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arXiv:1804.03413v1 [quant-ph] 10 Apr 2018 prtrfrteeigenstate the for cor- With the measure- eigenvector. is state the responding post-measurement is the the and outcome of ment measured one the that of states eigenvalues which postulate, Neu- projection von mann the to according measurement describes chanics Motivation: ere ffedmo h niomn r “summed degrees are observed remaining environment the the how determine of to over” freedom unobserved of the follow and degrees apparatus) Schr¨odinger evolution, measuring unitary environ- the the its includes and system (which the both ment projec- framework, sudden this In the that tion. of step interpolation important framework continuous The an provides is 2]. [1, decoherence process environmental this of of dynamics the ravel to states pure initially states. takes mixed evolution ensemble the and iin n rbblsi ntecoc f“ of choice repe- the on in consistent probabilistic irreversible, and sudden, tition, is change This oe.Teepoaiiisflo h onrule, out- Born the the follow for interpretation These ensemble comes. spec- an are outcomes requiring various ified, of spec- probabilities not is the run, only experimental ified; particular a in occur would prob vrteyas ayatmt aebe aet un- to made been have attempts many years, the Over ave Kumar, Parveen ( i = ) P i = h | uin rnmnqubit. 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[8, bounds parameters their only on or by exist untestable verified are positively they been capacity have experiments—either in- them limited theoretically of with not none are beings consistent, attempts human these Although by as [9–11]. prin- well in observable, as is what ciple and real philosophi- is with attempts what question other interactions cally Some ignored [5–8]. vari- or dynamics hidden dynamics known of novel introduction with ables e.g. physical, are look ensemble to us forces Schr¨odinger evolution. unitary that closed and the run), beyond experimental which in inrlto,Qatmtaetr,Sohsi evo- Stochastic trajectory, Quantum relation, tion 2, efcshr nteetnino unu quantum of extension the on here focus We quantum the of decompositions attempted the of Some ‡ .Vijayaraghavan, R. to sseict ahsystem-apparatus each to specific is ution nu rjcoydsrbto o weak for distribution trajectory antum eista trc h unu state quantum the attract that desics mtrwieniesohsi process. stochastic white-noise ameter eeaiga nebeo quantum of ensemble an generating , let naeigtednmc of dynamics the unraveling to clue l nosreteeouino quantum a of evolution the observe an tcue uniyi,addemonstrate and it, quantify itecture, h os n h trcint be to attraction the and noise the f nu theory. antum tosi etok.Te a be can They textbooks. in ctions doeao.Ti vlto can evolution This operator. ed aglr 602 India 560012, Bangalore , 2, § a ce, n pov Patel Apoorva and i iloccur will ” 1, ¶ ρ 2 the collapse basis as well as the system-apparatus inter- becomes a constant of evolution when gSξ =1 action strength can be varied by changing the control [14, 15], which we impose henceforth. The Itˆoform is parameters and without changing the apparatus mass or convenient for numerical simulations of quantum trajec- size or position. tories: Realisation of quantum measurement as a continu- ous stochastic process is tightly constrained by the well- dρ00 =2√g ρ00 ρ11 dW + (1/T1)ρ11 dt , (6) established properties of [14–16]. A where the Wiener increment satisfies dW (t) = 0 and precise combination of the attraction towards the eigen- 2 hh ii states and unbiased noise is needed to reproduce the Born (dW (t)) = dt. Although we are unable to integrate rule as a constant of evolution [15]. Two of us have em- Eq.(6) exactly, exact integrals of each of the two terms phasised recently that this is a fluctuation-dissipation re- on its right-hand-side are known. We simulate quantum lation [17]. It points to a common origin for the stochastic trajectories using a Gaussian distribution for dW and a symmetric Trotter-type integration scheme. The discreti- and the deterministic contributions to the measurement 2 evolution, analogous to both diffusion and viscous damp- sation error is then O((dt) ) for individual steps, and is ing arising from the same underlying molecular scatter- made negligible by making dt sufficiently small. ing in statistical physics. Moreover, for a binary mea- The distribution of the quantum trajecto- surement (i.e. when the measured operator has only two ries, p(ρ00,t), obeys the Fokker-Planck equation: eigenvalues), the complete quantum trajectory distribu- 2 ∂p(ρ00,t) ∂ 2 2 tion is predicted in terms of only a single dimensionless = 2g 2 ρ00ρ11p(ρ00,t) evolution parameter. ∂t ∂ ρ00   1 ∂ In this work, we experimentally observe quantum tra- ρ11p(ρ00,t) . (7) jectories for superconducting qubits, using weak mea- − T1 ∂ρ00   surements [18] that stretch out the evolution time from Its solution is easier to visualise after the map the initial state to the final projected state. Going be- yond previous experiments [19, 20] that deduced the tanh(z)=2ρ00 1= ρ00 ρ11 , (8) most likely evolution paths from the observed trajecto- − − ries, we quantitatively compare the entire observed tra- from ρ00 [0, 1] to z [ , ]. With the initial con- ∈ ∈ −∞ ∞ jectory distribution with the single parameter theoretical dition p(ρ00, 0) = δ(x), and in absence of any relaxation, prediction, to test the validity of the nonlinear stochastic the exact solution consists of two non-interfering parts evolution model. We also observe the quantum trajecto- with areas x and 1 x, monotonically travelling to the − ries for time scales going up to the relaxation time; the boundaries at ρ00 = 1 and 0 respectively [14, 15, 23]. In relaxation substantially alters the evolution, and we show terms of the variable z, these parts of p(z,t) are Gaus- −1 that the relaxation effects can be successfully described sians, with centres at z±(t) = tanh (2x 1) gt and by a simple modification of the theoretical model. common variance gt. relaxation− introduces± Theoretical Predictions: We consider evolution of a an additional drift in the evolution, which makes the part qubit undergoing binary in absence heading to ρ00 = 1 grow at the expense of the part head- of any driving Hamiltonian, and explicitly include the ing to ρ00 = 0. effect of a finite excited state relaxation time T1. In the Other than the unavoidable relaxation time T1, the continuous stochastic quantum measurement model, the entire quantum trajectory distribution is determined in evolution depends on the nature of the noise; we consider terms of the single dimensionless evolution parameter the particular case of white noise that is appropriate for t ′ ′ τ(g,t) 0 g(t )dt . A strong measurement, τ > 10 [24], weak measurements of [21]. In this quantum is essentially≡ R a projective measurement. In weak mea- diffusion scenario, with 0 and 1 as the measurement surement experiments on transmons, the coupling g is a | i | i eigenstates, the evolves according to [22]: tunable parameter, and the intervening stages between d 1 the initial state and the final projective outcome can be ρ00 =2g(w0 w1) ρ00 ρ11 + ρ11 . (4) observed as τ gradually increases. We next present the dt − T1 results of such an experiment. Here g(t) is the system-apparatus coupling, and wi(t) Experimental Results: Our experiments were carried are real weights representing evolution towards the two out on superconducting 3D transmon qubits, placed in- eigenstates. The weights satisfy w0 + w1 = 1, and side a microwave resonator cavity and dispersively cou- pled to it. The transmon is a nonlinear oscillator [25], w0 w1 = ρ00 ρ11 + Sξ ξ , (5) − − p consisting of a Josephson junction shunted by a capac- where the unbiased white noise with spectral density Sξ itor; the two lowest quantum levels are treated as a obeys ξ(t) = 0 and ξ(t)ξ(t′) = δ(t t′). qubit, which possesses good and is insensi- Thishh evolutionii is a stochastichh differentialii − process on the tive to charge noise. Measurements of the qubit are per- interval [0, 1], with perfectly absorbing boundaries. The formed in the circuit QED architecture [13], by probing 3 the cavity with a resonant microwave pulse; the ampli- estimated (1 e−∆ts/T1 ) as the uncertainty in the initial − tude of the microwave pulse controls the measurement state. The parameters I0, I1, σ were extracted by mak- strength. The scattered wave is amplified by a near- ing Gaussian fits to the current probability distributions quantum-limited Josephson parametric amplifier [26] op- for the ground and the excited states for each measure- erated in the phase-sensitive mode, so that only the ment strength (see Fig. 1). We found that the statistical quadrature containing information about the σz compo- errors in these parameters are small compared to their nent of the qubit is amplified [13]. The amplified single- systematic errors arising from their variations over time. quadrature signal is extracted as a measurement current To estimate the systematic errors, we monitored varia- using standard homodyne detection. We provide more tions in the parameters T1, I0, I1, σ for a duration of 3 details of the device and the experimental setup in the hours. (It took about 10 minutes to generate the tra- Supplementary Information. jectory ensemble for each choice of the system-apparatus The qubit eigenstates produce Gaussian distributions coupling and the initial state.) Then we created differ- for the measurement current, centred at I0 and I1, and ent trajectory distributions from the experimental cur- 2 with variance σ [27]. The measurements are weak when rent data, by varying the initial state, T1, I0, I1, one at ∆I = I0 I1 σ, as illustrated in Fig. 1. The inte- a time within their range of fluctuations, and added the grated| current− | measurement ≪ gives the quantum trajec- shifts in the trajectory distributions in quadrature to esti- tory evolution, according to the Bayesian prescription mate the total systematic error. (More details are given [21] (note that ρ11 =1 ρ00): in the Supplementary Information.) We did not worry − 2 2 about variations in σ, because they get absorbed in the ρ00(t + ∆t) ρ00(t) exp[ (Im(∆t) I0) /2σ ] value of the fit parameter τ(g,t) [30]. Overall, the domi- = − − 2 2 ,(9) ρ11(t + ∆t) ρ11(t) exp[ (Im(∆t) I1) /2σ ] nant sources of error were the uncertainties in the initial ∆t − − 1 ′ ′ state, I0 and I1. Im(∆t) = I(t ) dt . (10) ∆t Z0 Our observed evolution of the quantum trajectory dis- tribution, with the data divided into 100 histogram bins, Simultaneous relaxation of the excited state produces: is shown in Fig. 2, for a particular choice of the system- apparatus coupling and the initial qubit state. We have ρ11(t + ∆t)= ρ11(t) exp( ∆t/T1) . (11) − compared it with the simulated distribution of 107 tra- We construct complete quantum trajectories by combin- jectories, obtained by integrating Eq.(6) with τ(g,t) as ing these two evolutions in a symmetric Trotter-type the only fit parameter. (We generated the simulated dis- scheme. This construction has been shown to be fully tributions for many values of τ to find the best fit, and consistent with direct tomography at any we cross-checked that the simulated distributions agree time t [19]; even though the measurement extracts only with the solution of Eq.(7) obtained using a symmetric partial information, its back-action on the qubit is com- Trotter-type integration scheme.) This fitting of the en- pletely known, and the qubit evolution from a known starting state can be precisely constructed [19, 28]. We prepared the qubit by relaxation for 0.08 500µs, followed by a heralding strong measurement [29]. |0> After a 3µs delay to empty the measurement cavity of 0.04 |1> photons, the initial state in the XZ-subspace of the Bloch P(I) sphere was created by an excitation pulse at the qubit 0 transition frequency. The duration of the pulse was fixed 0.08 by demanding that an immediate strong measurement |0> gives outcomes 0 and 1 with the desired probabil- 0.04 |1> ities. Following| thei state| i preparation, weak measure- P(I) ments were performed to obtain 40µs long trajectories 0 100 110 120 130 140 150 160 with time step ∆t =0.5µs. The process was repeated to generate an ensemble of 106 trajectories. I We determined T1 from the decay rate of the ensem- ble averaged weak measurement current, after initialising FIG. 1: Measurement current distributions for the qubit eigenstates after evolution for 0.5µs, for an ensemble of 106 the qubit in the excited state. We observed that it de- trajectories. They are Gaussians to high accuracy. The bot- pended on the control parameters that fixed the system- tom figure corresponds to a weaker system-apparatus cou- apparatus coupling, i.e. the amplitude of the cavity drive. pling than the top one. The system-apparatus coupling in- Hence for each system-apparatus coupling, we extracted creases mostly by an increase in ∆I without much change in σ. a separate T1 and used that in Eqs.(4,7) to analyse the Gaussian fits give the parameters: (bottom) I0 = 128.443(2), quantum trajectories. Our strong measurements were I1 = 127.856(2), σ = 5.56(3), and (top) I0 = 128.919(2), I1 = 127.286(2), σ = 5.93(2). performed over a single time step ∆ts = 0.5µs, so we 4

12 4 4 t = 1µs t = 5µs t = 10µs τ = 0.011 τ = 0.129 τ = 0.368 9 3 3 χ2 = 188 χ2 = 135 χ2 = 299 ) ) )

00 6 00 2 00 2 ρ ρ ρ p( p( p( 3 1 1

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ρ ρ ρ 00 00 00

4 4 t = 15µs t = 20µs 8 t = 25µs τ = 0.646 τ = 0.881 τ = 1.183 3 2 3 2 2 χ = 220 χ = 378 6 χ = 375 ) ) )

00 2 00 2 00 ρ ρ ρ 4 p( p( p(

1 1 2

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 ρ ρ ρ 00 00 00

FIG. 2: Evolution of the quantum trajectory distribution for the weak Z-measurement of a transmon, with the initial state 6 ρ00 = 0.305(3). The histograms with bin width 0.01 (red) represent the experimental data for an ensemble of 10 trajectories. The trajectory parameters (with errors) were T1 = 45(4)µs, ∆t = 0.5µs, I0 = 128.44(2), I1 = 127.68(3), σ = 5.50(1). The blue curves are the best fits to the quantum diffusion model distribution including relaxation, with the evolution parameter τ ∈ [0, 1.2]; the green curves show the theoretical distributions with the same evolution parameter but with T1 set to infinity. tire trajectory distribution goes much beyond looking at mon qubits, we have obtained similar results in the range just the mean and the variance of the distribution. With τ(g,t) [0, 2]. 2 ∈ 100 data points and only one fit parameter, χ values Discussion: We note that several investigations in re- less than a few hundred indicate good fits, and that is cent years have observed quantum trajectories for trans- what we find. Also shown in Fig. 2 are the theoretically mons undergoing weak measurement; even the distribu- calculated trajectory distributions in absence of any re- tion of quantum trajectories has been qualitatively pre- laxation, i.e. T1 . We see that relaxation alters the → ∞ sented as grey-scale histograms in Ref. [20]. We have distributions substantially, even for t = 0.1T1, and the taken these observations to a quantitative level, demon- quantum diffusion model of Eq.(4) successfully accounts strating that the entire trajectory distribution can be de- for the changes. scribed in terms of a single evolution parameter that is We point out that the mismatch between theory and independent of the initial qubit state and the relaxation experiment grows with increasing evolution time, quite time. (Relaxation is unavoidable in any realistic evolu- likely due to magnification of small initial uncertainties tion, and we accounted for it as a simple exponential due to the iterative evolution. On closer inspection, we decay.) This result puts on a strong footing the quantum also observe certain systematic discrepancies, which are diffusion paradigm, which replaces the projection pos- likely due to experimental imperfections that we have not tulate by an attraction towards the measurement eigen- accounted for. They include transients caused by pho- states plus noise, both arising from the system-apparatus tons left in the cavity after the initial heralding pulse, interaction with precisely related magnitudes. This un- and contamination due to occupation of the higher ex- raveling of the measurement, singling out non-universal cited states of the transmon [31]. More stringent tests of quantum diffusion as an appropriate extension of quan- the quantum diffusion model would need to control and tum mechanics, opens a window on to quantum physics account for them. beyond the textbook description. Also, each quantum We have plotted the best fit τ values against t in Fig. 3, trajectory with its noise history is associated with an in- for two values of the system-apparatus coupling and for dividual experimental run, and understanding that can different qubit initial states. It is seen that they are help in improving quantum control and feedback mecha- essentially independent of the initial state, supporting nisms [32]. the assumption that the measurement evolution is gov- Looking at quantum trajectories as physical processes erned by the system-apparatus coupling alone. Using has important implications. First is the construction of different system-apparatus couplings and different trans- new physical from the distribution of quan- 5 tum trajectories. The trajectories are highly constrained, collapse would then amount to understanding why suf- with pure states (ρ2 = ρ, det(ρ) = 0) remaining pure ficiently amplified coherent states do not remain super- throughout measurement. With this restriction, any posed [33]. Amplification is a driven process, with built- power-series expandable function f(ρ) is a linear com- in time asymmetry, and so the dynamics of the amplifier bination of ρ and I, and so Tr(f(ρ)O) reduces to ensem- [34] would become a crucial ingredient for figuring out ble averages that define conventional expectation values. the quantum to classical cross-over. This is an open field Defining new physical observables that characterise the for future explorations. trajectory distribution is therefore a challenge. Acknowledgments: This work was supported by the Second, the fluctuation-dissipation relation for quan- Department of Atomic Energy of the Government of In- tum trajectories is a powerful clue for understanding the dia. PK acknowledges a CSIR research fellowship from dynamics of quantum measurement [17]. The measure- the Government of India. RV acknowledges funding from ment is specific to each system-apparatus pair, with a the Department of Science and Technology of the Gov- particular value of interaction coupling and a particular ernment of India via the Ramanujan Fellowship. type of noise. It implies that the quantum state collapse is not universal, and the environment can influence the measurement outcomes only via the apparatus and not directly. ∗ Electronic address: [email protected] Finally, while the quantum diffusion dynamics replaces † Electronic address: [email protected] the system-dependent Born rule by a system-independent ‡ Electronic address: [email protected] noise, the origin of the noise remains to be understood. § Electronic address: [email protected] That necessitates making a quantum model for the ap- ¶ Electronic address: [email protected] paratus. In our experiment, the apparatus pointer states [1] J.A. Wheeler and W.H. 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Supplementary Information no other effect on comparison between the experimen- tal data and the theoretical prediction. Working back- Experimental Details wards (see [30]), we estimate the amplifier efficiency to be about 0.2 for the weaker coupling, and about 0.35 for the The single junction 3D transmon device was fabri- stronger coupling (for the same two data sets presented cated using standard e-beam lithography and double an- in Figs. 1,3). gle evaporation of aluminum on an intrinsic silicon wafer. The extracted values of I0 and I1 showed a noticeable The device chip was placed inside a 3D aluminum cav- anomalous behaviour in the first 2µs; the former should ity with a resonant frequency ωc = 7.240 GHz and a be a constant and the latter should decay exponentially. linewidth κ = 3.05 MHz. The chip was positioned away This anomaly is likely due to the photons left in the cav- from the center of the cavity and the measured dispersive ity after the heralding/excitation pulse. To take care of coupling was gd = 81.76 MHz. The qubit frequency was it, we fitted the observed I0 and I1 by exponential func- measured to be ω01 =4.93521 GHz with an anharmonic- tions for t > 2µs. Then we constructed the evolution ity α = 331.6 MHz. The dispersive shift χ = 365 kHz is trajectories using the observed I0 and I1 for t 2µs, about an order of magnitude smaller than κ, so that the ≤ and the fitted values of I0(t ) and I1(2.5µs) for scattering phase shift and the average information per t> 2µs. → ∞ photon become small enough to enable operation in the To obtain χ2 values for fits of the experimental quan- weak measurement regime. tum trajectory histograms with the theoretical predic- The Josephson Parametric Amplifier (JPA) is also fab- tions, we needed estimates of errors for the binned his- ricated using similar techniques, but it is designed with a togram values. With sufficiently large trajectory ensem- SQUID to enable tuning of the amplifier center frequency. bles, the statistical errors are small compared to the sys- It is operated in the phase sensitive mode using double- tematic errors. The systematic errors were not directly pumping technique and amplifies only the quadrature available, because the histograms were produced after containing the information about the σz component of constructing the evolution trajectories as per Eqs.(9,11). the qubit. The measurement pulse, paramp pump and So we first estimated errors in the parameters T1, initial the demodulation signal are all generated from the same state, I0, I1 that are used in construction of the evolu- microwave source, and variable attenuators and phase tion trajectories. by direct analysis of the experimental shifters are used to control each tone. data. Then we generated different trajectory ensembles by shifting the parameter values by their errors, one pa- Data Analysis rameter at a time. Finally, assuming that the changes in histogram values obtained for individual parameter shifts were independent, they were added in quadrature The observed variance of the measurement current dis- to ascertain the total systematic error for the binned his- tributions, σ2, is modified from its ideal value due to lim- 2 2 2 togram values. ited amplifier efficiency: σ = σideal +σnoise. That affects the trajectory reconstruction, in the sense that the actual Overall, we have noticed that the agreement between collapse is faster than what is seen. As a result, the fit- experiment and theory improves with weaker couplings; ted value of τ(g,t) automatically includes the factor of we are yet to understand why. It may very well be that efficiency (the actual value of τ is different). Apart from the unaccounted sources of error have a larger influence this simple rescaling of τ, limited amplifier efficiency has at stronger couplings.