Shot-to-shot displacement noise in state-expansion protocols with inverted potentials

Optically levitated nanoparticles are promising candidates for the generation of macroscopic quantum states of mechanical motion. Protocols to generate such states expose the particle to a succession of different potentials. Limited reproducibility of the alignment of these potentials across experimental realizations introduces additional noise. Here, we experimentally investigate and model how such shot-to-shot noise limits the coherence length of a levitated nanoparticle during a state-expansion protocol using a dark, inverted electrical potential. We identify electric stray fields and mechanical instabilities as major sources of shot-to-shot fluctuations. We discuss the resulting experimental requirements for state expansion protocols exploiting inverted potentials.

💡 Research Summary

The authors present a systematic experimental study of shot‑to‑shot displacement noise in state‑expansion protocols that employ an inverted electrical potential for optically levitated nanoparticles. The work is motivated by the need to generate macroscopic quantum superpositions, which require expanding the motional wavefunction of a nanoparticle far beyond its ground‑state size while preserving quantum purity.

The experimental platform consists of a single silica nanoparticle (radius ≈78 nm, mass ≈4.4 fg, charge ≈100 e) trapped in high vacuum (≈2 mPa) by a tightly focused 1550 nm laser (≈600 mW). The optical trap provides harmonic confinement with frequencies (Ωz₀, Ωx₀, Ωy₀) = 2π × (44, 131, 150) kHz. Four planar electrodes are micro‑machined on a chip placed in the focal plane; a static voltage VDC creates an anti‑confining (inverted) harmonic potential along the z‑axis, U = ‑½ m Ωinv² z², while providing transverse confinement. The inverted frequency Ωinv is tunable up to ≈2π × 13 kHz by adjusting VDC.

The state‑expansion sequence has three phases: (i) simultaneous optical and inverted potentials are on, and feedback cooling prepares the particle in a Gaussian state with position variance σ₀² (σ₀ ranging from 150 pm to 384 pm depending on cooling gain). (ii) The optical trap is switched off within 200 ns, leaving only the inverted potential for a free‑expansion time τ (0–50 µs). (iii) The optical trap is switched back on, recapturing the particle; the recapture position z(τ) and momentum pz(τ) are inferred from a retro‑filter applied to the homodyne detection signal. Each τ is repeated 200 times to extract mean values, variances σz², σp², and covariance σzp.

In the absence of additional noise, the variance evolves as σz²(τ) = σcoh²(τ) + σwn²(τ), where σcoh² follows the deterministic exponential expansion governed by Ωinv and the ratio r = Ωz₀/Ωinv, while σwn² accounts for white‑noise heating at rate Γ (photon recoil plus gas collisions). Using an independently measured Γ = 2π × 554 kHz, the authors calculate the expected σz(τ) and the corresponding coherence length ξ = √(8P) σz, where P is the purity of the Gaussian state.

Experimental data show that σz grows faster than the model predicts, and more importantly, ξ does not converge to a universal value at long τ; instead, it depends on the initial σ₀. This discrepancy indicates the presence of a noise source not captured by white‑noise heating. The authors attribute it to shot‑to‑shot fluctuations of the relative alignment between the optical trap and the inverted electrical potential. They derive an additional variance term σshot² = σdisp²