Quantum trajectory of a spin oscillator, continuous retrodiction beyond the Heisenberg uncertainty relation

Abstract

The Heisenberg uncertainty relation in quantum mechanics limits the precision of predicting the outcome of measurements of any non-commuting observables. However, this restriction does not apply to retrodictive guessing of the outcomes of past measurements. Here, we experimentally demonstrate this using a spin oscillator consisting of $5.17\times 10^{10}$ rubidium atoms. Through weak continuous measurements, we monitor canonical spin observables ${\hat{x}}_{\mathrm{A}}$ and ${\hat{p}}_{\mathrm{A}}$ with $[{\hat{x}}_{\mathrm{A}},{\hat{p}}_{\mathrm{A}}]=i\hslash$. By combining information obtained before and after time $t$, we retrodict the outcomes of measurements of ${\hat{x}}_{\mathrm{A}}$ and ${\hat{p}}_{\mathrm{A}}$ at time $t$ with variances that remain, concurrently over 1.4 hours, $\sim$ 1.01 dB below the bound set by the Heisenberg uncertainty relation. We also infer and demonstrate the ability to retrodict, with the same precision, the measurement outcomes for arbitrary quadratures of the spin oscillator. This brings interesting prospects for further research on quantum measurements, and for quantum-enhanced sensing that combines trajectory and entanglement strategies.

arXiv comment8 pages, 4 figures and supplementary information