# Electric field correlation measurements on the electromagnetic vacuum state

## Abstract

Quantum mechanics ascribes to the ground state of the electromagnetic radiation1 zero-point electric field fluctuations that permeate empty space at all frequencies. No energy can be extracted from the ground state of a system, and therefore these fluctuations cannot be measured directly with an intensity detector. The experimental proof of their existence therefore came from more indirect evidence, such as the Lamb shift2,3,4, the Casimir force between close conductors5,6,7 or spontaneous emission1,8. A direct method of determining the spectral characteristics of vacuum field fluctuations has so far been missing. Here we perform a direct measurement of the field correlation on these fluctuations in the terahertz frequency range by using electro-optic detection9 in a nonlinear crystal placed in a cryogenic environment. We investigate their temporal and spatial coherence, which, at zero time delay and spatial distance, has a peak value of 6.2 × 10−2 volts squared per square metre, corresponding to a fluctuating vacuum field10,11 of 0.25 volts per metre. With this measurement, we determine the spectral components of the ground state of electromagnetic radiation within the bandwidth of our electro-optic detection.

## Access options

from\$8.99

All prices are NET prices.

## Data availability

The raw data associated with Figs. 2b, c, 3a–e and 4a, b are provided with the manuscript. Other data that support the findings of this study are available from the corresponding author on reasonable request.

## References

1. 1.

Loudon, R. The Quantum Theory of Light (Oxford Univ. Press, Oxford, 2000).

2. 2.

Lamb, W. E. Jr. & Retherford, R. C. Fine structure of the hydrogen atom by a microwave method. Phys. Rev. 72, 241–243 (1947).

3. 3.

Bethe, H. A. The electromagnetic shift of energy levels. Phys. Rev. 72, 339–341 (1947).

4. 4.

Fragner, A. et al. Resolving vacuum fluctuations in an electrical circuit by measuring the Lamb shift. Science 322, 1357–1360 (2008).

5. 5.

Casimir, H. B. G. On the attraction between two perfectly conducting plates. Indag. Math. 10, 261–263 (1948).

6. 6.

Moore, G. T. Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity. J. Math. Phys. 11, 2679–2691 (1970).

7. 7.

Wilson, C. M. et al. Observation of the dynamical Casimir effect in a superconducting circuit. Nature 479, 376–379 (2011).

8. 8.

Walls, D. & Milburn, G. Quantum Optics ch. 10 (Springer, 1994).

9. 9.

Wu, Q. & Zhang, X. C. Free-space electro-optic sampling of terahertz beams. Appl. Phys. Lett. 67, 3523–3525 (1995).

10. 10.

Riek, C. et al. Direct sampling of electric-field vacuum fluctuations. Science 350, 420–423 (2015).

11. 11.

Riek, C. et al. Subcycle quantum electrodynamics. Nature 541, 376–379 (2017).

12. 12.

Purcell, E. M. Spontaneous emission probabilities at radio frequencies. Phys. Rev. 69, 681 (1946).

13. 13.

Scalari, G. et al. Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial. Science 335, 1323–1326 (2012).

14. 14.

Keller, J. et al. Few-electron ultrastrong light–matter coupling at 300 GHz with nanogap hybrid LC microcavities. Nano Lett. 17, 7410–7415 (2017).

15. 15.

Bayer, A. et al. Terahertz light–matter interaction beyond unity coupling strength. Nano Lett. 17, 6340–6344 (2017).

16. 16.

Ciuti, C., Bastard, G. & Carusotto, I. Quantum vacuum properties of the intersubband cavity polariton field. Phys. Rev. B 72, 115303 (2005).

17. 17.

Günter, G. et al. Sub-cycle switch-on of ultrastrong light–matter interaction. Nature 458, 178 –181 (2009).

18. 18.

Benea-Chelmus, I.-C. et al. Subcycle measurement of intensity correlations in the terahertz frequency range. Phys. Rev. A 93, 043812–043819 (2016).

19. 19.

Benea-Chelmus, I.-C., Rösch, M., Scalari, G., Beck, M. & Faist, J. Intensity autocorrelation measurements of frequency combs in the terahertz range. Phys. Rev. A 96, 033821–033828 (2017).

20. 20.

Moskalenko, A. S., Riek, C., Seletskiy, D. V., Burkard, G. & Leitenstorfer, A. Paraxial theory of direct electro-optic sampling of the quantum vacuum. Phys. Rev. Lett. 115, 263601–263605 (2015).

21. 21.

Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014).

22. 22.

Khalili, F. Y. et al. Quantum back-action in measurements of zero-point mechanical oscillations. Phys. Rev. A 86, 033840 (2012).

23. 23.

Kwiat, P. G. et al. New high-intensity source of polarization-entangled photon pairs. Phys. Rev. Lett. 75, 4337–4341 (1995).

24. 24.

da Silva, M. P., Bozyigit, D., Wallraff, A. & Blais, A. Schemes for the observation of photon correlation functions in circuit QED with linear detectors. Phys. Rev. A 82, 043804 (2010).

25. 25.

Bozyigit, D. et al. Antibunching of microwave-frequency photons observed in correlation measurements using linear detectors. Nat. Phys. 7, 154–158 (2011).

26. 26.

Lähteenmäki, P., Paraoanu, G. S., Hassel, J. & Hakonen, P. J. Coherence and multimode correlations from vacuum fluctuations in a microwave superconducting cavity. Nat. Commun. 7, 12548 (2016).

27. 27.

Kizmann, M. et al. Subcycle squeezing of light from a time flow perspective. Preprint at https://arxiv.org/abs/1807.10519 (2018).

28. 28.

Benea-Chelmus, I.-C. et al. Three-dimensional phase modulator at telecom wavelength acting as a terahertz detector with an electro-optic bandwidth of 1.25 terahertz. ACS Photonics 5, 1398–1403 (2018).

29. 29.

Cong, K. et al. Dicke superradiance in solids. J. Opt. Soc. Am. B 33, C80–C101 (2016).

30. 30.

Hagenmüller, D., Schachenmayer, J., Schütz, S., Genes, C. & Pupillo, G. Cavity-enhanced transport of charge. Phys. Rev. Lett. 119, 223601 (2017).

31. 31.

Orgiu, E. et al. Conductivity in organic semiconductors hybridized with the vacuum field. Nat. Mater. 14, 1123–1129 (2015).

32. 32.

Rungsawang, R. et al. Intensity detection of terahertz quantum cascade laser radiation using electro-optic sampling. Appl. Phys. Lett. 93, 191111 (2008).

33. 33.

van Kolck, A. et al. Thermo-optic detection of terahertz radiation from a quantum cascade laser. Appl. Phys. Lett. 97, 251103 (2010).

## Acknowledgements

This work was funded by the European Research Council (Advanced Grant, Quantum Metamaterials in the Ultra Strong Coupling Regime) and the Swiss National Science Foundation (grant 165639). We acknowledge the mechanical workshop at ETHZ. We acknowledge the contribution of M. Ernzer to the noise analysis tools, E. Mavrona to the design of opto-mechanical components and the extraction of the refractive index of ZnTe, and A. Imamoglu for discussions.

## Author information

Authors

### Contributions

I.-C.B.-C. and J.F. conceived and designed the experiments. I.-C.B.-C., F.F.S and G.S. built the experimental setup. I.-C.B.-C. developed the data acquisition system and noise suppression protocols. I.-C.B.-C. and F.F.S. performed the measurements. I.-C.B.-C., F.F.S. and J.F. analysed and interpreted the data. I.-C.B.-C., F.F.S. and J.F. derived the theory. J.F. was the scientific supervisor of this work. All authors discussed the results and contributed to the writing of the manuscript.

### Corresponding authors

Correspondence to Ileana-Cristina Benea-Chelmus or Jérôme Faist.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Benea-Chelmus, I., Settembrini, F.F., Scalari, G. et al. Electric field correlation measurements on the electromagnetic vacuum state. Nature 568, 202–206 (2019). https://doi.org/10.1038/s41586-019-1083-9

• Accepted:

• Published:

• Issue Date:

• ### An integrated optical modulator operating at cryogenic temperatures

• Felix Eltes
• , Gerardo E. Villarreal-Garcia
• , Daniele Caimi
• , Heinz Siegwart
• , Antonio A. Gentile
• , Andy Hart
• , Pascal Stark
• , Graham D. Marshall
• , Mark G. Thompson
• , Jorge Barreto
• , Jean Fompeyrine
•  & Stefan Abel

Nature Materials (2020)

• ### Determination of the electric field and its Hilbert transform in femtosecond electro-optic sampling

• P. Sulzer
• , K. Oguchi
• , J. Huster
• , M. Kizmann
• , T. L. M. Guedes
• , A. Liehl
• , C. Beckh
• , A. S. Moskalenko
• , G. Burkard
• , D. V. Seletskiy
•  & A. Leitenstorfer

Physical Review A (2020)

• ### Vacuum-induced surface-acoustic-wave phonon blockade

• Jiangshan Tang
• , Yang Wu
• , Zhenkai Wang
• , Hui Sun
• , Lei Tang
• , Han Zhang
• , Tao Li
• , Yanqin Lu
• , Min Xiao
•  & Keyu Xia

Physical Review A (2020)

• ### A powder method for the high-efficacy evaluation of electro-optic crystals

• Feng Xu
• , Ge Zhang
• , Min Luo
• , Guang Peng
• , Yu Chen
• , Tao Yan
•  & Ning Ye

National Science Review (2020)

• ### Electro-optic interface for ultrasensitive intracavity electric field measurements at microwave and terahertz frequencies

• Ileana-Cristina Benea-Chelmus
• , Yannick Salamin
• , Francesca Fabiana Settembrini
• , Yuriy Fedoryshyn
• , Wolfgang Heni
• , Delwin L. Elder
• , Larry R. Dalton
• , Juerg Leuthold
•  & Jérôme Faist

Optica (2020)