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



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.

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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


  1. Department of Electrical and Computer Engineering, Rice University, Houston, TX, USA

    • Xinwei Li
    • , Weilu Gao
    • , Minhan Lou
    •  & Junichiro Kono
  2. Department of Materials Engineering Science, Osaka University, Osaka, Japan

    • Motoaki Bamba
  3. PRESTO, Japan Science and Technology Agency, Saitama, Japan

    • Motoaki Bamba
  4. Argonne National Laboratories, Lemont, IL, USA

    • Qi Zhang
  5. Department of Physics and Astronomy, Station Q Purdue, and Birck Nanotechnology Center, Purdue University, West Lafayette, IN, USA

    • Saeed Fallahi
    • , Geoff C. Gardner
    •  & Michael J. Manfra
  6. Department of Physics, Graduate School of Engineering, Yokohama National University, Yokohama, Japan

    • Katsumasa Yoshioka
  7. School of Materials Engineering and School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, USA

    • Michael J. Manfra
  8. Department of Material Science and NanoEngineering, Rice University, Houston, TX, USA

    • Junichiro Kono
  9. Department of Physics and Astronomy, Rice University, Houston, TX, USA

    • Junichiro Kono


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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.

Competing interests

The authors declare no competing interests.

Corresponding author

Correspondence to Junichiro Kono.

Supplementary information

  1. Supplementary Information

    Supplementary Sections 1–8.