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Mapping the optimal route between two quantum states


A central feature of quantum mechanics is that a measurement result is intrinsically probabilistic. Consequently, continuously monitoring a quantum system will randomly perturb its natural unitary evolution. The ability to control a quantum system in the presence of these fluctuations is of increasing importance in quantum information processing and finds application in fields ranging from nuclear magnetic resonance1 to chemical synthesis2. A detailed understanding of this stochastic evolution is essential for the development of optimized control methods. Here we reconstruct the individual quantum trajectories3,4,5 of a superconducting circuit that evolves under the competing influences of continuous weak measurement and Rabi drive. By tracking individual trajectories that evolve between any chosen initial and final states, we can deduce the most probable path through quantum state space. These pre- and post-selected quantum trajectories also reveal the optimal detector signal in the form of a smooth, time-continuous function that connects the desired boundary conditions. Our investigation reveals the rich interplay between measurement dynamics, typically associated with wavefunction collapse, and unitary evolution of the quantum state as described by the Schrödinger equation. These results and the underlying theory6, based on a principle of least action, reveal the optimal route from initial to final states, and may inform new quantum control methods for state steering and information processing.

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Figure 1: Set-up.
Figure 2: Quantum trajectories of the quantum state on the Bloch sphere are plotted against time.
Figure 3: Greyscale histograms of quantum trajectories in the undriven case.
Figure 4: Greyscale histograms of quantum trajectories in the driven case.


  1. Vandersypen, L. M. K. & Chuang, I. L. NMR techniques for quantum control and computation. Rev. Mod. Phys. 76, 1037–1069 (2005)

    ADS  Article  Google Scholar 

  2. Shapiro, M. & Brumer, P. Quantum Control of Molecular Processes (Wiley, 2012)

    MATH  Google Scholar 

  3. Carmichael, H. An Open Systems Approach to Quantum Optics (Springer, 1993)

    Book  Google Scholar 

  4. Carmichael, H. J., Kochan, P. & Tian, L. in Proc. Int. Symp. Coherent States: Past, Present, and Future (eds Feng, D. H, Strayer, M. R. & Klauder, J. R. ) 75–91 (World Scientific, 1994)

    Book  Google Scholar 

  5. Wiseman, H. & Milburn, G. Quantum Measurement and Control (Cambridge Univ. Press, 2010)

    MATH  Google Scholar 

  6. Chantasri, A., Dressel, J. & Jordan, A. N. Action principle for continuous quantum measurement. Phys. Rev. A 88, 042110 (2013)

    ADS  Article  Google Scholar 

  7. Guerlin, C. et al. Progressive field-state collapse and quantum non-demolition photon counting. Nature 448, 889–893 (2007)

    CAS  ADS  Article  Google Scholar 

  8. Sayrin, C. et al. Real-time quantum feedback prepares and stabilizes photon number states. Nature 477, 73–77 (2011)

    CAS  ADS  Article  Google Scholar 

  9. Hatridge, M. et al. Quantum back-action of an individual variable-strength measurement. Science 339, 178–181 (2013)

    CAS  ADS  MathSciNet  Article  Google Scholar 

  10. Vijay, R. et al. Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback. Nature 490, 77–80 (2012)

    CAS  ADS  Article  Google Scholar 

  11. Blok, M. et al. Manipulating a qubit through the backaction of sequential partial measurements and real-time feedback. Nature Phys. 10, 189–193 (2014)

    CAS  ADS  Article  Google Scholar 

  12. de Lange, G. et al. Reversing quantum trajectories with analog feedback. Phys. Rev. Lett. 112, 080501 (2014)

    ADS  Article  Google Scholar 

  13. Murch, K. W., Weber, S. J., Macklin, C. & Siddiqi, I. Observing single quantum trajectories of a superconducting qubit. Nature 502, 211–214 (2013)

    CAS  ADS  Article  Google Scholar 

  14. Jordan, A. N. Watching the wavefunction collapse. Nature 502, 177–178 (2103)

    Article  Google Scholar 

  15. Groen, J. P. et al. Partial-measurement backaction and nonclassical weak values in a superconducting circuit. Phys. Rev. Lett. 111, 090506 (2013)

    CAS  ADS  Article  Google Scholar 

  16. Campagne-Ibarcq, P. et al. Observing interferences between past and future quantum states in resonance fluorescence. Phys. Rev. Lett. 112, 180402 (2014)

    CAS  ADS  Article  Google Scholar 

  17. Ristè, D. et al. Deterministic entanglement of superconducting qubits by parity measurement and feedback. Nature 502, 350–354 (2013)

    ADS  Article  Google Scholar 

  18. Roch, N. et al. Observation of measurement-induced entanglement and quantum trajectories of remote superconducting qubits. Phys. Rev. Lett. 112, 170501 (2014)

    CAS  ADS  Article  Google Scholar 

  19. Koch, J. et al. Charge-insensitive qubit design derived from the Cooper pair box. Phys. Rev. A 76, 042319 (2007)

    ADS  Article  Google Scholar 

  20. Paik, H. et al. Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture. Phys. Rev. Lett. 107, 240501 (2011)

    ADS  Article  Google Scholar 

  21. Castellanos-Beltran, M. A., Irwin, K. D., Hilton, G. C., Vale, L. R. & Lehnert, K. W. Amplification and squeezing of quantum noise with a tunable Josephson metamaterial. Nature Phys. 4, 929–931 (2008)

    ADS  Article  Google Scholar 

  22. Hatridge, M., Vijay, R., Slichter, D. H., Clarke, J. & Siddiqi, I. Dispersive magnetometry with a quantum limited SQUID parametric amplifier. Phys. Rev. B 83, 134501 (2011)

    ADS  Article  Google Scholar 

  23. Korotkov, A. N. Quantum Bayesian approach to circuit QED measurement. Preprint at (2011)

  24. Korotkov, A. N. Continuous quantum measurement of a double dot. Phys. Rev. B 60, 5737–5742 (1999)

    CAS  ADS  Article  Google Scholar 

  25. Watanabe, S. Symmetry of physical laws. Part III. prediction and retrodiction. Rev. Mod. Phys. 27, 179–186 (1955)

    ADS  MathSciNet  Article  Google Scholar 

  26. Aharonov, Y., Bergmann, P. G. & Lebowitz, J. L. Time symmetry in the quantum process of measurement. Phys. Rev. 134, B1410–B1416 (1964)

    ADS  MathSciNet  Article  Google Scholar 

  27. Aharonov, Y., Albert, D. Z. & Vaidman, L. How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988)

    CAS  ADS  Article  Google Scholar 

  28. Williams, N. S. & Jordan, A. N. Weak values and the Leggett-Garg inequality in solid-state qubits. Phys. Rev. Lett. 100, 026804 (2008)

    ADS  Article  Google Scholar 

  29. Vijay, R., Slichter, D. H. & Siddiqi, I. Observation of quantum jumps in a superconducting artificial atom. Phys. Rev. Lett. 106, 110502 (2011)

    CAS  ADS  Article  Google Scholar 

  30. Gambetta, J. et al. Quantum trajectory approach to circuit QED: quantum jumps and the Zeno effect. Phys. Rev. A 77, 012112 (2008)

    ADS  Article  Google Scholar 

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We thank A. N. Korotkov, S. G. Rajeev, N. Roch and D. Toyli for discussions. This research was supported in part by the Army Research Office, Office of Naval Research and the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), through the Army Research Office. All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies of IARPA, the ODNI or the US government. A.N.J. acknowledges support from NSF grant no. DMR-0844899 (CAREER).

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Authors and Affiliations



S.J.W. and K.W.M. performed the experiment and analysed the experimental data. J.D. and A.C. wrote the trajectory simulation code. A.C., J.D. and A.N.J. contributed the theory. All work was carried out under the supervision of I.S. All authors contributed to writing the manuscript.

Corresponding author

Correspondence to K. W. Murch.

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The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Experimental schematic.

The weak measurement tone is always on. The projective readout tone is pulsed. The amplitude and phase of the signal displacement tone are adjusted to displace the measurement signals back to the origin of the X1X2 plane, which allows the LJPA to perform in the linear regime.

Extended Data Figure 2 Paths slightly varied from the optimal solution.

a, Overplotted x and z coordinates of 11 trajectories slightly varied from an optimized solution with boundary conditions (xIzI) = (0.88, 0), (xFzFTF) = (−0.683, −0.227, 0.464 µs) and the Rabi drive Ω/2π = 1.08 MHz. b, The corresponding conjugate variables px and pz. c, d, Plots of the unnormalized probability versus changes of the constant δ1 in the px differential equation (c) and the unnormalized probability versus changes of the constant δ2 in the pz differential equation (d). In this case, the optimized solution gives a maximum value of the path probability density.

Extended Data Figure 3 Optimal time between starting and destination states.

The probability density functions P(zF|zI = 0) plotted as functions of time T (solid curves) along with experimental data (dotted curves) with τ = 1.25 µs. The red, green and blue curves are the distribution functions P(zF = 0.2|zI = 0), P(zF = 0.4|zI = 0) and P(zF = 0.6|zI = 0), respectively. The optimized times Topt for the three cases are shown as the vertical black dashed lines with the labels T0.2, T0.4 and T0.6.

Extended Data Figure 4 Greyscale histograms of ensemble and post-selected trajectories for different Rabi frequencies and measurement strengths.

a, Ensemble and post-selected trajectories for Ω/2π = 1.08 MHz and τ = 1.25 µs. The post-selections for times {t1 = 464 ns, t2 = 944 ns, t3 = 1.424 µs} are (xFzF) = {(−0.78, −0.5), (0.7, −0.5), (−0.73, −0.5)} with a post selection window of ±0.08. b, Trajectories for Ω/2π = 1.08 MHz and τ = 315 ns with (xFzF) = {(−0.69, −0.5), (0.5, −0.5), (−0.73, −0.5)}. c, Trajectories for Ω/2π = 0.58 MHz and τ = 315 ns with (xFzF) = {(−0.35, −0.5), (−0.5, −0.5), (−0.56, −0.5)}. Note that all the trajectories use the same value of zF. The values of xF were chosen to give a large number of trajectories in the post-selected ensemble.

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Weber, S., Chantasri, A., Dressel, J. et al. Mapping the optimal route between two quantum states. Nature 511, 570–573 (2014).

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