Unravelling the emergence of quantum jumps in a monitored qubit

Abstract

Quantum jumps, the collapse of a quantum system upon measurement, are among the most striking consequences of observation in quantum mechanics. While recent experiments have revealed the continuous nature of individual jumps, the crossover from coherent dynamics to measurement-dominated behaviour has remained elusive. Here, we tune the measurement strength of a continuously monitored superconducting qubit, and observe that quantum jumps emerge not through a gradual crossover, but via a cascade of three distinct dynamical transitions. The first transition manifests as an exceptional point where coherent oscillations abruptly cease, giving way to jumps towards a stable eigenstate. The second transition marks the onset of dynamical state freezing, where the qubit's dwell time near the eigenstate diverges. A third threshold signals entry into the quantum Zeno regime, where stronger measurement paradoxically suppresses relaxation. Strikingly, we find that decoherence does not blur these transitions but rather fundamentally restructures the dynamical phase diagram, notably inverting their order. These results map measurement-induced transitions in a monitored qubit, revealing that the interplay between coherent driving, measurement, and decoherence gives rise to a hierarchy of distinct dynamical phases.

Figures (15)

• Figure 1: Monitored qubit with tunable measurement strength. The qubit is resonantly driven at a rate $\Omega_\mathrm{S}$ (blue arrows) while being continuously monitored by a detector. The detector produces clicks (red bars) at a rate $\alpha$ whenever the qubit is in the ground state $\left| 0 \right\rangle$. By tuning the dimensionless measurement strength $\lambda \equiv \alpha / (2\Omega_\mathrm{S})$, we drive a crossover in the qubit dynamics from Rabi oscillations to jump-like trajectories, illustrated by the excited-state population traces (blue and red shaded, respectively).
• Figure 2: Overview of dynamical transitions in the ideal monitored qubit model. As the dimensionless measurement strength $\lambda \equiv \alpha / (2 \Omega_\mathrm{S})$ increases, the qubit dynamics evolve through four distinct regimes (a--d), separated by three sharp transitions at $\lambda^c_{1-3}=\{1,\frac{2}{\sqrt{3}},2\}$. Each row displays simulated dynamics in the different regimes, showing (from top to bottom) a representative binary click record, the conditional excited state probability $P_{\left| 1 \right\rangle}^{\mathrm{cond}}$ of the corresponding trajectory, and the ensemble-averaged excited state probability $P_{\left| 1 \right\rangle}^\mathrm{ens}$. The Bloch spheres show the average qubit dwell time per unit angle $\tau_{\theta}(\theta)$ (orange curves), plotted as radial height above the unit circle (a.u.), alongside the time evolution of the ensemble-averaged state initialised at $\left| 0 \right\rangle$ (teal curves). a, For $\lambda<\lambda^{\mathrm{c}}_{1}$, trajectories exhibit periodic Rabi oscillations, intermittently reset to $\left| 0 \right\rangle$ by detector clicks, enabling the qubit to access the full range of polar angles. b, The first transition at $\lambda^{\mathrm{c}}_{1}$ marks the onset of smooth, deterministic no-click evolution towards a stable eigenstate at polar angle $\theta_+$ (dotted grey lines), creating a forbidden region on the Bloch sphere (grey shading). Detector clicks typically interrupt this no-click evolution before the qubit reaches the eigenstate. c, Beyond the second transition ($\lambda>\lambda^{\mathrm{c}}_{2}$), the qubit frequently reaches and freezes near the eigenstate before a click occurs, resulting in a divergent dwell time at $\theta_{+}$. d, At the third transition point ($\lambda=\lambda^{\mathrm{c}}_{3}$), the ensemble-averaged dynamics shift from oscillatory behaviour to overdamped decay with increasingly slow relaxation, marking the onset of quantum Zeno behaviour. In realistic experimental conditions, decoherence modifies both the locations and the ordering of these transitions.
• Figure 3: Observation of the transition to the quantum jump regime. Below the first transition (bottom Bloch sphere), coherent driving (blue arrows) dominates measurement-induced backaction (red arrows) across the entire Bloch sphere, resulting in oscillations. Above it (top Bloch sphere), measurement backaction precisely balances the coherent drive at a specific angle, causing oscillations to halt and giving rise to quantum jump dynamics. a, Tomographic data showing the excited state population $P_{\left| 1 \right\rangle}^{\mathrm{cond}}$ as a function of the no-click sequence duration and the measurement strength $\lambda$, revealing the shift from oscillatory to jump-like behaviour. The transition is observed at $\lambda^{\mathrm{obs}}_{1} = 0.99 \pm 0.01$ (dashed red line). Each data point is the average of 1500.0 experimental runs. b, One-dimensional cuts below (bottom panel, $\lambda=0.5$) and above (top panel, $\lambda=2.2$) the first transition at $\lambda$ values indicated by the dotted lines in a. The grey lines show the simulated no-click dynamics with independently-measured parameters. Shot-noise error bars are smaller than the marker size. c, Histograms of no-click durations further confirm this transition. We initialise the qubit in $\left| 1 \right\rangle$ (in contrast to $\left| 0 \right\rangle$ in a, b) to suppress the effect of errors in long no-click intervals. The grey solid lines show fits using a sum of three complex exponentials (Methods). Throughout this figure, the single-measurement integration time is $T_\text{int} = 320\,$ns and the coherent drive strength is set at $\Omega_\mathrm{S}/2\pi= 100\,$kHz. The tomographic measurement is preceded by a measurement to verify that the system is in the qubit manifold.
• Figure 4: Dwell time per unit angle $\tau_{\theta}(\theta)$ across the second transition.a, Ridge plot showing how the dwell time evolves with measurement strength $\lambda$. Purple dashed lines are fits to the analytical model $\tau_{\theta}(\theta) \sim \left[ \theta - \theta_{+}(\lambda) \right]^{\xi(\lambda)}$ (Methods), performed only when $\theta_+>-\pi/2$. Inset: zoom-in on four representative values of $\lambda$ illustrating the transition: at $\lambda=0.4$, the dwell time is nonzero across all angles; at $\lambda=0.57$, a forbidden region has formed, before the first transition at $\lambda^{\mathrm{obs}}_{1}$; at $\lambda=0.88$, the dwell time remains small near $\theta_+$; at $\lambda=1.17$, a pronounced peak develops. Vertical arrows in both the main panel and the inset indicate the scale. b, Fitted critical exponent $\xi$ versus $\lambda$. Error bars represent $1\sigma$ confidence intervals. The dashed black line is a fit to the analytical model, yielding $\lambda^{\mathrm{obs}}_{2}=0.92\pm0.01$ (vertical red line) at $\xi=0$ (horizontal grey line), marking the transition from finite to diverging dwell time near the forbidden region. Throughout, the coherent drive is $\Omega_\mathrm{S}/2\pi= 100\,$kHz and the angular bin width is $2\pi / 80$.
• Figure 5: Relaxation to the steady state across the third transition. At this transition, oscillatory decay gives way to overdamped relaxation; beyond it, stronger measurement slows relaxation, marking entry into the quantum Zeno regime. a, Ensemble-averaged excited state probability $P_{\left| 1 \right\rangle}^\mathrm{ens}$ for selected $\lambda$. Each data point is the average of 3000.0 experimental runs. Error bars are smaller than the marker size. Dashed lines show fits using a sum of three complex exponentials. b, Extracted relaxation rates (blue) and oscillation frequencies (purple) versus $\lambda$. Error bars indicate $1\sigma$ confidence intervals. The dashed line is a fit of the oscillation frequency to the predicted behaviour (Methods). The transition is observed at $\lambda^{\mathrm{obs}}_{3} = 1.09 \pm 0.01$ (dashed red line), below the ideal value $\lambda^{\mathrm{c}}_{3}=2$ owing to finite waiting time in the detector's excited state. Throughout, the coherent drive is $\Omega_\mathrm{S} / 2\pi = 100\,$kHz.