In solids, the high density of charged particles makes many-body interactions a pervasive principle governing optics and electronics1,2,3,4,5,6,7,8,9,10,11,12. However, Walter Kohn found in 1961 that the cyclotron resonance of Landau-quantized electrons is independent of the seemingly inescapable Coulomb interaction between electrons2. Although this surprising theorem has been exploited in sophisticated quantum phenomena13,14,15, such as ultrastrong light–matter coupling16, superradiance17 and coherent control18, the complete absence of nonlinearities excludes many intriguing possibilities, such as quantum-logic protocols19. Here, we use intense terahertz pulses to drive the cyclotron response of a two-dimensional electron gas beyond the protective limits of Kohn’s theorem. Anharmonic Landau ladder climbing and distinct terahertz four- and six-wave 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.
Controlling superpositions of electronic quantum states has been a paradigm of fundamental physics and quantum-information technology19. Sophisticated protocols have been implemented in atomic gases20, whereas solids represent a more challenging environment owing to complicated many-body interactions. Quantum manipulation has been fairly successful in atom-like single-particle systems, such as quantum dots21, whereas controlling interacting many-body systems is still in its infancy. Following Lev Landau’s suggestion to combine collective properties of an interacting particle system into quasiparticles1, researchers effectively analyse crystal electrons and holes4,5, excitons6,9, dropletons11, polarons7, magnons22 or Cooper pairs12. In particular, ultrashort pulses in the terahertz (1 THz = 1012 Hz) spectral range have become a powerful tool to probe4,6,7,12,17 and control8,18,22,23 quasiparticle excitations, such as intra-excitonic transitions6 or superconductor Higgs bosons12.
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 quantum-logic operations. However, the cyclotron resonance (CR) in a two-dimensional electron gas (2DEG) represents a unique exception2,13,14,15,16,17,18. 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 νc is unaffected by Coulomb scattering2. Although the absence of Coulomb complications warrants a long-lived 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 quantum-logic operations19.
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 six-wave mixing signals are observed. Although Kohn’s theorem has been violated via phonons24,25, non-parabolic electron dispersion26, or in a Bose–Einstein condensate close to a Feshbach resonance27, 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 30-nm wide n-doped GaAs quantum wells (QWs; see Methods), each containing a 2DEG. To calibrate the system, phase-locked low-amplitude () terahertz pulses are transmitted through the structure. Polarization components parallel () and perpendicular () to the incident pulses are separately detected by electro-optic 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, long-lived trailing oscillations emerge owing to the CR with νc = 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), re-emitting a circularly polarized terahertz wave (Fig. 1c, inset) whose coherence time τc = 9 ps is probably limited by superradiant decay17.
We then drive the calibrated system with intense terahertz pulses generated by tilted-pulse-front optical rectification23. The amplitude of the re-emitted 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 inter-LL polarization essentially follows the driving field, for , as the coherence decays almost instantly. The spectrum of the re-emitted field (Fig. 2b), which is initially centred at νc = 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 τc as a function of the terahertz amplitude . Starting at a low-field value of τc = 9 ps, the decay time drops abruptly for , and ultimately approaches τc = 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 modulation-doped 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 crystal28 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 cluster-expansion-based many-body computations3 including the non-parabolic dispersion of Landau electrons, their coupling to terahertz fields as well as phonons, and the Coulombic electron–electron and electron–ion interaction. Technically, aλ (aλ†) annihilates (creates) an electron in a LL λ such that pvλ = 〈aλ†av〉 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 Evλ is the transition energy between two LLs with a dipole dvλ, and is the Rabi energy of the terahertz field . The Coulomb interaction renormalizes both Evλ and Ωvλ, and introduces a contribution Cvλ 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 two-particle correlations3,29, inducing phonon scattering and excitation-induced dephasing (EID) stemming from Coulomb scattering of polarization with excited electrons3,29,30. Longitudinal optical (LO) phonons efficiently dephase polarization involving LLs with energy above ℏωLO = 36 meV, and the EID is described with an excitation-dependent 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 1010 Coulomb-sum 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 out-scattering 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 non-parabolic band and constant τc (shaded area). The weak-intensity 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 νc. 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 νc = 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 νc implying that the non-parabolic 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 threshold-like 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 terahertz-induced 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 many-body system and corresponding LLs should be well suited for coherent quantum control.
To test these perspectives systematically, two phase-locked terahertz pulses A and B, polarized in x- and y- directions, respectively, are focused onto the sample for field-resolved two-dimensional (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. Constant-phase 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 Δτ1 = 1/νc = 0.69 ps, evidencing that the nonlinear interaction is coherently mediated by νc. 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 Δτ2 = 2/νc = 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 single-well 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 νt and ντ associated with the EOS time and the relative delay between the pulses, respectively31,32. Several distinct maxima occur at integer multiples of νc: the peak located at (νt, ντ) = (νc, 0) (Fig. 3h, arrow ‘kp1’) represents a pump–probe signal where pulse A (B) acts as a pump (probe) pulse, whereas the maximum at (νc, −νc) (arrow ‘kp2’) is a pump–probe signal for which pulse A and B switch their roles. In addition, strong four-wave mixing (FWM, ‘k41’, ‘k42’) emerges. As is increased to 2.9 kV cm−1 (Fig. 3i), the FWM signal at k42 (red circle) surpasses the diagonal pump–probe signal kp2 in amplitude. Even six-wave mixing (SWM, ‘k61’, orange circle) occurs. For (Fig. 3j), FWM is the dominant contribution at non-zero ντ and explains the Δτ2-periodic response in the time-domain (Fig. 3d). Finally, the strongly reduced coherence time for (Fig. 3k) suppresses wave-mixing processes, and the incoherent pump–probe signal at kp1 dominates.
To identify the origin of these nonlinearities, we apply our many-body theory to the 2D scenario using the experimental terahertz waveforms. The time-domain data of the full calculation for weak (, Fig. 3f) and strong (, Fig. 3g) pulses explain our experiment, reproducing the Δτ1- and Δτ2-periods in , as well as their relative weights and decay. Correspondingly, the frequency maps (Fig. 3l, m) exhibit multi-wave 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 non-parabolic electron dispersion or intrinsic Coulomb-interaction effects. A switch-off 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 Δτ1-period dominates the dynamics of . When the non-parabolicity is switched off (red line), is strongly suppressed. However, eliminating Coulomb effects (dashed line) yields almost unchanged , revealing the non-parabolic 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 non-perturbative excitations of a Landau system opens the door to a rich spectrum of nonlinearities, fine-tuned by the terahertz driving field. High-order nonlinear processes promoted by the intrinsically large dipole moments of the CR hold the prospect for terahertz quantum control, even at low-field amplitudes. Future quantum-logic devices may combine cyclotron transitions with metamaterials to further lower the required field amplitude and the footprint of a single qubit. On-chip electronic terahertz sources may even pave the way towards scalable all-electrical quantum devices. Generally, the principle of accessing internal degrees of freedom in a many-body quantum system through Coulomb correlations suggests that the role of massive many-body interactions, thus far considered detrimental to quantum control, will have to be reassessed.
Our sample, grown by molecular beam epitaxy, hosts two 30-nm wide GaAs quantum wells separated by a 10-nm wide Al0.24Ga0.76As barrier. In the 2DEGs, a carrier density ρe = 1.6 × 1011 cm−2 per 2DEG is realized by two remote δ-doping layers, symmetrically enclosing the 2DEGs on both sides. Al0.24Ga0.76As 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 × 106 cm2 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 few-cycle terahertz fields by tilted-pulse-front optical rectification in a cryogenically cooled LiNbO3 crystal23. In a second optical branch, a small portion of the laser energy drives optical rectification in a 180-μm-thick (110)-cut GaP crystal. Both terahertz pulses are collinearly focused onto the sample, which is mounted in a magneto-optical 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 wire-grid polarizer and are focused onto a 0.5-mm-thick (110)-cut ZnTe crystal for polarization-resolved electro-optic 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 electro-optic sampling time t. Differential detection is employed through two mechanical choppers individually modulating the pulses at sub-harmonics 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).
We describe Landau electrons in the static magnetic field using the minimal substitution Hamiltonian where me is the effective electron mass and e denotes the elementary charge. The vector potential Astat 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 many-body theory. The full many-body Hamiltonian contains the non-interacting 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, an, l† and an, 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), ρion is the uniformly distributed positive charge density originating from the dopant ions, and rc is the radius of the classical Landau orbit. The Coulomb matrix elements V are at the core of the many-body dynamics of the highly excited Landau system and define the numerical challenge, linking electrons in Landau levels through 2.25 × 106 terahertz-induced 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 integro-differential equations with a fourth-order Runge–Kutta method, requiring more than 1010 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 parallel-computing implementation. To study the limitations of Kohn’s theorem, we also perform a classical calculation with a non-parabolic energy dispersion and a constant decay, the details of which are given in Supplementary Discussion 4.
The work in Regensburg was supported by the European Research Council through grant no. 305003 (QUANTUMsubCYCLE) and the Deutsche Forschungsgemeinschaft (LA 3307/1-1, HU 1598/2-1, BO 3140/3-1, 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 917/2-2 (M.K.), and the Alexander von Humboldt foundation (J.E.S.).
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Nature Photonics (2016)