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Quantum Trajectory Distribution for Weak Measurement of A Quantum Trajectory Distribution for Weak Measurement of a Superconducting Qubit: Experiment meets Theory Parveen Kumar,1, ∗ Suman Kundu,2, † Madhavi Chand,2, ‡ R. Vijayaraghavan,2, § and Apoorva Patel1, ¶ 1Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India 2Department of Condensed Matter Physics and Materials Science, Tata Institue of Fundamental Research, Mumbai 400005, India (Dated: April 11, 2018) Quantum measurements are described as instantaneous projections in textbooks. They can be stretched out in time using weak measurements, whereby one can observe the evolution of a quantum state as it heads towards one of the eigenstates of the measured operator. This evolution can be understood as a continuous nonlinear stochastic process, generating an ensemble of quantum trajectories, consisting of noisy fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. The rate of evolution is specific to each system-apparatus pair, and the Born rule constraint requires the magnitudes of the noise and the attraction to be precisely related. We experimentally observe the entire quantum trajectory distribution for weak measurements of a superconducting qubit in circuit QED architecture, quantify it, and demonstrate that it agrees very well with the predictions of a single-parameter white-noise stochastic process. This characterisation of quantum trajectories is a powerful clue to unraveling the dynamics of quantum measurement, beyond the conventional axiomatic quantum theory. PACS numbers: 03.65.Ta Keywords: Born rule, Density matrix, Fluctuation-dissipation relation, Quantum trajectory, Stochastic evo- lution, Transmon qubit. Motivation: The textbook formulation of quantum me- of freedom evolve. The result is still an ensemble descrip- chanics describes measurement according to the von Neu- tion, but it provides a quantitative understanding of how mann projection postulate, which states that one of the the off-diagonal elements of the reduced density matrix ρ eigenvalues of the measured observable is the measure- decay [3, 4]. Subsequently, solution of the “measurement ment outcome and the post-measurement state is the cor- problem” requires decomposing the ensemble into indi- responding eigenvector. With Pi denoting the projection vidual quantum contributions (i.e. which “i” will occur operator for the eigenstate i , in which experimental run), and that forces us to look | i beyond the closed unitary Schr¨odinger evolution. ψ P ψ / P ψ , (1) | i −→ i| i | i| i| Some of the attempted decompositions of the quantum † Pi = Pi , PiPj = Piδij , Pi = I. (2) ensemble are physical, e.g. introduction of hidden vari- Xi ables with novel dynamics or ignored interactions with known dynamics [5–8]. Some other attempts philosophi- This change is sudden, irreversible, consistent on repe- cally question what is real and what is observable, in prin- tition, and probabilistic in the choice of “i”. Which “i” ciple as well as by human beings with limited capacity would occur in a particular experimental run, is not spec- [9–11]. Although these attempts are not theoretically in- ified; only the probabilities of various outcomes are spec- consistent, none of them have been positively verified by ified, requiring an ensemble interpretation for the out- experiments—either they are untestable or only bounds comes. These probabilities follow the Born rule, arXiv:1804.03413v1 [quant-ph] 10 Apr 2018 exist on their parameters [8, 12]. We focus here on the extension of quantum mechanics prob(i)= ψ P ψ = T r(P ρ) , ρ P ρP , (3) h | i| i i −→ i i that describes the measurement as a continuous nonlinear Xi stochastic process. It is a particular case of the class of and the ensemble evolution takes initially pure states to stochastic collapse models that add a measurement driv- mixed states. ing term and a random noise term to the Schr¨odinger Over the years, many attempts have been made to un- evolution [8]. We look at these terms in an effective the- ravel the dynamics of this process [1, 2]. The framework ory approach, without assuming a specific collapse basis of environmental decoherence is an important step that (e.g. energy or position basis) or a specific collapse in- provides continuous interpolation of the sudden projec- teraction (e.g. gravity or some other universal interac- tion. In this framework, both the system and its environ- tion). In this expanded view, the collapse process can ment (which includes the measuring apparatus) follow be specific to each system-apparatus pair and need not the unitary Schr¨odinger evolution, and the unobserved be universal. Such a setting is necessary to understand degrees of freedom of the environment are “summed the quantum state evolution during continuous measure- over” to determine how the remaining observed degrees ments of superconducting transmon qubits [13], where 2 the collapse basis as well as the system-apparatus inter- Born rule 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 quantum dynamics [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 probability 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. Excited state relaxation− introduces± Theoretical Predictions: We consider evolution of a an additional drift in the evolution, which makes the part qubit undergoing binary weak measurement 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 transmons [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 density matrix 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′).
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