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

## Abstract

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.

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## References

1. 1

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

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).

3. 3

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

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).

5. 5

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

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).

7. 7

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

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).

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).

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).

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).

12. 12

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

13. 13

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

14. 14

Tasaki, H. Jarzynski relations for quantum systems and some applications. Preprint at http://arXiv.org/abs/cond-mat/0009244 (2000)

15. 15

Kurchan, J. A quantum fluctuation theorem. Preprint at http://arXiv.org/abs/cond-mat/0007360 (2000)

16. 16

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

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).

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).

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).

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).

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).

23. 23

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

24. 24

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

25. 25

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

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).

27. 27

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

28. 28

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

29. 29

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

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).

31. 31

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

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).

33. 33

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

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).

35. 35

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

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).

37. 37

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

38. 38

Mazonka, O. & Jarzynski, C. Exactly solvable model illustrating far-from-equilibrium predictions. Preprint at http://arXiv.org/abs/cond-mat/9912121 (1999)

39. 39

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

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).

41. 41

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

42. 42

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

43. 43

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

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).

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).

46. 46

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

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).

48. 48

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

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).

51. 51

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

## Acknowledgements

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

Authors

### Contributions

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)

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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). https://doi.org/10.1038/nphys3197

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