotted purple curves for the X and Y modes, respectively) are $2\pi \times 66.7(3)~\mathrm{s}^{-1}$ for the X mode and $2\pi \times 62.5(9)~\mathrm{s}^{-1}$ for the Y mode, both of which are close to the expected value of $2\pi \times 60~\mathrm{s}^{-1}$, confirming that the modulation is synchronous with the 60-Hz power-line signal.
• Figure 3: Active cancellation of sinusoidal secular-frequency modulation using phasor-based optimization. (a) Phasor diagram used for the least-squares optimization of the 60-Hz compensation signal. Blue arrows denote the injected compensation phasors $V_i \hat{u}_i$ for four trial settings. The red arrow indicates the extracted 60-Hz noise phasor $-V \hat{u}$. The extracted noise phasor has amplitude $V = 14(3)~\mathrm{mV}$ and phase $\arg(-\hat{u}) = 102(10)^{\circ}$, and the scale factor is $r = 0.38(4)~\mathrm{mV/Hz}$. Purple arrows show the resulting residual phasors $V_i \hat{u}_i - V \hat{u}$; black dots mark their endpoints, whose amplitudes are proportional to the measured residual modulation amplitude $A_i$ (after applying the scale factor $r$ defined in the text). (b) Echo-Ramsey signal of a radial motional mode with and without active cancellation. Blue circles and orange diamonds show the data without and with cancellation applied, respectively, with error bars indicating shot noise ($N = 150$ shots per point) for both datasets; the dashed black curve shows the independently simulated heating envelope (using an average heating rate $\dot{\bar{n}} = 15~\mathrm{s}^{-1}$). Active cancellation suppresses the 60-Hz modulation and extends the coherence time from $\sim 10~\mathrm{ms}$ to $\sim 35~\mathrm{ms}$, close to the heating-limited value.
• Figure 4: Schematic of the RF stabilization and compensation circuit. The set-point is combined with an externally injected compensation tone, and the resulting reference is compared with a rectified monitor signal derived from the trap RF output. The error signal drives a servo that controls the RF-drive amplitude via amplitude modulation of the RF source. After filtering by the helical resonator, the RF is delivered to the trap electrodes, and a fraction of the RF is fed back for rectification and stabilization.