Two-Mode Bosonic State Tomography with Single-Shot Joint-Parity Measurement of a Trapped Ion

PRX Quantum 6, 040352 (2025) · DOI: 10.1103/wsqr-j9f4 · arXiv: 2506.12628

License: CC BY 4.0.

Abstract

The full characterization of a continuous-variable quantum system is a challenging problem. For the trapped-ion system, a number of methods of measuring the quantum states have been developed, including the measurement of the Q quasiprobability function and the density-matrix elements in the Fock basis, but these approaches are often slow and difficult to scale to multimode states. Here, we demonstrate a novel and powerful scheme for measuring a continuous-variable quantum state that uses the direct single-shot measurement of the joint parity of the phonon states of a trapped ion. We drive a spin-dependent bichromatic beam-splitter interaction that coherently exchanges phonons between different harmonic oscillator modes of the ion. This interaction encodes the joint-parity information into the relative phase between the two spin states, enabling measurement of the combined phonon-number parity across multiple modes in a single shot. Leveraging this capability, we directly measure multimode Wigner quasiprobability distributions to perform quantum state tomography of an entangled coherent state, and calculate various quantum informational quantities with a model-based estimation of the density matrix. We further show that the single-shot joint-parity measurement can be used to detect parity-flip errors in real time. By postselecting the parity-measurement outcomes, we experimentally demonstrate the partial recovery of coherence, effectively implementing an error-mitigation technique. Lastly, we identify the various sources of error affecting the fidelity of the spin-dependent beam-splitter operation and study the feasibility of high-fidelity operations. The interaction studied in this work can be extended to more than two modes, and is highly relevant to continuous-variable quantum computing and quantum metrology.

Figures

23 panels with data across 12 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Illustrative figure, no extractable data. Shown in the paper PDF. Page 4 of the PDF.

Fig. 2

Illustrative figure, no extractable data. Shown in the paper PDF. Page 5 of the PDF.

Fig. 3

Illustrative figure, no extractable data. Shown in the paper PDF. Page 5 of the PDF.

Fig. 4

• panel (1): Spin population $P_{\uparrow}$ against time (ms) under the SBS interaction for the Fock states $|1\rangle|0\rangle$ (circles) and $|0\rangle|2\rangle$ (squares), with quantum projection noise error bars. Solid and dashed curves are simulations including motional heating, dotted curves the noiseless simulation; the dash-dotted line marks $t_\pi$. data.csv

Fig. 5

• panel (a): Measured Wigner distribution $W(\beta_1,\beta_2)$ of the two-mode Fock state $|1\rangle|1\rangle$ against $\mathrm{Re}(\beta_1)$ and $\mathrm{Re}(\beta_2)$, on a 61 by 61 grid. This is the Experiment column of printed panel (a); the Simulation column beside it is not shown. data.csv
• panel (b): Measured Wigner distribution $W(\beta_1,\beta_2)$ of the two-mode Fock state $|2\rangle|1\rangle$ against $\mathrm{Re}(\beta_1)$ and $\mathrm{Re}(\beta_2)$, on a 61 by 61 grid. This is the Experiment column of printed panel (b); the Simulation column beside it is not shown. data.csv

Fig. 6

• panel (a): Measured Wigner distribution $W(\beta_1,\beta_2)$ of the even entangled coherent state with $\alpha_1 = \alpha_2 = 1.2$ in the real-real plane, against $\mathrm{Re}(\beta_1)$ and $\mathrm{Re}(\beta_2)$. The points never recorded, black in the print, are left empty. The Simulation column and the dotted diagonal of the printed panel are not shown. data.csv
• panel (b): Real-real diagonal cut of the even entangled coherent state: parity (left axis) and $W(\beta_1,\beta_2)$ (right axis) against $\mathrm{Re}(\beta_1) = \mathrm{Re}(\beta_2)$. Exp markers are the measurement, with quantum projection noise error bars. The curves are Ideal, the perfect state, and G, GD and GDM, simulations with noise in the generation, displacement and measurement steps. data.csv
• panel (c): Measured Wigner distribution $W(\beta_1,\beta_2)$ of the even entangled coherent state with $\alpha_1 = \alpha_2 = 1.2$ in the imaginary-imaginary plane, against $\mathrm{Im}(\beta_1)$ and $\mathrm{Im}(\beta_2)$. The points never recorded, black in the print, are left empty. The Simulation column and the dotted diagonal of the printed panel are not shown. data.csv
• panel (d): Imaginary-imaginary diagonal cut of the even entangled coherent state: parity (left axis) and $W(\beta_1,\beta_2)$ (right axis) against $\mathrm{Im}(\beta_1) = \mathrm{Im}(\beta_2)$. Exp markers are the measurement, with quantum projection noise error bars. The curves are Ideal, the perfect state, and G, GD and GDM, simulations with noise in the generation, displacement and measurement steps. data.csv

Fig. 7

• panel (a): Measured Wigner distribution $W(\beta_1,\beta_2)$ of the odd entangled coherent state with $\alpha_1 = \alpha_2 = 1.2$ in the real-real plane, against $\mathrm{Re}(\beta_1)$ and $\mathrm{Re}(\beta_2)$. The Simulation column and the dotted diagonal of the printed panel are not shown. data.csv
• panel (b): Real-real diagonal cut of the odd entangled coherent state: parity (left axis) and $W(\beta_1,\beta_2)$ (right axis) against $\mathrm{Re}(\beta_1) = \mathrm{Re}(\beta_2)$. Exp markers are the measurement, with quantum projection noise error bars. The curves are Ideal, the perfect state, and G, GD and GDM, simulations with noise in the generation, displacement and measurement steps. data.csv
• panel (c): Measured Wigner distribution $W(\beta_1,\beta_2)$ of the odd entangled coherent state with $\alpha_1 = \alpha_2 = 1.2$ in the imaginary-imaginary plane, against $\mathrm{Im}(\beta_1)$ and $\mathrm{Im}(\beta_2)$. The Simulation column and the dotted diagonal of the printed panel are not shown. data.csv
• panel (d): Imaginary-imaginary diagonal cut of the odd entangled coherent state: parity (left axis) and $W(\beta_1,\beta_2)$ (right axis) against $\mathrm{Im}(\beta_1) = \mathrm{Im}(\beta_2)$. Exp markers are the measurement, with quantum projection noise error bars. The curves are Ideal, the perfect state, and G, GD and GDM, simulations with noise in the generation, displacement and measurement steps. data.csv

Fig. 8

• panel (a): Population against two-mode Fock-state index for the even entangled coherent state: bars for the Ideal Even ECS and for the Largest Pure State, the eigenvector of the estimated density matrix with the largest eigenvalue, whose error bars come from statistical bootstrapping. The print overlaps the two bar series; here they stand side by side. data.csv
• panel (b): Population against two-mode Fock-state index for the odd entangled coherent state: bars for the Ideal Odd ECS and for the Largest Pure State, the eigenvector of the estimated density matrix with the largest eigenvalue, whose error bars come from statistical bootstrapping. The print overlaps the two bar series; here they stand side by side. data.csv

Fig. 9

• panel (b): Measured one-dimensional Wigner slices $W(0,\beta_2)$ against $\mathrm{Re}(\beta_2)$ at three instances: A1 after preparation (crosses), A2 after 10 ms of thermalization (circles) and A3 after parity-based postselection (squares), with spin-projection noise error bars. The dotted, dashed and solid curves are the numerical-simulation results for A1, A2 and A3. data.csv

Fig. 10

The underlying data is not available. The figure is shown in the paper PDF. Page 14 of the PDF.

Fig. 11

• panel (b): One-dimensional Wigner slices of a cat state, Wigner function against displacement: B1 (before SBS) along $\mathrm{Im}(\beta_2)$ (crosses), and B2 (after SBS) along $\mathrm{Im}(\beta_2)$ (circles) and along $\mathrm{Re}(\beta_1)$ (squares), with spin-projection noise error bars. The solid and dashed curves are the theoretical slices for the ideal cat state and for vacuum. data.csv

Fig. 12

• panel (1): Spin population $P_{\uparrow}$ against time ($\mu$s) under the off-resonant carrier of the bichromatic beams, from the initial state $|\downarrow\rangle|0\rangle$, for $\phi_M = \pi/2$ (circles) and $\phi_M = \pi$ (squares), with quantum projection noise error bars. Solid and dashed curves are the theoretical curves; the dash-dotted line marks the ramp-up delay $t_0 = 0.55$ $\mu$s. data.csv

Fig. 13

• panel (1): Mean phonon number against wait time (ms) for the two radial modes, mode 1 (circles) and mode 2 (squares), with error bars for the fitting uncertainty. The solid and dashed lines are the heating-rate fits. The pale band the printed panel draws around each line is not shown. data.csv

Fig. 14

• panel (a): Single-mode blue-sideband Ramsey contrast against wait time (ms) for mode 1 (circles) and mode 2 (squares), with error bars. The solid and dashed curves are the Gaussian decay fits, each with the shaded band the printed panel draws around it. data.csv
• panel (b): Measured spin population $P_{\uparrow}$ (solid curve) against wall-clock time mod 60 Hz period (ms), for a blue-sideband Ramsey experiment on mode 1, with the number of shots in each bin on the right axis. The printed panel draws the shot counts as semitransparent bars; here they are a step outline. data.csv

Fig. 15

• panel (a): Beam-splitter Ramsey contrast against wait time (ms) for the states $|1\rangle|0\rangle$ (circles) and $|0\rangle|1\rangle$ (squares), with error bars. The three curves are numerical simulations under different noise conditions: Heating only (solid), Lindblad ($t_\gamma = 40$ ms) (dashed) and Sinusoid ($\Delta = 50$ Hz) (dotted). data.csv
• panel (b): Numerical simulations of the spin population $P_{\uparrow}$ against time (ms) under the SBS interaction, from $|1\rangle|0\rangle$, for five noise conditions: Heating only, Sinusoid ($\Delta$ = 50 Hz), Lindblad ($t_{\gamma}$ = 40 ms), Sinusoid ($\Delta$ = 200 Hz) and Lindblad ($t_{\gamma}$ = 10 ms). The dotted vertical line marks the $\pi$-time. data.csv

Fig. 16

• panel (a): Numerical simulation without the Lamb-Dicke approximation, against $\alpha$: the simulated $\sqrt{\bar{n}}$ (solid, left axis), the ideal relation $\bar{n} = |\alpha|^2$ (dotted diagonal) and the fidelity between the generated state and an ideal coherent state (dashed, right axis). data.csv
• panel (b): Numerically simulated spin population $P_{\uparrow}$ against time (ms) under the SBS interaction without the Lamb-Dicke approximation, for $|1\rangle|0\rangle$ (solid), $|7\rangle|0\rangle$ (dashed) and $|13\rangle|0\rangle$ (dash-dotted). The vertical dashed line marks the standard $\pi$-time of the $|1\rangle|0\rangle$ state. data.csv

Cite

Honggi Jeon, Jiyong Kang, Wonhyeong Choi, Kyunghye Kim, Jaehun You, Taehyun Kim. Two-Mode Bosonic State Tomography with Single-Shot Joint-Parity Measurement of a Trapped Ion. PRX Quantum 6, 040352 (2025). https://doi.org/10.1103/wsqr-j9f4

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