Sensitivity of Rydberg microwave electrometry limited by laser frequency noise

Bowen Yang, Yuhan Yan, Xuejie Li, Haojie Zhao, Ling Xiao, Xiaolin Li, Jianliao Deng, and Huadong Cheng

Phys. Rev. A 109, 032609 – Published 11 March 2024

Abstract

Laser frequency or phase noise is a main factor that limits the sensitivity of Rydberg microwave (MW) electrometry. In this study, we proposed a theoretical approach to estimate the effect of laser frequency noise on the atomic coherence of Rydberg sensors based on an atomic superheterodyne receiver. The noisy lasers were characterized using a phase-diffusion model. In particular, explicit formulas were derived to evaluate the noise-limited sensitivity for a given laser linewidth. Furthermore, the effect of Doppler broadening on the sensitivity of the Rydberg sensor at different atomic temperatures was estimated. Our theoretical results provide guidance for the development of Rydberg MW sensors and the method discussed herein can be applied to many other quantum systems.

Received 8 September 2023
Accepted 22 February 2024

DOI:https://doi.org/10.1103/PhysRevA.109.032609

Physics Subject Headings (PhySH)

Quantum sensing

Quantum Information, Science & Technology

Authors & Affiliations

Bowen Yang ^1,2,3, Yuhan Yan^1,2,4, Xuejie Li^1,2,4, Haojie Zhao^1,2,3, Ling Xiao^1,2, Xiaolin Li ⁵, Jianliao Deng^1,2,*, and Huadong Cheng ^1,2,3,†

¹Key Laboratory of Quantum Optics and Center of Cold Atom Physics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
²Wangzhijiang Innovation Center for Laser, Aerospace Laser Technology and System Department, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
³Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China
⁴Department of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
⁵School of Physics, East China University of Science and Technology, Shanghai 200237, China

^*jldeng@siom.ac.cn
^†chenghd@siom.ac.cn

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Vol. 109, Iss. 3 — March 2024

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Figure 1
(a) Typical experimental schematic for atomic superheterodyne receiver. The probe and coupling laser beams counterpropagate through the cell and pump atoms to the Rydberg state. With the dressing of a strong local MW, the energy perturbation induced by a small signal MW is measured by recording the transmission signal of the probe laser using a photodetector (PD). (b) Atomic energy level scheme involved. $Ω_{p}, Ω_{c}$ , and $Ω_{L}$ are the Rabi frequencies of the probe, coupling, and local MW fields, respectively. The perturbed Rabi frequency of the weak signal MW is given by $δ Ω e^{i δ_{s} t + ϕ_{s}}$ . The levels $| 1 〉$ and $| 2 〉$ denote ground and excited states, respectively, while the levels $| 3 〉$ and $| 4 〉$ are the two Rydberg states. (c) Example probe transmission signal $P$ as a function of coupling detuning $Δ_{c}$ in the presence of a strong local MW field. The strong resonant dressing of the local MW results in the appearance of AT splitting and two dressed states $| \pm 〉 = \frac{1}{\sqrt{2}} (| 3 〉 \pm | 4 〉)$ energetically separated by $ℏ Ω_{L}$ . The small energy shifts $E_{\pm} = \pm δ Ω cos (δ_{s} t + ϕ_{s})$ induced by the weak signal MW for the dressed states leads to a dynamic change $P_{s} (t)$ in the resonant probe transmission, which then can be used to measure the amplitude and phase of the signal MW.
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Figure 2
(a) Negative of the Doppler-averaged atomic coherence as a function of $Δ_{c}$ for different Rabi frequencies of local MW with $Ω_{c} = 2 π \times 3 MHz$ and $γ_{p} = γ_{c} = 2 π \times 100 kHz$ . The Rabi frequency of local MW $Ω_{L}$ varied from $Ω_{L} / 2 π = 0 MHz$ (solid gray line) to 3.2 MHz (dotted black line) to 30 MHz (dashed purple line). Panels (b), (c), and (d) correspond to the atomic coherence noise amplitude $S_{ξ}^{'} (0)$ transferred from laser frequency noise versus $Δ_{c}$ for $Ω_{L} = 0 MHz, Ω_{L} = 2 π \times 3.2 MHz$ , and $Ω_{L} = 2 π \times 30 MHz$ , respectively. The noise spectra of atomic coherence contributed by the different values of laser frequency noise in panels (b), (c), and (d) are represented by solid yellow ( $γ_{p} = γ_{c} = 2 π \times 100 kHz$ ), dotted blue ( $γ_{p} = 2 π \times 100 kHz$ and $γ_{c} = 0 kHz$ ), and dashed green ( $γ_{p} = 0 kHz$ and $γ_{c} = 2 π \times 100 kHz$ ) lines, respectively.
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Figure 3
The normalized noise PSD of atomic coherence $10 {log}_{10} (S_{ξ}^{'} (ω) / S_{ξ}^{'} (0))$ for $Ω_{L} = 0 MHz$ (solid gray line) and $Ω_{L} = 2 π \times 3.2 MHz$ (dotted black line) with $Ω_{c} = 2 π \times 3 MHz, Δ_{c} = 0$ , and $γ_{p} = γ_{c} = 2 π \times 100 kHz$ .
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Figure 4
Slope factor $d 〈 ξ^{'} 〉 / d Ω_{L}$ , noise PSD $S_{ξ}^{'} (0)$ , and noise-limited sensitivity $\sqrt{S_{δ E} (0)}$ versus laser linewidth $γ = γ_{p} = γ_{c}$ for coupling Rabi frequencies $Ω_{c} = 2 π \times 1 MHz$ (solid blue line), $Ω_{c} = 2 π \times 3 MHz$ (dotted yellowish-brown line), and $Ω_{c} = 2 π \times 5 MHz$ (dashed sky magenta line). (a) Slope factor. (b) Noise PSD. (c) Noise-limited sensitivity. For each coupling Rabi frequency $Ω_{c}$ and laser linewidth $γ$ , we chose an optimal Rabi frequency of the local MW $Ω_{L}$ which maximized the slope factor.
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Figure 5
Noise-limited sensitivity as a function of $γ$ contributed by different laser linewidths with $Ω_{c} = 2 π \times 3 MHz$ and optimal $Ω_{L}$ for each linewidth $γ$ . The solid yellow, dotted blue, and dashed green lines correspond to $γ = γ_{p} = γ_{c}, γ = γ_{p}$ , and $γ = γ_{c}$ , respectively.
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Figure 6
Slope factor $d 〈 ξ^{'} 〉 / d Ω_{L}$ , noise PSD $S_{ξ}^{'} (0)$ , and noise-limited sensitivity $\sqrt{S_{δ E} (0)}$ as a function of the atomic temperature $T$ with $Ω_{c} = 2 π \times 3 MHz, γ_{p} = γ_{c} = 2 π \times 100 kHz$ , and optimal $Ω_{L}$ maximizing $d 〈 ξ^{'} 〉 / d Ω_{L}$ for each temperature. (a) Slope factor. (b) Noise PSD. (c) Noise-limited sensitivity.
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