Abstract
Decoherence originates from the leakage of quantum information into external degrees of freedom. For a qubit, the two main decoherence channels are relaxation and dephasing. Here, we report an experiment on a superconducting qubit where we retrieve part of the lost information in both of these channels. We demonstrate that raw averaging the corresponding measurement records provides a full quantum tomography of the qubit state where all three components of the effective spin1/2 are simultaneously measured. From single realizations of the experiment, it is possible to infer the quantum trajectories followed by the qubit state conditioned on relaxation and/or dephasing channels. The incompatibility between these quantum measurements of the qubit leads to observable consequences in the statistics of quantum states. The high level of controllability of superconducting circuits enables us to explore many regimes from the Zeno effect to underdamped Rabi oscillations depending on the relative strengths of driving, dephasing, and relaxation.
Introduction
Decoherence can be understood as the result of measurement of a system by its environment. For a qubit, the two main sources of decoherence are relaxation by spontaneous emission and dephasing that can be modeled by unmonitored readout of coupled quantum systems (Fig. 1a). What becomes of the qubit state if, instead of disregarding the information leaking to the environment, we continuously monitor both decoherence channels? Owing to measurement backaction, the knowledge of the measurement record then leads to a stochastic quantum trajectory of the qubit state for each single realization of an experiment^{1,2,3}. Recently, diffusive quantum trajectories were observed following the continuous homodyne or heterodyne measurements of either a dephasing channel^{4,5,6,7,8,9} or a relaxation channel^{10,11}.
Here we report an experiment in which we have simultaneously monitored the spontaneous emission of a superconducting qubit by heterodyne measurement (relaxation channel) and the transmitted field through a dispersively coupled cavity by homodyne measurement (dephasing channel). We demonstrate that the average outcomes of these two nonprojective measurements are the three coordinates x, y, and z of the Bloch vector. It is remarkable that a full quantum tomography can be obtained at any time by simply raw averaging measurement outcomes of many realizations of a single experiment despite the incompatibility of the three observables that characterize a qubit state. For single realizations, the resulting quantum trajectories show signatures of the incompatibility between the measurement channels, therefore extending the previously explored case of two incompatible measurement outcomes^{10,12} to the case of three spin directions. By varying the drive amplitudes at the cavity and qubit transition frequencies, we are able to reach a variety of regimes corresponding to different configurations for Ω/Γ_{1} and Γ_{d}/Γ_{1}, where Ω is the Rabi frequency, Γ_{1} the fixed relaxation rate and Γ_{d} the dephasing rate. This work hence provides a textbook experimental demonstration of quantum measurement backaction on a qubit with incompatible and simultaneous measurements.
Results
Description of the experiment
Two parallel detection setups operate via spatially separated measurement lines (see Fig. 1b). The fluorescence heterodyne detection setup enables the measurement of both quadratures u(t) and v(t) of the spontaneously emitted field out of a 3D transmon qubit^{13}. The complex amplitude of the emitted field is on average proportional to the expectation of the qubit lowering operator σ_{−} = (σ_{x} − iσ_{y})/2 so that u and v are on average proportional to the expectations of σ_{x} and σ_{y}. Here, σ_{x} = \(\left g \right\rangle \left\langle e \right + \left e \right\rangle \left\langle g \right\), σ_{y} = \(i\left g \right\rangle \left\langle e \right  i\left e \right\rangle \left\langle g \right\) and σ_{z} = \(\left e \right\rangle \left\langle e \right  \left g \right\rangle \left\langle g \right\) are the qubit Pauli operators, where \(\left g \right\rangle\) and \(\left e \right\rangle\) are the ground and excited states. For a single realization, the measurement outcomes read^{10}
where Γ_{1} = (15 μs)^{−1} is the qubit relaxation rate, η_{f} = 0.14 is the total fluorescence measurement efficiency, x(t) = Tr(σ_{x}ρ_{t}) and y(t) = Tr(σ_{y}ρ_{t}) are the qubit Bloch coordinates corresponding to the density matrix ρ_{t} (see Fig. 1a) and W_{u} and W_{v} are two independent stochastic Wiener processes describing the measurement noise, which includes the zero point fluctuations, and such that dW^{2} = dt and dW is zero on average. Experimentally, the measurement takes a non infinitesimal time dt = 100 ns, which we chose smaller than the inverse measurement rates and compatible with the detection bandwidth (see Supplementary Note 4).
Similarly, the dispersive detection setup (see Fig. 1b) enables the measurement of a single quadrature w(t) of the transmitted field at frequency f_{d} = f_{r} − χ_{cq}/2, which is between the cavity resonance frequencies f_{r} and f_{r} − χ_{cq} respectively corresponding to a qubit in the ground and excited state (the qubit and cavity are in the dispersive regime and χ_{cq} = 5.1 MHz as explained in Supplementary Note 2). The phase of the measured quadrature in the homodyne measurement can then be chosen in such a way that^{8,14}
Importantly, the measurement induced dephasing rate Γ_{d} can be tuned arbitrarily as it is proportional to the drive power at f_{d}. Similarly to the notations above, η_{d} = 0.34 is the total dispersive measurement efficiency, z(t) = Tr(σ_{z}ρ_{t}) is the last of the three Bloch coordinates (see Fig. 1a) and W_{w} is another independent Wiener process.
Full tomography by direct averaging
As can be seen from Eqs. (1) and (2), taking a raw average of the outcomes (u, v, w) on a large number of realizations of the experiment directly leads to the Bloch coordinates (x,y,z) of the qubit. In Fig. 2, we show the direct averaging of the three outcomes in two configurations of the input drives: one in the regime of underdamped Rabi oscillations (Fig. 2a) and another in the regime of strong dispersive measurement rate, the socalled Zeno regime (Fig. 2b). The raw averaging of (u, v, w), once rescaled by the prefactors in Eqs. (1) and (2), agrees well with the average evolution of the qubit, as predicted by the solution of the master equation (Eq. (3) below without the last stochastic term). We thus demonstrate that performing a dispersive measurement and a measurement of fluorescence reveals information on all three components of a spin1/2. Such a direct full tomography cannot be done by measuring two records only^{10,12}. Note, however, that it is possible to perform an indirect tomography using a small number of records and maximum likelihood estimation^{15}. A comparison between our technique and the usual technique using a qubit rotation followed by a projective measurement is discussed in Supplementary Note 3.
Experiments of Fig. 2a and Fig. 2b differ by the relative rate of the dispersive readout Γ_{d} compared with the Rabi frequency Ω. For weak measurement rate Γ_{d}, Γ_{1} < Ω, the Rabi oscillations are underdamped, wherease they are overdamped when Γ_{d} >> Ω, Γ_{1} owing to the fact that the Zeno effect prevents any unitary evolution such as Rabi oscillations. For a single realization of the experiment though, the trajectory of the qubit state that one can infer from the measurement records u(t), v(t) and w(t) can strongly differ from this average behavior.
Single quantum trajectories
In order to determine this quantum trajectory, one can use the formalism of the stochastic master equation^{14}. The density matrix at time t + dt can be decomposed into ρ_{t+dt} = ρ_{t} + dρ_{t}, where
with the four Lindblad superoperators (k ∈ {u, v, w, φ})
and the measurement backaction superoperators
In these expressions, the jump operators corresponding to heterodyning fluorescence are \(L_u = \sqrt {{ {\Gamma }}_1{\mathrm{/}}2} \sigma _ \) and \(L_v = i\sqrt {{ {\Gamma }}_1{\mathrm{/}}2} \sigma _ \) and the jump operator corresponding to homodyning the dispersive measurement is \(L_w = \sqrt {{ {\Gamma }}_{\mathrm{d}}{\mathrm{/}}2} \sigma _z\). A fourth jump operator \(L_\varphi = \sqrt {{ {\Gamma }}_\varphi {\mathrm{/}}2} \sigma _z\) corresponds to the unread (η_{φ} = 0) pure dephasing of the qubit, so that the total decoherence rate \({\mathit{\Gamma }}_2 = \frac{{{\mathit{\Gamma }}_1}}{2} + {\mathit{\Gamma }}_\varphi + {\mathit{\Gamma }}_{\mathit{d}}\) can be tuned from Γ_{2} = (11.2 μs)^{−1} to higher arbitrary values depending on the power of the drive at frequency f_{d}. Interestingly, the two fluorescence measurement records u and v exert a different backaction but act identically on average (same Lindblad operators). The additional dispersive measurement that we introduced compared to ref.^{10} thus leads to a very different dynamics.
Using this formalism it is possible to reconstruct the quantum trajectory of the qubit state in time from any set of measurement records (see Fig. 1a, c in the case where Ω/2π = (5.2 μs)^{−1} and Γ_{d} = (0.9 μs)^{−1}). The validity of the reconstructed quantum trajectories can be tested independently by postselecting an ensemble of realizations of the experiment for which the trajectory predicts a given value x(T) = x_{traj} at a time T. If the trajectories are valid, then a strong measurement of σ_{x} at time T should give x_{traj} on average on this postselected ensemble of realizations (Fig. 3a). We have checked for any value of x_{traj}, y_{traj}, and z_{traj} (Fig. 3), and for 30 representative configurations of drives that the trajectories predict the strong measurement results (Supplementary Note 1). In fact, we found that the agreement is verified for efficiencies η_{f} and η_{d} within a confidence interval of ±0.02 for any of the 30 configurations.
Evolution of the distribution of states
Any measurement record is a stochastic process and the corresponding quantum trajectories follow a random walk in the Bloch sphere with a state dependent diffusion constant. The inherent backaction of a quantum measurement is thus better discussed by representing distributions of states at a given time^{4,5,8,10,11,12,16} or distributions of trajectories for a given duration^{7,9,17,18,19}. Figure 4 gives a different perspective to the Rabi oscillation of Fig. 2a by representing the distributions of the qubit states conditioned on the three measurement records u(t), v(t), and w(t) for 1.5 million realizations of the experiment. In the Supplementary Note 1, one can find movies of the distributions of 1.5 millions experimental realizations for each configuration of the Rabi frequency Ω and the dephasing rate Γ_{d} for a set of 30 different experimentally realized configurations. Evidently, the Rabi drive term −Ωσ_{y}/2 still provides an overall angular velocity in the x − z plane of the Bloch sphere. However, the measurement backaction is such that some trajectories are delayed, whereas others are advanced compared to the average evolution. As time increases the spread in the qubit states grows as a result of the cumulated effect of the stochastic measurement backaction at each time step.
The effect of decoherence under a strong Rabi drive corresponds to an average loss of purity, defined as Tr(ρ^{2}) = (1 + x^{2} + y^{2} + z^{2})/2 and it can be seen as a decreasing distance of the mean trajectory from the center of the Bloch sphere when time increases (solid line). When the dispersive measurement (dephasing channel) is measured in presence of the Rabi drive around σ_{y}, the corresponding distribution of states tends to be uniform in the x−z plane at long times (right panel in Fig. 4b), which is similar to what is obtained by simultaneously measuring σ_{x} and σ_{z} in an effectively undriven qubit^{12}. The experiment thus illustrates the fact that the average loss of purity corresponds to the statistical uncertainty on the quantum state when the decoherence channel is unread.
Interplay between detectors
Interestingly, although the average trajectory stays in the x − z plane with \(\left\langle {\sigma _y} \right\rangle = 0\), the backaction of the fluorescence measurement leads to a nonzero spread in the y direction of the Bloch sphere. This competition between the backaction of relaxation (fluorescence measurement) and dephasing (dispersive measurement) measurements can be better observed when decoherence dominates the dynamics. In Fig. 5, we show the distributions of qubit states at a long time τ = 6.5 μs after which the distribution is close to its steady state, whereas the qubit is both Rabi driven and dispersively measured at a strong measurement rate. The trajectories are determined using three sets of measurement records: dispersive only {w(t)}, fluorescence only {u(t), v(t)} or both. As in Fig. 2b, the Zeno effect then leads to the dampening of the Rabi oscillations and the average trajectory (solid line) quickly reaches its steady state.
In contrast, a trajectory corresponding to a single realization of the experiment where the dispersive measurement w(t) is recorded is found to consist in a series of stochastic jumps between two areas of the Bloch sphere that are close to the two eigenstate of the σ_{Z} measurement operator. In the distribution of states, this leads to two areas with high probability of occupation near the poles of the Bloch sphere. These areas can be interpreted as zones frozen by the Zeno effect. The rest of the Bloch sphere is still occupied with a lower probability (Fig. 5b) because of the finite time it takes for the jump to occur from one pole to the next under strong dispersive measurement rate^{20}. Note how the ensemble of trajectories can go from uniform for weak measurement rates (Fig. 4b rightest panel) to localized at the poles for strong measurement rates (Fig. 5b).
As can be seen from Fig. 5a,c, the dispersive measurement alone does not provide any backaction toward the y direction of the Bloch sphere so that the qubit states keeps a zero σ_{y} component during its evolution. This is in stark contrast with the trajectories corresponding to measurement records {u(t), v(t)} of the fluorescence (Fig. 5d–f), where at long times the qubit states spans a small ball in the Bloch sphere. Therefore, the combined action of Rabi drive and fluorescence measurement backaction leads to a uniform spread of the qubit state close to the most entropic state 1/2 at the center of the sphere. As expected, the quantum states that are conditioned on all measurement records {u(t), v(t), w(t)} are less entropic than with a single measurement. This can be seen in Fig. 5g–i where the spread of the distributions is larger than for the cases of single measurements Fig. 5a–f.
A clear asymmetry appears in the spread of the marginal distribution in the x − y plane of Fig. 5g between positive and negative values of x. This asymmetry originates from the fact that the fluorescence measurement is linked to the jump operator σ_{−}, for which \(\left g \right\rangle\) is the single pointer state. Indeed the measurement backaction is null when the qubit state is close to \(\left g \right\rangle\) (\({\cal M}_u\left( {\left g \right\rangle \left\langle g \right} \right)\) = \({\cal M}_v\left( {\left g \right\rangle \left\langle g \right} \right) = 0\)) while it is strongest when the qubit state is close to \(\left e \right\rangle\). As the Rabi drive correlates the ground state to positive x (red zone shifted to the right of the south pole in Fig. 5h) and the excited state to negative x, the spread in y is smaller for positive x than for negative x. This asymmetry highlights the profound difference between measuring both quadratures of fluorescence and measuring σ_{x} and σ_{y} simultaneously using dynamical states as in refs.^{12,21}. Although both methods lead to the same result on average, their backaction differs. The latter corresponds to quantum nondemolition measurements, whereas fluorescence does not. In the end, the asymmetry in the distributions of Fig. 5g, i results from the incompatibility between a dispersive measurement with no backaction on \(\left e \right\rangle\) and a fluorescence measurement with maximal backaction on \(\left e \right\rangle\).
In conclusion, we have shown quantum trajectories of a superconducting qubit reconstructed from three measurements originating from the simultaneous monitoring of its decoherence channels. It looks promising to test statistical properties of quantum trajectories^{22,23}, fluctuation relations in quantum thermodynamics^{24,25,26,27,28,29,30}, quantum smoothing protocols^{16,31,32,33,34,35,36}, and to perform parameter estimation^{37,38}.
Data availability
The experiment was carried out for 30 experimental configurations with Ω/2π ranging from 0 to (2 μs)^{−1} and Γ_{d} ranging from (30 μs)^{−1} to (300 ns)^{−1}. All the experimental results can be visualized in a small animated application available online at http://www.physinfo.fr/publications/Ficheux1710.html.
The measurement can be chosen to take into account the measurement records of the dispersive measurement only, the fluorescence measurement only or both. The movies are also available to download at https://doi.org/10.6084/m9.figshare.6127958.v1.
All raw data used in this study are available from the corresponding authors upon reasonable request.
Change history
13 July 2018
The original version of this Article omitted the following from the Acknowledgements: Z. Leghtas' primary affiliation is Centre Automatique et Systèmes, Mines ParisTech. This has been corrected in both the PDF and HTML versions of the Article.
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Acknowledgements
We thank Philippe CampagneIbarcq, Michel Devoret, Andrew Jordan, Raphaël Lescanne, Mazyar Mirrahimi, Klaus Mølmer, Pierre Rouchon, Alain Sarlette, Irfan Siddiqi, and Pierre Six for fruitful interactions. Nanofabrication has been made within the consortium Salle Blanche Paris Centre. This work was supported by the EMERGENCES grant QUMOTEL of Ville de Paris. Z. Leghtas’ primary affiliation is Centre Automatique et Systèmes, Mines ParisTech.
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Q.F., S.J. and B.H. designed research and performed research; S.J. and Q.F. analyzed data; Z.L. contributed to the experimental setup; all authors wrote the paper.
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Ficheux, Q., Jezouin, S., Leghtas, Z. et al. Dynamics of a qubit while simultaneously monitoring its relaxation and dephasing. Nat Commun 9, 1926 (2018). https://doi.org/10.1038/s41467018043729
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