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 
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 a cancellation scheme 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 the cancellation applied, 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.
Keywords: Power-line-induced noise, Noise cancellation, Motional-mode frequency modulation, Trapped-ion system
Subject terms: Engineering, Optics and photonics, Physics
Introduction
In trapped-ion quantum computers, the shared motional modes of the ions provide the quantum bus that mediates multi-qubit entangling gates1–4. The long-term stability of these modes is therefore a central requirement for scalable, high-fidelity quantum logic operations5–7. It is likewise essential for continuous-variable quantum computing and bosonic-encoding protocols that rely on precise phase-space control and phase accumulation of motional states8,9.
In many cases, dephasing in trapped-ion systems is modeled as stationary Gaussian noise and analyzed using the filter-function framework, where a control sequence acts as a frequency-domain filter on the noise power spectral density, enabling both noise spectroscopy and the systematic design of dephasing-robust operations10–13. More generally, this filter-function framework14 has been widely applied across diverse platforms, including nuclear magnetic resonance15, superconducting circuits16,17, nitrogen-vacancy (NV) centers in diamond18, and semiconductor quantum dots19.
For motional modes in particular, frequency fluctuations are often characterized by a power spectral density20–22. This spectral characterization has enabled substantial progress in mitigating dephasing from stochastic Gaussian noise through noise-robust control techniques such as pulse shaping13. However, trapped-ion experiments also face structured noise sources that are coherent and periodic, leading to deterministic modulation of the secular frequency with a well-defined phase relative to an external reference23,24. Because the phase of such modulation is not captured by a stationary noise power spectral density alone, a common practical workaround is to phase-synchronize each experimental shot to the underlying periodic reference.
A prominent example of such structured noise is power-line noise at 50/60 Hz, which is encountered across diverse platforms, including superconducting circuits25, NV centers in diamond26,27, and semiconductor quantum dots28. In trapped-ion systems, power-line noise likewise constitutes a well-known structured noise source that can limit qubit coherence via power-line-synchronous magnetic-field fluctuations29; accordingly, it is common practice to synchronize each experimental shot to a fixed phase of the line cycle (line triggering) to reduce its impact30–33. Building on this practice, several works have quantitatively characterized power-line-synchronous qubit-frequency modulation and demonstrated cancellation techniques to mitigate it34–36. Motional-mode secular frequencies can exhibit analogous power-line-synchronous modulation through electrical pathways, for example via RF-drive amplitude pickup, and this has motivated the use of line-triggered operation in practice23. Several works have further quantified this effect by characterizing the modulation24. However, in contrast to the qubit case, practical methods for canceling line-synchronous secular-frequency modulation of motional modes appear to have received little attention to date.
In this work, we investigate the effect of 60-Hz power-line noise on the motional modes of a single trapped 
ion, and we develop and demonstrate a systematic workflow for its characterization and cancellation. By extracting both the amplitude and phase of the induced secular-frequency modulation, we implement a noise-cancellation scheme based on direct injection into the trap control system. Our approach not only suppresses the dominant periodic component but also provides a practical template for identifying and compensating structured, coherent noise in trapped-ion systems. To the best of our knowledge, this constitutes the first experimental demonstration of cancellation of periodic motional-mode noise in a trapped-ion system, highlighting both its practical significance and its relevance for future large-scale quantum processors.
Results
Characterization of power-line noise
Amplitude measurement using echo-based Ramsey spectroscopy
We first benchmark motional coherence using Ramsey spectroscopy on the blue-sideband transition, starting from the qubit ground state and near the motional ground state, which is directly sensitive to secular-frequency fluctuations (see Supplementary Fig. S1 online for experimental details). The Ramsey signal as a function of the evolution time 
exhibits a Gaussian envelope over the first few milliseconds, 
, with a characteristic timescale 
associated with slow frequency fluctuations. Such a non-exponential decay is characteristic of dephasing dominated by slowly varying frequency fluctuations, whereas fast Markovian (Lindblad-type) dephasing leads to an exponential envelope14,16. We therefore infer that Markovian dephasing is subdominant on the timescales probed here, and that the remaining decoherence is dominated by low-frequency contributions such as coherent periodic modulation and slow drift. To suppress slow frequency wander on timescales of 
ms while preserving sensitivity to coherent periodic modulation, we employ an echo-type Ramsey sequence37 (a single 
pulse at 
) and multi-pulse Carr–Purcell (CP) sequences38 with evenly spaced 
pulses. Varying the number of 
pulses cross-validates the coherent-modulation signatures across different echo/CP orders while further suppressing slow drift.
In the presence of a sinusoidal modulation of the motional-mode secular frequency, the instantaneous mode frequency becomes 
with 
. Here A is the modulation amplitude (in rad/s, equivalently 
in Hz), 
is the modulation angular frequency, and 
is the modulation phase; for power-line-synchronous modulation one typically has 
. Although a model including higher harmonics can better reproduce some finer features of the measured Ramsey traces, the present data do not uniquely determine the harmonic content. We therefore use the single-tone model as the simplest robust description in the main text (see Supplementary Information, Sec. 5). This time-dependent secular-frequency shift periodically modulates the phase accumulated during free evolution (i.e., the azimuthal phase of the Bloch vector on the equator), giving rise to revival features in the measured Ramsey signal as a function of the total evolution time 
. In our data, spin-echo Ramsey exhibits revivals at 
and its integer multiples, while multi-pulse CP sequences show revivals only at specific fractions such as 
and 
.
We characterize the periodic modulation of a radial motional mode using CP sequences with 
refocusing 
pulses (
corresponding to echo Ramsey). The experiment is not synchronized to the 60-Hz power-line signal, so the measured Ramsey signals correspond to phase-averaged results. In the sinusoidal-modulation model, the signal 
for each n can be calculated analytically as (see Supplementary Information, Sec. 2 for the derivation)
 |
1 |
 |
2 |
 |
3 |
where 
is the Bessel function of the first kind of order zero, and 
is the total free-evolution time (sum of free-evolution intervals), excluding the finite durations of the 
and intermediate 
pulses. Fitting these expressions to the data, including a motional-heating envelope, yields 
and a heating rate 
for a radial mode with 
. While the analysis here focuses on coherent sinusoidal modulation, motional heating also contributes to the overall contrast decay. The consideration of heating effects and their interplay with sinusoidal noise is described in detail in the Supplementary Information, Sec. 3, where we use the Quantum Toolbox in Python (QuTiP)39.
Figure 1 shows the measured Ramsey signal as a function of the evolution time 
for 
, together with fits of the model above to the data. The revivals occur at evolution times set by the 60-Hz line cycle (e.g., for echo Ramsey at integer multiples of 
), consistent with a coherent line-synchronous modulation.
Fig. 1.
Amplitude characterization of sinusoidal secular-frequency modulation in a radial motional mode using CP sequences with (a) 
(echo Ramsey), (b) 
, and (c) 
refocusing pulses. Blue circles show the measured Ramsey signal 
with error bars indicating shot noise (
shots per point). Solid red curves show fits to the analytic sinusoidal-modulation model [Eqs. (1)–(3)] including heating effects. Best-fit parameters are (a) 
, 
; (b) 
, 
; (c) 
, 
. 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 
, and for multi-pulse CP at characteristic fractions (e.g., 
and 
), as predicted by the corresponding 
expressions.
Phase measurement using delay-swept spin-echo Ramsey spectroscopy
In this experiment, we operate in a phase-sensitive mode by synchronizing the start of each experimental shot to a fixed phase of the 60-Hz power-line signal. We then introduce a programmable delay 
before the pulse sequence, so that the spin-echo Ramsey sequence is executed at a controllable phase of the line cycle (see Supplementary Fig. S1 online for experimental details). Modeling the resulting secular-frequency shift as 
, the delay deterministically maps to the modulation phase as 
, where 
is the modulation phase at zero delay. For each delay (i.e., modulation-phase) setting, we perform a spin-echo Ramsey measurement and fit the analytic expression
 |
4 |
to the measured Ramsey signal to extract the modulation phase 
for that delay. Here 
is the phase accumulated during the echo sequence due to the sinusoidal secular-frequency shift.
We find that the two radial modes (X and Y) exhibit different modulation amplitude A while sharing essentially the same modulation phase 
as a function of the delay relative to the power-line trigger. This in-phase behavior suggests that the 60-Hz modulation is coupled through the trap RF drive, producing a common-mode secular-frequency modulation in both modes. Figure 2a shows the delay-dependent spin-echo Ramsey fringes, while Fig. 2b compares the extracted phases for the two modes.
Fig. 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 
. The data are plotted in the same format as in Fig. 1: the experiment (blue circles with error bars indicating shot noise, 
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 
, 
, and 
. Orange diamonds show the passive phase-correction verification with error bars indicating shot noise (
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 
for the X mode and 
for the Y mode, both of which are close to the expected value of 
, confirming that the modulation is synchronous with the 60-Hz power-line signal.
Verification via spin-echo Ramsey tracking
We verify the above characterization of the 60-Hz power-line noise by implementing a spin-echo Ramsey sequence in which the predicted phase accumulation during the free-evolution time is compensated. Using the estimated power-line phase extracted from the delay-swept measurements in Fig. 2, we calculate the expected phase accumulation during the free-evolution interval of the spin-echo Ramsey sequence. This predicted phase is then applied to the phase of the second 
pulse, thereby passively compensating the phase shift induced by the 60-Hz modulation. As shown by the orange diamond points in Fig. 2a, applying this post-phase correction largely eliminates the dependence on the power-line noise, indicating that the extracted noise parameters capture the dominant contribution to the observed secular-frequency modulation.
Cancellation of 60-Hz power-line noise
To cancel the 60-Hz power-line noise, we inject a sinusoidal compensation signal at the power-line frequency into the set point of the Proportional-Integral (PI) controller, as described in Methods. When the compensation signal has the same amplitude but an opposite phase relative to the ambient 60-Hz noise, the resulting periodic secular-frequency modulation is canceled.
The amplitude of the 60-Hz component is first estimated from the characterization results above. However, the phase of the compensation signal cannot be determined directly, because the injected 60-Hz tone is not phase-referenced to the experimental trigger; consequently, the relative phase between the injected tone and the ambient 60-Hz modulation is unknown a priori. To overcome this, we perform a phasor-fitting procedure in which compensation signals with various amplitudes (
) and phases (
) are sequentially injected into the controller. Here 
denotes a unit phasor (
) specifying the injected phase in the 60-Hz phasor plane. For each trial signal, the residual 60-Hz modulation amplitude (
) is measured using spin-echo Ramsey sequences. The relationship between the injected signal and the measured residual amplitude is then used to extract the optimal amplitude (V) and phase (
) of the compensation signal by minimizing
 |
5 |
Here r is a proportionality constant that converts the measured modulation amplitude 
(in frequency units) to the corresponding phasor magnitude (in voltage units), accounting for the unknown transfer gain between the injected set-point modulation and the resulting secular-frequency modulation. This optimization yields a vector representation of the 60-Hz noise phasor, allowing us to directly determine the opposite-phase compensation signal required for cancellation. Figure 3a shows the phasor plot of the injected compensation signals and the corresponding residual amplitudes, from which the noise phasor is extracted. Applying the resulting opposite-phase signal substantially suppresses the 60-Hz modulation in the spin-echo Ramsey measurements [Fig. 3b], demonstrating effective reduction of the dominant periodic noise component. As a result, the coherence time of the radial motional mode increases from approximately 10 ms to 35 ms, close to the limit expected from the heating rate (
). Notably, once optimized, the same compensation settings have continued to suppress the 60-Hz component for at least one month, highlighting the suitability of the method for long-term operation.
Fig. 3.
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 
for four trial settings. The red arrow indicates the extracted 60-Hz noise phasor 
. The extracted noise phasor has amplitude 
and phase 
, and the scale factor is 
. Purple arrows show the resulting residual phasors 
; black dots mark their endpoints, whose amplitudes are proportional to the measured residual modulation amplitude 
(after applying the scale factor r defined in the text). (b) Echo-Ramsey signal of a radial motional mode with and without cancellation. Blue circles and orange diamonds show the data without and with cancellation applied, respectively, with error bars indicating shot noise (
shots per point) for both datasets; the dashed black curve shows the independently simulated heating envelope (using an average heating rate 
). Cancellation suppresses the 60-Hz modulation and extends the coherence time from 
to 
, close to the heating-limited value.
Discussion
The present cancellation scheme, which relies on a PI controller and an externally generated 60-Hz compensation signal, effectively suppresses the dominant power-line-synchronous noise in our system. In our implementation, the compensation signal is optimized beforehand and then applied as a fixed waveform, which is sufficient because the dominant 60-Hz noise phasor remains stable over experimental timescales. The implementation could be simplified and made more efficient. Rather than using a dedicated signal generator, a 60-Hz reference could be derived directly from the power line using an electrically isolated sensing stage, followed by controllable phase shifting and amplitude scaling before injection into the controller set-point. Alternatively, the power-line reference could be digitized with an ADC and processed in real time, and the resulting compensation waveform could be synthesized with a DAC for injection into the set-point. Such an approach would reduce hardware complexity while preserving tunable amplitude and phase control for optimal cancellation.
Power-line-synchronous effects could in principle also arise from magnetic-field fluctuations via the second-order Zeeman shift of the qubit clock transition. However, a Ramsey measurement on the clock transition does not show a comparably strong 60-Hz-synchronous signature, indicating that such magnetic-field-induced contributions are subdominant in our parameter regime (see Supplementary Information, Sec. 4).
The physical origin of the 60-Hz modulation is likely multifaceted. In our setup, several essential devices—such as control electronics, RF amplifiers, and vacuum system components—must remain continuously connected to the power line. Together with possible ground loops and electromagnetic pickup in the laboratory wiring, these connections can provide pathways for periodic interference to couple into the ion-trap electronics and modulate the effective confinement.
Even after cancellation of the 60-Hz component, residual fluctuations persist in the motional-mode secular frequencies. In our measurements, the remaining noise is well described by a slow stochastic frequency drift with a characteristic scale on the order of tens of hertz (see Supplementary Information, Sec. 6). Because this residual drift is not phase-referenced to the power line, it is not addressed by the present scheme. Identifying the dominant mechanism and implementing complementary stabilization beyond cancellation of the power-line-synchronous component remain important directions for future work.
Methods
Experimental setup
We trap a single 
ion in a blade-type linear Paul trap, as described in our previous work40,41. The secular frequencies are 
; the axial mode at 
is not used in this study. The qubit is encoded in the hyperfine clock states 
and 
. State initialization and readout are performed using standard optical pumping and state-dependent fluorescence detection via the cycling transition, from which the population in 
is inferred42. We prepare the relevant motional modes close to their ground states using resolved sideband cooling prior to each experiment. The Raman beams used for spin-motion coupling are aligned such that the wavevector difference 
is perpendicular to the trap axis, thereby coupling predominantly to the radial motional modes41.
RF system and PI controller
The ion trap is driven by an RF source at approximately 
and applied to the trap electrodes via a helical resonator. The RF-drive amplitude is stabilized using a PI servo loop (New Focus LB1005 High-Speed Servo Controller), which adjusts the RF amplitude so that the measured amplitude tracks a set-point input; changes in the RF amplitude correspondingly shift the trap secular frequencies43. To enable cancellation of periodic noise, a periodic compensation signal is added to the set-point input of the PI controller, as illustrated schematically in Fig. 4.
Fig. 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.
Cancellation implementation
The compensation-signal amplitude is initially estimated relative to the PI controller set-point. In our configuration, maintaining a secular frequency of approximately 
requires a set-point voltage of about 
. Assuming a locally linear response between the set-point and the secular frequency, an unwanted 60-Hz modulation with an effective magnitude of 
corresponds to a set-point modulation on the order of 
. To generate a compensation signal at this level, we use a commercial signal generator in combination with fixed attenuation, adjusting the generator output amplitude such that a modulation on the order of tens of 
is applied at the set-point input.
In practice, the power-line frequency is not perfectly constant at 
and can drift by up to 
over the course of an experiment. Direct injection of a continuous 60-Hz tone would therefore lead to gradual phase slips between the compensation signal and the ambient 60-Hz modulation, reducing cancellation efficiency. To address this, we employ the burst mode of the signal generator, following a previously reported approach35. Using a trigger derived from the power line, the generator outputs a single-cycle sinusoidal burst at 
, i.e., one cycle at a slightly higher frequency than the nominal 
. This ensures that the output waveform terminates before the next trigger event, allowing the phase to be re-referenced on every cycle and preventing accumulation of phase error.
Supplementary Information
Author contributions
J.Y. designed, carried out, and analyzed the experiments. J.K. provided experimental guidance, particularly on the setup and control system. J.Y., J.K., K.K., and W.C. contributed to the construction of the experimental apparatus. T.K. supervised the project. All authors reviewed and discussed the experimental results and contributed to the manuscript.
Funding
This work was supported by the National Research Foundation of Korea (NRF) Grant (No. RS-2024-00442855, No. RS-2024-00413957) and the Institute for Information & Communications Technology Planning & Evaluation (IITP) Grant (No. RS-2022-II221040), all funded by the Korean government (MSIT).
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
The online version contains supplementary material available at 10.1038/s41598-026-49820-5.
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Supplementary Materials
Data Availability Statement
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.