Amplitude Sensing below the Zero-Point Fluctuations with a Two-Dimensional Trapped-Ion Mechanical Oscillator

We present a technique to measure the amplitude of a center-of-mass (c.m.) motion of a two-dimensional ion crystal of $\sim 100$ ions. By sensing motion at frequencies far from the c.m. resonance frequency, we experimentally determine the technique’s measurement imprecision. We resolve amplitudes as small as 50 pm, 40 times smaller than the c.m. mode zero-point fluctuations. The technique employs a spin-dependent, optical-dipole force to couple the mechanical oscillation to the electron spins of the trapped ions, enabling a measurement of one quadrature of the c.m. motion through a readout of the spin state. We demonstrate sensitivity limits set by spin projection noise and spin decoherence due to off-resonant light scattering. When performed on resonance with the c.m. mode frequency, the technique demonstrated here can enable the detection of extremely weak forces ($<1\mathrm{yN}$) and electric fields ($<1\mathrm{nV}/\mathrm{m}$), providing an opportunity to probe quantum sensing limits and search for physics beyond the standard model.

Figure 1

(a) Representation of ion spins arranged in a 2D triangular lattice, along with a cross-section illustration of the Penning trap, characterized by an axial magnetic field $B=4.45\mathrm{T}$ and an axial trap frequency ${\omega }_z=2\pi \times 1.57\mathrm{MHz}$. The blue dots represent ions. Cylindrical electrodes (yellow) generate a harmonic confining potential along the $\widehat{z}$ axis. Radial confinement is provided by the Lorentz force from ($\overrightarrow{E}\times \overrightarrow{B}$)-induced rotation in the axial magnetic field. The beams generating the spin-dependent optical-dipole force (the green arrows) cross the ion plane at $\pm 10°$, forming a 1D traveling-wave potential (the green lines) with $\delta k=2\pi /(0.9\mu \mathrm{m})$. An ac voltage source is connected to the trap end cap and used to drive an axial oscillation with calibrated amplitude $Z_c$. (b) Quantum lock-in CPMG sequence used to detect spin precession produced by c.m. motion resonant with the ODF. Doppler cooling and $|\uparrow {⟩}_N$ spin-state preparation occur before the sequence, and spin-state detection after. The grey blocks with solid borders represent microwave $\pi /2$ rotations around $\widehat{y}$ and $\pi$ rotations around $\widehat{x}$. The orange blocks with dashed borders represent ODF pulses. The ODF phase is advanced by $\mathrm{\Delta }\phi$ in a modulation scheme discussed in Ref. [23], where $\mathrm{\Delta }\phi =\pi$ for $\omega =\mu$. The dashed vertical lines indicate the $m$ segments of the sequence (here, $m=2$). We make use of an $m=8$ sequence for Figs. 2, 3, 4.

Figure 2

Line shape of the spin-precession signal for amplitudes $Z_c$ of 500 pm (the red diamonds), 1 nm (the blue triangles), 2 nm (the green squares), and 5 nm (the orange circles) for $\tau =20\mathrm{ms}$. The black triangles are the background, with the drive turned off. The dashed lines are predictions with no free parameters. The error bars represent standard error. Here $N=90$ ions and $F_0=7.9\mathrm{yN}$.

Figure 3

(Top panels) Bloch sphere representation [29] of spin dephasing for $Z_c=485\mathrm{pm}$. Each blue vector represents an experimental trial with a different phase $\delta$ (see the text). From left to right, the spread in the blue vectors corresponds to ${\theta }_{\max }=0.470$, 1.41, 3.62 rad and $F_0/F_{0M}=0.1$, 0.3, 0.77, where $F_{0M}$ is the maximum optical-dipole force. Our experiment measures the length of the Bloch vector averaged over many trials, denoted by the thick red vector. (Main plot) As a function of ODF strength, the background (the black diamonds) with no applied drive and signal (the blue points) for a 485 pm amplitude and total ODF interaction time $\tau =24\mathrm{ms}$ is shown. The red dashed line is a fit to the background. The black dashed line is the prediction with no free parameters, given the background fit. Here, $N=75$ ions and $F_{0M}=41.3\mathrm{yN}$. (Inset) The black points are experimentally determined values for $Z_c^2$. The red dashed line is the calibrated value of $Z_c^2$. The error bars represent standard error.

Figure 4

Amplitude sensing limits for $N=85$. The black points are the experimentally measured signal-to-noise ratio for determinations of $Z_c^2$ from single pairs of $P_{\uparrow }$, $P_{\uparrow ,\mathrm{bck}}$ measurements as a function of the experimentally imposed $Z_c$. Our measurement for $Z_c=25\mathrm{pm}$ is consistent with zero. The red dashed line is the prediction for the signal-to-noise ratio, including projection noise and the random c.m. mode quadrature measured each trial. The blue solid line is the predicted limiting signal-to-noise ratio for small amplitudes [Eq. (5)], assuming only projection noise and parameters relevant for our setup. The error bars represent standard error.