Quantum Non-Gaussianity of Multiphonon States of a Single Atom

L. Podhora, L. Lachman, T. Pham, A. Lešundák, O. Číp, L. Slodička, and R. Filip

Phys. Rev. Lett. 129, 013602 – Published 28 June 2022

Abstract

Quantum non-Gaussian mechanical states are already required in a range of applications. The discrete building blocks of such states are the energy eigenstates—Fock states. Despite progress in their preparation, the remaining imperfections can still invisibly cause loss of the aspects critical for their applications. We derive and apply the most challenging hierarchy of quantum non-Gaussian criteria on the characterization of single trapped-ion oscillator mechanical Fock states with up to 10 phonons. We analyze the depth of these quantum non-Gaussian features under intrinsic mechanical heating and predict their requirement for reaching quantum advantage in the sensing of a mechanical force.

Received 18 November 2021
Revised 22 March 2022
Accepted 25 May 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.013602

Physics Subject Headings (PhySH)

Photon statistics Quantum optics Quantum states of light

Ions

Atom & ion cooling

Atomic, Molecular & Optical

Authors & Affiliations

L. Podhora ¹, L. Lachman¹, T. Pham², A. Lešundák², O. Číp ², L. Slodička ^1,*, and R. Filip^1,†

¹Department of Optics, Palacký University, 17. listopadu 12, 77146 Olomouc, Czech Republic
²Institute of Scientific Instruments of the Czech Academy of Sciences, Královopolská 147, 612 64 Brno, Czech Republic

^*slodicka@optics.upol.cz
^†filip@optics.upol.cz

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Vol. 129, Iss. 1 — 1 July 2022

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Images

Figure 1
(a) The mechanical oscillator corresponds to axial harmonic motion of a single ${}^{40}{Ca}^{+}$ ion localized in a linear Paul trap. The generation and analysis of states approaching idealized Fock states illustrated by their corresponding wave functions is implemented through interaction of the electronic ground $| g ⟩$ and metastable $| e ⟩$ states with the quantized harmonic motion on the first motional sidebands. (b) The sequence for preparation of genuine QNG states includes initialization to the electronic and motional ground state $| g, 0 ⟩$ followed by deterministic transfer of the accumulated population to number states by repetitive coherent excitation of the blue and red sidebands depicted as blue and red arrows, respectively. The spectroscopic analysis of phonon number populations $P_{n}$ around the target Fock state implements an unambiguous identification of the genuine QNG features. (c) Characterization of the Fock states of mechanical motion. The yellow points represent the measured populations $P_{n}$ for the experimentally generated states. The thresholds for genuine $n$ -phonon QNG are represented by blue points with the corresponding black numbers showing their numerical value [24, 26]. The associated blue numbers quantify the thermal depth of genuine $n$ -phonon QNG. Similarly, the red points identify thresholds for observation of basic QNG aspects [35] and the associated red numbers determine their thermal depth. The green bars depict the force estimation capability of a specific model of noisy Fock states, where the probability $P_{n}$ exceeding the presented threshold values certifies a metrological advantage [12] against the previous ideal Fock state $| n - 1 ⟩$ , while the corresponding numbers quantify the thermal depth of this advantage for the measured states.
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Figure 2
Measurements of the QNG depth for states approaching Fock states $| 2 ⟩$ , $| 5 ⟩$ , and $| 10 ⟩$ . Each graph shows the probability $P_{n}$ and the overall probability $P_{n - 1} + P_{n + 1}$ relevant for the evaluation of the thermalization. The red points represent the measured states after several heating steps with an increasing length of the 397 nm laser pulse. The red dashed lines correspond to the theoretical model of ion heating by random photon recoils, with blue points giving a scale of the applied thermalization using an equivalent mean thermal phonon number ${\bar{n}}_{th}$ when applied to the vacuum state. The green dashed line in a graph for state approaching $| 10 ⟩$ shows the simulated trajectory above the Doppler cooling limit and horizontal blue lines depict thresholds of the $n$ th order genuine QNG [24, 26] given by (1) and (2). For $n = 2$ and 5 the green dashed lines coincide with the red dashed one.
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Figure 3
Estimation of the metrological advantage of realized states for sensing of a small force. The horizontal axis quantifies the amplitude in the phase space that the force causes. The vertical axis shows the minimal standard deviation $σ$ estimated by optimization of the Fisher information in Eq. (3) normalized to $σ_{0}$ resulting from sensing using a motional ground state. The black lines show $σ / σ_{0}$ for ideal Fock states. The brown, blue, orange, and green solid curves correspond to prepared states approaching the Fock states $| n ⟩ = | 1 ⟩, | 2 ⟩, | 5 ⟩$ , and $| 8 ⟩$ , respectively. The colored regions show $σ / σ_{0}$ for states with the phonon-number distributions within experimental error bars.
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