Phase statistics of a single qubit emission as a direct probe of its coherence

Abstract

The emission of photon from an individual atom encodes the phase of its initialized quantum state. Using single-shot heterodyne detection, we measure the phase distribution of the emission from a superconducting transmon qubit in an open waveguide configuration and track its evolution over time. We demonstrate that the presence of a quantum superposition is encoded in the phase statistics of the emission and remains resolvable despite a high noise level. These phase statistics serve as a quantitative probe of the qubit coherence. The decay of the emission envelope with increasing integration time reveals the energy relaxation rate of the emitted wavepacket, while phase distribution broadening tracks pure dephasing processes. We thereby establish a direct link between the decoherence dynamics of an open quantum system and the statistical properties of its radiated field.

Figures (4)

• Figure 1: Time-resolved emission of a qubit under controlled drive pulses. a, The optical image of the device showing two waveguides coupled to a transmon qubit with a zoomed image on the transmon region obtained with scanning electron microscope. b, Example pulse sequence used to prepare the qubit with pulse amplitude $D$, duration of 180 ns per repetition, and 4 $\mu$s spacing between repetitions. c, The signal amplitudes: $\langle |S| \rangle$ (dashed lines) represents the shot-averaged magnitude, dominated by background noise, while $|\langle S \rangle|$ (solid lines) reveals coherent emission emerging from complex-plane averaging, canceling incoherent background; the vertical double-headed arrow indicates the signal-to-noise ratio (SNR) scale, and the horizontal arrow marks the optimal time window for analysis. d, Phase of the shot-averaged signal, $\arg \langle S \rangle$, exhibiting minimal information and a slight drift, demonstrating residual readout-related instabilities. e, Mean phase, $\langle \arg(S) \rangle$, showing the main result: coherent emission with phase evolution dependent on qubit preparation angle, illustrating the temporal signature of the emitted radiation.
• Figure 2: Phase probability density of the emitted microwave field for different qubit preparation angles $\theta$. The probability density $\mathcal{P}(\varphi)$ is constructed from repeated measurements of the field phase $\varphi \equiv \arg(S)$, obtained by integrating the emission over the optimal 36 ns time window indicated in Fig. \ref{['fig:emission_time_domain']}c. The heatmap shows $\mathcal{P}(\varphi)$, with vertical lines marking the three preparation angles $\theta=\pi/2$ (blue, oriented along $+x$), $\pi$ (red, $+z$), and $3\pi/2$ (orange, $-x$), corresponding to the Bloch vectors depicted on the adjacent Bloch spheres. For $\theta=\pi/2$ and $\theta=3\pi/2$, the distributions exhibit pronounced peaks separated by $\pi$, reflecting the sign change of the coherent emission amplitude. For $\theta=\pi$, the distribution is nearly uniform, consistent with the absence of a well-defined emission phase. These results show that the phase statistics of the emitted field faithfully reflect the prepared qubit superposition.
• Figure 3: Dependence of qubit emission amplitude and phase uncertainty on the qubit preparation angle $\theta$ and integration time window $T$. a, Holevo variance $V_H$ of the emitted field, quantifying phase uncertainty, as a function of $\theta$. $V_H$ is calculated from repeated measurements of the emission phase $\varphi \equiv \arg(S)$, integrated over the indicated temporal windows (solid curves). Maxima of $V_H$ occur at integer multiples of $\pi$, reflecting maximal phase uncertainty when the emitted field lacks a well-defined phase. b, Mean emission amplitude corresponding to the same dataset. Minima appear at odd multiples of $\pi$, indicating incoherent emission, and at even multiples of $\pi$, corresponding to the qubit prepared in the ground state. Dashed horizontal lines in both panels denote reference levels measured prior to qubit manipulation. Different solid curves correspond to distinct integration times, highlighting the influence of temporal window choice. The anticorrelated behavior of amplitude and Holevo variance reflects the inverse relationship between coherent emission strength and phase uncertainty, while deviations at larger $\theta$ indicate intrinsic qubit nonlinearities and non-stationary processes within the integration window.
• Figure 4: Mean resultant length $R$ of the emitted qubit radiation, quantifying the sharpness of the phase distribution. a, $R$ as a function of the number of averages $M$ for integration times of 4 ns and 36 ns, demonstrating how averaging over repeated measurements increases phase concentration. b, $R$ versus integration time $T$ for different observation start times $t_{\rm start}$. Solid lines are fits using the phenomenological model of Eq. (\ref{['eq:Sharpness_1']}), capturing the combined effects of wavepacket decay and phase decoherence. c, $R$ as a function of $t_{\rm start}$ for several fixed integration times, showing the decay of phase coherence along the wavepacket envelope. These panels together illustrate how the detected phase sharpness depends on averaging, observation window, and temporal position along the emission, revealing both the wavepacket envelope and the influence of finite phase coherence on the measured signal.