Device fabrication and parameters

The emitter system consists of a transmon circuit characterized by charging energy E C /h=270 MHz and Josephson energy E J /h=24.6 GHz. The circuit was fabricated by double-angle evaporation of aluminium on a high resistivity silicon substrate. The circuit was then placed at the centre of a waveguide cavity (dimensions 34.15 × 27.9 × 5.25 mm) machined from 6061 aluminium. The cavity geometry was chosen to be resonant with the lowest energy transition of the transmon circuit. The resonant interaction between the circuit and the cavity (characterized by coupling rate g/2π=136 MHz) results in hybrid states, as described by the Jaynes–Cummings Hamiltonian. The cavity is deliberately coupled to two 50 Ω cables: one weakly coupled port, characterized by coupling quality factor Q c ≃105, is used to drive the system, while a more strongly coupled port Q c ≃104 sets the total radiative decay time of the system. This configuration results in an effectively ‘one dimensional atom’, where all of the radiative decay is captured by the strongly coupled cable16. Spontaneous emission from this ‘artificial atom’ is amplified by a near-quantum-limited Josephson parametric amplifier, consisting of a 1.5 pF capacitor, shunted by a superconducting quantum interference device (SQUID) composed of two I 0 =1 μA Josephson junctions. The amplifier is operated with negligible flux threading the SQUID loop and produces 20 dB of gain with an instantaneous 3-dB-bandwidth of 20 MHz.

We used standard techniques to measure the energy decay time T 1 =430 ns and Ramsey decay time , indicating that the emitter experiences a negligibly small amount of pure dephasing. We also examined the equilibrium state populations of the emitter using a Rabi driving technique30, and found the excited state population to be less than 3%.

State tracking

We use a master equation (equivalent to equations (2, 3, 4)) to propagate the density matrix for the emitter’s state conditioned on the detected homodyne signal. The signal is digitized in 20 ns steps, and scaled such that its variance is γdt. At each time step, we update the density matrix components ρ 11 [i] and ρ 01 [i] based on the detected measurement signal dV[i], where z≡1–2ρ 11 and x≡2Re[ρ 01 ]. Our state update is consistent with the Itô formulation of stochastic calculus.

Ensemble dynamics

Based on 9 × 105 repetitions of the experiment and associated quantum trajectories, we can examine ensemble dynamics of the paths on the Bloch sphere taken by our decaying emitter. The behaviour of single trajectories characterizes the dynamics of spontaneous decay subject to homodyne detection, and is distinctly different than the full ensemble behaviour that decays deterministically towards the ground state.

Figure 5 displays greyscale histograms of the state at different points in time for two different initial conditions. For trajectories initialized in −z (Fig. 5a), these histograms demonstrate how the decay paths are restricted to a deterministic arc in the Bloch sphere. Curiously enough, a state prepared in a traditional eigenstate of spontaneous emission will develop some quantum coherence when monitored under homodyne detection. The x-components of such trajectories may be pinned to the edges of this arc on the X-axis, or instead may oscillate about the central value of x=0. We note that though the trajectories exhibit an immediate diffusive behaviour for short timescales, the decay of coherence takes over at longer timescales, indicated by a decreasing upper bound on the stochastically acquired coherence. Examining behaviour along the Z-axis, we see that though some trajectories may decay by more quickly approaching the ground state, no trajectory may decay more slowly in z than a specific lower bound at each time step.

Figure 5: State histograms. Greyscale histograms represent the distribution for values of x and z at each time point. The greyscale shading is normalized such that the most frequent value is 1 at each time point. (a) Histograms of the state when the emitter is initialized in the state −z with a few sample trajectories shown in colour. (b) Histograms associated with decay from the state +x. Full size image

On the other hand, when the emitter is initialized along +x in a superposition of its excited and ground states, the histograms of the Bloch sphere coordinates show different behaviour (Fig. 5b). The x-component of the trajectory encounters a decreasing upper bound on its maximum value, once more illustrating motion along a shrinking deterministic arc. The z-component, however, can exhibit extremely varied behaviour. In addition to following the average decay path, the state may also stochastically excite, or it may rapidly decay in z while approaching the surface of the Bloch sphere. Currently, it is these states that rapidly decay that have the highest purity on average, retaining the most information about the state. In comparison, due to our limited measurement efficiency, stochastically excited trajectories become more mixed as they diffuse towards the excited state. We note that for η=1, all of our trajectories, regardless of dynamics, would describe pure states confined to move only on the surface on the Bloch sphere.

In fact, we expect the ensemble ratio of stochastically excited trajectories to increase with increasing η. As mentioned in the main text, trajectories experience dz<0 when the Weiner increment obtained from the measurement record satisfies . Recall that dW t is a zero-mean random variable distributed with variance dt, and consider the back-action experienced by trajectories initialized with x=1. Naively, the probability of stochastic excitation is then given by the integral,

As η increases, so does the value of this integral. For η=1 and a time step dt=20 ns, the probability for stochastic excitation for our system reaches a maximum value of ∼41.5%. For our measured quantum efficiency of η=0.3, we expect ∼35% of trajectories to excite in the first time step.

Tomography and readout calibration

All tomography results are corrected for imperfect state preparation and readout fidelities. We perform state readout by first applying a resonant pulse at 6.73 GHz to transfer the excited state population to a higher excited state, and then proceed to drive the bare cavity resonance at 6.95 GHz at high power to conduct the Jaynes–Cummings high-power readout technique31. Tomography for y and x is achieved by first applying a 40 ns π/2 rotation about the X or Y axes. The combined state preparation and readout fidelity (80%) was determined from the contrast of resonant Rabi oscillations. Each experimental sequence includes separate calibration measurements used to determine the readout level of the ground state and the prepared excited state. These levels are used to scale the tomography results. Figure 6a,b shows the ensemble decay curves for the state preparations −z and +x.

Figure 6: Tomography calibrations. The ensemble decay as determined by projective measurements for initial states −z (a) and +x (b). (c) Tomographic validation for the ensemble of trajectories shows the average tomography values (x, z) versus the values obtained from individual trajectories. Full size image

The emitter’s state is characterized by expectation values (x, z). To characterize accuracy of the state tracking, we compare the expectation values that are calculated for a single iteration of the experiment to the values obtained from an ensemble of projective measurements. In Fig. 4 we show this comparison to reconstruct an individual trajectory. To accomplish this, we denote an individual trajectory (Note that ). At each time point, we perform several experiments of total duration t′, followed by one of three tomography and readout sequences. For each of these experiments, we calculate (x(t′), z(t′)); if x(t′) and z(t′) are within ±0.12 of and , then the subsequent tomography result is included in the tomographic validation at t′. We follow this process for each t′ along the trajectory, resulting in a tomographic reconstruction of the trajectory.

We can further test the predictions given by the individual trajectories for all runs of the experiment at all times. Figure 6c displays the average projective measurement outcomes conditioned on the values of or compared with the values or showing good agreement between the individual trajectories and the projective measurements.

Phase-sensitive back-action

When the emitter is initialized in +y the state dynamics are not confined to the X–Z plane. Figure 7 displays the state conditioned on the integrated homodyne signal and shows how the y-component does not acquire a correlation with the measurement signal. This may be understood as a result of phase-sensitive amplification with ϕ=0. When we perform our homodyne measurement of the real part of σ − , we de-amplify the quadrature containing information on the imaginary part of σ − , corresponding to σ y on the Bloch sphere. The de-amplification of this orthogonal signal suppresses the magnitude of its quantum fluctuations, effectively eliminating the information associated with the σ y quadrature of the emitter’s dipole. Therefore we do not perform weak measurements of σ y , and we do not observe quantum dynamics such as stochastic excitation.

Figure 7: Spontaneous decay from the state +y. The emitter’s state at different times conditioned on the integrated homodyne measurement signal. The decay times are 80, 160, 320, 640 and 960 ns, and the data correspond to what is depicted in Fig. 2f. The X–Z plane plotted in Fig. 2f is highlighted in blue. Full size image

We may also understand this phenomenon by examining the dz and dy segments of the stochastic master equation provided in the main text. The presence of an xy coefficient on the measurement term in equation (4), means the stochastic back-action has no effect on the state when it is in an eigenstate of σ x or σ y , limiting dynamics to a deterministic reduction in y. Meanwhile, if we examine equation (3) after factoring out a common factor of (1−z), which serves to push the trajectory towards the ground state, we see the measurement term is proportional only to x. Therefore, for a state prepared with y=±1, there will be no initial stochastic excitation, and the state will begin its decay by deterministically approaching the ground state. However, once fluctuations in the measurement signal cause the state to acquire a nonzero x value, the trajectory’s dynamics will cease to be trivial.

Experimental set-up

Figure 8 displays a simplified schematic of the experimental set-up. A single generator is used for qubit rotations, the amplifier pump, and demodulation of the amplified signal. The parametric amplifier is pumped by two sidebands that are equally separated from the carrier by 550 MHz, allowing for phase-sensitive amplification without leakage at the emitter’s transition frequency. The experimental repetition rate is 8 kHz.

Figure 8: Experimental set-up. A single microwave frequency generator is used for the qubit rotations, the amplifier pump and the local oscillator for demodulation. A second generator, operating at the high-power resonance frequency of the cavity is used for state readout for tomography. The third generator is used to transfer state population to a higher excited state of the circuit-cavity system to enhance the readout contrast. Full size image

Data availability

The data that support the findings of this study are available from the corresponding author on request.