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Weak-to-strong transition of quantum measurement in a trapped-ion system


Quantum measurement remains a puzzle through its stormy history from the birth of quantum mechanics to state-of-the-art quantum technologies. Two complementary measurement schemes have been widely investigated in a variety of quantum systems: von Neumann’s projective ‘strong’ measurement and Aharonov’s weak measurement. Here, we report the observation of a weak-to-strong measurement transition in a single trapped 40Ca+ ion system. The transition is realized by tuning the interaction strength between the ion’s internal electronic state and its vibrational motion, which play the roles of the measured system and the measuring pointer, respectively. By pre- and post-selecting the internal state, a pointer state composed of two of the ion’s motional wavepackets is obtained, and its central-position shift, which corresponds to the measurement outcome, demonstrates the transition from the weak-value asymptotes to the expectation-value asymptotes. Quantitatively, the weak-to-strong measurement transition is characterized by a universal transition factor \({e}^{-{\varGamma }^{2}/2}\), where Γ is a dimensionless parameter related to the system–apparatus coupling. This transition, which continuously connects weak measurements and strong measurements, may open new experimental possibilities to test quantum foundations and prompt us to re-examine and improve the measurement schemes of related quantum technologies.

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Fig. 1: The weak-to-strong quantum measurement set-up in a trapped-ion system and its weak-value to expectation-value transition.
Fig. 2: The weak-to-strong measurement transition is exactly characterized by the overlapping extent of the motional cat state’s superposed wavepackets \(\langle \phi (z+{\gamma }_{0}t)| \phi (z-{\gamma }_{0}t)\rangle ={e}^{-{\varGamma }^{2}/2}\).
Fig. 3: The measurement regimes of 〈δzθ/(γ0t) in the full parameter space (Γ, θ).

Data availability

Data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.


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This work was supported in part by DIP (German–Israeli Project Cooperation) and by the I-CORE Israel Center of Research Excellence programme of the ISF and by the Crown Photonics Center, and was also supported by the National Basic Research Program of China under grant no. 2016YFA0301903 and the National Natural Science Foundation of China under grant nos. 61632021 and 11574398. E.C. acknowledges support from the Israeli Innovation Authority under project no. 70002 and from the Quantum Science and Technology Program of the Israeli Council of Higher Education.

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Authors and Affiliations



Y.P. and E.C. proposed the concept and the modelling. Y.P., J.Z., C.W. and P.C. contributed to the theoretical analysis, design and setting up of the experiments. J.Z. and C.W. performed the atom experiment, and analysed the data together with Y.P. and E.C. All authors contributed to the writing and revision of the manuscript.

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Correspondence to Yiming Pan or Chun-wang Wu.

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Supplementary information

Supplementary Information

Supplementary Figs. A1–A3, experimental set-up details and data analysis method.

Supplementary Data 1

Source data for Supplementary Fig. 2. Probability distribution of motional wavepacket with Γ = 2.9, θ = 0.02.

Supplementary Data 2

Source data for Supplementary Fig. 3. Fitting method for extracting the averaged position of the motional wavepacket with the parameters Γ = 0.4, θ = 0.2.

Source data

Source Data Fig. 2

Data for characterizing the overlapping extent of the cat state’s superposed wavepackets.

Source Data Fig. 3

The measurement regimes in the full parameter space (Γ, θ).

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Pan, Y., Zhang, J., Cohen, E. et al. Weak-to-strong transition of quantum measurement in a trapped-ion system. Nat. Phys. 16, 1206–1210 (2020).

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