Dynamics of a qubit while simultaneously monitoring its relaxation and dephasing

Q. Ficheux,1, 2 S. Jezouin,2 Z. Leghtas,3, 2, 4 and B. Huard1, 2, ∗

1Universite Lyon, ENS de Lyon, Universite Claude Bernard,CNRS, Laboratoire de Physique, F-69342 Lyon, France

2Laboratoire Pierre Aigrain, Departement de physique de l’ENS,Ecole normale superieure, PSL Research University,

Universite Paris Diderot, Sorbonne Paris Cite, Sorbonne Universites,UPMC Univ. Paris 06, CNRS, 75005 Paris, France

3Centre Automatique et Systemes, Mines ParisTech, PSL Research University,60 Boulevard Saint-Michel, 75272 Paris Cedex 6, France.

4QUANTIC team, INRIA de Paris, 2 Rue Simone Iff, 75012 Paris, France(Dated: April 26, 2018)

AbstractDecoherence 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 anexperiment on a superconducting qubit where we retrieve part of the lost information in both of thesechannels. We demonstrate that raw averaging the corresponding measurement records provides afull quantum tomography of the qubit state where all three components of the effective spin-1/2are simultaneously measured. From single realizations of the experiment, it is possible to inferthe quantum trajectories followed by the qubit state conditioned on relaxation and/or dephasingchannels. The incompatibility between these quantum measurements of the qubit leads to observableconsequences in the statistics of quantum states. The high level of controllability of superconductingcircuits enables us to explore many regimes from the Zeno effect to underdamped Rabi oscillationsdepending on the relative strengths of driving, dephasing and relaxation.

Introduction

Decoherence can be understood as the result of mea-surement of a system by its environment. For a qubit,the two main sources of decoherence are relaxation byspontaneous emission and dephasing that can be mod-eled by unmonitored readout of coupled quantum sys-tems (Fig. 1a). What becomes of the qubit state if,instead of disregarding the information leaking to theenvironment, we continuously monitor both decoherencechannels? Owing to measurement backaction, the knowl-edge of the measurement record then leads to a stochas-tic quantum trajectory of the qubit state for each singlerealization of an experiment [1–3]. Recently, diffusivequantum trajectories were observed following the contin-uous homodyne or heterodyne measurements of either adephasing channel [4–9] or a relaxation channel [10, 11].

Here we report an experiment in which we have si-multaneously monitored the spontaneous emission of asuperconducting qubit by heterodyne measurement (re-laxation channel) and the transmitted field through adispersively coupled cavity by homodyne measurement(dephasing channel). We demonstrate that the averageoutcomes of these two non-projective measurements arethe three coordinates x, y and z of the Bloch vector. Itis remarkable that a full quantum tomography can beobtained at any time by simply raw averaging measure-ment outcomes of many realizations of a single experi-ment despite the incompatibility of the three observablesthat characterize a qubit state. For single realizations theresulting quantum trajectories show signatures of the in-compatibility between the measurement channels, there-

fore extending the previously explored case of two in-compatible measurement outcomes [10, 12] to the case ofthree spin directions. By varying the drive amplitudes atthe cavity and qubit transition frequencies, we are able toreach a variety of regimes corresponding to different con-figurations for Ω/Γ1 and Γd/Γ1, where Ω is the Rabi fre-quency, Γ1 the fixed relaxation rate and Γd the dephasingrate. This work hence provides a textbook experimen-tal demonstration of quantum measurement backactionon a qubit with incompatible and simultaneous measure-ments.ResultsDescription of the experiment. Two parallel de-

tection setups operate via spatially separated measure-ment lines (see Fig. 1b). The fluorescence heterodynedetection setup enables the measurement of both quadra-tures u(t) and v(t) of the spontaneously emitted field outof a 3D transmon qubit [13]. The complex amplitude ofthe emitted field is on average proportional to the expec-tation of the qubit lowering operator σ− = (σx − iσy)/2so that u and v are on average proportional to the ex-pectations of σx and σy. Here, σx = |g〉 〈e| + |e〉 〈g|,σy = i |g〉 〈e| − i |e〉 〈g| and σz = |e〉 〈e| − |g〉 〈g| are thequbit Pauli operators, where |g〉 and |e〉 are the groundand excited states. For a single realization, the measure-ment outcomes read [10]

u(t)dt =√ηfΓ1/2x(t)dt+ dWu(t)

v(t)dt =√ηfΓ1/2y(t)dt+ dWv(t)

, (1)

where Γ1 = (15 µs)−1 is the qubit relaxation rate,ηf = 0.14 is the total fluorescence measurement efficiency,

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FIG. 1: Measurement setup and quantum trajectory result-ing from its outputs. a Bloch vector representation of aqubit whose state is described by a density matrix ρt =(1 + x(t)σx + y(t)σy + z(t)σz) /2. A quantum trajectory ρtis represented as a blue line. The qubit decoherence can bemodeled as originating from a relaxation channel at a rateΓ1 and a dispersive measurement channel at a rate Γd. b Asuperconducting qubit in a cavity is driven by two microwavesignals at the weakly coupled input. The one at qubit fre-quency fq = 5.353 GHz (orange) induces Rabi oscillationsof the qubit at frequency Ω. The one at cavity frequencyfd = 7.761 GHz (purple) leads to a dispersive measurementof the qubit state along σz. A diplexer at the strongly cou-pled output port separates the outgoing signals depending ontheir frequency. The radiation at fq that is spontaneouslyemitted by the qubit is processed by a Josephson Paramet-ric Converter (JPC) [14, 15] so that a following heterodynemeasurement reveals the two quadratures u(t) and v(t) of thefluorescence field [10, 16, 17]. The transmitted signal at fdis processed by a doubly pumped Josephson Parametric Am-plifier (JPA) [4, 18] with a pump phase such that a followinghomodyne measurement reveals the quadrature w(t) of thefield at fd. c Measurement records u (blue), v (red) and w(yellow) as a function of time for one realization of the exper-iment. These records feed the stochastic master equation (3),which leads to the trajectory in a

x(t) = Tr(σxρt) and y(t) = Tr(σyρt) are the qubit Blochcoordinates corresponding to the density matrix ρt (seeFig. 1a) and Wu and Wv are two independent stochas-tic Wiener processes describing the measurement noise,which includes the zero point fluctuations, and such thatdW 2 = dt and dW is zero on average. Experimen-tally, the measurement takes a non infinitesimal timedt = 100 ns, which we chose smaller than the inversemeasurement rates and compatible with the detectionbandwidth (see Supplementary Note 4).

Similarly, the dispersive detection setup (see Fig. 1b)enables the measurement of a single quadrature w(t) ofthe transmitted field at frequency fd = fr−χcq/2, whichis between the cavity resonance frequencies fr and fr−χcq

respectively corresponding to a qubit in the ground and

excited state (the qubit and cavity are in the dispersiveregime and χcq = 5.1 MHz as explained in Supplemen-tary Note 2). The phase of the measured quadrature inthe homodyne measurement can then be chosen in sucha way that [8, 17]

w(t)dt =√

2ηdΓdz(t)dt+ dWw(t). (2)

Importantly, the measurement induced dephasing rateΓd can be tuned arbitrarily as it is proportional to thedrive power at fd. Similarly to the notations above, ηd =0.34 is the total dispersive measurement efficiency, z(t) =Tr(σzρt) is the last of the three Bloch coordinates (seeFig. 1a) and Ww is another independent Wiener process.

Full tomography by direct averaging. As canbe seen from Eqs. (1,2), taking a raw average of theoutcomes (u, v, w) on a large number of realizations ofthe experiment directly leads to the Bloch coordinates(x, y, z) of the qubit. In Fig. 2, we show the direct aver-aging of the three outcomes in two configurations of theinput drives: one in the regime of underdamped Rabi os-cillations (Fig. 2a) and another in the regime of strongdispersive measurement rate, the so-called Zeno regime(Fig. 2b). The raw averaging of (u, v, w), once rescaled bythe prefactors in Eqs. (1,2), agrees well with the averageevolution of the qubit, as predicted by the solution of themaster equation (Eq. 3 below without the last stochasticterm). We thus demonstrate that performing a dispersivemeasurement and a measurement of fluorescence revealsinformation on all three components of a spin-1/2. Such adirect full tomography cannot be done by measuring tworecords only [10, 12]. Note however that it is possible toperform an indirect tomography using a small number ofrecords and maximum likelihood estimation [19]. A com-parison between our technique and the usual techniqueusing a qubit rotation followed by a projective measure-ment is discussed in Supplementary Note 3.

Experiments of Figs. 2a and 2b differ by the relativerate of the dispersive readout Γd compared to the Rabifrequency Ω. For weak measurement rate Γd,Γ1 < Ω,the Rabi oscillations are underdamped, while they areoverdamped when Γd Ω,Γ1 owing to the fact that theZeno effect prevents any unitary evolution such as Rabioscillations. For a single realization of the experimentthough, the trajectory of the qubit state that one caninfer from the measurement records u(t), v(t) and w(t)can strongly differ from this average behavior.

Single quantum trajectories. In order to determinethis quantum trajectory, one can use the formalism of thestochastic master equation [17]. The density matrix attime t + dt can be decomposed into ρt+dt = ρt + dρt,where

dρt = i[Ω

2σy, ρt]dt+

∑k

Dk(ρt)dt+∑k

√ηkMk(ρt)dWk(t),

(3)

3

FIG. 2: Direct averaging of the three measurement records.a Dots: Rescaled average of the measurement recordsu(t) = u(t)/

√ηfΓ1/2, v(t) = v(t)/

√ηfΓ1/2 and w(t) =

w(t)/√

2ηdΓd for 1.5 106 realizations of an experiment wherethe qubit starts in |g〉 at time 0 and is driven so that it rotatesat a Rabi frequency Ω/2π = (2 µs)−1 around σy and enduresa measurement induced dephasing rate Γd = (5 µs)−1. Lines:Calculated coordinates of the Bloch vector x(t), y(t) and z(t)from the master equation (Eq. 3 with ηi = 0). b Same figurein the Zeno regime with a drive such that Ω/2π = (16 µs)−1

and Γd = (0.9 µs)−1.

with the four Lindblad superoperator (k ∈ u, v, w, ϕ)

Dk(ρt) = LkρtL†k −

1

2ρtL†kLk −

1

2L†kLkρt, (4)

and the measurement backaction superoperator

Mk(ρt) = Lkρt + ρtL†k − Tr(Lkρt + ρtL

†k)ρt. (5)

In these expressions, the jump operators correspondingto heterodyning fluorescence are Lu =

√Γ1/2σ− and

Lv = i√

Γ1/2σ− and the jump operator correspond-ing to homodyning the dispersive measurement is Lw =√

Γd/2σz. A fourth jump operator Lϕ =√

Γϕ/2σzcorresponds to the unread (ηϕ = 0) pure dephasingof the qubit, so that the total decoherence rate Γ2 =Γ1

2 + Γϕ + Γd can be tuned from Γ2 = (11.2 µs)−1 tohigher arbitrary values depending on the power of thedrive at frequency fd. Interestingly, the two fluorescencemeasurement records u and v exert a different backac-tion but act identically on average (same Lindblad oper-ators). The additional dispersive measurement that weintroduced compared to Ref. [10] thus leads to a verydifferent dynamics.

Using this formalism it is possible to reconstruct thequantum trajectory of the qubit state in time from anyset of measurement records (see Fig. 1a,c in the casewhere Ω/2π = (5.2 µs)−1 and Γd = (0.9 µs)−1). Thevalidity of the reconstructed quantum trajectories canbe tested independently by post-selecting an ensemble of

FIG. 3: Tomographic validation of the quantum trajectories.a,b,c Correlations between the coordinates (xtraj, ytraj, ztraj)of the trajectories after 19.8 µs of evolution and an indepen-dent tomography on the dataset corresponding to the experi-ment of Fig. 2a. Each panel reprensents the average value ofthe tomography results for the subset of trajectories endingup less than 0.01 away from a given value of xtraj (a), ytraj(b) or ztraj (c). The error bars are given by the standard de-viation of the tomography results divided by the square rootof the number of trajectories in the subset (out of a totalnumber of 1.5 million trajectories per panel). The agreementbetween the tomography and the coordinates of the trajec-tories demonstrates the validity of the quantum trajectories.d Bloch sphere representation of 3 quantum trajectories thatend up with 0.74 < xtraj < 0.76 (red dashed line) after 19.8 µscorresponding to one bin of the histogram in a.

realizations of the experiment for which the trajectorypredicts a given value x(T ) = xtraj at a time T . If thetrajectories are valid, then a strong measurement of σx attime T should give xtraj on average on this post-selectedensemble of realizations (Fig. 3a). We have checked forany value of xtraj, ytraj and ztraj (Fig. 3), and for 30 rep-resentative configurations of drives that the trajectoriespredict the strong measurement results (SupplementaryNote 1). In fact, we found that the agreement is verifiedfor efficiencies ηf and ηd within a confidence interval of±0.02 for any of the 30 configurations.

Evolution of the distribution of states. Any mea-surement record is a stochastic process and the corre-sponding quantum trajectories follow a random walk inthe Bloch sphere with a state dependent diffusion con-stant. The inherent backaction of a quantum measure-ment is thus better discussed by representing distribu-tions of states at a given time [4, 5, 8, 10–12, 20] ordistributions of trajectories for a given duration [7, 21–24]. Figure 4 gives a different perspective to the Rabioscillation of Fig. 2a by representing the distributions ofthe qubit states conditioned on the three measurementrecords u(t), v(t) and w(t) for 1.5 million realizationsof the experiment. In the Supplementary Note 1, one

4

a

b

c

# trajectories1 10 100 1000

FIG. 4: Evolution of the distribution of quantum states. a,b,c Colored dots: Each frame represents the marginal distribution,in the x− y (a), x− z (Fig. b) and y− z (Fig. c) planes of the Bloch sphere, of the states of the qubit at a given time τ for 1.5million realizations of the experiment, in the same experimental conditions as Fig. 2a. Each state (x, y, z) is reconstructed fromthe measurement records u(t), v(t), w(t) from time t between 0 to τ using Eq. (3). Time τ is increasing from 0.3 µs to 19 µsfrom left to right as indicated at the bottom of the figure. For each figure, the surrounding black circles represent the purestates of the plane (e.g. z = 0 for a). Solid lines: average projection of all 1.5 millions of quantum trajectories x(t), y(t), z(t)for 0.2 µs < t < τ .

can find movies of the distributions of 1.5 millions ex-perimental realizations for each configuration of the Rabifrequency Ω and the dephasing rate Γd for a set of 30 dif-ferent experimentally realized configurations. Evidently,the Rabi drive term −Ωσy/2 still provides an overall an-gular velocity in the x − z plane of the Bloch sphere.However, the measurement backaction is such that sometrajectories are delayed while others are advanced com-pared to the average evolution. As time increases thespread in the qubit states grows as a result of the cumu-lated effect of the stochastic measurement backaction ateach time step.

The effect of decoherence under a strong Rabi drivecorresponds to an average loss of purity, defined asTr(ρ2) = (1 + x2 + y2 + z2)/2 and it can be seen as adecreasing distance of the mean trajectory from the cen-ter of the Bloch sphere when time increases (solid line).When the dispersive measurement (dephasing channel)is measured in presence of the Rabi drive around σy, thecorresponding distribution of states tends to be uniformin the x− z plane at long times (right panel in Fig. 4b),which is similar to what is obtained by simultaneouslymeasuring σx and σz in an effectively undriven qubit [12].The experiment thus illustrates the fact that the averageloss of purity corresponds to the statistical uncertaintyon the quantum state when the decoherence channel isunread.

Interplay between detectors. Interestingly, whilethe average trajectory stays in the x − z plane with〈σy〉 = 0, the backaction of the fluorescence measure-ment leads to a nonzero spread in the y direction of theBloch sphere. This competition between the backactionof relaxation (fluorescence measurement) and dephasing(dispersive measurement) measurements can be betterobserved when decoherence dominates the dynamics. InFig. 5, we show the distributions of qubit states at a longtime τ = 6.5 µs after which the distribution is close to itssteady state while the qubit is both Rabi driven and dis-persively measured at a strong measurement rate. Thetrajectories are determined using three sets of measure-ment records: dispersive only w(t), fluorescence onlyu(t), v(t) or both. As in Fig. 2b, the Zeno effect thenleads to the dampening of the Rabi oscillations and theaverage trajectory (solid line) quickly reaches its steadystate.

In contrast, a trajectory corresponding to a single real-ization of the experiment where the dispersive measure-ment w(t) is recorded is found to consist in a series ofstochastic jumps between two areas of the Bloch spherethat are close to the two eigenstate of the σZ measure-ment operator. In the distribution of states, this leadsto two areas with high probability of occupation nearthe poles of the Bloch sphere. These areas can be in-terpreted as zones frozen by the Zeno effect. The rest of

5

a b c

d e f

ihg

dispersive measurement

fluorescence measurement

dispersive and fluorescence measurement

# trajectories1 10 100 1000

FIG. 5: Impact of the type of detector on the distribution ofquantum states. a,b,c Marginal distribution in the x, y (a),x − z (b) and y − z (c) planes of the Bloch sphere of thequbit states ρτ corresponding to 1.5 millions of measurementrecords at the cavity frequency only w(t) from time t be-tween 0 and τ = 6.5 µs. The information about u(t), v(t)is here discarded (ηf = 0). All panels in the figure correspondto the Zeno regime (Ω/2π = (5.2 µs)−1 Γd = (0.9 µs)−1)As in Fig. 4, the boundary of the Bloch sphere is representedas a black circle and the average quantum trajectory as asolid line. d,e,f Case where the states are conditioned onfluorescence records u(t), v(t) instead while discarding theinformation on w(t) (ηd = 0). g,h,i Case where the statesare conditioned on both fluorescence and dispersive measure-ment records u(t), v(t), w(t).

the Bloch sphere is still occupied with a lower probability(Fig. 5b) because of the finite time it takes for the jumpto occur from one pole to the next under strong disper-sive measurement rate [25]. Note how the ensemble oftrajectories can go from uniform for weak measurementrates (Fig. 4b rightest panel) to localized at the poles forstrong measurement rates (Fig. 5b).

As can be seen from Figs. 5a,c, the dispersive measure-ment alone does not provide any backaction towards they direction of the Bloch sphere so that the qubit stateskeeps a zero σy component during its evolution. Thisis in stark contrast with the trajectories correspondingto measurement records u(t), v(t) of the fluorescence

(Figs. 5d-f), where at long times the qubit states spansa small ball in the Bloch sphere. Therefore, the com-bined action of Rabi drive and fluorescence measurementbackaction leads to a uniform spread of the qubit stateclose to the most entropic state 1/2 at the center of thesphere. As expected, the quantum states that are condi-tioned on all measurement records u(t), v(t), w(t) areless entropic than with a single measurement. This canbe seen in Figs. 5g-i where the spread of the distribu-tions is larger than for the cases of single measurementsFigs. 5a-f.

A clear asymmetry appears in the spread of themarginal distribution in the x − y plane of Fig. 5g be-tween positive and negative values of x. This asymme-try originates from the fact that the fluorescence mea-surement is linked to the jump operator σ− for which|g〉 is the single pointer state. Indeed the measurementbackaction is null when the qubit state is close to |g〉(Mu(|g〉〈g|) =Mv(|g〉〈g|) = 0) while it is strongest whenthe qubit state is close to |e〉. Since the Rabi drive cor-relates the ground state to positive x (red zone shiftedto the right of the south pole in Fig. 5h) and the excitedstate to negative x, the spread in y is smaller for posi-tive x than for negative x. This asymmetry highlightsthe profound difference between measuring both quadra-tures of fluorescence and measuring σx and σy simulta-neously using dynamical states as in Ref. [12, 26]. Whileboth methods lead to the same result on average, theirbackaction differs. The latter corresponds to quantumnon-demolition measurements, while fluorescence doesnot. In the end, the asymmetry in the distributions ofFigs. 5g,i results from the incompatibility between a dis-persive measurement with no backaction on |e〉 and afluorescence measurement with maximal backaction on|e〉.

In conclusion, we have shown quantum trajectories ofa superconducting qubit reconstructed from three mea-surements originating from the simultaneous monitoringof its decoherence channels. It looks promising to teststatistical properties of quantum trajectories [27, 28],fluctuation relations in quantum thermodynamics [29–35], quantum smoothing protocols [20, 36–41], and toperform parameter estimation [42, 43].

Data availability The experiment was carried out for 30 exper-imental configurations with Ω/2π ranging from 0 to (2 µs)−1 andΓd ranging from (30 µs)−1 to (300 ns)−1. All the experimentalresults can be visualized in a small animated application availableonline at

http://www.physinfo.fr/publications/Ficheux1710.html.

The measurement can be chosen to take into account the mea-surement records of the dispersive measurement only, the fluores-cence measurement only or both. The movies are also available todownload at

https://doi.org/10.6084/m9.figshare.6127958.v1.

All raw data used in this study are available from the corre-sponding authors upon reasonable request.

Author contributions Q.F. and S.J. contributed equally tothis work. Q.F., S.J. and B.H. designed research and performedresearch; S.J. and Q.F. analyzed data; ZL contributed to the ex-

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perimental setup; All authors wrote the paper.

Acknowledgements We thank Philippe Campagne-Ibarcq,Michel Devoret, Andrew Jordan, Raphael Lescanne, Mazyar Mir-rahimi, Klaus Mølmer, Pierre Rouchon, Alain Sarlette, Irfan Sid-diqi and Pierre Six for fruitful interactions. Nanofabrication hasbeen made within the consortium Salle Blanche Paris Centre. Thiswork was supported by the EMERGENCES grant QUMOTEL ofVille de Paris.

Competing Interests The Authors declare no Competing Fi-nancial or Non-Financial Interests.

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