Retrodiction beyond the Heisenberg uncertainty relation

Han Bao, Shenchao Jin, Junlei Duan, Suotang Jia, Klaus Mølmer (), Heng Shen () and Yanhong Xiao ()
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Han Bao: State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University
Shenchao Jin: Fudan University
Junlei Duan: Fudan University
Suotang Jia: State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University
Klaus Mølmer: Aarhus University
Heng Shen: Collaborative research center on Quantum optics and extreme optics, Shanxi University
Yanhong Xiao: State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University

Nature Communications, 2020, vol. 11, issue 1, 1-7

Abstract: Abstract In quantum mechanics, the Heisenberg uncertainty relation presents an ultimate limit to the precision by which one can predict the outcome of position and momentum measurements on a particle. Heisenberg explicitly stated this relation for the prediction of “hypothetical future measurements”, and it does not describe the situation where knowledge is available about the system both earlier and later than the time of the measurement. Here, we study what happens under such circumstances with an atomic ensemble containing 1011 rubidium atoms, initiated nearly in the ground state in the presence of a magnetic field. The collective spin observables of the atoms are then well described by canonical position and momentum observables, $${\hat{x}}_{\text{A}}$$ x ̂ A and $${\hat{p}}_{\text{A}}$$ p ̂ A that satisfy $$[{\hat{x}}_{\text{A}},{\hat{p}}_{\text{A}}]=i\hslash$$ [ x ̂ A , p ̂ A ] = i ℏ . Quantum non-demolition measurements of $${\hat{p}}_{\text{A}}$$ p ̂ A before and of $${\hat{x}}_{\text{A}}$$ x ̂ A after time t allow precise estimates of both observables at time t. By means of the past quantum state formalism, we demonstrate that outcomes of measurements of both the $${\hat{x}}_{\text{A}}$$ x ̂ A and $${\hat{p}}_{A}$$ p ̂ A observables can be inferred with errors below the standard quantum limit. The capability of assigning precise values to multiple observables and to observe their variation during physical processes may have implications in quantum state estimation and sensing.

Date: 2020
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DOI: 10.1038/s41467-020-19495-1

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