Spin-Locking Spectroscopy of Harmonic Motion

TL;DR

This work introduces motional spin-locking spectroscopy to directly measure the noise spectrum of the motional degree of freedom in a trapped-ion system. By continuously driving a spin-locked transition and analyzing the decay and coherent oscillations of the qubit, they map phase-noise spectral density to observable magnetization dynamics, achieving coverage from about $200$ Hz to $5$ kHz and resolving two orders of magnitude in noise power, with a relative frequency sensitivity near $10^{-6}$. They distinguish motion-induced noise from laser and magnetic noise by comparing carrier and blue-sideband transitions, revealing both stochastic and coherent modulations of the trap frequency, up to Δν/ν_x ≈ 2×10^-5, and demonstrating a method to extract Sν(ω) from the decay and the damped-oscillation fits. The work identifies practical limitations and discusses improvements, including Raman-driven schemes to cancel laser noise and employing a ground-state qubit to lift the lifetime floor, making the approach broadly applicable to other external-degree-of-freedom platforms.

Abstract

Characterization of noise of a quantum harmonic oscillator is important for many experimental platforms. We experimentally demonstrate motional spin-locking spectroscopy, a method that allows us to directly measure the motional noise spectrum of a quantum harmonic oscillator. We measure motional noise of a single trapped ion in a frequency range from 200 Hz to 5 kHz with a power spectral density that resolves noise over two orders of magnitude. Coherent modulations in the oscillation frequency of the oscillator can be probed with a relative frequency sensitivity at the $10^{-6}$ level.

Figures (4)

• Figure 1: Principle of spin locking. (a) In spin locking, a qubit is prepared in a superposition $\left| + \right>_x = (\left|\downarrow \right> + \left| \uparrow \right>)/\sqrt{2}$ by application of a $\pi/2$ pulse, continuously rotated around the $x$ axis and finally $\left< \sigma_x \right>$ is measured. If the phase of the rotation is modulated with a frequency that matches the Rabi frequency, then (b) the state of the qubit evolves along a spiral. (c) Under noise, the orientation and the inclination of the trajectory is changing with each realization of the experiment, which leads to a depolarization of the state. (d) The depolarization is described by an exponential decay with time $t$, Eq. \ref{['eq:exponential_decay']}, with a rate proportional to the phase noise power spectral density $S_\phi$ at the Rabi frequency $\Omega$.
• Figure 2: Motional spin-locking spectroscopy. (a) The carrier transition is affected only by noise on the laser or the qubit, for all Fock states $\left| n \right>$. The spin-locking spectroscopy protocol consists of a $\pi/2$ pulse around the $-y$ axis, followed by a continuous rotation $R_x$ around the $x$ axis for a duration $t$. The expectation value $\left< \sigma_x(t) \right>$ is measured by a final analysis $\pi/2$ pulse. (b) Relative coupling strength to the carrier transition as a function of the phonon number. The gray peak shows the phonon distribution for the ground-state cooled state used in the experiment. The experimental data shows the coupling strength for a coherent motional state which approximates the coupling strength of a Fock state for large phonon numbers. The Lamb-Dicke parameter is $\eta = 0.038$. (c) Noise probed on the carrier transition shows a dominant noise peak at $60kHz$, which originates from the laser lock feedback loop. (d) Motional noise is affecting the transition frequency between two Fock states $\left| n \right>$ and $\left| n+1 \right>$. By probing the sideband transition, motional noise is detected. In the experiment, a coherent motional state is created by applying a resonant radio-frequency drive (RF) in order to maximize the coupling to the blue sindeband. (e) The relative coupling strength to the blue sideband has a maximum at $\bar{n} = 610$ phonons. The phonon distribution of the coherent state is narrow compared to the variation of coupling strength around the maximum. (f) The spin-locking signal obtained from probing the blue sideband transition shows motional noise in the low frequency region below approximately $5kHz$.
• Figure 3: Frequency noise spectrum of a quantum harmonic oscillator. (a) The frequency noise power spectral density $S_\nu$ of the blue sideband transition and of the carrier transition are shown. The dotted line is a power-law function as a guide to the eye. Spin-locking spectroscopy is limited by the spontaneous decay rate $\Gamma = 0.9\per\s$ of the $3d{}^2D_{5/2}$ level to noise above $S_{\nu,\mathrm{lim}} = \Gamma$, as indicated by the dash-dotted line. (b) The peaks highlighted in subpanel (a) show a considerable degree of coherence, which leads to damped oscillations of the magnetization $\left< \sigma_x(t) \right>$. The solid lines are fits to Eq. \ref{['eq:damped_oscillations']}, which gives the frequency modulation (orange) and the noise spectral density (blue) for a series of peaks that are shown in subpanel (c).
• Figure 4: Spin-locking spectroscopy under coherent phase modulation. A superposition $(\left| 1 \right> + \left| 0 \right>) / \sqrt{2}$ on the carrier transition is driven while the light is phase modulated at $\Omega = 2\pi \times 5\kHz$ with a modulation index of $\beta = 200mrad$. The Rabi frequency of the 1$^\mathrm{st}$ order modulation sideband is $\Omega^{(1)} \approx \frac{1}{2} \beta \Omega$. (a) The evolution of the state is described by a spiral on the Bloch sphere. (b) Magnetization $\left< \sigma_\alpha \right>$ ($\alpha = x,y,z$) as a function of the spin-locking time. Points represent experimental data averaged over 150 repetitions, solid lines are theory curves using Eqs. \ref{['eq:mag_X_coherent']}-\ref{['eq:mag_Z_coherent']}. The modulation phase offset $\delta$ is obtained from a least-squares fit to the data.