Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Synthesizing arbitrary quantum states in a superconducting resonator


The superposition principle is a fundamental tenet of quantum mechanics. It allows a quantum system to be ‘in two places at the same time’, because the quantum state of a physical system can simultaneously include measurably different physical states. The preparation and use of such superposed states forms the basis of quantum computation and simulation1. The creation of complex superpositions in harmonic systems (such as the motional state of trapped ions2, microwave resonators3,4,5 or optical cavities6) has presented a significant challenge because it cannot be achieved with classical control signals. Here we demonstrate the preparation and measurement of arbitrary quantum states in an electromagnetic resonator, superposing states with different numbers of photons in a completely controlled and deterministic manner. We synthesize the states using a superconducting phase qubit to phase-coherently pump photons into the resonator, making use of an algorithm7 that generalizes a previously demonstrated method of generating photon number (Fock) states in a resonator8. We completely characterize the resonator quantum state using Wigner tomography, which is equivalent to measuring the resonator’s full density matrix.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Circuit diagram and one photon Rabi-swap oscillations between qubit and resonator.
Figure 2: Sequence to synthesize an arbitrary resonator state.
Figure 3: Wigner tomography of superpositions of resonator Fock states |0〉 + | N 〉.
Figure 4: Wigner tomography of the states |0〉 + e ik π/8 |3〉 + |6〉 for five values of phase k = 0 to 4.


  1. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000)

    MATH  Google Scholar 

  2. Ben-Kish, A. et al. Experimental demonstration of a technique to generate arbitrary quantum superposition states of a harmonically bound spin-1/2 particle. Phys. Rev. Lett. 90, 037902 (2003)

    Article  ADS  CAS  Google Scholar 

  3. Deléglise, S. et al. Reconstruction of non-classical cavity field states with snapshots of their decoherence. Nature 455, 510–514 (2008)

    Article  ADS  Google Scholar 

  4. Houck, A. A. et al. Generating single microwave photons in a circuit. Nature 449, 328–331 (2007)

    Article  ADS  CAS  Google Scholar 

  5. Sillanpää, M. A., Park, J. I. & Simmonds, R. W. Coherent quantum state storage and transfer between two phase qubits via a resonant cavity. Nature 449, 438–442 (2007)

    Article  ADS  Google Scholar 

  6. Boozer, A. D., Boca, A., Miller, R., Northup, T. E. & Kimble, H. J. Reversible state transfer between light and a single trapped atom. Phys. Rev. Lett. 98, 193601 (2007)

    Article  ADS  CAS  Google Scholar 

  7. Law, C. K. & Eberly, J. H. Arbitrary control of a quantum electromagnetic field. Phys. Rev. Lett. 76, 1055–1058 (1996)

    Article  ADS  CAS  Google Scholar 

  8. Hofheinz, M. et al. Generation of Fock states in a superconducting quantum circuit. Nature 454, 310–314 (2008)

    Article  ADS  CAS  Google Scholar 

  9. Martinis, J. M., Devoret, M. H. & Clarke, J. Energy-level quantization in the zero-voltage state of a current-biased Josephson junction. Phys. Rev. Lett. 55, 1543–1546 (1985)

    Article  ADS  CAS  Google Scholar 

  10. Clarke, J. & Wilhelm, F. K. Superconducting quantum bits. Nature 453, 1031–1042 (2008)

    Article  ADS  CAS  Google Scholar 

  11. Vion, D. et al. Manipulating the quantum state of an electrical circuit. Science 296, 886–889 (2002)

    Article  ADS  CAS  Google Scholar 

  12. Niskanen, A. O. et al. Quantum coherent tunable coupling of superconducting qubits. Science 316, 723–726 (2007)

    Article  ADS  CAS  Google Scholar 

  13. Plantenberg, J. H., de Groot, P. C., Harmans, C. J. P. M. & Mooij, J. E. Demonstration of controlled-NOT quantum gates on a pair of superconducting quantum bits. Nature 447, 836–839 (2007)

    Article  ADS  CAS  Google Scholar 

  14. Fink, J. M. et al. Climbing the Jaynes-Cummings ladder and observing its nonlinearity in a cavity QED system. Nature 454, 315–318 (2008)

    Article  ADS  CAS  Google Scholar 

  15. Steffen, M. et al. State tomography of capacitively shunted phase qubits with high fidelity. Phys. Rev. Lett. 97, 050502 (2006)

    Article  ADS  Google Scholar 

  16. Liu, Y. X. et al. Generation of non-classical photon states using a superconducting qubit in a quantum electrodynamic microcavity. Europhys. Lett. 67, 941–947 (2004)

    Article  ADS  CAS  Google Scholar 

  17. Vogel, K., Akulin, V. M. & Schleich, W. P. Quantum state engineering of the radiation field. Phys. Rev. Lett. 71, 1816–1819 (1993)

    Article  ADS  CAS  Google Scholar 

  18. Smithey, D. T., Beck, M., Raymer, M. G. & Faridani, A. Measurement of the Wigner distribution and the density matrix of a light mode using optical homodyne tomography: Application to squeezed states and the vacuum. Phys. Rev. Lett. 70, 1244–1247 (1993)

    Article  ADS  CAS  Google Scholar 

  19. Banaszek, K. & Wódkiewicz, K. Direct probing of quantum phase space by photon counting. Phys. Rev. Lett. 76, 4344–4347 (1996)

    Article  ADS  CAS  Google Scholar 

  20. Lutterbach, L. G. & Davidovich, L. Method for direct measurement of the Wigner function in cavity QED and ion traps. Phys. Rev. Lett. 78, 2547–2550 (1997)

    Article  ADS  CAS  Google Scholar 

  21. Banaszek, K., Radzewicz, C., Wódkiewicz, K. & Krasiński, J. S. Direct measurement of the Wigner function by photon counting. Phys. Rev. A 60, 674–677 (1999)

    Article  ADS  CAS  Google Scholar 

  22. Bertet, P. et al. Direct measurement of the Wigner function of a one-photon Fock state in a cavity. Phys. Rev. Lett. 89, 200402 (2002)

    Article  ADS  CAS  Google Scholar 

  23. Wigner, E. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 40, 749–759 (1932)

    Article  ADS  CAS  Google Scholar 

  24. Haroche, S. & Raimond, J.-M. Exploring the Quantum — Atoms, Cavities and Photons (Oxford Univ. Press, 2006)

    Book  Google Scholar 

  25. Neeley, M. et al. Transformed dissipation in superconducting quantum circuits. Phys. Rev. B 77, 180508 (2008)

    Article  ADS  Google Scholar 

  26. Breitenbach, G., Schiller, S. & Mlynek, J. Measurement of the quantum states of squeezed light. Nature 387, 471–475 (1997)

    Article  ADS  CAS  Google Scholar 

  27. Lvovsky, A. I. & Babichev, S. A. Synthesis and tomographic characterization of the displaced Fock state of light. Phys. Rev. A 66, 011801 (2002)

    Article  ADS  Google Scholar 

  28. Leibfried, D. et al. Experimental determination of the motional quantum state of a trapped atom. Phys. Rev. Lett. 77, 4281–4285 (1996)

    Article  ADS  CAS  Google Scholar 

  29. Wang, H. et al. Measurement of the decay of Fock states in a superconducting quantum circuit. Phys. Rev. Lett. 101, 240401 (2008)

    Article  ADS  CAS  Google Scholar 

Download references


Devices were made at the UCSB Nanofabrication Facility, a part of the NSF-funded National Nanotechnology Infrastructure Network. We thank M. Geller for discussions. This work was supported by IARPA (grant W911NF-04-1-0204) and by the NSF (grant CCF-0507227).

Author Contributions M.H. performed the experiments and analysed the data. H.W. improved the resonator design and fabricated the sample. J.M.M. and E.L. designed the custom electronics and M.H. developed the calibrations for it. M.A. and M.N. provided software infrastructure. All authors contributed to the fabrication process, qubit design or experimental set-up. M.H., J.M.M. and A.N.C. conceived the experiment and co-wrote the paper.

Author information

Authors and Affiliations


Corresponding author

Correspondence to A. N. Cleland.

Supplementary information

Supplementary Information

This file contains Supplementary Methods and Data, Supplementary Figures S1-S3 with Legends and Supplementary References. (PDF 693 kb)

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hofheinz, M., Wang, H., Ansmann, M. et al. Synthesizing arbitrary quantum states in a superconducting resonator. Nature 459, 546–549 (2009).

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing