Dynamical birefringence: Electron-hole recollisions as probes of Berry curvature
Hunter B. Banks, Qile Wu, Darren C. Valovcin, Shawn Mack, Arthur C. Gossard, Loren Pfeiffer, Ren-Bao Liu, Mark S. Sherwin
aa r X i v : . [ c ond - m a t . m t r l - s c i ] O c t Dynamical birefringence: Electron-hole recollisions as probes of Berry curvature
Hunter B. Banks,
1, 2
Qile Wu,
3, 4
Darren C. Valovcin,
1, 2
Shawn Mack, Arthur C. Gossard, Loren Pfeiffer, Ren-Bao Liu, and Mark S. Sherwin
1, 2 Physics Department, University of California, Santa Barbara, USA Institute for Terahertz Science and Technology, University of California, Santa Barbara, USA Department of Physics, The Chinese University of Hong Kong, Hong Kong, China Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China U.S. Naval Research Laboratory, Washington, DC, USA Materials Department, University of California, Santa Barbara, USA Electrical Engineering Department, Princeton University, Princeton, NJ, USA (Dated: October 4, 2018)The direct measurement of Berry phases is still a great challenge in condensed matter systems.The bottleneck has been the ability to adiabatically drive an electron coherently across a largeportion of the Brillouin zone in a solid where the scattering is strong and complicated. We breakthrough this bottleneck and show that high-order sideband generation (HSG) in semiconductors isintimately affected by Berry phases. Electron-hole recollisions and HSG occur when a near-band gaplaser beam excites a semiconductor that is driven by sufficiently strong terahertz (THz)-frequencyelectric fields. We carried out experimental and theoretical studies of HSG from three GaAs/AlGaAsquantum wells. The observed HSG spectra contain sidebands up to the 90th order, to our knowledgethe highest-order optical nonlinearity reported in solids. The highest-order sidebands are associatedwith electron-hole pairs driven coherently across roughly 10% of the Brillouin zone around the Γpoint. The principal experimental claim is a dynamical birefringence: the intensity and polarizationof the sidebands depend on the relative polarization of the exciting near-infrared (NIR) and the THzelectric fields, as well as on the relative orientation of the laser fields with the crystal. We explaindynamical birefringence by generalizing the three-step model for high-order harmonic generation.The hole accumulates Berry phases due to variation of its internal state as the quasi-momentumchanges under the THz field. Dynamical birefringence arises from quantum interference betweentime-reversed pairs of electron-hole recollision pathways. We propose a method to use dynamicalbirefringence to measure Berry curvature in solids.
I. INTRODUCTION
When parameters in a quantum system change adia-batically, the quantum states of the system accumulateboth dynamic and Berry phases [1]. Dynamic phasesare associated with the energy eigenvalues of the system,while Berry phases are associated with adiabatic changesin wave functions of the system through a quantity calledthe Berry curvature. Berry phases are of fundamental im-portance in many branches of physics, such as quantumfield theories [2], optics [3], ultra-cold atoms [4], quan-tum computing [5], and condensed matter physics [6, 7].In condensed matter systems, Berry phases are accumu-lated when a Bloch electron moves along a path in quasi-momentum space [6, 7]. Many manifestations of Berryphases in condensed matter physics have been observed,such as quantum Hall effects, anomalous Hall effects, andFaraday rotations [8–10]. In materials that exhibit theseand related phenomena, the Berry curvature of an energyband is as important as its dispersion relation. How-ever, although there has been recent progress in ultra-cold atoms [11–13] and optical systems [14], Berry cur-vature has largely resisted direct experimental measure-ment in solids [15, 16] because it is difficult to coherentlyand adiabatically drive an electron across a large portionof the Brillouin zone without the quantum pathway beingdestroyed by scattering. Strong laser fields can accelerate Bloch electrons acrossa substantial fraction of the Brillouin zone in timesshorter than typical scattering times, and hence provideopportunities to probe Berry curvature. In solids, high-order harmonic generation (HHG) results when laserfields in the V/˚A (100 MV/cm) range at wavelengthslonger than 3 µ m (photon energy less than 0 . [24]. Recent theoretical work has pointed to theimportance of Berry curvature to HSG [10, 35, 36]. TheBerry phases arise due to variation of the internal statesof the electron and hole with varying quasi-momentumin the Brillouin zone. The polarizations of high-ordersidebands are affected by quantum interferences betweentime-reversed pairs of quantum trajectories that have op-posite Berry phases [10, 35, 36].Previous measurements of HSG showed that low-ordersidebands from InGaAs QWs were slightly weaker whenthe electric field of the NIR laser was polarized perpendic-ular to the electric field of the THz beam than when thefields were parallel [31]. This observation is surprising:within the three-step model, and given the cubic symme-try of GaAs, why should the polarization of the NIR laserthat created the electron-hole pairs affect the intensity ofthe sidebands caused by their recollision? A subsequenttheoretical investigation predicted that in bulk GaAs thehighest-order sidebands generated when NIR and THzlaser polarizations were perpendicular should be strongerthan when they were parallel [37].In this paper, we carried out systematic experimentaland theoretical studies of HSG from three GaAs/AlGaAsquantum wells driven by 40 ns pulses of linearly-polarized540 GHz radiation with a strength of 35 kV/cm in thequantum wells. The observed HSG spectra containedsidebands up to the 90th order, to our knowledge thehighest-order optical nonlinearity reported in solids. Thehighest-order sidebands were associated with electron-hole pairs driven coherently across roughly 10% of theBrillouin zone around the Γ point, making the Berryphase effects especially relevant. Although GaAs ex-hibits neither birefringence (polarization-dependent re-fraction) nor dichroism (polarization-dependent absorp-tion), we observed surprising polarization-dependent ef-fects in HSG that we call “dynamical birefringence”: atsufficiently high orders, the sideband intensities are usu-ally larger when the exciting NIR and the THz electricfields are polarized perpendicular than when they areparallel, and also depend on the angle between the THzelectric field and the crystal axes of the GaAs quantumwells; and sideband polarizations exhibit significant el-lipticity that increases with increasing order even thoughthe polarizations of both the exciting NIR laser and theTHz field are nearly linear.To understand dynamical birefringence we generalizethe three-step model for HHG to the case of HSG, in-cluding both the effects of band structure and Berry curvature. The hole accumulates Berry phases due tovariation of its internal state as the quasi-momentumchanges under the THz field. Dynamical birefringencearises from quantum interference between time-reversedpairs of electron-hole recollision pathways, which are as-sociated with different Berry phases. The observationand theoretical understanding of the dynamical birefrin-gence in HSG open the door to direct measurements ofcomplete electronic structures of semiconductors and in-sulators near the Γ point, including band structures, scat-tering rates, and Berry curvatures. II. EXPERIMENTAL RESULTS
High-order sideband generation (HSG) experimentswere performed on three samples with different degrees ofquantum confinement and disorder. All the samples con-tained multiple Al x Ga As quantum wells (QWs) sep-arated by Al Ga As barriers grown on (100) GaAssubstrates [38]. The sample with the strongest quantumconfinement contained twenty 5 nm GaAs QWs. Fluctu-ations in the widths of such narrow quantum wells causefluctuations in the 2D band gap that manifest themselvesin the widths of the excitonic absorption peaks [39] (seeAppendix A for the absorption spectra from all threesamples). The second sample contained twenty 10 nmAl Ga As QWs. In this sample, well-width fluctu-ations are smaller than for the 5 nm QWs, but alloydisorder due to local fluctuations in the concentration ofaluminum atoms in the well region causes fluctuationsin the 2D band gap, which are manifested in the signif-icant broadenings of the excitonic absorption peaks ofthis sample. The third sample was the least disordered,and contained ten 10 nm GaAs QWs that were grownwith special care to ensure the smoothest possible side-walls, and hence excitonic absorption peaks much nar-rower than the other two samples.To generate high-order sidebands, a continuous-waveNIR laser was tuned just above the frequency of the low-est exciton absorption line (heavy hole) while the sampleswere driven by 40 ns pulses of 540 GHz radiation fromthe UC Santa Barbara MM-Wave Free-Electron Laser.The THz electric field in the QWs was 35 ± A. High-order sideband generation
During simultaneous THz and NIR illumination, theNIR radiation transmitted through the quantum wellscontained dozens of sidebands at sideband frequencies f SB = f NIR + nf THz , where f NIR is the frequency ofthe NIR laser, f THz is the frequency of the THz field,and n is the sideband order. The HSG spectra for the FIG. 1. Optical setup and HSG spectrum. (a) Details ofthe sample. The sample consists of epitaxially grown quan-tum wells transferred to a sapphire wafer. A thin film ofITO on the back side of the sapphire reflects the THz lightback towards the quantum wells while transmitting radiationwith NIR wavelengths. See Supplementary Material [40] formore details. For sideband measurements, the NIR and THzlasers are focused collinearly on the same spot on the sample,and propagate normal to the surface. (b) Full HSG spec-trum spanning 106 orders from the 5 nm GaAs sample. Thesidebands (solid lines) decorate the NIR laser represented bythick dashed red line. The NIR and THz laser polarizationsare parallel to each other, and the [011] direction of the latticemakes a 55 ◦ angle with the THz polarization. three samples were all similar to the spectrum shown inFig. 1(b). Because these samples were grown on (100)GaAs, a plane with inversion symmetry, only sidebandswith even n were observed. At wavelengths longer thanthe NIR laser line, sidebands with n ≤ − n . These sidebands areassociated with perturbative nonlinear optical processes[41, 42]. At wavelengths shorter than the NIR laser line,sidebands in Fig. 1(b) are visible with n up to 90, morethan three times the highest order previously observed inexperiments done on the same sample [33]. The increasein the number of observable sidebands is due to a dra-matically improved detection scheme and a stronger THzfield. All three samples showed sidebands with order upto at least 60, and each spectrum spanned more than 150meV.The large number of sidebands observed in the HSGspectra reported here enable systematic testing of athree-step model of high-order sideband generation. Insuch a model, each sideband is associated with anelectron-hole pair that recollides and recombines after ac- celeration through the band structure of a quantum well.Thus, as sideband order increases, so does the fraction ofthe Brillouin zone explored by the electron and hole, en-hancing sensitivity to non-parabolic features of the bandstructure and to mixing between subbands. B. Dynamical birefringence
In all three samples, the intensities of sidebands de-pend strongly on the relative polarization of the NIRand THz lasers at sufficiently high sideband offset en-ergy (or order). The sideband offset energy is the sum ofthe kinetic energy of the electron and hole at recollisionand a 5-10 meV detuning of the NIR laser below the 2Dband gap (see Supplementary Material [40]). In the 5 nmGaAs sample (Fig. 2, top panel), the onset of polarizationdependence was at about 70 meV ( n ≈ n ≈ i.e. , the real part of the index of refraction) de-pends on the angle of electric field of light with respectto an optical axis, and dichroism if the absorption oflight ( i.e. , the imaginary part of the index of refraction)depends on said angle [43]. However, GaAs, having a cu-bic lattice, has no optical axis and is not birefringent ordichroic. As we will show, the strong THz field definesa dynamical optical axis, and the polarization responseshown in Fig. 2 is a manifestation of what we call “dy-namical birefringence”. For simplicity, we define dynam-ical birefringence to encompass both effects analogous torefraction (which primarily affect the relative phases of x-and y-components of a sideband’s electric field, and hencethe sideband’s polarization) and effects analogous to ab-sorption or emission (which primarily affect the relativeamplitudes of sidebands excited with different polariza-tions that is shown in Fig. 2). The correlation betweenthe onset of dynamical birefringence and the well widthsuggests that quantum confinement effects on the bandstructure influence this phenomenon.Rotating the samples with respect to the THz polar-ization reveals that HSG is sensitive to the band struc-ture along the direction of electron and hole motion inthe 5 and 10 nm GaAs QWs, see Fig. 3 [44]. On theleft side of Fig. 3, the orientation of the crystal latticeis defined by the angle θ between the [011] axis of theGaAs lattice and the THz electric field. The remain-ing panels of Fig. 3 plot sideband conversion efficiencies -10 -8 -6 -4 -10 -8 -6 -4 -50 0 50 100 150 20010 -10 -8 -6 -4 pe r pend i c u l a r pa r a ll e l NIR polarization: S i deband c on v e r s i on e ff i c i en cy
10 nm AlGaAs QWsTHz polarization: pe r pend i c u l a r pa r a ll e l Sideband offset energy (meV)10 nm GaAs QWs pe r pend i c u l a r pa r a ll e l -20 0 20 40 60 80Sideband order FIG. 2. Sideband conversion efficiencies for NIR laser fieldpolarized parallel and perpendicular to THz field. The an-gles between the THz polarizations and the [011] directionare 85 ◦ , 91 ◦ , and 93 ◦ for the 5 nm GaAs sample, the 10nm AlGaAs sample and the 10 nm GaAs sample respectively.The sideband offset energy is the difference of the sidebandphoton energy and the NIR laser photon energy. The side-band conversion efficiency is the power in the sideband di-vided by the power of the NIR laser incident on the sample.The perpendicular-excited sidebands are stronger than theparallel-excited sidebands above an energy offset of roughly70 meV for the 5 nm GaAs sample and 30 meV for the two10 nm samples. The perpendicularly-excited sidebands falloff more slowly above this offset than below it. In both the5 nm GaAs and 10 nm AlGaAs samples, the sideband in-tensities fall off almost exponentially as the sideband orderincreases. Sideband intensities from the 10 nm GaAs sam-ple, however, show a more complicated relationship with thesideband order, perhaps because of weaker scattering in thissample. The non-exponential decay of negative-order side-bands in this sample may be due to a relatively thick GaAslayer outside of the QWs (see Supplementary Material [40]). for parallel- and perpendicular-excited sidebands fromall three samples with different values of θ . The lowestconduction subband is approximately parabolic aroundthe band minimum, so instead we focus on the valencesubbands. The dispersion relations for the three high-est valence subbands are plotted below the data on side- band conversion efficiency for each sample and value of θ measured (see Supplementary Material [45] for the bandstructure calculation). While comparing HSG spectrawith features in the hole dispersion relations, it is im-portant to note that holes carry only 10-30% of the totalkinetic energy of an electron-hole pair at a given quasi-momentum [46].The 5 nm GaAs sample has the strongest quantumconfinement, which is correlated with having the largestenergy required for the onset of dynamical birefringence.The spacing between the two highest heavy hole (HH)subbands, HH1 and HH2, is approximately 25 meV atzero momentum, as shown in the bottom left of Fig. 3.The subbands along the 55 ◦ and 85 ◦ directions are nearlyindistinguishable in HH1 and HH2 subbands until a ∼ /a where a is the lattice constant of bulkGaAs. Above 0.1 1 /a , the gap between the HH1 and HH2subbands is larger for the 85 ◦ than for the 55 ◦ orienta-tion. The sideband conversion efficiencies for the parallel-excited sidebands are nearly indistinguishable until asideband offset energy of about 100 meV. Above this side-band offset energy, the sideband conversion efficiency islarger for the orientation with the smaller avoided cross-ing (55 ◦ ). The sideband conversion efficiencies for theperpendicular-excited sidebands show much weaker de-pendence on sample orientations.The 10 nm GaAs sample has much weaker quantumconfinement compared to the 5 nm GaAs sample, with a ∼
10 meV spacing between HH1 and HH2 subbands atzero momentum, as shown in the bottom right of Fig. 3.As in the 5 nm GaAs sample, the HH1 and HH2 subbandsalong the two directions here are nearly indistinguishableuntil an avoided crossing with a ∼ /a . Above 0.08 1 /a , the gap between HH1 and HH2subbands is slightly larger for the 91 ◦ than for the 63 ◦ orientation. The sideband conversion efficiencies for boththe parallel- and perpendicular-excited sidebands showa very strong dependence on lattice orientation abovesideband offset energy of about 20 meV, and are largerfor the 63 ◦ orientation.The 10 nm AlGaAs sample has nearly identical sub-band spacings as the 10 nm GaAs sample, but muchstronger quenched disorder because of alloy scatteringthat is not present in either GaAs sample. The influenceof disorder is represented as broadenings of the subbandsin the lower center part of Fig. 3. Interestingly, while the10 nm AlGaAs and 10 nm GaAs samples showed bire-fringence above roughly the same offset energy, the side-band conversion efficiencies in the 10 nm AlGaAs sam-ple depend very little on the orientations of the lattice.The persistence of dynamical birefringence in the face ofalloy disorder is striking, and suggests that dynamicalbirefringence is related to local electronic structure, andnot simply to crystal symmetry. We speculate that, atthe length scale of a recollision (a few tens of nm), local FIG. 3. HSG spectra from all three samples for different lattice orientations, and calculated valence subband dispersion relations.The THz polarization was kept horizontal. The angle θ between the [011] direction and the THz field is shown on the left side,with the propagation direction into the page. Experimental data shows sideband offset energy versus sideband (conversion)efficiency. The empty black squares are replotted from Fig. 2. The valence subband dispersion relations are plotted along thedirections of the THz electric field to elucidate the relation between the hole subbands and the sideband spectra. : The relatively large avoided crossing between HH1 and HH2 is correlated with a relatively large energy required forthe onset of dynamical birefringence (see Fig. 2). The dependence of HSG spectra on lattice orientation is relatively weak,except for the parallel-excited spectrum above 100 meV sideband offset energy.
10 nm AlGaAs QWs : The avoided crossingbetween HH1 and HH2 is much smaller than for the 5 nm GaAs QW. Broadening caused by alloy disorder is represented asa shaded strip on each curve in the dispersion relation. The sideband conversion efficiencies show little dependence on latticeorientations.
10 nm GaAs QWs : The hole dispersion relations are nearly identical to those for the 10 nm AlGaAs QW, butbroadening from alloy disorder is absent in this sample, which has smaller disorder than the other two samples (see Fig. 7 andthe related discussions). The sideband spectra depend substantially on angle θ . avoided crossings persist in the presence of alloy disorder,so the THz-induced birefringence is similar in the 10 nmAlGaAs and GaAs samples. At the length scale of the200 µ m NIR spot, however, the ensemble of recollisionssamples many different local band gaps. In the ensem-ble, the four-fold symmetry of the band structure maybe averaged out to nearly cylindrical symmetry withoutmasking the structure of the Bloch wave functions.The experimental results of this section are consistentwith the notion that the direction of the THz electricfield defines the birefringent axis, and that the effectsof this dynamical birefringence are related to the bandstructure. The 5 nm GaAs sample, with the narrower,more strongly confined wells, shows the weakest dynam-ical birefringence: the energy required for the onset ofdynamical birefringence is the highest, and the contrast between the intensities of parallel- and perpendicular-excited sidebands is the weakest. The two 10 nm samples,despite having very different dependences of HSG spectraon crystal orientations, have similar degrees of dynami-cal birefringence. Before investigating the experimentalresults further, we consider possible physical mechanismsfor this dynamical birefringence and develop a theoreticalmodel to describe the phenomenon. III. THEORETICAL ANALYSISA. Berry physics in HSG
In a three-step model of high-order sideband genera-tion (HSG), the first step is the creation of electron-holepairs whose initial state is determined by the polariza-tion of the excitation laser; the second step is the accel-eration of electrons and holes along k -space trajectoriesin the conduction and valence bands; and the third stepis the emission of a photon whose polarization state isdetermined by the final state of the electron-hole pair.Therefore, the polarization dependence of HSG is closelyrelated to the variation of the internal states (includ-ing spin and orbital states) in the Brillouin zone, orthe Berry connection. The Berry connection is definedas ~R mn = h u m, k | i∂ k | u n, k i , where | u n, k i is the cellularfunction (internal state) of the n th band or subband atquasi-momentum k . The Berry connection is in generalnon-Abelian and is Abelian if ~R mn = for all m = n .As an excitation moves in the reciprocal space, Berryconnections are accumulated as a Berry phase. Beinggauge-dependent, the Berry connection is not a physicalquantity. A gauge-independent quantity that character-izes the variation of cellular functions is the Berry cur-vature, which is defined as F γ = ε αβγ F αβ with F αβ = i (cid:2) D k α , D k β (cid:3) , where D k = ∂ k − i ~R ( α, β, γ = x, y, z ). Inthe Abelian case, the relation between the Berry connec-tion and Berry curvature reduces to ~ F = ∂ k × ~R , similarto that between the vector potential and magnetic fieldin electromagnetism.In a band insulator with both time-reversal and in-version symmetries, and in-plane dipole matrix elementsbeing cylindrically symmetric at a quasi-momentum k and nonzero only between valence and conduction bands,there should be no dynamical birefringence in HSG if theBerry connection is zero. It is convenient to describe theradiation on the basis σ ± ( σ ± correspond to photons withangular momentum ± σ ± photons. If the Berry connection iszero, the cellular functions for each energy band and thedipole matrix elements will be the same for all Bloch wavevectors. Thus, assuming zero intraband, inter-valence-band and inter-conduction-band dipole matrix elements,electron-hole pairs associated with different cellular func-tions will be completely decoupled. In this case, in theacceleration process, an electron-hole pair can only accu-mulate a dynamic phase and a dephasing factor, but noBerry phase. Without Berry phases, the recombinationof an electron-hole pair created by a σ + NIR photon canonly produce a σ + sideband photon, which carries the dy-namic phase and the dephasing factor of the electron-holepair. It is similar for a σ − NIR photon. Due to time-reversal and inversion symmetries, for each electron-holepair created by a σ + NIR photon, there is always an-other one that can be created by a σ − NIR photon withthe same dynamic phase and dephasing factor. There-fore, in the band insulator described above, a zero Berryconnection implies that the amplitudes of the sidebandsare proportional to the exciting NIR laser, which meansrotating a linearly polarized NIR laser has no effect onthe sideband intensity. For the mathematical details andgeneralizations, see Appendix B 1. The (100) GaAs QWsare band insulators that satisfy the conditions outlined above [47], so a nonzero Berry connection is essential toexplain the observation of dynamical birefringence.Dynamical birefringence arises as a result of quantuminterference between electron-hole recollision pathwaysassociated with different Berry phases in systems like the(100) GaAs QWs because of nonzero Berry connection.In the THz field, an electron-hole pair excited by a σ + NIR photon can evolve into a state whose in-plane tran-sition dipole moment is a linear combination of σ + and σ − . Thus, a sideband generated from a σ + NIR photonis of the form aσ + + bσ − , where the energy levels, thedephasing rates and the Berry curvatures are coded inthe coefficients a and b , which depend on the directionof the electron and hole motion. Similarly, for the sameorder of sideband generated from a σ − NIR photon, theradiation has the form a ′ σ − + b ′ σ + . With nonzero Berryconnection, the polarization states of the sidebands canbe quite different from that of the NIR laser. For an NIRlaser linearly polarized along e − i σ + − σ − , each side-band is of the form ( e − i a − b ′ ) σ + + ( e − i b − a ′ ) σ − ,a sum of interfering quantum amplitudes. The normof that sum—the electric field amplitude of a particu-lar sideband—depends on the polarization angle Ψ, asdo the experimentally-observed sidebands. This analysisalso suggests that sidebands should depend on the lat-tice orientation with respect to the THz field as observed,and, in general, should be elliptically polarized. B. Semiclassical picture
To construct a physical picture of dynamical birefrin-gence, we establish a semiclassical theory with a non-Abelian Berry connection using the saddle-point method,which has been successfully used to construct semiclassi-cal theories for both HHG [48] and HSG [10, 35, 36, 49].We first model the band structure of the GaAs QWs byusing the envelope function approximation based on thesix-band Kane Hamiltonian [50]. Thus the basis we useto describe the electronic states contains combinations ofenvelope functions and the following cellular functions | u i = | S, ↑i , | u i = | S, ↓i , (1) | u i = − √ | ( X + iY ) , ↑i = (cid:12)(cid:12)(cid:12)(cid:12) , + 32 (cid:29) , (2) | u i = − √ | ( X + iY ) , ↓i − | Z, ↑i ] = (cid:12)(cid:12)(cid:12)(cid:12) , + 12 (cid:29) , (3) | u i = 1 √ | ( X − iY ) , ↑i + 2 | Z, ↓i ] = (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) , (4) | u i = 1 √ | ( X − iY ) , ↓i = (cid:12)(cid:12)(cid:12)(cid:12) , − (cid:29) , (5)where | S i belongs to the irreducible representation Γ of T d symmetry group, | X i , | Y i , | Z i belong to Γ , and |↑i , |↓i are eigenvectors of Pauli matrix σ z in spin space.The eigenstates | u i , | u i are usually called the electroncomponents. The eigenstates | u i , | u i , | u i , | u i canalso be labeled by the quantum numbers of spin- . Theeigenstates | u i , | u i are usually called the heavy-holecomponents, while | u i , | u i are the so-called light-holecomponents. In bulk GaAs, the energy gap is about 1 . f n ( z ) = r L ( cos (cid:0) m − L πz (cid:1) : n = 2 m − (cid:0) mL πz (cid:1) : n = 2 m , (6)where m = 1 , , ... , L is the well width, and f n ( z ) isodd/even as a function of z when n is even/odd. We de-note the basis as f n | S, ↑i , f n | S, ↓i for the electron com-ponents, and f n (cid:12)(cid:12) , ± (cid:11) , f n (cid:12)(cid:12) , ± (cid:11) for the hole compo-nents. In our model, zero Berry connection is assumedfor the conduction subbands, while heavy hole-light holecoupling induces a non-Abelian Berry connection in thevalence subbands. See Supplementary Material [45] formore details about band structure calculations.To study the simplest model that is expected to cap-ture the main physics of HSG in the GaAs QWs, wewill consider only the lowest conduction subband andthe highest two valence subbands. As the QWs haveboth time-reversal and inversion symmetries, we choosethe eigenstates in each subband as pairs related by atime-reversal transformation and an inversion. The cel-lular functions of the lowest conduction subband (E1 sub-band) are denoted as | E , ↑ i ≡ f | S, ↑i and | E , ↓ i ≡ f | S, ↓i , which are k -independent under the assump-tion of zero Berry connection. For the j th highestvalence subband (HH j subband, j = 1 , | HH j, ↑ i k and | HH j, ↓ i k .With non-Abelian Berry connection, the states | HH , ↑ i k and | HH , ↓ i k are linear combinations of f m − (cid:12)(cid:12) , + (cid:11) , f m (cid:12)(cid:12) , − (cid:11) , f m − (cid:12)(cid:12) , − (cid:11) , f m (cid:12)(cid:12) , + (cid:11) ( m = 1 , , ... ),with k -dependent coefficients. Thus there are only fourtypes of electron-hole pairs involved that have nonzerotransition dipole moments: f | S, ↑i - f (cid:12)(cid:12) , + (cid:11) , f | S, ↑i - f (cid:12)(cid:12) , − (cid:11) , f | S, ↓i - f (cid:12)(cid:12) , − (cid:11) and f | S, ↓i - f (cid:12)(cid:12) , + (cid:11) (see Supplementary Material [51]). The electron-holepairs involving a f | S, ↑i component are decoupled withthose containing f | S, ↓i , and we can divide the electron-hole pairs into two groups, which are related to each otherby a time-reversal transformation and an inversion. Forone group, we have electron-hole pairs | E , ↑ i - | HH , ↑ i k , | E , ↑ i - | HH , ↓ i k , and for the other group, we have | E , ↓ i - | HH , ↓ i k , | E , ↓ i - | HH , ↑ i k .With the band model established, we start the saddle-point analysis by writing the amplitude of the n th ordersideband in the following standard path integral form P n = X s = ↑ , ↓ Z + ∞−∞ dt Z t −∞ dt ′ Z d P (2 π ) Z D [ φ † s , φ s ] D † s [ k ( t )] φ s [ k ( t )] e i ~ S s φ † s [ k ( t ′ )] D s [ k ( t ′ )] · F NIR , (7) where s = ↑ , ↓ labels the two groups of electron-hole pairs, ~ k ( t ) = ~ P − e A ( t ) is the kinetic momentum with ~ P being the canonical momentum and − ˙ A = E THz ( t ) = F THz cos( ωt ) the THz electric field with frequency ω , E NIR ( t ) = F NIR e − i Ω t is the electric field of the NIR laserunder the rotating wave approximation with frequency Ω, D ↑ [ k ( t )] and φ ↑ are respectively a two-component dipolevector and an SU(2) functional field corresponding toelectron-hole pairs | E , ↑ i - | HH , ↑ i k and | E , ↑ i - | HH , ↓ i k , S ↑ = R tt ′ L ↑ [ k ( t ′′ ) , ˙ k ( t ′′ ) , φ ↑ , ˙ φ ↑ ] dt ′′ + ~ Ω( t − t ′ ) + n ~ ωt ,with L ↑ = i ~ φ †↑ ˙ φ ↑ − φ †↑ [Λ( k ) − e E THz ( t ) · ~R ↑ ( k )] φ ↑ ,Λ( k ) = diag { E cv , ( k ) , E cv , ( k ) } is a diagonal matrixwith E cv ,j ( k ) being the energy level difference betweenE1 subband and HHj subband, ~R ↑ ( k ) is the non-AbelianBerry connection, and D [ φ †↑ , φ ↑ ] is the functional mea-sure (similar for s = ↓ ) (see Supplementary Material [51]for the derivation). The dephasing rate is neglected tosimplify the picture and will be discussed in later sec-tions. Variations with respect to k and φ † s respectivelygive the following saddle-point equations0 = Z tt ′ { φ † s [ k ( t ′′ )] 1 ~ [ D s k , Λ[ k ( t ′′ )]] φ s [ k ( t ′′ )] − ˙ k ( t ′′ ) × φ † s [ k ( t ′′ )] ~ F s [ k ( t ′′ )] φ s [ k ( t ′′ )] } dt ′′ , (8) i ~ dφ s dt = Λ[ k ( t )] φ s − e E THz ( t ) · ~R s [ k ( t )] φ s , (9)where [ D s k , Λ( k )] / ~ is the covariant relative group veloc-ity of the electron-hole pairs with D s k = ∂ k − i ~R s thecovariant derivative, and ~ F s is the non-Abelian Berrycurvature matrix defined as F γs = ε αβγ F αβs with F αβs = i h D sk α , D sk β i ( α, β, γ = x, y, z ).Another two saddle-point equations concerning conser-vation of energy are obtained by variations with respectto t and t ′ :Re " F NIR · D † s ( t ′ )Λ[ k ( t ′ )] φ s ( t ′ ) F NIR · D † s ( t ′ ) φ s ( t ′ ) = ~ Ω , (10)Re " ˆ E l · D † s ( t )Λ[ k ( t )] φ s ( t )ˆ E l · D † s ( t ) φ s ( t ) = ~ (Ω + nω ) , (11)which require that the weighted average energy ofthe electron-hole pair equals the photon energy of theNIR laser at the moment of excitation, and reachesthe sideband photon energy at the moment of recol-lision. The weights are proportional to the transi-tion dipole moments. For instance, if we write φ ↑ =( φ ↑ , , φ ↑ , ) T , and define | φ ↑ , i = φ ↑ , | E , ↑ i | HH , ↑ i k , | φ ↑ , i = φ ↑ , | E , ↑ i | HH , ↓ i k , then the average energy canbe cast in the form Re[ h µ | φ ↑ , i E cv, + h µ | φ ↑ , i E cv, h µ | φ ↑ , i + h µ | φ ↑ , i ], where h µ | = h g | er with | g i being the ground state and r theprojection of the radius vector along a certain direction.Armed with Eqs. 8-11, a semiclassical picture can beconstructed following the three-step sequence of the side-band generation process. This picture is shown schemati-cally in Fig. 4. For a detailed look at calculated sidebandtrajectories, see Appendix C, and for the method of solv-ing these equations, see Supplementary Material [52].First, an incoming NIR photon resonant with the en-ergy gap is decomposed into circularly polarized compo-nents, σ ± ≡ ± ( ˆ X ± i ˆ Y ) / √ X and ˆ Y are unit vectorsalong [010] and [001], respectively). The σ − componentexcites an f (cid:12)(cid:12) , + (cid:11) electron from the highest valencesubband to the state f | S, ↑i in the lowest conductionsubband, creating a spin-up electron wave packet and ahole wave packet with angular momentum − / σ + component creates the time-reversalcounterpart electron-hole pair.In real space, as shown on the left of Fig. 4, after be-ing created by the NIR laser, the electron and hole wavepackets first move along opposite directions under theTHz field, then are driven back and finally recollide witheach other to generate sidebands. This recollision processis described by the first saddle point equation, Eq. 8, inwhich the integrand can be written as the relative ve-locity of the electron and hole wave packets. With zeroBerry connection, the velocity of the electron is the or-dinary group velocity. For the hole, with non-AbelianBerry connection, the ordinary group velocity is replacedby a covariant group velocity, and there is a velocity com-ponent perpendicular to the THz field, which looks likethe Lorentz force resulting from a k -space magnetic field(see Ref. [53, 54]). Here, the Lorentz-like velocity of thehole is neglected for simplicity, because it is small for side-bands of order n <
60 (see animations for n = 20 , , ~ ˙ k = e E THz ( t ). The dy-namics of the pseudo-spin φ s is described by the secondsaddle-point equation, Eq. 9. With zero Berry connec-tion, the spin state of the electron remains the same.For the holes, we can consider φ s as a pseudo-spin onthe basis {| HH , ↑ i k , | HH , ↓ i k } , or {| HH , ↓ i k , | HH , ↑ i k } .During the acceleration process, a hole acquires a Berryphase, which is non-Abelian and induces Landau-Zenertunneling between the hole subbands. In the non-Abeliancase, the dynamic phase and Berry phase are insepara-ble. In the language of spinors, the pseudo-spin φ s isinitially in the spin-up state, and precesses because ofthe non-Abelian Berry curvature.Finally, for the third step, consider the electron-holepair created by a σ − NIR photon. At recollision, theelectron is still in the state f | S, ↑i while the holehas evolved into a superposition P n,j η n,j f n | u j i , where η n,j = φ ↑ , α ↑ , n,j + φ ↑ , α ↑ , n,j with α ↑ ,mn,j being the coefficientof f n | u j i component in the cellular function of the m thsubband. The non-Abelian geometric phase is carried bythe pseudo-spin ( φ ↑ , , φ ↑ , ) T . We have fixed the gauge at k = and the gauge is smoothed over the Brillouin zone,so α ↑ ,mn,j is uniquely determined. Except for the heavy-hole component f (cid:12)(cid:12) , + (cid:11) and the light-hole component f (cid:12)(cid:12) , − (cid:11) , all other components in the hole wave packetcannot recombine with the electron state. Recombinationof f | S, ↑i electron with f (cid:12)(cid:12) , + (cid:11) hole and f (cid:12)(cid:12) , − (cid:11) hole respectively produces sideband components σ − and σ + . Similarly, σ + photons can produce both σ − and σ + sideband photons.Based on this picture, we explain the observed dynam-ical birefringence as a consequence of quantum interfer-ence between electron-hole recollision pathways injectedwith opposite spins. For an NIR laser linearly polarizedalong cos Ψ ˆ X + sin Ψ ˆ Y , both σ + and σ − components arepresent with a definite relative phase π − σ + , which can be produced byboth σ + and σ − NIR photons with different amplitudesdenoted by a and b . The sideband strength of the σ + component is proportional to | e i ( π − a + b | . In ourmodel, a and √ b are respectively the amplitudes of theheavy hole component f (cid:12)(cid:12) , − (cid:11) and light hole compo-nent f (cid:12)(cid:12) , + (cid:11) at recollision. We can immediately seethat with a nonzero b , the sideband intensity should de-pend on the polarization of the NIR laser, because of theheavy hole-light hole coupling, or the nonzero Berry cur-vature. For simplicity, suppose the THz field is polarizedalong the [010] direction of the lattice and the NIR laseris linearly polarized parallel or perpendicular to the THzfield. For the parallel case, the NIR laser is polarizedalong ˆ X ∝ σ + − σ − , so the sideband strength for thecomponent σ + is I x, + ∝ | − a + b | , while for the per-pendicular case, since the NIR laser is polarized alongˆ Y ∝ σ + + σ − , the corresponding sideband strength is I y, + ∝ | a + b | . When the relative phase of a and b lieswithin ( − π , π ), as in our cases, there will be I y, + > I x, + .For these special configurations, the sidebands are almostlinearly polarized along the NIR laser, and the strengthof each sideband is proportional to the one of the corre-sponding σ + component.Because of the Landau-Zener tunneling induced by thenon-Abelian Berry curvature, the initially time-reversedelectron-hole pairs can have nonzero total angular mo-mentum at recollisions. This provides a mechanism fora linearly polarized NIR laser to produce elliptically po-larized sidebands. In contrast, when both electrons andholes are restricted to a single subband, a linearly polar-ized NIR laser can only produce linearly-polarized side-bands, which may be rotated with respect to the polar-ization of the NIR laser by the Berry connection, which isAbelian in this case [56]. Note that Abelian Berry phasescan induce elliptically polarized HSG from linearly po-larized NIR laser, say, in the case when the electron-holepair is created at more than one wave vector k [49]. SeeAppendices B 2 and B 3 for the mathematical details andgeneralizations. C. Quantum simulation
To compare with experimental data, we then numer-ically simulate the evolution of interband polarizations,
FIG. 4. Schematic representation of 3-step model for high-order sideband generation in position space (left) and momentumspace (right). The figures represent a quantum interference process, with the upper (lower) arms associated with creation anddynamics of electron-hole pairs with total spin -1 (+1). The linearly-polarized NIR excitation light is a superposition of a rightand a left circularly-polarized component, σ ± NIR . In step (1), a quantum superposition of an electron-hole pair with total spin-1 (electron spin +1 /
2, heavy-hole spin − /
2) and an electron-hole pair with total spin angular momentum +1 (electron spin − /
2, heavy hole spin +3 /
2) is created. In step (2), the electrons and holes are accelerated by the THz field. In position space,the electron spinors (vertical arrows at 4 different times along “electron” trajectories, defined in Sect. III C) do not rotatebecause Berry curvature is negligible, while the hole spinors rotate substantially under the influence of non-Abelian Berrycurvature. In momentum space, the electron state remains confined to a single band during its acceleration, while the holestate mixes into nearby bands during its acceleration. In step (3), electrons and holes recollide. Upon recollision, the hole stateis a superposition of states (depicted by ovals in the right figure) in several different hole subbands with coefficients determinedby the non-Abelian Berry curvature and the initial NIR excitation. Some of these states have allowed dipole transitions withthe electron, and so create photons σ ± HSG that then interfere to generate the HSG emission at frequency f HSG with ellipticalpolarization. Rotating the NIR polarization changes the relative phase of the σ +NIR and σ − NIR photons and influences the outputby changing how the emitted sideband photon states interfere, resulting in dynamical birefringence. whose Fourier transforms give the HSG spectra. In theHeisenberg picture, when the Coulomb interaction is ne-glected, the dynamics of the electron-hole pairs is gov-erned by i ~ dη k ( t ) ,s dt = Λ[ k ( t )] η k ( t ) ,s − e E THz ( t ) · ~R s [ k ( t )] η k ( t ) ,s − D s [ k ( t )] · E NIR ( t ) − i ~ γ [ k ( t )] η k ( t ) ,s , (12)where η k ( t ) , ↑ = ( c † H1 , ↑ , k ( t ) c E , ↑ , k ( t ) , c † H2 , ↓ , k ( t ) c E , ↑ , k ( t ) ) T ,with c † H1 , ↑ , k ( t ) and c † H2 , ↓ , k ( t ) being creation opera-tors corresponding to Bloch states e i k · r | HH , ↑ i k and e i k · r | HH , ↓ i k , and c E , ↑ , k an annihilation operator for anelectron in the Bloch state e i k · r | E , ↑ i (similar for η k ( t ) , ↓ ), γ ( k ) = diag { γ , ( k ) , γ , ( k ) } is a diagonal matrix witheach matrix element being a momentum-dependent de-phasing rate due to phonon and impurity scattering. Lowelectron and hole densities are assumed, so that eachelectron-hole operator in η k ( t ) ,s is approximately bosonic.Since the THz photon energy is much smaller than theenergy gap and the NIR laser field is much weaker thanthe THz field, we neglect the THz field in the initial opti-cal excitation process, while the NIR laser field is ignored when the electron-hole pairs are accelerated. After Eq. 12is solved, the interband polarization is obtained as the ex-pectation value of ~ P ( t ) = P s D † s [ k ( t )] η k ( t ) ,s + H.c. . In thenumerical integration, we combine the leap-frog methodwith the Crank-Nicolson method, and only consider res-onantly excited electron-hole pairs.In the next section, experiment and theory are com-pared. In the hard-wall approximation used in the calcu-lation, the height of the barrier is assumed to be infiniteand the well-widths are enlarged. In order to reproduceboth the measured exciton peaks and the heavy hole-light hole exciton splitting, effective well-widths in theconduction and valence bands are assumed to be differ-ent. The effective well widths used in the calculation ofthe valence subbands (Fig. 3) are 10.9 nm, 15.6 nm, 15.9nm for the 5 nm GaAs QWs, the 10 nm AlGaAs QWs,and the 10 nm GaAs QWs respectively (see Supplemen-tary Material [45] for more details). Given the relativelysevere approximations, and the small number of param-eters adjusted, detailed quantitative agreement betweenexperiment and theory is not expected, but trends shouldbe reproduced. See Supplementary Material [51] for thederivation of the dynamical equation and Supplementary0
FIG. 5. Polarization state of the sidebands in terms of thetwo ellipticity angles. (a) Definition of α and γ for the po-larization ellipse. (b) Experimental and theoretical valuesfor the polarization angles. Experimental measurements arefilled scatter points, theoretical calculations are empty scat-ter points. The shaded region is the error in the calculationpropagated from error in the measured NIR laser polariza-tion state. Polarization state measurements were performedon both 10 nm samples for both excitation geometries, and thenearly-perpendicular measurements were performed twice, la-beled trials 1 and 2. For sideband polarimetry for the 10 nmGaAs sample, see Supplementary Material [40]. The angle ofthe [011] direction relative to the THz field is 93 ◦ . The NIRlaser polarization state is plotted as the order zero sidebandand circled in black for each measurement. Overall, both the-ory and experiment agree that the polarization state of a givensideband is extremely sensitive to the NIR laser polarizationstate. Material [57] for the numerical method.
IV. COMPARISON OF EXPERIMENT ANDTHEORY
Both experiment and theory show that the polariza-tion states of the sidebands are in general different fromthe polarization of the NIR laser, and change systemat-ically with increasing sideband order. We measured thepolarization states of the NIR laser and of the sidebandsusing a home-built Stokes polarimeter (see Supplemen-tary Material [40] for experimental details). In all cases,we found the sidebands to be perfectly polarized, so thatthe polarization ellipse completely describes their polar-ization states. The polarization ellipse is parameterizedby two angles: α is the angle the major axis makes with the dynamical optical axis defined by the direction ofTHz polarization, and γ is the arctangent of the ratio ofthe semi-minor to semi-major axes, see Fig. 5(a). Withthe measured NIR laser polarization as an input (esti-mated error ± ◦ ), we also calculated the two ellipticityangles α and γ associated with each sideband to comparewith experiment.For the 10 nm AlGaAs sample, with the NIR lasernearly linearly polarized parallel to the dynamical opticalaxis ( γ ≈
0, and α = − ◦ ), both α and γ of the sidebandsrotate clockwise with increasing sideband order. The cal-culated and measured α and γ show the same trends andagree within the experimental error (See Fig. 5b, left).With the NIR laser within 10 degrees perpendicularto the dynamical optical axis, two measurement trials,together with associated calculations, show that the po-larization states of the sidebands can depend sensitivelyon the polarization of the NIR laser (See Fig. 5b, right).In trial 1, the NIR laser was nearly perfectly linearly po-larized ( γ ≈
0) at α =91 ◦ , which was oriented only 2 ◦ from the [011] direction. Theory predicts, in this specialcase, a nearly null effect–that the sidebands should allhave polarizations that are extremely close to the polar-ization of the NIR laser. This case is analogous to thecase of linearly polarized light that is nearly parallel orperpendicular to the optical axis of a birefringent crys-tal, in which case the transmitted beam’s polarizationis nearly unchanged. Indeed, measured values for γ areclose to calculated ones, showing nearly linear polariza-tion at all measured orders ( | γ | < ◦ ). The measured α in trial 1 are within experimental error of the theoreticalprediction up to about n = 20, but are about 1 standarddeviation above theory between 20 and 40. We note that,for the nearly perpendicular case, the error in calculatedsideband polarization, which is propagated from the er-ror in measured NIR laser polarization, decreases slightlywith increasing order, making the comparison of experi-ment and theory in these cases more sensitive to approx-imations and systematic experimental errors than for thenearly parallel case (see Supplementary Material [58] forestimation of error propagation, and Supplementary Ma-terial [40] for a discussion of systematic errors).In trial 2, the NIR laser was nearly linearly polarized( γ =0) at an angle of α =81 ◦ with respect to the dy-namical optical axis. Experiment and theory are in goodagreement in this case: the sidebands in trial 2 remainnearly linearly polarized, and their polarization rotatescounter-clockwise with increasing order, reaching nearly90 ◦ at n = 40.Measurements and calculations were also performedfor the 10 nm GaAs samples (see Supplementary Ma-terial [40]). The sideband polarization states actuallychange more strongly with order than for the 10 nm Al-GaAs sample. However, perhaps because details of theband structure are more important in this cleaner sam-ple, the quantitative agreement between experiment andtheory is not as good as for the 10 nm AlGaAs sample.In addition to accounting for dependence of sideband1polarization on sideband order, the theory also accountsfor the dependence of sideband intensity on NIR laserpolarization. In order to largely factor out the effects ofscattering on the intensities of high-order sidebands, wecompare the theoretically-calculated and experimentally-measured ratios of the sideband intensities I ⊥ /I k , where I ⊥ is for the case when the NIR laser field is perpen-dicular to the THz field and I k for the case when thetwo fields are parallel, see Fig. 6. For the 5 nm GaAssample at 85 ◦ orientation, for sidebands of order n . ≈
100 meV), the calculation and the experiment havean almost perfect match, increasing monotonically at thesame rate. Above 100 meV, the experimentally measuredratio continues to increase, while the calculated one de-creases. The trends for both theory and experiment forthe 10 nm GaAs and AlGaAs QWs for all sample orien-tations are similar to those for the 5 nm GaAs QW at85 ◦ , except that the experimentally-measured sidebandratios for the 10 nm GaAs QW, the cleanest sample, showsome non-monotonic structure that is not present in thetheory. The experimental measurements for the 5 nmGaAs sample mounted at 55 ◦ are quite different from allthe others. The measured ratios are all close to 1—thereis, in this orientation, almost no dynamical birefringence!The calculated sideband ratios for this orientation do notagree with the measured ones.We do not understand the deviations between experi-ment and theory for the 5 nm GaAs QW mounted at 55 ◦ .One possible cause is the hard-wall approximation. Thenon-Abelian geometric phases depend on the band energyand the non-Abelian Berry connection due to heavy hole-light hole coupling, both determined by the well width(see Supplementary Material [45]). As the effective wellwidth decreases, the splitting of the valence subbands be-comes larger, holes are more likely to remain in a singlesubband, and the Berry connection tends to zero. Asdiscussed earlier, with a cylindrically symmetric dipolevector and zero Berry connection, the sideband polar-ization would be independent of the polarization of theNIR laser. Therefore, the deviation of the ratio I ⊥ /I k for the 5 nm GaAs QWs at 55 ◦ could be explained byan overestimation of Berry connection in the calculation(see Appendix D for more details). However, a smallerBerry connection would also reduce the dynamical bire-fringence predicted at 85 ◦ , increasing deviations betweenexperiment and theory along that direction. A secondpossible cause is anisotropic scattering from fluctuationsin well-width. The 5 nm GaAs QW is actually more ac-curately described as consisting of rectangle-like islandsthat are 9, 10 or 11 monolayers thick, with the long di-rection parallel to the [¯110] direction [59]. Scatteringfrom these islands, which are likely smaller than the 10nm excitonic Bohr radius because the sample was grownwithout pauses at GaAs/AlGaAs interfaces [60], may bedifferent along the two directions measured. Further in-vestigations will be required to understand dynamicalbirefringence from such narrow QWs. Experiment Calculation S i deband r a t i o ( I / I // ) FIG. 6. Comparison of dynamical birefringence in experimentand theory. The ratio I ⊥ /I k is compared in all three quantumwells at different lattice angles. The nonzero Berry connectionis responsible for deviations from unity. The quantum theoryof sidebands only includes macroscopic dephasing, where theonly k -dependence arises when the electrons (or holes) haveenough energy to be elastically scattered into other bands (i.e.the dephasing rates are step functions). Comparing intensityratios instead of absolute intensities largely cancels out thespin-independent scattering effect, which is assumed to bethe dominant mechanism of scattering. The blue line is theresult if the Berry connection is assumed to be zero. V. DISCUSSIONA. What about the “plateau”?
The classical three-step model predicts the existenceof a “plateau” in which the strengths of high-order side-bands [30] or harmonics [48] depend relatively weakly onsideband order up to a cutoff. In atomic HHG, if the elec-tron is launched on a valid recollision trajectory, thereis little to stop it from recolliding with its parent ion,leading to high-order harmonics whose intensity varieslittle with order below the cutoff. In HSG, however, theelectron and hole must interact with the lattice. Previ-ously, scattering and dephasing were posited as the dom-inant mechanisms for the decrease in sideband strengthwith increasing sideband order [24, 33]. The general-ized three-step model presented here suggests that theBerry curvature should also contribute. When a holethat is initially in the HH1 state mixes into the nearbysubbands upon acceleration, that hole is less likely to ra-diatively recombine with the electron upon recollision. In2the GaAs QWs studied here, the probability of Landau-Zener tunneling between subbands increases dramaticallywith increasing sideband order (see Appendix C, Fig. 8),and so the proportion of HSG-active holes decreases withincreasing sideband order even in the absence of scatter-ing. In general, even Abelian Berry curvature can causeelectron-hole pairs to become a mixture of many com-ponents, most of which could be not HSG-active. Wesuggest that a clear plateau in HSG, one similar to theplateau observed in HHG from atoms, should only beexpected in cases with nearly zero Berry curvature andweak scattering.
B. Why do disordered samples generate strongsidebands?
When we began this study, we assumed that dephasingand decoherence were the dominant factors attenuatinghigh-order sideband generation [33], and we expected thehighest sideband conversion efficiency to come from theleast disordered sample. A careful look at Fig. 2 willshow that the 10 nm GaAs sample, which was grown tohave very smooth walls and has no Al atoms to causeband gap fluctuations, generates fewer and weaker side-bands than the more disordered 10 nm AlGaAs sample.We speculate that the hole Landau-Zener tunneling intodark states may explain this difference. In clean mate-rial, once the hole mixes into the HH2 subband, it is verylikely to remain in that subband and be unable to radia-tively recombine. In dirtier material, the disorder maysuppress the coherent Landau-Zener tunneling and henceleave a larger component of the hole in the HH1 subband,from which it can radiatively recombine. Further theo-retical work is necessary in order to fully understand therole of disorder in HSG.
C. A proposal for measuring band structure andBerry curvature
We conclude the discussion section of this paper by ex-plaining how the generalized three-step model presentedhere, in combination with experimental measurementslike those presented here, can be used to determine bandstructures, Berry curvatures, and dephasing rates in aself-consistent way. This is significant because, althoughthe dispersion relations of energy bands can be mea-sured by Angle-Resolved Photoemission Spectroscopy(ARPES) and magnetotransport [61], and ARPES hasbeen used to measure Berry phase in a special situa-tion [62], these techniques are not sensitive, to our knowl-edge, to Berry curvature. We note that it has been sug-gested that HHG may be used to measure Berry curva-ture [23].To clarify the role of polarization in this proposed tech-nique, we introduce a simple linear formalism that relatesthe electric field of the n th sideband to that of the NIR laser with a complex, 2 × J n ( T n ) for a linearly (circularly) polarizedbasis, see Appendix E for details. It is straightforwardto measure this matrix experimentally. In fact, the ratio I ⊥ /I k plotted in Fig. 6 is simply | J yy,n /J xx,n | , althoughseveral more measurements are required to fully deter-mine J n . Theoretically, it is more natural to work in thecircularly-polarized basis, and semiclassical or quantumtheory can be used to calculate T n . The matrices T n and J n are related by a unitary transformation. Good agree-ment between the experimental and theoretical values ofthis matrix then confirm a convincing understanding ofthe host material.There are likely many ways to solve the inverse prob-lem of extracting band structures, Berry curvatures, anddephasing rates by comparing experimentally-measuredand theoretically-calculated dynamical Jones matrices.One method is to establish a trial band model [22] anda trial dephasing model from which to calculate T n . Thecalculated T n is then compared with the measured T n andthe band model is modified iteratively until the measured T n is reproduced. We can start from low-order sidebandswith a tight-binding model extended from the k · p theoryas is used in this paper, in which the low energy physics iswell described. For high-order sidebands, more high en-ergy terms might be needed to better describe the large- k behavior.The theoretical model that can reproduce the mea-sured dynamical Jones matrix is not unique unless allcomponents of the wave functions are optically active inthe experiments. For example, in our experiments, theprobability of the hole being in the HH2 subband is quitesmall near the zone center, where the Berry connection inthis subband is hardly accumulated and is irrelevant. Inorder to measure the Berry connection for HH2 subbandnear k = , a stronger THz field or an NIR frequencyresonant with that exciton should be used.Several experimental techniques can be used to im-prove the results of the self-consistent algorithm. Forexample, exciting with purely circularly polarized lightwill isolate any polarization changes to just one electronand hole species, measuring the non-Abelian Berry cur-vature more directly by eliminating the complex interfer-ence generated by linearly polarized light. Then, by tun-ing the THz frequency and field strength, the timescalesfor sidebands of the same order or offset energy can bechanged to measure scattering rates with different timeconstants. As is done in Ref. [24], one can also use NIRpulses instead of continuous waves. The advantage ofusing pulses is that it is possible to inject electron-holepairs at particular phases of the THz field so as to initiatedesignated quantum trajectories.When the semiclassical theory works well or there isonly a single quantum path for each spin sector, we canalready measure the Berry phase in the Abelian case,where the dynamical Jones matrix can be approximated3as T n ∝ e − Γ d e i Γ D e i Γ B α H √ e − i Γ B α ∗ L √ e i Γ B α L e − i Γ B α ∗ H ! , (13)where Γ d , Γ D ,and Γ B are respectively the dephasing fac-tor, dynamic phase and the open-path Berry phase forthe electron-hole pair created by an σ + NIR photon. α H and α L are the coefficients in the cellular functionfor the components f | , − i and f | , + i . We canchoose the gauge that α H is real at k = , which meansthrough gauge smoothing, α H can be made real all overthe whole Brillouin zone. Since T n can be determinedfrom the techniques in this paper to within a phase fac-tor, T −− ,n /T ++ ,n can be measured, which is just e − i Γ B . VI. CONCLUSION
In conclusion, we have studied how the interplay be-tween the relative orientations of the NIR laser polar-ization, THz polarization, and lattice affects HSG. Wehave measured HSG spectra of up to the 90th order andspanning over 200 meV, a bandwidth of over 12% of theNIR laser frequency, by manipulating those relative ori-entations. We have shown conclusively that electronsand holes accelerate coherently through the lattice be-fore recolliding. This coherence allows for interferencebetween different electron and hole pathways, initializedby the NIR laser polarization and caused by non-AbelianBerry curvature in the hole subbands, and leads to largechanges in sideband strength and sideband polarizationstate.In the next experiments, the observations discussedhere should lead to a new generation of complete bandstructure measurement because HSG is inherently sen-sitive to both elements of the Bloch wavefunction, e i k · x | u k i . By clever control of the NIR laser polarization,the THz frequency and field strength, and lattice orien-tation, a self-consistent algorithm can be developed forthe direct measurement of the electron and hole disper-sion relations, non-Abelian Berry curvatures, and even k - and t -dependent scattering rates in a broad class ofmaterials. VII. ACKNOWLEDGMENTS
We would like to thank Garrett Cole, David Follman,and Paula Heu of Crystalline Mirror Solutions for teach-ing us the epitaxial transfer technique; John Leonard forteaching us ITO deposition techniques; Andrew Piercefor developing the methodology to measure the field-enhancement factor of the QW-sapphire-ITO system;and David Enyeart for maintaining, repairing, and as-sisting with the operation of the UCSB FEL. The UCSBFEL upgrade that made this work possible was fundedby a Major Research Instrumentation (MRI) grant from the National Science Foundation, nsf-dmr 1126894. HB,DV and MSS, were funded by nsf-dmr 1405964. DV andMSS received additional support from the Office of NavalResearch under grant N00014-13-1-0806. QW and RBLwere supported by Hong Kong RGC and CUHK VC’sOne-Off Discretionary Fund.HBB and DCV conducted the experiments underMSS’s supervision. QW developed the theory and didall calculations under RBL’s supervision. SM grew the10 nm AlGaAs and 5 nm GaAs samples under ACG’s su-pervision. LP grew the 10 nm GaAs sample. MSS, QW,HBB, DCV, and RBL wrote the manuscript.
Appendix A: Sample absorption
Near-IR absorption measurements were performed onall samples using methods described in SupplementaryMaterials [40]. Several excitonic features are apparentin the absorption spectrum for each sample (see Fig. 7).The lower and higher energy peaks are assigned to theheavy-hole exciton (HHX) and light-hole exciton (LHX),respectively. The splitting between the HHX and LHXpeaks arises because quantum confinement breaks theHH-LH degeneracy at the top of the valence band. Inthe 5 nm QWs, the HHX-LHX splitting is 25 meV. Inboth 10 nm QW samples, the HHX-LHX splitting is 10meV, since larger well widths lead to smaller splitting dueto weaker quantum confinement. The two 10 nm sam-ples have different aluminum concentrations in the wellregion leading to the absorption differences. For the 10nm AlGaAs sample, the 5% Al content increases the 2Dband gap so that the HHX absorption line is blue-shiftedup to coincide with the HHX absorption line from the 5nm GaAs sample. The 5 nm GaAs and 10 nm AlGaAssamples were produced by the same epitaxial growth assamples studied in Ref. [33].The linewidths of the exciton lines probe the quencheddisorder of the samples and so depend on the composi-tion and growth conditions. The full width at half max(FWHM) of the HHX line in the 10 nm GaAs sampleis 2.0 meV. In both the 5 nm GaAs sample and the 10nm AlGaAs sample, it is 6.3 meV. There are two im-portant sources of the inhomogeneous broadenings, well-width fluctuations and alloy disorder in the well region.Well-width fluctuations lead to a broadening that is in-versely proportional to the well width, and so it is thedominant source of the inhomogeneous broadening forthe 5 nm GaAs sample. If alloy disorder is modeled byPoisson-distributed aluminum content fluctuations overthe 10-nm-diameter exciton wavefunction leading to lo-cal fluctuations in the band edge, we should expect aninhomogeneous excitonic linewidth broadening of about5 meV for the 10 nm AlGaAs sample with 5% aluminumconcentration, consistent with the measured linewidth.The 10 nm GaAs sample, with wider and un-alloyedQWs, has the narrowest linewidth.4 A b s o r p t i on ( d B ) Photon frequency (eV)10 nm GaAs *800 790 770 760Wavelength (nm)
FIG. 7. Optical absorption spectra of the three samples mea-sured by differential transmission. In all three samples, theheavy hole exciton lines (starred peaks) and the light holeexciton lines (slightly weaker, blue-shifted by 10 or 25 meVdepending on the well widths) are both clearly resolved. Themeasurements are performed at 15 K. In the 5 nm GaAs sam-ple, the onset of the heavy hole 2D band gap is apparent atabout 1.630 eV. The onset of the light hole 2D band gap isalso evident (not pictured). The smaller heavy hole-light holesplitting in the 10 nm samples masks the heavy hole 2D bandgap in them, but, in the 10 nm GaAs sample, the onset ofthe light hole 2D band gap is apparent at 1.570 eV. The NIRlaser wavelength used for the HSG experiments are shown asdark red arrows.
Appendix B: Mathematics in Berry physics
To mathematically study the Berry physics in HSG,we investigate the amplitude of the n th order sidebandin the form P n = i ~ Z + ∞−∞ dt Z t −∞ dt ′ Z d P (2 π ) d e i (Ω+ nω ) t D † [ k ( t )] e − i ~ S D [ k ( t ′ )] · E NIR ( t ′ ) , (B1) e − i ~ S = ˆ T exp {− i ~ Z tt ′ H [ k ( t ′′ )] dt ′′ } , (B2)where d is the dimension of the system, H ( k ) = Λ( k ) − e E THz ( t ) · ~R ( k )+ E THz ( t ) · D int ( k ) − i ~ γ ( k ), ˆ T denotes thetime-ordering operator in the integration, and D int ( k )is a matrix describing the intraband, inter-valence-bandand inter-conduction-band dipole matrix elements. Wehave assumed that the dephasing rates only depend onthe band index and the quasi-momentum. The symbolsused here are similar to those in Eq. 12 but in a moregeneral sense that there could be more energy bands in-cluded. See Supplementary Material [51] for more details.
1. Zero Berry connection
In a band insulator with zero intraband dipole matrixelements, if there are only one conduction band and onevalence band involved in HSG, a zero Berry connectionwill imply that all sidebands should have the same polar-ization state, and the HSG spectra will be independentof the NIR laser polarization, except for an overall fac-tor that is uniform for all orders of sidebands (the back-ground optical birefringence of the sample). If the Berryconnection is zero, then the dipole matrix elements arethe same for all Bloch wave vectors. Since there are onlyone conduction band and one valence band involved, withzero intraband dipole matrix elements, we have D int = ,so that H ( k ) is diagonal. In this case, the sideband am-plitude (Eq. B1) can be simplified to P n = i ~ Z + ∞−∞ dt Z t −∞ dt ′ Z d P (2 π ) d e i (Ω+ nω ) t X j,g j e − i ~ R tt ′ H j [ k ( t ′′ )] dt ′′ d ∗ g j d g j · E NIR ( t ′ ) , (B3)summing over the electron-hole pairs from different bandsand degenerate states (labeled by j and g j respectively),with H j ( k ) = E cv ,j ( k ) − i ~ γ ,j ( k ) and d g j being a con-stant dipole vector. If there are only one conduction bandand one valence band, the label j has only one valueand the dynamic phases and dephasing factors associ-ated with each k -space trajectory (fix t , t ′ and canonicalmomentum P in Eq. B3) are the same for all electron-hole pairs. Thus, we have P n ∝ P g j d ∗ g j d g j · F NIR , whichmeans all sidebands have the same polarization state. Inan HSG spectrum with logarithmic scales, a variationof the NIR laser polarization will only induce an overallchange of sideband intensity.In a band insulator with more than two bands in-volved in HSG, even if the Berry connection is zero,sideband polarization states and degrees of dynamicalbirefringence can depend on the sideband order. In thiscase, electron-hole pairs can be created directly by theNIR laser from more than two bands, or can be firstcreated by the NIR laser from two bands and then tun-nel to other bands through inter-valence-band or inter-conduction-band transition dipole moments. For simplic-ity, we discuss the case when D int = . Suppose associ-ated with each sideband, there are two electron-hole pairsfrom different energy bands with different dipole vectors(labeled by j = 1 ,
2) and a k -space trajectory. If theBerry connection is zero, we can still apply Eq. B3, fromwhich the amplitude of a sideband generated by the twoelectron-hole pairs has the form P n ∝ P j =1 , exp[ i Γ D,j − Γ d,j ] d ∗ j d j · F NIR , where Γ
D,j = − R tt ′ E cv,j [ k ( t ′′ )] dt ′′ / ~ and Γ d,j = R tt ′ γ ,j [ k ( t ′′ )] dt ′′ ( j = 1 ,
2) are the dynamicphase and dephasing factor, which are in general not thesame for different bands and depend on the sideband or-der. Therefore, the sideband polarization states and thedegrees of dynamical birefringence should, in general, de-5pend on the sideband order.In a band insulator with both time-reversal and in-version symmetries, and in-plane dipole matrix elementsbeing cylindrically symmetric at a quasi-momentum k and nonzero only between valence and conduction bands( D int = ), there should be no dynamical birefringencein HSG if the Berry connection is zero. As discussedabove, Eq. B3 is valid in this case, and in addition,all in-plane transition dipole moments are cylindricallysymmetric, i.e., d g j ∝ σ + or σ − . Consider a recolli-sion pathway along a k -space trajectory from k ( t ′ ) to k ( t ) for an electron-hole pair with transition dipole mo-ment d ∝ σ + . The contribution of this pathway to thesideband amplitude is C exp[ i Ω( t − t ′ ) + inωt + i Γ D, − Γ d, ] σ ∗ + σ + · F NIR , where Γ D, and Γ d, are the dynamicphase and dephasing factor respectively as defined in pre-vious paragraph, and C is a constant that does not de-pend on the choices of recollision pathways. Due to time-reversal and inversion symmetries, there is another rec-ollision pathway for an electron-hole pair with the same k -space trajectory, band energy difference E cv , ( k ) = E cv , ( k ) and dephasing rate γ , ( k ) = γ , ( k ) but a com-plex conjugate dipole moment d = d ∗ . The contribu-tion of this second pathway to the sideband amplitude is C exp[ i Ω( t − t ′ ) + inωt + i Γ D, − Γ d, ] σ ∗− σ − · F NIR . Thesum of the contributions from these two recollision path-ways is proportional to − ( σ + σ − + σ − σ + ) · F NIR = F NIR .Thus, for such a band insulator, a zero Berry connectionimplies that the amplitudes of the sidebands are pro-portional to the exciting NIR laser, which means rotat-ing a linearly polarized NIR laser has no effect on thesideband intensity, i.e., no dynamical birefringence. Theproof above does not require the laser fields to be contin-uous waves (CWs).If the laser fields are CWs, then the statement inthe previous paragraph is still valid for even order side-bands in the absence of inversion symmetry. Consider arecollision pathway along k -space trajectory k ( t ′′ ) from k ( t ′ ) = k to k ( t ) = k e for an electron-hole pair withtransition dipole moment d ∝ σ + . The contribu-tion of this pathway to the sideband amplitude is still C exp[ i Ω( t − t ′ ) + inωt + i Γ D, − Γ d, ] σ ∗ + σ + · F NIR , with C , Γ D, and Γ d, defined the same as above. By atime-reversal transformation, we can find another rec-ollision pathway for an electron-hole pair with transi-tion dipole moment d = d ∗ along k -space trajectory¯ k (¯ t ) = − k ( t ′′ ) from − k to − k e , where ¯ t = t ′′ + π/ω ,since a CW THz field changes its sign every half a period.The band energy difference and dephasing rate for thissecond electron-hole pair satisfy E cv, ( k ) = E cv, ( − k )and γ cv, ( k ) = γ cv, ( − k ), so the dynamic phases and thedephasing factors associated with the two time-reversedpathways are the same. The contribution of this secondpathway to the sideband amplitude is ( − n C exp[ i Ω( t − t ′ ) + inωt + i Γ D, − Γ d, ] σ ∗− σ − · F NIR . The sum of thecontributions from these two recollision pathways is pro-portional to − [( − n σ + σ − + σ − σ + ] · F NIR . Therefore,even order sideband amplitudes are proportional to the NIR laser polarization, while, all odd order sidebandshave the same polarization that is a mirror image of theNIR laser polarization, apart from the different intensi-ties.
2. Abelian Berry connection
In a band insulator with time-reversal and inversionsymmetries, and in-plane dipole matrix elements be-ing nonzero only between valence and conduction bands( D int = ), when there are only two electron-hole recol-lision pathways (related by time-reversal and inversionsymmetries with the same k -space trajectory) associ-ated with each sideband, an Abelian Berry connectioncan only induce rotations of linear polarizations. If theBerry connection is Abelian, the matrices ~R ( k ), Λ( k ) and γ ,j ( k ) are all diagonal and commute with each other.Taking into account only two electron-hole recollisionpathways related by time-reversal and inversion symme-tries with a k -space trajectory from k ( t ′ ) to k ( t ), we havefrom Eq. B1 P n ∝ e i [Ω( t − t ′ )+ nωt ] X j =1 , e − i ~ R tt ′ H j [ k ( t ′′ )] dt ′′ × e i Γ j [ k ( t ) , k ( t ′ )] d ∗ j [ k ( t )] d j [ k ( t ′ )] · F NIR , (B4)where Γ j ( k , k ′ ) = R kk ′ ~R jj ( k ) · d k is the Berry phase for theelectron-hole pair labeled by j , ~R jj ( k ) is the correspond-ing Abelian Berry connection and d j is a momentum-dependent dipole vector. As discussed in the case ofzero Berry connection, the dynamic phases Γ D and de-phasing factors Γ d are the same for these two pathways.With a suitable gauge choice, we make the dipole vec-tors for these two pathways be complex conjugates, andmeanwhile the Berry phases be opposite [35]. Denote thedipole vector at t for the first pathways as d [ k ( t )] = a t σ + + b t σ − , and the Berry phase it gains along the k -space trajectory as Γ B . For an NIR laser linearly po-larized along F NIR = e − i Ψ σ + + e i Ψ σ − ∝ sin Ψ ˆ X +cos Ψ ˆ Y ,the dipole coupling is Q ≡ d [ k ( t ′ )] · F NIR = − ( e i Ψ a t ′ + e − i Ψ b t ′ ) for the first pathway, and d ∗ [ k ( t ′ )] · F NIR = − Q ∗ for the other. Thus, from Eq. B4, we have P n ∝ e i Γ B Q ( a t σ + + b t σ − ) − c.c. , i.e., P n ∝ ρ ( e − iϕ σ + + e iϕ σ − ),where ρe − iϕ = Q a t e i Γ B + Q ∗ b ∗ t e − i Γ B . Therefore, thesidebands are linearly polarized.If the laser fields are CWs, the statement in the previ-ous paragraph is still valid for sidebands of all orders inthe absence of inversion symmetry. We choose the sametime-reversed two electron-hole recollision pathways asin the case of zero Berry connection, but with dipolevectors d [ k ( t )] = a t σ + + b t σ − = d ∗ [ − k ( t )]. Dueto time-reversal symmetry, the dynamic phases and de-phasing factors for the two pathways are the same, whilethe Berry phases are opposite. So from Eq. B4, we have P n ∝ Q − ( − n Q ∗ , where Q = e i Γ B Q ( a t σ + + b t σ − ),i.e., P n ∝ ρ [ e − iϕ σ + + ( − n e iϕ σ − ]. Thus, all sidebandsare linearly polarized.6In a band insulator with time-reversal and inversionsymmetries, and in-plane dipole matrix elements be-ing nonzero only between valence and conduction bands( D inter = ), in general, an Abelian Berry connectioncan induce ellipticity from linear polarizations. Sup-pose, associated with each sideband, there are electron-hole pairs from different energy bands or there is morethan one k -space trajectory. In this case, there is morethan one pair of electron-hole recollision pathways re-lated by time-reversal and inversion symmetries. As dis-cussed above, each pair of recollision pathways contributea linearly polarized amplitude to a sideband. In general,dynamic phases and Berry phases, obtained by electron-hole pairs from different energy bands or along different k -space trajectories, are not the same. The phase factorexp[ i Ω( t − t ′ ) + inωt ] also depends on the k -space trajec-tory. Therefore, even if the Berry connection is Abelian,each sideband, as a sum of linear polarizations with dif-ferent phases and polarization angles, can be ellipticallypolarized.
3. Non-Abelian Berry connection
In a band insulator with time-reversal and inversionsymmetries, even if there is only one k -space trajectoryassociated with each sideband, through inter-valence-band or inter-conduction-band transitions (which canbe induced by a non-Abelian Berry connection, ornonzero inter-valence-band/inter-conduction-band tran-sition dipole moments), time-reversed electron-hole pairsinjected by a linearly polarized NIR laser can havenonzero total angular momentum at recollisions. We dis-cuss the case of inter-valence-band transitions induced bya non-Abelian Berry connection. The discussion for thecase of inter-valence-band/inter-conduction-band dipoletransitions is similar. Consider the (100) GaAs QWswith only the lowest conduction subband and the high-est two valence subbands as discussed in Section III B.For simplicity, we neglect the dephasing effects, and fur-ther assume that the cellular functions for the valencesubbands only involve f (cid:12)(cid:12) , + (cid:11) , f (cid:12)(cid:12) , − (cid:11) , f (cid:12)(cid:12) , − (cid:11) and f (cid:12)(cid:12) , + (cid:11) . At a certain quasi-momentum k , wecan choose the cellular functions as related by time-reversal and inversion symmetries, so that the Berry con-nection matrices of the two spin sectors satisfy ~R ↓ = − ~R ∗↑ . When the QWs are resonantly excited by a lin-early polarized NIR laser, the initial hole spinor statesare φ s = g s (1 , T ( g s is a constant with modulus 1).Angular momentum conservation law requires that thetotal angular momentum of the holes is zero initially.Right before inter-valence-band tunneling happens, thehole spinor states are still of the form φ s = g ′ s (1 , T ( g ′ s is a constant containing the dynamic phase andAbelian Berry phase with modulus 1), and the totalangular momentum of the holes remains zero, as dis-cussed in the case of Abelian Berry connection. To seehow non-Abelian Berry connection induce angular mo- mentum changes of the holes, we calculate the spinorstate φ ↑ from the dynamical equation, Eq. 9, with ini-tial condition φ ↑ (0) = g ′↑ (1 , T in two steps, consideringonly the off-diagonal elements of the Berry connectionfor the first step, and the dynamic phases and AbelianBerry phases for the second. In the first step, the holespinor φ ↑ (0) evolves to φ ↑ (∆ t ) = g ′↑ (1 , iλ ∆ t ) T , where λ = ˙ k (0) · ~R ∗↑ , [ k (0)]. Denote the cellular functions at k (∆ t ) as | HH , ↑ i k (∆ t ) = f ( a (cid:12)(cid:12) , + (cid:11) + b (cid:12)(cid:12) , − (cid:11) ), and | HH , ↓ i k (∆ t ) = f ( c (cid:12)(cid:12) , + (cid:11) + d (cid:12)(cid:12) , − (cid:11) ). The angularmomentum of the hole spinor φ ↑ (∆ t ) can be calculatedas J ↑ = (3 / | a + iλ ∆ tc | − (1 / | b + iλ ∆ td | . (B5)Similarly, for φ ↓ (∆ t ) with initial state φ ↓ (0) = g ′↓ (1 , T , we have J ↓ = ( − / | a ∗ − iλ ∗ ∆ tc ∗ | +(1 / | b ∗ − iλ ∗ ∆ td ∗ | . (B6)The total angular momentum of the holes is J ↑ + J ↓ = 0.In the second step, the hole spinor φ ↑ (∆ t ) evolves into φ ↑ (2∆ t ) = g ′↑ ( e i (Γ D, +Γ B, ) , e i (Γ D, +Γ B, ) iλ ∆ t ) T , whereΓ D,j = − E cv,j [ k (∆ t )]∆ t/ ~ , Γ B,j = ~R ↑ ,jj [ k (∆ t )] · ˙ k (∆ t )∆ t ( j = 1 ,
2) are the dynamic phase and AbelianBerry phase respectively. The hole spinor φ ↑ (∆ t ) getthe same dynamic phases but opposite Berry phases,i.e., φ ↓ (2∆ t ) = g ′↓ ( e i (Γ D, − Γ B, , e i (Γ D, − Γ B, ) iλ ∆ t ) T . De-note the cellular functions at k (2∆ t ) as | HH , ↑ i k (2∆ t ) = f ( a (cid:12)(cid:12) , + (cid:11) + b (cid:12)(cid:12) , − (cid:11) ), and | HH , ↓ i k (2∆ t ) = f ( c (cid:12)(cid:12) , + (cid:11) + d (cid:12)(cid:12) , − (cid:11) ). After some algebra, we canget the total angular momentum for the hole spinors at2∆ t as J = 2∆ t sin(∆Γ D ) ℜ [( b ∗ d − a ∗ c ) λ ∗ e i ∆Γ B ] , (B7)where ∆Γ D = Γ D, − Γ D, and ∆Γ B = Γ B, − Γ B, .Thus, a nonzero total angular momentum is inducedby electron-hole pairs associated with different energybands. Appendix C: Semiclassical carrier dynamics
To demonstrate how the non-Abelian Berry connec-tion affects the dynamics of the electron-hole pairs, twosemiclassical trajectories with different relative phasesbetween the NIR and THz laser fields are shown in Fig. 8.As can be seen in Fig. 2, the relative strength of the par-allel and perpendicular sidebands of 20th and 60th or-ders are very different for the 10 nm AlGaAs QWs. Thephases of the THz field for ionization and recollision forthe two orders of HSG are labeled in the upper rightof Fig. 8. The trajectory associated with the 60th or-der sideband is about 100 fs longer than the trajectoryassociated with the 20th order sideband.The details of the trajectory associated with the 20thorder sideband are plotted in the lower left two graphs7
FIG. 8. Semiclassical trajectories for the n = 20 and n = 60 sidebands in the 10 nm AlGaAs sample. (upper left) Fullperpendicular HSG spectrum experimentally measured in the 10 nm AlGaAs sample oriented at 47 ◦ . (upper right) Time traceof the THz electric field. The arrows point to the time instants of ionization, i n , and recollision, r n , for the two sidebandsconsidered here. (middle left) The real-space trajectories of the electron and hole for the 20th order sideband. The spin stateis ~η e = [ ↑ , ↓ ] T for the electron, and ~η h = [ | HH1 i , | HH2 i ] T for the hole. Each arrow represents a spin/pseudo-spin in xz-planefor each of seven instants throughout the trajectory. (lower left) The location of the hole in the valence subbands at each ofthose seven instants with the area of each maroon circle representing the relative weight in either subband. (middle right) Thereal-space trajectories of the electron and hole for the 60th order sideband, with the spinor directions drawn for eight instants.The hole spinor almost entirely flips to the HH2 state at recollision. (bottom right) The location of the hole in the valencesubbands at each of the eight instants. For both cases, the electron is always in spin-down state and the y-component of thehole pseudo-spin is approximately zero. Comparing the two cases shows the scale of the effects of non-Abelian Berry curvature. in Fig. 8. Over the course of the acceleration step, theelectron and hole paths separate by almost 30 nm attheir farthest. The spinor states of the two particles areplotted at seven different time instants (the spinor di-rections are chosen for excitation by a σ + NIR photon).The spin state of the electron does not change, but thehole pseudo-spin rotates by a large amount in the last 50fs. Notice that the position at recollision is not at ex-actly zero, but slightly positive. This translation resultsmainly from the non-parabolic nature of the hole sub-bands. If the masses of the electron and hole remainedthe same in the entire process, the recollision would oc-cur at exactly zero. The hole states at the same seven in-stants are shown in the band structure underneath, wherethe area of each circle represents the probability ampli-tude of being in the subband. At instant −
7, the hole spinor rotates significantly asit passes through the avoided crossing point of HH1 and HH2 subbands. The majority of the spinor weight is thenin the HH2 subband.The details of the 60th order trajectory are plotted inthe lower right two graphs in Fig. 8. The non-AbelianBerry curvature has a much stronger effect. The electronand hole travel much further apart for this trajectory, andthe hole spinor rotates more substantially, shown now ateight instants in the trajectory. The location of recolli-sion is shifted almost 10 nm away from the origin. At t = 120 fs, instant Sideband offset energy (meV) S i deband R a t i o ( I / I ) FIG. 9. Theoretical calculation of sideband intensity ratio I ⊥ /I k for the 5 nm GaAs sample at 55 ◦ with two differentwell widths for the conduction band, 5 nm (red curve) and8.13 nm (black curve replotted from Fig. 6). The blue line isthe result if the Berry connection is assumed to be zero. Appendix D: Effects of well width
We show that the deviation of the ratio I ⊥ /I k betweenexperiment and theory for the 5 nm GaAs QWs at 55 ◦ could be explained by the overestimation of Berry con-nection in the calculation. In Fig. 6, the effective wellwidth for the conduction band is taken to be L e = 8 . L e = 5 nm without changing all other parameters, weobtain a sideband ratio close to 1, as observed in theexperiment. The conduction subbands have higher en-ergy levels for narrower QWs. If the effective mass of theconduction subbands do not depend on the effective wellwidth L e , as in our calculation, the k -space region, inwhich the conduction band energy of the QWs lies belowthat of the AlGaAs barriers, will be smaller for narrowerQWs. For QWs with L e = 5 nm, at 55 ◦ , the largest mo-mentum that an electron can have before it steps into thebarrier is calculated to be about 0.08 1/a. To generatea sideband, the electron and hole should have the samemomentum, so the relevant k -space region in HSG for thevalence subbands lies between ± .
08 1/a along the lat-tice direction at 55 ◦ . In this k -space region, the highestvalence subband is nearly parabolic (see Fig. 3) and theBerry curvature is close to zero, which implies that thereis almost no dynamical birefringence. For QWs with L e = 8 .
13 nm, a hole has chances to pass the avoidedcrossing point, where the Berry curvature is relativelylarge, which could induce a larger degree of dynamicalbirefringence. Therefore, we might have had an overesti-mation of the Berry connection by using a larger effectivewell width in the calculation in Fig. 6.
Appendix E: Dynamical Jones Calculus
The Jones calculus is a convenient formalism for de-scribing the propagation of perfectly (or fully) polarizedlight through linear optical media and components [63].The Jones calculus manipulates the Jones vector, a com-plex two-component vector that can only describe per-fectly polarized light. This formalism handles interfer-ence phenomena naturally and can be used with any or-thogonal polarization state basis, such as linear or circu-lar. Because the NIR laser and sidebands are perfectlypolarized, as was shown in Sect. IV, and interference iscentral to our model, we generalize the Jones calculus tothe nonlinear optical phenomenon of HSG.Familiar linear optical elements, like polarizers andwave plates, can each be associated with a conventionalJones matrix. In the dynamical Jones calculus, the THz-driven quantum well acts as the optical element, and theJones matrix relates the polarization state of each side-band to that of the incident NIR laser. For each sideband,we assign a Jones matrix J n = J ij,n , which is defined as (cid:18) E x, HSG E y, HSG (cid:19) n = (cid:18) J xx J xy J yx J yy (cid:19) n (cid:18) E x, NIR E y, NIR (cid:19) . (E1)Following the input of a NIR laser field described byJones vector ( E x, NIR , E y, NIR ) T , the THz-driven quantumwell produces HSG with Jones vector ( E x, HSG , E y, HSG ) T .The elements of dynamical and conventional Jones vec-tors and Jones matrices are in general complex. If HSGwere an isotropic effect, J n would be proportional to theidentity matrix. Dynamical linear birefringence is the dy-namical analog to the familiar linear birefringence thatis observed in a material like calcite. In the case of puredynamical linear birefringence, J n is diagonal, with di-agonal elements having different complex phases.The J n can be determined experimentally. Measure-ments of the polarization angles α and γ for more thanthree different polarization states of the NIR laser de-termine J xy,n /J xx,n , J yx,n /J xx,n , J yy,n /J xx,n . Togetherwith one measurement of sideband intensity for a certainNIR laser field, the dynamical Jones matrix can be de-termined to within an overall phase factor, which can bemeasured through time-resolved experiments (see Sup-plementary Material [58] for more details).In the experiment, the polarization angles of the side-bands and their intensities are measured independently.However, both the polarization state and the intensityof a sideband are determined by the dynamical Jonesmatrix J n . Using the J n formulation, the sideband in-tensity ratio I ⊥ /I k can be derived from the polarizationellipse measurements. This provides a way to check theconsistency of the experiments.Fig. 10 compares the ratio I ⊥ /I k calculated from thepolarization state measurements (from Fig. 5(b)) withthe one calculated from the sideband intensity measure-ments (from center left of Fig. 6). To calculate the ratiofrom the polarization states, Eq. E1 was used with the9 FIG. 10. Dynamical birefringence ratio I ⊥ /I k from two in-dependent measurements. The black squares are from theintensity measurements (center left, Fig. 6). The red squaresare obtained from the unitary Jones matrices J n calculatedfrom the polarization state angle measurements from Fig. 5. three data sets from Fig. 5(b) to calculate J xy,n /J xx,n , J yx,n /J xx,n , and J yy,n /J xx,n . The values were then usedin Eq. E1 to calculate the expected sideband intensityratio given an input NIR laser polarization state. Itshould be noted that no information of the intensitiesof the sidebands is used, only the polarization states ofthe sidebands and NIR laser.The polarization state measurements remarkably wellwith the intensity measurements, matching the trend ofincreasing sideband intensity ratio with increasing order,as well as the overall scale. Agreement between tech-niques is not as good for the GaAs sample (see Supple-mentary Material [40]). The deviation between the twomethods is likely due to a systematic error in the Stokes polarimeter used to measure the NIR laser and sidebandpolarization states. The polarimeter is sensitive to theexact retardance value of the quarter wave plate used(see Supplementary Material [40] for details). Improvingthe accuracy of these measurements is outside the scopeof the current work. The internal consistency betweenthe sideband intensity measurements and the polariza-tion state measurements—which is independent of anyinputs to simulation—strongly supports the validity ofour theoretical approach.Jones matrices can also be computed from the theoryand compared with experiment. In the context of theory,it is natural to express the Jones vectors and matrices onthe basis of circularly polarized states σ ± . With thisbasis, we define the Jones matrices T n = T ij,n as (cid:18) σ +HSG σ − HSG (cid:19) n = (cid:18) T ++ T + − T − + T −− (cid:19) n (cid:18) σ +NIR σ − NIR (cid:19) . (E2)Following the input of a NIR laser field described byJones vector ( σ +NIR , σ − NIR ) T , the THz-driven quantumwell produces HSG with Jones vector ( σ +HSG , σ − HSG ) T .The Jones matrices T n and J n are related by a uni-tary transformation. 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