Noise spectroscopy through dynamical decoupling with a superconducting flux qubit

Journal name:
Nature Physics
Year published:
Published online


Quantum coherence in natural and artificial spin systems is fundamental to applications ranging from quantum information science to magnetic-resonance imaging and identification. Several multipulse control sequences targeting generalized noise models have been developed to extend coherence by dynamically decoupling a spin system from its noisy environment. In any particular implementation, however, the efficacy of these methods is sensitive to the specific frequency distribution of the noise, suggesting that these same pulse sequences could also be used to probe the noise spectrum directly. Here we demonstrate noise spectroscopy by means of dynamical decoupling using a superconducting qubit with energy-relaxation time T1=12μs. We first demonstrate that dynamical decoupling improves the coherence time T2 in this system up to the T2=2T1 limit (pure dephasing times exceeding 100μs), and then leverage its filtering properties to probe the environmental noise over a frequency (f) range 0.2–20MHz, observing a 1/fα distribution with α<1. The characterization of environmental noise has broad utility for spin-resonance applications, enabling the design of optimized coherent-control methods, promoting device and materials engineering, and generally improving coherence.

At a glance


  1. Qubit device and characterization.
    Figure 1: Qubit device and characterization.

    a, Device and biasing schematic representation: an aluminium superconducting loop interrupted by Josephson junctions (crosses) with a readout d.c. SQUID. b, Frequency spectroscopy of the qubit’s |0right fenceright arrow|1right fence transition. c, Spectroscopy at Φb=0 (arrow in b). d, Echo decay (blue triangles) and relaxation from the excited state (black circles) at Φb=0. In the insets, τ is a time delay and XΘ symbolizes a rotation of the Bloch vector by the angle Θ around the axis . The red squares indicate the readout. e, Free-induction decay (Ramsey fringe) at Φb=0. f, Rabi oscillations at Φb=0.

  2. Dynamical-decoupling pulse sequences.
    Figure 2: Dynamical-decoupling pulse sequences.

    a, Schematic representation of the CP(MG) sequence; the π -pulses are shifted 0° (Xπ) and 90° (Y π) from the π/2 -pulses (Xπ/2) for CP and CPMG, respectively. b, The Nth UDD pulse (Y π) has the relative position δj=sin2(πj/(2N+2)). c, CP(MG) (solid lines) and UDD (dashed) filter functions gN(ω,τ), equation (3), for a total pulse-sequence length τ=1μs (identical for N=0,1,2). d, Single-π -pulse (N=1) filter function for various total pulse lengths τ. e, An illustration using the filter function gN(ω,τ) to sample the noise PSD for a particular N and τ corresponding to an angular frequency ω′. S(ω) is assumed constant within the filter’s bandwidth B.

  3. Dephasing under the CPMG sequence.
    Figure 3: Dephasing under the CPMG sequence.

    a, Decay rates (inverse of 1/e times) versus flux detuning: free induction (Ramsey, N=0, green squares) and CPMG (coloured circles) with N=1, 2, 4, 6, 8, 10, 16, 20, 24, 30, 36, 42, 48 (colours correspond to those in Fig. 2c). Solid lines are calculations using equation (2) with parameters in Table 1. b, 1/e decay time under N-pulse CPMG, CP and UDD sequences at Φb=−0.4mΦ0 (ε/2π=430MHz). The simulation (red line) assumes ideal pulses and noise acting during the free evolution.

  4. Decoherence during driven dynamics.
    Figure 4: Decoherence during driven dynamics.

    a, Rabi oscillations with ΩR/2π=2MHz at ε/2π=225MHz. The red line envelope is a fitting using ζ(t) and ΓR. The black line shows the ΓR decay only, and the green line the ζ(t)-envelope contribution. b, Rabi-decay rate ΓR versus flux detuning at ΩR/2π=2MHz, with a parabolic fit to equation (4) used to obtain Sε(ΩR)=(1/π)ΓΩ(ε).

  5. Noise-power spectral density (PSD).
    Figure 5: Noise-power spectral density (PSD).

    Multicoloured dots, δε-noise PSD (0.2–20MHz) derived from CPMG data at Φb=−0.4mΦ0 (see the text). Colours correspond to the various N in Fig. 3a; grey dots for data up to N=250. Yellow squares, δε -noise PSD (2–20MHz) derived from Rabi spectroscopy (Fig. 4b and Supplementary Information J). Diagonal, dashed lines, Estimated flux (red) and δΔ (blue) 1/f noises, converted to frequency, inferred from the Ramsey and echo measurements (see AΦ,ic and ωhighΔ in Table 1). Solid, red line, Power-law dependence, Sε(2πf)=κε2AΦ/(2πf)α, extrapolated beyond the qubit’s frequency, Δ/2π. Fitting the low-frequency, linear portion of the measured PSD derived from CPMG data (before the slight upturn beyond 2MHz) yields the parameters AΦ=(0.8μΦ0)2 and α=0.9 (see Supplementary Information I.3). The shaded area covers α±0.05. Green dots, High-frequency transverse noise is purely δε -noise at Δ/2π=5.4GHz and becomes predominantly δΔ -noise at higher frequency. Purple line, Guide to indicate linearly increasing Nyquist (quantum) noise, including the eigenbasis rotation (see the inset); circles indicate transverse δε - (red) and δΔ- (blue) noises. Dash–dotted line, Expected thermal and quantum noise (equation (6)). Inset, Graphic representation of the quantization axis (grey arrows of fixed length) with the qubit’s (Z′) eigenstate tilted from the ‘laboratory’ frame (Z) by the angle θ. Fields ε(Φb) and Δ point in the X and Z directions, respectively. Red and blue double arrows indicate transverse noise.


  1. Hahn, E. L. Spin echoes. Phys. Rev. 80, 580594 (1950).
  2. Carr, H. Y. & Purcell, E. M. Effects of diffusion on free precession in nuclear magnetic resonance experiments. Phys. Rev. 94, 630638 (1954).
  3. Meiboom, S. & Gill, D. Modified spin-echo method for measuring nuclear relaxation times. Rev. Sci. Instrum. 29, 688691 (1958).
  4. Slichter, C. P. Principles of Nuclear Magnetic Resonance 3rd edn (Springer, 1990).
  5. Biercuk, M. J. et al. Optimized dynamical decoupling in a model quantum memory. Nature 458, 9961000 (2009).
  6. Biercuk, M. J. et al. Experimental Uhrig dynamical decoupling using trapped ions. Phys. Rev. A 79, 062324 (2009).
  7. Sagi, Y., Almog, I. & Davidson, N. Process tomography of dynamical decoupling in a dense cold atomic ensemble. Phys. Rev. Lett. 105, 053201 (2010).
  8. Szwer, D. J., Webster, S. C., Steane, A. M. & Lucas, D. M. Keeping a single qubit alive by experimental dynamic decoupling. J. Phys. B 44, 025501 (2011).
  9. Du, J. et al. Preserving electron spin coherence in solids by optimal dynamical decoupling. Nature 461, 12651268 (2009).
  10. Barthel, C., Medford, J., Marcus, C. M., Hanson, M. P. & Gossard, A. C. Interlaced dynamical decoupling and coherent operation of a singlet–triplet qubit. Phys. Rev. Lett. 105, 266808 (2010).
  11. Bluhm, H. et al. Dephasing time of GaAs electron-spin qubits coupled to a nuclear bath exceeding 200μs. Nature Phys. 7, 109113 (2011).
  12. de Lange, G., Wang, Z. H., Riste, D., Dobrovitski, V. V. & Hanson, R. Universal dynamical decoupling of a single solid-state spin from a spin bath. Science 330, 6063 (2010).
  13. Ryan, C. A., Hodges, J. S. & Cory, D. G. Robust decoupling techniques to extend quantum coherence in diamond. Phys. Rev. Lett. 105, 200402 (2010).
  14. Viola, L. & Lloyd, S. Dynamical suppression of decoherence in two-state quantum systems. Phys. Rev. A 58, 27332744 (1998).
  15. Faoro, L. & Viola, L. Dynamical suppression of 1/f noise processes in qubit systems. Phys. Rev. Lett. 92, 117905 (2004).
  16. Falci, G., D’Arrigo, A., Mastellone, A. & Paladino, E. Dynamical suppression of telegraph and 1/f noise due to quantum bistable fluctuators. Phys. Rev. A 70, 040101 (2004).
  17. Uhrig, G. S. Keeping a quantum bit alive by optimized π -pulse sequences. Phys. Rev. Lett. 98, 100504 (2007).
  18. Uhrig, G. S. Exact results on dynamical decoupling by π pulses in quantum information processes. New J. Phys. 10, 083024 (2008).
  19. Cywiński, L., Lutchyn, R. M., Nave, C. P. & Das Sarma, S. How to enhance dephasing time in superconducting qubits. Phys. Rev. B 77, 174509 (2008).
  20. Pasini, S. & Uhrig, G. S. Optimized dynamical decoupling for power-law noise spectra. Phys. Rev. A 81, 012309 (2010).
  21. Clarke, J. & Wilhelm, F. K. Superconducting quantum bits. Nature 453, 10311042 (2008).
  22. Lasic, S., Stepisnik, J. & Mohoric, A. Displacement power spectrum measurement by CPMG in constant gradient. J. Magn. Reson. 182, 208214 (2006).
  23. Jenista, E. R., Stokes, A. M., Branca, R. T. & Warren, W. S. Optimized, unequal pulse spacing in multiple echo sequences improves refocusing in magnetic resonance. J. Chem. Phys. 131, 204510 (2009).
  24. Orlando, T. et al. Superconducting persistent-current qubit. Phys. Rev. B 60, 1539815413 (1999).
  25. Mooij, J. E. et al. Josephson persistent-current qubit. Science 285, 10361039 (1999).
  26. Astafiev, O., Pashkin, Y. A., Nakamura, Y., Yamamoto, T. & Tsai, J. S. Quantum noise in the Josephson charge qubit. Phys. Rev. Lett. 93, 267007 (2004).
  27. Averin, D. V. Quantum computing and quantum measurement with mesoscopic Josephson junctions. Fortschr. Phys. 48, 10551074 (2000).
  28. Makhlin, Y., Schön, G. & Shnirman, A. Quantum-state engineering with Josephson-junction devices. Rev. Mod. Phys. 73, 357400 (2001).
  29. Ithier, G. et al. Decoherence in a superconducting quantum bit circuit. Phys. Rev. B 72, 134519 (2005).
  30. Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F. & Schoelkopf, R. J. Introduction to quantum noise, measurement, and amplification. Rev. Mod. Phys. 82, 11551208 (2010).
  31. Yoshihara, F., Harrabi, K., Niskanen, A. O., Nakamura, Y. & Tsai, J. S. Decoherence of flux qubits due to 1/f flux noise. Phys. Rev. Lett. 97, 167001 (2006).
  32. Martinis, J. M., Nam, S., Aumentado, J., Lang, K. M. & Urbina, C. Decoherence of a superconducting qubit due to bias noise. Phys. Rev. B 67, 094510 (2003).
  33. Borneman, T. W., Hurlimann, M. D. & Cory, D. G. Application of optimal control to CPMG refocusing pulse design. J. Magn. Reson. 207, 220233 (2010).
  34. Wellstood, F. C., Urbina, C. & Clarke, J. Low-frequency noise in dc superconducting quantum interference devices below 1K. Appl. Phys. Lett. 50, 772774 (1987).
  35. Geva, E., Kosloff, R. & Skinner, J. L. On the relaxation of a two-level system driven by a strong electromagnetic field. J. Chem. Phys. 102, 85418561 (1995).
  36. Van der Wal, C. H., Wilhelm, F. K., Harmans, C. J. P. M. & Mooij, J. E. Engineering decoherence in Josephson persistent-current qubits. Eur. Phys. J. B 31, 111123 (2003).
  37. Shnirman, A., Schön, G., Martin, I. & Makhlin, Y. Low- and high-frequency noise from coherent two-level systems. Phys. Rev. Lett. 94, 127002 (2005).
  38. Kerman, A. J. & Oliver, W. D. High-fidelity quantum operations on superconducting qubits in the presence of noise. Phys. Rev. Lett. 101, 070501 (2008).
  39. Van Harlingen, D. J. et al. Decoherence in Josephson-junction qubits due to critical-current fluctuations. Phys. Rev. B 70, 064517 (2004).

Download references

Author information


  1. Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • Jonas Bylander,
    • Simon Gustavsson &
    • William D. Oliver
  2. Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

    • Fei Yan &
    • David G. Cory
  3. The Institute of Physical and Chemical Research (RIKEN), Wako, Saitama 351-0198, Japan

    • Fumiki Yoshihara,
    • Khalil Harrabi,
    • Yasunobu Nakamura &
    • Jaw-Shen Tsai
  4. MIT Lincoln Laboratory, 244 Wood Street, Lexington, Massachusetts 02420, USA

    • George Fitch &
    • William D. Oliver
  5. Institute for Quantum Computing and Department of Chemistry, University of Waterloo, Ontario, N2L 3G1, Canada

    • David G. Cory
  6. Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2J 2W9, Canada

    • David G. Cory
  7. Green Innovation Research Laboratories, NEC Corporation, Tsukuba, Ibaraki 305-8501, Japan

    • Yasunobu Nakamura &
    • Jaw-Shen Tsai
  8. Present address: Physics Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia

    • Khalil Harrabi


F. Yoshihara, K.H., Y.N. and J-S.T. designed and fabricated the device. J.B. and S.G. carried out the experiments. G.F., S.G. and J.B. contributed to the software infrastructure. J.B., F. Yan, W.D.O. and Y.N. analysed the data and F. Yoshihara and D.G.C. provided feedback. J.B. and W.D.O. wrote the paper with feedback from all authors. W.D.O. supervised the project. All authors contributed to discussions during the conception, execution and interpretation of the experiments.

Competing financial interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to:

Author details

Supplementary information

PDF files

  1. Supplementary Information (950k)

    Supplementary Information

Additional data