Multimode Fock-State Measurements using Dispersive Shifts in a Trapped Ion

TL;DR

The paper addresses the challenge of efficiently characterizing multimode bosonic registers in trapped-ion systems where motional modes outnumber available spin qubits. It proposes a unified SDR-based Ramsey protocol that uses dispersive shifts $H_{\mathrm{disp}} \approx \hbar\,\hat{\sigma}_z \sum_{j=1}^M \chi_j (\hat{n}_j+\tfrac{1}{2})$ to imprint phonon-number-dependent phases on a single spin, with a selective decoupling scheme to cancel carrier AC-Stark shifts. The method enables extraction of multimode Fock-state populations $p_{\boldsymbol{n}}$ by fitting spin-population dynamics $P_\uparrow(t)$, performs parity-based filtering to generate Schrödinger-cat states and entangled coherent states, and realizes single-shot nondestructive Fock-state measurements through iterative filtering. The approach scales to many motional modes, including a multi-ion generalization that increases measurement constraints, with potential applications in quantum metrology and bosonic error correction.

Abstract

Trapped ions naturally host multiple motional modes alongside long-lived spin qubits, providing a scalable multimode bosonic register. Efficiently characterizing such bosonic registers requires the ability to access many motional modes with limited spin resources. Here we introduce a single-spin, multimode measurement primitive using dispersive shifts in the far-detuned multimode Jaynes-Cummings interaction. We implement a Ramsey sequence that maps phonon-number-dependent phases onto the spin, thereby realizing a multimode spin-dependent rotation (SDR). We also introduce a selective-decoupling scheme that cancels the phase induced by the carrier AC-Stark shift while preserving the phonon-number-dependent phase induced by the dispersive shift. Using this SDR-based Ramsey sequence on a single trapped ion, we experimentally extract two-mode Fock-state distributions, perform parity-based filtering of two-mode motional states, and realize a nondestructive single-shot measurement of a single-mode Fock state via repeated filtering steps.

Figures (4)

• Figure 1: (a) Laser detunings and sideband frequencies relative to the carrier transition in the two-mode $\operatorname{SDR}$. The sign of detunings relative to both sidebands is flipped in step $2$ to make the constructive phonon-number-dependent phase. (b) Ramsey sequence with selective decoupling used to apply dispersive shifts for filtering and measurement. The sequence is divided into two step segments separated by a microwave $\pi$ pulse, which acts as a spin echo. This cancels the carrier-induced AC-Stark shift while preserving the dispersive shift. (c) Sequence of the parity-based filtering and Fock-state population measurement. The protocol in (b) is applied twice, where the first run with a measurement implements the filtering and the second extracts the Fock-state populations.
• Figure 2: Fock-state population fitting for the single-mode case. The coherent state $\ket{\alpha}$ is filtered based on the single-mode parity. (a), (b) Time evolutions of the Ramsey sequence for even- and odd-parity-selected states, respectively, where $\theta=\chi_{\mathrm{eff},1}\sum_k t^{(k)}$. The markers denote experimental data, the solid curves are fits, and the orange dashed curves show the expected curves for ideal cat states. (c), (d) Corresponding Fock-state populations extracted from the fits (black open rectangles). The orange bars show the Fock-state distributions of the best-fit even and odd cat states, respectively.
• Figure 3: Fock-state population fitting for the two-mode case. The two-mode coherent state $\ket{\alpha,\alpha}$ is filtered based on the joint parity. (a), (b) Time evolutions of the Ramsey sequence for even- and odd-joint-parity-selected states, respectively, where $\theta_\mathrm{min}=\mathrm{min}(\chi_{\mathrm{eff},1}, \chi_{\mathrm{eff},2})\sum_k t^{(k)}$. The markers denote experimental data and the solid curves are fits. The legend labels indicate the ratio $\chi_{\mathrm{eff},1}:\chi_{\mathrm{eff},2}$. (c), (d) Corresponding two-mode Fock-state populations extracted from fits to the measured time-evolution data, shown as a $5\times5$ grid of bar plots (black open rectangles). The orange bars show the best-fit even and odd ECS distributions, respectively. The population axis in each cell is scaled to 0.4.
• Figure 4: (a) Quantum circuit for single-shot Fock state measurement. In the sequence we alternate dispersive shift operations $\hat{V} (\boldsymbol{\theta}_\ell, \phi_\ell)$ with mid-circuit spin measurements. See main text for specific rotation angles $\boldsymbol{\theta}_\ell$ and phases $\phi_\ell$. (b) Experimental readout of Fock state population inferred from the mid-circuit outcomes. $n_\mathrm{prepare}$ and $n_\mathrm{measure}$ denote the Fock numbers of the prepared Fock state and the target Fock state for the measurement. The population of each cell is calculated from 500 repetitions of the full sequence.