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.

You are viewing this page in draft mode.

Experimental test of the quantum Jarzynski equality with a trapped-ion system


The Jarzynski equality relates the free-energy difference between two equilibrium states to the work done on a system through far-from-equilibrium processes—a milestone that builds on the pioneering work of Clausius and Kelvin. Although experimental tests of the equality have been performed in the classical regime, the quantum Jarzynski equality has not yet been fully verified owing to experimental challenges in measuring work and work distributions in a quantum system. Here, we report an experimental test of the quantum Jarzynski equality with a single 171Yb+ ion trapped in a harmonic potential. We perform projective measurements to obtain phonon distributions of the initial thermal state. We then apply a laser-induced force to the projected energy eigenstate and find transition probabilities to final energy eigenstates after the work is done. By varying the speed with which we apply the force from the equilibrium to the far-from-equilibrium regime, we verify the quantum Jarzynski equality in an isolated system.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.


All prices are NET prices.

Figure 1: Experimental set-up for testing the Jarzynski equality and equilibrium and non-equilibrium work processes.
Figure 2: Thermal state preparation of the phonon and measurement results.
Figure 3: Experimental scheme for the projective measurement of the phonon state.
Figure 4: Dissipated work Wdiss = WΔ F = νΔ n and probabilities for three different ramping speeds of the force.


  1. 1

    Jarzynski, C. Nonequilibrium equality for free energy differences. Phys. Rev. Lett. 78, 2690–2693 (1997).

    ADS  Article  Google Scholar 

  2. 2

    Crooks, G. E. Entropy production fluctuation theorem and the nonequilibrium work relation for free-energy differences. Phys. Rev. E 60, 2721–2726 (1999).

    ADS  Article  Google Scholar 

  3. 3

    Hummer, G. & Szabo, A. Free energy reconstruction from nonequilibrium single-molecule pulling experiments. Proc. Natl Acad. Sci. USA 98, 3658–3661 (2001).

    ADS  Article  Google Scholar 

  4. 4

    Liphardt, J., Dumont, S., Smith, S. B., Tinoco, I. J. & Bustamante, C. Equilibrium information from nonequilibrium measurements in an experimental test of the Jarzynski equality. Science 296, 1832–1835 (2002).

    ADS  Article  Google Scholar 

  5. 5

    Collin, D. et al. Verification of the Crooks fluctuation theorem and recovery of RNA folding free energies. Nature 437, 231–234 (2005).

    ADS  Article  Google Scholar 

  6. 6

    Douarche, F., Ciliberto, S., Petrosyan, A. & Rabbiosi, I. An experimental test of the Jarzynski equality in a mechanical experiment. Europhys. Lett. 70, 593–599 (2005).

    ADS  Article  Google Scholar 

  7. 7

    Bustamante, C., Liphardt, J. & Ritort, F. The nonequilibrium thermodynamics of small systems. Phys. Today 58, 43–48 (July, 2005).

    Article  Google Scholar 

  8. 8

    Blickle, V., Speck, T., Helden, L., Seifert, U. & Bechinger, C. Thermodynamics of a colloidal particle in a time-dependent nonharmonic potential. Phys. Rev. Lett. 96, 070603 (2006).

    ADS  Article  Google Scholar 

  9. 9

    Harris, N. C., Song, Y. & Kiang, C-H. Experimental free energy surface reconstruction from single-molecule force spectroscopy using Jarzynski’s equality. Phys. Rev. Lett. 99, 068101 (2007).

    ADS  Article  Google Scholar 

  10. 10

    Saira, O-P. et al. Test of the Jarzynski and Crooks fluctuation relations in an electronic system. Phys. Rev. Lett. 109, 180601 (2012).

    ADS  Article  Google Scholar 

  11. 11

    Jarzynski, C. Equalities and inequalities: Irreversibility and the second law of thermodynamics at the nanoscale. Annu. Rev. Condens. Matter Phys. 2, 329–351 (2011).

    ADS  Article  Google Scholar 

  12. 12

    Seifert, U. Stochastic thermodynamics, fluctuation theorems and molecular machines. Rep. Prog. Phys. 75, 126001 (2012).

    ADS  Article  Google Scholar 

  13. 13

    Talkner, P., Lutz, E. & Hänggi, P. Fluctuation theorems: Work is not an observable. Phys. Rev. E 75, 050102(R) (2007).

    ADS  Article  Google Scholar 

  14. 14

    Tasaki, H. Jarzynski relations for quantum systems and some applications. Preprint at (2000)

  15. 15

    Kurchan, J. A quantum fluctuation theorem. Preprint at (2000)

  16. 16

    Mukamel, S. Quantum extension of the Jarzynski relation: Analogy with stochastic dephasing. Phys. Rev. Lett. 90, 170604 (2003).

    ADS  Article  Google Scholar 

  17. 17

    Esposito, M., Harbola, U. & Mukamel, S. Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Rev. Mod. Phys. 81, 1665–1702 (2009).

    ADS  MathSciNet  Article  Google Scholar 

  18. 18

    Huber, G., Schmidt-Kaler, F., Deffner, S. & Lutz, E. Employing trapped cold ions to verify the quantum Jarzynzki equality. Phys. Rev. Lett. 101, 070403 (2008).

    ADS  Article  Google Scholar 

  19. 19

    Huber, G. T. Quantum Thermodynamics with Trapped Ions PhD thesis, Univ. Ulm (2010)

  20. 20

    Campisi, M., Hänggi, P. & Talkner, P. Colloquium: Quantum fluctuation relations: Foundations and applications. Rev. Mod. Phys. 83, 771–791 (2011).

    ADS  Article  Google Scholar 

  21. 21

    Batalhao, T. et al. Experimental reconstruction of work distribution and verification of fluctuation relations at the full quantum level. Phys. Rev. Lett. 113, 140601 (2014).

    ADS  Article  Google Scholar 

  22. 22

    Heyl, M. & Kehrein, S. Crooks relation in optical spectra: Universality in work distributions for weak local quenches. Phys. Rev. Lett. 108, 190601 (2012).

    ADS  Article  Google Scholar 

  23. 23

    Dorner, R. et al. Extracting quantum work statistics and fluctuation theorems by single qubit interferometry. Phys. Rev. Lett. 110, 230601 (2013).

    ADS  Article  Google Scholar 

  24. 24

    Mazzola, L., Chiara, G. D. & Paternostro, M. Measuring the characteristic function of the work distribution. Phys. Rev. Lett. 110, 230602 (2013).

    ADS  Article  Google Scholar 

  25. 25

    Shen, C., Zhang, Z. & Duan, L-M. Scalable implementation of boson sampling with trapped ions. Phys. Rev. Lett. 112, 050504 (2014).

    ADS  Article  Google Scholar 

  26. 26

    Meekhof, D., Monroe, C., King, B., Itano, W. & Wineland, D. Generation of nonclassical motional states of a trapped atom. Phys. Rev. Lett. 76, 1796–1799 (1996).

    ADS  Article  Google Scholar 

  27. 27

    Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. Quantum dynamics of single trapped ions. Rev. Mod. Phys. 75, 281–324 (2003).

    ADS  Article  Google Scholar 

  28. 28

    Walther, A. et al. Controlling fast transport of cold trapped ions. Phys. Rev. Lett. 109, 080501 (2012).

    ADS  Article  Google Scholar 

  29. 29

    Bowler, R. et al. Coherent diabatic ion transport and separation in a multizone trap array. Phys. Rev. Lett. 109, 080502 (2012).

    ADS  Article  Google Scholar 

  30. 30

    Haljan, P. C., Brickman, K-A., Deslauriers, L., Lee, P. J. & Monroe, C. Spin-dependent forces on trapped ions for phase-stable quantum gates and entangled states of spin and motion. Phys. Rev. Lett. 94, 153602 (2005).

    ADS  Article  Google Scholar 

  31. 31

    Lee, P. J. et al. Phase control of trapped ion quantum gates. J. Opt. B 7, S371–S383 (2005).

    Article  Google Scholar 

  32. 32

    Monroe, C. et al. Resolved-sideband Raman cooling of a bound atom to the 3d zero-point energy. Phys. Rev. Lett. 75, 4011–4014 (1995).

    ADS  MathSciNet  Article  Google Scholar 

  33. 33

    Turchette, Q. A. et al. Heating of trapped ions from the quantum ground state. Phys. Rev. A 61, 063418 (2000).

    ADS  Article  Google Scholar 

  34. 34

    Turchette, Q. et al. Decoherence and decay of motional quantum states of a trapped atom coupled to engineered reservoirs. Phys. Rev. A 62, 053807 (2000).

    ADS  Article  Google Scholar 

  35. 35

    Myatt, C. J. et al. Decoherence of quantum superpositions through coupling to engineered reservoirs. Nature 403, 269–273 (2000).

    ADS  Article  Google Scholar 

  36. 36

    Intravaia, F., Maniscalcoa, S., Piilob, J. & Messina, A. Quantum theory of heating of a single trapped ion. Phys. Lett. A 308, 6–10 (2003).

    ADS  Article  Google Scholar 

  37. 37

    Zhang, J., Zhang, J., Zhang, X. & Kim, K. Realization of geometric Landau–Zener–Stückelberg interferometry. Phys. Rev. A 89, 013608 (2014).

    ADS  Article  Google Scholar 

  38. 38

    Mazonka, O. & Jarzynski, C. Exactly solvable model illustrating far-from-equilibrium predictions. Preprint at (1999)

  39. 39

    Speck, T. & Seifert, U. Distribution of work in isothermal nonequilibrium processes. Phys. Rev. E 70, 066112 (2004).

    ADS  Article  Google Scholar 

  40. 40

    Imparato, A., Peliti, L., Pesce, G., Rusciano, G. & Sasso, A. Work and heat probability distribution of an optically driven Brownian particle: Theory and experiments. Phys. Rev. E 76, 050101(R) (2007).

    ADS  Article  Google Scholar 

  41. 41

    Talkner, P., Burada, P. S. & Hänggi, P. Statistics of work performed on a forced quantum oscillator. Phys. Rev. E 78, 011115 (2008).

    ADS  Article  Google Scholar 

  42. 42

    Hendrix, D. A. & Jarzynski, C. A “fast growth” method of computing free energy differences. J. Chem. Phys. 114, 5974–5981 (2001).

    ADS  Article  Google Scholar 

  43. 43

    Wineland, D. Nobel Lecture: Superposition, entanglement, and raising Schrodinger’s cat. Rev. Mod. Phys. 85, 1103–1114 (2013).

    ADS  Article  Google Scholar 

  44. 44

    Haroche, S. Nobel Lecture: Controlling photons in a box and exploring the quantum to classical boundary. Rev. Mod. Phys. 85, 1083–1102 (2013).

    ADS  Article  Google Scholar 

  45. 45

    Quan, H. T., Liu, Y. X., Sun, C. P. & Nori, F. Quantum thermodynamic cycles and quantum heat engines. Phys. Rev. E 76, 031105 (2007).

    ADS  MathSciNet  Article  Google Scholar 

  46. 46

    Abah, O. et al. Single-ion heat engine at maximum power. Phys. Rev. Lett. 109, 203006 (2012).

    ADS  Article  Google Scholar 

  47. 47

    Aaronson, S. & Arkhipov, A. in Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (eds Fortnow, L. & Vadhan, S.) 333–342 (ACM, 2011).

    MATH  Google Scholar 

  48. 48

    Shen, C. & Duan, L. M. Correcting detection errors in quantum state engineering through data processing. New J. Phys. 14, 053053 (2012).

    ADS  Article  Google Scholar 

  49. 49

    Poschinger, U. Quantum Optics Experiments in a Microstructured Ion Trap PhD thesis, Univ. Ulm (2011)

  50. 50

    Olmschenk, S. et al. Manipulation and detection of a trapped Yb+ hyperfine qubit. Phys. Rev. A 76, 052314 (2007).

    ADS  Article  Google Scholar 

  51. 51

    Zhang, X. et al. State-independent experimental tests of quantum contextuality in a three dimensional system. Phys. Rev. Lett. 110, 070401 (2013).

    ADS  Article  Google Scholar 

Download references


We thank C. Jarzynski for careful reading of the manuscript and helpful comments. This work was supported in part by the National Basic Research Program of China Grants 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grants 61073174, 61033001, 61061130540, 11374178, 11375012 and 11105136. K.K. and H.T.Q. acknowledge the recruitment program of Global Youth Experts of China.

Author information




S.A., M.U., D.L. and Y.L. developed the experimental system and performed the experiments. Y.L. and S.A. analysed the results. J.Z. developed the adiabatic protocol for the projective measurement. J-N.Z. provided the concrete model for the experiment. J-N.Z., Z-Q.Y. and H.T.Q. provided theoretical support to the project. K.K. supervised the project. All authors contributed to writing the manuscript.

Corresponding authors

Correspondence to H. T. Quan or Kihwan Kim.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Information (PDF 898 kb)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

An, S., Zhang, JN., Um, M. et al. Experimental test of the quantum Jarzynski equality with a trapped-ion system. Nature Phys 11, 193–199 (2015).

Download citation

Further reading


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