Present address: ]Raymond and Beverly Sackler School of Physics and Astronomy,
Tel Aviv University, Tel Aviv 69978, Israel

# All-optical probe of three-dimensional topological insulators based on high-
harmonic generation by circularly-polarized laser fields

Denitsa Baykusheva<EMAIL_ADDRESS>Stanford PULSE Institute, SLAC
National Accelerator Laboratory, Menlo Park, California 94025, USA  Alexis
Chacón  Center for Nonlinear Studies and Theoretical Division, Los Alamos
National Laboratory, Los Alamos, New Mexico 87545, USA  Department of Physics
and Center for Attosecond Science and Technology, POSTECH, 7 Pohang 37673,
South Korea  Max Planck POSTECH/KOREA Research Initiative, Pohang 37673, South
Korea  Jian Lu  Present address: Department of Physics and Astronomy,
University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA
19104-6396, USA  Stanford PULSE Institute, SLAC National Accelerator
Laboratory, Menlo Park, California 94025, USA  Trevor P. Bailey  Department of
Physics, University of Michigan, Ann Arbor, Michigan 48109, USA  Jonathan A.
Sobota  Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, Menlo Park, California 94025, USA  Hadas Soifer  [
Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, Menlo Park, California 94025, USA  Patrick S.
Kirchmann  Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, Menlo Park, California 94025, USA  Costel R. Rotundu
Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, Menlo Park, California 94025, USA  Ctirad Uher
Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
Tony F. Heinz  Stanford PULSE Institute, SLAC National Accelerator Laboratory,
Menlo Park, California 94025, USA  David A. Reis  Stanford PULSE Institute,
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
Shambhu Ghimire<EMAIL_ADDRESS>Stanford PULSE Institute, SLAC
National Accelerator Laboratory, Menlo Park, California 94025, USA

###### Abstract

We report the observation of a novel nonlinear optical response from the
prototypical three-dimensional topological insulator Bi2Se3 through the
process of high-order harmonic generation. We find that the generation
efficiency increases as the laser polarization is changed from linear to
elliptical, and it becomes maximum for circular polarization. With the aid of
a microscopic theory and a detailed analysis of the measured spectra, we
reveal that such anomalous enhancement encodes the characteristic topology of
the band structure that originates from the interplay of strong spin-orbit
coupling and time-reversal symmetry protection. Our study reveals a new
platform for chiral strong-field physics and presents a novel, contact-free,
all-optical approach for the spectroscopy of topological insulators. The
implications are in ultrafast probing of topological phase transitions, light-
field driven dissipationless electronics, and quantum computation.

###### pacs:

## Introduction

High-harmonic spectroscopy is an all-optical method to probe the structure and
ultrafast dynamics of atoms and molecules subjected to strong laser fields [1,
2]. The basic idea is based on the underlying microscopic mechanism of the
high harmonic generation (HHG) process, which can be largely understood with
the aid of a three-step recollision model [3, 4] comprising sub-cycle
tunneling of electrons, their laser-field acceleration in the vacuum, and
their subsequent returns with possible recombination to parent ions. The
recombination is accompanied by the emission of high-order harmonics whose
amplitude and phase depends on the recollision angle and time, providing HHG
with an atomic-scale spatial and temporal resolution [5]. Analogous processes
are also being investigated in solid-state HHG [6, 7, 8, 9, 10], leading to
the possibility of compact attosecond light sources [11, 12, 13, 14] and to
extend these all-optical spectroscopic abilities to solid materials such as
semiconductors [15], dielectrics [8] and to a range of quantum materials
(QMs). Recent experimental efforts on high-harmonic spectroscopy of QMs
include the observation of non-perturbative high-harmonics from graphene [16],
monolayer transition metal dicacogenides (TMDC) [17], and topological
insulators (TIs) [18, 19, 20] subjected to intense laser fields without
damage.

Three-dimensional TIs (3D TIs) are characterized by the presence of insulating
bulk bands and conducting surface bands forming an odd number of Dirac cones
within the bulk band gap as a result of band inversion owing to the strong
spin-orbit coupling [21, 22]. Due to a multitude of emergent phenomena,
ranging from the unique spin texture [23] supporting helical currents [24] to
dissipationless transport [25], TIs are also anticipated to serve as building
blocks for future technology. In particular TIs may host the fastest optical
transients [26], and have applications in spintronics, in quantum sensing and
metrology, as well as in quantum computation [27]. Therefore, it is desirable
to develop advanced methods that can probe TIs including their non-equilibrium
states. Currently, scanning tunnelling microsocpy (STM) and angle-resolved
photoemission spectroscopy (ARPES) [28] are the most common methods to probe
TIs, although they require stringent sample environments and are not
compatible with the use of strong light-fields. Therefore, the development of
purely optical methods that are endowed with the required temporal resolution
and are compatible with a variety of sample environments including external
magnetic fields [29] and ambient conditions is highly desired.

To this aim, in addition to the demonstration of symmetry breaking at the
surface through the measurement of even-order harmonics [18, 19, 20], it is
crucial to elucidate the imprint of characteristic features of 2D TIs on the
high harmonic spectra, particularly the strong spin-orbit coupling (SOC). The
SOC gives rise to the band inversion and the crossing of the surface valence
and conduction bands at or near the Fermi level. In a recent theoretical work,
we predicted a unique enhancement of high harmonics for a circularly-polarized
driving laser fields from the surface state of a prototypical topological
insulator system Bi2Se3 [30]. Here, we report experimental results. We use
sub-band-gap, ultrashort, intense mid-infrared (MIR) laser pulses to induce
non-perturbative high-harmonic generation in ultra-thin single-crystals using
transmission geometry, as well as in freshly-cleaved bulk crystals in
reflection mode by avoiding material’s self absorption of high-energy photons.
First, we discuss results from linearly polarized laser fields, and analyse
the intensity and polarization of high harmonics with respect to the crystal
orientation. This allows us to identify spectral features that we attribute to
the contribution from the surface. Then, we present main results, which
include the anomalous high-harmonic response of the topological material to
circularly polarized laser fields.

## Results and Discussion

### Non-perturbative high-harmonic generation driven by ultrashort MIR pulses

Bi2Se3 (space group $R\overline{3}m$, $D_{3d}^{5}$, $\sharp$166) [31] has a
layered rhombohedral structure composed of covalently bonded quintuple layers
(QLs) stacked along the trigonal axis and stabilized by weak van der Waals
interactions (Fig. 1 a). The bulk has a moderately large band gap ($\sim 0.3$
eV), which makes it suitable for non-resonant, below-band-gap excitation in
the MIR wavelength range. Our experimental setup, outlined in detail in the
Materials and Methods section below as well as in Sec. LABEL:sec:setup in the
SM, allows for the generation of highly tunable MIR laser pulses spanning the
range between 5 and 10 $\mathrm{\mu m}$ (0.12 – 0.25 eV). The MIR laser is
focused to a spot size of $\sim 300\times 300$ $\mu$m2, yielding vacuum peak
intensities of up to $I_{0}\sim 0.03$ TW/cm2 (field strength in the sample
$\sim 0.03$ V/Å), see supplementary material (SM) for details.

The band structure of the surface states (Fig. 1 b) features a single Dirac
point characterized by nearly linear dispersion. In our experiments, we
generate high harmonics in transmission, at normal incidence, from thin films
of sub–100-nm thickness deposited on a (111)-oriented, optically isotropic
BaF2-substrates. To confirm that the emission is from the pristine sample, we
also perform similar measurements in freshly cleaved (in air) bulk Bi2Se3, but
using specular-reflection-geometry to avoid absorption losses (see Materials
and Methods section). A typical MIR-driven HHG spectrum comprising both even
and odd harmonic orders is shown in panel d of Fig. 1, displaying distinct
harmonic peaks spanning the range from above-band-gap harmonic order (HO) 13
($\sim 1.85$ eV) to 17 ($\sim 2.41$ eV). The experimentally determined
dependence of the harmonic yield on the driving field intensity presented in
Fig. 1 e clearly demonstrates the non-perturbative response. All harmonic
orders exhibit a behavior markedly different from the predictions of the
perturbative response ($I_{\mathrm{HHG}}\propto I_{0}^{h}$, whereby $h$ is the
harmonic order).

Figure 1: High harmonics from Bi2Se3: (a): Real space crystal structure
Bi2Se3. The crystal is composed of alternating plane layers of Bi and Se
atoms, aligned along the $z$-direction. The building unit of the lattice is
formed by quintuple layers (QL). Each QL features five atoms: 2 symmetry-
equivalent Bi sites, 2 equivalent Se sites (Se(1)), and a third Se atom
(Se(2), which coincides with the inversion center). (b): 3D-representation of
the energy dispersion ($k_{\parallel}\leq 0.4$ Å-1) of the surface states.
(c): Schematics of a crystal-orientation dependence study, where HHG yield is
recorded as a function of the azimuthal angle $\varphi$, defined as the offset
between the mirror plane of the crystal (dashed line) and the plane of
incidence. (d): Representative harmonic spectrum from a bulk sample, recorded
in reflection geometry ($\lambda_{\mathrm{MIR}}=8.7$ $\mathrm{\mu m}$). The
peak marked with an asterisk is an artefact caused by stray light. Due to the
limitations in our detection system, lower-order harmonics are not accessible.
(e): Intensity dependence of the HHG yield from a thin film sample in
transmission geometry ($\lambda_{\mathrm{MIR}}=5.2$ $\mathrm{\mu m}$). The
yellow dashed line indicates the power dependence of HO 7 predicted by the
perturbative scaling law. Crystal structure schemes are adopted from [30].

### Polarization-resolved analysis of the high-harmonic response

Bulk Bi2Se3 is centrosymmetric, and exhibits a three-fold rotational symmetry
($C_{3}$) along the $z$-axis. However, at the surface ($xy$-plane in Fig. 1 c,
$P3m1$ wallpaper group ($C_{3v}$)), inversion symmetry is necessarily broken
and there is a mirror axis along the $y$-direction. This distinction allows us
to attribute the observed even-harmonic generation (cp. Fig. 1 d as well as
Fig. LABEL:fig:orient_integ in the SM) to the participation of surface
electronic states in the HHG process. In what follows, we concentrate our
analysis on the three representative harmonic orders: HO 7, 8 and 9 of $7.5$
$\mathrm{\mu m}$ (see SM, Fig. LABEL:fig:orient_integ for spectra).

Panels a and b of Figure 2 show the crystal-orientation-dependent HHG yield
($I_{\mathrm{HHG}}$), disentangled in components parallel
($\vec{E}_{\mathrm{HHG}}\parallel\vec{E}_{\mathrm{MIR}}$) and perpendicular
($\vec{E}_{\mathrm{HHG}}\perp\vec{E}_{\mathrm{MIR}}$) with respect to the
driving laser field ($\vec{E}_{\mathrm{MIR}}$). The three-fold crystal
symmetry is reflected in the six-fold periodicity of the HHG signal as a
function of the azimuthal angle $\varphi$ defined in Fig. 1 c. The parallel
component (panel a) has local maxima along
($\varphi=0^{\circ},60^{\circ}\ldots$) and at $30^{\circ}$
$(\varphi=30^{\circ},90^{\circ}\ldots)$ with respect to the mirror axis. We
note that in the reduced Brilloiuin zone (BZ) of the surface, these two
directions correspond to the high-symmetry directions $\overline{\Gamma M}$
and $\overline{\Gamma K}$, respectively (s. Fig. LABEL:fig:BZ a in the SM). In
contrast, the perpendicular
($\vec{E}_{\mathrm{HHG}}\perp\vec{E}_{\mathrm{MIR}}$) components exhibit
minima along the mirror axes.

Panels c and d of Fig. 2 display the polarization ellipses of HOs 7 to 9 when
the mirror axis is along ($\varphi=0^{\circ}$) and perpendicular
($\varphi=30^{\circ}$) to the laser field, respectively. It is seen that in
both cases, the polarization of the odd orders follows the driving field
($\vec{E}_{\mathrm{HHG}}^{\mathrm{odd}}\parallel\vec{E}_{\mathrm{MIR}}$),
whereas that of the even order (HO 8) depends on the orientation: it is
parallel to the laser field vector when the latter is fixed at
$\varphi=0^{\circ}$, and perpendicular to it for the case
$\varphi=30^{\circ}$. These results can be rationalized with the aid of
symmetry arguments as introduced in Ref. [32] and elaborated for the case of
Bi2Se3 in Ref. [30]. We note that perpendicularly-polarized even orders have
been reported in other non-centrosymmetric materials such as $\alpha$-quartz
[33] and polytype $\varepsilon$-GaSe [34] but the special aspect regarding
Bi2Se3 is that the inversion symmetry is broken only at the surface [35, 19].

Figure 2: HHG orientation and polarization analysis: Measured crystal-
orientation dependent HHG yields for HOs 7, 8, and 9 for fixed polarization
anlges, $\theta_{\mathrm{pol}}$, along (a) and perpendicular (b) to the
polarization direction of MIR laser field. The relative orientation between
the MIR laser polarization (horizontal) and the crystal’s mirror axis (dashed
line) is quantified by $\varphi$, as shown schematically on the left side in
each panel. The curves for various harmonics are separated by an offset of 1
for clarity. Polarization measurement of HOs 7, 8, and 9 for MIR laser field
along (c) and perpendicular (d) to the crystal’s mirror axis. The non-
vanishing signal for HO 8 at $\theta_{\mathrm{pol}}=0^{\circ}$ in panel d is
due to spectral contamination from the strong HO 7 and HO 9. Figure 3:
Calculated HHG response: (a), (b): Calculated HHG spectrum for the bulk system
subject to a linearly polarized MIR pulse with a center wavelength of
$\lambda_{\mathrm{MIR}}=7.5$ $\mu$m and comprising 12 optical cycles with an
intensity of $I_{0}=0.0025$ TW/cm2. The polarization vector is parallel to the
mirror plane ($\varphi=0^{\circ}$) in panel (a) and orthogonal to it in panel
(b) ($\varphi=30^{\circ}$). Panels (c) and (d) show an analogous calculation
for the surface state, whereby the perpendicular component is shown in red.

For comparison, we show calculated HHG spectra for a linearly polarized laser
field oriented along ($\varphi=0^{\circ}$) and perpendicular
($\varphi=30^{\circ}$) to the mirror plane, separately for the bulk and the
surface states in panels a-d of Fig. 3. These calculations follow the standard
semiconductor Bloch equations (SBE) approach [36] and include geometrical
effects, such as (non-Abelian) Berry curvature and Berry connections that are
particularly important in the presence of gapless Dirac bands. The band
structure was obtained using $4\times 4$-tight-binding-model (TBM) Hamiltonian
formulated in Ref. [37] and the topological surface states (TSSs) were deduced
by applying open boundary conditions [38, 39]. The detail calculation results
and theoretical methodology can be found in Ref. [30]. It can be seen that
bulk states produce odd-order harmonics only, while surface state produce both
odd and even-order harmonics.

In particular, when the laser field is oriented along the $\varphi=30^{\circ}$
direction (orthogonal to the mirror plane), the orthogonally polarized even-
order harmonics become significantly more intense. The odd orders in the
direction orthogonal to the driving field vanish for both configurations.
These results capture the experimental observations discussed above on a
qualitative level. The rest of the manuscript is devoted to the novel
ellipticity-dependent response of the material and its connection to the
specific features in the band structure. For this aim, we study how the
harmonic intensity varies with the polarization state of the driving MIR laser
field as the latter is changed from linear to elliptical and fully circular.

### Non-trivial ellipticity response

We study the response to the ellipticity of the laser field by varying the
polarization state of the laser from right-circularly polarized (RCP,
$\epsilon=+1$) through linear (LIN, $\epsilon=0$) to left-circularly polarized
(LCP, $\epsilon=-1$) in small increments. We keep the major polarization axis
fixed along $\varphi=30^{\circ}$ throughout the scan. Figure 4 shows
representative results, and we refer to the supplementary information section
(Sec. LABEL:sec:ell) for additional data sets. Evidently, the intensity of HO
7 increases as the ellipticity increases from $\epsilon=0$ to $\epsilon=\pm
1$, in stark contrast to the response of isotropic gas media or other solid-
state systems [6, 11, 17], where the harmonic signal drops at high
ellipticities. Additionally, HO 7 (cp. Fig. LABEL:fig:ell_supp of the SM) has
the same handedness as the fundamental circularly polarized laser field, and
so does HO 13, which is in accordance with the selection rules for HHG in
circularly polarized fields (cp. Ref. [32]). Both HO 7 and HO 13 show strong
enhancement, approaching an order of magnitude
($I_{\mathrm{HHG}}^{\mathrm{(CPL)}}:I_{\mathrm{HHG}}^{\mathrm{(LIN)}}\approx
6$ for HO 7 and
$I_{\mathrm{HHG}}^{\mathrm{(CPL)}}:I_{\mathrm{HHG}}^{\mathrm{(LIN)}}\approx
10$ for HO 13), compared to the linear polarization for the same peak
intensity.

For CPL fields, selection rules for crystalline media with three-fold
rotational symmetry restrict HHG from the surface to harmonics fulfilling the
requirement $\omega=(3n\pm 1)\omega_{0}$ (with $n\in\mathcal{N}$ and
$\omega_{0}$ being the fundamental frequency) [40, 32]. The presence of
inversion symmetry in the bulk forbids even harmonic generation, hence the
corresponding selection rule reads $\omega=(6n\pm 1)\omega_{0}$. Calculated
harmonic spectra of bulk and surface states driven by a left-circularly
polarized field are displayed in Figs. 5 b and c, respectively. The
experimentally observed even-harmonic emission (HO 8, cp. Fig. 4 b) is only
present in the surface-state spectrum (Fig. 4 e). HO 9 is forbidden for both
the bulk and surface states, which is also consistent with the experimental
observation
($I_{\mathrm{HHG}}^{\mathrm{(CPL)}}:I_{\mathrm{HHG}}^{\mathrm{(LIN)}}\leq
0.15$), as can be discerned in Figs. 5 a and 4 c.

Figure 4: HHG ellipticity dependence: (a), (b), (c): Experimental results;
ellipticity dependent response of HHG measured at normal incidence using an
ultrathin sample. The inset shows the experimental geometry, where the major
axis of the MIR ellipse (indicated by the orange arrow) is kept orthogonal to
the mirror axis (dashed black line) of the crystal throughout the measurement.
(d), (e), (f): Calculated ellipticity response (for a 12-cycle Gaussian pulse,
$I_{0}=0.0075$ TW/cm2 at $\epsilon=0$). The contributions of the bulk (“B”,
dashed lines, empty symbols) and the the surface (“S”, solid lines, filled
symbols) are shown separately. Crystal symmetry prohibits HO 9 for circular
polarization.

In our theoretical analysis, we calculate the HHG response in terms of the
contributions from the bulk and the surface states as described in detail in
Ref. [30]. For all harmonics orders considered, the bulk contribution
decreases monotonically as the ellipticity increases from $\epsilon=0$ to
$|\epsilon|=1$ ( Fig. 4 d, e, f, dashed lines, empty diamonds), whereas the
contribution from the surface (solid curve, filled symbols) allows for
significant yields for fully circular fields. For example, the theory
correctly reproduces the enhancement of the harmonic yield of HO 7 at high
ellipticities as well as the local maxima at $|\epsilon|\approx 0.5$ for HO 9
and the features at intermediate ellipticities for HO 8. There are, however,
some discrepancies in the detailed features such as at around $\epsilon=0$ in
HO 7. Although slightly unusual ellipticity dependence has been reported in
other solid-state systems such as MgO [41] and graphene [16], a substantial
selective enhancement of the co-rotating odd-order harmonics for circularly
polarized laser field represents a truly distinct feature that is reported
here. Here, the origin of non-trivial ellipticity dependent response is unique
to the TSSs, which is in contrast to the response from the bulk states of the
same material. We now turn to elucidating links of such a markedly different
ellipticity-dependent behavior of high harmonics with the physical parameters
of the the surface states in Bi2Se3.

Figure 5: Mechanism of HHG from $\mathrm{Bi_{2}Se_{3}}$ in CPL fields: (a)
Integrated harmonic intensity generated with a CPL MIR driver, normalized with
respect to the average of the signal over the entire ellipticity scan for the
data presented in Fig. 4 a-c. The CPL signal is averaged over the left/right
helicity contributions, and the error bars correspond to one standard
deviation. Panels (b) and (c) illustrate the CPL selection rules for bulk and
surface separately, where $(3n+1)\omega_{0}$ and $(6n+1)\omega_{0}$-orders are
co-rotating (LCP, (+), red), $(3n+2)\omega_{0}$ and $(6n+2)\omega_{0}$-orders
have the opposite helicity (RCP, (-), blue) with respect to the left-
circularly polarized MIR driver . (d) and (e): Dipole matrix element
$\vec{d}_{cv}(\vec{k}_{\parallel})$ connecting the lower and the upper part of
the Dirac cone. The color quantifies the absolute magnitude; the real (panel
(d)) and imaginary (panel (e)) parts of the $\hat{x}$\- and the
$\hat{y}$-components are visualized as a vector field (white arrows) and give
rise to a helical vortex structure around the Dirac point. In panels b, c, d,
and e, primary data from Figs. 3 and 5 of our related work Ref. [30] has been
used.

Panels d and e of Fig. 5 display the transition dipole matrix element
connecting the lower $(v)$ and the upper $(c)$ Dirac cones in reciprocal
space, decomposed in terms of its real (panel d) and imaginary (panel e)
components. In the region of low momenta, the highly localized and singular
nature of the dipole vector forms a helical vortex-like pattern encircling the
Dirac point, as illustrated by the white stream lines in Fig. 5 d. At higher
momenta, hexagonal “ridge-like” regions of increasing dipole amplitude can be
clearly distinguished. These structures are mediated by the higher-order (i.e.
$\mathcal{O}(k_{\parallel}^{3})$), “warping” terms in the effective
Hamiltonian, which cause the hexagonal deformation of the surface energy
surface and also represent the counterpart of the Dresselhaus spin-orbit
interaction in rhombohedral structures [42]. These terms are responsible for
the out-of-plane spin polarization of the surface bands. By varying these
specific parameters in the tight-binding model Hamiltonian, we find that the
enhanced response of relatively low-order harmonics has its origin in the
vortex-like pattern around the Dirac point, whereas the yield of higher-order
harmonics is predominantly mediated by the parameters related to the spin-
orbit coupling and out-of-plane spin polarization [30]. In contrast, the
transition dipole matrix elements of the bulk bands (plotted in panels c and d
of Fig. LABEL:fig:BZ in the SM) have negligible amplitudes in the low momentum
range and exhibit annular maxima only in the momentum range where the energy
separation of the inverted bands (cp. Fig. LABEL:fig:BZ b in the SM) is
smallest. Hence, the demonstrated non-trivial response of CPL field-driven HHG
can be linked to the topology of the band structure that originates in the
interplay of strong spin-orbit couplings and time-reversal symmetry
protection, and it underlines the potential of our technique to probe such
features of electronic structure in a purely optical setting.

Finally, we note that quantitative features, such as the exact positions of
the maxima of HOs 8 and 9 as well as the exact enhancement ratio for HO 7 at
$|\epsilon|=1$, are not captured correctly by our model. This is most likely
due to the simplicity of our model, as we use a $4\times 4$ TBM Hamiltonian
and the adopted parametrization does not reproduce the band structure of
Bi2Se3 with quantitative accuracy, especially in the high-momentum regions of
the band structure. Further, couplings between surface and bulk bands are
neglected. Strictly speaking, the wavefunctions and the dispersions of the
surface bands are accurate only in the low-energy limit (i.e. up to $20~{}\%$
of the BZ away from the zone center). Also, details of the macroscopic
propagation of the emitted HHG are not taken into consideration within the
current theoretical framework. Nevertheless, our model captures the salient
features of the non-linear strong-field response observed experimentally and
thus can provide useful insights into the complex microscopic physics of
3D-TIs in strong laser fields.

## Conclusion and Outlook

In conclusion, we measured an anomalous laser-ellipticity dependence of high
harmonics from the prototypical topological insulator Bi2Se3. It is manifested
in a significant enhancement of co-rotating high-harmonic orders for
circularly-polarized laser fields, which is attributed to the non-trivial
topology of the surface Bloch bands, in particular the dipole couplings and
Berry connections close to the $\Gamma$-point and the influence of the high-
order “warping” terms away from the zone center. Our interpretation is based
on theoretical analysis combining a quantum-mechanical treatment of the HHG
process in the framework of the semiconductor Bloch equations formalism with a
minimal $4\times 4$-tight-binding Hamiltonian for 3D topological insulators.
Identification of relevant experimental observables and their relations to the
topological properties of the system lays the foundation for forthcoming (and
possibly time-resolved) studies of topological phase transitions based on HHG.
For instance, by tracking the efficiency of high-harmonics generated by the
elliptically-polarized laser fields in a pump-probe setup one could monitor
the inter-conversion between two distinct topological phases if the underlying
crystalline symmetries are different, such as in the Weyl semimetal compound
WTe2 [43]. The anomalous ellipticity behavior can be exploited to track the
influence of the spin-orbit coupling in strong-field light-matter interactions
by means of doping Bi2Se3 with lighter elements [44] or magnetic impurities
[45] in a controlled manner. The unique strengths of the HHG-based approach is
the unprecedented access to sub-cycle strong-field driven dynamics [46]. The
reported highly non-trivial response to circularly polarized driving fields
provides novel opportunities for the study of chiral phenomena such as helical
currents on the surfaces of topological materials, with implications for both
attosecond metrology and light-wave-driven nanoelectronics [47, 48, 26, 49]. A
forthcoming theoretical challenge consists in the development of a methodology
for the extraction of topological invariants [50, 51, 46] from experimentally
accessible observables.

## Materials and Methods

### Experimental setup

Femtosecond pulses in the MIR spectral range (5–10 $\mathrm{\mu m}$) are
generated in a setup (see Fig. LABEL:fig:exp in the Supplementary Material)
based on a 1 kHz regenerative amplifier and optical parametric amplification
followed by difference-frequency generation in GaSe. The output of the Ti:Sa
amplifier (Coherent, 7 mJ, 45 fs, with a center wavelength of $\sim$790 nm) is
used to pump a commercial optical parametric amplifier (TOPAS-HE, Light
Conversion) in order to generate signal and idler pulses of $\sim$48–55 fs
typical duration with central wavelengths tuned between 1415–1435 nm and
1715–1820 nm, respectively. Non-collinear type-I-phase-matched difference-
frequency-mixing of the OPA output in a 500 $\mathrm{\mu m}$ thick, $z$-cut
GaSe crystal (Eksma optics) yields MIR pulses tunable in the range 5–10
$\mathrm{\mu m}$ reaching typical energies of 80 $\mathrm{\mu J}$ at $7.5$
$\mathrm{\mu m}$. A 3 mm Ge filter is used to separate the MIR from the
residual signal and idler beams. A thin Ge (0.5 mm) Brewster plate is used to
clean the polarization of the horizontally polarized DFG output. The frequency
of the driving MIR field is quantified using Michelson interferometry, whereas
the pulse duration is estimated using SHG-autocorrelation measurements in a
AgGaS2 crystal. The findings presented in the main text have been obtained
with MIR pulses tunable in the range from $5$ $\mathrm{\mu m}$ to $10$
$\mathrm{\mu m}$ and $\approx 300$ fs in duration. The parameters for the
various datasets presented in the main text and in the SM are summarised in
Tab. S1 in the SM.

The MIR is focused onto the sample using a silver-coated ($90^{\circ}$) off-
axis parabolic mirror of 10 cm focal length. In practice, the sample is
positioned few mm away from the focal point, the resulting combination of a
loose-focusing geometry and a large beam spot on the sample surface minimizes
damage while ensuring sampling over a larger surface area. The beam waist
($1/e^{2}$) at the sample position is 300 $\mathrm{\mu m}$, as determined with
a knife-edge scan measurement. The polarization state of the driving field is
varied with the aid of tunable half- (HWP, $\lambda/2$) and quarter-wave (QWP,
$\lambda/4$) plates designed for the spectral region 1–19 $\mathrm{\mu m}$ (5
mm CdSe, Alphalas). The ellipticity scan measurements presented in Fig. 4 of
the main text as well as Fig. LABEL:fig:ellscan_60 in the SM were performed by
rotating the quarter-wave plate while simultaneously rotating the crystal by
the same angle in order to keep the major axis of the polarization ellipse
aligned to the selected high-symmetry direction. The design of the tunable
waveplate allows for pre-compensating the phase delay introduced by the
different Fresnel coefficients of the S/P components as well as the
distortions introduced by the parabolic mirror by adjusting the tilt of the
waveplate. The polarization of the MIR in the focal plane was confirmed to be
nearly circular, with ellipticities reaching 0.95 (cp. Fig. LABEL:fig:ell_supp
a in the SM).

The harmonic emission is re-collimated, spectrally dispersed in a
monochromator (Acton series, Princeton instruments) and imaged onto a CCD
camera (Andor DO440-BN). All experimental measurements have been conducted at
ambient conditions.

Due to limitations imposed by the instrument response, lower-order (than HO 7)
harmonic orders lie outside of the detection range.

### Sample growth and characterization

Thin Bi2Se3 films of sub-100-nm thickness are grown by molecular beam epitaxy
(MBE) on lattice-matched BaF2 (111) substrates. BaF2 is a wide-band gap
insulator (band gap of $\sim 11$ eV [52]), transparent and optically
isotropic, making it particularly suitable for optical spectroscopy. BaF2
substrates (1 cm2 $\times$ 0.5 mm, polished on both sides) were cleaned in
boiling acetone and ethanol and subsequently baked at $\sim 800^{\circ}$ C for
30 minutes. The substrates were held at $\sim 225^{\circ}$ C during growth and
annealing. A Se:Bi flux ratio of 7 was used during the growth, as calibrated
by the in-situ crystal monitor. The films were annealed for 90 minutes in Se
flux in order to reduce Se vacancy formation. The films were capped by a 2 nm
protective Te layer. The orientation of the Bi2Se3 structure (along the
$c$-axis) as well as the high crystallinity of the films were confirmed by
standard laboratory X-ray diffraction measurements.

## Acknowledgments

We acknowledge Marc Welch for technical assistance. Theoretical calculations
have been performed using the Sherlock HPC cluster at Stanford University as
well as using computational resources provided by the Fundación Pública Galega
Centro Tecnolóxico de Supercomputación de Galicia (CESGA).

## Funding

This work was primarily supported by the US Department of Energy, Office of
Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and
Biosciences Division through the Early Career Research Program. D.B.
gratefully acknowledges support from the Swiss National Science Foundation
(SNSF) through project No: P2EZP2_184255. A.C. acknowledges financial support
by the Max Planck POSTECH/KOREA Research Initiative Program (Grant No.
2016K1A4A4A0192202) through the National Research Foundation of Korea (NRF)
funded by the Ministry of Science, ICT and Future Planning, Korea Institute
for Advancement of Technology (KIAT) grant funded by the Korea Government
(MOTIE) (P0008763, The Competency Development Program for Industry
Specialists) and LANL LDRD project.

## Author contributions

D.B., S.G., and A.C. conceived the project. D.B. and J.L. constructed the
experimental setup. D.B. recorded and analyzed the data. D.B. and A.C.
developed the theoretical model and performed the numerical calculations.
C.R.R. and T.P.B. synthesized and characterized the samples. H.S., J.A.S.,
P.S.K., T.F.H., and D.A.R. contributed in interpreting data. All authors
discussed the manuscript.

## Competing interests

The authors declare no competing interests.

## Data and materials availability

All data needed to evaluate the conclusions in the paper are present in the
paper and/or the Supplementary Materials. Raw data sets used for generating
Figures 1-5 as well as the additional experimental results shown in the
supplementary material will be made available on Zenodo.

## Supplementary materials

Supplementary Text
Figs. S1 to S6
Table S1

## References

  * Itatani  _et al._ [2004] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pépin, J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, Tomographic imaging of molecular orbitals, Nature 432, 867 (2004).
  * Peng  _et al._ [2019] P. Peng, C. Marceau, and D. M. Villeneuve, Attosecond imaging of molecules using high harmonic spectroscopy, Nature Reviews Physics 1, 144 (2019).
  * Kulander  _et al._ [1993] K. C. Kulander, K. J. Schafer, and J. L. Krause, Dynamics of Short-Pulse Excitation, Ionization and Harmonic Conversion BT - Super-Intense Laser-Atom Physics (Springer US, Boston, MA, 1993) pp. 95–110.
  * Corkum [1993] P. B. Corkum, Plasma perspective on strong field multiphoton ionization, Physical Review Letters 71, 1994 (1993).
  * Uzan  _et al._ [2020] A. J. Uzan, H. Soifer, O. Pedatzur, A. Clergerie, S. Larroque, B. D. Bruner, B. Pons, M. Ivanov, O. Smirnova, and N. Dudovich, Spatial molecular interferometry via multidimensional high-harmonic spectroscopy, Nature Photonics 14, 188 (2020).
  * Ghimire  _et al._ [2011] S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Observation of high-order harmonic generation in a bulk crystal, Nature Physics 7, 138 (2011).
  * Vampa  _et al._ [2015a] G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, and P. B. Corkum, Linking high harmonics from gases and solids, Nature 522, 462 (2015a).
  * Luu  _et al._ [2015] T. T. Luu, M. Garg, S. Yu. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, Extreme ultraviolet high-harmonic spectroscopy of solids, Nature 521, 498 (2015), arXiv:1604.03768 .
  * Tancogne-Dejean  _et al._ [2018] N. Tancogne-Dejean, M. A. Sentef, and A. Rubio, Ultrafast Modification of Hubbard $U$ in a Strongly Correlated Material: Ab initio High-Harmonic Generation in NiO, Phys. Rev. Lett. 121, 097402 (2018).
  * Ghimire and Reis [2018] S. Ghimire and D. A. Reis, High-harmonic generation from solids, Nature Physics 10.1038/s41567-018-0315-5 (2018).
  * Ndabashimiye  _et al._ [2016] G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Solid-state harmonics beyond the atomic limit, Nature 534, 520 (2016).
  * Garg  _et al._ [2018] M. Garg, H. Y. Kim, and E. Goulielmakis, Ultimate waveform reproducibility of extreme-ultraviolet pulses by high-harmonic generation in quartz, Nature Photonics 10.1038/s41566-018-0123-6 (2018).
  * Li  _et al._ [2020] J. Li, J. Lu, A. Chew, S. Han, J. Li, Y. Wu, H. Wang, S. Ghimire, and Z. Chang, Attosecond science based on high harmonic generation from gases and solids, Nature Communications 11, 2748 (2020).
  * Nourbakhsh  _et al._ [2021] Z. Nourbakhsh, N. Tancogne-Dejean, H. Merdji, and A. Rubio, High harmonics and isolated attosecond pulses from $\mathrm{Mg}\mathrm{O}$, Phys. Rev. Applied 15, 014013 (2021).
  * Vampa  _et al._ [2015b] G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, All-Optical Reconstruction of Crystal Band Structure, Physical Review Letters 115, 193603 (2015b).
  * Yoshikawa  _et al._ [2017] N. Yoshikawa, T. Tamaya, and K. Tanaka, High-harmonic generation in graphene enhanced by elliptically polarized light excitation, Science 356, 736 (2017).
  * Liu  _et al._ [2017] H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, High-harmonic generation from an atomically thin semiconductor, Nature Physics 13, 262 (2017).
  * Baykusheva  _et al._ [2019] D. Baykusheva, J. Lu, J. A. Sobota, H. Soifer, C. R. Rotundu, P. S. Kirchmann, D. A. Reis, and S. Ghimire, High-harmonic generation from topological insulators, in  _2019 Conference on Lasers and Electro-Optics (CLEO)_ (2019) pp. 1–2.
  * Bai  _et al._ [2020] Y. Bai, F. Fei, S. Wang, N. Li, X. Li, F. Song, R. Li, Z. Xu, and P. Liu, High-harmonic generation from topological surface states, Nature Physics 10.1038/s41567-020-01052-8 (2020).
  * Schmid  _et al._ [2021] C. P. Schmid, L. Weigl, P. Grössing, V. Junk, C. Gorini, S. Schlauderer, S. Ito, M. Meierhofer, N. Hofmann, D. Afanasiev, J. Crewse, K. A. Kokh, O. E. Tereshchenko, J. Güdde, F. Evers, J. Wilhelm, K. Richter, U. Höfer, and R. Huber, Tunable non-integer high-harmonic generation in a topological insulator, Nature 593, 385 (2021).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Reviews of Modern Physics 82, 3045 (2010), arXiv:1002.3895 .
  * Qi and Zhang [2011] X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Reviews of Modern Physics 83, 1057 (2011), arXiv:1008.2026 .
  * Hsieh  _et al._ [2009] D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of unvonventional quantum spin textures in topological insulators, Science 323, 915 (2009).
  * McIver  _et al._ [2012] J. W. McIver, D. Hsieh, H. Steinberg, P. Jarillo-Herrero, and N. Gedik, Control over topological insulator photocurrents with light polarization, Nature Nanotechnology 7, 96 (2012), arXiv:1111.3694 .
  * Roushan  _et al._ [2009] P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh, D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and A. Yazdani, Topological surface states protected from backscattering by chiral spin texture, Nature 460, 1106 (2009), arXiv:0908.1247 .
  * Oliaei Motlagh  _et al._ [2018] S. A. Oliaei Motlagh, J. S. Wu, V. Apalkov, and M. I. Stockman, Fundamentally fastest optical processes at the surface of a topological insulator, Physical Review B 98, 125410 (2018).
  * Moore [2010] J. E. Moore, The birth of topological insulators, Nature 464, 194 (2010), arXiv:1011.5462v1 .
  * Xia  _et al._ [2009] Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, and M. Z. Hasan, Observation of a large-gap topological-insulator class with a single Dirac cone on the surface, Nature Physics 5, 398 (2009), arXiv:0908.3513 .
  * Wu  _et al._ [2016] L. Wu, M. Salehi, N. Koirala, J. Moon, S. Oh, and N. P. Armitage, Quantized Faraday and Kerr rotation and axion electrodynamics ofa 3D topological insulator, Science 354, 1124 (2016).
  * Baykusheva  _et al._ [2021] D. Baykusheva, A. Chacón, D. Kim, D. E. Kim, D. A. Reis, and S. Ghimire, Strong-field physics in three-dimensional topological insulators, Phys. Rev. A 103, 23101 (2021).
  * Zhang  _et al._ [2009] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Topological insulators in Bi2Se3, Bi2Te3 and Sb2Te3 with a single Dirac cone on the surface, Nature Physics 5, 438 (2009), arXiv:1405.2036 .
  * Neufeld  _et al._ [2019] O. Neufeld, D. Podolsky, and O. Cohen, Floquet group theory and its application to selection rules in harmonic generation, Nature Communications 10, 405 (2019).
  * Luu and Wörner [2018] T. T. Luu and H. J. Wörner, Measurement of the Berry curvature of solids using high-harmonic spectroscopy, Nature Communications 9, 916 (2018).
  * Langer  _et al._ [2017] F. Langer, M. Hohenleutner, U. Huttner, S. W. Koch, M. Kira, and R. Huber, Symmetry-controlled temporal structure of high-harmonic carrier fields from a bulk crystal, Nature Photonics 11, 227 (2017).
  * Hsieh  _et al._ [2011] D. Hsieh, J. W. McIver, D. H. Torchinsky, D. R. Gardner, Y. S. Lee, and N. Gedik, Nonlinear optical probe of tunable surface electrons on a topological insulator, Physical Review Letters 106, 057401 (2011).
  * Kira and Koch [2011] M. Kira and S. W. Koch,  _Semiconductor Quantum Optics_ (Cambridge University Press, Cambridge, 2011).
  * Mao  _et al._ [2011] S. Mao, A. Yamakage, and Y. Kuramoto, Tight-binding model for topological insulators: Analysis of helical surface modes over the whole Brillouin zone, Physical Review B - Condensed Matter and Materials Physics 84, 115413 (2011).
  * Shan  _et al._ [2010] W. Y. Shan, H. Z. Lu, and S. Q. Shen, Effective continuous model for surface states and thin films of three-dimensional topological insulators, New Journal of Physics 12, 043048 (2010), arXiv:0911.3706 .
  * Liu  _et al._ [2010] C. X. Liu, X. L. Qi, H. Zhang, X. Dai, Z. Fang, and S. C. Zhang, Model Hamiltonian for topological insulators, Physical Review B - Condensed Matter and Materials Physics 82, 045122 (2010).
  * Saito  _et al._ [2017] N. Saito, P. Xia, F. Lu, T. Kanai, J. Itatani, and N. Ishii, Observation of selection rules for circularly polarized fields in high-harmonic generation from a crystalline solid, Optica 4, 1333 (2017).
  * You  _et al._ [2017] Y. S. You, D. A. Reis, and S. Ghimire, Anisotropic high-harmonic generation in bulk crystals, Nature Physics 13, 345 (2017).
  * Fu [2009] L. Fu, Hexagonal warping effects in the surface states of the topological insulator Bi2Te3, Physical Review Letters 103, 266801 (2009).
  * Sie  _et al._ [2019] E. J. Sie, C. M. Nyby, C. D. Pemmaraju, S. J. Park, X. Shen, J. Yang, M. C. Hoffmann, B. K. Ofori-Okai, R. Li, A. H. Reid, S. Weathersby, E. Mannebach, N. Finney, D. Rhodes, D. Chenet, A. Antony, L. Balicas, J. Hone, T. P. Devereaux, T. F. Heinz, X. Wang, and A. M. Lindenberg, An ultrafast symmetry switch in a Weyl semimetal, Nature 565, 61 (2019).
  * Brahlek  _et al._ [2012] M. Brahlek, N. Bansal, N. Koirala, S. Y. Xu, M. Neupane, C. Liu, M. Z. Hasan, and S. Oh, Topological-metal to band-insulator transition in (Bi1-xInx)2Se3 thin films, Physical Review Letters 109, 186403 (2012).
  * Chang  _et al._ [2013] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator, Science 340, 167 (2013), arXiv:1212.4783 .
  * Silva  _et al._ [2019] R. E. Silva, Jiménez-Galán, B. Amorim, O. Smirnova, and M. Ivanov, Topological strong-field physics on sub-laser-cycle timescale, Nature Photonics 13, 849 (2019), arXiv:1806.11232 .
  * Garg  _et al._ [2016] M. Garg, M. Zhan, T. T. Luu, H. Lakhotia, T. Klostermann, A. Guggenmos, and E. Goulielmakis, Multi-petahertz electronic metrology, Nature 538, 359 (2016).
  * Reimann  _et al._ [2018] J. Reimann, S. Schlauderer, C. P. Schmid, F. Langer, S. Baierl, K. A. Kokh, O. E. Tereshchenko, A. Kimura, C. Lange, J. Güdde, U. Höfer, and R. Huber, Subcycle observation of lightwave-driven Dirac currents in a topological surface band, Nature 562, 396 (2018).
  * Hafez  _et al._ [2018] H. A. Hafez, S. Kovalev, J.-c. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-j. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions, Nature 561, 507 (2018).
  * Chacón  _et al._ [2018] A. Chacón, W. Zhu, S. P. Kelly, A. Dauphin, E. Pisanty, A. Picón, C. Ticknor, M. F. Ciappina, A. Saxena, and M. Lewenstein, Observing Topological Phase Transitions with High Harmonic Generation, Arxiv Preprint , 1807.01616v1 (2018), arXiv:1807.01616 .
  * Bauer and Hansen [2018] D. Bauer and K. K. Hansen, High-Harmonic Generation in Solids with and without Topological Edge States, Physical Review Letters 120, 177401 (2018), arXiv:1711.05783 .
  * Rubloff [1972] G. W. Rubloff, Far-Ultraviolet Reflectance Spectra and the Electronic Structure of Ionic Crystals, Physical Review B 5, 662 (1972).