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Precision bounds for frequency estimation under collective dephasing and open-loop control

Francisco Riberi, Gerardo Paz-Silva, Lorenza Viola

Abstract

Dephasing noise is a ubiquitous source of decoherence in current atomic sensors. We address the problem of entanglement-assisted frequency estimation subject to classical dephasing noise with full spatial correlations (collective) and arbitrary temporal correlations. Our contributions are threefold. (i) We derive rigorous, state-independent bounds on the achievable estimation precision, showing how they are entirely determined by the short-time behavior of the decoherence function. For temporally uncorrelated (Markovian) dephasing, precision is limited by a probe-independent constant. For temporally correlated stationary noise, the bound approaches the noiseless limit for classical states, precluding any asymptotic quantum advantage. (ii) We show that these scaling bounds are tight, by constructing generalized Ramsey protocols that saturate them. These optimal protocols use squeezing at the input and before readout, both of which are available in state-of-the-art atomic interferometers. Implementing a perfect-echo protocol, which reaches Heisenberg scaling in the absence of noise, remains optimal in this noisy setting, irrespective of the noise temporal correlations. (iii) We prove that arbitrary collective open-loop control cannot lift the no-go for super-classical precision scaling under either Markovian or colored stationary noise, highlighting the detrimental nature of full spatial correlations. In the latter case, temporal correlations may nonetheless enable constant-factor improvements over the standard quantum limit, which may still be important in practical metrological scenarios.

Precision bounds for frequency estimation under collective dephasing and open-loop control

Abstract

Dephasing noise is a ubiquitous source of decoherence in current atomic sensors. We address the problem of entanglement-assisted frequency estimation subject to classical dephasing noise with full spatial correlations (collective) and arbitrary temporal correlations. Our contributions are threefold. (i) We derive rigorous, state-independent bounds on the achievable estimation precision, showing how they are entirely determined by the short-time behavior of the decoherence function. For temporally uncorrelated (Markovian) dephasing, precision is limited by a probe-independent constant. For temporally correlated stationary noise, the bound approaches the noiseless limit for classical states, precluding any asymptotic quantum advantage. (ii) We show that these scaling bounds are tight, by constructing generalized Ramsey protocols that saturate them. These optimal protocols use squeezing at the input and before readout, both of which are available in state-of-the-art atomic interferometers. Implementing a perfect-echo protocol, which reaches Heisenberg scaling in the absence of noise, remains optimal in this noisy setting, irrespective of the noise temporal correlations. (iii) We prove that arbitrary collective open-loop control cannot lift the no-go for super-classical precision scaling under either Markovian or colored stationary noise, highlighting the detrimental nature of full spatial correlations. In the latter case, temporal correlations may nonetheless enable constant-factor improvements over the standard quantum limit, which may still be important in practical metrological scenarios.
Paper Structure (35 sections, 5 theorems, 134 equations, 3 figures, 1 table)

This paper contains 35 sections, 5 theorems, 134 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Frequency estimation by Ramsey interferometry
  4. Classical collective dephasing
  5. Noise temporal correlations and short-time coherence decay
  6. Fundamental precision bounds under collective dephasing
  7. QFI growth under collective dephasing
  8. Time optimization and asymptotic precision bounds
  9. Markovian noise ($n=1$).
  10. Colored noise ($n\geq2$).
  11. Saturating the precision bounds with optimal protocols
  12. Interferometry with GHZ states
  13. Interferometry with one-axis twisted spin-squeezed states
  14. Holstein-Primakoff description of OATS inputs.
  15. Noise-averaged evolution and covariance matrix.
  16. ...and 20 more sections

Key Result

Theorem 5.1

Let $0< t_1<t_2 <\ldots <t_{ Q}<t$ be the segmentation of an arbitrary sequence of instantaneous collective pulses whose output state is given by Eq. (eq:compressed_multilabelRU). Then, the single-shot QFI is upper-bounded by a quadratic form as follows: where $\Sigma \succ 0 \in \mathbb{R}^{Q' \times Q'}$ and $\Delta \vec{t} \in \mathbb {R}^{Q'}$ are the uncompressed covariance matrix of the ran

Figures (3)

  • Figure 1: Performance of GHZ and spin-squeezed inputs subject to collective dephasing Left: Optimal precision of a GHZ state, Eq. (\ref{['dbGHZ']}), as function of probe number for different short-time behavior of the decay coefficient, $\chi(t)=\chi_0^n (\omega_c t)^n$. The precision saturates the scaling bound in Eq. (\ref{['eq:QFI_summary_bounds']}), including $N$-independent precision for Markovian noise (solid, red), SQL bound for colored, stationary noise (solid, blue), with HS achieved in the noiseless case (black, dot-dashed). Inset: Uncertainty $\Delta \hat{b}(t)$ as a function of time for fixed system size $N=50$. Right: Same plot and inset for the optimal perfect echo protocol using OATS, with input squeezing and rotation strength as in Monika, see Eq. (\ref{['eq:QFI_OATS_opt']}) and Table \ref{['table:0']}. Besides being less sensitive to preparation errors, OATS outperform GHZ inputs by a constant factor. Parameters: $\omega_c=1$, $\chi_0^2=1$.
  • Figure 2: Optimal interferometric protocol using spin-squeezed states subject to collective dephasing. Top row: Noisy generalized Ramsey protocol using $N=2J$ quantum sensors, with precision saturating the scaling bound in Eq. (\ref{['eq:QFI_summary_bounds']}). The input state is prepared by successively applying a squeezing $e^{-i \mu J_x^2}$$(\mu=\sqrt{N})$ and rotation $e^{-i \frac{\pi}{2}J_z}$ to a CSS, resulting in the input state $U_{\rm sq} |\rm CSS\rangle_{\hat{z}}=|\rm OATS\rangle_{\hat{z}}$. The signal is imprinted after evolution under the noisy Hamiltonian $H(t)= J_x (b+\xi(t))$ for an encoding time $t$. Fluctuations $\xi(t)$ induce diffusion of strength $J \chi_0^n (\omega_c t)^n$ along the $J_y$ axis. Finally, $U_{\rm sq}^{\dagger}$ un-does the initial squeezing, resulting in output $\rho_b(t)$. Measurement of the collective spin component $J_y$ is optimal. The second and third rows represent the evolution of the state in phase space before and after performing the HP transformation, respectively. Parameters: $J=50$, $b=1$, $n=2$, $\omega_c=1$, $\chi_0^2=\frac{1}{4}$, $t=0.1$.
  • Figure 3: Constant prefactor in the QFI upper-bound as a function of pulse number. Left: Behavior of the constant $K_{Q'}$ entering $F_Q^{\rm tot}[\rho_b(t)]$ in Eq. (\ref{['eq:compressed_QFIperf']}) as a function of pulse number $Q$ for a colored spectrum with a Gaussian profile, Eq. (\ref{['Gausspec']}). The prefactor grows as $Q^{(s+1)/2}$. The colored curves join the exact $K_{Q'}$ values, represented by dots, whereas the black solid lines describe the coarse asymptotic behavior. Right: Estimation lower bound as a function of pulse number. Sub-SQL performance is still precluded, but the prefactor $\Delta \hat{b}_{\rm bnd}^2 N \sim K_{Q'}^{-1}$ in Eq. (\ref{['eq:compressed_QFIperf']}) may improve significantly. Parameters: $\alpha=1$, $\omega_c=1$, $T=1$.

Theorems & Definitions (10)

  • Theorem 5.1: Upper bound to the QFI
  • proof
  • Lemma 5.1: Monotonicity of the Gaussian quadratic form under linear compression
  • Corollary 5.1
  • proof
  • Corollary 5.2
  • proof
  • Lemma 5.2: Quadratic form bound in the short-time regime
  • proof
  • proof