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Length scale estimation of excited quantum oscillators

Tyler Volkoff, Giri Gopalan

TL;DR

This paper analyzes quantum-enhanced estimation of the length scale $d$ of massive quantum oscillators, showing that excited eigenstates achieve Heisenberg scaling in $d$-sensitivity with excitation number $n$, and that entangled two-oscillator probes can match the advantage of adding another mode. It develops a rigorous QFI framework, identifies practical $q$-quadrature readouts that saturate the bound for real-wavefunction states, and extends the advantage to multipartite, GHZ-like entangled probes yielding $\text{QFI}=O(N^{2}n^{2}/d^{2})$. The authors also examine the impact of noise, connecting the length-scale estimation problem to a scale-model covariant approach with generator $A=\frac{1}{4}(i a^{2}-i a^{\dagger 2})$, and provide estimation-theoretic analyses (MOM, MLE, Bayesian) that guide practical implementations. Experimentally, trapped-ion platforms are identified as promising routes, with challenges including motional dephasing and non-Gaussian state preparation, and the work offers a blueprint for quantum-enhanced sensing of dynamical length- or scale-related parameters in continuous-variable systems.

Abstract

Massive quantum oscillators are finding increasing applications in proposals for high-precision quantum sensors and interferometric detection of weak forces. Although optimal estimation of certain properties of massive quantum oscillators such as phase shifts and displacements have strict counterparts in the theory of quantum estimation of the electromagnetic field, the phase space anisotropy of the massive oscillator is characterized by a length scale parameter that is an independent target for quantum estimation methods. We show that displaced squeezed states and excited eigenstates of a massive oscillator exhibit Heisenberg scaling of the quantum Fisher information for the length scale with respect to excitation number, and discuss asymptotically unbiased and efficient estimation allowing to achieve the predicted sensitivity. We construct a sequence of entangled states of two massive oscillators that provides a boost in length scale sensitivity equivalent to appending a third massive oscillator to a non-entangled system, and a state of $N$ oscillators exhibiting Heisenberg scaling with respect to the total energy.

Length scale estimation of excited quantum oscillators

TL;DR

This paper analyzes quantum-enhanced estimation of the length scale of massive quantum oscillators, showing that excited eigenstates achieve Heisenberg scaling in -sensitivity with excitation number , and that entangled two-oscillator probes can match the advantage of adding another mode. It develops a rigorous QFI framework, identifies practical -quadrature readouts that saturate the bound for real-wavefunction states, and extends the advantage to multipartite, GHZ-like entangled probes yielding . The authors also examine the impact of noise, connecting the length-scale estimation problem to a scale-model covariant approach with generator , and provide estimation-theoretic analyses (MOM, MLE, Bayesian) that guide practical implementations. Experimentally, trapped-ion platforms are identified as promising routes, with challenges including motional dephasing and non-Gaussian state preparation, and the work offers a blueprint for quantum-enhanced sensing of dynamical length- or scale-related parameters in continuous-variable systems.

Abstract

Massive quantum oscillators are finding increasing applications in proposals for high-precision quantum sensors and interferometric detection of weak forces. Although optimal estimation of certain properties of massive quantum oscillators such as phase shifts and displacements have strict counterparts in the theory of quantum estimation of the electromagnetic field, the phase space anisotropy of the massive oscillator is characterized by a length scale parameter that is an independent target for quantum estimation methods. We show that displaced squeezed states and excited eigenstates of a massive oscillator exhibit Heisenberg scaling of the quantum Fisher information for the length scale with respect to excitation number, and discuss asymptotically unbiased and efficient estimation allowing to achieve the predicted sensitivity. We construct a sequence of entangled states of two massive oscillators that provides a boost in length scale sensitivity equivalent to appending a third massive oscillator to a non-entangled system, and a state of oscillators exhibiting Heisenberg scaling with respect to the total energy.

Paper Structure

This paper contains 11 sections, 4 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Benchmarks: classical and Gaussian states of the massive quantum oscillator
  4. Heisenberg scaling
  5. Loss of Heisenberg scaling under noisy quantum channels
  6. Relation to quantum estimation in the scale model
  7. Estimation theory
  8. Length scale estimation with multiple massive oscillators
  9. Experimental directions and challenges
  10. Discussion
  11. Statistical theory for first excited state

Key Result

Lemma 1

For any $n\in \mathbb{N}_{\ge 2}$, The derivatives of the ground state and first excited state are, respectively,

Theorems & Definitions (8)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof