Time-domain terahertz spectroscopy in high magnetic fields

Abstract

There are a variety of elementary and collective terahertz-frequency excitations in condensed matter whose magnetic field dependence contains significant insight into the states and dynamics of the electrons involved. Often, determining the frequency, temperature, and magnetic field dependence of the optical conductivity tensor, especially in high magnetic fields, can clarify the microscopic physics behind complex many-body behaviors of solids. While there are advanced terahertz spectroscopy techniques as well as high magnetic field generation techniques available, a combination of the two has only been realized relatively recently. Here, we review the current state of terahertz time-domain spectroscopy (THz-TDS) experiments in high magnetic fields. We start with an overview of time-domain terahertz detection schemes with a special focus on how they have been incorporated into optically accessible high-field magnets. Advantages and disadvantages of different types of magnets in performing THz-TDS experiments are also discussed. Finally, we highlight some of the new fascinating physical phenomena that have been revealed by THz-TDS in high magnetic fields.

Graphical abstract

Keywords

high magnetic field / terahertz time-domain spectroscopy (THz-TDS)

References

[1] Nuss M C, Orenstein J. Terahertz time-domain spectroscopy. In: Grüner G, ed. Millimeter and Sub-millimeter Wave Spectroscopy of Solids. Berlin: Springer-Verlag, 1998, Chap. 2, 7–50

[2] Schmuttenmaer C A. Exploring dynamics in the far-infrared with terahertz spectroscopy. Chemical Reviews, 2004, 104(4): 1759–1780

[3] Lee Y S. Principles of Terahertz Science and Technology, vol. 170. Berlin: Springer, 2009

[4] Jepsen P U, Cooke D G, Koch M. Terahertz spectroscopy and imaging–modern techniques and applications. Laser & Photonics Reviews, 2011, 5(1): 124–166

[5] Ulbricht R, Hendry E, Shan J, Heinz T F, Bonn M. Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy. Reviews of Modern Physics, 2011, 83(2): 543–586

[6] Neu J, Schmuttenmaer C A. Tutorial: an introduction to terahertz time domain spectroscopy (THz-TDS). Journal of Applied Physics, 2018, 124(23): 231101

[7] Cong K, Noe G T II, Kono J. Excitons in Magnetic Fields. Oxford: Elsevier, 2018, 63–81

[8] MacDonald A H, Rezayi E H. Fractional quantum Hall effect in a two-dimensional electron-hole fluid. Physical Review B: Condensed Matter and Materials Physics, 1990, 42(5): 3224–3227

[9] Dzyubenko A B, Lozovik Y E. Symmetry of Hamiltonians of quantum two-component systems: condensate of composite particles as an exact eigenstate. Journal of Physics A, Mathematical and General, 1991, 24(2): 415–424

[10] Apal’kov V M, Rashba E I. Magnetospectroscopy of 2D electron-gas: cusps in emission-spectra and Coulomb gaps. JETP Letters, 1991, 53: 442–448

[11] Rashba E I, Sturge M D, Yoon H W, Pfeiffer L N. Hidden symmetry and the magnetically induced “Mott transition” in quantum wells containing an electron gas. Solid State Communications, 2000, 114(11): 593–596

[12] Proust C, Taillefer L. The remarkable underlying ground states of cuprate superconductors. Annual Review of Condensed Matter Physics, 2019, 10(1): 409–429

[13] Shi Z, Baity P G, Sasagawa T, Popović D. Vortex phase diagram and the normal state of cuprates with charge and spin orders. Science Advances, 2020, 6(7): eaay8946

[14] Ran S, Liu I L, Eo Y S, Campbell D J, Neves P M, Fuhrman W T, Saha S R, Eckberg C, Kim H, Graf D, Balakirev F, Singleton J, Paglione J, Butch N P. Extreme magnetic field-boosted superconductivity. Nature Physics, 2019, 15(12): 1250–1254

[15] Dean C R, Young A F, Cadden-Zimansky P, Wang L, Ren H, Watanabe K, Taniguchi T, Kim P, Hone J, Shepard K L. Multicomponent fractional quantum Hall effect in graphene. Nature Physics, 2011, 7(9): 693–696

[16] Moll P J, Potter A C, Nair N L, Ramshaw B J, Modic K A, Riggs S, Zeng B, Ghimire N J, Bauer E D, Kealhofer R, Ronning F, Analytis J G. Magnetic torque anomaly in the quantum limit of Weyl semimetals. Nature Communications, 2016, 7(1): 12492

[17] Wu Q, Zhang X C. Free-space electro-optic sampling of terahertz beams. Applied Physics Letters, 1995, 67(24): 3523–3525

[18] Wu Q, Zhang X C. Ultrafast electro-optic field sensors. Applied Physics Letters, 1996, 68(12): 1604–1606

[19] Nahata A, Weling A S, Heinz T F. A wide-band coherent terahertz spectroscopy system using optical rectification and electro-optic sampling. Applied Physics Letters, 1996, 69(16): 2321–2323

[20] Wu Q, Zhang X C. 7 terahertz broadband GaP electro-optic sensor. Applied Physics Letters, 1997, 70(14): 1784–1786

[21] Huber R, Brodschelm A, Tauser F, Leitenstorfer A. Generation and field-resolved detection of femtosecond electromagnetic pulses tunable up to 41 THz. Applied Physics Letters, 2000, 76(22): 3191–3193

[22] Liu K, Xu J, Zhang X C. GaSe crystals for broadband terahertz wave detection. Applied Physics Letters, 2004, 85(6): 863–865

[23] Smith P R, Auston D H, Nuss M C. Subpicosecond photoconducting dipole antennas. IEEE Journal of Quantum Electronics, 1988, 24(2): 255–260

[24] Lu X, Karpowicz N, Zhang X C. Broadband terahertz detection with selected gases. Journal of the Optical Society of America B, Optical Physics, 2009, 26(9): A66–A73

[25] Elzinga P A, Kneisler R J, Lytle F E, Jiang Y, King G B, Laurendeau N M. Pump/probe method for fast analysis of visible spectral signatures utilizing asynchronous optical sampling. Applied Optics, 1987, 26(19): 4303–4309

[26] Janke C, Först M, Nagel M, Kurz H, Bartels A. Asynchronous optical sampling for high-speed characterization of integrated resonant terahertz sensors. Optics Letters, 2005, 30(11): 1405–1407

[27] Yasui T, Saneyoshi E, Araki T. Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition. Applied Physics Letters, 2005, 87(6): 061101

[28] Bartels A, Cerna R, Kistner C, Thoma A, Hudert F, Janke C, Dekorsy T. Ultrafast time-domain spectroscopy based on high-speed asynchronous optical sampling. Review of Scientific Instruments, 2007, 78(3): 035107

[29] Spencer B, Smith W F, Hibberd M T, Dawson P, Beck M, Bartels A, Guiney I, Humphreys C J, Graham D M. Terahertz cyclotron resonance spectroscopy of an AlGaN/GaN heterostructure using a high-field pulsed magnet and an asynchronous optical sampling technique. Applied Physics Letters, 2016, 108(21): 212101

[30] Tauser F, Rausch C, Posthumus J H, Lison F. Electronically controlled optical sampling using 100 MHz repetition rate fiber lasers. In: Proceedings of Commercial and Biomedical Applications of Ultrafast Lasers VIII. San Jose: SPIE, 2008, 68810O

[31] Kim Y, Yee D S. High-speed terahertz time-domain spectroscopy based on electronically controlled optical sampling. Optics Letters, 2010, 35(22): 3715–3717

[32] Liu J, Mbonye M K, Mendis R, Mittleman D M. Measurement of terahertz pulses using electronically controlled optical sampling (ECOPS). In: Proceedings of CLEO/QELS: 2010 Laser Science to Photonic Applications. San Jose: IEEE, 2010, 1–2

[33] Noe G T II, Zhang Q, Lee J, Kato E, Woods G L, Nojiri H, Kono J. Rapid scanning terahertz time-domain magnetospectroscopy with a table-top repetitive pulsed magnet. Applied Optics, 2014, 53(26): 5850–5855

[34] Molter D, Ellrich F, Weinland T, George S, Goiran M, Keilmann F, Beigang R, Léotin J. High-speed terahertz time-domain spectroscopy of cyclotron resonance in pulsed magnetic field. Optics Express, 2010, 18(25): 26163–26168

[35] Teo S M, Ofori-Okai B K, Werley C A, Nelson K A. Single-shot THz detection techniques optimized for multidimensional THz spectroscopy. Review of Scientific Instruments, 2015, 86(5): 051301

[36] Minami Y, Hayashi Y, Takeda J, Katayama I. Single-shot measurement of a terahertz electric-field waveform using a reflective echelon mirror. Applied Physics Letters, 2013, 103(5): 051103

[37] Topp M, Rentzepis P, Jones R. Time-resolved absorption spectroscopy in the 10–12-sec range. Journal of Applied Physics, 1971, 42(9): 3415–3419

[38] Topp M, Rentzepis P, Jones R. Time resolved picosecond emission spectroscopy of organic dye lasers. Chemical Physics Letters, 1971, 9(1): 1–5

[39] Kim K Y, Yellampalle B, Taylor A J, Rodriguez G, Glownia J H. Single-shot terahertz pulse characterization via two-dimensional electro-optic imaging with dual echelons. Optics Letters, 2007, 32(14): 1968–1970

[40] Katayama I, Sakaibara H, Takeda J. Real-time time-frequency imaging of ultrashort laser pulses using an echelon mirror. Japanese Journal of Applied Physics, 2011, 50(10): 102701

[41] Noe G T II, Katayama I, Katsutani F, Allred J J, Horowitz J A, Sullivan D M, Zhang Q, Sekiguchi F, Woods G L, Hoffmann M C, Nojiri H, Takeda J, Kono J. Single-shot terahertz time-domain spectroscopy in pulsed high magnetic fields. Optics Express, 2016, 24(26): 30328–30337

[42] Makihara T, Hayashida K, Noe G T II, Li X, Kono J. Magnonic quantum simulator of antiresonant ultrastrong light-matter coupling. 2020, arXiv:2008:10721

[43] Jiang Z, Zhang X C. Electro-optic measurement of THz field pulses with a chirped optical beam. Applied Physics Letters, 1998, 72(16): 1945–1947

[44] Jiang Z, Zhang X C. Single-shot spatiotemporal terahertz field imaging. Optics Letters, 1998, 23(14): 1114–1116

[45] Matlis N, Plateau G, van Tilborg J, Leemans W. Single-shot spatiotemporal measurements of ultrashort THz waveforms using temporal electric-field cross correlation. Journal of the Optical Society of America B, Optical Physics, 2011, 28(1): 23–27

[46] Noe G T II, Zhang Q, Lee J, Kato E, Woods G L, Nojiri H, Kono J. Rapid scanning terahertz time-domain magnetospectroscopy with a table-top repetitive pulsed magnet. Applied Optics, 2014, 53(26): 5850–5855

[47] Walecki W, Some D, Kozlov V, Nurmikko A. Terahertz electromagnetic transients as probes of a two-dimensional electron gas. Applied Physics Letters, 1993, 63(13): 1809–1811

[48] Some D, Nurmikko A V. Real-time electron cyclotron oscillations observed by terahertz techniques in semiconductor heterostructures. Applied Physics Letters, 1994, 65(26): 3377–3379

[49] Some D, Nurmikko A V. Coherent transient cyclotron emission from photoexcited GaAs. Physical Review B: Condensed Matter and Materials Physics, 1994, 50(8): 5783–5786

[50] Some D, Nurmikko A V. Ultrafast photoexcited cyclotron emission: contributions from real and virtual excitations. Physical Review B: Condensed Matter and Materials Physics, 1996, 53(20): R13295–R13298

[51] Crooker S A. Fiber-coupled antennas for ultrafast coherent terahertz spectroscopy in low temperatures and high magnetic fields. Review of Scientific Instruments, 2002, 73(9): 3258–3264

[52] Wang X, Hilton D J, Ren L, Mittleman D M, Kono J, Reno J L. Terahertz time-domain magnetospectroscopy of a high-mobility two-dimensional electron gas. Optics Letters, 2007, 32(13): 1845–1847

[53] Sumikura H, Nagashima T, Kitahara H, Hangyo M. Development of a cryogen-free terahertz time-domain magnetooptical measurement system. Japanese Journal of Applied Physics, 2007, 46(4A): 1739–1744

[54] Ikebe Y, Shimano R. Characterization of doped silicon in low carrier density region by terahertz frequency Faraday effect. Applied Physics Letters, 2008, 92(1): 012111

[55] Scalari G, Maissen C, Turcinková D, Hagenmüller D, De Liberato S, Ciuti C, Reichl C, Schuh D, Wegscheider W, Beck M, Faist J. Ultrastrong coupling of the cyclotron transition of a 2D electron gas to a THz metamaterial. Science, 2012, 335(6074): 1323–1326

[56] George D K, Stier A V, Ellis C T, McCombe B D, Černe J, Markelz A G. Terahertz magneto-optical polarization modulation spectroscopy. Journal of the Optical Society of America. B, Optical Physics, 2012, 29(6): 1406–1412

[57] Wood C D, Mistry D, Li L H, Cunningham J E, Linfield E H, Davies A G. On-chip terahertz spectroscopic techniques for measuring mesoscopic quantum systems. Review of Scientific Instruments, 2013, 84(8): 085101

[58] Wu L, Salehi M, Koirala N, Moon J, Oh S, Armitage N P. Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator. Science, 2016, 354(6316): 1124–1127

[59] Crooker S. Fiber-coupled antennas for ultrafast coherent terahertz spectroscopy in low temperatures and high magnetic fields. Review of Scientific Instruments, 2002, 73(9): 3258–3264

[60] Wang X, Belyanin A A, Crooker S A, Mittleman D M, Kono J. Interference-induced terahertz transparency in a semiconductor magneto-plasma. Nature Physics, 2010, 6(2): 126–130

[61] Arikawa T, Wang X, Hilton D J, Reno J L, Pan W, Kono J. Quantum control of a Landau-quantized two-dimensional electron gas in a GaAs quantum well using coherent terahertz pulses. Physical Review B: Condensed Matter and Materials Physics, 2011, 84(24): 241307

[62] Arikawa T, Wang X, Belyanin A A, Kono J. Giant tunable Faraday effect in a semiconductor magneto-plasma for broadband terahertz polarization optics. Optics Express, 2012, 20(17): 19484–19492

[63] Zhang Q, Arikawa T, Kato E, Reno J L, Pan W, Watson J D, Manfra M J, Zudov M A, Tokman M, Erukhimova M, Belyanin A, Kono J. Superradiant decay of cyclotron resonance of two-dimensional electron gases. Physical Review Letters, 2014, 113(4): 047601

[64] Zhang Q, Lou M, Li X, Reno J L, Pan W, Watson J D, Manfra M J, Kono J. Collective non-perturbative coupling of 2D electrons with high-quality-factor terahertz cavity photons. Nature Physics, 2016, 12(11): 1005–1011

[65] Li X, Bamba M, Zhang Q, Fallahi S, Gardner G C, Gao W, Lou M, Yoshioka K, Manfra M J, Kono J. Vacuum Bloch–Siegert shift in Landau polaritons with ultra-high cooperativity. Nature Photonics, 2018, 12(6): 324–329

[66] Li X, Bamba M, Yuan N, Zhang Q, Zhao Y, Xiang M, Xu K, Jin Z, Ren W, Ma G, Cao S, Turchinovich D, Kono J. Observation of Dicke cooperativity in magnetic interactions. Science, 2018, 361(6404): 794–797

[67] Toth J, Bird M D, Bole S, O’Reilly J W. Fabrication and assembly of the NHMFL 25 T resistive split magnet. IEEE Transactions on Applied Superconductivity, 2012, 22(3): 4301604

[68] Curtis J A, Burch A D, Barman B, Linn A G, McClintock L M, O’Beirne A L, Stiles M J, Reno J L, McGill S A, Karaiskaj D, Hilton D J. Broadband ultrafast terahertz spectroscopy in the 25 T Split Florida-Helix. Review of Scientific Instruments, 2018, 89(7): 073901

[69] Curtis J A, Tokumoto T, Nolan N K, McClintock L M, Cherian J G, McGill S A, Hilton D J. Ultrafast pump-probe spectroscopy in gallium arsenide at 25 T. Optics Letters, 2014, 39(19): 5772–5775

[70] Paul J, Stevens C E, Smith R P, Dey P, Mapara V, Semenov D, McGill S A, Kaindl R A, Hilton D J, Karaiskaj D. Coherent two-dimensional Fourier transform spectroscopy using a 25 Tesla resistive magnet. Review of Scientific Instruments, 2019, 90(6): 063901

[71] Kim K Y, Taylor A J, Glownia J H, Rodriguez G. Coherent control of terahertz supercontinuum generation in ultrafast laser-gas interactions. Nature Photonics, 2008, 2(10): 605–609

[72] Kress M, Löffler T, Eden S, Thomson M, Roskos H G. Terahertz-pulse generation by photoionization of air with laser pulses composed of both fundamental and second-harmonic waves. Optics Letters, 2004, 29(10): 1120–1122

[73] Molter D, Torosyan G, Ballon G, Drigo L, Beigang R, Léotin J. Step-scan time-domain terahertz magneto-spectroscopy. Optics Express, 2012, 20(6): 5993–6002

[74] Noe G T II, Nojiri H, Lee J, Woods G L, Léotin J, Kono J. A table-top, repetitive pulsed magnet for nonlinear and ultrafast spectroscopy in high magnetic fields up to 30 T. Review of Scientific Instruments, 2013, 84(12): 123906

[75] Post K W, Legros A, Rickel D G, Singleton J, McDonald R D, He X, Bozovic I, Xu X, Shi X, Armitage N P, Crooker S A. Observation of cyclotron resonance and measurement of the hole mass in optimally-doped La_2−xSr_xCuO₄. 2020, arXiv:2006.09131

[76] Hebling J, Yeh K L, Hoffmann M C, Bartal B, Nelson K A. Generation of high-power terahertz pulses by tilted-pulse-front excitation and their application possibilities. Journal of the Optical Society of America B, Optical Physics, 2008, 25(7): B6–B19

[77] McCombe B D, Wagner R J. Intraband magneto-optical studies of semiconductors in the far-infrared. I. In: Marton L, ed. Advances in Electronics and Electron Physics, vol. 37. New York: Academic Press, 1975, 1–78

[78] Mittleman D M. Sensing with Terahertz Radiation. Berlin: Springer, 2003

[79] Basov D N, Averitt R D, VanDerMarel D, Dressel M, Haule K. Electrodynamics of correlated electron materials. Reviews of Modern Physics, 2011, 83(2): 471–541

[80] Dexheimer S L. Terahertz Spectroscopy: Principles and Applications. Boca Raton, Florida: CRC press, 2017

[81] Kézsmárki I, Szaller D, Bordács S, Kocsis V, Tokunaga Y, Taguchi Y, Murakawa H, Tokura Y, Engelkamp H, Rõõm T, Nagel U. One-way transparency of four-coloured spin-wave excitations in multiferroic materials. Nature Communications, 2014, 5(1): 3203

[82] Bordács S, Kézsmárki I, Szaller D, Demkó L, Kida N, Murakawa H, Onose Y, Shimano R, Rõõm T, Nagel U, Miyahara S, Furukawa N, Tokura Y. Chirality of matter shows up via spin excitations. Nature Physics, 2012, 8(10): 734–738

[83] Penc K, Romhányi J, Rõõm T, Nagel U, Antal A, Fehér T, Jánossy A, Engelkamp H, Murakawa H, Tokura Y, Szaller D, Bordács S, Kézsmárki I. Spin-stretching modes in anisotropic magnets: spin-wave excitations in the multiferroic Ba₂CoGe₂O₇. Physical Review Letters, 2012, 108(25): 257203

[84] Peedu L, Kocsis V, Szaller D, Viirok J, Nagel U, Rõõm T, Farkas D G, Bordács S, Kamenskyi D L, Zeitler U, Tokunaga Y, Taguchi Y, Tokura Y, Kézsmárki I. Spin excitations of magnetoelectric LiNiPO₄ in multiple magnetic phases. Physical Review B: Condensed Matter and Materials Physics, 2019, 100(2): 024406

[85] Talbayev D, LaForge A D, Trugman S A, Hur N, Taylor A J, Averitt R D, Basov D N. Magnetic exchange interaction between rare-earth and Mn ions in multiferroic hexagonal manganites. Physical Review Letters, 2008, 101(24): 247601

[86] Mihály L, Talbayev D, Kiss L F, Zhou J, Fehér T, Jánossy A. Field-frequency mapping of the electron spin resonance in the paramagnetic and antiferromagnetic states of LaMnO₃. Physical Review B: Condensed Matter and Materials Physics, 2004, 69(2): 024414

[87] Mihály L, Fehér T, Dóra B, Náfrádi B, Berger H, Forró L. Spin resonance in the ordered magnetic state of Ni₅(TeO₃)₄Cl₂. Physical Review B: Condensed Matter and Materials Physics, 2006, 74(17): 174403

[88] Kézsmárki I, Nagel U, Bordács S, Fishman R S, Lee J H, Yi H T, Cheong S W, Rõõm T. Optical diode effect at spin-wave excitations of the room-temperature multiferroic BiFeO₃. Physical Review Letters, 2015, 115(12): 127203

[89] Autore M, Engelkamp H, D’Apuzzo F, Gaspare A D, Pietro P D, Vecchio I L, Brahlek M, Koirala N, Oh S, Lupi S. Observation of magnetoplasmons in Bi₂Se₃ topological insulator. ACS Photonics, 2015, 2(9): 1231–1235

[90] Wang Z, Reschke S, Hüvonen D, Do S H, Choi K Y, Gensch M, Nagel U, Rõõm T, Loidl A. Magnetic excitations and continuum of a possibly field-induced quantum spin liquid in α-RuCl₃. Physical Review Letters, 2017, 119(22): 227202

[91] Sahasrabudhe A, Kaib D A S, Reschke S, German R, Koethe T C, Buhot J, Kamenskyi D, Hickey C, Becker P, Tsurkan V, Loidl A, Do S H, Choi K Y, Grüninger M, Winter S M, Wang Z, Valentí R, van Loosdrecht P H M. High-field quantum disordered state in α-RuCl₃: spin flips, bound states, and multi-particle continuum. Physical Review B: Condensed Matter and Materials Physics, 2020, 101(14): 140410

[92] LaForge A D, Frenzel A, Pursley B C, Lin T, Liu X, Shi J, Basov D N. Optical characterization of Bi₂Se₃ in a magnetic field: Infrared evidence for magnetoelectric coupling in a topological insulator material. Physical Review B: Condensed Matter and Materials Physics, 2010, 81(12): 125120

[93] Schafgans A, Post K W, Taskin A A, Ando Y, Qi X L, Chapler B C, Basov D N. Landau level spectroscopy of surface states in the topological insulator Bi_0.91Sb_0.09 via magneto-optics. Physical Review B: Condensed Matter and Materials Physics, 2012, 85(19): 195440

[94] Schafgans A A, LaForge A D, Dordevic S V, Qazilbash M M, Padilla W J, Burch K S, Li Z Q, Komiya S, Ando Y, Basov D N. Towards a two-dimensional superconducting state of La_2−xSr_x-CuO₄ in a moderate external magnetic field. Physical Review Letters, 2010, 104(15): 157002

[95] Dresselhaus G, Kip A F, Kittel C. Cyclotron resonance of electrons and holes in silicon and germanium crystals. Physical Review, 1955, 98(2): 368–384

[96] Lax B, Mavroides J G. Cyclotron resonance. In: Seitz F, Turnbull D, eds. Solid State Physics, vol. 11. New York: Academic Press, 1960, 261–400

[97] McCombe B D, Wagner R J. Intraband magneto-optical studies of semiconductors in the far-infrared. II. In: Marton L, ed. Advances in Electronics and Electron Physics, vol. 38. New York: Academic Press, 1975, 1–53

[98] Kono J. Cyclotron resonance. In: Kaufmann E N, et al. (eds.) Methods in Materials Research. New York: John Wiley & Sons, 2001, Chap. 9b.2

[99] Kono J, Miura N. Cyclotron resonance in high magnetic fields. In: Miura N, Herlach F, eds. High Magnetic Fields: Science and Technology, Volume III. Singapore: World Scientific, 2006, 61–90

[100] Hilton D J, Arikawa T, Kono J. Cyclotron resonance. In: Kaufmann E N, ed. Characterization of Materials, 2nd edition. New York: John Wiley & Sons, Inc., 2012, 1–15

[101] Wang X, Hilton D J, Reno J L, Mittleman D M, Kono J. Direct measurement of cyclotron coherence times of high-mobility two-dimensional electron gases. Optics Express, 2010, 18(12): 12354–12361

[102] Dicke R H. Coherence in spontaneous radiation processes. Physical Review, 1954, 93(1): 99–110

[103] Miura N, Yokoi H, Kono J, Sasaki S. High field cyclotron resonance and the electron effective masses in AlAs. Solid State Communications, 1991, 79(12): 1039–1042

[104] Kono J, Miura N, Takeyama S, Yokoi H, Fujimori N, Nishibayashi Y, Nakajima T, Tsuji K, Yamanaka M. Observation of cyclotron resonance in low-mobility semiconductors using pulsed ultra-high magnetic fields. Physica B, Condensed Matter, 1993, 184(1–4): 178–183

[105] Kono J, Takeyama S, Takamasu T, Miura N, Fujimori N, Nishibayashi Y, Nakajima T, Tsuji K. High-field cyclotron resonance and valence-band structure in semiconducting diamond. Physical Review B: Condensed Matter and Materials Physics, 1993, 48(15): 10917–10925

[106] Kono J, Takeyama S, Yokoi H, Miura N, Yamanaka M, Shinohara M, Ikoma K. High-field cyclotron resonance and impurity transition in n-type and p-type 3C-SiC at magnetic fields up to 175 T. Physical Review B: Condensed Matter and Materials Physics, 1993, 48(15): 10909–10916

[107] Knap W, Contreras S, Alause H, Skierbiszewski C, Camassel J, Dyakonov M, Robert J L, Yang J, Chen Q, Asif Khan M, Sadowski M L, Huant S, Yang F H, Goiran M, Leotin J, Shur M S. Cyclotron resonance and quantum hall effect studies of the two-dimensional electron gas confined at the GaN/AlGaN interface. Applied Physics Letters, 1997, 70(16): 2123–2125

[108] Wang Y, Kaplan R, Ng H K, Doverspike K, Gaskill D K, Ikedo T, Akasaki I, Amono H. Magneto-optical studies of GaN and GaN/Al_xGa_1−xN: Donor Zeeman spectroscopy and two dimensional electron gas cyclotron resonance. Journal of Applied Physics, 1996, 79(10): 8007–8010

[109] Cheng B, Taylor P, Folkes P, Rong C, Armitage N P. Magnetoterahertz response and Faraday rotation from massive dirac fermions in the topological crystalline insulator Pb_0.5Sn_0.5Te. Physical Review Letters, 2019, 122(9): 097401

[110] Jeffries C D. Electron-hole condensation in semiconductors: electrons and holes condense into freely moving liquid metallic droplets, a plasma phase with novel properties. Science, 1975, 189(4207): 955–964

[111] Zhang Q, Wang Y, Gao W, Long Z, Watson J D, Manfra M J, Belyanin A, Kono J. Stability of high-density two-dimensional excitons against a Mott transition in high magnetic fields probed by coherent terahertz spectroscopy. Physical Review Letters, 2016, 117(20): 207402

[112] Li X, Yoshioka K, Zhang Q, Marquez Peraca N, Katsutani F, Gao W, Noe G T II, Watson J D, Manfra M J, Katayama I, Takeda J, Kono J. Observation of terahertz gain in two-dimensional magnetoexcitons. 2020, arXiv:2004.11459

[113] Hangyo M, Tani M, Nagashima T. Terahertz time-domain spectroscopy of solids: a review. International Journal of Infrared and Millimeter Waves, 2005, 26(12): 1661–1690

[114] von Klitzing K, Dorda G, Pepper M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters, 1980, 45(6): 494–497

[115] Ikebe Y, Morimoto T, Masutomi R, Okamoto T, Aoki H, Shimano R. Optical Hall effect in the integer quantum Hall regime. Physical Review Letters, 2010, 104(25): 256802

[116] Shimano R, Yumoto G, Yoo J Y, Matsunaga R, Tanabe S, Hibino H, Morimoto T, Aoki H. Quantum Faraday and Kerr rotations in graphene. Nature Communications, 2013, 4(1): 1841

[117] Fiebig M. Revival of the magnetoelectric effect. Journal of Physics D, Applied Physics, 2005, 38(8): R123–R152

[118] Yu S, Dhanasekhar C, Adyam V, Deckoff-Jones S, Man M K L, Madéo J, Wong E L, Harada T, Murali Krishna M B, Dani K M, Talbayev D. Terahertz-frequency magnetoelectric effect in Ni-doped CaBaCo₄O₇. Physical Review B, 2017, 96(9): 094421

[119] Armitage N P, Wu L. On the matter of topological insulators as magnetoelectrics. SciPost Physics, 2019, 6: 046

[120] Essin A M, Moore J E, Vanderbilt D. Magnetoelectric polarizability and axion electrodynamics in crystalline insulators. Physical Review Letters, 2009, 102(14): 146805

[121] Maciejko J, Qi X L, Drew H D, Zhang S C. Topological quantization in units of the fine structure constant. Physical Review Letters, 2010, 105(16): 166803

[122] Morimoto T, Furusaki A, Nagaosa N. Topological magnetoelectric effects in thin films of topological insulators. Physical Review B: Condensed Matter and Materials Physics, 2015, 92(8): 085113

[123] Qi X L, Hughes T L, Zhang S C. Topological field theory of time-reversal invariant insulators. Physical Review B: Condensed Matter and Materials Physics, 2008, 78(19): 195424

[124] Tse W K, MacDonald A H. Giant magneto-optical Kerr effect and universal Faraday effect in thin-film topological insulators. Physical Review Letters, 2010, 105(5): 057401

[125] Tse W K, MacDonald A H. Magneto-optical Faraday and Kerr effects in topological insulator films and in other layered quantized Hall systems. Physical Review B: Condensed Matter and Materials Physics, 2011, 84(20): 205327

[126] Wang J, Lian B, Qi X L, Zhang S C. Quantized topological magnetoelectric effect of the zero-plateau quantum anomalous Hall state. Physical Review B: Condensed Matter and Materials Physics, 2015, 92(8): 081107

[127] Zhang D, Shi M, Zhu T, Xing D, Zhang H, Wang J. Topological axion states in the magnetic insulator MnBi₂Te₄ with the quantized magnetoelectric effect. Physical Review Letters, 2019, 122(20): 206401

[128] Wilczek F. Two applications of axion electrodynamics. Physical Review Letters, 1987, 58(18): 1799–1802

[129] Hancock J N, van Mechelen J L, Kuzmenko A B, van der Marel D, Brüne C, Novik E G, Astakhov G V, Buhmann H, Molenkamp L W. Surface state charge dynamics of a high-mobility three-dimensional topological insulator. Physical Review Letters, 2011, 107(13): 136803

[130] Jenkins G S, Sushkov A B, Schmadel D C, Butch N P, Syers P, Paglione J, Drew H D. Terahertz Kerr and reflectivity measurements on the topological insulator Bi₂Se₃. Physical Review B: Condensed Matter and Materials Physics, 2010, 82(12): 125120

[131] Valdés Aguilar R, Stier A V, Liu W, Bilbro L S, George D K, Bansal N, Wu L, Cerne J, Markelz A G, Oh S, Armitage N P. Terahertz response and colossal Kerr rotation from the surface states of the topological insulator Bi₂Se₃. Physical Review Letters, 2012, 108(8): 087403

[132] Wu L, Tse W K, Brahlek M, Morris C M, Aguilar R V, Koirala N, Oh S, Armitage N P. High-resolution Faraday rotation and electron-phonon coupling in surface states of the bulk-insulating topological insulator Cu_0.02Bi₂Se₃. Physical Review Letters, 2015, 115(21): 217602

[133] Dziom V, Shuvaev A, Pimenov A, Astakhov G V, Ames C, Bendias K, Böttcher J, Tkachov G, Hankiewicz E M, Brüne C, Buhmann H, Molenkamp L W. Observation of the universal magnetoelectric effect in a 3D topological insulator. Nature Communications, 2017, 8(1): 15197

[134] Li X, Yoshioka K, Xie M, Noe G T, Lee W, Marquez Peraca N, Gao W, Hagiwara T, Handegård Ø S, Nien L W, Nagao T, Kitajima M, Nojiri H, Shih C K, MacDonald A H, Katayama I, Takeda J, Fiete G A, Kono J. Terahertz Faraday and Kerr rotation spectroscopy of Bi_1−xSb_x films in high magnetic fields up to 30 Tesla. Physical Review B: Condensed Matter and Materials Physics, 2019, 100(11): 115145

[135] Okada K N, Takahashi Y, Mogi M, Yoshimi R, Tsukazaki A, Takahashi K S, Ogawa N, Kawasaki M, Tokura Y. Terahertz spectroscopy on Faraday and Kerr rotations in a quantum anomalous Hall state. Nature Communications, 2016, 7(1): 12245

[136] Morris C M, Valdés Aguilar R, Ghosh A, Koohpayeh S M, Krizan J, Cava R J, Tchernyshyov O, McQueen T M, Armitage N P. Hierarchy of bound states in the one-dimensional ferromagnetic Ising chain CoNb₂O₆ investigated by high-resolution time-domain terahertz spectroscopy. Physical Review Letters, 2014, 112(13): 137403

[137] Little A, Wu L, Lampen-Kelley P, Banerjee A, Patankar S, Rees D, Bridges C A, Yan J Q, Mandrus D, Nagler S E, Orenstein J. Antiferromagnetic resonance and terahertz continuum in α-RuCl₃. Physical Review Letters, 2017, 119(22): 227201

[138] Wu L, Little A, Aldape E E, Rees D, Thewalt E, Lampen-Kelley P, Banerjee A, Bridges C A, Yan J Q, Boone D, Patankar S, Goldhaber-Gordon D, Mandrus D, Nagler S E, Altman E, Orenstein J. Field evolution of magnons in α-RuCl₃ by high-resolution polarized terahertz spectroscopy. Physical Review. B, 2018, 98(9): 094425

[139] Ozel I O, Belvin C A, Baldini E, Kimchi I, Do S, Choi K Y, Gedik N. Magnetic field-dependent low-energy magnon dynamics in α-RuCl₃. Physical Review. B, 2019, 100(8): 085108

[140] Shi L, Liu Y Q, Lin T, Zhang M Y, Zhang S J, Wang L, Shi Y G, Dong T, Wang N L. Field-induced magnon excitation and in-gap absorption in the Kitaev candidate RuCl₃. Physical Review B: Condensed Matter and Materials Physics, 2018, 98(9): 094414

[141] Yu S, Gao B, Kim J W, Cheong S W, Man M K L, Madéo J, Dani K M, Talbayev D. High-temperature terahertz optical diode effect without magnetic order in polar FeZnMo₃O₈. Physical Review Letters, 2018, 120(3): 037601

[142] Forn-Díaz P, Lamata L, Rico E, Kono J, Solano E. Ultrastrong coupling regimes of light-matter interaction. Reviews of Modern Physics, 2019, 91(2): 025005

[143] Kockum A F, Miranowicz A, De Liberato S, Savasta S, Nori F. Ultrastrong coupling between light and matter. Nature Reviews Physics, 2019, 1(1): 19–40

[144] Hagenmüller D, De Liberato S, Ciuti C. Ultra-strong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor. Physical Review B: Condensed Matter and Materials Physics, 2010, 81(23): 235303

[145] Herrmann G. Resonance and high frequency susceptibility in canted antiferromagnetic substances. Journal of Physics and Chemistry of Solids, 1963, 24(5): 597–606

[146] Artoni M, Birman J L. Polaritonsqueezing: theory and proposed experiment. Quantum Optics: Journal of the European Optical Society Part B, 1989, 1(2): 91–97

[147] Schwendimann P, Quattropani A. Nonclassical properties of polariton states. Europhysics Letters, 1992, 17(4): 355–358

[148] Ciuti C, Bastard G, Carusotto I. Quantumvacuum properties of the intersubband cavity polariton field. Physical Review B: Condensed Matter and Materials Physics, 2005, 72(11): 115303

Acknowledgements

J. K. acknowledges support from the U.S. Army Research Office (W911NF-17-1-0259), the U.S. National Science Foundation (NSF MRSEC DMR-1720595), the U.S. Department of Energy (DEFG02-06ER46308), and the Robert A. Welch Foundation (C-1509).

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