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.
Subscribe to Journal
Get full journal access for 1 year
only $14.08 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Allen, L. & Eberly, J. H. Optical Resonance and Two-Level Atoms (John Wiley & Sons, New York, 1975).
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).
Bloch, F. & Siegert, A. Magnetic resonance for nonrotating fields. Phys. Rev. 57, 522–527 (1940).
Shirley, J. H. Solution of the Schrödinger equation with a Hamiltonian periodic in time. Phys. Rev. 138, B979–B987 (1965).
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).
AbragamA. The Principles of Nuclear Magnetism (Oxford Univ. Press, Oxford, 1961).
Tuorila, J. et al. Stark effect and generalized Bloch-Siegert shift in a strongly driven two-level system. Phys. Rev. Lett. 105, 257003 (2010).
Pietikäinen, I. et al. Observation of the Bloch-Siegert shift in a driven quantum-to-classical transition. Phys. Rev. B 96, 020501 (2017).
Sie, E. J. et al. Large, valley-exclusive Bloch-Siegert shift in monolayer WS2. Science 355, 1066–1069 (2017).
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).
Bayer, A. et al. Terahertz light–matter interaction beyond unity coupling strength. Nano. Lett. 17, 6340–6344 (2017).
Anappara, A. A. et al. Signatures of the ultrastrong light-matter coupling regime. Phys. Rev. B 79(20), 201303 (R) (2009).
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).
Zhang, Q. et al. Collective non-perturbative coupling of 2D electrons with high-quality-factor terahertz cavity photons. Nat. Phys. 12, 1005–1011 (2016).
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).
Niemczyk, T. et al. Circuit quantum electrodynamics in the ultrastrong-coupling regime. Nat. Phys. 6, 772–776 (2010).
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).
Yoshihara, F. et al. Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime. Nat. Phys. 13, 44–47 (2017).
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).
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).
Zhang, Q. et al. Superradiant decay of cyclotron resonance of two-dimensional electron gases. Phys. Rev. Lett. 113, 047601 (2014).
Maissen, C. et al. Ultrastrong coupling in the near field of complementary split-ring resonators. Phys. Rev. B 90, 205309 (2014).
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).
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).
Kawada, Y. et al. Achromatic prism-type wave plate for broadband terahertz pulses. Opt. Lett. 39, 2794–2797 (2014).
Scalari, G. et al. Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial. Science 335, 1323–1326 (2012).
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).
Ciuti, C., Bastard, G. & Carusotto, I. Quantum vacuum properties of the intersubband cavity polariton field. Phys. Rev. B 72, 115303 (2005).
Rzażewski, K., Wódkiewicz, K. & Żakowicz, W. Phase transitions, two-level atoms, and the A 2 term. Phys. Rev. Lett. 35, 432–434 (1975).
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).
Bamba, M. & Ogawa, T. Stability of polarizable materials against superradiant phase transition. Phys. Rev. A 90, 063825 (2014).
Hagenmüller, D. & Ciuti, C. Cavity QED of the graphene cyclotron transition. Phys. Rev. Lett. 109, 267403 (2012).
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).
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.
The authors declare no competing interests.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
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
Physical Review B (2021)
Nature Communications (2021)
Quantum Science and Technology (2021)
Physical Review Research (2020)