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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

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.

Correspondence to Junichiro Kono.

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    Supplementary Sections 1–8.

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Further reading

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.