Skip to main content

Thank you for visiting nature.com. 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.

Vacuum Bloch–Siegert shift in Landau polaritons with ultra-high cooperativity

Abstract

A two-level system resonantly interacting with an a.c. magnetic or electric field constitutes the physical basis of diverse phenomena and technologies. However, Schrödinger’s equation for this seemingly simple system can be solved exactly only under the rotating-wave approximation, which neglects the counter-rotating field component. When the a.c. field is sufficiently strong, this approximation fails, leading to a resonance-frequency shift known as the Bloch–Siegert shift. Here, we report the vacuum Bloch–Siegert shift, which is induced by the ultra-strong coupling of matter with the counter-rotating component of the vacuum fluctuation field in a cavity. Specifically, an ultra-high-mobility two-dimensional electron gas inside a high-Q terahertz cavity in a quantizing magnetic field revealed ultra-narrow Landau polaritons, which exhibited a vacuum Bloch–Siegert shift up to 40 GHz. This shift, clearly distinguishable from the photon-field self-interaction effect, represents a unique manifestation of a strong-field phenomenon without a strong field.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Fig. 1: CR cavity QED setup.
Fig. 2: Landau polaritons in the USC regime with ultra-high cooperativity.
Fig. 3: Landau polariton dispersions as a function of B.
Fig. 4: Distinction between the VBS shift due to the CRTs and the shift due to the A2 terms in the USC regime.

References

  1. 1.

    Allen, L. & Eberly, J. H. Optical Resonance and Two-Level Atoms (John Wiley & Sons, New York, 1975).

  2. 2.

    Feynman, R. P., Vernon, F. L. & Hellwarth, R. W. Geometrical representation of the Schrödinger equation for solving maser problems. J. Appl. Phys. 28, 49–52 (1957).

    ADS  Article  Google Scholar 

  3. 3.

    Bloch, F. & Siegert, A. Magnetic resonance for nonrotating fields. Phys. Rev. 57, 522–527 (1940).

    ADS  Article  Google Scholar 

  4. 4.

    Shirley, J. H. Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev. 138, B979–B987 (1965).

    ADS  Article  Google Scholar 

  5. 5.

    Cohen-Tannoudji, C., Dupont-Roc, J. & Fabre, C. A quantum calculation of the higher order terms in the Bloch-Siegert shift. J. Phys. B 6, L214–L217 (1973).

    ADS  Article  Google Scholar 

  6. 6.

    AbragamA. The Principles of Nuclear Magnetism (Oxford Univ. Press, Oxford, 1961).

    Google Scholar 

  7. 7.

    Tuorila, J. et al. Stark effect and generalized Bloch-Siegert shift in a strongly driven two-level system. Phys. Rev. Lett. 105, 257003 (2010).

    ADS  Article  Google Scholar 

  8. 8.

    Pietikäinen, I. et al. Observation of the Bloch-Siegert shift in a driven quantum-to-classical transition. Phys. Rev. B 96, 020501 (2017).

    ADS  Article  Google Scholar 

  9. 9.

    Sie, E. J. et al. Large, valley-exclusive Bloch-Siegert shift in monolayer WS2. Science 355, 1066–1069 (2017).

    ADS  Article  Google Scholar 

  10. 10.

    De Zela, F., Solano, E. & Gago, A. Micromaser without the rotating-wave approximation: the Bloch-Siegert shift and related effects. Opt. Commun. 142, 106–118 (1997).

    ADS  Article  Google Scholar 

  11. 11.

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

    ADS  Article  Google Scholar 

  12. 12.

    Anappara, A. A. et al. Signatures of the ultrastrong light-matter coupling regime. Phys. Rev. B 79(20), 201303 (R) (2009).

    ADS  Article  Google Scholar 

  13. 13.

    Muravev, V. M., Andreev, I. V., Kukushkin, I. V., Schmult, S. & Dietsche, W. Observation of hybrid plasmon-photon modes in microwave transmission of coplanar microresonators. Phys. Rev. B 83, 075309 (2011).

    ADS  Article  Google Scholar 

  14. 14.

    Zhang, Q. et al. Collective non-perturbative coupling of 2D electrons with high-quality-factor terahertz cavity photons. Nat. Phys. 12, 1005–1011 (2016).

    Article  Google Scholar 

  15. 15.

    Forn-Daz, P. et al. Observation of the Bloch-Siegert shift in a qubit-oscillator system in the ultrastrong coupling regime. Phys. Rev. Lett. 105, 237001 (2010).

    ADS  Article  Google Scholar 

  16. 16.

    Niemczyk, T. et al. Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat. Phys. 6, 772–776 (2010).

    Article  Google Scholar 

  17. 17.

    Forn-Daz, P., Romero, G., Harmans, C. J. P. M., Solano, E. & Mooij, J. E. Broken selection rule in the quantum Rabi model. Sci. Rep. 6, 26720 (2016).

    ADS  Article  Google Scholar 

  18. 18.

    Yoshihara, F. et al. Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime. Nat. Phys. 13, 44–47 (2017).

    Article  Google Scholar 

  19. 19.

    Forn-Diaz, P. et al. Ultrastrong coupling of a single artificial atom to an electromagnetic continuum in the nonperturbative regime. Nat. Phys. 13, 39–43 (2017).

    Article  Google Scholar 

  20. 20.

    Hagenmüller, D., De Liberato, S. & Ciuti, C. Ultrastrong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor. Phys. Rev. B 81, 235303 (2010).

    ADS  Article  Google Scholar 

  21. 21.

    Zhang, Q. et al. Superradiant decay of cyclotron resonance of two-dimensional electron gases. Phys. Rev. Lett. 113, 047601 (2014).

    ADS  Article  Google Scholar 

  22. 22.

    Maissen, C. et al. Ultrastrong coupling in the near field of complementary split-ring resonators. Phys. Rev. B 90, 205309 (2014).

    ADS  Article  Google Scholar 

  23. 23.

    Maissen, C., Scalari, G., Beck, M. & Faist, J. Asymmetry in polariton dispersion as function of light and matter frequencies in the ultrastrong coupling regime. New J. Phys. 19, 043022 (2017).

    ADS  Article  Google Scholar 

  24. 24.

    Bamba, M. & Ogawa, T. System-environment coupling derived by Maxwell’s boundary conditions from the weak to the ultrastrong light-matter-coupling regime. Phys. Rev. A 88, 013814 (2013).

    ADS  Article  Google Scholar 

  25. 25.

    Kawada, Y. et al. Achromatic prism-type wave plate for broadband terahertz pulses. Opt. Lett. 39, 2794–2797 (2014).

    ADS  Article  Google Scholar 

  26. 26.

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

    ADS  Article  Google Scholar 

  27. 27.

    Ashhab, S. & Nori, F. Qubit-oscillator systems in the ultrastrong-coupling regime and their potential for preparing nonclassical states. Phys. Rev. A 81, 042311 (2010).

    ADS  Article  Google Scholar 

  28. 28.

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

    ADS  Article  Google Scholar 

  29. 29.

    Rzażewski, K., Wódkiewicz, K. & Żakowicz, W. Phase transitions, two-level atoms, and the A 2 term. Phys. Rev. Lett. 35, 432–434 (1975).

    ADS  Article  Google Scholar 

  30. 30.

    Nataf, P. & Ciuti, C. No-go theorem for superradiant quantum phase transitions in cavity QED and counter-example in circuit QED. Nat. Commun. 1, 72 (2010).

    ADS  Article  Google Scholar 

  31. 31.

    Bamba, M. & Ogawa, T. Stability of polarizable materials against superradiant phase transition. Phys. Rev. A 90, 063825 (2014).

    ADS  Article  Google Scholar 

  32. 32.

    Hagenmüller, D. & Ciuti, C. Cavity QED of the graphene cyclotron transition. Phys. Rev. Lett. 109, 267403 (2012).

    ADS  Article  Google Scholar 

  33. 33.

    Chirolli, L., Polini, M., Giovannetti, V. & MacDonald, A. H. Drude weight, cyclotron resonance, and the Dicke model of graphene cavity QED. Phys. Rev. Lett. 109, 267404 (2012).

    ADS  Article  Google Scholar 

Download references

Acknowledgements

We thank K. Hazzard, Y. Todorov and C. Sirtori for discussions. We thank Y. Kawada, H. Takahashi and Hamamatsu Photonics K.K. for fabricating the achromatic terahertz quarter-wave plate. J.K. acknowledges support from the Army Research Office (grant W911NF-17-1-0259) for terahertz magneto-spectroscopy measurements and the National Science Foundation (grant DMR-1310138) for cavity fabrication. M.B. acknowledges support from JST PRESTO (grant JPMJPR1767), KAKENHI (grant 26287087), and ImPACT Program of Council for Science, Technology and Innovation (cabinet office, government of Japan). The work at Purdue was supported by the Department of Energy, Office of Basic Energy Sciences, under Award DE-SC0006671.

Author information

Affiliations

Authors

Contributions

X.L. fabricated terahertz cavity devices, performed all measurements, analysed all experimental data and performed semiclassical simulations under the supervision and guidance of Q.Z. and J.K. M.B. performed quantum mechanical and semiclassical calculations. S.F., G.C.G. and M.J.M. grew the 2DEG sample. Q.Z., W.G., M.L. and K.Y. assisted X.L. with cavity sample preparation and measurements. X.L., M.B. and J.K. wrote the manuscript. All authors discussed the results and commented on the manuscript.

Corresponding author

Correspondence to Junichiro Kono.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

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

Supplementary information

Supplementary Information

Supplementary Sections 1–8.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, X., Bamba, M., Zhang, Q. et al. Vacuum Bloch–Siegert shift in Landau polaritons with ultra-high cooperativity. Nature Photon 12, 324–329 (2018). https://doi.org/10.1038/s41566-018-0153-0

Download citation

Further reading

Search

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