Squeezed Superradiance Enables Robust Entanglement-Enhanced Metrology Even with Highly Imperfect Readout

Martin Koppenhöfer, Peter Groszkowski, and A. A. Clerk

Phys. Rev. Lett. 131, 060802 – Published 8 August 2023

Abstract

Quantum metrology protocols using entangled states of large spin ensembles attempt to achieve measurement sensitivities surpassing the standard quantum limit (SQL), but in many cases they are severely limited by even small amounts of technical noise associated with imperfect sensor readout. Amplification strategies based on time-reversed coherent spin-squeezing dynamics have been devised to mitigate this issue, but are unfortunately very sensitive to dissipation, requiring a large single-spin cooperativity to be effective. Here, we propose a new dissipative protocol that combines amplification and squeezed fluctuations. It enables the use of entangled spin states for sensing well beyond the SQL even in the presence of significant readout noise. Further, it has a strong resilience against undesired single-spin dissipation, requiring only a large collective cooperativity to be effective.

Received 25 April 2023
Revised 20 June 2023
Accepted 14 July 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.060802

Physics Subject Headings (PhySH)

Hybrid quantum systems Quantum metrology Quantum sensing Spin dynamics Squeezing of quantum noise

Atomic ensemble Nitrogen vacancy centers in diamond Superconducting devices Trapped ions

Mean field theory Quantum master equation

Quantum Information, Science & Technology

Authors & Affiliations

Martin Koppenhöfer ¹, Peter Groszkowski ^1,2, and A. A. Clerk ¹

¹Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA
²National Center for Computational Sciences, Oak Ridge National Laboratory, Tennessee 37831, USA

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Vol. 131, Iss. 6 — 11 August 2023

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Images

Figure 1
Sketch of a dissipative spin-amplification protocol with sensitivity below the SQL. A small parameter $ϕ$ to be measured is encoded in the transverse polarization of an entangled state of $N$ spins with spin-squeezing parameter $r_{0}$ . Readout of the state adds an amount $Σ_{\det}^{2} = Ξ_{\det}^{2} N / 4$ of detection noise. The transient dynamics of SSR decay, i.e., collective decay to a squeezed bath with squeezing parameter $r$ and decay rate $Γ$ , reduces the detrimental impact of detection noise and allows one to reach an estimation error $(Δ ϕ)_{amp}^{2}$ below the SQL.
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Figure 2
Minimum squared estimation error $(Δ ϕ)_{amp}^{2}$ after amplification [defined in Eq. (1)], relative to the SQL value $(Δ ϕ)_{SQL}^{2} = 1 / N$ , (a) as a function of the number of spins for fixed single-spin cooperativities $η_{ϕ}$ , $η_{rel}$ and (b) as a function of the single-spin cooperativity for fixed $N = 50$ . In both plots, the signal is $ϕ = 10^{- 5}$ , and the detection noise is at the level of the projection noise of a CSS, $Ξ_{\det}^{2} = 1$ . In the gray shaded region, the estimation error is worse than the SQL. (a) The green (blue) markers correspond to SSR amplification in the presence of single-spin dephasing (relaxation) with a single-spin cooperativity $η_{ϕ} = Γ / γ_{ϕ} = 0.5$ ( $η_{rel} = Γ / γ_{rel} = 0.5$ ). The red (orange) markers show the corresponding minimum $(Δ ϕ)_{amp}^{2}$ for a unitary amplification scheme using OAT dynamics, which is more than an order of magnitude worse than the SSR result and fails to surpass the SQL. The SSR data points have been obtained from a numerically exact solution of the QME shown in Eq. (3), and an optimization of the amplification time $t$ and the squeezing parameters $r_{0}$ , $r$ . The OAT dynamics is implemented using a Tavis-Cummings coupling to a detuned bosonic mode (see Ref. [29]). The solid lines are MFT simulations for the same parameters as the markers of the corresponding color (see Ref. [29]). (b) The green and blue (red and orange) markers show the results for SSR-based (OAT-based) amplification in the presence of undesired single-spin dephasing and relaxation, respectively {for SSR, this is modeled by the QME [Eq. (4)]}. The dashed lines show the performance of the SSR scheme if single-spin dissipation degrades the preparation of the squeezed initial state (see Ref. [29] for details). In both plots, the green (red) dotted line corresponds to the ideal performance of SSR-based (OAT-based) amplification in the absence of undesired single-spin (single-spin and collective) dissipation.
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Figure 3
(a) Minimum squared estimation error $(Δ ϕ)_{amp}^{2}$ after amplification, relative to the SQL value $(Δ ϕ)_{SQL}^{2} = 1 / N$ , as a function of the number of spins $N$ and the level $Ξ_{\det}^{2}$ of readout noise for $ϕ = 10^{- 5}$ . The data are linearly interpolated based on an equidistant grid of 25 (7) data points along the $Ξ_{\det}^{2}$ ( $N$ ) direction. The contour lines indicate the maximum level of readout noise tolerable if the estimation error should stay the indicated factor $f$ below the SQL. For large $N$ , the contour lines scale as $0.13 f^{2.17} N$ . (b) Optimal values of the gain $G$ and squeezing parameter $r$ ( $r_{0}$ ) of the SSR decay (initial state) for $N = 400$ [indicated by the gray vertical line in (a)]. In the green shaded area on the right, the amplification reduces the estimation error by more than a factor of 2 compared with no amplification. In the gray shaded area on the left, amplification is not used since it does not improve the estimation error.
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