Sensitivity, quantum limits, and quantum enhancement of noise spectroscopies

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Sensitivity, quantum limits, and quantum enhancement of noise spectroscopies

Vito Giovanni Lucivero, Aleksandra Dimic, Jia Kong, Ricardo Jiménez-Martínez, and Morgan W. Mitchell

Phys. Rev. A 95, 041803(R) – Published 21 April 2017

Abstract

We study the fundamental limits of noise spectroscopy using estimation theory, Faraday rotation probing of an atomic spin system, and squeezed light. We find a simple and general expression for the Fisher information, which quantifies the sensitivity to spectral parameters such as resonance frequency and linewidth. For optically detected spin noise spectroscopy, we find that shot noise imposes “local” standard quantum limits for any given probe power and atom number, and also “global” standard quantum limits when the probe power and atom number are taken as free parameters. We confirm these estimation theory results using nondestructive Faraday rotation probing of hot Rb vapor, observing the predicted optima, and finding good quantitative agreement with a first-principles calculation of the spin noise spectra. Finally, we show sensitivity beyond the atom- and photon-number-optimized global standard quantum limit using squeezed light.

Received 7 October 2016

DOI:https://doi.org/10.1103/PhysRevA.95.041803

Physics Subject Headings (PhySH)

Atomic & molecular processes in external fields Atomic spectra Fluctuation theorems Fluctuations & noise Magneto-optical spectra Noise Photon statistics Quantum fluctuations & noise Quantum optics

Atomic, Molecular & OpticalQuantum Information, Science & TechnologyStatistical Physics & Thermodynamics

Authors & Affiliations

Vito Giovanni Lucivero^1,*, Aleksandra Dimic^1,2, Jia Kong¹, Ricardo Jiménez-Martínez¹, and Morgan W. Mitchell^1,3

¹ICFO–Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
²Faculty of Physics, University of Belgrade, Studentski Trg 12-16, 11000 Belgrade, Serbia
³ICREA–Institució Catalana de Recerca i Estudis Avançats, 08015 Barcelona, Spain

^*Corresponding author: lucivero@princeton.edu; present address: Department of Physics, Princeton University, Princeton, New Jersey 08544, USA.

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Issue

Vol. 95, Iss. 4 — April 2017

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Images

Figure 1
Representative noise spectra showing spin noise resonances of ${}^{85}{Rb}$ at 42.6 kHz and ${}^{87}{Rb}$ at 64.0 kHz. Upper spectrum (blue) with coherent-state probing, lower spectrum (green) with polarization-squeezed probe light. Each spectrum is an average of $N_{ave} = 5$ acquisitions with coarse graining of $N_{bin} = 20$ . Data acquired with $n = 4.23 \times 10^{12} c m^{- 3}$ and $P = 2.0 m W$ . It is clearly seen that the noisiness $var [\bar{S} (ν)]$ of the spectrum increases with increasing $〈\bar{S} (ν)〉$ . The squeezed probe reduces the shot-noise background, with a beneficial effect on both the signal-to-noise ratio and the precision of spectral parameter estimation. The orange bar below the spectra shows the fit region, and the blue and green curves show fits of Eq. (4) to the coherent and squeezed spectra, respectively. Inset: Principle of spin noise measurement. Polarized light experiences Faraday rotation by an angle $ϕ$ proportional to the on-axis magnetization of the atomic ensemble, and is detected with a polarimeter (not shown).
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Figure 2
$Γ_{22}^{th}$ (left), variance in the estimated line center, and $Γ_{44}^{th}$ (right), variance in estimated linewidth as predicted by Eq. (3) as a function of number density and probe power. Both in units of ${Hz}^{2}$ . The respective optima are $Γ_{22}^{th} = 1190 {Hz}^{2}$ and $Γ_{44}^{th} = 10914 {Hz}^{2}$ .
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Figure 3
Sensitivity of spin noise spectroscopy vs atomic density in theory and experiment. The optical power is $P = 2$ mW throughout. (a) The lower blue curve shows $Γ_{22}$ , the variance of the Larmor frequency estimate, computed by Eq. (3) and from experiment (blue open circles), on the left (blue) axis. The upper green curve shows $Γ_{44}$ , the variance of the linewidth estimate, and observed variance (green solid circles), on the right (green) axis. (b) The upper red curve shows $Γ_{11}$ , the variance of the shot-noise estimate, from theory and from experiment (red solid circles), on the left (red) axis. The lower orange curve shows $Γ_{33}$ , the variance of the spin noise estimate, and observed variance (orange open circles), on the right (orange) axis. Error bars show plus/minus one standard error (see SM [43]).
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Figure 4
Sensitivity of spin noise spectroscopy vs probe power, and sensitivity enhancement by squeezing. The atom number density is $n = 7.65 \times 10^{12}$ ${cm}^{- 3}$ throughout. Main plot: Variance $Γ_{44}^{\exp}$ of the linewidth estimate, vs power for a coherent-state probe by Eq. (3) (upper, blue curve) and experiment (blue solid circles). Same quantities with a polarization squeezed probe with $2.6 dB$ of squeezing by Eq. (3) (green, lower curve) and experiment (green open circles). Error bars show plus/minus one standard error. The inset shows the same quantities for $Γ_{22}^{\exp}$ , the variance of the Larmor frequency estimate.
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