Experimental Reconstruction of Bloch wavefunctions
J.B. Costello, S.D. O'Hara, Q. Wu, D. C. Valovcin, L. N. Pfeiffer, K.W. West, M.S. Sherwin
1 Title: Experimental Reconstruction of Bloch wavefunctions
Authors:
J. B. Costello *, S. D. O’Hara *, Q. Wu *, D. C. Valovcin , L. N. Pfeiffer , K. W. West , and M. S. Sherwin Affiliations: Physics Department and Institute for Terahertz Science and Technology, University of California, Santa Barbara, CA 93106. Mathworks, 1 Apple Hill Drive, Natick, MA 01760. Department of Electrical Engineering, Princeton University, Princeton, NJ 08544. *These authors contributed equally to this work. † e-mail: [email protected] Anticipated breakthroughs in solid-state quantum computing will rely on achieving unprecedented control over the wave-like states of electrons in crystalline materials. For example, an international effort to build a quantum computer that is topologically protected from decoherence is focusing on carefully engineering the wave-like states of electrons in hybrid devices that proximatize an elemental superconductor and a semiconductor nanostructure . However, more than 90 years after Bloch derived the functional forms of electronic waves in crystals (now known as Bloch wavefunction) rapid scattering processes have so far prevented their direct experimental reconstruction, even in bulk materials. In high-order sideband generation (HSG) , electrons and holes generated in semiconductors by a near-infrared (NIR) laser are accelerated to high kinetic energy by a strong terahertz field, and recollide to emit NIR sidebands before they are scattered. Here we reconstruct the Bloch wavefunctions of two types of holes in gallium arsenide by experimentally measuring sideband polarizations and introducing an elegant theory that ties those polarizations to quantum interference between different recollision pathways. Because HSG can, in principle, be observed from any direct-gap semiconductor or insulator, we expect the method introduced in this Article can be used to reconstruct Bloch wavefunctions in a large class of bulk and nanostructured materials, accelerating the development of topologically-protected quantum computers as well as other important electronic and optical technologies. Bloch’s theorem tells us both the Bloch wavefunctions, and the spectrum of electronic energies (the "band structure") . Knowledge about both the Bloch wavefunctions and band structure are essential to calculating the response of crystalline solids to most external stimuli. The band structure of many crystalline materials can be experimentally reconstructed from angle-resolved photoemission spectroscopy (ARPES) of electrons emitted from their surfaces . ARPES enables determination of the energies of electronic waves as functions of their wavelengths and directions of propagation. However, there are no comparably direct methods to reconstruct Bloch wavefunctions. As a result, estimates of Bloch wavefunctions typically depend on parameters derived from fits of complex models to a few pieces of experimental data—such as the orbital frequency of an electron in a magnetic field (cyclotron resonance) —that are sensitive only to averages over a range of electronic wave propagation directions and wavelengths. A key obstacle to directly probing Bloch wavefunctions in solids has been that, unlike molecules, where reconstructed electron wavefunctions have been reported , electronic waves in solids are typically distorted in a few picoseconds by scattering. Recently, strong laser fields have been used to significantly accelerate electronic waves in solids before they are scattered. For example, high-harmonic generation (HHG) has been demonstrated in solids and led to an alternative method to probe band structures . However, complicated interference between quantum pathways of electronic waves across multiple bands in HHG experiments hinders the reconstruction of Bloch wavefunctions. High-order sideband generation
In this paper, we present a direct experimental reconstruction of Bloch wavefunctions of holes in bulk gallium arsenide (GaAs) using high-order sideband generation (HSG) . In HSG, a relatively weak near-infrared (NIR) laser with frequency 𝑓 NIR and a strong laser with terahertz (THz) frequency 𝑓 THz simultaneously interact with a semiconductor, resulting in the emission of sideband photons with frequencies 𝑓 SB = 𝑓 NIR + 𝑛 ∙ 𝑓
THz where 𝑛 is an integer. An HSG spectrum from bulk GaAs at 60 K is shown in Fig. 1a. In the experiment, a 100 mW NIR laser, and 2.01±0.13 mJ, 40 nanosecond, 0.447 ± 0.001 THz pulses generated by UCSB’s MM-Wave 3 Free-Electron Laser were linearly polarized and collinearly focused on a 500 nm GaAs epilayer (see Fig. 1a inset). The THz electric field strength in the epilayer was 70±2 kV/cm (Methods). In GaAs, HSG can be described by the following three-step process (see Fig. 1b, Methods). First, electrons (E) and two species of holes—light holes (LH) and heavy holes (HH)—are created by the NIR laser. Second, the E-LH and E-HH pairs are driven apart and then back towards each other along the direction defined by the THz field (Fig. 1b). Crucially, during this acceleration phase, the Bloch waves associated with the E-LH and E-HH pairs interfere with each other. Third, they recollide with significant kinetic energy and emit sideband photons. Information about the Bloch wavefunctions sampled by electrons and holes on their journeys through the Brillouin zone is imprinted on the polarization state of each sideband , which we measured by Stokes polarimetry (Methods). Because light from a particular sideband can be associated with a quasi-momentum that is controlled by the THz field, its polarization will be different from the incoherent light emitted at the same energy in the absence of a strong THz field (photoluminescence), which is a superposition of emission from electron-hole pairs with all quasimomenta satisfying energy conservation. The linear orientation angle, α , and the ellipticity angle, , for each sideband are shown in Fig. 1c for four different NIR polarizations. The polarizations of sidebands depend on the sideband index 𝑛 and the NIR polarization in a manifestation of dynamical birefringence . Although sideband intensities have a highly nonlinear dependence on THz power, they are proportional to the NIR power if it is sufficiently small . All data reported here were taken in this regime of linear NIR response. In this linear regime, the sideband polarization can be mapped onto the polarization state of the NIR laser by a dynamical Jones matrix 𝑇 , defined in a basis of circularly-polarized fields 𝜎 ± (with helicity ± 1) as (𝐸 +,𝑛 𝐸 −,𝑛 ) = (𝑇 ++,𝑛 𝑇 +−,𝑛 𝑇 −+,𝑛 𝑇 −−,𝑛 ) (𝐸 +,𝑁𝐼𝑅 𝐸 −,𝑁𝐼𝑅 ) , where 𝐸 ±,𝑛 and 𝐸 ±,NIR respectively denote the 𝜎 ± components of the electric field associated with the n -th sideband and NIR laser, and T ±±,n denote the dynamical Jones matrix elements associated with the n -th sideband. 𝑇 -matrix elements were determined by measuring the sideband polarizations for four different linear NIR laser polarizations. The Luttinger Hamiltonian and Dynamical Jones Matrices
In order to understand the physics underlying each 𝑇 -matrix element, it is necessary to consider the spins of electrons and holes. The four recollision pathways from the excitations generated by the 𝜎 NIR− component of the NIR laser, are shown in Fig. 2. The electrons have spin 1/2, while the heavy holes (HHs) and light holes (LHs) have total spin 3/2. Driven by the THz field, an electron-hole pair acquires a dynamic phase 𝐴 𝐻𝐻(𝐿𝐻) (𝑡 ′ , 𝑡) = − ∫ 𝑑𝑡′′(𝐸 𝑐 [𝒌(𝑡 ′′ )] − 𝐸 𝐻𝐻(𝐿𝐻) [𝒌(𝑡 ′′ )])/ℏ 𝑡𝑡 ′ , where 𝐸 𝑐 and 𝐸 𝐻𝐻(𝐿𝐻) are the energies associated with the E and HH (LH) bands shown schematically in Fig. 1b, 𝒌 is the quasimomentum, and ℏ is the reduced Planck’s constant (Methods, Supplementary Discussion). The spin ±1/2 of the electron does not change during acceleration. Because the Bloch wavefunctions in both HH and LH bands are superpositions of states with spin ±1/2 and ±3/2, the 𝜎 NIR− component can generate sidebands with either 𝜎 HSG− or 𝜎 HSG+ while satisfying angular-momentum conservation, giving rise to dynamical Jones matrix elements 𝑇 −−,𝑛 and 𝑇 +−,𝑛 respectively. Similar recollision pathways follow from the excitations generated by the 𝜎 NIR+ component, giving rise to 𝑇 −+,𝑛 and 𝑇 ++,𝑛 (Methods, Extended data Fig. 4). 4 Properties of dynamical Jones matrices can be derived from the Luttinger Hamiltonian , which describes the HH and LH states. We tune the NIR laser just below the band gap and direct the THz field to propagate along the z -axis to ensure that the electrons and holes have no z -component of quasimomentum 𝒌 . In this case the Luttinger Hamiltonian takes a block diagonal form 𝐻 𝑣± (𝒌) = − ℏ 𝑘 [𝛾 𝜏 − 2𝛾 𝒏 ± ∙ 𝝉] =− ℏ 𝑘 ( 𝛾 + 𝛾 −√3(𝛾 sin(2𝜃) ± 𝑖𝛾 cos(2𝜃))−√3(𝛾 sin(2𝜃) ∓ 𝑖𝛾 cos(2𝜃)) 𝛾 − 𝛾 ) (Eq. 1) where 𝜏 is the identity matrix, 𝝉 is the vector of Pauli matrices, 𝜃 is the angle between the THz field and the [110] crystal direction (Fig. 3d inset), 𝛾 , 𝛾 , and 𝛾 are the scalar Luttinger parameters, 𝑚 is the electron rest mass, and 𝒏 ± is 𝒏 ± = ( √32 sin(2𝜃) , ∓ √3𝛾 cos(2𝜃) , − ) (Eq. 2) The Bloch wavefunctions are found by diagonalizing Eq. 1 and only depend on the 𝒏 ± ∙ 𝝉 term because the first term is proportional to the identity. Since 𝒏 ± depends only on crystal angle 𝜃 and 𝛾 𝛾 ⁄ , an experimental measurement of 𝛾 𝛾 ⁄ allows the reconstruction of the Bloch wavefunctions. Even for 𝑘 𝑧 ≠ 0 , although the Luttinger Hamiltonian is not block diagonal, knowing 𝛾 𝛾 ⁄ is still sufficient to reconstruct the Bloch wavefunctions. Extraction of 𝜸 𝜸 ⁄ and Reconstruction of Bloch Wavefunctions We use ratios of T -matrix elements to check the validity of the theory and measure 𝛾 𝛾 ⁄ . Because the diagonal elements of Eq. 1 are real, when sideband and NIR laser polarizations are the same, for each pathway that produces a 𝜎 HSG+ photon there is an equivalent pathway that produces a 𝜎 HSG− photon (Fig. 2, Methods and Extended data Fig. 4) through states related by time-reversal symmetry. Therefore, the ratio of diagonal dynamical Jones matrix elements for all sideband indices and crystal angles is 𝑇 ++,𝑛 (𝜃)𝑇 −−,𝑛 (𝜃) ≡ 𝜉 𝑛 (𝜃) = 1 (Eq. 3) Because the off-diagonal elements of Eq. 1 are complex, when sideband and NIR laser polarizations are different, each pathway that produces a 𝜎 HSG+ photon has an equivalent pathway that produces a 𝜎 HSG− photon with a complex-conjugated phase factor (Methods). Therefore, the ratio of off-diagonal dynamical Jones matrix elements for all sideband indices is 𝑇 +−,𝑛 (𝜃)𝑇 −+,𝑛 (𝜃) ≡ 𝜒 𝑛 (𝜃) = 𝛾 sin(2𝜃)−𝑖𝛾 cos(2𝜃)𝛾 sin(2𝜃)+𝑖𝛾 cos(2𝜃) (Eq. 4) The magnitude of 𝜒 𝑛 (𝜃) is 1 for all angles, but the argument depends on 𝛾 𝛾 ⁄ and 𝜃 . The experimentally measured values 𝜒 𝑛 (𝜃) and 𝜉 𝑛 (𝜃) at various 𝜃 are compared with the predictions of Eqs. 3 and 4 in Fig. 3 and Extended Data Fig. 5 using values for 𝛾 and 𝛾 recommended in . Within experimental error, |𝜒 𝑛 (𝜃)| and |𝜉 𝑛 (𝜃)| are 1, as predicted by Eqs. 3 and 4 (Fig. 3a). The arguments of 𝜒 𝑛 (𝜃) for eight different 𝜃 are independent of 𝑛 (Fig. 3b), lying within of the constant values predicted by Eq. 4 (dashed lines) for all 𝜃 except −45° . The values of 𝜒(𝜃) ≡ 〈𝜒 𝑛 (𝜃)〉 and 𝜉(𝜃) ≡ 〈𝜉 𝑛 (𝜃)〉 , where averages are over 𝑛 , are plotted at each 𝜃 in Figs. 3c and d. The magnitudes |𝜒(𝜃)| and |𝜉(𝜃)| are independent of 𝜃 , with a value of 1, as predicted by Eqs. 3 and 4 (Fig. 3c). The argument of 𝜒(𝜃) is plotted with respect to 𝜃 in Fig. 3d, and is close to the prediction provided by Eq. 4. Averaging the argument of 𝜒(𝜃) over experimentally sampled 𝜃 gives 𝛾 𝛾 ⁄ = 1.47 ± 0.48 , within experimental error of the value 1.42 recommended in . We attribute the deviations in measured 𝜒 𝑛 (𝜃) and 𝜉 𝑛 (𝜃) from 5 theoretical predictions, as well as much of the experimental error in the determination of 𝛾 𝛾 ⁄ , to small inhomogeneous strain in the GaAs membrane (Methods, Extended data Fig. 2). From 𝛾 𝛾 ⁄ , we reconstruct the Bloch wavefunctions of the Luttinger Hamiltonian in GaAs. For two coupled bands, the Bloch wavefunctions can be represented as spinors on a Bloch sphere . In the 𝑘 𝑧 = 0 plane, each block of the Luttinger Hamiltonian is a two-by-two matrix, whose eigenfunctions—the Bloch wavefunctions—depend on 𝜃 but not on |𝒌| = 𝑘 . Thus, in the 𝑘 𝑧 = 0 plane, for any 𝜃 , a single point on the Bloch sphere represents the Bloch wavefunctions for arbitrary 𝑘 . The closed black curves in the Northern and Southern hemispheres of the Bloch sphere in Fig. 4 represent the most likely Bloch wavefunctions consistent with our measured 𝛾 𝛾 ⁄ for the LH and HH, respectively. The North and South poles represent the states with spin -3/2 and +1/2, respectively. The Bloch wavefunctions for the degenerate partners of those represented in Fig. 4 are related by time-reversal symmetry. Discussion: dynamical and Berry phases and topological quantum computing.
Recent approaches to topological quantum computing rely on a detailed understanding of Bloch wavefunctions in semiconductors . One approach requires semiconductors or insulators with intrinsic topological properties defined by their so-called Berry connection and Berry curvature. Although Berry curvature measurements have been reported in systems with symmetries connecting other observables to the local Berry physics , measurements of these quantities remain challenging in most materials. Since the Bloch wavefunctions define the Berry connection and curvature , we are able to reconstruct the Berry physics of GaAs (Supplementary Discussion). In order to simplify the reconstruction of Bloch wavefunctions, experimental conditions for this Article were chosen such that the Bloch wavefunctions of holes did not change along their trajectories, which were straight lines through the Gamma point oriented along the constant angle 𝜃 defined by the linearly polarized THz electric field. Along these special trajectories, electron-hole pairs acquired dynamic phases 𝐴 𝐻𝐻(𝐿𝐻) but no Berry phases (Supplementary Discussion). Excitation above the band gap, or with elliptically-polarized THz radiation, will result in hole trajectories that sample variations of Bloch wavefunctions, and may result in signatures of Berry phases in HSG from GaAs . A second approach to topological quantum computation is to design and build devices in which a conventional superconductor is deposited on a semiconductor nanowire in which spin-orbit coupling is strong . Under delicately balanced conditions that depend critically on the Bloch wavefunctions in the nanowire, the ends of such a superconductor-semiconductor structure are predicted to host topological excitations called Majorana zero modes. Hole-doped GaAs quantum wires have been proposed as candidates for the semiconductor host . Direct reconstruction of Bloch wavefunctions by the method we have demonstrated—perhaps in the host nanowires themselves-- may be helpful in selecting an optimal semiconductor structure to host Majorana zero modes. The complete electronic structure of a crystalline solid should include both its band structure and Bloch wavefunctions. We have reconstructed Bloch wavefunctions in GaAs from polarimetry of high-order sideband spectra. GaAs is one of the most widely studied semiconductors, and the consistency of our results with the vast body of complementary previous work validates the novel method we have presented in this Article. Because HSG can, in principle, be observed from any direct-gap semiconductor or insulator, we expect the method described in this paper can be extended to reconstruct Bloch wavefunctions in a large class of materials that will form a foundation for the second quantum revolution . 6 References Cited
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High-order sideband generation (HSG) in bulk GaAs. a , An HSG spectrum (blue). The red dash-dot line indicates the photon energy of the NIR laser. The sidebands (peaks) are spaced by twice the THz frequency.
Inset , Schematic of the experimental setup. b, Three-step model of HSG in GaAs. Electron-heavy hole (E-HH) and electron-light hole (E-LH) recollision trajectories are denoted in red and blue, respectively, and can be thought of as classical representations of interfering quantum pathways. The y-axis (into page) corresponds to time. Bottom plane: real-space trajectories of E-HH and E-LH pairs. 3-D mesh plot: 𝑘 -space trajectories of the same pairs, where the 𝑧 -axis corresponds to energy, and the 𝑥 -axis corresponds to dimensionless quasimomentum 𝑘𝑎 , where 𝑎 is the lattice constant. Step 1: creation of E-HH and E-LH pairs by NIR laser (up arrows of equal length). Step 2: acceleration by the THz field. Electrons and holes begin at 𝑘 = 0 , and recollide with substantial excess kinetic energy at |𝑘𝑎| > 0.1 . Step 3: recombination of E-HH and E-LH pairs and emission of sideband (down arrows of equal length). Top line: linearly-polarized NIR laser photons (black double arrow) lead to emission of E-HH and E-LH sideband components (red and blue double arrows) with rotated linear polarizations and different phases, which combine to emit an elliptically-polarized sideband. Calculations of trajectories are classical (methods) and for the 24 th order sideband. c, Sideband linear orientation angle 𝛼 and ellipticity angle 𝛾 as functions of sideband index for NIR laser linear orientation 𝛼 𝑁𝐼𝑅 = (cyan), (red), (orange), and −45° (green) defined in upper inset. NIR laser polarization angles are plotted as sidebands with 𝑛 = 0 . The lower inset defines 𝛼 and 𝛾 with respect to the linearly-polarized THz field. The measured polarization at each sideband index is displayed directly below the corresponding peak in the sideband spectra displayed in a . The error bars denote the standard deviation. Fig. 2 | Quantum interference leading to sideband polarization.
A photon from the NIR laser is decomposed into components 𝜎 NIR± , with helicity ±1. (1) A 𝜎 NIR− photon excites either a spin-up electron and hole of spin -3/2 or a spin-down electron and hole of spin -1/2. (2) Driven by the THz field, an electron-hole pair accumulates dynamic phase 𝐴 𝐻𝐻 or 𝐴 𝐿𝐻 , depending on the band of the hole state (HH or LH). The electron spin is unchanged, while the hole states originating from the spin -3/2 state are superpositions of spin -3/2 and +1/2 states and the states originating from the spin -1/2 state are superpositions of spin -1/2 and +3/2 states. (3) Upon recollision, either 𝜎 HSG+ or 𝜎 HSG− photons are produced following angular momentum conservation—for example, a spin +3/2 hole recombining with a spin-down (-1/2) electron produces a 𝜎 HSG+ photon with helicity +3/2 -1/2 = +1. The interference of the evolution pathways from 𝜎 NIR− to 𝜎 HSG+ ( 𝜎 HSG− ) produces the dynamical Jones matrix element 𝑇 +− (𝑇 −− ) . Photons with 𝜎 NIR+ result in similar pathways to produce to 𝑇 −+ and 𝑇 ++ (Methods, Extended Data Fig. 4) . Fig. 3 | Ratios of Jones matrix elements, 𝝃 𝒏 (𝜽) ≡ 𝑻 ++,𝒏 (𝜽) 𝑻 −−,𝒏 (𝜽)⁄ and 𝝌 𝒏 (𝜽) ≡ 𝑻 +−,𝒏 (𝜽) 𝑻 −+,𝒏 (𝜽)⁄ , measured by Stokes polarimetry. a , The magnitude of 𝜒 𝑛 (𝜃) and 𝜉 𝑛 (𝜃) at 𝜃 = 39° . The dash-dot line marks the magnitude of 1 predicted by Eq. 3. For other 𝜃 , see Extended Data Fig. 5 . b, The argument of 𝜒 𝑛 (𝜃) as a function of sideband index 𝑛 , at various 𝜃 . The dash-dot lines mark the expected values (noted on the right) from Eq. 4, using values of 𝛾 and 𝛾 recommended in . c , The magnitudes of 𝜒(𝜃) ≡ 〈𝜒 𝑛 (𝜃)〉 and 𝜉(𝜃) ≡ 〈𝜉 𝑛 (𝜃)〉 averaged over sideband index 𝑛 . The dash-dot line indicates the magnitude of 1 predicted by Eqs. 3 and 4. d . The argument of 𝜒(𝜃) at each experimentally probed 𝜃 . The solid blue line is the expected argument from Eq. 4 using the values of 𝛾 and 𝛾 recommended in . Inset , The definition of 𝜃 , using the GaAs crystal lattice and the THz electric field. Error bars denote the standard deviation. Fig. 4 | Reconstruction of the Bloch wavefunctions for 𝒌 𝒛 = 𝟎 . The Bloch wavefunctions of heavy hole (HH) and light hole (LH) bands associated with 𝐻 + in Eq (1) are plotted as black lines. The orange shaded area corresponds to the uncertainty in the wavefunction associated with one standard deviation in the measurement of 𝛾 /𝛾 . For a given 𝜃 , each wavefunction is represented by a point on the Bloch sphere. The arrows within the Bloch sphere point from the origin to the LH and HH Bloch wavefunctions for the values of 𝜃 defined by their Miller indices in the inset below. The poles correspond to the spin -3/2 and spin +1/2 states. The wavefunctions for 𝐻 − are paths reflected across the xz -plane on a Bloch sphere with poles representing the spin +3/2 and spin -1/2 states. Methods
Fabrication of GaAs Sample
A 500 nm thick GaAs epilayer was grown via molecular beam epitaxy (MBE) and then transferred onto a sapphire substrate through Van-der-Waals bonding . The sapphire was transparent to both near-infrared (NIR) and terahertz (THz) radiation. The fact that the thermal expansion coefficients of sapphire and GaAs are closely matched ensures relatively small strains in the GaAs epilayer upon thermal cycling. In order to make the strain as small and homogeneous as possible, the GaAs epilayer was etched to be circular. A layer of indium tin oxide (ITO)—which reflects THz while transmitting NIR radiation—was grown on the sapphire surface that was opposite to the GaAs epilayer to create a low-Q cavity that enhanced the THz field in GaAs at selected THz frequencies. At the 447 GHz frequency used in this Article, the THz field is enhanced by a factor of 1.5 from the ITO layer (Extended Data Fig. 1). A silicon dioxide (SiO ) anti-reflection coating was grown on top of the ITO to minimize its NIR reflection and avoid NIR Fabry-Perot oscillations in the sideband spectra. See Supplementary Methods for a step-by-step fabrication procedure. Absorbance spectra of the GaAs epilayer were measured in a cryogenic chamber as a preliminary characterization on strains, as well as the excitation gap, which motivated our choice of NIR laser wavelengths for HSG experiments. Extended Data Fig. 2 shows an absorbance spectrum measured at a sample temperature of 60 K using a white light source, and calculated as A=-10 Log (Transmitted power with sample in cryostat/Transmitted power with cryostat (and sample) removed). The sharp peaks are assigned to exciton resonances associated with band-edge states with different angular momenta. These peaks are separated by 2.6 meV. A recent study has associated a similar splitting with a strain of order 0.1% . The absorbance spectra in the immediate neighborhood of the illuminated spot chosen for the HSG experiments in this Article showed little variation. Optical methods
The NIR laser was generated from an M Squared laser cavity, with a 7 W, 532 nm Sprout laser as the pump. The M Squared cavity is tunable via piezoelectric response, with a precision of 0.01 Å output NIR wavelength, measured in real time by a WS6-600 wavemeter. An acousto-optic modulator was used to direct the NIR laser onto the sample for 1 s at a 0.0001% duty cycle, synchronized with the THz output pulse from the FEL. After the modulation, only the 1 st order beam propagated through the rest of the optical elements. The polarization of the NIR laser beam incident onto the GaAs epilayer was set with a quarter-wave plate and a half-wave plate, and measured by a Thorlabs PAX polarimeter. The NIR beam was focused down to ~500 m at the GaAs epilayer using a 500 mm lens. The THz radiation was generated from the cavity-dumped UCSB MM-wave free-electron laser . Most of the variance in the output frequency was due to variance in the terminal voltage of the electrostatic accelerator that drives the FEL . The THz beam output from an optical transport system was split into two beam paths. 10% of the THz output power was directed into a fast-response pyroelectric reference detector, which measured the output power of each FEL pulse. The other 90% of the THz output power was directed onto the cryostat containing the GaAs epilayer. A 12.5 cm, gold-coated off-axis parabolic mirror was used to focus the THz beam into a 1.2 mm diameter spot. An ITO slide, which was transmissive in NIR but reflective in THz range, was used to adjust the THz beam spot on the GaAs epilayer and make sure the NIR and THz fields were collinear. The pulse energy was measured on each day 13 prior to the HSG experiments using a Thomas Keating absolute power/energy meter placed after the beam splitter but before the parabolic mirror. The pulse energy measured by the Thomas Keating power meter was used to calibrate the fast pyroelectric reference detector. The THz field strength in the GaAs epilayer is estimated to be 70±2 kV/cm. This high THz field strength resulted in the large number of sidebands reported in this Article. However, the dependence of sideband polarization angles on NIR laser polarization angles measured at fields as low as 35 kV/cm was similar to the dependence reported in Fig. 1c. In the calculation of THz field strength, we assume that the gold-coated off axis parabolic mirror is 100% reflective, the ITO slide is 70% reflective, the cryostat window is 95% transmissive, and the ITO coating on the sapphire provides a 150% field enhancement on the sample. The sidebands generated from the GaAs sample were first transmitted through a Stokes polarimeter, which includes a rotating quarter-wave plate (RQWP) and a horizontal linear polarizer. The Stokes polarimeter was calibrated by measuring the NIR laser polarizations with the Thorlabs PAX polarimeter, which was impractical for Stokes polarimetry of the sidebands because it is optimized for use with a cw laser beam at a single frequency. The intensity of each sideband was measured either by a photomultiplier tube (PMT) or a charge-coupled device (CCD), each coupled to a dedicated monochromator . The PMT measured the lowest-order sidebands, while the CCD imaged many higher-order sidebands simultaneously. To optimize the efficiencies of the diffraction gratings, a half-wave plate was placed after the Stokes polarimeter to rotate the sideband polarizations. Extraction of 𝛄 /𝛄 from Stokes polarimetry We characterize the polarization of each sideband using the four Stokes parameters defined as 𝑆 = 𝐼 , 𝑆 = 𝐼𝑝 cos 2𝛼 cos 2𝛾 , 𝑆 = 𝐼𝑝 sin 2𝛼 cos 2𝛾 , and 𝑆 = 𝐼𝑝 sin 2𝛾 , where 𝐼 is the total intensity, 𝑝 is the degree of polarization, and the orientation angle 𝛼 and ellipticity angle 𝛾 are defined in the inset of Extended Data Fig. 3b. After a sideband passes through the RQWP and horizontal linear polarizer, the intensity of the outgoing light, 𝑆 𝑜𝑢𝑡 (𝜙) , can be expressed as 𝑆 𝑜𝑢𝑡 (𝜙) = 𝑆 where 𝜙 is the angle between the fast-axis of the RQWP and the horizontal. By measuring 𝑆 𝑜𝑢𝑡 as a function of 𝜙 , the four Stokes parameters can be extracted from the Fourier transform ℱ 𝑚 =∫ 𝑆 𝑜𝑢𝑡 (𝜙)𝑒 −𝑖𝑚𝜙 𝑑𝜙 /2𝜋 : 𝑆 = 2ℱ − 4Re(ℱ ) , 𝑆 = 8Re(ℱ ) , 𝑆 = −8Im(ℱ ) , 𝑆 =4Im(ℱ ) . We sampled the intensities of each sideband at 16 different angles 𝜙 . We define plots of 𝑆 𝑜𝑢𝑡 as functions of the angle 𝜙 as “polaragrams” (see Extended Data Fig. 3a and c for examples). For each angle 𝜙 , four CCD scans were taken to establish the variance of the intensity 𝑆 𝑜𝑢𝑡 . From the Stokes parameters of the n -th order sideband, 𝑆 𝑖,𝑛 , the polarization state of the sideband can be extracted by calculating the angles 𝛼 𝑛 and 𝛾 𝑛 from relations tan(2𝛼 𝑛 ) = 𝑆 𝑆 tan(2𝛾 𝑛 ) = 𝑆 √𝑆 + 𝑆 Examples of extracted polarization states of sidebands are shown in Extended Data Fig. 3b and d. To reconstruct the dynamical Jones matrices, the polarization states of the sidebands were measured for four different polarization states of the NIR laser. All polarizations of the NIR laser were linear ( 𝛾 NIR = 0° ) with orientation angles 𝛼 NIR = 0° , , , and −45° , respectively. 14 Each dynamical Jones matrix 𝒥 connects the electric fields of the NIR laser and a sideband through (𝐸 𝑥,𝑛 𝐸 𝑦,𝑛 ) = (𝒥 𝑥𝑥,𝑛 𝒥 𝑥𝑦,𝑛 𝒥 𝑦𝑥,𝑛 𝒥 𝑦𝑦,𝑛 ) (𝐸 𝑥,NIR 𝐸 𝑦,NIR ) where (𝐸 𝑥,𝑛 𝐸 𝑦,𝑛 ) = (cos 𝛼 𝑛 − sin 𝛼 𝑛 sin 𝛼 𝑛 cos 𝛼 𝑛 ) ( cos 𝛾 𝑛 𝑖 sin 𝛾 𝑛 ) ≡ ( cos 𝛽 𝑛 𝑒 𝑖𝛿 𝑛 sin 𝛽 𝑛 ) 𝑒 𝑖𝜁 𝑛 (𝐸 𝑥,NIR 𝐸 𝑦,NIR ) = (cos 𝛼 NIR − sin 𝛼
NIR sin 𝛼
NIR cos 𝛼
NIR ) ( cos 𝛾
NIR 𝑖 sin 𝛾
NIR ) ≡ ( cos 𝛽
NIR 𝑒 𝑖𝛿 NIR sin 𝛽
NIR ) 𝑒 𝑖𝜁 NIR
The ratio 𝐸 𝑦,𝑛 /𝐸 𝑥,𝑛 yields the equation cos 𝛽 𝑛 (𝒥 𝑦𝑥,𝑛 𝒥 𝑥𝑥,𝑛 cos 𝛽 NIR + 𝒥 𝑦𝑦,𝑛 𝒥 𝑥𝑥,𝑛 𝑒 𝑖𝛿 NIR sin 𝛽
NIR ) − 𝑒 𝑖𝛿 𝑛 sin 𝛽 𝑛 (cos 𝛽 NIR + 𝒥 𝑥𝑦,𝑛 𝒥 𝑥𝑥,𝑛 𝑒 𝑖𝛿 NIR sin 𝛽
NIR )= 0 which is linear with respect to the ratios 𝒥 𝑦𝑥,𝑛 /𝒥 𝑥𝑥,𝑛 , 𝒥 𝑦𝑦,𝑛 /𝒥 𝑥𝑥,𝑛 , and 𝒥 𝑥𝑦,𝑛 /𝒥 𝑥𝑥,𝑛 . Measurements for three polarization states of the NIR laser give three such linear equations, which uniquely determine the ratios between the dynamical Jones matrix elements. From the measurements for the four NIR polarizations, we obtained four linear equations, which were solved by the method of least squares. The absolute values of the dynamical Jones matrix elements, which are not concerns of this Article, can be determined through the absolute values of the Stokes parameters. Each dynamical Jones matrix 𝒥 was converted to the T -matrix in a basis of circular polarizations through unitary transformation 𝑇 = 𝑈 † 𝒥𝑈 , where 𝑈 = 1√2 ( 𝑒 −𝑖𝜑 −𝑒 𝑖𝜑 𝑖𝑒 −𝑖𝜑 𝑖𝑒 𝑖𝜑 ) Here, 𝜑 is the angle between the THz polarization and the [100] crystal direction. From Eq. 4 in the main text, with the measured T -matrix, the ratio 𝛾 /𝛾 can be calculated as 𝛾 𝛾 = | tan 2𝜃 |√1 − cos 𝐴𝑟𝑔(𝑇 +−,𝑛 /𝑇 −+,𝑛 )1 + cos 𝐴𝑟𝑔(𝑇 +−,𝑛 /𝑇 −+,𝑛 ) where 𝐴𝑟𝑔(𝑇 +−,𝑛 /𝑇 −+,𝑛 ) is the argument of 𝑇 +−,𝑛 /𝑇 −+,𝑛 , and 𝜃 is the angle between the THz polarization and the [110] crystal direction. From each angle 𝜃 ( sin 2𝜃, cos 2𝜃 ≠ 0 ) and the ratio 𝑇 +−,𝑛 /𝑇 −+,𝑛 for each sideband, one value of 𝛾 /𝛾 was obtained. An average over sideband index n and angle 𝜃 yields 𝛾 /𝛾 = 1.47 ± 0.48 , where the quoted error is the standard deviation of 𝛾 /𝛾 . A Monte Carlo simulation was performed to estimate the errors in the dynamical Jones matrix elements from two sources which were added in quadrature: (1) the variance in the sideband intensity measurements and (2) the deviation 𝛿𝜂 of RQWP retardance from its ideal value π/2. A small deviation 𝛿𝜂 in the RQWP retardance modifies the relations between the Fourier transforms ℱ 𝑚 and Stokes parameters as: 𝑆 = 2ℱ − 4Re(ℱ )(1 − 𝛿𝜂)/(1 + 𝛿𝜂) , 𝑆 =8Re(ℱ )/(1 + 𝛿𝜂) , 𝑆 = −8Im(ℱ )/(1 + 𝛿𝜂) , 𝑆 = 4Im(ℱ )/(1 − 𝛿𝜂 /2) . The deviation 𝛿𝜂 was calibrated to be in the range [−𝜋/36, 𝜋/36] . The angles 𝛼 𝑛 and 𝛾 𝑛 of the n -th order sideband were randomly sampled from normal distributions, with the mean and standard deviation set as the measured mean values and errors as shown in Extended Data Fig. 6a. Each 15 set of 𝛼 𝑛 and 𝛾 𝑛 were sampled 10,000 times, generating 10,000 values for each of the dynamical Jones matrix elements. As an example, Extended Data Fig. 6b shows the distribution of 1000 sets of 𝛼 and 𝛾 for four polarization states of the NIR laser. The value and error of each dynamical Jones matrix element was calculated as the mean and the standard deviation of the generated distribution, respectively. Note that the dynamical Jones matrix elements are complex valued, and we set 𝒥 𝑥𝑥,𝑛 = 1 in this Article. Extended Data Fig. 6c shows the distributions of the dynamical Jones matrix elements produced from the distributions of 𝛼 and 𝛾 in Extended Data Fig. 6b. Interference of Bloch waves
We consider the case where the photon energy of the NIR laser lies just below the bandgap and assume that the sideband amplitudes are dominantly determined by electron-hole pairs created at 𝒌 = 𝟎 . Under an approximation of free electrons and holes, the amplitude of the n -th sideband can be written as (Supplementary Discussion) ℙ 𝑛 = ∑ 𝑖𝜔2𝜋𝑉ℏ ∫ 𝑑𝑡𝑒 𝑖(𝛺+𝑛𝜔)𝑡 ∫ 𝑑𝑡 ′ 𝔻 𝑠† 𝑅 𝑠 (𝑒 𝑖𝐴 𝐻𝐻 (𝑡 ′ ,𝑡)
00 𝑒 𝑖𝐴 𝐿𝐻 (𝑡 ′ ,𝑡) ) 𝑅 𝑠† 𝔻 𝑠 ⋅ 𝑬 NIR (𝑡 ′ ) 𝑡−∞2𝜋/𝜔0𝑠=± where 𝜔 is the angular frequency of the THz field, 𝑉 is the volume of the material, 𝑬 NIR (𝑡) =𝑭
NIR 𝑒 −𝑖𝛺𝑡 is the electric field of the NIR laser under the rotating wave approximation, the two components of 𝔻 + = −𝑑(𝝈 − , 𝝈 + /√3) 𝑇 ( 𝔻 − = −𝑑(𝝈 + , 𝝈 − /√3) 𝑇 ) are dipole matrix elements between spin-down (spin-up) electron and hole states with spin +3/2 (-3/2) and spin -1/2 (+1/2) respectively ( 𝑑 is a constant dipole matrix element), and 𝑅 ± is a two-by-two unitary matrix that diagonalizes the hole Hamiltonian [𝐻 𝑣± (𝒌)] ∗ through 𝑅 ±† (𝒏̂ ± ∙ 𝝉 ∗ )𝑅 ± = 𝜏 𝑧 with 𝒏̂ ± ≡ 𝒏 ± /|𝒏 ± | . The first and second column of 𝑅 + ( 𝑅 − ) respectively represent the wavefunction of heavy-hole and light-hole on the basis of hole states with spin +3/2 (-3/2) and -1/2 (+1/2). The first (second) component of the quantity 𝑅 ±† 𝔻 ± ≡ (𝕯 𝐻𝐻,± , 𝕯
𝐿𝐻,± ) 𝑇 represents the dipole matrix elements between E and HH (LH) bands. The acceleration process is described by the dynamic phase 𝐴 𝐻𝐻(𝐿𝐻) (𝑡 ′ , 𝑡) , which contains the quasimomentum 𝒌 𝑡′ (𝑡′′) = 𝑒𝑭 THz (sin 𝜔𝑡 ′ − sin 𝜔𝑡 ′′) satisfying the initial condition 𝒌 𝑡 ′ (𝑡 ′ ) = 𝟎 indicated by the subscript 𝑡 ′ and ℏ𝜕 𝑡′′ 𝒌 𝑡 ′ (𝑡′′) =−𝑒𝑬 THz (𝑡′′) , with 𝑒 being the elementary charge and 𝑬 THz (𝑡) = 𝑭
THz cos 𝜔𝑡 the THz electric field. The three-step process of HSG can thus be described as interference of the following recollision pathways: a Bloch wave associated with an electron-hole pair E-HH (E-LH) is first created by the NIR laser with amplitude proportional to 𝕯 𝐻𝐻(𝐿𝐻),± ⋅ 𝑬
NIR (𝑡′) , acquires a dynamic phase 𝐴 𝐻𝐻(𝐿𝐻) (𝑡 ′ , 𝑡) during the acceleration phase from 𝑡’ to 𝑡 , and generates sidebands through the dipole vector 𝕯 𝐻𝐻(𝐿𝐻),± . The major contribution to the sideband amplitudes comes from the recollision pathways around the saddle-points ( 𝑡’, 𝑡 ) given by the stationary-phase conditions: −ℏ 𝜕𝐴
𝐻𝐻(𝐿𝐻) (𝑡 ′ , 𝑡)𝜕𝑡 ′ + ℏ𝛺 = ∫ 𝑑𝑡′′ 𝜕𝒌 𝑡′ (𝑡 ′ ′)𝜕𝑡 ′ ⋅ (𝛻 𝒌 𝐸 𝑐 [𝒌 𝑡′ (𝑡 ′ ′)] − 𝛻 𝒌 𝐸 𝐻𝐻(𝐿𝐻) [𝒌 𝑡′ (𝑡 ′′ )]) 𝑡𝑡 ′ = 0 −ℏ 𝜕𝐴 𝐻𝐻(𝐿𝐻) (𝑡 ′ , 𝑡)𝜕𝑡 = 𝐸 𝑐 [𝒌 𝑡′ (𝑡)] − 𝐸 𝐻𝐻(𝐿𝐻) [𝒌 𝑡′ (𝑡)] = ℏ𝛺 + 𝑛ℏ𝜔 We have used the condition 𝐸 𝑐 (𝟎) − 𝐸 𝐻𝐻(𝐿𝐻) (𝟎) = 𝐸 𝑔 = ℏΩ , where 𝐸 𝑔 is the bandgap. Subtituting the energy dispersion relations 𝐸 𝑐 (𝒌) = 𝐸 𝑔 + ℏ 𝑘 𝑐 ( 𝑚 𝑐 is the effective mass of the conduction band), 𝐸 𝐻𝐻 (𝒌) = − ℏ 𝑘 𝐻𝐻 and 𝐸 𝐿𝐻 (𝒌) = − ℏ 𝑘 𝐿𝐻 ( 𝑚 𝐻𝐻 = 𝑚 (𝛾 − 2𝛾 |𝒏 ± |) −1 and 16 𝑚 𝐿𝐻 = 𝑚 (𝛾 + 2𝛾 |𝒏 ± |) −1 are respectively the effective masses of HH and LH bands) into the stationary-phase conditions, we obtain ∫ 𝑑𝑡 ′′ 𝒌 𝑡′ (𝑡′′) = 𝟎 𝑡𝑡 ′ ℏ 𝒌 𝑡 ′ (𝑡)2𝜇 𝑐ℎ(𝑐𝑙) = 𝑛ℏ𝜔 where 𝜇 𝑐ℎ = (𝑚 𝑐−1 + 𝛾 −2𝛾 |𝒏 ± |𝑚 ) −1 ( 𝜇 𝑐ℎ = (𝑚 𝑐−1 + 𝛾 +2𝛾 |𝒏 ± |𝑚 ) −1 ) is the reduced mass of the E-HH (E-LH) pair. The first equation has the meaning that the electron-hole pairs return to the position where they are created. The second equation states energy conservation at recollision. For each sideband order n , these two equations can be solved for the saddle-points ( 𝑡’, 𝑡 ), which determine k-space trajectories 𝒌 𝑡′ (𝑡 ′′ ) , as well as classical real-space trajectories with the velocities of E, HH, and LH given by ℏ𝒌 𝑡 ′ (𝑡 ′′ )/𝑚 𝑐 , −ℏ𝒌 𝑡 ′ (𝑡 ′′ )/𝑚 𝐻𝐻 , and −ℏ𝒌 𝑡 ′ (𝑡 ′′ )/𝑚 𝐿𝐻 . Fig. 1b shows the shortest trajectory for the 24 th order sideband, and parameters 𝑚 𝑐 = 0.067𝑚 , 𝑚 𝐻𝐻 = 0.711𝑚 , and 𝑚 𝐿𝐻 = 0.081𝑚 are used in the calculation. Relation between T -matrix and 𝛄 /𝛄 Each dynamical Jones matrix 𝑇 connects the sideband amplitude ℙ 𝑛 and the NIR field 𝑬 NIR (𝑡) through relation (𝐸 +,𝑛 𝐸 −,𝑛 ) = (𝑇 ++,𝑛 𝑇 +−,𝑛 𝑇 −+,𝑛 𝑇 −−,𝑛 ) (𝐸 +,NIR 𝐸 −,NIR ) or, ℙ 𝑛 = −(𝝈 + 𝝈 − ) (𝑇 ++,𝑛 𝑇 +−,𝑛 𝑇 −+,𝑛 𝑇 −−,𝑛 ) (𝝈 − 𝝈 + ) ⋅ 𝑭 NIR where ℙ 𝑛 = 𝐸 +,𝑛 𝝈 + + 𝐸 −,𝑛 𝝈 − , and 𝑭 NIR = 𝐸 +,NIR 𝝈 + + 𝐸 −,NIR 𝝈 − . To calculate the T -matrices, we rewrite the theoretical relation between ℙ 𝑛 and 𝑭 NIR as ℙ 𝑛 = 1𝑑 ∑ 𝔻 𝑠† 𝑅 𝑠 (ℚ 𝑛𝐻𝐻
00 ℚ 𝑛𝐿𝐻 ) 𝑅 𝑠† 𝔻 𝑠 ⋅ 𝑭 NIR𝑠=± with (ℚ 𝑛𝐻𝐻
00 ℚ 𝑛𝐿𝐻 ) = 𝑖𝜔𝑑 𝑖(𝛺+𝑛𝜔)𝑡 ∫ 𝑑𝑡 ′ (𝑒 𝑖𝐴 𝐻𝐻 (𝑡 ′ ,𝑡)
00 𝑒 𝑖𝐴 𝐿𝐻 (𝑡 ′ ,𝑡) ) 𝑒 −𝑖𝛺𝑡 ′ 𝑡−∞2𝜋/𝜔0 Using the relation 𝑅 𝑠† (𝒏̂ 𝑠 ∙ 𝝉 ∗ )𝑅 𝑠 = 𝜏 𝑧 that defines the unitary matrix 𝑅 𝑠 , we have ℙ 𝑛 = ∑ − (𝝈 𝑠 , 𝝈 −𝑠 √3 ) [ℚ 𝑛𝐻𝐻 + ℚ 𝑛𝐿𝐻 + ℚ 𝑛𝐻𝐻 − ℚ 𝑛𝐿𝐻 𝑠 ∙ 𝝉 ∗ ] (𝝈 −𝑠 𝝈 𝑠 √3 ) ⋅ 𝑭 NIR𝑠=± which gives 𝑇 ++,𝑛 = 𝑇 −−,𝑛 = 23 (ℚ 𝑛𝐻𝐻 + ℚ 𝑛𝐿𝐻 ) + 𝑛 𝑧 𝑛𝐻𝐻 − ℚ 𝑛𝐿𝐻 ) 𝑇 +−,𝑛 = 𝑛 𝑥 + 𝑖𝑛 𝑦 √3 (ℚ 𝑛𝐻𝐻 − ℚ 𝑛𝐿𝐻 ) 𝑇 −+,𝑛 = 𝑛 𝑥 − 𝑖𝑛 𝑦 √3 (ℚ 𝑛𝐻𝐻 − ℚ 𝑛𝐿𝐻 )
17 where 𝒏̂ + ≡ (𝑛 𝑥 , 𝑛 𝑦 , 𝑛 𝑧 ) . Note that these forms of T -matrix elements remain the same if the dynamic phase factor 𝑒 𝑖𝐴 𝐻𝐻(𝐿𝐻) (𝑡 ′ ,𝑡) is modified as 𝑒 𝑖𝐴 𝐻𝐻(𝐿𝐻) (𝑡 ′ ,𝑡)−𝐺 𝐻𝐻(𝐿𝐻) (𝑡 ′ ,𝑡) by introducing a band-dependent dephasing factor 𝐺 𝐻𝐻(𝐿𝐻) (𝑡 ′ , 𝑡) = ∫ 𝑑𝑡 ′′ 𝛤 𝐻𝐻(𝐿𝐻) [𝒌(𝑡 ′′ )] 𝑡𝑡 ′ , where 𝛤 𝐻𝐻(𝐿𝐻) (𝒌) is a momentum-dependent dephasing rate. Since the unit vector 𝒏̂ + defines the hole wavefunctions, these relations provide connections between the microscopic wavefunctions and the macroscopically measurable T -matrices. The values of the quantities ℚ 𝑛𝐻𝐻 and ℚ 𝑛𝐿𝐻 depend on details of the materials, which can vary from one experiment to another. By calculating the following ratios, we eliminate these quantities and obtain two universal relations: 𝑇 +−,𝑛 (𝜃)𝑇 −+,𝑛 (𝜃) = 𝑛 𝑥 + 𝑖𝑛 𝑦 𝑛 𝑥 − 𝑖𝑛 𝑦 = 𝛾 sin(2𝜃) − 𝑖𝛾 cos(2𝜃)𝛾 sin(2𝜃) + 𝑖𝛾 cos(2𝜃) ≡ 𝜒 𝑛 (𝜃) 𝑇 ++,𝑛 (𝜃)𝑇 −−,𝑛 (𝜃) = 1 ≡ 𝜉 𝑛 (𝜃) Representation of Bloch wavefunctions
The wavefunctions of the Hamiltonian 𝐻 𝑣± (𝒌) are eigenfunctions of 𝒏̂ ± ∙ 𝝉 , which is defined on the basis of spin ∓ ± 𝒏̂ ± = (sin Θ cos Φ , sin Θ sin Φ , cos Θ) as a point on a Bloch sphere with polar angle Θ and azimuthal angle Φ , and write the eigenfunctions of 𝐻 𝑣± (𝒌) as: |𝐻𝐻 + ⟩ = (cos (Θ2) 𝑒 𝑖Φ sin (Θ2)) |𝐿𝐻 + ⟩ = (−sin (Θ2) 𝑒 𝑖Φ cos (Θ2)) The point on the Bloch sphere with coordinates 𝒏̂ ± ( −𝒏̂ ± ) represents the state |𝐻𝐻 ± ⟩ ( |𝐿𝐻 ± ⟩ ) for the HH (LH) band. The angles Θ and Φ are determined from the measured γ /γ and angle 𝜃 through the definition 𝒏 ± = ( √32 sin(2𝜃) , ∓ √3𝛾 cos(2𝜃) , − ) . Data availability:
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Code availability:
The codes used in the data analysis are available from the corresponding author on reasonable request.
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50. S. Takahashi, G. Ramian, M. S. Sherwin, Cavity dumping of an injection-locked free-electron laser.
Applied Physics Letters , 234102 (2009). 51. D. C. Valovcin et al. , Optical frequency combs from high-order sideband generation. Opt Express , 29807-29816 (2018). Acknowledgments
We gratefully acknowledge Profs. Renbao Liu and Mackillo Kira for reading an earlier version of the manuscript; Dr. Garrett Cole and Paula Heu for assistance with GaAs membrane fabrication; Mr. Alex Peñaloza for assistance with design and fabrication of cryostat modifications; Mr. Cameron Cannon for implementing software for Monte Carlo error estimation; Mr. David Enyeart and Dr. Nikolay Agladze for assistance with maintaining and operating the UCSB MM-wave free-electron laser; and Jerry Meyer and Igor Vurgaftman for a discussion. The portion of this research conducted at UCSB was funded by NSF-DMR 1710639 and NSF-DMR 2004995. Upgrades to the UCSB Terahertz facility that was used for this research were funded by NSF-DMR 1626681 and NSF-DMR 1126894. The portion of this research conducted at Princeton was funded in part by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF9615 to L. N. Pfeiffer, and by the National Science Foundation MRSEC grant DMR 1420541.
Author contributions
Performing experiments, data collection, and analysis: S. O. and J. C. Software: J. C., D. V. and S. O. Theory: Q. W. Conceptualization: M. S. S. and Q. W. Resources (GaAs epilayer growth): K. W. W. and L. P. Development of broad-band polarimetry: D. V. Writing: M. S. S., Q. W., S. O. and J. C. Supervision, funding acquisition and project administration: M. S. S.
Competing interests:
The authors declare no competing interests.
Additional information Supplementary information is available for this paper.
Correspondence and requests for materials should be addressed to M. S. S. 19
Extended Data Fig. 1.
Field enhancement at the GaAs epilayer from the ITO-coated sapphire substrate.
The field enhancement is calculated as |1 + 𝑟(𝑓
THz )| with the complex reflection coefficient 𝑟(𝑓 THz ) measured by a Vector Network Analyzer . Extended Data Extended Data Fig. 2. An absorbance spectrum of the GaAs epilayer mounted on the ITO-coated sapphire substrate.
The measurement was taken at the spot illuminated by a white light source (left inset). The right inset shows a zoom-in of the spectrum, with the bandgap and the photon energy of the NIR laser denoted by dash-dot blue and red lines, respectively. The two peaks are strain-split exciton resonances associated with band-edge states with different angular momenta. The temperature was 60K. 21
Extended Data Fig. 3. Stokes polarimetry with linearly polarized NIR laser ( 𝜸 𝐍𝐈𝐑 = 𝟎° ). ( a ) Polaragrams for sideband index 𝑛 = 12 and orientation angle of the NIR laser 𝛼 NIR = 0° . ( b ) The polarization state of the sideband extracted from the polaragrams in ( a ). ( c ) Polaragrams for sideband index 𝑛 = 24 and orientation angle of the NIR laser 𝛼 NIR = 45° . ( d ) The polarization state of the sideband extracted from the polaragrams in ( c ). In ( a ) and ( c ), the black dots show the measured polaragrams, with error bars showing the standard deviation over 4 measurements, and the red solid lines are the reconstructed polaragram through Fourier transform, with the red dotted lines showing the bounds. In ( b ) and ( d ), the polarization states of the sidebands are represented as trajectories of the tips of the electric field vectors (𝐸 𝑥 , 𝐸 𝑦 ) over time. The orientation angle 𝛼 and ellipticity angle 𝛾 are defined in the inset in ( b ). 22 Extended Data Fig. 4. Quantum interference in three-step model of HSG leading to sideband polarization.
A photon from the NIR laser is decomposed into components 𝜎 NIR± , with helicity ±1. (1) A 𝜎 NIR− photon excites either a spin-up electron and hole of spin -3/2 or a spin-down electron and hole of spin -1/2. A 𝜎 NIR+ photon excites either a spin-up electron and hole of spin +1/2 or a spin-down electron and hole of spin +3/2. (2) Driven by the THz field, an electron-hole pair accumulates dynamic phase 𝐴 𝐻𝐻 or 𝐴 𝐿𝐻 , depending on the band of the hole state (HH or LH). The electron spin is unchanged, while the hole states originating from the spin -3/2 state are superpositions of spin -3/2 and +1/2 states and the states originating from the spin -1/2 state are superpositions of spin -1/2 and +3/2 states. (3) Upon recollision, either 𝜎 HSG+ or 𝜎 HSG− photons are produced following angular momentum conservation—for example, a spin +3/2 hole recombining with a spin-down (-1/2) electron produces a 𝜎 HSG+ photon with helicity +3/2 -1/2=+1. The interference of the evolution pathways from 𝜎 NIR± to 𝜎 HSG+ ( 𝜎 HSG− ) produces the dynamical Jones matrix element 𝑇 +± (𝑇 −± ) . Extended Data Fig. 5. Additional data for ratios of Jones matrix elements, 𝝃 𝒏 (𝜽) and 𝝌 𝒏 (𝜽) . ( A ) The argument of 𝝃 𝒏 (𝜽) . The dash-dot line marks the expected value of 0. ( B ) The magnitude of 𝝃 𝒏 (𝜽) . The dash-dot line marks the expected value of 1. ( C ) The magnitude of 𝝌 𝒏 (𝜽) . The dash-dot line marks the expected value of 1. All quantities are presented as functions of sideband index n for eight values of angle 𝜽 . (Inset) The definition of 𝜽 by using the GaAs crystal lattice and the THz electric field. -15-10-5051015200.51.01.52.0
45 15 39 00 29 -22 22 -45 [110]GaAs lattice THzpolarization | χ n | | ξ n | A r gu m en t ( ξ n ) ( D eg r ee s ) A)B)C) Extended Data Fig. 6. Monte Carlo simulation in calculating the dynamical Jones matrices. ( a ) The polarization state of the 𝑛 = 12 sideband ( 𝜃 = 23° ) for all 4 initial NIR polarizations (i- 𝛼 NIR = 0° , ii- 𝛼 NIR = 45° , iii- 𝛼 NIR = 90° , iv- 𝛼 NIR = −45° ). The horizontal and vertical axes represent 𝛼 and 𝛾 , respectively. Dashed ovals correspond to confidence intervals in the measurement of 𝛼 and 𝛾 . ( b ) Histograms of 𝛼 and 𝛾 for the 4 measured sidebands polarizations. Normal distributions of 𝛼 and 𝛾 were sampled, with the central value and standard deviation of the distributions set by the measured values. In this figure, 1,000 iterations are shown, but the results of this paper are calculated from 10,000 iterations. ( c ) The complex 𝒥 -matrix elements resulting from the 𝛼 and 𝛾 in ( b ). The horizontal and vertical axes represent the real and imaginary part, respectively. Each red dashed line shows one standard deviation of the distribution of each 𝒥 -matrix element resulting from the Monte Carlo simulation. All three plots have the same scale. The value of 𝒥 𝑥𝑥,𝑛 is set as 1 in these calculations. 25 Extended Data Fig. 7. Berry connection matrix element 𝓐 𝐻𝐻 + ,𝐻𝐻 + in the 𝒌 𝒛 = 𝟎 plane of the Brillouin zone. The double-headed black dotted arrow represents a path of a hole accelerated by a linearly polarized THz field, which is perpendicular to the Berry connection (color arrows) at all points. The Berry connection is plotted in units of 𝑎 , which is the lattice constant of GaAs. 26 Supplementary Information for
Experimental Reconstruction of Bloch Wavefunctions J. B. Costello † , S. D. O’Hara † , Q. Wu † , D. C. Valovcin, L. N. Pfeiffer, K. West, and M. S. Sherwin* Correspondence to: [email protected]
Table of Contents
SUPPLEMENTARY METHODS: FABRICATION OF GAAS SAMPLE ........................................ 27 a) Sample growth ........................................................................................................................................... 27 b) Epitaxial transfer ........................................................................................................................................ 27 i. Sapphire substrate preparation and characterization ............................................................................................. 27 ii.
GaAs wafer preparation .......................................................................................................................................... 27 iii.
Van-der-Waals bonding ......................................................................................................................................... 28
SUPPLEMENTARY DISCUSSION: THEORY OF HSG POLARIMETRY AND BLOCH WAVEFUCNTION RECONSTRUCTION ............................................................................................ 28 T HE B AND MODEL ............................................................................................................................................... 28 S IDEBAND AMPLITUDES ........................................................................................................................................ 29 B ERRY PHYSICS .................................................................................................................................................... 32 Supplementary Methods: Fabrication of GaAs Sample a) Sample growth
The 500 nm GaAs epilayer was grown via molecular beam epitaxy (MBE) at Princeton University. A 100 nm lattice-match layer was grown on top of a 500 m GaAs substrate first, followed by a 300 nm thick Al Ga As (AlGaAs) etch-stop layer. Then the GaAs epilayer was grown on top of the etch-stop layer. b) Epitaxial transfer
The epilayer was transferred onto the sapphire substrate through Van-der-Waals (VDW) bonding . The fabrication steps to transfer the circular epilayer onto the prepared sapphire substrate are as follows: i. Sapphire substrate preparation and characterization
The ITO layer was deposited onto a clean 488 m thick c-axis sapphire wafer in a heated electron-beam (E-beam) deposition environment. To ensure high surface quality, the ITO layer was deposited as slowly as the beam current would allow, on the order of one atomic layer per few seconds, until reaching the final thickness of 250 nm. After the growth of the ITO layer, the sheet resistance of the substrate was measured to be 12 Ω per square, which was constant throughout the substrate. The thickness of the deposited ITO layer was measured on a silicon wafer after the deposition process with a Filmetrics thin film interference measurement. We characterized the enhancement of the THz radiation at the surface of the sapphire/ITO substrate by measuring the complex reflection coefficient 𝑟(𝑓 THz ) using a Vector Network Analyzer . Extended Data Fig. 1 displays the field enhancement factor, |1 + 𝑟(𝑓 THz )| , as a function of frequency. Values greater than 1 (less than 1) indicate field enhancement (suppression) from constructive (destructive) interference between the incident and reflected THz waves. At the 447 GHz frequency we use in this Article, the field enhancement factor is 1.5 and atmospheric transmission is high. Due to the impedance difference between the substrate and air, there are significant Fabry-Pérot oscillations in the NIR range. To counteract this, a SiO anti-reflection coating was grown on top of the ITO, again through E-beam lithography. The thickness of the deposited SiO was 150 nm, which was measured after the deposition process through a Filmetrics thin film measurement on a silicon wafer. With the ITO and SiO deposited, the sapphire was then cut into appropriately sized pieces by a diamond dicing saw at the UCSB nanofabrication facility. ii. GaAs wafer preparation
The GaAs wafer was initially cleaved, with one flat edge providing the crystal orientation, which was marked onto the backside of the wafer for future reference. The wafer was cleaved into a 10 x 10 mm square, and cleaned with solvents for mesa etching. Negative photoresist was spin-coated onto the epilayer side of the wafer. With a circular pattern (7 mm inner diameter washer) placed on top of the wafer, the photoresist was flood- exposed with UV light. Upon photoresist development, a circular region of photoresist remained to define the epilayer mesa. To prevent undercutting of the photoresist and the epilayer mesa, H PO /H O /H O (1:1:1) was used as the etchant, which etched quickly and anisotropically through the [001] direction of GaAs and AlGaAs. A 7 m high mesa formed after the wafer was submerged in the etchant for 45 seconds. The height of this mesa was confirmed by a Confocal Microscopy measurement. At this stage, the 500 nm thick GaAs epilayer formed the top of the circular mesa. 28 The backside of the wafer was then removed to only leave behind a smooth, circular GaAs epilayer, which was protected by Apiezon Wax W (commonly called Black Wax). The Black Wax was attached to a maximum-tackiness Gel Pak for handling purposes. First the wafer was etched with 10:1 H O /NH OH for 50 minutes. Then the sample was placed in a more dilute 30:1 H O /NH OH etchant, which was selective to GaAs relative to the AlGaAs etch-stop layer. This etching process lasted on the order of one hour, and the wafer was removed from the etchant when there was a visible interference pattern from the thin GaAs membrane underneath the AlGaAs. Finally, the AlGaAs etch-stop layer was removed by 49% HF and only the 500 nm GaAs epilayer was left. iii.
Van-der-Waals bonding
The sapphire substrate, prepared in parallel with the GaAs epilayer, was free of all visible particle defects after solvent cleaning and a 200 W O descum. The substrate was placed on top of a Berkshire fiberless wipe to maintain cleanliness, and the epilayer was gently brought into contact with the substrate. This started the process of VDW bonding, which lasted overnight. After that, the attachment of the rigid Black Wax was softened by heating up to a temperature close to, but below, its melting point, allowing the whole epilayer to be VDW-bonded with the substrate. When the epilayer was cooled and it was confirmed that the epilayer was bonded with the sapphire, the Black Wax was removed with chloroform. The sample was cleaned with solvents and O -descumed before it was brought into a cryogenic chamber. Supplementary Discussion: Theory of HSG Polarimetry and Bloch Wavefunction Reconstruction The Band model
To establish the relationship between the Bloch wavefunctions and the measured dynamical Jones matrices, we start from a three-band model. In this model, the electron (E) band is parabolic with energy dispersion 𝐸 𝑐 = 𝐸 𝑔 + ℏ 𝑐 𝑘 where 𝐸 𝑔 is the bandgap, ℏ is the reduced Planck’s constant, and 𝑚 𝑐 is the effective mass for the conduction band. The corresponding Bloch wavefunctions have no structure except for a two-fold spin degeneracy with basis {|𝑆 ↑⟩, |𝑆 ↓⟩} , where |𝑆⟩ is the band-edge Bloch state belonging to the scalar representation of the 𝑇 𝑑 point group, and |↑⟩ and |↓⟩ are eigenstates of the Pauli matrix 𝜎 𝑧 in the spin space of the electron. The heavy-hole (HH) and light-hole (LH) bands, both doubly degenerate, are described by the Luttinger Hamiltonian , 𝐻 𝑣 = − ℏ [(𝛾 + 52 𝛾 ) 𝑘 − 2𝛾 (𝒌 ∙ 𝑱) + 2(𝛾 − 𝛾 )(𝑘 𝑥2 𝐽 𝑥2 + 𝑘 𝑦2 𝐽 𝑦2 + 𝑘 𝑧2 𝐽 𝑧2 )] where 𝑚 is the electron rest mass , 𝛾 , 𝛾 , and 𝛾 are Luttinger parameters that determine the electronic structure of the valence bands, and the components of 𝑱 , 𝐽 𝑥 , 𝐽 𝑦 , and 𝐽 𝑧 , are spin-3/2 matrices. The Bloch wavefunctions we aim to reconstruct experimentally are the eigenfunctions of this Luttinger Hamiltonian with the basis consisting of band-edge Bloch states | + 32⟩ = − |(𝑋 + 𝑖𝑌) ↑⟩√2 | + 12⟩ = − |(𝑋 + 𝑖𝑌) ↓⟩ − 2|𝑍 ↑⟩√6 | − 12⟩ = |(𝑋 − 𝑖𝑌) ↑⟩ + 2|𝑍 ↓⟩√6 | − 32⟩ = |(𝑋 − 𝑖𝑌) ↓⟩√2 where |𝑋⟩ , |𝑌⟩ , and |𝑍⟩ belong to the vector representation of the 𝑇 𝑑 point group. The valence band structure is determined by all three Luttinger parameters, while the eigenfunctions depend only on the ratio 𝛾 𝛾 ⁄ . Sideband amplitudes
The amplitudes of the sidebands can be derived following the three-step process of electron-hole recollisions, as shown in Extended Data Fig. 4.
In the first step, the NIR laser operating in the regime of linear response excites an electron from the valence bands to the conduction band, leaving a hole in the valence bands. Based on the three-band model, the state of the electron in the conduction band is a linear combination of states |𝑆 ↑⟩ and |𝑆 ↓⟩ , and the hole in the valence band is a superposition of states |− ⟩ ℎ , |− ⟩ ℎ , |+ ⟩ ℎ , and |+ ⟩ ℎ , where the spin- 𝑚 hole state |𝑚⟩ ℎ describes a missing electron of state | − 𝑚⟩ from the filled valence bands. These band-edge states form an eight-dimensional basis {|− ⟩ ℎ , |− ⟩ ℎ , |+ ⟩ ℎ , |+ ⟩ ℎ }⨂{|𝑆 ↑⟩, |𝑆 ↓⟩} for the electron-hole pairs. From the tetrahedral symmetry of GaAs, the dipole vectors corresponding to these electron-hole states can be calculated as 𝔻 = (𝑫 , 𝑫 , 𝑫 , 𝑫 , 𝑫 , 𝑫 , 𝑫 , 𝑫 ) 𝑇 = −𝑑 (𝝈 + , 0, 𝝈 − √3 , 0, 0, 𝝈 + √3 , 0, 𝝈 − ) 𝑇 where 𝑫 , 𝑫 , … , 𝑫 are the dipole vectors for the states, |𝑆 ↑⟩ |− ⟩ ℎ , |𝑆 ↓⟩ |− ⟩ ℎ , |𝑆 ↑⟩ |− ⟩ ℎ , |𝑆 ↓⟩ |− ⟩ ℎ , |𝑆 ↑⟩ |+ ⟩ ℎ , |𝑆 ↓⟩ |+ ⟩ ℎ , |𝑆 ↑⟩ |+ ⟩ ℎ , and |𝑆 ↓⟩ |+ ⟩ ℎ , respectively, 𝑑 = ⟨𝑆|𝑒𝑥|𝑋⟩ is the dipole matrix element along the [100] crystal direction with e being the charge of a free electron, 𝝈 ± = ±(𝑋̂ ± 𝑖𝑌̂ )/√2 are basis vectors for circular polarizations ( 𝑋̂ and 𝑌̂ are unit vectors along [100] and [010] crystal directions, respectively). The nonzero components 𝑫 𝑗 (𝑗 =1,3,6,8) indicate that four of the band-edge electron-hole states can be created through linear optical transition. As shown in Extend Data Fig. 4 (1), the 𝜎 NIR+ component (with helicity +1) of the NIR laser creates states |𝑆 ↑⟩ |+ ⟩ ℎ and |𝑆 ↓⟩ |+ ⟩ ℎ , while the 𝜎 NIR− component (with helicity -1) creates states |𝑆 ↑⟩ |− ⟩ ℎ and |𝑆 ↓⟩ |− ⟩ ℎ . The THz field is negligible in this step since the THz photon energy is much smaller than the bandgap. In the second step, the electron-hole pairs created by the NIR laser are accelerated by the strong THz field. Previous HSG models neglecting the Coulomb interaction have achieved reasonable agreement between experiment and theory . For simplicity, we thus adopt a free-electron and hole approximation. The evolution of the electron-hole pairs is governed by Hamiltonian 𝐻[𝒌(𝑡)] = 𝟏 ⨂𝐸 𝑐 [𝒌(𝑡)] − 𝐻 𝑣∗ [𝒌(𝑡)]⨂𝟏 , where 𝑛 is an n-dimensional identity matrix, ℏ𝒌(𝑡) = ℏ𝑷 − 𝑒𝑨(𝑡) is the kinetic momentum with ℏ𝑷 being the canonical momentum, 30 and 𝑨(𝑡) the vector potential of the THz electric field. The long THz pulse is treated as a continuous wave of the form 𝑬 THz (𝑡) = 𝑭
THz cos(𝜔𝑡) satisfying −𝑨̇ = 𝑬
THz (𝑡) . The low-power NIR laser is neglected in this step. The acceleration of the electron-hole pairs can be described by the dynamics of the interband density matrix elements as an eight-dimensional vector, 𝝆 𝒌(𝑡) ≡ 〈(𝑏 − ,𝒌(𝑡) , 𝑏 − ,𝒌(𝑡) , 𝑏 + ,𝒌(𝑡) , 𝑏 + ,𝒌(𝑡) ) 𝑇 ⨂(𝑎 ↑,𝒌(𝑡) , 𝑎 ↓,𝒌(𝑡) ) 𝑇 〉 where 𝑏 𝑚,𝒌(𝑡) and 𝑎 𝜁,𝒌(𝑡) are annihilation operators for hole state 𝑒 −𝑖𝒌(𝑡)⋅𝒓 |𝑚⟩ ℎ and electron state 𝑒 𝑖𝒌(𝑡)⋅𝒓 |𝑆𝜁⟩ , respectively. Following ( ), the vector 𝝆 𝒌(𝑡) satisfies the equation 𝑖ℏ 𝑑𝑑𝑡 𝝆 𝒌(𝑡) = 𝐻[𝒌(𝑡)]𝝆 𝒌(𝑡) − 𝔻 ⋅ 𝑬 NIR (𝑡) where 𝑬 NIR (𝑡) = 𝑭
NIR 𝑒 −𝑖𝛺𝑡 is the electric field of the NIR laser under the rotating wave approximation, and the source term 𝔻 ⋅ 𝑬
NIR (𝑡) is an eight-dimensional vector with the j -th component calculated as 𝑫 𝑗 ⋅ 𝑬 NIR (𝑡) . For linearly polarized THz field propagating along [001] crystal direction, each k-space trajectory 𝒌(𝑡) is a straight line in the Brillouin zone with constant 𝑘 𝑧 . Along a general trajectory 𝒌(𝑡) , all four hole band-edge states couple with each other. To simplify the physical picture and facilitate the extraction of the ratio 𝛾 𝛾 ⁄ , we consider the case where the photon of the NIR laser is just below the bandgap and assume that the sideband amplitudes are dominantly determined by electron-hole pairs created at 𝒌 = 𝟎 . Each trajectory 𝒌(𝑡) for such electron-hole pairs can be expressed as 𝒌(𝑡) = 𝑘(𝑡)(cos (𝜋4 − 𝜃) , sin (𝜋4 − 𝜃) , 0) where 𝜃 is the angle between the THz electric field and the [110] crystal direction. Under this assumption, the hole states are described by block diagonal Luttinger Hamiltonian with Blocks 𝐻 𝑣± (𝒌) = − ℏ 𝑘 [𝛾 𝜏 − 2𝛾 𝒏 ± ∙ 𝝉] where τ is the identity matrix, the components of 𝝉 , 𝜏 𝑥 , 𝜏 𝑦 , and 𝜏 𝑧 are the Pauli matrices written on the basis {| ∓ ⟩ , | ± ⟩} , and 𝒏 ± = (√32 sin 2𝜃 , ∓ √3𝛾 cos 2𝜃 , − 12) Only four electron-hole pairs are involved in this sideband generation process, in which the state |𝑆 ↓⟩ |+ ⟩ ℎ ( |𝑆 ↑⟩ |− ⟩ ℎ ) only couples with |𝑆 ↓⟩ |− ⟩ ℎ (|𝑆 ↑⟩ |+ ⟩ ℎ ) through Hamiltonian 𝐻 ± [𝒌(𝑡)] = 𝟏 ⨂𝐸 𝑐 [𝒌(𝑡)] − 𝐻 𝑣±∗ [𝒌(𝑡)] . Thus, the interband density matrix elements decouple into 𝝆 +,𝒌(𝑡) ≡ 〈(𝑏 + ,𝒌(𝑡) , 𝑏 − ,𝒌(𝑡) ) 𝑇 ⨂𝑎 ↓,𝒌(𝑡) 〉 and 𝝆 −,𝒌(𝑡) ≡ 〈(𝑏 − ,𝒌(𝑡) , 𝑏 + ,𝒌(𝑡) ) 𝑇 ⨂𝑎 ↑,𝒌(𝑡) 〉 , which satisfy the equation 𝑖ℏ 𝑑𝑑𝑡 𝝆 𝑠,𝒌(𝑡) = 𝐻 𝑠 [𝒌(𝑡)]𝝆 𝑠,𝒌(𝑡) − 𝔻 𝑠 ⋅ 𝑬 NIR (𝑡) where 𝑠 = ± and 𝔻 𝑠 = −𝑑 (𝝈 −𝑠 , 𝝈 𝑠 √3 ) 𝑇 . In this special case, the equation can be solved by introducing the SU(2) matrix 𝑅 𝑠 , which diagonalizes 𝒏̂ 𝑠 ∙ 𝝉 ∗ ( 𝒏̂ 𝑠 ≡ 𝒏 𝑠 /|𝒏 𝑠 | ) as 𝑅 𝑠† (𝒏̂ 𝑠 ∙ 𝝉 ∗ )𝑅 𝑠 =𝜏 𝑧 . The SU(2) rotation transforms the equation of 𝝆 𝑠,𝒌(𝑡) into 31 𝑖ℏ 𝑑𝑅 𝑠† 𝝆 𝑠,𝒌(𝑡) 𝑑𝑡 = (𝐸 𝑐 [𝒌(𝑡)] − 𝐸 𝐻𝐻 [𝒌(𝑡)] 00 𝐸 𝑐 [𝒌(𝑡)] − 𝐸 𝐿𝐻 [𝒌(𝑡)]) 𝑅 𝑠† 𝝆 𝑠,𝒌(𝑡) − 𝑅 𝑠† 𝔻 𝑠 ⋅ 𝑬 NIR (𝑡) where 𝐸 𝐻𝐻 (𝒌) = − ℏ 𝑘 [𝛾 𝜏 − 2𝛾 |𝒏 𝑠 |] and 𝐸 𝐿𝐻 (𝒌) = − ℏ 𝑘 [𝛾 𝜏 + 2𝛾 |𝒏 𝑠 |] are the band structure of the HH and LH bands, respectively. Each HH or LH state, is a linear combination of states |+ ⟩ ℎ and |− ⟩ ℎ ( 𝑠 = + ), or states |− ⟩ ℎ and |+ ⟩ ℎ ( 𝑠 = − ). The two components in 𝑅 𝑠† 𝝆 𝑠,𝒌(𝑡) represent the density matrix elements for E-HH and E-LH pairs, respectively. The solution in this HH-LH representation can be written as 𝑅 𝑠† 𝝆 𝑠,𝒌(𝑡) = 𝑖ℏ ∫ 𝑑𝑡 ′ (𝑒 𝑖𝐴 𝐻𝐻 (𝑡 ′ ,𝑡)
00 𝑒 𝑖𝐴 𝐿𝐻 (𝑡 ′ ,𝑡) ) 𝑅 𝑠† 𝔻 𝑠 ⋅ 𝑬 NIR (𝑡′) 𝑡−∞ where 𝐴 𝐻𝐻(𝐿𝐻) (𝑡 ′ , 𝑡) = − ∫ 𝑑𝑡′′(𝐸 𝑐 [𝒌(𝑡 ′′ )] − 𝐸 𝐻𝐻(𝐿𝐻) [𝒌(𝑡 ′′ )])/ℏ 𝑡𝑡 ′ is the dynamic phase acquired by an electron-hole pair created at 𝑡′ and accelerated to 𝑡 when the hole resides in the HH (LH) band. As the electron-hole pairs are assumed to have zero quasimomenta when created, we take 𝒌(𝑡 ′ ) = 𝟎 in the integrand. As shown in Extended Data Fig. 4 (2), each of the four electron-hole states, |𝑆 ↓⟩ |+ ⟩ ℎ , |𝑆 ↓⟩ |− ⟩ ℎ , |𝑆 ↑⟩ |− ⟩ ℎ , and |𝑆 ↑⟩ |+ ⟩ ℎ , is a component of an E-HH (or E-LH) pair, which acquires a dynamic phase 𝐴 𝐻𝐻 (or 𝐴 𝐿𝐻 ) in the acceleration process. No Berry connections are present because the eigenstates of both HH and LH bands do not change their spins along a straight-line trajectory 𝒌(𝑡) passing 𝒌 = 𝟎 . In the third step, the electrons and holes recombine and emit sidebands. The sideband amplitudes can be calculated from the interband polarization, which is determined by the dipoles of the electron-hole pairs when they recollide. From the density matrix elements in 𝝆 ±,𝒌(𝑡) , the interband polarization can be written as ℙ(𝑡) = 1𝑉 ∑ 𝔻 𝑠† 𝝆 𝑠,𝒌(𝑡)𝒌(𝑡),𝑠=± = ∑ 𝑖𝑉ℏ ∫ 𝑑𝑡 ′ 𝔻 𝑠† 𝑅 𝑠 (𝑒 𝑖𝐴 𝐻𝐻 (𝑡 ′ ,𝑡)
00 𝑒 𝑖𝐴 𝐿𝐻 (𝑡 ′ ,𝑡) ) 𝑅 𝑠† 𝔻 𝑠 ⋅ 𝑭 NIR 𝑒 −𝑖𝛺𝑡 ′ 𝑡−∞𝑠=± where 𝑉 is the volume of the material and only the trajectory 𝒌(𝑡) starting from 𝒌 = 𝟎 is summed over. The function ℙ(𝑡)𝑒 𝑖𝛺𝑡 has the same periodicity as the THz field such that the interband polarization
ℙ(𝑡) can be decomposed into discrete Fourier components. Each component lies at a frequency
𝛺 + 𝑛𝜔 ( 𝑛 is an integer) corresponding to the n-th order sideband, which has the amplitude ℙ 𝑛 = ∑ 𝑖𝜔2𝜋𝑉ℏ ∫ 𝑑𝑡𝑒 𝑖(𝛺+𝑛𝜔)𝑡 ∫ 𝑑𝑡 ′ 𝔻 𝑠† 𝑅 𝑠 (𝑒 𝑖𝐴 𝐻𝐻 (𝑡 ′ ,𝑡)
00 𝑒 𝑖𝐴 𝐿𝐻 (𝑡 ′ ,𝑡) ) 𝑅 𝑠† 𝔻 𝑠 ⋅ 𝑭 NIR 𝑒 −𝑖𝛺𝑡 ′ 𝑡−∞2𝜋/𝜔0𝑠=± Extended Data Fig. 4 (3) shows the contributions to the sideband amplitudes from the four electron-hole states. Consider a recollision pathway starting from the creation of the state |𝑆 ↑⟩ |− ⟩ ℎ ( |𝑆 ↑⟩ |+ ⟩ ℎ ), which evolves in the THz field as a component of an E-HH or E-LH pair. Since the E-HH or E-LH pair contains both states |𝑆 ↑⟩ |− ⟩ ℎ and |𝑆 ↑⟩ |+ ⟩ ℎ , this recollision pathway contributes both 𝜎 HSG− and 𝜎 HSG+ components in the sideband emission. Similarly, a recollision pathway involving the states |𝑆 ↓⟩ |+ ⟩ ℎ and |𝑆 ↓⟩ |− ⟩ ℎ contributes 𝜎 HSG+ and 𝜎 HSG− components through the states |𝑆 ↓⟩ |+ ⟩ ℎ and |𝑆 ↓⟩ |− ⟩ ℎ , respectively. 32 Collecting the contributions from recollision pathways starting from 𝜎 NIR𝑎 to 𝜎 HSG𝑏 yields the dynamical Jones matrix element 𝑇 𝑏𝑎 (a , 𝑏 = ± ). Berry physics
Although the sideband experiments in this Article do not involve Berry phases, from the reconstructed Bloch wavefunctions in the last section, the Berry phase of a hole moving in the 𝑘 𝑧 = 0 plane can be derived. For instance, the Berry connection matrix for the states |𝐻𝐻 + ⟩ and |𝐿𝐻 + ⟩ can be calculated as 𝓐 = (𝓐 𝐻𝐻 + ,𝐻𝐻 + 𝓐 𝐻𝐻 + ,𝐿𝐻 + 𝓐 𝐿𝐻 + ,𝐻𝐻 + 𝓐 𝐿𝐻 + ,𝐿𝐻 + ) = (⟨𝐻𝐻 + |𝑖𝜕 𝒌 |𝐻𝐻 + ⟩ ⟨𝐻𝐻 + |𝑖𝜕 𝒌 |𝐿𝐻 + ⟩⟨𝐿𝐻 + |𝑖𝜕 𝒌 |𝐻𝐻 + ⟩ ⟨𝐿𝐻 + |𝑖𝜕 𝒌 |𝐿𝐻 + ⟩ ) The elements 𝓐 𝐻𝐻 + ,𝐻𝐻 + and 𝓐 𝐿𝐻 + ,𝐻𝐻 + ( 𝓐 𝐻𝐻 + ,𝐿𝐻 + and 𝓐 𝐻𝐻 + ,𝐿𝐻 + ) describe the variation of the state |𝐻𝐻 + ⟩ ( |𝐿𝐻 + ⟩ ) in the Brillouin zone, satisfying 𝓐 𝐻𝐻 + ,𝐿𝐻 + = 𝓐 𝐿𝐻 + ,𝐻𝐻 + ∗ . An electron in the state |𝐻𝐻 + ⟩ ( |𝐿𝐻 + ⟩ ) acquires a Berry phase Λ HH+ = ∮ 𝓐 𝐻𝐻 + ,𝐻𝐻 + ⋅ 𝑑𝒌 ⊥ ( Λ LH+ = ∮ 𝓐 𝐿𝐻 + ,𝐿𝐻 + ⋅𝑑𝒌 ⊥ ), upon completing a closed trajectory in the 𝑘 𝑧 = 0 plane if there is no tunneling between the HH and LH band. In general, the nonzero off-diagonal element 𝓐 𝐻𝐻 + ,𝐿𝐻 + ( 𝓐 𝐿𝐻 + ,𝐻𝐻 + ) induces tunneling from LH (HH) to HH (LH) band, and along an infinitesimal increment 𝑑𝒌 ⊥ ( 𝒌 ⊥ = (𝑘 𝑥 , 𝑘 𝑦 )) in a trajectory in the 𝑘 𝑧 = 0 plane, an electron in the wavefunction space spanned by |𝐻𝐻 + ⟩ and |𝐿𝐻 + ⟩ acquires a non-abelian geometric phase factor exp (𝑖𝓐 ⋅ 𝑑𝒌 ⊥ ) (a two-by-two matrix). In the cylindrical coordinate system ( 𝑘 ⊥ , 𝜋4 − 𝜃, 𝑘 𝑧 ), where 𝑘 ⊥ is the magnitude of the vector 𝒌 ⊥ , the Berry connection matrix elements in the 𝑘 𝑧 = 0 plane can be calculated as 𝓐 𝐻𝐻 + ,𝐻𝐻 + = − 1𝑘 ⊥ ⟨𝐻𝐻 + |𝑖𝜕 𝜃 |𝐻𝐻 + ⟩𝜃̂ 𝓐 𝐻𝐻 + ,𝐿𝐻 + = 𝓐 𝐿𝐻 + ,𝐻𝐻 + ∗ = − 1𝑘 ⊥ ⟨𝐿𝐻|𝑖𝜕 𝜃 |𝐿𝐻 + ⟩𝜃̂ 𝓐 𝐿𝐻 + ,𝐿𝐻 + = − 1𝑘 ⊥ ⟨𝐿𝐻 + |𝑖𝜕 𝜃 |𝐿𝐻 + ⟩𝜃̂ where 𝜃̂ is the unit vector perpendicular to 𝒌 ⊥ . Because the states |𝐻𝐻 + ⟩ and |𝐿𝐻 + ⟩ are independent of the magnitude of 𝒌 ⊥ , the Berry connection matrix does not have a radial component. From the relations cos Θ = − 1√3(sin
2𝜃 + (𝛾 𝛾 ⁄ ) cos tan Φ = − (𝛾 𝛾 ⁄ ) cot 2𝜃 we can calculate, for example, the element 𝓐 𝐻𝐻 + ,𝐻𝐻 + as 𝓐 𝐻𝐻 + ,𝐻𝐻 + = − 1𝑘 ⊥ cos Θ − 12 𝜕 𝜃 Φ = 1𝑘 ⊥ γ γ (1 + 1√3f (θ, γ γ ) + 1)/f (θ, γ γ ) 𝜃̂ where f (θ, γ γ ) = sin
2𝜃 + (𝛾 𝛾 ⁄ ) cos . Extended Data Fig. 7 shows the Berry connection matrix element 𝓐 𝐻𝐻 + ,𝐻𝐻 + in the 𝑘 𝑧 = 0 plane. Along a straight-line trajectory in the 𝑘 𝑧 = 0 plane passing 𝒌 = 𝟎 (double-headed arrow in Extended Data Fig. 7), the Berry connection vector 𝓐 𝐻𝐻 + ,𝐻𝐻 + is always perpendicular to the trajectory, and so are the other Berry connection matrix elements. For a hole created at 𝒌 = 𝟎 and driven by a linearly polarized THz field, its k-space trajectory 𝒌(𝑡) is always perpendicular the Berry connection matrix 𝓐 so that the hole acquires 33 no geometric phase, i.e., 𝓐 ⋅ 𝑑𝒌 ⊥ = 0 . For a hole circling around 𝒌 = 0 in the 𝑘 𝑧 = 0 plane, there can be nonzero Berry phase. References Cited
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