Abstract
In solids, the high density of charged particles makes manybody interactions a pervasive principle governing optics and electronics^{1,2,3,4,5,6,7,8,9,10,11,12}. However, Walter Kohn found in 1961 that the cyclotron resonance of Landauquantized electrons is independent of the seemingly inescapable Coulomb interaction between electrons^{2}. Although this surprising theorem has been exploited in sophisticated quantum phenomena^{13,14,15}, such as ultrastrong light–matter coupling^{16}, superradiance^{17} and coherent control^{18}, the complete absence of nonlinearities excludes many intriguing possibilities, such as quantumlogic protocols^{19}. Here, we use intense terahertz pulses to drive the cyclotron response of a twodimensional electron gas beyond the protective limits of Kohn’s theorem. Anharmonic Landau ladder climbing and distinct terahertz four and sixwave mixing signatures occur, which our theory links to dynamic Coulomb effects between electrons and the positively charged ion background. This new context for Kohn’s theorem unveils previously inaccessible internal degrees of freedom of Landau electrons, opening up new realms of ultrafast quantum control for electrons.
Main
Controlling superpositions of electronic quantum states has been a paradigm of fundamental physics and quantuminformation technology^{19}. Sophisticated protocols have been implemented in atomic gases^{20}, whereas solids represent a more challenging environment owing to complicated manybody interactions. Quantum manipulation has been fairly successful in atomlike singleparticle systems, such as quantum dots^{21}, whereas controlling interacting manybody systems is still in its infancy. Following Lev Landau’s suggestion to combine collective properties of an interacting particle system into quasiparticles^{1}, researchers effectively analyse crystal electrons and holes^{4,5}, excitons^{6,9}, dropletons^{11}, polarons^{7}, magnons^{22} or Cooper pairs^{12}. In particular, ultrashort pulses in the terahertz (1 THz = 10^{12} Hz) spectral range have become a powerful tool to probe^{4,6,7,12,17} and control^{8,18,22,23} quasiparticle excitations, such as intraexcitonic transitions^{6} or superconductor Higgs bosons^{12}.
For most quasiparticle excitations, Coulomb scattering leads to coherence times in the range of a few to a few hundred femtoseconds, which is too short for most quantumlogic operations. However, the cyclotron resonance (CR) in a twodimensional electron gas (2DEG) represents a unique exception^{2,13,14,15,16,17,18}. In a magnetically biased 2DEG, Landau electrons emerge as elementary quasiparticles. According to Kohn’s theorem the quantization energy of the harmonic spectrum of Landau levels (LLs) defined by the cyclotron frequency ν_{c} is unaffected by Coulomb scattering^{2}. Although the absence of Coulomb complications warrants a longlived CR coherence, the accompanying perfectly harmonic behaviour has an apparent downside: multiphoton excitations drive perfect Landau electrons only to climb up the LLs, preventing Rabi flopping, which is the desired elementary process in many quantumlogic operations^{19}.
Here, we investigate the possibility of inducing a distinctly anharmonic CR response with the goal of coherently controlling transitions among a few selected LLs. Intense terahertz pulses coherently drive the state of a 2DEG up the Landau ladder by as much as six rungs within a single cycle of the carrier wave, and induce anharmonicities which render LL transitions distinguishable. Furthermore, strong coherent nonlinearities, such as four and sixwave mixing signals are observed. Although Kohn’s theorem has been violated via phonons^{24,25}, nonparabolic electron dispersion^{26}, or in a Bose–Einstein condensate close to a Feshbach resonance^{27}, our experiment–theory comparison unambiguously reveals a nonlinear response that stems from the Coulomb interaction itself, between the electrons and the ionic background.
Our sample hosts two 30nm wide ndoped GaAs quantum wells (QWs; see Methods), each containing a 2DEG. To calibrate the system, phaselocked lowamplitude () terahertz pulses are transmitted through the structure. Polarization components parallel () and perpendicular () to the incident pulses are separately detected by electrooptic sampling (EOS), as a function of delay time t (Fig. 1a). Without magnetic bias, (Fig. 1c, black curve) closely follows the incident waveform (Supplementary Discussion 1). When a magnetic bias B = 3.5 T is applied perpendicularly to the 2DEG plane, longlived trailing oscillations emerge owing to the CR with ν_{c} = 1.45 THz (Fig. 1c, blue curve)^{18}. Here the weak terahertz field induces a polarization between LLs n = 0〉 (filling factor f = 0.95) and n = 1〉 (Fig. 1b), reemitting a circularly polarized terahertz wave (Fig. 1c, inset) whose coherence time τ_{c} = 9 ps is probably limited by superradiant decay^{17}.
We then drive the calibrated system with intense terahertz pulses generated by tiltedpulsefront optical rectification^{23}. The amplitude of the reemitted field (Fig. 2a) increases with , but the temporal waveform remains similar up to . For , the trailing oscillations slow down and decay more rapidly until the interLL polarization essentially follows the driving field, for , as the coherence decays almost instantly. The spectrum of the reemitted field (Fig. 2b), which is initially centred at ν_{c} = 1.45 THz, redshifts and broadens with increasing until its shape converges to the spectrum of the driving pulse. This pronounced field dependence is irreconcilable with the linear response of a harmonic oscillator, indicating that the cyclotron transition is driven into a strongly nonlinear regime by the intense terahertz transients. Figure 2c shows the dephasing time τ_{c} as a function of the terahertz amplitude . Starting at a lowfield value of τ_{c} = 9 ps, the decay time drops abruptly for , and ultimately approaches τ_{c} = 0.5 ps.
Kohn’s theorem implies that the purely repulsive electron–electron Coulomb interaction cannot produce observable nonlinearities in a 2DEG. However, in a real modulationdoped system, the 2DEG always resides on a positive background charge that holds the electrons together as a homogeneous gas by preventing the formation of a Wigner crystal^{28} or the escape of electrons to the edges of the sample. As the attractive electron–ion interaction is not limited by Kohn’s theorem, it can indeed induce CR nonlinearities.
The experiments are analysed with clusterexpansionbased manybody computations^{3} including the nonparabolic dispersion of Landau electrons, their coupling to terahertz fields as well as phonons, and the Coulombic electron–electron and electron–ion interaction. Technically, a_{λ} (a_{λ}^{†}) annihilates (creates) an electron in a LL λ such that p_{v}^{λ} = 〈a_{λ}^{†}a_{v}〉 identifies both microscopic polarization (λ ≠ v) and electron distributions (λ = v) among different LLs. The terahertz excitation yields an exact quantum dynamics defined by the semiconductor Bloch equations (SBEs; ref. 3) where E_{v}^{λ} is the transition energy between two LLs with a dipole d_{v}^{λ}, and is the Rabi energy of the terahertz field . The Coulomb interaction renormalizes both E_{v}^{λ} and Ω_{v}^{λ}, and introduces a contribution C_{v}^{λ} whose electron–electron (electron–ion) part contains a repulsive (attractive) Coulomb matrix element V (W) between four LLs (LLs and ions), yielding Coulomb sums of type and Besides these Hartree–Fock contributions, the SBEs couple to twoparticle correlations^{3,29}, inducing phonon scattering and excitationinduced dephasing (EID) stemming from Coulomb scattering of polarization with excited electrons^{3,29,30}. Longitudinal optical (LO) phonons efficiently dephase polarization involving LLs with energy above ℏω_{LO} = 36 meV, and the EID is described with an excitationdependent model explained in Supplementary Discussion 10.
The principal quantum number n of a LL λ = (n, l) defines its energy, while an additional quantum number l defines angular momentum and introduces infinite degeneracy among LLs. We use 1,500 LLs in solving equation (1), which is shown to yield a converged macroscopic response (Supplementary Discussion 8). The SBEs then contain 10^{10} Coulombsum calculations at each time step, rendering simulations extremely demanding. Nevertheless, our computations include even the principal EID effects; at low excitations, all l states are occupied evenly such that the Coulombic in and outscattering fully compensate each other, yielding vanishing EID. However, strong terahertz excitations generate peaks in the l distribution, unbalancing the Coulomb scattering and yielding detectable EID proportional to the density of excited electrons.
Figure 2d compares the transmitted field for and 5.7 kV cm^{−1} as obtained from our full theory (solid curves) with the results computed without Coulomb interaction (dashed curves) and a classical calculation with a nonparabolic band and constant τ_{c} (shaded area). The weakintensity result () is well reproduced by all theoretical models, verifying the applicability of Kohn’s theorem in this case. However, for the stronger field , only the full quantum theory can reproduce the experimental decay of and the redshift of ν_{c}. As in the experiment, the indicated oscillation minimum of (red vertical line) is delayed by 0.1 ps with respect to the position expected for constant ν_{c} = 1.45 THz (blue vertical line), which manifests a violation of Kohn’s theorem. Even though the simplified models neglect Coulomb interaction, they do yield a change of ν_{c} implying that the nonparabolic dispersion contributes to the softening of the CR. However, they predict incorrect decay dynamics because the classical computation lacks the phase diffusion among LLs, present in the quantum calculations.
Our full theory also explains the thresholdlike onset of dephasing above (Fig. 2c, red solid line), attributing this effect to an efficient population transfer. Whereas low terahertz amplitudes (0.7 kV cm^{−1}) excite only a few percent of charge carriers into LL n = 1〉 (Fig. 2e, blue bars), strong pulses drive coherent ladder climbing. At (violet bars), more than half of the electrons are excited and distributed up to n = 6〉, where rapid dephasing and relaxation by LO phonon emission sets in. The terahertzinduced population transfer is faster than relaxation such that even LLs above the LO phonon energy are populated. Remarkably, terahertz transients with (red bars) largely depopulate n = 0〉 and prepare a compact superposition of eigenstates, peaking at n = 5〉. This wavepacket rapidly loses its coherence because 25% of its weight is located above the phonon energy. Thus, the abrupt onset of dephasing above arises from the combination of a compact wavepacket distribution and a sharp threshold for phonon scattering. Furthermore, the experiment corroborates that coherent state inversion works even for a massive manybody system and corresponding LLs should be well suited for coherent quantum control.
To test these perspectives systematically, two phaselocked terahertz pulses A and B, polarized in x and y directions, respectively, are focused onto the sample for fieldresolved twodimensional (2D) terahertz spectroscopy. The transmitted total terahertz field is resolved in amplitude and phase as a function of EOS time, t, and the delay between the two incident pulses, τ (Fig. 3a). Subsequently, the response to individual pulses A and B is subtracted to isolate the nonlinear polarization induced by both pulses (refs 31,32), which vanishes for a strictly harmonic CR. In contrast, Fig. 3b–e shows strong nonlinear signals for all peak amplitudes between and 5.7 kV cm^{−1} while is kept constant. Constantphase lines of pulse B appear vertically in the 2D data maps, whereas the phase fronts of pulse A occur under a 45° angle (see Supplementary Discussion 3).
For (Fig. 3b), is strongly modulated along τ with a period Δτ_{1} = 1/ν_{c} = 0.69 ps, evidencing that the nonlinear interaction is coherently mediated by ν_{c}. This feature persists even when the incident pulses do not overlap in time because the coherence is stored in the LL system. For larger (Fig. 3c–e), an additional modulation with a period of Δτ_{2} = 2/ν_{c} = 0.34 ps, corresponding to twice the cyclotron frequency, emerges and becomes dominant for . Increasing the field to (Fig. 3e) accelerates decoherence, leaving only weak modulations during temporal overlap of both pulses, around t + τ = 0. Supplementary Discussion 2 shows similar results for a singlewell structure, demonstrating the universality of these nonlinearities.
A 2D Fourier transformation allows us to disentangle different nonlinear optical processes contributing to . Figure 3h–k shows the spectral amplitude as a function of the frequencies ν_{t} and ν_{τ} associated with the EOS time and the relative delay between the pulses, respectively^{31,32}. Several distinct maxima occur at integer multiples of ν_{c}: the peak located at (ν_{t}, ν_{τ}) = (ν_{c}, 0) (Fig. 3h, arrow ‘k_{p1}’) represents a pump–probe signal where pulse A (B) acts as a pump (probe) pulse, whereas the maximum at (ν_{c}, −ν_{c}) (arrow ‘k_{p2}’) is a pump–probe signal for which pulse A and B switch their roles. In addition, strong fourwave mixing (FWM, ‘k_{41}’, ‘k_{42}’) emerges. As is increased to 2.9 kV cm^{−1} (Fig. 3i), the FWM signal at k_{42} (red circle) surpasses the diagonal pump–probe signal k_{p2} in amplitude. Even sixwave mixing (SWM, ‘k_{61}’, orange circle) occurs. For (Fig. 3j), FWM is the dominant contribution at nonzero ν_{τ} and explains the Δτ_{2}periodic response in the timedomain (Fig. 3d). Finally, the strongly reduced coherence time for (Fig. 3k) suppresses wavemixing processes, and the incoherent pump–probe signal at k_{p1} dominates.
To identify the origin of these nonlinearities, we apply our manybody theory to the 2D scenario using the experimental terahertz waveforms. The timedomain data of the full calculation for weak (, Fig. 3f) and strong (, Fig. 3g) pulses explain our experiment, reproducing the Δτ_{1} and Δτ_{2}periods in , as well as their relative weights and decay. Correspondingly, the frequency maps (Fig. 3l, m) exhibit multiwave mixing features in quantitative agreement with the experiment.
The nonlinearities are a consequence of the breakdown of the translational invariance and thus a violation of Kohn’s theorem which follows either from the nonparabolic electron dispersion or intrinsic Coulombinteraction effects. A switchoff analysis allows us to determine the microscopic origin of nonlinearities. Figure 4 shows slices of at constant t = 3.3 ps (see Fig. 3f, g, vertical lines). For (Fig. 4a), the Δτ_{1}period dominates the dynamics of . When the nonparabolicity is switched off (red line), is strongly suppressed. However, eliminating Coulomb effects (dashed line) yields almost unchanged , revealing the nonparabolic band structure as the dominant nonlinearity for low fields. For (Fig. 4b), the nonlinear response is almost one order of magnitude stronger than for . Now the parabolic band approximation (red line) produces virtually the same result as the full computation (black line), showing that the Coulomb interaction dominates for strong fields. Further insights into the intriguing Coulomb effects are gained by computing without EID (dashed line, see Supplementary Discussion 10); only the computation with EID explains the experiment, which indicates new possibilities to utilize ultrafast nonlinear switching and decay channels.
Lifting the protection of Kohn’s theorem through nonperturbative excitations of a Landau system opens the door to a rich spectrum of nonlinearities, finetuned by the terahertz driving field. Highorder nonlinear processes promoted by the intrinsically large dipole moments of the CR hold the prospect for terahertz quantum control, even at lowfield amplitudes. Future quantumlogic devices may combine cyclotron transitions with metamaterials to further lower the required field amplitude and the footprint of a single qubit. Onchip electronic terahertz sources may even pave the way towards scalable allelectrical quantum devices. Generally, the principle of accessing internal degrees of freedom in a manybody quantum system through Coulomb correlations suggests that the role of massive manybody interactions, thus far considered detrimental to quantum control, will have to be reassessed.
Methods
Experiment.
Our sample, grown by molecular beam epitaxy, hosts two 30nm wide GaAs quantum wells separated by a 10nm wide Al_{0.24}Ga_{0.76}As barrier. In the 2DEGs, a carrier density ρ_{e} = 1.6 × 10^{11} cm^{−2} per 2DEG is realized by two remote δdoping layers, symmetrically enclosing the 2DEGs on both sides. Al_{0.24}Ga_{0.76}As spacers of 72 nm thickness serve as barriers between the 2DEGs and the doping layers (see Supplementary Discussion 2). The resulting high confinement potential for the electrons of 220 meV and the symmetric layout yield two independent electron gases with a high electron mobility of μ = 4.6 × 10^{6} cm^{2} V^{−1} s^{−1}, as verified through van der Pauw measurements.
A Ti:sapphire laser amplifier (centre wavelength: 800 nm, pulse energy: 5.5 mJ, repetition rate: 3 kHz, pulse duration: 33 fs) is used to generate intense fewcycle terahertz fields by tiltedpulsefront optical rectification in a cryogenically cooled LiNbO_{3} crystal^{23}. In a second optical branch, a small portion of the laser energy drives optical rectification in a 180μmthick (110)cut GaP crystal. Both terahertz pulses are collinearly focused onto the sample, which is mounted in a magnetooptical cryostat at a constant temperature of 4.3 K. The superconducting magnet provides homogeneous fields which are polarized perpendicularly to the quantum well plane and are tunable between 0 and 5 T. The transmitted terahertz pulses pass a rotatable wiregrid polarizer and are focused onto a 0.5mmthick (110)cut ZnTe crystal for polarizationresolved electrooptic detection covering the frequency window between 0.1 and 3.0 THz. Two mechanical delay stages allow us to independently vary the delay time τ between the maxima of the two terahertz transients labelled A and B, and the electrooptic sampling time t. Differential detection is employed through two mechanical choppers individually modulating the pulses at subharmonics of the laser repetition rate, which allows us to clearly isolate feeble terahertz electric fields of less than 0.1 V cm^{−1} from a background of the order of 10 kV cm^{−1} (see Supplementary Discussion 3 for details).
Theory.
We describe Landau electrons in the static magnetic field using the minimal substitution Hamiltonian where m_{e} is the effective electron mass and e denotes the elementary charge. The vector potential A_{stat} is employed in the symmetric gauge and contains the magnetic bias perpendicular to the sample surface. The Hamiltonian (2) yields the Landau wavefunctions, cyclotron frequency, and characteristic length scales on which we base our manybody theory. The full manybody Hamiltonian contains the noninteracting contributions , into which we incorporate trivial bandstructure effects, and the interaction term describing the light–matter coupling. The complex electron–electron and electron–ion interactions, fully described in Supplementary Discussion 4, are implemented through Here, a_{n, l}^{†} and a_{n, l} represent the creation and annihilation, respectively, of Landau electrons with principal quantum number n and angular momentum l in the basis determined through equation (3), ρ_{ion} is the uniformly distributed positive charge density originating from the dopant ions, and r_{c} is the radius of the classical Landau orbit. The Coulomb matrix elements V are at the core of the manybody dynamics of the highly excited Landau system and define the numerical challenge, linking electrons in Landau levels through 2.25 × 10^{6} terahertzinduced transition amplitudes. We solve these dynamics via the semiconductor Bloch equations (SBEs; ref. 3) which yield the relevant experimental quantities, such as the polarization and occupation densities, accounting for LO phonon scattering as well as EID, as fully detailed in Supplementary Discussion 10 together with the explicit form of the SBEs. We propagate the related integrodifferential equations with a fourthorder Runge–Kutta method, requiring more than 10^{10} calculations owing to the Coulomb coupling for each time step, which makes the simulations extremely demanding. Nevertheless, this formidable challenge is executable via an efficient parallelcomputing implementation. To study the limitations of Kohn’s theorem, we also perform a classical calculation with a nonparabolic energy dispersion and a constant decay, the details of which are given in Supplementary Discussion 4.
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Acknowledgements
The work in Regensburg was supported by the European Research Council through grant no. 305003 (QUANTUMsubCYCLE) and the Deutsche Forschungsgemeinschaft (LA 3307/11, HU 1598/21, BO 3140/31, and Collaborative Research Center SFB 689). The work at the University of Marburg was supported by the Deutsche Forschungsgemeinschaft through SFB 1083 and grant KI 917/22 (M.K.), and the Alexander von Humboldt foundation (J.E.S.).
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Department of Physics, University of Regensburg, 93040 Regensburg, Germany
 T. Maag
 , A. Bayer
 , S. Baierl
 , M. Hohenleutner
 , T. Korn
 , C. Schüller
 , D. Schuh
 , D. Bougeard
 , C. Lange
 & R. Huber
Department of Physics, University of Marburg, 35032 Marburg, Germany
 M. Mootz
 , J. E. Sipe
 , S. W. Koch
 & M. Kira
Department of Physics and Institute for Optical Sciences, University of Toronto, 60 St George St., Toronto, Ontario M5S 1A7, Canada
 J. E. Sipe
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Contributions
T.M., A.B., M.M. and C.L. contributed equally to this work. C.L., T.M., M.K., S.W.K. and R.H. conceived the study. T.M., C.L., A.B., S.B., M.H., T.K., C.S., D.B. and R.H. carried out the experiment and analysed the data. A.B., D.S. and D.B. prepared the sample. M.M., J.E.S., S.W.K. and M.K. developed the quantummechanical model and carried out the computations. C.L., T.M., M.M., S.W.K., M.K. and R.H. wrote the manuscript. All authors discussed the results.
Competing interests
The authors declare no competing financial interests.
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Correspondence to C. Lange.
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