††thanks: E.L. and J.H. contributed equally††thanks: E.L. and J.H. contributed
equally

# Quantum control of exciton wavefunctions in 2D semiconductors

Jenny Hu  Department of Applied Physics, Stanford University, Stanford, CA
94305, USA  SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
Etienne Lorchat  NTT Research, Inc. Physics & Informatics Laboratories, 940
Stewart Dr, Sunnyvale, CA 94085  Xueqi Chen  Department of Physics, Stanford
University, Stanford, CA 94305, USA  SLAC National Accelerator Laboratory,
Menlo Park, CA 94025, USA  Kenji Watanabe  Research Center for Functional
Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba
305-0044, Japan  Takashi Taniguchi  International Center for Materials
Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan  Tony F. Heinz  Department of Applied Physics,
Stanford University, Stanford, CA 94305, USA  SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  Puneet A. Murthy<EMAIL_ADDRESS>NTT
Research, Inc. Physics & Informatics Laboratories, 940 Stewart Dr, Sunnyvale,
CA 94085  Institute for Quantum Electronics, ETH Zürich, CH-8093 Zürich,
Switzerland  Thibault Chervy<EMAIL_ADDRESS>NTT Research,
Inc. Physics & Informatics Laboratories, 940 Stewart Dr, Sunnyvale, CA 94085

(August 11, 2023)

Excitons –bound electron-hole pairs– play a central role in light-matter
interaction phenomena, and are crucial for wide-ranging applications from
light harvesting and generation to quantum information processing. A long-
standing challenge in solid-state optics has been to achieve precise and
scalable control over the quantum mechanical state of excitons in
semiconductor heterostructures. Here, we demonstrate a technique for creating
tailored and tunable potential landscapes for optically active excitons in 2D
semiconductors that enables in-situ wavefunction shaping at the nanoscopic
lengthscale. Using nanostructured gate electrodes, we create localized
electrostatic traps for excitons in diverse geometries such as quantum dots
and rings, and arrays thereof. We show independent spectral tuning of multiple
spatially separated quantum dots, which allows us to bring them to degeneracy
despite material disorder. Owing to the strong light-matter coupling of
excitons in 2D semiconductors, we observe unambiguous signatures of confined
exciton wavefunctions in optical reflection and photoluminescence
measurements. Our work introduces a new approach to engineering exciton
dynamics and interactions at the nanometer scale, with implications for novel
optoelectronic devices, topological photonics, and many-body quantum nonlinear
optics.

### Introduction

Figure 1: Quantum excitonics in 2D semiconductors. Schematic illustration of
our approach. Upper layer: Nanostructured gate electrodes are positioned in
the vicinity ($10$s of nm) of a 2D semiconductor. Middle layer Voltages
applied to the gates define lateral distributions of in-plane electric fields
$\vec{F}(\mathbf{r})$ and charge densities $\rho(\mathbf{r})$ for itinerant
holes (red) and electrons (blue) in the 2D semiconductor. Lower layer Exciton
confinement occurs due to a combination of electric field-induced Stark shift
and exciton-charge interactions (see Eq. 1). This enables in-situ control of
the center-of-mass exciton wavefunction in arbitrary traps such as quantum
rings, quantum dots and scalable arrays.

The ability to trap and manipulate the quantum mechanical state of particles
is a central pillar across various disciplines of quantum science, including
ultracold atoms [1, 2], ion traps [3] and superconducting qubits [4]. In
semiconductor optics, the traditional approach to confinement of optical
excitations involves nanostructures such as self-assembled or colloidal
quantum dots, which are typically fabricated through growth or implantation
techniques. These methods lead to ensembles of quantum dots (or rings) with
random positions and broad energy distribution with limited local control [5,
6, 7]. The resulting lack of scalability has hindered their widespread
technological implementation. In addition to these methods, alternative
material modulation approaches have recently been explored, such as strain
engineering [8, 9, 10, 11, 12], electron and ion beam irradiation [13, 14],
and moiré potential engineering [15, 16, 17, 18], which offer varying degrees
of control on the positions, energies or number of emitters. Furthermore,
electrostatic confinement of spatially indirect excitons in semiconductor
heterostructures has been reported [19, 20]; however, their vanishing coupling
to light limits potential photonics applications. The ability to precisely
control the individual quantum states of optically active excitons with strong
light-matter coupling has thus remained a major experimental challenge.

Recently, a novel technique has been reported for confining direct excitons
purely using electric fields and charge density gradients in 2D semiconductor
heterostructures [21]. The method relies on the fact that in-plane electric
fields ($\vec{F}(\mathbf{r})$) induce a quadratic dc Stark shift of the
excitons, and charge density ($\rho(\mathbf{r})$) leads to an interaction-
induced density-dependent energy shift, according to

$\Delta E=-\frac{1}{2}\alpha|\vec{F}(\mathbf{r})|^{2}+\beta\rho(\mathbf{r}),$
(1)

where $\alpha$ is the dc polarizability of excitons and $\beta$ is the
exciton-electron coupling constant. Spatial variations of doping densities and
electric fields can thus be used to trap excitons. Based on this effect,
quantum confinement was demonstrated in a gate-defined lateral p-i-n junction.
However, as the trapping occurs along the gate edges, the technique has so far
been limited to excitonic 1D quantum wires.

Here, we demonstrate scalable and tunable electrostatic traps of arbitrary
shapes for direct excitons in monolayer Transition Metal Dichalcogenide (TMD)
semiconductors. We illustrate our approach in Fig. 1. The crux of our work
lies in using lithographically nano-structured gate electrodes in proximity to
the 2D semiconductor plane (upper layer). This allows to precisely define the
lateral distribution of in-plane electric fields $\vec{F}(\mathbf{r})$ and
density $\rho(\mathbf{r})$ of itinerant holes (red) and electrons (blue) in
the 2D semiconductor, with a resolution down to a few tens of nanometers
(middle layer). This in turn enables tailor-made landscapes for excitons with
a high degree of control on the wavefunction profile (lower layer). To
highlight the versatility of our technique, we focus on two exemplary trap
geometries, namely quantum rings and quantum dots, that may serve as building
blocks for more extended landscapes. Furthermore, we demonstrate the
unprecedented scalability of our approach by realizing arrays of quantum rings
and independently controlled quantum dots which is of particular relevance for
future optoelectronics and photonics technologies.

The basic structure of our exciton trapping devices consists of a monolayer
TMD semiconductor, such as MoSe2, encapsulated by hexagonal boron nitride
(hBN) spacers of appropriate thickness. This heterostructure is stacked on a
Si/SiO2 substrate with top (TG) and bottom (BG) gate electrodes, which can
either be graphene (Gr) or metallic thin films. We nanostructure one of the
gate electrodes through a combination of electron beam lithography and dry
etching, which allows for patterning resolutions of around 50 nm. A global BG
is used to define the charge configuration of the entire monolayer, whereas
the nanostructured TGs enable local control of charge densities and fields.
Even though either of the gates can be patterned without losing functionality,
we choose to define the structures on the TGs and keep the BG for global
doping, for ease of fabrication. Our devices are cooled down to $\sim 5\,$K in
a closed cycle dry cryostat with optical access. To characterize the excitonic
states in our system, we mainly rely on optical broadband reflection and
photoluminescence spectroscopy with a spatial resolution of
$0.7\,\mu\textrm{m}$. A detailed account of our fabrication and experimental
procedures is described in the Supplementary Information (SI .3,.1).

Figure 2: Ring traps for excitons. (a) Dual-gated TMD heterostructure where a
nano-hole etched in the top gate defines quantum ring confinement for
excitons. (b) Finite element electrostatic simulations of the device for a
$600\,$nm gate hole with gate voltage configuration
$(V_{\mathrm{{BG}}},V_{\mathrm{{TG}}})=(-5.0\,\mathrm{V},9.5\,\mathrm{V})$;
Charge distribution shows a globally electron doped semiconductor with holes
trapped in the center of the nano-hole, surrounded by a ring-shaped neutral
region. (c) Schematic showing radially dipolar excitons quantum confined in a
ring. (d) Finite element simulations of the charge distribution (_upper
panel_), field distribution (_middle panel_) and exciton trapping potential
(_lower panel_) for a 600 nm diameter hole device. The horizontal gray lines
in the lower panel are the c.o.m eigenstates of the 2D Schrödinger equation.
Gate voltage configuration is the same as in (b). (e) First-derivative
reflectance contrast spectra as a function of $V_{\mathrm{{BG}}}$, for fixed
$V_{\mathrm{{TG}}}=9.5\,$V. Charging configurations for each regime (I - V)
are illustrated above. (f) Spatially resolved scan of PL emission integrated
over the confined states acquired on a 1 $\mu$m diameter hole. (g) Spectral
cross cut through the center of the hole ($y=0$), showing the trion emission
from regions A and B and neutral exciton emission of ring states. (h,i) Stokes
vector analysis of the confined state emission map in the linear polarization
basis, which demonstrates that PL emission of ring states is polarized in the
radial-azimuthal basis. Arrows represent the local linear Stokes vector
$\textbf{S}=(S_{1},S_{2})$.

### Excitonic quantum rings

In Fig. 2, we describe our scheme to create ring traps for excitons. An
annular confinement geometry can be obtained simply by patterning circular
holes in the TG, as shown schematically in Fig. 2 a. In order to obtain the
strongest confinement, we apply opposite voltages on the top and bottom gates
to reach the annular p-i-n regime, where a central p-doped puddle is enclosed
in a Fermi sea of electrons with a ring-shaped neutral region separating them,
as seen in Fig. 2 b. The combination of in-plane fields and charge gradients
leads to tight confinement of excitons in this annular region (1).
Furthermore, the in-plane fields lead to radially polarized in-plane dipolar
excitons as illustrated in Fig. 2 c.

In Fig. 2 d, we show results of finite element simulations of the charge and
field distributions in the annular p-i-n configuration for a top gate hole
with diameter $600\,$nm, which provides a quantitative understanding of the
electrostatics of the device. We calculate the confinement potential according
to Eq. 1, which shows the expected Mexican hat-like profile. We note that the
dominant contribution to the potential comes from the exciton-electron
interaction (second term in Eq. 1) whereas the Stark shift has a relatively
mild effect. Solving the 2D Schrödinger equation in the center-of-mass (c.o.m)
frame of excitons, we obtain discrete radial trap levels, with a level
separation of $\sim 0.3-0.8\,$meV. We present a detailed account of the trap
profile as a function of voltage in the SI (section .6).

In order to verify the nature of excitonic states in this system, we now move
on to optical measurements of the device. In monolayer MoSe2, the onset of
electron or hole doping is associated with the emergence of a bound trion
state and a density-dependent blue shifting exciton state (repulsive polaron)
[22, 23, 24]. Therefore, measuring reflectance spectra as a function of
$V_{\mathrm{{TG}}}$ and $V_{\mathrm{{BG}}}$ allows to precisely distinguish
doping configurations inside and outside the TG hole. Region A (inside the
hole) is primarily affected by the BG, whereas region B (outside the hole) can
be doped using both TG and BG.

In Fig. 2 e, we present the measured reflectance spectra obtained from a
$600\,$nm diameter hole as a function of $V_{\mathrm{{BG}}}$, for a fixed
$V_{\mathrm{{TG}}}=9.5\,$V. In order to highlight the faint signatures of
quantum confined states, the reflectance contrast $\Delta R/R_{0}$ is
differentiated with respect to energy as detailed in SI .2. Five distinct
regimes can be identified as the applied back gate voltage $V_{\mathrm{{BG}}}$
is swept. For positive $V_{\mathrm{{BG}}}$, the action of the back gate
reinforces that of the top gate, leading to overall n-doping of the device
(regime V in Fig. 2 e). As expected, we observe a trion branch at $\sim
1.610\,$eV, and a repulsive polaron branch at $\sim 1.635\,$eV. Decreasing
$V_{\mathrm{{BG}}}$ reduces the electron density, until we reach the i-i-n
regime where region A is neutral (regime IV), as evidenced by the transfer of
oscillator strength from the trion branch to the neutral exciton state. As we
decrease $V_{\mathrm{{BG}}}$ further, we reach the p-i-n regime where region A
is p-doped and region B is n-doped (regime III), with an annular neutral
region in between. Here, we observe repulsive polaron branches from both A and
B regions, associated to the two types of carriers. In addition, we observe
narrow and discrete resonances emerging from the neutral exciton state. We
attribute these new resonances to 1D quantum ring states of different radial
mode orders existing at the periphery of the nanohole. Upon decreasing the
bottom gate voltage even further, the confined exciton states merge back with
the 2D exciton continuum, as the region B outside the dot reaches charge
neutrality (regime II). The energy separation between the discrete states is
approximately $0.5\,$meV, in agreement with the simulations, which corresponds
to a trapping length of $\sim 10\,$nm.

To investigate the spatial profile of the trap, we now turn to a larger TG
hole of diameter $1\,\mu$m, which is large enough to optically resolve the
different regions A and B with a confocal microscope setup. Fig. 2 f shows a
spatially resolved photoluminescence (PL) map of the hole in the radial p-i-n
configuration ($V_{\mathrm{{BG}}}=-2.5\,$V, $V_{\mathrm{{TG}}}=+9.0\,$V). The
emission is filtered at the energy of the confined states. As expected, we
observe a clear doughnut-shaped emission profile. The PL spectra taken along a
cross-cut through the center of the hole (Fig. 2 g) shows distinct trionic
emission from regions A and B, and confined states emission from the annular
neutral region.

Further insights into the quantum ring states can be obtained by studying the
polarization texture of the emitted photons. Previous studies on 1D confined
states have shown that electron-hole exchange interactions, coupled with tight
confinement of the excitonic wavefunction along one direction, leads to strong
polarization splitting in a linear basis [21]. Applying this argument to our
circular geometry, the ring confinement should induce polarization splitting
in the radial-azimuthal basis. To investigate this behavior, we record the PL
emission intensity from the lowest confined exciton as a function of position,
for different polarization states (see SI .1). From these measurements, the
normalized linear polarization Stokes vectors
$\textbf{S}\equiv(S_{1},S_{2})=(I_{x}-I_{y},I_{d}-I_{ad})/I_{0}$ can be
reconstructed as a function of position on the ring, where $I_{x}$, $I_{y}$,
$I_{d}$ and $I_{ad}$ respectively correspond to PL intensity along $x$, $y$,
diagonal ($45^{\circ}$), and anti-diagonal ($-45^{\circ}$) directions, and
$I_{0}$ is the total PL intensity. In Fig. 2 h and i, we show $S_{1}$ and
$S_{2}$ for the lowest confined state which both exhibit two-fold symmetric
patterns as expected for an azimuthally polarized emitter. These observations
clearly demonstrate the quantum confinement of excitons in ring-shaped
potentials, and constitute the first major result of this work.

Figure 3: Bow tie traps for excitons. (a) Dual-gated TMD heterostructure with
top gate structured as a bow tie. Inset: AFM map of a $35\,$nm bow tie gate.
(b - d) Electrostatic simulations with
$(V_{\mathrm{{L}}},V_{\mathrm{{R}}},V_{\mathrm{{BG}}})\equiv(5.0\,\text{V},-5.0\,\text{V},0\,\text{V})$.
(b) Top view ($xy$) of charge density and in-plane electric field at the TMD
plane, showing n-doped (blue) and p-doped (red) regions. (c) Side view ($xz$)
of the electric field at a vertical cut along $y=0$. (d) Confining potential
for excitons in the gap region. (e) Calculated dependence of resonance energy
of ladder of states as a function of $\Delta
V=V_{\mathrm{{L}}}-V_{\mathrm{{R}}}$. (f) First-derivative reflectance
contrast spectroscopy of the bow tie gap region, as a function of $\Delta V$.
The inset shows vertical cross-cuts from $\Delta V=11\,$V (dark blue) to
$\Delta V=13\,$V (light blue). The red line is a guide to the eye following
the confined state. (g) First-derivative reflectance contrast of a $100\,$nm
gap bow tie, as a function of $\Delta V$, showing a ladder of confined states.
(h) Spatial dependence of the first-derivative reflectance contrast,
integrated over the 0D exciton signal at $\Delta V=9.9\,$V. Here,
$V_{\mathrm{{BG}}}=1.5\,$V is applied in order to further energetically
separate the 0D state from the X2D signal. This leads to a slight elongation
of the states towards the left electrode (SI .8).

### Tunable quantum dots

We now turn to the demonstration of tunable quantum dot-like confinement,
which has been a major goal in quantum photonics [25, 26, 27]. To generate
electrostatic 0D nanotraps, we need a gate geometry that allows for tight
confinement in both lateral directions. While the gate hole structure
described in Fig. 2 may also be used for this purpose, the tightness of
confinement and energy tunability is significantly limited by fabrication
constraints. To achieve tunable 0D traps, we design a bow tie electrode
structure as illustrated in Fig. 3 a, which concentrates electric fields in
the nanoscopic gap between the electrodes. These bow ties are fabricated using
electron beam lithography and deposition of $10\,$nm Au, with gap sizes
ranging from $30-100\,$nm. An atomic force microscope micrograph of a bow tie
($\sim 35\,$ nm gap) is shown in the inset.

In this geometry, we operate primarily in the neutral regime, where the
semiconductor is globally set to neutrality using the bottom gate, and a bias
voltage $\Delta V=V_{\mathrm{{L}}}-V_{\mathrm{{R}}}$ is applied between the
bow tie electrodes to obtain in-plane fields in the nanogap. An electrostatic
simulation of the charge and field distributions in the device is shown in
Fig. 3 b and c (top and side view, respectively; red: holes, blue: electrons).
In general, the top hBN thickness should be as small as possible ($\lesssim
20\,$nm) in order to maximize the in-plane component of the electric field on
the TMD plane in the gap region.

In Fig. 3 d, we show the expected 2D trapping potential for excitons in bow
ties computed from Eq. 1. First, we note that in contrast to the ring traps,
the dominant contribution to confinement in the bow tie geometry comes from
the electric field-induced Stark shift (first term in Eq. 1). Further, the
trapping potential features $x-y$ anisotropy, which can be adjusted by varying
the width of the electrode tip and the gap size. By solving the 2D Schrödinger
equation for the exciton c.o.m motion in such a potential, we obtain the
expected dependence of the quantum confined states energies on $\Delta V$,
shown in Fig. 3 e.

To experimentally observe quantum dot states in the bow tie structure, we
measure reflectance spectra in the gap region as a function of $\Delta V$.
These spectra are acquired from a bow tie with a gap size of $\sim 35\,$nm.
Since the gap size is much smaller than the diffraction limit, we expect that
any confined states will have a strongly diminished oscillator strength as
compared to the 2D exciton resonance. In Fig. 3 f, we show the first-
derivative reflectance contrast spectra $d(\Delta R/R_{0})/dE$ as a function
of $\Delta V$. For simplicity, we apply symmetric voltages on the two bow tie
electrodes; eg. to obtain $\Delta V=20\,$V, we apply $V_{\mathrm{{L}}}=-10\,$V
and $V_{\mathrm{{R}}}=+10\,$V. In addition to the broad neutral 2D exciton
background (centered around $X^{2D}\sim 1.638\,$eV) and the blue-shifting
polaron branches from the gated regions, we observe much narrower resonances
emerging below the neutral exciton (linewidth $\Gamma\lesssim 300\,\mu$eV).
These states clearly red shift with $\Delta V$, with a similar dependence to
Fig. 3 e. This observed voltage dependence is in excellent quantitative
agreement with our simulations (see SI .7). Line cuts at different $\Delta V$
are shown in the inset. The energy dependence of these states on the electric
field across the bow tie is strong evidence that they occur in the gap region
(see SI .7, .8). A further compelling signature of quantum confinement in such
electrical traps is the emergence of a ladder of states corresponding to
discrete energy levels in the trap [21]. In Fig. 3 f, such signatures are
barely apparent possibly due to the small gap size leading to vanishing
oscillator strength of excited states. To confirm this, we perform the same
reflectance measurement on a bow tie with a larger gap size ($\sim 100\,$nm),
as shown in Fig. 3 g. Here we do observe a clear ladder of excitonic states
emerging below the 2D exciton continuum, which supports our hypothesis.

To further confirm that the observed resonances indeed originate in the gap
region, we perform a spatially resolved scan of reflectivity in the vicinity
of the bow tie. For this measurement, a small voltage
$V_{\mathrm{{BG}}}=1.5\,$V is applied in order to further energetically
separate the 0D state from the X2D background, which facilitates the analysis
(see SI .8). The first-derivative reflectance contrast map, integrated over
the narrow exciton resonance, is shown in Fig. 3 h which demonstrates that
this resonance only appears close to the gap region and vanishes as we move
away. We have reproduced these observations across different bow ties in the
same device, as well as on several different devices (see SI .9), which
underlines the robustness of our technique. Taken together, our measurements
unambiguously demonstrate electrically tunable 0D quantum confinement in bow
tie electrode structures, which constitutes the second key result of this
work.

Figure 4: Scalable arrays. (a) Optical micrograph of a van der Waals
heterostructure featuring Au bow tie electrodes (red dashed lines) and
patterned few layer graphene top gate (black dashed lines). The blue dashed
line indicates the $400\,$nm diameter hole array, and the green one highlights
the 1 $\mu$m diameter hole studied in Fig. 2. The gray dashed lines indicate
the contacts to the TMD. (b) AFM topography of the 400 nm hole array. (c) PL
map of the hole array, filtered at the confined state emission energy. (d) AFM
topography of a bow tie array. BT1, BT2 and BT3 are three independent
electrode while the counter electrode BT0 is common to the three bow ties. (e)
Fitted energies of the 2D exciton (blue dots) and 0D confined states (brown,
red, orange dots) for a fixed counter electrode voltage
$V_{\mathrm{BT0}}=15\,$V, as a function of the independent electrode voltages
($V_{\mathrm{BT1}}$, $V_{\mathrm{BT2}}$, $V_{\mathrm{BT3}}$). The shaded blue
area indicates the FWHM of X2D. Rightmost panel: Confined states spectra
acquired under resonance tuning conditions ($V_{\mathrm{BT0}}=15\,$V,
$V_{\mathrm{BT1}}=-11.2\,$V, $V_{\mathrm{BT2}}=-4.6\,$V,
$V_{\mathrm{BT3}}=-3.5\,$V)

### Scalable arrays of quantum dots and rings

The most important advantage of our approach for electrical nanoscale control
of exciton wavefunctions is the potential for scaling up to more complex
structures. The quantum dot and ring trapping schemes that we presented above
can be seen as the building blocks for larger systems. Now, we demonstrate how
a variety of nano-structures can be realized in a single TMD heterostructure
simply by lithographically defining the desired gate patterns (Fig. 4). Figure
4 ashows a micrograph of a patterned heterostructure, demonstrating how
different gate geometries can be defined on the same device. In Fig. 4 b, we
show an AFM scan of a $3\times 3$ array of $400\,$nm diameter holes, with a
$600\,$nm pitch that allows to optically resolve individual lattice sites. A
spatially resolved scan of the PL emission, integrated over the quantum ring
state resonances, is shown in Fig. 4 c. We clearly observe the emission
profile of the ring array that matches excellently with the gate pattern.
Since the hole diameters are smaller than the diffraction limit, we do not
resolve the ring shape of the emission patterns (see Fig. 2 ffor comparison).
This shows that ring traps can be scaled up arbitrarily in a single monolayer.

Next, we demonstrate the scalability of electrically tunable quantum dots. An
important motivation for our work is the realization of multiple quantum dots
with identical energies, which is a crucial and basic ingredient for several
applications, including photonic quantum information processing and quantum
communications [28]. This has so far remained a hurdle to achieve since
existing material modulation approaches are drastically limited by material
disorder and process variation [29]. We address this problem by fabricating an
array of bow ties with independent control for each. An AFM scan of our
structure with three bow ties with approximate gap size of $50\,$nm and a
separation of $1\,\mu$m is shown in Fig. 4 d. We electrically short the left-
hand side electrodes of the three bow ties (BT0), while maintaining individual
control on each of the right-hand side electrodes (BT1, BT2, BT3). This allows
to reduce the number of control gates per quantum dot to only one, thus
enhancing scalability without compromising on the control over each site. As
shown in SI .9, we observe the quantum dot confinement signatures described in
Fig. 3 f in each of the bow ties.

In Fig. 4 e, we report the fitted energy of the 0D states as a function of
individual gate voltages $V_{\mathrm{BT1}}$, $V_{\mathrm{BT2}}$ and
$V_{\mathrm{BT3}}$, while keeping the common counter electrode at
$V_{\mathrm{BT0}}=15\,$V. The energy of the 2D exciton $X^{2D}$ is shown with
blue dots and the linewidth is indicated by the blue shaded regions. The three
bow ties show different dependence with voltage, possibly due to material
disorder and variations due to fabrication uncertainties. Nevertheless, the
three quantum dots can be simultaneously tuned to degeneracy (horizontal
dashed line) by applying the suitable voltages across the bow ties (vertical
dashed lines). This is clearly evident in the spectra shown in Fig. 4 f, taken
at $V_{\mathrm{BT1}}=-11.2\,$V, $V_{\mathrm{BT2}}=-4.6\,$V, and
$V_{\mathrm{BT3}}=-3.5\,$V respectively, which show three quantum dot states
resonantly tuned in energy. The ability to combine position-controlled
lithographically defined quantum dots, with simultaneous and independent
energy tunability into scalable arrays of quantum sites, constitutes the third
main result of this work.

### Outlook

We have demonstrated new fundamental building blocks for quantum excitonics.
Our approach enables continuous control of the c.o.m wavefunctions, from
micron sized extended states all the way down to quantum confinement into
nanoscale dots and rings. Our approach is intrinsically scalable to
independently tunable arrays, which is an important feature for future
applications. A key advantage of this method is that arbitrary landscapes for
excitons can be defined in a non-intrusive manner while retaining the pristine
properties of the active material. Hence it can be extended to different
semiconductors, including perovskites [30] and metal chalcogenide compounds
[31]. Furthermore, improvements in fabrication techniques - in particular in
lithography - will enable even smaller trapping length scales and better
spatial control.

Our work opens up several avenues for further research. On the fundamental
level, an important question is the nature of the confined excitonic state in
the relative coordinates, which may be investigated by applying a combination
of strong electric and magnetic fields [32]. Moreover, a detailed study of
lifetime and coherence time will provide insights into exciton dynamics in
such potentials. The fact that excitons confined in such nanoscopic traps
still retain substantial oscillator strength is a major advantage, as this
will enable strong coupling to light in microcavities [33] with expected Rabi
splitting in the meV range. Arrays of such electrically confined quantum dots
and rings may be combined with microcavity arrays to realize Bose-Hubbard
model of photons [34]. The potentially enhanced optical nonlinearity of
confined exciton systems may allow to go beyond the mean field regime and
explore the strong correlations regime, where exotic phases such as Mott
insulators are expected. Along the same lines, trapping radially dipolar
excitons in ring potentials may have immediate relevance for effecting
artificial gauge fields for optical excitations [35, 36], which is key to
realizing topological effects such as a photonic fractional quantum Hall
state. From the technological perspective, these configurable exciton
landscapes could be of relevance for development of active photonic
metamaterials [37] and novel light sources [38].

Acknowledgements. We are grateful to Patrick Knüppel, Deepankur Thureja and
Andrea Bergschneider for fruitful discussions. We thank Deepankur Thureja for
assistance with finite element simulations, and Elie Vandoolaeghe for help
with the experimental setup. Funding: Support for the Stanford/SLAC
researchers was provided primarily by the Q-NEXT Quantum Center, a U.S.
Department of Energy (DOE), Office of Science, National Quantum Information
Science Research Center, with additional support for device fabrication from
the DOE, Office of Science, Office of Basic Energy Sciences (BES), Materials
Sciences and Engineering Division and the Gordon and Betty Moore Foundation’s
EPiQS Initiative through grant number GBMF9462. J.H. acknowledges support from
NTT Research Fellowship. K.W. and T.T. acknowledge support from the JSPS
KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).

Author contributions P.A.M. and T.C. conceptualized the work. J.H., X.C. and
E.L. fabricated the devices. E.L., J.H. and T.C. performed the measurements
and analysed the data. J.H. performed the simulations. P.A.M and T.C wrote the
manuscript. K.W. and T.T. provided the hBN crystals. T.F.H., P.A.M and T.C.
supervised the project.

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## Supplementary Information

###  .1 Experimental setup

Figure S1: Schematic of the optical setup.

Figure S1 depicts our experimental setup. The sample is kept at $\sim 5$K
using an attoDRY800 closed-loop cryostat from Attocube. The sample is fixed on
a custom made PCB board, which is mounted on a positioning piezo stack
composed of three piezo stepper (two Attocube ANPx101/RES/LT and one
ANPz102/RES/LT) and a piezo scanner (ANSxyz100std/LT). A cryo-, vacuum-
compatible, high numerical aperture (NA = 0.81) objective (Attocube LT-
APO/NIR/0.81) is used within the cryostat space. Voltages are applied to the
sample gates using a Source Measure Unit (SMU) from Hewlett Packard (HP4142B)
fitted with HP 41422B (41420A) modules.

The white light reflectivity measurements are performed using a
Superluminescent Light Emitting Diode (SLED Exalos: EXS210065-01). A $720\,$nm
Ti:Sa continuous wave tunable laser (Matisse C from Spectra Physics), is used
for photoluminescence measurement. The signal beam is collected through a
single mode optical fiber (Thorlabs 630HP) that we use as a confocality pin
hole. The corresponding collection area on the sample surface has a FWHM of
$0.7\,\mu$m which sets the spatial resolution of our optical setup. The fiber
output is directed to an imaging spectrometer (Andor SR-750-D1 with
SR5-GRT-0600-0500, 600 line per mm grating) equipped with a Peltier cooled CCD
camera (Andor DU940P-UV).

Polarization resolved experiments are performed using a set of broadband
polarization optics in the excitation and detection arms, both consisting of
linear polarizers (Thorlabs LPVIS100-MP2), half wave plates (Thorlabs
AHWP10M-980), and quarter wave plates (Thorlabs AQWP10M-980). Linear Stokes
vector maps are obtained by recording PL spectra in different linear
polarization basis at each point on the sample:

$\displaystyle S_{0}=\langle E_{x}^{2}\rangle+\langle E_{y}^{2}\rangle=\langle
E_{d}^{2}\rangle+\langle E_{a}^{2}\rangle$ $\displaystyle S_{1}=(\langle
E_{x}^{2}\rangle-\langle E_{y}^{2}\rangle)/S_{0}$ $\displaystyle
S_{2}=(\langle E_{d}^{2}\rangle-\langle E_{a}^{2}\rangle)/S_{0}$

where $\langle E_{x(y)}^{2}\rangle$ is the intensity of light polarized along
$x$ ($y$) direction, and $\langle E_{d(a)}^{2}\rangle$ is the intensity of
light polarized along the diagonal (antidiagonal) direction. $S_{0}$ is the
total intensity of the signal.

AFM topographies are obtained in tapping mode, using ScanAsyst-air AFM probes
on a MultiMode-8-HR AFM from Bruker.

###  .2 Differential reflectivity data analysis

Differential reflectivity spectra $\Delta R/R_{0}=(R-R_{0})/R_{0}$ are
obtained from the measured reflectivity $R$ by subtracting and dividing by the
background spectrum $R_{0}$, measured at the same spot with strong homogeneous
doping. This is achieved by keeping the local top gates at 0 V and applying 10
V to the global back gate. Where needed, the differential reflectivity spectra
are numerically differentiate with respect to energy ($d(\Delta R/R_{0})/dE$)
in order to increase the visibility of the confined states resonances.

In Fig. 4 e, the energy of the 2D exciton and 0D exciton are shown. Those are
extracted from fits of $d(\Delta R/R_{0})/dE$ series. The reflectivity
spectral lineshape ($S(E)$) of heterostructures as studied here are well
approximated by the sum of a pure and dispersive Lorentzian lineshapes
($L_{0}$ and $L_{D}$ respectively), defined as follow:

$\displaystyle
L_{0}(E)=\frac{A\Gamma}{2\left[(E-E_{0})^{2}+(\Gamma/2)^{2}\right]}$
$\displaystyle
L_{D}(E)=\frac{A(E-E_{0})}{2\left[(E-E_{0})^{2}+(\Gamma/2)^{2}\right]}$
$\displaystyle S(E)=\cos(\theta)L_{0}(E)+\sin(\theta)L_{D}(E)$

with $A$ the Lorentzian amplitude, $\Gamma$ the Full Width at Half Maximum
(FWHM), $E_{0}$ the central energy of the mode and $\theta$ the phase between
the pure and dispersive Lorentzian. This Lorentzian is then numerically
differentiate and fitted to the data, using the least_squares routine from the
python library scipy.optimize.

The spectra present different features: the trapped exciton signal and a broad
resonance for the 2D exciton. Three complex Lorentzian are used to fit the
spectra, using the same phase parameter $\theta$. Two Lorentzians are needed
to fit the 2D exciton, most likely due to local strain in the TMD layer. Fig.
S2 present an example of fit for the three different bow ties.

Figure S2: Representative fitting results for the three different bow ties.
The blue curve is the differential reflectivity spectrum $\Delta R/R_{0}$, the
green one is its energy derivative, $d(\Delta R/R_{0}$)/dE, and the red one is
the triple dispersive Lorentzian fit. The thin grey lines represent each
individual derivated Lorentzian used to fit the derivative differential
reflectivity.

###  .3 Fabrication method

All hBN, graphite, and MoSe2 flakes used in this work are produced with
mechanical exfoliation using scotch tape. The high-quality hBN crystal is from
NIMS, and the MoSe2 crystal is from HQ graphene. The vdW heterostructure is
assembled using the standard PC (polycarbonate) dry transfer method in an
ambient environment. The heterostructure is then dropped down to a SiO2/Si
substrate at 180 C. Multilayer graphite flakes are picked up as contacts to
the MoSe2 monolayer.

Nanometer-scale features are generated with e-beam lithography (Raith Voyager,
50 keV) using a thin PMMA layer of 100 nm thick. For the quantum ring samples,
the vdW heterostructure consists of hBN capping layer/top Gr gate/top
hBN/MoSe2 monolayer/bottom hBN/bottom Gr gate. After e-beam patterning, the
hBN capping layer and the graphite top gate are etched by reactive ion etching
(Oxford Plasma Pro 80) sequentially. 10 sccm CHF3/10 sccm Ar/2 sccm O2 is used
to etch hBN, and 10 sccm O2 is used to etch graphite. For the bow tie samples,
the heterostructure consists of top hBN/MoSe2 monolayer/bottom hBN/bottom Gr
gate. The same e-beam lithography process is used to pattern the nanogap, then
a thin layer of metal (3 nm Ti/10 nm Au) is deposited as the top gates with a
Kurt J. Lesker high vacuum e-beam evaporator.

The leads and bonding pads to the patterned gates and the sample contacts are
formed by 5 nm Ti/45 nm Au, with patterns generated by either e-beam
lithography (with 200 nm PMMA) or photo-lithography (ML3 MicroWriter).

The thicknesses of the top and bottom hBN of the devices discussed in the main
text is listed in table S1.

Table S1:  Top and bottom hBN thicknesses of the devices discussed in the main text. Relevant figure | top hBN [nm] | bottom hBN [nm]
---|---|---
Fig. 2 b, d, e | 40 | 40
Fig. 2 f\- i, Fig. 4 a-c | 40 | 20
Fig. 3, Fig. 4 d, e | 15 | 25

###  .4 Electrostatic simulation method

The electrostatic simulation is done with COMSOL multiphysics electrostatic
modeling on a 3D grid using finite element method. The TMD monolayer is
modeled as a semiconductor, sandwiched in two insulating hBN slabs. The bottom
gate is global and the top gate is set to be certain geometry. See ref.[21]
for detailed parameters.

After the trapping potential is obtained from COMSOL simulation, the confined
state energy and wavefunction is obtained by solving the 2D Schrodinger
equation numerically in the c.o.m frame.

###  .5 Doping diagram

Figure S3: Doping diagram for a dual gated TMD with a hole in the top gate.
Each region separated by a solid black line correspond to a different doping
state of the TMD. This doping state is sketched in the squares, that are
charge density map, from hole doped in red to electron doped in blue, passing
by white at neutrality. The thick red triangles highlight the PiNiP and NiPiN
region, where the TMD 2D exciton can be trapped into a ring. The bold Roman
numerals correspond to the doping regime in Fig. 2 of the main text. The two
red triangles correspond to the region where ring confinement occurs.

Fig. S3 presents a sketch of the top and bottom gate voltage dependence of the
doping for a nanohole structure. First, the sample can be separated into two
different region, the dual gated region outside the hole (region B in the main
text), and the single gated region inside the hole (region A in the main
text).

In region A, only the bottom gate acts to set the charge density of the TMD.
Therefore, within the hole, the doping will switch from p- to n-doped due to
the sole bottom gate. This result in the vertical lines on the doping diagram
(Fig. S3), which are the bottom gate voltage at which the TMD switch from
neutral to p- or n-doped.

The dual gated region however, one can tune separately the vertical electric
field and the TMD doping. An opposite voltage between the top and bottom gate
will result in an out of plane electric field and no doping, which is the
central diagonal line of Fig. S3. On each side of this diagonal, p- and
n-doped region of the outter hole appears.

Since the trapping of 1D edge states occurs in the p-i-n regions, the
excitonic ring trap only appear in the upper left and bottom right red
triangles (Fig. S3), where the doping in the hole is of a different kind than
the doping outside it.

In addition, a weaker trap can be present in theory for the repulsive polaron,
when both top gate and bottom gate are positive or negative. In this case, the
inner part of the hole would be less doped, which will results in a lower
energy polaron in the hole, and a higher one outside. The dashed line in Fig.
S3 shows the limit at which the top gate voltage changes it sign which will
induce a lower charge density outside of the hole and a larger one within.
This latter configuration lead to an anti-trap for the repulsive polaron.

###  .6 Trapping potential vs $V_{\mathrm{{BG}}}$ in the 600 nm hole

Fig. S4 shows the simulated trapping potential at different
$V_{\mathrm{{BG}}}$ in a 600 nm hole. Here $V_{\mathrm{{TG}}}$ is set to be
$9.5\,$V, the same as in Fig. 2 e. A ring-shaped trap is only formed when
$V_{\mathrm{{BG}}}$ is from $-9\,$V to $-1\,$V, which agrees with our
observation in Fig. 2 e that the trapped states only show up in regime III.

Figure S4: Electrostatic simulation of the trapping potential at different
$V_{\mathrm{{BG}}}$ in the 600 nm hole. $V_{\mathrm{{TG}}}$ is set to be
$9.5\,$V, the same as in Fig. 2e.

###  .7 Simulation results for tunable 0D states in the bow tie

Electrostatic simulation for a nano-gap structure made with thin metal layers
can be tricky. Due to the small size of the electrode and reduced
conductivity, the voltage applied from the source-measurement unit might not
reflect the actual voltage drop across the bow tie. This results in a much
larger energy dispersion in the simulation compared to the measurements.

Here we present a way to calibrate the actual $\Delta V$ with respect to the
applied $\Delta V$. In the reflectivity spectra in Fig. 3 f, we observe both
the 0D states and the blue-shifting repulsive polaron (RP) emerging from
charging the TMD below the left and right electrodes. The blueshift of the RP
corresponds to the second term in Eq.1, which can be calculated from the
simulated charge density. By comparing it to the measured RP blueshift, we
determine a reduction factor that overlaps the simulated polaron shift to the
measured polaron shift, as shown in Fig. S5 (here $\Delta
V_{\text{actual}}=0.33\Delta V_{\text{applied}}$). Then we use the same
reduction factor to calculate the trapping potential and solve the energy
dispersion of the 0D states (shown in red dots in Fig. S5). It agrees
reasonably well with the measurement.

Figure S5: Comparison of the calculated 0D trapped state dispersion and the
measurement. The $\Delta V$ for the simulation input is calibrated using the
blue shifted repulsive polaron state.

###  .8 Transition from 0D confinement to 1D confinement in the bow tie

Figure S6: Simulation results for the evolution of the potential landscape
with the backgate voltage at
$(V_{\mathrm{{L}}},V_{\mathrm{{R}}})=(5\,\text{V},-5\,\text{V})$. It shows a
continuous transition from the 0D dot confinement ($V_{\mathrm{{BG}}}=0\,$V)
to 1D edge confinement wrapped around the right (left) electrode when the
backgate is tuned to negative (positive) voltage.  Figure S7: Reflectivity
spectra in the bow tie structure as a function of $V_{\mathrm{{BG}}}$ at
$(V_{\mathrm{{L}}},V_{\mathrm{{R}}})=(5\,\text{V},-5\,\text{V})$.

Here in Fig. S6, we show the simulation results for the potential landscape in
the bow tie structure, as a function of the backgate voltage. The left and
right bow tie electrodes are at +5 V and -5 V respectively, i.e. $\Delta
V=10\,$V. It demonstrates a continuous transition from the 0D dot confinement
($V_{\mathrm{{BG}}}=0\,$V) to 1D edge confinement wrapped around the right
(left) electrode when $V_{\mathrm{{BG}}}$ is tuned to negative (positive)
values. This 0D to 1D transition is observed experimentally as well, see Fig.
S7.

Fig. S8 shows a $V_{\mathrm{{L}}}-V_{\mathrm{{R}}}$ dual gate reflection
contrast scan at $V_{\mathrm{{BG}}}=0\,$V. Panel a is the integrated
reflection map, and the vertical (horizontal) stripe shows the gates range
where the 2D exciton under the right (left) bow tie electrode is kept neutral.
Panel b is the $\Delta R/R_{0}$ line cut along the solid line in panel a which
corresponds to applying $\Delta V$ across the bow tie. As expected, signature
for 0D trapped states is present. Panel c is the $\Delta R/R_{0}$ line cut
along the dashed line in panel a, corresponding to $\Delta V=0\,$V. No 0D
state is observed because no trap is formed when the same voltage is applied
to both bow tie electrodes.

Figure S8: $V_{\mathrm{{L}}}-V_{\mathrm{{R}}}$ dual gate scan map of
reflectivity at $V_{\mathrm{{BG}}}=0\,$V. (a) The integrated reflection map,
the vertical (horizontal) stripe shows the gate range where the 2D exciton
under the right (left) bow tie electrode is neutralized. (b) The $\Delta R/R$
line cut along the solid line in (a) which corresponds to applying $\Delta V$
across the bow tie. As expected, signature for 0D trapped state is present.
(c) The $\Delta R/R$ line cut along the dashed line in (a) corresponding to
$\Delta V=0\,$V, where no trapped state is observed.

Fig. S9 shows a $V_{\mathrm{{L}}}-V_{\mathrm{{R}}}$ dual gate scan map of
reflectivity at $V_{\mathrm{{BG}}}=10\,$V. Panel a is the integrated
reflection map, and the vertical (horizontal) stripe indicates the gates range
where the 2D exciton under the right (left) bow tie electrode is tuned to
neutrality. This is at a negative voltage now because $V_{\mathrm{{BG}}}$ is
set to be positive. Panel b is the $\Delta R/R_{0}$ line cut along the solid
line at $V_{\mathrm{{R}}}=0\,$V, showing 1D confined states around the left
bow tie electrode (as shown in the inset). Panel c is the $\Delta R/R_{0}$
line cut along the dashed line at $V_{\mathrm{{L}}}=0\,$V, showing 1D confined
states around the right bow tie electrode. Panel d plots the line cut along
the dashdot line, displaying two groups of 1D confined states around both bow
tie electrodes. The two sets of 1D confined states can also be tuned to
resonance. This shows that bow tie structures are very versatile and can host
both 0D and 1D tunable confined exciton.

###  .9 0D trapped states in multiple samples

The 0D trapped state is consistently observed in various bow tie structures on
multiple samples, as shown in Fig. S11. For some bow ties, we only see 0D
state on one side of $\Delta V$, possibly due to charge inhomogeneity around
the nano-gap region.

Figure S9: $V_{\mathrm{{L}}}-V_{\mathrm{{R}}}$ dual gate scan map of
reflectivity at $V_{\mathrm{{BG}}}=10\,$V. (a) The integrated reflection map,
the vertical (horizontal) stripes shows the gates range where the 2D exciton
under the right (left) bow tie electrode is neutralized. (b) The $\Delta R/R$
line cut along the solid line at $V_{\mathrm{{R}}}=0\,$V, displaying 1D
confined states around the left bow tie electrode (as shown in the inset). (c)
The $\Delta R/R$ line cut along the dashed line at $V_{\mathrm{{L}}}=0\,$V,
displaying 1D confined states around the right bow tie electrode. (d) The
$\Delta R/R$ line cut along the dashdot line, showing 1D confined states
around both bow tie electrode.

###  .10 Polarization dependence of the 0D states

In Fig. S10, we show the polarization dependence of the 0D trapped states in
the bow tie structure. The trapped state follows a linear polarization basis,
consistent with the anisotropy of the trapping potential shown in Fig. 3 d.

Figure S10: Polarization dependence of the 0D trapped states in the bow tie
structure. Figure S11: 0D confined states measured in bow tie structures on
multiple samples.