Observing Quantum State Diffusion by Heterodyne Detection of Fluorescence

A qubit can relax by fluorescence, which prompts the release of a photon into its electromagnetic environment. By counting the emitted photons, discrete quantum jumps of the qubit state can be observed. The succession of states occupied by the qubit in a single experiment, its quantum trajectory, depends in fact on the kind of detector. How are the quantum trajectories modified if one measures continuously the amplitude of the fluorescence field instead? Using a superconducting parametric amplifier, we perform heterodyne detection of the fluorescence of a superconducting qubit. For each realization of the measurement record, we can reconstruct a different quantum trajectory for the qubit. The observed evolution obeys quantum state diffusion, which is characteristic of quantum measurements subject to zero-point fluctuations. Independent projective measurements of the qubit at various times provide a quantitative verification of the reconstructed trajectories. By exploring the statistics of quantum trajectories, we demonstrate that the qubit states span a deterministic surface in the Bloch sphere at each time in the evolution. Additionally, we show that when monitoring fluorescence field quadratures, coherent superpositions are generated during the decay from excited to ground state. Counterintuitively, measuring light emitted during relaxation can give rise to trajectories with increased excitation probability.

Popular Summary

Light emitted via fluorescence is associated with matter decaying in energy, and this light can be viewed as a probe that carries information about the state of its emitter. When this information is lost, the fragile quantum properties of the emitter are destroyed, a process known as decoherence. Using a superconducting qubit, we demonstrate how the sole measurement of fluorescence makes it possible to accurately track the quantum state in time. The observed evolution is erratic, which is expected based on the random backaction of measurements in quantum mechanics.

We continuously measure the amplitude of the fluorescence field emitted by a superconducting qubit using an amplifier close to the quantum limit; our measurements are obtained at cryogenic temperatures. From each fluorescence record, we can reconstruct a quantum trajectory, which is the succession of states the qubit occupies on a single relaxation event. We collect independent measurements of the qubit state at an arbitrary time during relaxation. These measurements follow the statistics that are expected from the quantum trajectories, thereby verifying the reconstructed quantum states. By repeating the experiment millions of times, we are able to determine the distribution of quantum trajectories. Strikingly, monitoring fluorescence can generate a superposition of states and counterintuitively lead to a temporary increase in the qubit excitation probability.

Our work provides an experimental demonstration of the quantum-state diffusion associated with spontaneous emission that triggered the field of quantum trajectories in the 1990s. We expect that our findings, which enlighten the correspondence between decoherence and measurement by the environment, will contribute to the progress of quantum error correction.

Figure 1

Scheme of the experiment. (a) The fluorescence field of a superconducting qubit in an off-resonant cavity is recorded using a heterodyne detection setup from time 0 to $T$. Following amplification by a Josephson parametric converter (JPC), the signal is down-converted to $f_h=100\mathrm{MHz}$ and numerically demodulated and integrated on time steps $dt=200\mathrm{ns}$ into its two quadratures $dI$ and $dQ$. The rectangular symbols represent the correction of finite detection bandwidth in the setup [28]. The quantum trajectory $\{{\rho }_t\}$ is then computed using Eq. (2). A single local oscillator at $f_q+f_h$ is used for qubit manipulation and down-conversion of the fluorescence signal. The transmission at cavity frequency $f_c=7.8\mathrm{GHz}$ is used to independently read-out the qubit at time $T$ [28]. (b) Pulse sequence. At time 0, the qubit is prepared in $|+x⟩$ ($|+e⟩$) with a 52 ns (104 ns) rotation pulse around ${\sigma }_y$. The fluorescence record is acquired for a duration $T$ ranging from 0 to $10\mu \mathrm{s}$. The state is then projectively read-out along one of the three Pauli operators using a pulse at frequency $f_c$ preceded by a rotation pulse around ${\sigma }_y$, ${\sigma }_x$, or no pulse.

Figure 2

Quantum trajectories. (a) Each panel displays two different measurement records of one quadrature of the fluorescence field for a qubit initially in $|+x⟩$. Actual measurements are shown as red and green dots linked by straight lines for clarity. The blue dots correspond to the average record on all experiments, and the blue line corresponds to an exponential fit of these dots in $e^{-{\gamma }_{1}t/2-{\gamma }_{\varphi }t}$. A zoom-in is shown as an inset. Measurements of $dI$ and $dQ$ are normalized by an overall prefactor so that their variance is $dt$ [28]. (b) Quantum trajectories followed by the qubit under relaxation represented in the Bloch sphere for the measurement records shown in (a). The Bloch vector $(x,y,z)$ corresponds to the state $\rho =(\mathbf{1}+x{\sigma }_x+y{\sigma }_y+z{\sigma }_z)/2$ and the black circles set the scale of the Bloch sphere extrema. The trajectories are obtained by solving numerically Eq. (2) and are shown as dots linked by straight lines both in the sphere and in the three projections along $x$, $y$, and $z$. Colors identify which record of (a) is used. A blue line shows the average evolution without monitoring ($\eta =0$). (c), (d) Same representations for two realizations starting from $|e⟩$. They are arbitrarily selected to end up in a state with similar Bloch coordinate $x_{\mathrm{traj}}=0.42\pm 0.02$, as indicated by a dashed red line.

Figure 3

Tomography versus quantum trajectories. (a) For each value of $x_{\mathrm{traj}}$ a red dot indicates the average value $x_{\mathrm{tomo}}$ of the final measurement of ${\sigma }_x$ on the subset of experiments starting from $|+x⟩$ and for which the quantum trajectory ends up at time $T=4\mu \mathrm{s}$ in a state such that $⟨{\sigma }_x{⟩}_{{\rho }_T}=x_{\mathrm{traj}}\pm 0.02$. Error bars represent the statistical uncertainty and the solid line has a slope 1. Subsets with less than 40 experiments are not shown. The final projective measurement infidelity is corrected for [28]. Similar dots represent the case of $y$ (blue) and $z$ (green). (b) Distribution of the final Bloch coordinate $i_{\mathrm{traj}}$ of the ${10}^6$ quantum trajectories starting in $|+x⟩$ for $T=4\mu \mathrm{s}$, and followed by a ${\sigma }_i$ projective measurement. The color code and the bin width match those in (a). (c),(d) Same as (a) and (b) for experiments starting in $|e⟩$.

Figure 4

Statistics of quantum trajectories. Distributions of the qubit states along $10\text{−}\mu \mathrm{s}$-long trajectories for a qubit initially in (a) $|+x⟩$ and (b) $|e⟩$. The number of trajectories reaching each cubic cell of side 0.04 is encoded in color, out of a total of $3\times {10}^6$. White arrows go from $-1$ to $+1$ along ${\sigma }_{x,y,z}$ and the Bloch sphere is colored in gray. Meridians of the spheroid spanned by Eq. (5) are also shown.