Quantum Spectrometry for Arbitrary Noise

We present a technique for recovering the spectrum of a non-Markovian bosonic bath and/or non-Markovian noises coupled to a harmonic oscillator. The treatment is valid under the conditions that the environment is large and hot compared to the oscillator, and that its temporal autocorrelation functions are symmetric with respect to time translation and reflection—criteria which we consider fairly minimal. We model a demonstration of the technique as deployed in the experimental scenario of a nanosphere levitated in a Paul trap, and show that it would effectively probe the spectrum of an electric field noise source from $10^2$ to $10^6\mathrm{Hz}$ with a resolution inversely proportional to the measurement time. This technique may be deployed in quantum sensing, metrology, computing, and in experimental probes of foundational questions.

Figure 1

Simulated performance of the spectrometer as enacted on a levitated nanosphere system with realistic parameters, deployed to measure a fictional and nontrivial structure to the electric field noise. The dashed blue line gives the noise spectrum to be measured, $\tilde{C}(\nu )$. An ideal spectrometer would capture this perfectly. The three solid lines show what an experimenter using our protocol would reconstruct, with the purple, red, and green lines showing reconstructions using measurement times of $t={10}^{-4}\mathrm{s}$, $t={10}^{-3}\mathrm{s}$, and $t={10}^{-2}\mathrm{s}$, respectively. The longer measurement times allow for a higher fidelity reconstruction of the objective function—in particular, the features at lower frequencies. The parameters utilized for the model are achievable with present-day technology, and the scaling of the $E$ field noise has been chosen such that even with the noise density structure shown it would yield heating rates commensurate with those typically reported in the literature. Simulation parameters and the theoretical treatment of the noise sources can be found in Appendix 3 of the Supplemental Material [23].