Qubit-parity interference despite unknown interaction phases

TL;DR

The paper tackles observing quantum interference between a low-dimensional qubit and a high-dimensional oscillator when interaction phases are stable but unknown. It demonstrates this phase-insensitive interference by preparing a Schrödinger-cat–like state in a single trapped $^{40}$Ca$^+$ ion using alternating blue- and red-sideband pulses, enforcing a robust qubit–parity correlation $| ext{g},2n angle$ vs. $| ext{e},2n-1 angle$ and achieving a mean phonon number $\langle \hat{n} \rangle \approx 4$. A minimal two-pulse interferometric sequence isolates qubit–oscillator coherence (controlled by $\theta$) and internal oscillator coherence (controlled by $\Theta$) via phase sums and differences, enabling measurement of higher-order coherences up to $|n-m|\le 3$ with visibilities around $0.20$ for qubit–oscillator and $0.40$ for internal oscillator, consistent with theory and a coherence factor $w \approx 0.9$. This phase-insensitive scheme provides a scalable, tomography-free witness of complex quantum coherence in high-dimensional states and lays groundwork for robust quantum control in non-ideal environments and potential multimode extensions.

Abstract

Quantum interference between interacting systems is fundamental to basic science and quantum technology, but it typically requires precise control of the interaction phases of lasers or microwave generators. Can interference be observed if those interaction phases are stable but unknown, usually prohibitive for complex state without active control? Here, we answer this question by experimentally preparing a Schrödinger-cat-like state of an internal qubit and a motional oscillator of a trapped $^{40}$Ca$^{+}$ ion, and its robustness to such uncontrolled phase. By applying alternating red and blue sideband pulses, we enforce a strict qubit-parity correlation and interference inherently insensitive to stable but unknown phases of the driving laser. For this qubit-parity interference, we use a minimal two-pulse interferometric sequence to demonstrate characteristic visibilities of $20\%$ and $40\%$, which approach the theoretical visibility limit, providing a scalable coherence witness without full state tomography for high-dimensional states.

Figures (12)

• Figure 1: Creating and verifying qubit-oscillator interference and internal oscillator coherence in a trapped ion despite unknown interaction phases. a Visualization of the qubit-parity correlation in the state $\ket{\Psi}_{\bm{\varphi}}$ (\ref{['eq:psi']}), where the qubit state dictates the parity of the motional state. b The state preparation sequence, consisting of alternating blue- and red-sideband (BSB, RSB) pulses producing Jaynes-Cummings and anti-Jaynes-Cummings interactions with stable but unknown interaction phases $\varphi_i$. c The measured phonon number distribution $P(n)$ obtained via Rabi flopping (described in Appendix \ref{['appendix:experiment']}), showing a significant mean phonon number ($\langle\hat{n}\rangle=4.09$) and standard deviation. The BSB pulse duration $t_\mathrm{B}$ is varied, while the pulse phase $\phi_\mathrm{B}$ is fixed. d Measurement of the qubit-parity correlation, confirming that the ground (excited) qubit state is associated with even (odd) phonon numbers (Figure \ref{['fig:new_fig2']}c,d). The RSB pulse duration $t_\mathrm{R}$ is varied, while the pulse phase $\phi_\mathrm{R}$ is fixed. e A two-pulse verification sequence with experimentally scanned phases $\phi_1$ and $\phi_2$ used to measure interference fringes in the final qubit ground state population $P_g$, from the state in c and d ($N=8, \langle \hat{n} \rangle \approx 4$). Solid lines represent fits to experimental data; different colors indicate distinct phase constraints. Without full state tomography and active control of local phases $\theta$ and $\Theta$, this measurement directly distinguishes between the two fundamental types of coherence: Qubit-oscillator interference (locally controllable by the qubit phase $\theta$) represents the coherence between the qubit's state and the motion's parity. The lowest order qubit-oscillator interference can be sufficiently measured using a single red pulse measurement. Its higher order interference is probed by scanning the phase sum of the two verification pulses, $\frac{\phi_1 + \phi_2}{2}$ while the phase difference $\frac{\phi_1 - \phi_2}{2}$ is constrained by different values. Internal oscillator coherence (locally controllable by the oscillator phase $\Theta$) representing coherence between different motional components within a single parity subspace, is probed by scanning the phase difference $\frac{\phi_1 - \phi_2}{2}$, while the phase sum is held constant by different values. The contrast and visibility of all these curves are averaged over the phases $\varphi_i$ for $i=1,...,N$.
• Figure 2: Oscillator statistics.a Phonon number distribution $P(n)$ for a state prepared with $N=8$ pulses. The plot compares the experimentally measured distribution (red) against the theoretical prediction (light blue, see Appendix \ref{['app:reconstruction']}), showing the creation of a dispersed motional state. b Growth of the mean phonon number $\langle \hat{n} \rangle$ and its standard deviation $\Delta \hat{n}$ with the number of preparation pulses $N$. This confirms the generation of increasingly populated and adequately delocalized motional states. c Conditional phonon number distributions demonstrating the strong qubit-parity correlation in the dispersed $P(n)$. Deviations between experiment and theory are color-coded by sign (orange: positive, light blue: negative). (Top) the distribution when the qubit is measured in the ground state $|g\rangle$, which is composed of dominantly even phonon numbers. (Bottom) The distribution for the excited state $|e\rangle$, composed of dominantly odd phonon numbers. d Measurement of the qubit-parity correlation. The average motional parity $\langle(-1)^{\hat{n}}\rangle$ as a function of the preparation pulse number, $N$. The data confirms that the ground (excited) qubit state is correlated with even (odd) phonon numbers, demonstrating the imprinting of the desired state structure.
• Figure 3: Interference by single-pulse measurement.a Interference fringe from a single-pulse verification measurement for $N=8$. The final ground state population, $P_g$, is plotted as a function of the verification pulse phase, $\phi_1$. The theoretical curve represents a fit to this specific dataset, using the unknown preparation phases as parameters to validate the accuracy of our model for a single realization. b Contrast and visibility of the final qubit population modulation $P_g$ from a single-pulse verification scan. The theoretical prediction curves, which require no fitting, are averaged over all possible unknown stable phases to show the expected performance. This non-zero contrast confirms the presence of coherence in the prepared state, distinguishing it from a simple incoherent mixture (\ref{['eq:incoherentDM2']}) or (\ref{['eq:incoherentDM']}).
• Figure 4: Two-dimensional interference landscape for a$N=8$ and b$N=12$ pulses. Experimental data (points) are overlaid with a theoretical model where the interaction phases have been fitted to these specific runs. The excellent agreement validates the model's ability to describe individual experimental outcomes.
• Figure 5: Interference by two-pulse measurements. (a, c) Interference fringes for $N=8$, showing agreement between experimental data and our theoretical model when the unknown interaction phases $\bm{\varphi}$ are fitted to the specific dataset. (b, d) Average interferometric metrics versus preparation pulse number $N$. Experimental data is compared against parameter-free ideal theoretical predictions that have been averaged over randomly sampled all possible interaction phases. The close agreement with the theoretical visibility fundamentally bounded (to the theoretical value $\lesssim 0.5$) validates the general robustness to unknown phases. This effect is present in both the experimental data and the theoretical predictions.