# Optical pumping of electronic quantum Hall states with vortex light

Deric Session  These authors contributed equally to this work  Joint Quantum
Institute (JQI), University of Maryland, College Park, MD 20742, USA  Mahmoud
Jalali Mehrabad<EMAIL_ADDRESS>Joint Quantum Institute (JQI), University of
Maryland, College Park, MD 20742, USA  Nikil Paithanker  L-NESS, Department of
Physics, Politecnico di Milano, Via Anani 42, 22100 Como, Italy  Tobias Grass
DIPC - Donostia International Physics Center, Paseo Manuel de Lardizábal 4,
20018, San Sebastián, Spain  Ikerbasque - Basque Foundation for Science, Maria
Diaz de Haro 3, 48013, Bilbao, Spain  Christian Eckhardt  Institut für Theorie
der Statistischen Physik, RWTH Aachen University and JARA-Fundamentals of
Future Information Technology, 52056 Aachen, Germany  Max Planck Institute for
the Structure and Dynamics of Matter, Center for Free-Electron Laser Science
(CFEL), Luruper Chaussee 149, 22761 Hamburg, Germany  Bin Cao  Joint Quantum
Institute (JQI), University of Maryland, College Park, MD 20742, USA  Daniel
Gustavo Suárez Forero  Joint Quantum Institute (JQI), University of Maryland,
College Park, MD 20742, USA  Kevin Li  Joint Quantum Institute (JQI),
University of Maryland, College Park, MD 20742, USA  Mohammad S. Alam  Joint
Quantum Institute (JQI), University of Maryland, College Park, MD 20742, USA
Glenn S. Solomon  Department of Physics University of Adelaide Adelaide, SA,
AU  Nathan Schine  Joint Quantum Institute (JQI), University of Maryland,
College Park, MD 20742, USA  Jay Sau  Joint Quantum Institute (JQI),
University of Maryland, College Park, MD 20742, USA  Condensed Matter Theory
Center, University of Maryland, College Park, MD 20742, USA  Roman Sordan
L-NESS, Department of Physics, Politecnico di Milano, Via Anani 42, 22100
Como, Italy  Mohammad Hafezi<EMAIL_ADDRESS>Joint Quantum Institute (JQI),
University of Maryland, College Park, MD 20742, USA

###### Abstract

A fundamental requirement for quantum technologies is the ability to
coherently control the interaction between electrons and photons. However, in
many scenarios involving the interaction between light and matter, the
exchange of linear or angular momentum between electrons and photons is not
feasible, a condition known as the dipole-approximation limit. An example of a
case beyond this limit that has remained experimentally elusive is when the
interplay between chiral electrons and vortex light is considered, where the
orbital angular momentum of light can be transferred to electrons. Here, we
present a novel mechanism for such an orbital angular momentum transfer from
optical vortex beams to electronic quantum Hall states. Specifically, we
identify a robust contribution to the radial photocurrent, in an annular
graphene sample within the quantum Hall regime, that depends on the vorticity
of light. This phenomenon can be interpreted as an optical pumping scheme,
where the angular momentum of photons is transferred to electrons, generating
a radial current, and the current’s direction is determined by the light’s
vorticity. Our findings offer fundamental insights into the optical probing
and manipulation of quantum coherence, with wide-ranging implications for
advancing quantum coherent optoelectronics.

Coherent manipulation of light-matter hybrids plays a crucial role in
advancing future quantum technologies and optoelectronics [1, 2]. Particularly
desirable is the control over the spatial degree of freedom in light-matter
interactions. Typically, due to the presence of disorder or Coulomb binding,
electronic wavefunctions are much more spatially confined than the wavelengths
of associated optical transitions. Consequently, the light-matter interaction
occurs locally, and neither the spatial profile of the optical field nor the
spatial extent of the electron wavefunction has a significant influence on
these interactions, a regime known as the dipole approximation. In other
words, in this regime only direct optical transitions are accessible, and the
transfer of linear and angular momentum, which enables optical control of the
spatial degrees of electrons, is not possible.

To understand this, one can consider a simplified model of a hydrogen-like
atom, where the typical Bohr radius ($a_{\text{B}}$) is much smaller than the
corresponding optical transition wavelength ($\lambda$). Therefore, the next-
order quadrupole transition is weaker by a factor of
$(a_{\text{B}}/\lambda)^{2}$ than the dipole transition, yet it is still
observable in experiments [3]. One approach to enhance such effects is to
shrink the wavelength of the electromagnetic field, which can be achieved by
using plasmonic effects [4, 5]. Alternatively, if the electronic wave function
is coherently extended over the associated optical transition wavelength, and
electrons are more itinerant than bound, then a gross violation of dipole
approximation is expected. A striking example is the quantum Hall system,
where the electrons in two dimensions are subject to a strong out-of-plane
magnetic field. Consequently, the kinetic energy is suppressed and electrons
exhibit cyclotron motions with chiral characteristics that make it a promising
system to investigate the interplay of the chirality of electrons and photons
and the transfer of angular momentum in between [6, 7, 8, 9, 10].

In particular, there has been a growing interest in investigating chiral and
topological effects in photonic systems and also light-matter hybrids [11, 12,
13, 9, 14]. Such topological features can be either in the momentum domain and
lead to Chern bands, or simply in the spatial degrees of freedom, such as
optical vortex beams. Specifically, in addition to spin, in the form of
polarization, light can also carry orbital angular momentum (OAM) [15, 16].
Such an OAM is quantized and given by $\hbar\ell$, where $\ell$ is the mode
number and determines the phase winding of a vortex beam. The interaction of
such vortex beams with the material has led to a plethora of exciting
phenomena [13], such as the orbital photogalvanic effect [17].

In this work, we experimentally demonstrate the transfer of OAM to electrons
in a quantum Hall graphene device with annular geometry using optical vortex
beams. In particular, harnessing non-conventional optical selection rules of
the LLs described in Fig. 1c, we show a vorticity-selective light-matter
interaction between twisted light and the electronic wavefunctions manifesting
as a radial photocurrent (PC). We show that this radial photocurrent only
depends on the vorticity of light as a direct indication of spatially coherent
light-matter interaction. We provide further evidence of the robustness of
this mechanism by comparison with circularly polarized light. Specifically, we
find that the photocurrent contribution from OAM is at least one order of
magnitude larger than the contribution of spin angular momentum
(polarization), allowing us to confirm the significant role of the beam’s
spatial topology, and its ability to control the spatial degree of electrons.

##  I Orbital angular momentum pumping of electrons

Figure 1: Concept of orbital angular momentum pumping: (a) Schematic of LLs in
the annular disc geometry subject to the vertically irradiated vortex beam.
The green surfaces show representative states within a single LL. The electron
distributions in the lowest LLs and directions of the induced radial
photocurrents are indicated by arrows. (b) Optical transitions and
photocurrent generation between inner and outer contacts using $\ell=+1$ (blue
arrows) and $\ell=-1$ (red arrows) for $\sigma^{+}$ polarized light. For
$\sigma^{-}$ transitions see the supplementary information (SI). Light
carrying OAM $\ell=+1$ ($\ell=-1$) increases (decreases) $m$ and hence leads
to expansion (shrinking) of the spatial extent of the electronic wavefunction.
In this way, the shown scheme realizes an analog to optical pumping. (c)
Schematic of the photoexcitation transitions between LLs with negative (red
arrow) vortex beam in the presence of an upward (top) and downward (bottom)
pointing magnetic field. In the top (bottom) panel, the magnetic field is
anti-parallel (parallel) to the helicity of light, leading to the shrinking
(expansion) of the wavefunction’s radius. In (c), only the electron relaxation
is considered.

To present the motivation and a basic understanding of our experiment, we
discuss spatially-dependent light-matter interactions that can manipulate the
spatial degrees of freedom of electrons within a quantum Hall system. In
particular, using a Landau-level picture, we observe how the transfer of
orbital angular momentum from photons to electrons results in a radial
current, where the current’s direction is determined by the light’s vorticity.
As shown in Fig. 1, we consider optical transitions between two Landau levels,
in the presence of rotational symmetry perpendicular to the plane of the
quantum Hall sample. In this scenario, the sample is irradiated by an optical
vortex where each photon is carrying an orbital angular momentum $\hbar\ell$
[6, 8].

During the excitation process, the orbital angular momentum of $\hbar\ell$ is
transferred to electrons [6, 8]. As the radius of the electronic wavefunction
increases monotonically with angular momentum, this optical transfer of
angular momentum causes a radial change in the electronic wavefunction, which
is solely determined by the light’s vorticity. The subsequent relaxation
process conserves OAM on average and therefore maintains the OAM transfer from
the original excitation. This concept is in direct analogy to optical pumping
in atomic systems, wherein cyclical pumping among different hyperfine states
of bound electrons within an atom transfers them to a specific quantum state
[18].

Note that despite the absence of rotational symmetry in the presence of
disorder, the optical pumping model continues to hold true [8]. Moreover, the
optical pumping picture also provides a simple estimate of the resulting
photocurrent: Assuming that OAM pumping was the only mechanism of charge
transport, the OAM needed to carry one electron through the sample equals the
number $M$ of orbitals in a Landau level, which is given by $M=A/(\pi
l_{B}^{2})$, where $A$ is the area of the sample, and $l_{B}$ is the magnetic
length. The estimated transported charge, $\delta Q$, during the time interval
$\delta t$ by $N_{\rm ph}$ photons, is $\delta Q=e\frac{N_{\rm ph}\ell}{M}$.
Therefore, for photons with energy $\hbar\omega$, the photocurrent obtained
from the laser power $P=\hbar\omega\times\frac{N_{\rm ph}}{\delta t}$ is
$I=\frac{\delta Q}{\delta t}=e\frac{P\ell}{\hbar\omega M}$. This picture also
implies that, upon inverting the direction of the magnetic field, the
direction of the photocurrent flips sign, as illustrated in Fig. 1c.

Figure 2: OAM-selective photocurrent generation: (a) Optical-microscope image
of the sample, showing two Corbino devices labeled D1 and D2, one with a
smaller diameter ($\sim 2\ \rm{\mu m}$) and the other with a larger diameter
($\sim 4\ \rm{\mu m}$). D1 is the device used for the data in the main text of
this article, for which the inner and outer contacts are marked. The gray
rectangle shows the metallic back gate. (b) (i) Sample schematics, (ii-iii)
Spatially-resolved PC difference $\Delta_{PC}$ for $\ell=\pm 2$, at $B=9\
\rm{T}$ and $B=-9\ \rm{T}$, respectively. Bias voltage $V_{b}$ for both cases
was $4\ \rm{mV}$. (c), (d) Measured PC as a function of bias voltage $V_{b}$
generated using light carrying $\ell=+2$ (blue) and $\ell=-2$ (red) at (c)
$B=9\ \rm{T}$ and (d) $B=-9\ \rm{T}$. (e-g) Shows the $\Delta_{PC}$ for $B=9\
\rm{T}$ (green) and $B=-9\ \rm{T}$ (brown) for $\ell=\pm 2,\pm 1,0$,
respectively. In (g), at each magnetic field, the $\Delta_{PC}$ is calculated
by subtracting the PC generated using two Gaussian beams with opposite
circular polarization ($\sigma^{\pm}$). In all the panels, the gate voltage
$V_{g}$ and average pump power were $1.78\ \rm{V}$ and $10\ \rm{\mu W}$,
respectively (see SI). Relative differences of curves within and between
panels e and f depend on system parameters, which are described in SI.

##  II Bias voltage dependence of photocurrent

To experimentally demonstrate this mechanism, we use a device consisting of a
hexagonal boron nitride (hBN) encapsulated monolayer graphene in an annular
(Corbino) geometry as shown in Fig. 2a. The inner and outer contacts are used
to apply an in-plane electric field and also measure the generated
photocurrent, while the back-gate voltage $V_{g}$ controls the Fermi level. We
apply an out-of-plane external magnetic field up to 9 Tesla at 4.2 Kelvin to
be in the quantum Hall regime. The optical vortex beams of different vortices
$\ell$ are generated by a spatial light modulator (SLM) and are concentrically
focused on the Corbino device (see the SI for the sample and optical setup).
This enables us to excite the carriers in our device which undergo vorticity-
selective optical transitions, shown in Fig. 1c. For all the measurements, we
choose $V_{g}=\text{1.78 V}$, which sets the Fermi energy near filling factor
$\nu=6$ (see Fig. 4 and associated text for discussion about this choice).

The generated photocurrents for various optical vortices $\pm\ell$ are
independently measured, while the beam is spatially scanned over the sample.
Fig. 2b shows PC difference (subtracted), $\Delta_{PC}$, where the vortex beam
$\ell=\pm 2$ is spatially scanned over the sample (shown schematically in
(i)), at $B=9\ \rm{T}$ (ii) and $B=-9\ \rm{T}$ (iii). It can be clearly
observed that the $\Delta_{PC}$ flips sign when the magnetic field’s direction
is reversed. This remarkable observation corroborates with the earlier optical
pumping picture of Fig. 1a, where depending on the vortices of the optical
beam the radial extent of the electrons either shrink or expand during the
optical excitation. Note that the sign of the observed photocurrent difference
depends solely on the phase winding of the optical vortex beam, and not the
intensity, therefore, one can not associate this (beyond the dipole-
approximation) process with heating.

In order to rule out the origin of this radial photocurrent to an in-plane
electric field, we apply a bias voltage $\rm{V}_{b}$ between the inner and
outer contacts to create a controllable potential gradient in the radial
direction. Fig. 2c-d show the measured PC as a function of $\rm{V}_{b}$ for
$B=9\ \rm{T}$ and $B=-9\ \rm{T}$, respectively (for $B=4\ \rm{T}$ and $B=-4\
\rm{T}$ see SI). Remarkably, we observe multiple unambiguous signatures of the
vorticity-selective light-matter interaction. First, as shown in Fig. 2c-d,
for $\ell=+2$ (blue) and $\ell=-2$ (red) at each magnetic field, we observe a
consistent and significant difference in the generated PC for a wide range of
$\rm{V}_{b}$ ($-10\leq{V}_{b}\leq 10$ mV). In other words, radially tilting
the electric potential can change the total photocurrent, however, the
photocurrent difference $\Delta_{PC}$ remains relatively constant. Second, for
the opposite magnetic field, we observe a clear sign flip for the PC
difference (Fig. 2e). Specifically, since the OAM is defined relative to the
magnetic field, inverting the latter effectively inverts the OAM and should
therefore lead to a sign-change of the observed radial photocurrent, which is
clearly observed in Fig. 2e. In an ideal case, the amplitude of this flipped
current should be the same, however, due to slightly different spatial
alignment for different magnetic fields, the magnitude of the PC is different
(See Fig. 2b (ii) and (iii))). Third, to investigate the effect of the degree
of vorticity on the generated photocurrent, we illuminate the sample with
$\ell=\pm 1$ beams. As shown in Fig. 2, robust photocurrent difference
$\Delta_{PC}^{\ell=\pm 1}$ is observed across a wide range voltage bias, and
the sign reversal with the magnetic field is present. For a large sample
subject to the optical vortex, one expects that $\Delta_{PC}$ to increase with
the vorticity degree $\ell$ [8]. However, since our sample and optical vortex
spatial profile is comparable, and in particular, the spatial overlap for
$\ell=\pm 2$ is smaller than that of $\ell=\pm 1$, we observe a reduced
$\Delta_{PC}$ for the larger value of $|\ell|$. Fifth, in order to rule out
the origin of $\Delta_{PC}$ based on circular polarization, the sample is
illuminated with $\ell=0$ beams consisting of two Gaussian beams with opposite
circular polarization ($\sigma^{+}$ and $\sigma^{-}$). As shown in Fig. 2g,
the difference photocurrent for different circular polarization and $\ell=0$
is at least an order of magnitude smaller than the non-zero $\ell$ cases.
Therefore, we associate the non-zero radial current with the orbital angular
momentum of light, rather than the spin angular momentum.

##  III Polarization dependence of photocurrent

Figure 3: Polarization-resolved and power-dependent photocurrent measurements:
(a) polarization of the pump beam as a function of the QWP rotation, shown
schematically at the top. $\sigma^{+}$ and $\sigma^{-}$ denote positive and
negative-circular polarization. Between each consecutive circular
polarization, continuous rotation of the QWP changes the polarization
periodically from circular to elliptical, to linear polarization. PC measured
as a function of QWP rotation using light carrying $\ell=+2$ (blue) and
$\ell=-2$ (red) at (b) $B=9\ \rm{T}$ and (c) $B=-9\ \rm{T}$, respectively. The
green vertical dashed lines indicated the consecutive $\sigma^{+}$ and
$\sigma^{-}$ polarization. All measurements are done at a fixed $V_{g}=1.78\
\rm{V}$, $V_{b}=-4\ \rm{mV}$, and average pump-power of $10\ \rm{\mu W}$. (d),
(e) Pump-power-dependence of PC: photocurrent difference $\Delta_{PC}$
measured as a function of average pump-power using light carrying (d)
$\ell=\pm 2$ and (e) $\ell=\pm 1$ orbital angular momentum at $B=9\ \rm{T}$
(green circles) and $B=-9\ \rm{T}$ (brown circles), respectively. The black
lines show the least-squares linear fit. In all of the data presented in the
main text, the average pump power was set to 10 $\rm{\mu W}$, at which a
linear behavior is observed.

To further decouple the role of the polarization of the optical field from OAM
in our experiments, we perform polarization-resolved PC generation using OAM
of light by using a variable quarter-wave plate (QWP) in the excitation path.
We rotate the QWP such that the polarization continuously changes from linear
to elliptical to circular while measuring the PC (Fig. 3a). The measured PC
for $\ell=+2$ (blue) and $\ell=-2$ (red) at $B=9\ \rm{T}$ and $B=-9\ \rm{T}$
is shown in Fig. 3b-c.

Here, there are three clear observations confirming the robustness of OAM-
induced PC generation. First, the OAM-induced PC difference is almost an order
of magnitude larger than the amplitude of the current oscillations induced by
QWP rotation. Second, the OAM-induced PC difference never changes sign as a
function of QWP rotation, further confirming the domination of the observed
vorticity-selective PC. Finally, this measurement sheds light on the role of
focusing the optical beam on its polarization properties. This is crucial
since in our measurements, we focus the beam onto the sample using an aspheric
lens with a numerical aperture (NA) of 0.68. This large NA may distort the
polarization beyond the paraxial approximation. However, Fig. 3b-c shows that
this distortion of polarization is relatively insignificant, and it does not
affect the validity of our results.

##  IV Power dependence of photocurrent

We also investigate the power dependence of the OAM-induced PC in our
measurements as a function of optical pump power by varying the average power
of our excitation beam within the range 0.1 to 18 $\rm{\mu W}$. Fig. 3d-e
shows that the generated PC increases linearly with the pump intensity. From
the above optical pumping estimate, we have $l_{B}\approx\frac{26\ {\rm
nm}}{\sqrt{B/{1\ \rm Tesla}}}\approx 9$ nm, and $M=A/(\pi l_{B}^{2})\approx
3\times 10^{4}$. The photocurrent is estimated to be $I\approx(0.1\ {\rm
nA}/{\rm\mu W})P$. Interestingly, the slope of the experimental power
dependence ($\sim 1\ {\rm nA}/{\rm\mu W}$) significantly exceeds this
estimate. As we discuss in more detail below, the PC signal also exhibits a
strong gate voltage dependence due to relaxation effects, and the experimental
values are taken at local PC difference maxima, where significant carrier
multiplication can be expected [19]. In addition, the donut-shaped intensity
profile of the light, not taken into account in the estimate, enhances the
current flow into the outer contact.

Additionally, the OAM-induced PC differences increase linearly with pump
power. This is in apparent contrast to the theoretical prediction of sub-
linear 2/3 power law due to the Pauli-blockade in Ref. [8]. However, there are
two distinctions. In Ref. [8], the transition between two consecutive Landau
levels ($\text{LL}_{0}$ and $\text{LL}_{1}$) was considered, while in our
experiment, our excitation energy is much higher than the cyclotron energy
$\approx 100$ meV and the optical wavelength of 940 nm corresponds to
transitions between Landau level of about $n\approx 40$ (see SI for details).
Moreover, the weak average excitation pump power used in all of our
experiments ($\approx 10\ \rm{\mu W}$) places our result away from any
nonlinear regime, indicating that Pauli blocking only has an insignificant
effect in our observations.

##  V Gate voltage dependence of photocurrent

Next, we investigate the role of the Fermi energy ($E_{\text{F}}$) in the
generated vorticity-selective PC. By changing the gate voltage $V_{g}$, we
tune $E_{\text{F}}$ between LLs, that is, we tune the LL filling factor $\nu$.
As shown in Fig. 4b, the subtracted PC (for $\ell=\pm 2$) changes sign as a
function of $V_{g}$, with the PC’s direction changing sign twice between each
consecutive conductance peak. In Fig. 4b, green (brown) denotes the subtracted
PC at $B=9\ \rm{T}$ ($B=-9\ \rm{T}$), respectively. Recently, an effect known
as the “bottleneck effect”, was developed to describe the observation of a
similar PC direction change as a function of $V_{g}$ in a rectangular geometry
[19]. This picture describes this $V_{g}$-dependent sign based on the relative
position of the electrons and holes compared to the Fermi energy and their
propagation through the edge states. However, in contrast to graphene in a
rectangular geometry, in our Corbino device, edge states do not contribute to
the measured PC. Therefore, our measurements are only sensitive to bulk
physics. More discussion about this can be found in the SI.

Figure 4: Gate voltage dependence of photocurrent: (a) Schematic of graphene’s
band structure near the Dirac point with LLs shown in yellow and purple for
the electron and holes, respectively. The inset shows the density of states
(DOS) as a function of energy with the Fermi energy. (b) Measured PC
difference as a function of gate voltage using light carrying $\ell=\pm 2$ at
$B=9\ \rm{T}$ (green) and $B=-9\ \rm{T}$ (brown), respectively. The top x-axis
shows the calculated integer filling factors (see SI). The two-terminal
longitudinal conductance measured at $B=9\ \rm{T}$ (black circles) is also
shown. This measurement was done at $V_{b}=-4\ \rm{mV}$, and an average pump-
power of $10\ \rm{\mu W}$. (c) Landau fan measured via magneto-transport as a
function of the magnetic field, gate voltage (carrier density marked on the
top x-axis). The conductance is normalized for each magnetic field value. The
white dashed lines denote integer filling factors calculated from the
conversion of gate voltage to carrier density (see SI).

##  VI Outlook

In summary, we demonstrated a novel mechanism to impart angular momentum from
photons to electrons. This was made possible by the fact that the electronic
wavefunction experiences the spatial phase variation of the optical field,
using an optical vortex field. However, the idea of spatially coherent-light
matter interaction can be explored using other optical patterns [20, 16], and
other sample sizes and geometries. This could allow one to fully understand
the role of spatial coherence of the electronic wavefunction.

Moreover, it is intriguing to explore whether such a transfer of momentum
could be exploited as a way to coherently image and eventually manipulate the
wavefunction of electrons. One potential direction is to use THz fields,
instead of optical in our case, to drive the two nearest LLs [21]. The
advantage is that the cascade relaxation is absent in that case, however, the
electromagnetic phase variation occurs on a larger length scale. Nevertheless,
the indication of gradient fields on the quantum Hall system has recently been
observed in the THz domain which makes this platform a promising candidate
[22].

Our results can be understood in a single-particle framework but it may pave
the way for intriguing possibilities in the many-body regime to use twisted
light for the realization of anyons in fractional quantum Hall systems [23,
24], Floquet engineering of topological insulators [25], and producing
electronic skyrmions and vortices [26, 26, 27, 20]. More generally, our system
can host exotic experiments including the investigation of the topological
interplay between electrons and photons [14, 1].

##  VII Acknowledgements

The authors acknowledge fruitful discussions with C. Dean and I. Ahmadabadi.
This work was supported by ONR N00014-20-1-2325, AFOSR FA95502010223, ARO
W911NF1920181, MURI FA9550-19-1-0399, NSF IMOD DMR-2019444, ARL W911NF1920181,
Simons and Minta Martin foundations, and EU Horizon 2020 project Graphene
Flagship Core 3 (grant agreement ID 881603). T.G. acknowledges funding by BBVA
Foundation (Beca Leonardo a Investigadores en Física 2023) and Gipuzkoa
Provincial Council (QUAN-000021-01).

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##  VIII Supplementary Information

##  IX Device fabrication

Figure 5:  Sample fabrication: (a) Schematic of the graphene Corbino sample.
(b) Exploded view of (a).

Graphene was exfoliated from natural graphite crystals (HQ Graphene) and hBN
was exfoliated from commercial (HQ Graphene) synthetic crystals. Monolayer
graphene and hBN were identified based on their color contrast with an optical
microscope, while the thickness of the hBN layers used as a gate insulator was
measured by atomic force microscopy (AFM). A hot pickup technique [28] was
used to stack the hBN/graphene/hBN heterostructure, which was then transferred
onto a pre-patterned local metallic back-gate made of 3 nm of Cr and 2 nm of
Pt. The area for the outer contact was etched using a selective reactive ion
etching (RIE) which was followed by the evaporation of 50 nm of Au to form the
outer contact [29]. Next, a third layer of hBN was dropped on top of the
heterostructure to act as an insulating layer between the overlapping parts of
the contacts. The same selective etch was used to expose the area for the
inner contact and 100 nm of Au was evaporated to make the contact. The
fabricated devices were wire-bonded to chip carriers for the electrical and PC
measurements.

Figure 6:  SdH oscillation measurement: (a) Longitudinal conductance measured
as a function of $1/B$ at $V_{g}=2.5\ V$. Using a Fourier transform to extract
the frequency, the carrier density was calculated to be $1.887\times 10^{12}\
\rm{cm}^{-2}$. (b) A linear fit to the extracted carrier densities as a
function of gate voltage. The slope of the least-squares linear fit is
$(0.614\pm 0.009)\times 10^{12}\ \rm{cm}^{-2}/\rm{V}$ and the y-intercept is
$(0.34\pm 0.02)\times 10^{12}\ \rm{cm}^{-2}$.

##  X Transport measurements

Transport measurements are carried out with low-frequency lock-in techniques
either by biasing with a current of 20 nA or with a voltage at a frequency of
13 Hz. Gate voltage sweeping is performed using a DC source measure unit
(Keithley 2450). The two-terminal longitudinal conductance for various
magnetic fields was measured to obtain the Landau fan (Fig. 4c in the main
text). These measurements are performed using the circuits depicted in Fig.
7a-b.

To calibrate the conversion between gate voltage and carrier density, we
measured Shubnikov-de Haas (SdH) oscillations by setting the gate voltage and
sweeping the magnetic field. The carrier density is extracted from the
frequency of the oscillation ($f$) by $n=4ef/h$ [30]. We extracted the carrier
density for gate voltages between -3 V and 2.5 V and used a linear fit as a
conversion between gate voltage and carrier density, see Fig. 6a-b.

Figure 7:  Experimental setup: (a) The current bias circuit used for transport
measurements. A sinusoidal voltage is provided by the lock-in amplifier. The
constant resistor ($10\ \rm{M\Omega}$) is in series with the sample $R_{S}$.
This larger resistor is used to current bias the sample. The voltage across
the sample is measured through the inputs of the lock-in amplifier. (b) The
voltage bias circuit is alternatively used for transport measurements. We use
the same source as in (a), however, there is no large resistor to convert the
voltage signal to current. In this case, the resulting current from the sample
is passed through a trans-impedance amplifier (TIA), and the resulting voltage
is measured with the lock-in amplifier. (c) The effective circuit for
measuring photocurrent. In this circuit, the sample is modeled as a current
source, $I(P,V_{g})$, which is in series with a DC voltage source, and in
parallel with a resistor (sample resistance). The TIA is in series with the
sample and is connected to both terminals of the lock-in amplifier. (d) The
effective circuit used to gate the sample: in this circuit, the sample is
treated as a capacitor with capacitance $C_{g}$. It is in series with two
$1\rm{G}\Omega$ resistors to decrease the current in the circuit.  Figure 8:
The optical setup used for the photocurrent generation.

##  XI Photocurrent measurements

The sample, inside a variable temperature insert (VTI), is mounted on top of a
piezo-electric stack (scanners (ANSxy100) and positioners (ANPx101, ANPz201)).
It is cooled down to 4.2 K using liquid helium cryogenic with a magnetic field
up to 9 T. The VTI has an optical window on top and a confocal microscope is
built above to optically resolve the sample. The excitation laser is
illuminated through the same window and the laser spot’s alignment to the
sample can be monitored with the microscope. During measurements the laser
power is constantly monitored with a power meter right before the beam enters
the optical window and a feedback is given to a PID loop controlling a laser
power control module. The power control module consists of a DC voltage
applied to a Thorlabs electronic variable optical attenuator.

The two sample contacts are connected to a custom-built trans-impedance
amplifier (TIA) outside of the cryostat [19]. The outputs of the TIA are
connected to a lock-in amplifier (SRS SR860). The excitation laser is chopped
at a frequency of 308 Hz and the lock-in is frequency-locked to the chopper.

We sweep the gate in the same way as in the transport measurement. However, in
the PC measurements, we apply a DC voltage bias across the sample using an SRS
SIM928. The PC measurements are performed using the circuit depicted in Fig.
7c and the optical setup depicted in Fig. 8.

Figure 9:  Beam profiles are imaged with a beam profiler for $\ell=+1$,
$\ell=-1$, $\ell=+2$, and $\ell=-2$. Images are taken of the collimated beam
diverted from the sample path.

##  XII Generation and optimization of OAM beams

To generate beams with orbital angular momentum a Gaussian beam is diffracted
off of a phase-only spatial light modulator (SLM). Ideally, the pattern
displayed to achieve this would be

$h(r,\theta)=\frac{1}{2\pi}\rm{mod}\left(\ell\theta-\frac{2\pi}{D}r\cos(\theta),\
2\pi\right).$ (1)

This blazed hologram maximizes the diffraction efficiency [31, 32]. The
obtained diffraction modes are a superposition of Laguerre-Gaussian (LG) modes
with the same $\ell$ but various $p$ which is the transverse index of the LG
mode [33]. A Meadowlark optics 1920x1152 XY Phase series SLM is used. The
performance of the pixels in the SLM is not homogeneous. The phase hologram
can be modified to compensate for this. The phase hologram used in the
experiment is

$h(r,\theta)=\mathcal{M}(r,\theta)\rm{mod}\left(\mathcal{F}(r,\theta)+\frac{2\pi}{D}r\cos(\theta),\
2\pi\right),$ (2)

where $\mathcal{M}=1+\frac{1}{\pi}\rm{sinc}^{-1}(A(r,\theta))$ and
$\mathcal{F}=\Phi-\pi\mathcal{M}$. The beam can be optimized by setting
$\Phi=\ell\theta$ and manually adjusting the function $A(r,\theta)$ until a
satisfactory beam is reached [34, 35]. The OAM beams after optimization are
shown in Fig. 9.

To verify that the beams used in the experiment have the correct OAM, the
beams’ reflection of the sample was interfered with a Gaussian reference beam.
The results can be seen in Fig. 10. The interference patterns show the same
pitchfork but in opposite directions, verifying that one has $\ell=+1$ and the
other has $\ell=-1$.

###  XII.1 Profile Intensity Comparison

To minimize the beam profile intensity mismatch effects in the photocurrent
measurements, beam optimization was performed to make the beams with opposite
OAMs as similar as possible. However, naturally, the beams will not be
perfectly identical, as can be seen in Fig. 9. This is important since
differing beam profiles can cause a difference in generated photocurrent. To
quantify and demonstrate the insignificance of this effect in our experiments,
the overlap of the beams is calculated using the following function:

$O[I(x,y),J(x,y)]=\frac{\left|\int\int
I(x,y)J(x,y)dxdy\right|^{2}}{\int\int\left|I(x,y)\right|^{2}dxdy\int\int\left|J(x,y)\right|^{2}dxdy},$
(3)

where $I(x,y)$ and $J(x,y)$ are the two beam profiles with opposite OAMs. The
maximum possible overlap is one. The calculated overlap for the $\ell=\pm 1$
and $\ell=\pm 2$ beams is 0.96 and 0.92 respectively. In other words, the
$\ell=\pm 1$ beams have a 4% difference and the $\ell=\pm 2$ beams have an 8%
intensity profile difference. The minimum percent difference between the
average total current and the subtracted current for $\ell=\pm 2$ at $B=9\
\rm{T}$ is 60%. For $B=-9\ \rm{T}$ it is 134%. For $\ell=\pm 1$ the minimum
percent differences are 78% and 75% respectively. This confirms that the beam
profiles’ intensity mismatch cannot be affecting our observations.

Moreover, the beam profiles do not change with flipping the magnetic field,
further confirming that the intensity profile mismatch cannot cause the sign
of the photocurrent difference to flip for the opposite magnetic field.

Figure 10:  (a) Grating displayed on SLM to generate $\ell=+1$ as the first
diffraction order. (b) The same as (a), but for $\ell=-1$. (c) Interference
pattern when the $\ell=+1$ beam interferes with a Gaussian beam on the sample
substrate. (d) The same as (c), but with $\ell=-1$.

##  XIII Gate voltage sweep measurements for different bias

The gate voltage dependence for different bias voltages was also measured. It
is observed that the gate voltage dependence changes very little with bias
voltage and the flip with magnetic field stays the same.

Figure 11:  Measured PC difference as a function of gate voltage for different
$V_{b}$ at $B=9\ \rm{T}$: (a) Subtraction of PC for $\ell=\pm 1$. (b)
Subtraction of PC for $\ell=\pm 2$

##  XIV $B=\pm 4\ \rm{T}$ Measurements

All measurements in the main text were taken at $B=\pm 9\ \rm{T}$. This is the
maximum accessible magnetic field. To verify that these results are consistent
with $B$, measurements were also taken at $B=\pm 4\ \rm{T}$. The results for
the bias voltage-dependent measurements for $\ell=\pm 1$ are shown in Fig. 12.
It is observed that these results are consistent with the $B=\pm 9\ \rm{T}$
results in the main text. For the $B=\pm 4\ \rm{T}$ measurements the gate
voltage was set such that the Fermi energy was near $\nu=6$ just like the
$B=9\ \rm{T}$ data.

Figure 12:  (a), (b) Measured PC as a function of bias voltage $V_{b}$
generated using light carrying $\ell=+1$ (blue) and $\ell=-1$ (red) at (c)
$B=4\ \rm{T}$ and (d) $B=-4\ \rm{T}$. (c) Shows the $\Delta_{PC}$ for $B=4\
\rm{T}$ (green) and $B=-4\ \rm{T}$ (brown) for $\ell=\pm 1$, respectively. In
all the panels, the gate voltage $V_{g}$ and average pump power were $0.5\
\rm{V}$ and $10\ \rm{\mu W}$, respectively.

##  XV Selection Rules

We discuss the selection rules of exciting electrons with light from the lower
to the upper band in graphene in the Quantum Hall regime, considering the
possibility that the light might carry non-zero OAM. We consider the
electromagnetic vector potential $\vec{A}$ of incident light coupled via
Peierls substitution $\vec{k}\rightarrow\vec{k}-\mathrm{e}\vec{A}$ and compute
transition matrix elements as

$M(n_{\mathrm{f}},m_{\mathrm{f}},n_{\mathrm{i}},m_{\mathrm{i}})=-\mathrm{e}v_{\rm
F}\langle n_{\rm f},m_{\rm f}|\vec{A}(\vec{r})\cdot\vec{\sigma}|n_{\rm
i},m_{\rm i}\rangle$ (4)

where $|n_{\rm x},m_{\rm x}\rangle$ denotes initial (x = i) and final (x = f)
states. $n_{\rm x}$ denotes the Landau level and $m_{\rm x}$ the quantum
number related to angular momentum. The expected angular momentum in
$z$-direction can be computed as $\langle L_{z}\rangle=\hbar(m-n)$. We compute
the matrix elements in the spinor representation of the wave-function limiting
to transitions from the lower band of graphene (negative $n$) to the upper
band (positive $n$) and assuming $n\neq 0$. For $\sigma_{+}$ polarized light,
$\vec{A}(\vec{r})\cdot\vec{\sigma}=A(\vec{r})\sigma_{+}$, we get

$\displaystyle
M^{+}(n_{\mathrm{f}},m_{\mathrm{f}},n_{\mathrm{i}},m_{\mathrm{i}})$
$\displaystyle=-\frac{\mathrm{e}v_{\rm F}}{2}\begin{pmatrix}\langle n_{\rm
f}-1,m_{\rm f}|,&\langle n_{\rm f},m_{\rm
f}|\end{pmatrix}A(\vec{r})\sigma_{+}\begin{pmatrix}|n_{\rm i}-1,m_{\rm
i}\rangle\\\ -|n_{\rm i},m_{\rm i}\rangle\end{pmatrix}$ (5)
$\displaystyle=\frac{\mathrm{e}v_{\rm F}}{2}\langle n_{\rm f}-1,m_{\rm
f}|A(\vec{r})|n_{\rm i},m_{\rm i}\rangle$

while for $\sigma_{-}$ polarized light we get

$M^{-}(n_{\mathrm{f}},m_{\mathrm{f}},n_{\mathrm{i}},m_{\mathrm{i}})=-\frac{\mathrm{e}v_{\rm
F}}{2}\langle n_{\rm f},m_{\rm f}|A(\vec{r})|n_{\rm i}-1,m_{\rm i}\rangle.$
(6)

Figure 13:  (a) Matrix element $\mu^{+}(\ell,\delta,n,m)$ as defined in Eq.
(10) as a function of $\delta$ for $\ell=0,1,2$ as in the experiment (limiting
to positive OAM). $n=37$ is fixed as in the experiment and $m=5000$ which
corresponds to a state with expected radial position $\langle
r\rangle_{n=37,m=5000}\approx 1\ \mu\mathrm{m}$ which is somewhere in the
middle of the sample, with radius $R=2\ \mu\mathrm{m}$. The inset illustrates
the effect of $\delta$ in the case of $\ell=1$: $\delta=0$ corresponds to the
situation where the OAM of the light only changes the quantum-number $m$ while
$\delta\neq 0$ considers further transitions where $n$ and $m$ are
simultaneously changed. Only the $\delta=0$ transition has substantial weight.
(b) Same as in $a.)$ just with $m=500$ which corresponds to a radial position
of the electron of $\langle r\rangle_{n=37,m=500}\approx 300\ \mathrm{nm}$
hence close to the inner contact. The $\delta=0$ transitions also dominate
here but for smaller $m$, $\delta\neq 0$ transitions have an increased weight.
c.) Illustration of selection rules considering only $\delta=0$ that have the
dominant weight (see (a) and (b)).

For light carrying OAM, we write $A(\vec{r})=Ae^{i\varphi\ell}$, $\ell=0,\pm
1,\pm 2,\dots$ assuming no radial dependence of the light field. We compute
the dimensionless and normalized matrix elements for $\sigma^{+}$ polarized
light

$\mu^{+}=\langle n_{\rm f}-1,m_{\rm f}|e^{i\varphi\ell}|n_{\rm i},m_{\rm
i}\rangle$ (7)

and $\sigma^{-}$ polarized light

$\mu^{-}=\langle n_{\rm f},m_{\rm f}|e^{i\varphi\ell}|n_{\rm i}-1,m_{\rm
i}\rangle$ (8)

using the wave-functions

$\langle r,\varphi|n,m\rangle=\frac{i^{(m-n)}}{\sqrt{2\pi}\ell_{\rm
B}}\sqrt{\frac{n!}{m!}}e^{-\frac{r^{2}}{4\ell_{\rm
B}^{2}}}e^{i\varphi(m-n)}r^{(m-n)}L_{n}^{(m-n)}\left(\frac{r^{2}}{2\ell_{\rm
B}^{2}}\right).$ (9)

Here $L_{a}^{b}(x)$ are the associated Laguerre polynomials. We assume
$n<m-1$, since this is the relevant case for the experiment. In what follows,
we will focus on $\sigma^{+}$ polarized light computing $\mu^{+}$. However,
analogous steps can be performed to compute the matrix elements $\mu^{-}$.
Inserting the wave function [36, 37] Eq. (9) into Eq. (7) we get

$\displaystyle\mu^{+}=$ $\displaystyle i^{(m_{\rm i}-m_{\rm f}-n_{\rm
i}+n_{\rm f}-1)}\sqrt{\frac{n_{\rm i}!(n_{\rm f}-1)!}{m_{\rm i}!m_{\rm
f}!}}\frac{1}{2\pi}\int_{0}^{2\pi}\mathrm{d}\varphi\,e^{i\varphi(m_{\rm
i}-m_{\rm f}-n_{\rm i}+n_{\rm f}-1+\ell)}$ (10)
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\,2xe^{-x^{2}}x^{m_{\rm i}+m_{\rm
f}-n_{\rm i}-n_{\rm f}+1}L_{n_{\rm i}}^{m_{\rm{}_{i}}-n_{\rm
i}}(x^{2})L_{n_{\rm f}-1}^{m_{\rm{}_{f}}-n_{\rm f}+1}(x^{2}).$

The angular integral enforces angular momentum conservation. We introduce
$n:=n_{i}$ and $m:=m_{i}$ and define

$\displaystyle n_{\rm f}$ $\displaystyle=:n+\delta+1$ (11) $\displaystyle
m_{\rm f}$ $\displaystyle=:m+\ell+\delta$

such that angular momentum conservation is fulfilled for any choice of $n$,
$m$, $\ell$, and $\delta$. Note that Refs. [6, 19] only considered $\delta=0$
case. With this

$\mu^{+}(\ell,\delta,n,m)=i^{\ell}\sqrt{\frac{n!(n+\delta)!}{m!(m+\delta+\ell)!}}\int_{0}^{\infty}\mathrm{d}x\,2xe^{-x^{2}}x^{2(m-n)+\ell}L_{n}^{m-n}(x^{2})L_{n+\delta}^{m-n+\ell}(x^{2}).$
(12)

We plot $\mu^{+}$ as function of $\delta$ for $\ell=0,{+}1,{+}2$ in Fig. 13
for $n=37$ and $m=5000$ (a) and $m=500$ (b). For $m=5000$ the expected radial
position of the electron is $\langle r\rangle_{n=37,m=5000}\approx 1\
\mu\mathrm{m}$ hence somewhat in the radial middle of the sample that has an
overall radius of $R\approx 2\ \mu\mathrm{m}$. For $m=500$ we have $\langle
r\rangle_{n=37,m=500}\approx 300\ \mathrm{nm}$, so close to the inner contact.
For $\ell=0$ the orthogonality of the Laguerre polynomials enforces $\delta=0$
such that we get the well known selection rules [37]

$\mu^{+}(\ell=0,\delta,n,m)=\delta_{\delta,0}\Rightarrow\mu^{+}=\delta_{n_{\rm
f}-1,n_{\rm i}}\delta_{m_{\rm f},m_{\rm i}}$ (13)

and for $\mu^{-}$

$\mu^{-}(\ell=0,\delta,n,m)=\delta_{\delta,0}\Rightarrow\mu^{-}=\delta_{n_{\rm
f},n_{\rm i}-1}\delta_{m_{\rm f},m_{\rm i}}.$ (14)

For $\ell\neq 0$ we don’t get strict Kronecker-Delta-like selection rules but
instead, multiple transitions are in principle allowed. However, as can be
seen in Fig. 13, transitions with $\delta\neq 0$ are suppressed compared to
the $\delta=0$ transitions. We illustrate the selection rules, only
considering $\delta=0$, in Fig. 13 (c).

In the main text, as a control experiment, we have inverted the $B$-field
$\vec{B}\to-\vec{B}$. It is therefore interesting to consider how the
selection rules change in this case. For deriving the selection rules as
above, we have used the $B$-field to define the $z$-axis of the system.
Therefore the sign of the OAM of light as well as the handedness of the
polarization is determined by whether they are aligned or anti-aligned with
the magnetic field. Inverting the $B$-field is therefore equivalent to
inverting the orbital angular momentum and the polarization. Hence we can
relate the transition matrix elements at magnetic field $\vec{B}$ and
$-\vec{B}$ as

$\displaystyle\mu^{+}(\ell,\delta,n,m)\Big{|}_{\vec{B}}=\mu^{-}(-\ell,\delta,n,m)\Big{|}_{-\vec{B}}.$
(15)

Based on this observation, when measuring the subtracted photo-current at
magnetic field $\vec{B}$

$\Delta
I\Big{|}_{\vec{B}}=I^{+\ell}\Big{|}_{\vec{B}}-I^{-\ell}\Big{|}_{\vec{B}}$ (16)

one expects this to change sign when inverting the magnetic field hence

$\Delta I\Big{|}_{\vec{B}}=-\Delta I\Big{|}_{-\vec{B}}.$ (17)

This is what we have used as a control experiment in the main text and indeed
we observe the same sign reversal in the experiment.

##  XVI Relation between expected radius and quantum numbers $n$ and $m$

In this part, we compute the expected radius of the electronic wavefunctions
in graphene in the quantum Hall regime. In spinor representation, they can be
computed as

$\langle r\rangle=\begin{cases}\langle n=0,m|r|n=0,m\rangle\,;\,\,\,n=0\\\
\frac{1}{2}\left(\langle n,m|r|n,m\rangle+\langle
n-1,m|r|n-1,m\rangle\right)\,;\,\,\,n\neq 0\end{cases}.$ (18)

The wave functions $\phi_{n,m}(r)=\langle r,\phi|n,m\rangle$ read [37, 36]

$\phi_{n,m}(r,\varphi)=\frac{i^{|n-m|}}{\sqrt{2\pi}\ell_{\rm
B}}\sqrt{\frac{\text{min}(n,m)!}{\text{max}(n,m)!}}e^{-\frac{r^{2}}{4\ell_{\rm
B}^{2}}}e^{i\varphi|n-m|}L_{\text{min}(n,m)}^{|n-m|}r^{|n-m|}\left(\frac{r^{2}}{2\ell_{\rm
B}}\right)$ (19)

and are normalized according to

$\int_{\mathbb{R}^{2}}\text{d}\vec{r}\,\phi_{n,m}(\vec{r})\phi^{*}_{n^{\prime},m^{\prime}}(\vec{r})=\delta_{n,n^{\prime}}\delta_{m,m^{\prime}}.$
(20)

We show the expected radii as a function of the quantum number $m$ for
different values of $n$ in Fig. 14. For $n=0$ the expected position
approximately coincides with the position of the guiding center
$r_{\mathrm{Guide}}=\ell_{\rm B}\sqrt{2m+1}$ while for larger $n$ one gets
larger radii as expected.

Figure 14: Expected radial position $\langle r\rangle$ according to Eq. (18)
as a function of quantum number $m$ in different Landau levels $n$.

##  XVII Ideal interacting quantum Hall OAM response

The above argument can be generalized to understand the OAM photocurrent in
the presence of Coulomb interactions. Let us consider photons incident on
disorder-free Graphene in a perfect Corbino geometry with a magnetic field and
Coulomb interactions. The Coulomb interaction is key to helping the carriers
to thermalize and our discussion includes relaxation processes such as Auger.
Let us further assume that the bulk is gapped and there are two edges i.e. an
inner and outer edge, which for now we will assume to be isolated from
contacts.

Given the circular symmetry, angular momentum is a key quantum number to think
about. In fact, in this ideal situation, energy $E$ and angular momentum $L$
are the only two conserved quantities. For a gapped bulk where the system
thermalizes to a state with a low density of bulk quasiparticles. This means
that the edge charge $Q$ is also a conserved quantity. As a consequence, the
final state (after the absorption of some number of photons) is characterized
by edges with quantum numbers $L$, $E$, and $Q$.

The edge theory is more conventionally described in terms of momentum $k$
rather than angular momentum. Note that in a magnetic field, momentum is
shifted by the vector potential at radius $R$ of the edge is $k_{F}=A=BR/2$.
Transferring charge $Q$ to the edge changes momentum by $Qk_{F}$. Adding
energy through particle-hole excitations changes momentum by $E/v$ where $v$
is the edge velocity. The change in angular momentum is
$L=R(Qk_{F}+E/v)=R(QBR/2+E/v)$. This is consistent with the wave-functions of
Landau levels at angular momentum $m$ seen in the last section being at a
radius $\sim\sqrt{2m/B}$.

Photons supply both OAM angular momentum $l$ and energy $E_{ph}$. Considering
angular momentum and ignoring the contribution of the photon energy, the total
charge transferred is $\delta Q_{L}=2N_{ph}l/BR^{2}=2N_{ph}l(l_{B}/R)^{2}$.
The photon energy contribution turns out to be independent of OAM and
therefore even in magnetic field. The charge transfer expression matches that
from the LL wave-function picture in the previous section and shows that
interactions (at least in the absence of phonons and disorder) do not affect
the OAM photocurrent, which is a result of angular momentum pumping.

Here, the claim is that angular momentum is equal to charge transfer as long
as the excitation thermalizes. Therefore, even though the photon is an
excitation at an eV scale, one expects a universal charge transport. This is
because the angular momentum of a quantum Hall system in a Corbino geometry is
the difference in occupations of the chiral edge states. To prove this simply
consider the insertion of a flux quantum into the hole of the Corbino. This
generates an electric field $E=\Phi/2\pi r\delta t$. The orbital angular
momentum introduced $\delta L=\delta t\rho\int 2\pi
rErdr=\rho\pi(R_{1}^{2}-R_{2}^{2})(\Phi/2\pi)=N\Phi/2\pi$, where $N$ is the
total charge in the Corbino. The total charge transferred between the edges
$\delta Q=\Phi=2\pi\delta L/N$, in quantitative agreement with the estimate of
the main text. But this derivation applies in the presence of arbitrary
interactions, which are important to allow the system to thermalize, relax,
Auger, etc.

##  XVIII Photocurrent sign oscillation: Landau levels and injection current
picture

LLs picture: As described in a simplified picture earlier (Fig. 1 main text),
in the quantum Hall regime, light with non-zero OAM can induce optical
transitions between the LLs such that the spatial extent of these states
depends on the associated angular quantum number $m$, where the radius of LLs
increases with $m$. In our Corbino device, depending on the helicity of light,
this spatial expansion (shrinking) of the carriers’ wavefunction manifests
itself as an outward or inward radial current.

This intuitive picture for the PC generation due to the radial modulation of
the electron wavefunction caused by the optical vortex does not immediately
suggest PC oscillations as a function of gate voltage ($V_{g}$), which are
seen in Fig. 4 of the main text. To understand this effect, one must note that
the PC is actually the result of optical excitation and subsequent carrier
relaxation. As known from earlier studies [19, 38] the position of $E_{F}$
within a LL, controlled by $V_{g}$, gives rise to a relaxation bottleneck for
either electrons or holes. Due to the opposite charges of the different
carriers (electrons and holes), this leads to a change of PC polarity. That is
PC oscillations upon sweeping through a Landau level. While in the rectangular
geometry experiment of Ref.[19] this bottleneck argument relied on equal
chirality of electrons and holes at the edge of the sample, in the present
scenario the direction of transport is determined by the relative helicity of
twisted light and magnetic field. Naively, one might expect that the OAM would
lead to a relative shift of electron and hole position, i.e. to opposite
shifts for the two types of carriers. However, Coulomb attraction ensures that
the shift is experienced by the center of mass of the electron-hole-pair (see
also Ref. [39] for the case of excitons). Hence, electrons and holes are moved
in the same direction which explains the observed current oscillations.

Unfortunately, formalizing this picture, i.e., via optical Bloch equations as
has been done in Ref. [19], is extremely complicated. Not only would it be
necessary to explicitly account for Landau level and orbital degrees of
freedom, but the picture also suggests that properly accounting for the
Coulomb attraction between electrons and holes will be crucial to capture the
oscillations. However, an alternative description is able to provide a
formalism which is suited to explain the observed behavior.

Injection current picture: In our system, since the photon frequency is in the
optical regime (pump wavelength $940\ \rm{nm}=1.32\ \rm{eV}$ and
$(E_{n=37}-E_{n=-36})=1.315\ \rm{eV}$), one expects the LL spacing
($E_{n=37}-E_{n=36}=9\ \rm{meV}$) to be much smaller than the lifetime
broadening of excited electrons and holes. Therefore, we can consider
optically excited electrons and holes as approximately free Dirac particles.
Beyond the dipole approximation, the OAM of the pump photons locally imparts
momentum to the electrons and holes in the tangential direction (in the polar
coordinates of the Corbino disk). Moreover, the Lorentz force from the
external out-of-plane magnetic field imparts radial components to the velocity
of the electron and holes. As detailed in the following section, the radial
current of both the electrons and holes in this scenario is therefore given by
the following

$I_{\text{radial}}=-\frac{Bq}{k^{4}}\cos(2\theta)(\tau_{e}+\tau_{h})$ (21)

where $B$ is the external out-of-plane magnetic field, $q$ is the local OAM-
induced momentum, $\theta$ is the angle between the electron and hole’s
momentum $k$, and $\tau_{e}$ and $\tau_{h}$ are the scattering times of the
electrons and holes, respectively. From the Dirac equation
$k=\omega/(2v_{D})$. The total induced current is

$I_{\text{meas}}=\int{d\theta W(\theta)I_{\text{radial}}(\theta)}$ (22)

where $W(\theta)$ is the weight of the current contribution from the electrons
and holes moving at the angle $\theta$. In an ideal case, $W$ should be
$\theta$-independent, leading to a vanishing of the measured current in our
system. However, as explained later in the following section, $W$ can be a
function of both $\theta$ and the longitudinal conductivity, which is a
function of the carrier density (and therefore gate voltage $V_{g}$) and $B$.
This can potentially explain the observed $V_{g}$-dependent polarity change of
the measured PC.

##  XIX Theory of OAM photoresponse in the strong scattering regime

The OAM field from light with frequency $\omega$ can be written as
$A(\vec{r})\propto e^{il\theta}$, where $(r,\theta)$ which can then be
expanded in terms of a tangential coordinate $x_{t}=r\theta$ as
$A(\vec{r})\propto e^{ilx_{t}/r}=e^{iqx_{t}}$ where $q=l/r$. Locally, in the
Corbino geometry, we can approximate the OAM as a plane wave with wave-vector
$q$.

Figure 15:  Optical excitation by finite wave-vector light in a band structure
of Dirac electrons in graphene.

Considering the absorption of the photon by Dirac electrons, at rate $R$ per
unit area, shown in Fig. 15 leads to an electron-hole pair with momenta
$k_{e}=k+q/2$ and $-k_{h}=-(k-q/2)$ respectively where $|k|=\omega/2v_{D}$ is
independent of $q$. To check that this is correct note that the energy of such
a pair is $|k+q/2|+|k-q/2|=2|k|+o(q^{2})$.

Next, we consider (seen in Fig. 16) the generated electron-hole pair in
momentum space along with the velocity of each particle as well as the
Lorentz-force induced momentum change. For a quasiparticle at momentum $k$
(note that we designate holes to be at the momentum of the missing electron),
the velocity is $u_{e,h}(k)=\pm v_{D}\vec{k}/k$. The Lorentz-force induced
momentum change is $B\times u_{e,h}(k)\delta\tau_{e,h}$ where $\tau_{e}$
($\tau_{h}$) is the scattering time of electrons (holes). The average (over a
scattering time) change in velocity induced by the Lorentz force turns out to
be $\delta u_{e,h}(k)=v_{D}^{2}\tau_{e,h}\frac{B\times\vec{k}}{2k^{2}}$, which
is the same for electrons and holes. The sign is independent of electrons or
holes because both the velocity as well as the momentum change changes sign
for both electrons or holes.

Figure 16:  Wave-vector and velocity structure of electron-hole pairs in a
magnetic field excited by light at wave-vector $q$, which is tangential in
direction relative to the Corbino geometry. The dotted line shows the wave-
vector $|k|=\omega/2v_{D}$ where the optical excitations originally occur.

The Lorentz-force-induced current for electrons and holes at wave-vector $k$
would cancel (since the velocities are the same) except for the relative wave-
vector $q$ induced by the OAM. This leads to a OAM-B-field induced velocity
$\delta_{2}u(k)=(q\cdot\nabla_{k})\delta u(k)=\frac{qv_{D}^{2}\tau
B}{k^{2}}(\sin{2\theta},-\cos{2\theta})$ where $\theta$ is the angle of
$\vec{k}$ relative to the tangential direction $\vec{q}$. This velocity is
opposite for electrons and holes so that the electron and hole components of
the current now add up. The relevant current in the Corbino geometry is the
radial current
$I_{rad}=\frac{q_{e}^{2}Rv_{D}^{2}(\tau_{e}^{2}+\tau_{h}^{2})qB}{k^{2}}\cos{2\theta}$
$=q_{e}(Rl_{B}^{2})(2\pi
l)\\{(\tau_{e}^{2}+\tau_{h}^{2})\omega_{c}^{2}\\}\cos{2\theta}$. The prefactor
$2q_{e}\pi l(Rl_{B}^{2})$ is the ideal expectation based on angular momentum.

As shown in the figure, the radial component of the current flips sign with
$\theta$ i.e. $I_{rad}(\theta\sim 0,\pi)>0$ while
$I_{rad}(\theta\sim\pm\pi/2)<0$. In fact, the straightforward average over
$\cos{2\theta}$ would vanish. However, for reasons we describe below, the
total measured radial current $I_{meas}=\int d\theta W(\theta)I_{rad}(\theta)$
is a weighted average of the radial current where the presence of a
$\cos{2\theta}$ Fourier component of $W(\theta)$ would lead to a non-vanishing
measured current.

To understand the origin of a non-trivial weight function $W(\theta)$, we note
that $\theta$ represents the angle of the wave-vector $k$ (and hence velocity)
relative to the tangential direction in the Corbino geometry at which the
electron-hole pair is excited. Electron-hole pairs at angles
$\theta\sim\pm\pi/2$ are dominantly moving in the radial direction either
towards or away from the Corbino edge. In contrast pairs at $\theta\sim 0,\pi$
are moving tangentially. As clear from Fig. 17 the formal set of electron/hole
pairs and the associated current can hit the edge on a lower time scale. This
reduces the scattering time $\tau_{e,h}$ for such quasiparticles. Thus the
weight function in the radial direction
$W(\theta\sim\pm\pi/2)\sim(\tau_{e}^{(r)}+\tau_{h}^{(r)})/(\tau_{e}+\tau_{h})$,
where $\tau^{(r)}_{e,h}$ is the quasiparticle scattering time for particles
moving in the radial direction and $\tau_{e,h}$ is the scattering time of the
quasi-particles averaged over the Fermi surface. In contrast, the tangentially
moving quasiparticles have an averaged scattering time, but have a
contribution to the edge conductance that is suppressed by the longitudinal
conductivity $\sigma_{xx,W}$ of warm carriers. Note that the relevant
conductivity is that of electron-hole pairs that are relaxing from the high
optical energy scale to the ground state. The relevant weight function in the
tangential direction, which arises from a current divider effect between the
bulk and contact resistance, is $W(\theta\sim 0,\pi)\propto
R_{contact}\sigma_{xx,W}$, where $R_{contact}$ is the contact resistance and
$\sigma_{xx,W}$ is the two dimensional conductivity (which has the same units
as $R_{contact}^{-1}$). This is a rough qualitative estimate that may be
refined by more detailed modeling of the conductivity. However, we expect the
key feature of dependence on bulk conductivity to be robust.

Figure 17:  Scattering in Corbino geometry. Quasiparticles moving radially
towards the edge i.e. $\theta\sim\pm\pi/2$ have a shorter scattering time
resulting in smaller B-field current though can more efficiently reach the
edge because of ballistic transport relative to those moving tangentially i.e.
$\theta\sim 0,\pi$.

The form of the weight function discussed above can predict a switching of
sign with changing gate voltage of the measured B-dependent part of the OAM
photocurrent. As the gate voltage changes the longitudinal conductance
$\sigma_{xx,W}$ is expected to change as one passes through Landau levels.
This leads to modulation of $W(\theta\sim\pm\pi/2)$, while one can expect
$W(\theta\sim 0)\lesssim 1$ to have minor variations. At the same time
$I_{rad}(\theta\sim 0,\pi)\sim-I_{rad}(\theta\sim\pi/2)$. Therefore
$I_{meas}\sim I_{rad}(\theta\sim 0,\pi)[W(\theta\sim
0,\pi)-W(\theta\sim\pm\pi/2)]$ changes sign when the bulk conductivity changes
from low to high or vice-versa as the gate voltage sweeps through Landau
levels.