# Probing inter-valley coupling via exciton hybridization in a charge-tunable GaAs/AlAs quantum dot

Arnault Raymond, Paola Atkinson, Antoine Chapuis, Savvas Germanis,
Florent Margaillan, Mathieu Bernard, Valia Voliotis, Benoit Eble

► To cite this version:

Arnault Raymond, Paola Atkinson, Antoine Chapuis, Savvas Germanis, Florent Margaillan, et al. Probing inter-valley coupling via exciton hybridization in a charge-tunable GaAs/AlAs quantum dot. Physical Review B, 2025, 112 (12), pp.125302. 10.1103/qlvx-wpbf. hal-05280128

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Distributed under a Creative Commons Attribution 4.0 International License
Probing inter-valley coupling via exciton hybridization in a charge-tunable
GaAs/AlAs quantum dot

Arnault Raymond,\* Paola Atkinson, Antoine Chapuis, Savvas Germanis,
Florent Margaillan, Mathieu Bernard, Valia Voliotis, and Benoit Eble\d

Sorbonne Université, CNRS, Institut des Nanosciences de Paris, 4 Place Jussieu, 75252 Paris, France

(Dated: September 23, 2025)

We study a type-II GaAs/AlAs quantum dot system that enables inter-valley coupling between electronic states in the $\Gamma$ and $X_z$ valleys, probed optically at the single-dot level. This structure intrinsically supports both direct and indirect excitons, with their spatial separation tunable via an external electric field. Stark shift spectroscopy reveals a clear double anticrossing in the neutral exciton line as a direct signature of inter-valley coupling. Experimental observations, including measurements of static dipole moments and electron g-factors, confirm electron confinement within the AlAs barrier and highlight the involvement of barrier-localized states. We further propose that these states may contribute to the formation of multicharged excitonic complexes. A simplified model allows us to estimate the interfacial coupling strength from the measured anticrossing amplitudes.

I. INTRODUCTION

In the context of quantum information processing, semiconductor quantum dot molecules (QDMs), comprising two quantum dots (QDs) separated by a nanometer-scale tunneling barrier, have emerged as promising platforms due to their capacity to mediate interactions between direct and indirect excitons [1]. Direct excitons, formed by electron-hole (e-h) pairs localized within the same dot, are characterized by strong light-matter coupling and short radiative lifetimes. In contrast, indirect excitons, in which the electron and hole are spatially separated across the two dots, exhibit longer lifetimes, weaker optical transitions, and a stronger interaction with external electric fields owing to their large static dipole moment, enabling substantial energy tunability.

Understanding the mechanisms that govern the cou-
pling between direct and indirect excitons is therefore
critical for the realization of quantum technologies based
on charge and spin degrees of freedom [2, 3].

Here, we present an alternative to QDMs: a single
GaAs QD embedded between upper and lower AlAs bar-
riers. These AlAs barriers define the confinement po-
tential for the GaAs QD via the conduction band edge
at the Γ point, while simultaneously lowering the con-
duction band minimum at the X point relative to the
Γ point in AlAs. As a result, electron states associ-
ated with the X-valley become energetically favorable
and spatially stabilized within the AlAs barriers due to
quantum confinement[4, 5]. These X-valley electrons can
couple to Γ-valley electrons in the QD via Γ-X inter-
valley coupling at the interface [6–11], playing a role
analogous to the tunnel coupling in QDMs. This struc-
ture bears similarities to type-II GaAs/AlAs quantum
wells [7, 12–15], including short-period superlattices ex-
tensively studied in the 1990s [16–18]. However, it is

distinctive in that it enables optical addressing of a sin-
gle e-h pair state. This state consists of a hole confined
within the GaAs dot and an electron that may reside
either within the dot forming a direct exciton, both spa-
tially and in *k*-space or in the barrier, forming a spatially
and momentum-space indirect exciton. We will demon-
strate a key feature of this system: both the upper and
lower AlAs barriers, separated by only a few nanometers,
contribute to the formation and stabilization of indirect
excitons whose static dipole orientation can be controlled
via an external electric field.

Historically, QD optics dominated by X-valley elec-
tron confinement have mostly been realized in the In-
AlAs/GaAs material system [19–24]. In such systems,
both the X-valley electron and the hole remain confined
within the same dot, corresponding to a type-I band
alignment. Although this configuration generally yields
high emission efficiency due to good wavefunction over-
lap, it lacks the crucial QDM-like property of coupling
between two spatially separated electronic states. More-
over, these InAlAs/GaAs dots are typically studied in
ensembles, in contrast to the GaAs/AlAs QDs used in
this work, which are grown via epitaxial nanohole filling.
This technique enables ultra-low dot densities, making
single-dot spectroscopy possible.

To render *k*-space indirect excitons optically active, they must be coherently mixed with direct excitons in order to satisfy momentum conservation during photon emission [13]. In the following, we present a direct and quantitative approach to estimate the inter-valley coupling strength by leveraging the electric field tunability of a QD embedded in a Schottky diode structure with gate-controlled bias.

II. SAMPLE DESCRIPTION AND
EXPERIMENTAL RESULTS

The sample consists of GaAs/AlAs QDs fabricated
by nanohole infilling within an n-i Schottky structure

<footnote>* araymond@insp.jussieu.fr</footnote>

<footnote>† benoit.eble@insp.jussieu.fr</footnote>
<figure><img src="image_2.png" /></figure>

FIG. 1. Map of the luminescence intensity in log scale as a function of the applied electric field and energy. The red and green colormaps are associated with different linear polarization of detection, respectively X-polarization and Y-polarization, aligned with the two neutral exciton components. Fig. 1(a) covers a wide range of electric fields, providing a comprehensive view of the coexistence of direct and indirect transitions, which are characterized by a strong Stark effect. The different excitonic charge states, as well as the charge domains of the QD, are labeled. Fig. 1 (b) to (g) present zoomed-in views highlighting the presence of anti-crossings between direct and indirect transitions. Dashes lines are included as visual guides to emphasize these anti-crossings. Specifically, Fig. 1(b) reveals an anti-crossing for the $X^{+,*}$ state. Fig. 1 (c), (d), and (e), (f) illustrate the two anti-crossings experienced by the neutral exciton at different electric field values, $F_z = -100$ kV/cm and $F_z = -40$ kV/cm. Finally, Fig. 1(g) shows the anti-crossing for the $X^-$ state. The three arrows marking the $X^0$ lines also indicate the electric field values ($F_z = -105, -80$ and $-30$ kV/cm) at which the fine structure of the neutral exciton was measured in Fig. 2 (a), (b), and (c).

with a 168-nm-wide intrinsic region (hereafter labelled d). The nanoholes, formed via Al droplet deposition followed by annealing under As4, typically measure 8 to 10 nm in depth and approximately 60 nm along [110] to 90 nm in width. Overgrowth with a 1-nm-thick GaAs layer, combined with net material migration toward the bottom of the hole during a brief 15-second growth interrupt, results in QDs that are 2 to 4 nm in height. Nanohole etching is performed after the deposition of an 18-nm $Al_{0.45}Ga_{0.55}As$ intrinsic spacer atop the n-doped contact, positioning the QDs nominally around 10 nm from the n-contact. The QD density is extremely low, at just one or two QDs per square micrometer, enabling easy isolation of individual QDs for optical spectroscopy experiments. The QD charge is controlled by applying a bias between the semitransparent Schottky gate on the top surface of the sample and the diffusive Ohmic back contact. The Schottky diode structure is ideal for imple-

menting Stark shift spectroscopy because it provides a reliable estimation of the electric field $F_z = -(V - V_s)/d$ within the sample, especially since the threshold voltage $V_s \sim 0.8$ V is determined from the I-V characteristics in dark condition. The linear relationship between electric field and applied voltage, as well as the value of the threshold voltage $V_s$, remain unchanged under low-power optical excitation, with minimal screening effects due to photo-generated charges induced by the 635 nm laser, in the vicinity of the QD.

Fig. 1 shows the cryogenic temperature (T=3K) Stark spectroscopy of a QD whose luminescence centered at 1.748 eV is dispersed by a double spectrometer with a spectral resolution of 15 µeV. The different charge states of the QD are highlighted and appear clearly in successive electric field regions [25, 26], which are delimited by the vertical dotted lines in Fig. 1. At high electric fields, we can label the positive trion $X^+$, as
well as the hot trion $X^{+,*}$ (with a hole in a P-shell), then the neutral exciton $X^0$ and the neutral biexciton $2X^0$. The dot becomes stable in the one-electron regime where the luminescence of the negative trion $X^-$ is detected above -50 kV/cm. The log-scale is used as the vertical scale, revealing low intensity lines characterized by a strong Stark shift, mostly with a positive slope, but some with a negative slope. These signatures were observed repeatedly in this sample (See Suppl. Mat.). Let us focus on transitions exhibiting a pronounced Stark effect. The magnitude of the Stark shift corresponds to a static dipole of $\mu/e = +1.58$ nm for the upslope and $\mu/e = -0.5$ nm for the downslope. This straightforward analysis rules out Mahan excitations[27, 28] as the origin of these lines, thereby excluding any significant role of the Fermi sea located 10 nm above the lowest point of the nanohole. If coupling between the QD and the delocalized electronic continuum of the Fermi sea were present, it would produce, on one hand, an optical signature consistent with a static dipole of 10 nm and on the other hand, an absence of anti-crossing behavior, as the continuum of states would couple to a QD discrete state. However, experimental evidence does not support the existence of such features. Instead, the effective electrostatic dipoles' lengths align with the nominal distance between the center of the QD and the center of the barriers. This observation indicates the localization of electrons within the barriers, thereby demonstrating the role of quantum confinement within the AlAs X-valley (See Fig. 5(b)).

The dipole length estimation excludes also the configuration in which the electron remains localized outside the AlAs barrier at the Γ-point of the AlGaAs matrix and stabilized by the Coulomb attraction of the hole in the dot, forming an interface exciton[29]. It is important to note that the absence of a blocking barrier between the QD AlAs barrier and the Fermi sea prevents the stabilization of such an interface exciton in the strong electric field regime. Instead, it allows for efficient electron tunneling. However, experimental results indicate that indirect lines begin to become detectable in the strong band tilting regime, i.e., for electric fields reaching -120 kV/cm. Consequently, confinement within the AlAs barrier is essential to prevent electron escape at a rate faster than the radiative lifetime.

### III. LEVEL ANTI-CROSSING SPECTROSCOPY IN ABSENCE OF MAGNETIC FIELD

In order to better understand the emission of these indirect transitions, let's focus on $X^0$, whose luminescence is visible from -110 kV/cm to -20 kV/cm. The direct excitonic branch exhibits two distinct intersections with an indirect transition: one near -40 kV/cm where the latter exhibits a negative slope, and another around -110 kV/cm with a positive slope. Well-resolved anti-crossings

emerge clearly.

To validate this observation, Fig. 2 displays the photoluminescence (PL) intensity as a function of the linear polarization analysis angle, recorded at three electric field values indicated by the arrows in Fig. 1: one prior to the first anti-crossing at $F_z = -110$ kV/cm (Fig. 1(c)), another before the second anti-crossing at $F_z = -80$ kV/cm, and a third following the second anti-crossing at $F_z = -30$ kV/cm (Fig. 1(e)). Despite variations in overall PL intensity, all three maps exhibit a comparable fine structure splitting amplitude and a consistent asymmetry between the orthogonally polarized excitonic components. This strongly suggests that the same excitonic state is being probed on both sides of the two anti-crossings.

The interaction responsible for the anti-crossing also mediates the transfer of oscillator strength between the direct and indirect transitions. This effect is particularly evident when examining the evolution of the direct and indirect lines in Fig. 1(c) and (d). As shown in Fig. 2(d), the linear dependence of the PL intensity ratio measured under a fixed linear polarization, $I_{IND}/I_{X^0}$, as a function of the inverse square of the energy splitting $\Delta E$ between the direct and indirect branches, is consistent with first-order perturbation theory. In this framework,

<figure><img src="image_3.png" /></figure>

FIG. 2. Fig.(a), (b), and (c) show the luminescence of the direct transition as a function of the linear polarization analysis direction at $F_z=-105$, -80 and -30 kV/cm. These correspond, respectively, to the regions before the first anti-crossing (see Fig. 1(c) and (d)), in between, and after the second anti-crossing (see Fig. 1(e) and (f)). The measurements reveal the same fine structure splitting amplitude and phase, as well as an identical asymmetry in the intensity ratio of the two orthogonal polarization components. Fig.(d) shows the intensity ratio of the direct and indirect transitions measured in the X-polarization configuration as a function of the inverse squared splitting between the direct and indirect lines. Fig.(e) presents the evolution of the linewidth as a function of $F_z - F_{AC}$ along the lowest energy branch of the anti-crossing, where $F_{AC}$ corresponds to the first anti-crossing field at -100 kV/cm.
$I_{IND}/I_{X^0} \propto |W|^2/\Delta E^2$ where W is the coupling matrix element. This dependence holds for both orthogonal linear polarizations.

As shown in Fig. 1, four distinct anti-crossings can be identified. Two of them involve the direct transition of the neutral exciton ($X^0$), one of which is accompanied by an indirect branch exhibiting an inverted Stark shift at $F_z = -35$ kV/cm, as previously discussed. The two remaining anti-crossings are attributed to the hot positive trion ($X^{+,*}$) at $F_z = -115$ kV/cm (Fig. 1(b)) and the negative trion ($X^-$) at $F_z = -40$ kV/cm (Fig. 1(g)). As illustrated in Fig. 1(b)–(g), a common feature of these anti-crossings is observed: when the indirect transition energy exceeds that of the direct transition, the corresponding PL signal rapidly decreases. This phenomenon has been previously reported in AlGaAs/AlAs type-II superlattices [30, 31], and is attributed to fast thermalization from the X- to Γ-valley. The oscillator strength of the indirect transition is insufficient to compensate for the population loss due to this thermal redistribution, when it comes to favor the Γ-valley minimum. The thermal relaxation time back to equilibrium has been measured in type-II superlattices and found to be on the order of picoseconds [32, 33]. This ultrafast intervalley charge transfer may occur faster than the exciton formation time, effectively suppressing radiative recombination via the indirect channel. Such a transfer process is likely phonon-assisted [34] and strongly dependent on the crystallographic quality of the interfaces, including interface roughness and alloy disorder [35, 36]. It is therefore reasonable to assume that a similar thermalization mechanism is active in the GaAs/AlAs QD system. This process may also contribute to the broadening of the indirect emission lines, particularly away from the anti-crossing regions.

As far as the width of indirect lines is concerned, the role of spectral diffusion due to electrical noise in the vicinity of the dot cannot be excluded, especially since it has already been shown to contribute significantly to the broadening of direct transitions in the same type of sample [37]. The phenomenon of spectral diffusion is all the more relevant in this case because the static dipole is an order of magnitude larger than that of the direct transitions (estimated to be ~0.1 nm). Fig. 2(e) illustrates the evolution of the spectral width of the indirect transition of the neutral exciton, showing a plateau far from the anti-crossing, where the linewidth is ~10 times larger than for the direct exciton. The width progressively narrows, approaching the width associated with the direct exciton. This spectral width transfer is also an expected signature of the anti-crossing phenomenon. The observed anti-crossing amplitudes, ranging from 200 to 250 µeV, are somewhat smaller than those typically reported for GaAs/AlAs type-II superlattices and quantum wells, which are generally on the order of 1–2 meV [16].

We can now provide a more detailed description of the X-type electron. In the bulk, the X-valley is three

times degenerate, $X_x$, $X_y$ and $X_z$, positioned at the edge of the first Brillouin zone at $\vec{q}_x = (\pm\pi/a_0, 0, 0,)$, $\vec{q}_y = (0, \pm\pi/a_0, 0)$ and $\vec{q}_z = (0, 0, \pm\pi/a_0)$, $a_0$ being the monolayer thickness ($a_0 = 2.83Å$). Confinement along the z-axis and potentially residual strain lift the degeneracy between $X_{x,y}$ and $X_z$ valleys. It has been demonstrated in GaAs/AlAs quantum wells [38] that the conduction ground state in the X-valley should be considered as that arising from the $X_z$-valley, for AlAs barrier thicknesses ranging from 15 Å to 45 Å. This range includes typical values for AlAs barrier thicknesses for the QDs used here. We then assume that the electronic states of the barrier are of $X_z$-type. Electronic states in the $X_{x,y}$ valleys are some 30 meV higher [15].

The sign of the slopes for the indirect transitions is, of course, correlated with the location of the electron, which is confined either in the lower barrier for positive slopes or in the upper barrier for the opposite case (See Fig. 5(b)). This is illustrated in Fig. 1 with the two anti-crossing zones undergone by $X^0$. Within the framework of a simple model that will be discussed in the last section, we can show that the high or low location of the electron depends not only on the geometry of the {dot + barriers} system, but is also driven by the Coulomb interaction.

In the vicinity of the anti-crossing region, the two exciton eigenstates describing the upper (+) and lower (-) branches are written:

$$|\Psi_{\pm}\rangle = \alpha_{\pm,\Gamma} |\tilde{\Phi}_{\text{env}}^{\Gamma} \Gamma^e \Gamma^h\rangle + \alpha_{\pm,X_z} |\tilde{\Phi}_{\text{env}}^{X_z} X_z \Gamma^h\rangle, \quad (1)$$

where $\langle\vec{r}_e|\Gamma^e\rangle = u_{\Gamma_6}(\vec{r}_e)$ and $\langle\vec{r}_h|\Gamma^h\rangle = u_{\Gamma_8}(\vec{r}_h)$ are respectively the Bloch functions in the center of the Brillouin zone for the conduction and heavy-hole band, whereas $\langle\vec{r}_e|X_z\rangle = u_{X_1z}(\vec{r}_e)$ is the Bloch function of the lowest energy X1 valley at the $X_z$-point. $\langle\vec{r}_e, \vec{r}_h|\tilde{\Phi}_{\text{env}}^{\Gamma}$ and $\langle\vec{r}_e, \vec{r}_h|\tilde{\Phi}_{\text{env}}^{X_z}$ are the excitonic envelope wavefunctions of the direct and indirect exciton taking into account the e-h Coulomb interaction. Finally, $\alpha_{\pm,\Gamma}$ and $\alpha_{\pm,X_z}$ are complex weights dependent on the interaction strength between the valleys. This process converts an $X_z$-type electron wavefunction in AlAs into a Γ-type wavefunction in GaAs as it crosses the heterointerface. This is a coupling between Bloch functions. More precisely, Volkov et. al. in Ref. [11] demonstrated in the context of $\vec{k} \cdot \vec{p}$ theory that this interaction<fnref n="1" />, for a stratified heterostructure along the [001], direction carried by $\vec{u}_z$, could be expressed as:

$$\hat{W}(\vec{r}_e) = \sum_{i,s_z,s'_z} \beta_i^{s_z,s'_z} \delta(\vec{r}_e \cdot \vec{u}_z - z_i) e^{-i\vec{q}_z \cdot \vec{r}_e} |X_z, s_z\rangle \langle \Gamma^e, s'_z| + \text{h.c.}, \quad (2)$$

<footnote n="1">¹ This work validates earlier empirical approaches. See Ref. 6</footnote>
where $\vec{r}_e$ is the electron position and $\delta(z)$ is the Dirac distribution. $s_z = \pm 1/2$ represents the electron spin projection. $\beta_i$ is a quantity that expresses the strength of coherent inter-valley mixing at the i-th heterointerface, located at $z_i$. It depends on the difference in atomic potential between an Al atom in the AlAs matrix and a Ga atom in the GaAs matrix. By convention, the phase factor is taken as negative for electronic wave transfer from GaAs to AlAs. The above expression applies to a crystallographically perfect planar interface. However, for a real interface that may exhibit local atomic disorder, we extend this expression under the assumption that the Hamiltonian in Eq. 2 reliably describes the coupling within a coherence volume that is sufficiently large compared to the characteristic size of the system under study.

In expression (2), we refine the formulation of the interface valley coupling to include electron spin dependence. Specifically, we distinguish between $\beta_{s.c.} = 2\beta_i^{s_z, s_z}$ representing the spin-conserving component, and $\beta_{f.f.} = 2\beta_i^{s_z, -s_z}$, corresponding to the flip-flop component of the interface coupling. The latter may originate from bulk inversion asymmetry in III-V semiconductors, or from interfaces lacking a common shared atom system, an effect in principle absent from the GaAs-AlAs interface [39]. As we will see, our experiments provide a unique means to quantify the relative contribution of the flip-flop term compared to the spin-conserving one, confirming that the former is expected to be weak.

Let us first examine the spin properties of the inter-valley coupling. Previous studies have demonstrated that dark excitons (DE) can become optically active due to symmetry-breaking-induced mixing with bright excitons [37, 40, 41]. It is noteworthy that although the DE is barely visible in Fig. 1 230 µeV below the bright exciton (BE) lines (corresponding to the dark-bright splitting $\Delta_{DB}$), the PL energy evolution with the bias directly probes the influence of the flip-flop component, which couples it to the bright configuration of the indirect exciton (See Eq. 2). Consequently, as the DE approaches the anti-crossing region, it should also exhibit an avoided crossing with the indirect transition of bright type. However, this avoiding behavior has not been clearly observed in the immediate vicinity of the anti-crossing (see Fig. 3). In contrast, the DE energy, as well as the position of the indirect transition, remain essentially unaffected in the vicinity of the anticrossing with the bright exciton. This observation strongly supports the condition $|\beta_{s.c.}| \gg |\beta_{f.f.}|$. A simple model that evaluates the DE energy as a function of electric field yields an upper bound of 10% for the ratio $|\beta_{f.f.}/\beta_{s.c.}|$. The accuracy of this estimate is intrinsically limited by the low visibility of the DE, which becomes increasingly obscured near the indirect transition due to its broad spectral linewidth. Nonetheless, to ensure compatibility with the experimentally obtained results, the extrapolated DE position as a function of electric

field, calculated under the assumption of a finite $\beta_{f.f.}$, must remain within the experimental error bars of the indirect exciton position. This constraint allows us to confidently exclude ratios exceeding 10% (See fig. 3). Thus, the inter-valley coupling conserves the electron spin and, by construction, does not affect the hole spin. As a result, it has no influence on the linear dichroism of the exciton. This is illustrated in Figs. 1(c) and (d) on one hand, and (e) and (f) on the other, which map the interaction with the neutral exciton $X^0$. Two distinct anti-crossings are observed, corresponding to the two linearly polarized components of the neutral exciton. The anti-crossing centers depend on the orientation of the linear polarization analysis and are shifted according to the fine structure splitting amplitude $\delta_1$, consistent with the behavior of the two orthogonal excitonic eigenstates.

These final considerations offer a clearer framework for

<figure><img src="image_4.png" /><figcaption>FIG. 3. Evolution of the low-energy bright exciton component and the dark exciton as a function of $F_z - F_{AC}$, where $F_{AC} = -100$ kV/cm denotes the electric field at the anticrossing. The light blue region indicates the field range over which the bright exciton exhibits an avoided crossing. Blue error bars reflect the uncertainty in the measured energy positions, while a red shaded area provides a visual guide to the envelope of these uncertainties around the data points. The dark exciton exhibits very low intensity, and its energy position is extracted directly from CCD camera data without fitting. Consequently, error bars are conservatively set to $\pm 1/2$ pixel around the central pixel. Dotted black lines represent the calculated energies of the hybridized DE and indirect exciton in the presence of a flip-flop coupling term $\beta_{f.f.}$ for different ratios $|\beta_{f.f.}/\beta_{s.c.}|$. In the expected crossing region between the DE and the indirect exciton, only anticrossing gaps corresponding to a ratio $|\beta_{f.f.}/\beta_{s.c.}| < 10\%$ remain consistent with the experimental error bars on the indirect exciton energy, thereby allowing larger ratios to be confidently ruled out.</figcaption></figure>
analyzing the anti-crossing associated with the negative trion $X^-$ (see Fig. 1(g)). Notably, the anti-crossing amplitude is larger in the case of $X^-$, suggesting that the hole, due to its heavier effective mass, is preferentially localized at the bottom of the QD. This localization tends to attract both electrons toward the same region, enhancing their spatial overlap at the heterointerface and thereby increasing inter-valley mixing.

In particular, this coupling efficiently mixes the direct negative trion, which forms a spin singlet state as both electrons occupy the same orbital, with its indirect counterpart. Since the inter-valley interaction is predominantly spin-conserving, the spin multiplicity of the two-electron system is preserved. As a result, the indirect trion is formed in the singlet configuration $|S_0\rangle = 1/\sqrt{2} (|\uparrow_\Gamma \downarrow^{X_z}\rangle - |\downarrow_\Gamma \uparrow^{X_z}\rangle)$. Using the notation commonly employed in QDMs spectroscopy [1], the indirect optical transition corresponds to: $|\frac{1}{-1}\frac{1}{1}\rangle_{S_0} \approx |\frac{0}{-0}\frac{1}{1}\rangle$, where the first row indicates the number of electrons and the second row the number of holes, in the bottom barrier and in the dot respectively<fnref n="2" /> (left to right). Only the bottom barrier needs to be considered in this case, as the positive slope of the indirect line confirms electron localization in that region.

At the exit of the anti-crossing, an additional emission line emerges, positioned approximately 200µeV above the direct component of the trion (See Fig. 1(g)). As the electric field exceeds -40 kV/cm, approaching the flat-band regime, the intensity of this new line increases, while the emission from the trion correspondingly decreases. These observations indicate a correlation between this line and the occupation of an electronic state located in the AlAs barrier, rather than in the quantum dot. Further insight comes from the Zeeman structure of this line, which exhibits only two branches with an energy separation governed by an effective g-factor similar in magnitude to that of the neutral exciton (not shown). Taken together, these features strongly suggest that the observed line is not associated with the $X^{2-}$ state, even though such a charge configuration would be compatible with the electric field regime under study (i.e., the dot remains stable with two electrons). Notably, no fine structure is resolved at zero magnetic field, and no additional spectral lines emerge under magnetic field, both of which contradict the expected behavior of a genuine $X^{2-}$ [42].

Instead, a more consistent explanation involves such an optical transition: $|\frac{1}{-1}\frac{2}{1}\rangle \approx |\frac{1}{-0}\frac{1}{1}\rangle$, a configuration with a singlet electron pair confined in the quantum dot and a third, unpaired electron occupying a state in the AlAs barrier in the excited state. This scenario supports

experimental findings under realistic assumptions:

(i) The electron-hole exchange interaction in this excited configuration is negligible, due to the vanishing overlap between the unpaired electron and the confined hole, resulting in an absence of measurable fine structure splitting.

(ii) The energy difference between the $\Gamma$- and $X_z$-valley electrons in the final state, after photon emission, is estimated to be on the order of 10 meV, which exceeds both the inter-valley coupling and the exchange interaction between the electrons. This results in two well-separated lines in Zeeman spectroscopy, especially in the regime where $|g_X^e| \gg |g_\Gamma^e|$, as confirmed experimentally.

(iii) Finally, the energy of this recombination lies close to the expected position of the $X^{2-}$, further supporting the interpretation of a $X^{2-}$-like state: its emission energy proximity arises because the Coulomb interaction energies involved in the excited state $|\frac{1}{-1}\frac{2}{1}\rangle$ and its corresponding final state $|\frac{1}{-0}\frac{1}{1}\rangle$ are quantitatively similar to those of the $X^{2-}$ and two-electron in-dot configurations. This is due to the spatial proximity of the X-electron localized in the bottom AlAs barrier to the quantum dot, which leads to a comparable Coulomb renormalisation.

#### IV. STARK SPECTROSCOPY IN A FARADAY MAGNETIC FIELD

Other measurements support the role of the AlAs barrier, as previously demonstrated by Stark spectroscopy, particularly in magneto-optical measurements. It is well established that the electron g-factor exhibits a significant material dependence, with a value of -0.44 in bulk GaAs and approximately ~+2 in the minium energy valley of bulk AlAs, while remaining positive but decreasing in ternary compounds with sufficient Al concentration. This change in sign has been used to explain the weak values of the electron g-factor and its possible sign reversal in GaAs/AlGaAs dots[43]. This is because the measured g-factor is averaged by the weight of the envelope function spreading into the AlAs barrier. Additionally, transport measurements have quantified the electron g-factor of 3.8 in 2D electron gas filling the X-valleys of a AlAs quantum well [44]. In general, experimental determination of the electronic g-factor of the X-valley in the AlAs system indicates a value significantly higher than that of the GaAs system in the $\Gamma$-valley, and this is even more pronounced as the electron-X is confined.

To confirm the important role of the AlAs X-valley, an external magnetic field up to 6 T is then applied in the Faraday configuration, lifting the degeneracy of the circularly polarized transitions. Fig. 4(a) maps the circular polarization rate of the luminescence<fnref n="3" />, defined as

<footnote n="2">² Note that the formalism employed here can be readily extended to include the upper AlAs barrier by adding a third column. The symbol “-” indicates that no hole is present in the AlAs barrier. The underlined traits are used to identify the carriers that recombine.</footnote>

<footnote n="3">³ Note that $P_c$ differs from the polarization rate calculated using the total intensity of both components. $P_c$ provides local infor-</footnote>
<figure><img src="image_5.png" /><figcaption>FIG. 4. a) Circular polarization rate as a function of photoluminescence energy and electric field, for an applied Faraday magnetic field of $B_z = 4$T. Exciton species luminescence energies are superimposed and labeled, with the neutral exciton represented by marker symbols, where the diamond (resp. cross) indicates the lower (resp. upper) branch of the anti-crossing, and red (resp. blue) corresponds to the $\sigma^+$ (resp. $\sigma^-$) polarization. b) Zeeman splitting of the lower and upper branches of the anti-crossing for a $B_z = 4$T magnetic field as a function of electric field. The dotted lines represent the measured splitting values for the g-factors associated to the direct and indirect excitons at the corresponding electric field magnitudes. c) Zeeman splitting for the direct (at $F = -90$ kV/cm) and indirect ($F = -108$ kV/cm) excitons, far from the anti-crossing. The dotted line are numerical fits, yielding the displayed values for the g-factors of the direct and indirect exciton. Note that, for the magnetic field range used here, the Zeeman splitting is larger than the exciton fine structure splitting. We therefore neglect the latter for the g-factor estimations.</figcaption></figure>

$P_C(E, F_z) = (I_+ - I_-)/(I_+ + I_-)$, under a magnetic field of 4T, with $I_+$ (resp. $I_-$) is the measured $\sigma^+$ (resp. $\sigma^-$) intensity at energy $E$ and electric field $F_z$. Additional information is overlaid, including the energy positions of all the excitonic species previously identified in Fig. 1 for the same electric field range. They are represented using symbol markers for the neutral exciton, and dashed lines for other excitonic states.

Before delving into the case of the neutral exciton, we will qualitatively examine the behaviour of the positively charged trion $X^+$. It is important to note that due to the partial overlap of spectra involving counter-polarized broad lines, the energy positions do not exactly coincide with extrema of the circular polarization rate. For convenience, the Zeeman splitting is here defined as the energy difference between the $\sigma^+$ and $\sigma^-$ component energy positions. This allows for a meaningful interpretation of the exciton g-factor sign, which is taken to be positive for the direct exciton[37, 43]. We thus observe that the Zeeman splitting is positive for $X^+$ (purple dashed lines) with a value of 260 µeV, and significantly larger for the hot trion state $X^{+,*}$ (green dashed lines), reaching 350 µeV. The neutral exciton

case presents greater complexity due to the presence of an anti-crossing already observable at zero magnetic field (see Fig. 1(c) and (d)). When a magnetic field is applied, this configuration evolves into four distinct branches, as shown in Fig. 4(a). We refer to the two Zeeman-split branches associated with the indirect exciton, which appear at lower energy for large absolute electric fields, as the *lower branches*. Conversely, the *upper branches* correspond to the Zeeman components of the direct exciton. A detailed determination of the difference in diamagnetic shifts between the direct and indirect excitons, based on the evolution of the anticrossing electric field with applied magnetic field, is presented in the Supplemental Material[45]. Away from the anti-crossing region centered at $F_z = -102$ kV/cm, and on either side i.e. $F_z > -95$ kV/cm and $F_z < -110$ kV/cm, a positive Zeeman splitting of 320 µeV is observed for the direct exciton. It should be noted that the $\sigma^+$ component of the direct exciton energy position is observed in a region of negative circular polarization, as a consequence of proximity to the more intense $\sigma^-$ component of the hot trion.

As illustrated in Fig. 4(a), the two circularly polarized components associated with the indirect transitions of the upper branch vanish as soon as they reach an energy slightly higher than that of their direct counterpart.

<footnote>mation at a specific point $(E, F_z)$ on the map.</footnote>
This has already been seen in the absence of a magnetic field. Let us now examine indirect transitions far from the anti-crossing zone. For the lower energy branch, these components remain measurable for electric fields below -100 kV/cm. However, a significant distinction emerges when comparing these transitions with direct transitions, since the $\sigma^-$ component is positioned above the $\sigma^+$ one, which yields a negative Zeeman splitting of -720 µeV. This results in a reversal of the g-factor sign, and a continuous electric field-control of the Zeeman splitting from a negative to a positive values, as displayed in Fig. 4(b) (See Suppl. Mat.) The minor discrepancy in the g-factor direct exciton observed beyond and on both sides of the anticrossing region may be attributable to the weak g-factor electric field dependence[46], which is independent of inter-valley mixing. A notable feature is the existence of a critical electric field at which the Zeeman splitting vanishes. The dependence of this critical field on the applied magnetic field is further discussed in the Supplemental Material[45].

Fig. 4(c) presents the evolution of the Zeeman splitting as a function of $B_z$, measured for both direct and indirect excitons for a fixed applied electric field. The g-factors can be readily extracted, yielding for the direct exciton $g_{dir}^z = 1.28 \pm 0.03$ and for the indirect exciton $g_{ind}^z = -2.8 \pm 0.1$. As the same hole is shared for both transitions, the relation between the g-factors of the $\Gamma$-valley and $X_z$-valley electrons is as follows:

$$g_{X_z}^e - g_{\Gamma}^e = g_{dir}^z - g_{ind}^z, \quad (3)$$

leading to $g_{X_z}^e > g_{\Gamma}^e$. Studies on the same type of QDs[37], where the dark exciton emission could be followed in the whole range of magnetic fields (which is not the case for the dot in Fig. 1, where its signal is too weak), have allowed us to estimate a typical value of +0.2 for $g_{\Gamma}^e$, leading to $g_{X_z}^e$ of approximately +4.1. This order of magnitude can also be applicable to a secondary indirect transition, skimming the end of the line associated with $X^+$ in Fig. 1 and labeled $X_{ind2}$ in Fig. 4 (a). These findings definitively establish the AlAs barriers as the origin of the indirect transitions. Note that the estimated electron g-factor is substantially larger than its bulk value, highlighting the impact of quantum confinement effects[44, 47]. As observed for electrons in the $\Gamma$-valley, the g-factor magnitude is expected to vary significantly due to residual strain[48] and confinement-induced modifications of the electronic wavefunction[49].

## V. ESTIMATION OF THE INTER-VALLEY COUPLING

In this section, we present a quantitative analysis of excitonic states formed by localizing the electron in the $X_z$ valley and comprising a single electron-hole pair. More

complex configurations, such as charged excitons, are beyond the scope of this work and will be investigated in future studies. Our primary objective is to elucidate the double anti-crossing observed in the neutral exciton line ($X^0$), especially the appearance of two indirect transitions with opposite slopes. Rather than aiming for exact quantitative agreement with experiment, we focus on capturing the essential physical mechanisms using a minimal and tractable model. Additionally, we provide

<figure><img src="image_6.png" /><figcaption>FIG. 5. (a) Schematic representation of a GaAs/AlAs quantum dot grown by nanohole infilling. The nanohole has a depth of 86 Å and a radius of 300 Å. The GaAs dot itself is 38 Å high, while the lower and upper AlAs barriers are 26 Å and 32 Å thick, respectively. This specific geometry yields a calculated direct exciton photoluminescence energy that closely matches the value measured in Fig. 1. The same dot is used as the model system to define the spatial potential in Eq. (4). The discretization of the lower and upper interfaces is illustrated on the right side of the dot with $N_{max}^+ = 19$ and $N_{max}^- = 8$. $N_{max}^\pm$ denotes the total number of atomic monolayers used to model the discretized lower and upper interfaces, respectively, as introduced in the main text. Owing to the $l = 0$ symmetry, the primary contribution to the envelope wavefunction overlap and thus to the evaluation of the interfacial coupling strength, originates from the central region of the dot. (b) Diagram of the conduction and valence band profiles of a GaAs/AlAs quantum dot near the flat-band condition. The diagram illustrates the coexistence of both type-I and type-II excitons, involving electron localization within the GaAs dot and the surrounding AlAs barriers. Band offsets are indicated in meV.</figcaption></figure>
a simple estimate of $\beta_{s.c.}$, assumed uniform across all AlAs/GaAs interfaces.

To this end, we adopt the QD geometry shown in Fig. 5(a), featuring upper and lower AlAs barriers embedded within an AlGaAs matrix to replicate the experimental heterostructure. The QD is assumed to possess rotational symmetry, simplifying the calculations while remaining consistent with observed behavior. Detailed design parameters (QD radius = 300 Å, height = 38 Å, nanohole depth = 86 Å) are taken from AFM measurements [37]. Barrier thicknesses are tuned so that, in the absence of an external field, the direct exciton energy lies near 1.748 eV.

Assuming that the exciton can be adequately described using only the heavy-hole band in GaAs, and neglecting any $\vec{k} \cdot \vec{p}$-induced coupling with higher-energy X-valley states [11, 50], its energy and envelope wavefunction are governed by an effective two-valley Hamiltonian [51, 52].

$$ \hat{H} = -\frac{\hbar^2}{2} [\nabla_{\vec{r}_e} \overline{\overline{m}}_{X_z}^{-1} \nabla_{\vec{r}_e} + \nabla_{\vec{r}_h} \overline{\overline{m}}_{\Gamma_s}^{-1} \nabla_{\vec{r}_h}] + V_e(\vec{r}_e) + V_h(\vec{r}_h) - \frac{e^2 |\vec{r}_e - \vec{r}_h|^{-1}}{4\pi\epsilon_0\epsilon_r} - eF_z[z_e - z_h]. \quad (4) $$

Here, $\overline{\overline{m}}_{\Gamma_s}$ and $\overline{\overline{m}}_{X_z}$ are the effective-mass tensors of the hole and the $X_z$ electron, respectively, while $V_e$ and $V_h$ describe the conduction- and valence-band potentials in the heterostructure (see Fig. 5(a) and (b)). An electric field $F_z$ is applied along the growth axis. The symbols $e$, $\epsilon_0$, and $\epsilon_r$ denote the elementary charge, vacuum permittivity, and relative permittivity. All physical parameters required to solve Eq. 4 are provided in the Supplemental Material [45] (see also references [53, 54] therein). Eq. 4 is solved via the Finite Element Method (FEM), which provides a numerically convenient and easily implementable framework. This approach yields a self-consistent solution of the coupled Schrödinger-Poisson equations derived from Eq. 4 and allows straightforward computation of the ground and excited excitonic states, denoted by $\{E_{n,l}, |\Phi_{env}^{X_z}(n,l)\}_{n>0,l}$, where n and l are the principal and azimuthal quantum numbers reflecting the system's rotational symmetry. Given that the effective mass of the $X_z$ electron is roughly three times heavier than the heavy hole in GaAs, the density of states is substantially enhanced. We therefore compute the luminescence energies of the first seven bound excitonic states (n, l) as a function of $F_z$. The results appear in Fig. 6.

A salient feature in Fig. 6(a) is the non-monotonic variation of bound-state energies with $F_z$, exhibiting slope reversals. These reversals correspond to shifts in the field-induced stabilization of the $X_z$ electron toward either the lower or upper AlAs barrier, which in turn flips the equivalent electrostatic dipole direction while the hole remains in the GaAs dot (See Fig. 5(b)). Fig. 6(b) confirms this behavior by plotting the $X_z$ electron position versus $F_z$, showing perfect correlation with the slope changes in Fig. 6(a).

<figure><img src="image_7.png" /><figcaption>FIG. 6. (a) Simulated evolution of the luminescence energy of indirect excitons, labeled by the quantum numbers $(n, l)$, as a function of the applied electric field, in the absence of a magnetic field. For $l \neq 0$, the states $(n, l)$ and $(n, -l)$ are degenerate. Avoided crossings between states of identical symmetry (same $l$) are observed, notably between (2, 0) and (3, 0) around -60 kV/cm, and between (1, 0) and (2, 0) near -10 kV/cm. The black curve represents the calculated luminescence energy of the direct exciton, which intersects the indirect (2, 0) branch twice at -90 kV/cm and -30 kV/cm each time with a slope inversion, consistent with the experimental observations shown in Fig. 1. (b) Evolution of the electron position along the growth axis as a function of electric field, illustrating the spatial jumps responsible for the slope changes observed in (a). The light gray regions indicate the positions of the AlAs barriers.</figcaption></figure>

These localization jumps are controlled by the relative confinement potentials of the barriers and by the QD thickness, which acts as a tunneling barrier between AlAs layers. This is evidenced by the avoided level crossings between states of identical azimuthal symmetry. The interplay between the electric field's tendency to trap the electron in the lower barrier and the Coulomb attraction to the hole in the GaAs dot ultimately dictates the localization outcome.

The calculated energy of the direct exciton as a function of $F_z$ is plotted in Fig. 6(a), and it intersects the energy of the $(n = 2, l = 0)$ indirect state at two distinct electric field values: $F_z = -95$ kV/cm and $F_z = -30$ kV/cm. These crossings coincide with a reversal in the sign of the Stark shift for the indirect exciton, thereby reproducing the experimentally observed double anti-crossing reported in Fig. 1(a) for the $X^0$ case. It is worth noting that, within the framework of
this model characterized by cylindrical symmetry, only indirect states with $l = 0$ can couple to the direct exciton via the interaction term $\hat{W}$, as both share the same azimuthal quantum number. Moreover, we observe an avoided crossing between the (2,0) and (1,0) states in the region of weak electric field. This effect may explain the reduction in the effective dipole moment measured for the second anti-crossing at $F_z = -40$ kV/cm.

These simulations, which reveal a high density of type-II excitonic states, also provide qualitative insight into two key experimental observations. First, the appearance of additional indirect transitions (labeled $X_{ind2}$ in Figs. 1 and 5, and further discussed in the Supplemental Material[45]), which may become optically active through coupling with the direct exciton over specific ranges of electric field, particularly in the presence of broken rotational symmetry, as often occurs in real systems due to shape anisotropy or interface disorder. In such cases, new eigenstates emerge as linear combinations of states with azimuthal quantum numbers $l = \dots, -2, -1, 0, 1, 2, \dots$, with their degeneracy fully lifted. This symmetry breaking enhances the likelihood of anti-crossings by enabling couplings that are otherwise forbidden under perfect cylindrical symmetry. Second, since the direct exciton couples preferentially to excited type-II states via the inter-valley mixing, this mechanism could also account for the broad linewidths observed experimentally for the indirect transitions, potentially originating from intra-valley relaxation processes.

Finally, this simplified model can also be used to estimate the coupling parameter $\beta_{s,c}$, as the unperturbed excitonic wavefunctions $|\tilde{\Phi}_{\text{env}}^{\Gamma e}\Gamma^{h}\rangle$ and $|\tilde{\Phi}_{\text{env}}^{X_z}\Gamma^{h}\rangle$ are computed independently, without accounting for heterointerface mixing, following the approach introduced in Ref. 33. The coupling is then extracted from the experimentally determined anti-crossing amplitude, given by $\sim 2|\langle\tilde{\Phi}_{\text{env}}^{X_z}\Gamma^{h}|\hat{W}|\tilde{\Phi}_{\text{env}}^{\Gamma e}\Gamma^{h}\rangle|$, which is found to be on the order of 0.2 meV (see Fig. 1). Since this interaction is small compared to the confinement and Coulomb energies, the interface coupling can be treated as a perturbation, enabling an estimate of $\beta_{s,c}$, provided that the GaAs/AlAs interface can be approximated as a discrete series of atomic steps of thickness $a_0$. Given the geometric parameters of the quantum dot model used to reproduce the experimental data shown in Fig. 1, the lower and upper AlAs barriers are modeled with thicknesses of 9 and 11 atomic monolayers, respectively, centered around a GaAs quantum dot composed of 13 monolayers. To discretize the lower (+) and upper (-) interfaces, the heterointerface is modeled as a set of concentric annular atomic steps. Each step lies at height $z_n^{\pm}$ and is defined by an inner radius $r_n^{\pm}$ and an outer radius $r_{n+1}^{\pm}$. The next step lies at $z_{n+1}^{\pm} = z_n^{\pm} + a_0$ and begins at the outer radius of the previous one, continuing this way up to $N_{\text{max}}^{\pm} - 1$ steps per interface. The matrix element $\langle\tilde{\Phi}_{\text{env}}^{X_z}\Gamma^{h}|\hat{W}|\tilde{\Phi}_{\text{env}}^{\Gamma e}\Gamma^{h}\rangle$ can thus be approximated as the sum of two contributions, $W^+$ and $W^-$, correspond-

ing to the lower and upper interfaces, respectively. Using cylindrical coordinates, i.e. $\vec{r}_e = (r_e, z_e)$, these terms are given by:

$$W^{\pm} = 2\pi\beta_{s,c} \sum_{n=0}^{N_{\text{max}}^{\pm}-1} \int_{\mathbb{R}^3} d^3\vec{r}_h \int_{r_n^{\pm}}^{r_{n+1}^{\pm}} r_e dr_e e^{\pm iq_z z_n^{\pm}} \times \tilde{\Phi}_{\text{env}}^{X_z}(2,0)[\vec{r}_h; (r_e, z_n^{\pm})] \tilde{\Phi}_{\text{env}}^{\Gamma}[\vec{r}_h; (r_e, z_n^{\pm})]^*, \quad (5)$$

The validity of this discretized-interface approximation relies on the condition that the envelope functions vary slowly on the atomic scale. This is fulfilled when the following inequality holds:

$$a_0 \left| \partial_{z_e} \ln \int_{\mathbb{R}^3} d^3 \vec{r}_h \left| \tilde{\Phi}_{\text{env}}[\vec{r}_h; (r_e, z_e)] \right|^2 \right| \ll 1, \text{ which is indeed satisfied in our case.}$$

Note the sign reversal in the phase factor of $W^-$, which arises from the inversion of the upper interface relative to the lower one. It is also worth emphasizing that $W^+$ dominates over $W^-$ in the anti-crossing associated with a positive slope, while the opposite holds for the second anti-crossing, which exhibits a negative slope. Taking the anti-crossing observed at $-100$ kV/cm as a reference and using its amplitude as a scaling benchmark, the computed value of $|W^+ + W^-|$ at this field yields an estimated transfer potential $|\beta_{s,c}| \approx 20$ meV·Å. This estimated value can be compared to the inter-valley transfer potentials previously measured in type-II GaAs/AlAs superlattices, which typically yield values on the order of 100 meV·Å, across a wild range of material thicknesses [17]. Although our estimation for GaAs/AlAs QDs is somewhat lower by roughly a factor of five, it remains within the same order of magnitude. This discrepancy may be attributed to the differences in interface geometry: planar interfaces in superlattices are generally more favorable for enhancing the $\Gamma-X_z$ coupling.

## VI. CONCLUSION

Inter-valley coupling in semiconductor heterostructures, a topic extensively studied in the 1990s, finds renewed relevance within the framework of quantum dots. In the present work, we have demonstrated that such coupling between the $\Gamma$ and $X_z$ valleys can be effectively observed and quantified in a single GaAs/AlAs quantum dot structure. This system simultaneously hosts both direct and indirect excitons, making it an alternative platform to quantum dot molecules (QDMs), but with enhanced simplicity and electrostatic control.

One of the key insights from our study is the critical role played by electron confinement in the AlAs barrier. This was revealed experimentally through two independent observations: (i) the measurement of the static dipole associated with the indirect exciton, and (ii) the extraction of a large electron g-factor, consistent with $X_z$-valley localization. Moreover, we have shown that the presence of
both upper and lower AlAs barriers allows deterministic control over the dipole direction of the indirect exciton. In particular, it enables spatial tuning of the electron position between the two barriers, while the hole remains confined within the dot. This spatial configurability is directly linked to the ability to reverse the exciton's dipole moment via an applied electric field.

Our results also suggest that electronic states in the barrier may play a role in the formation of multicharged excitonic complexes. This opens the possibility of initial-izing and manipulating an electron spin qubit localized in the barrier, a regime distinct from standard quantum dot configurations. Such spatially extended excitonic states may therefore serve as building blocks for new quantum protocols.

Several open questions remain, particularly regarding the broad spectral linewidth observed in indirect transitions. While several relaxation mechanisms could be responsible, a definitive understanding is still lacking. A promising direction involves engineering the QD geometry (specifically the barrier thicknesses and dot height) to optimize the overlap and coupling between direct and indirect excitons. In parallel, strain-tuning experiments could be conducted to control the relative energy align-ment of the different $X_z$ and $X_{x,y}$ valleys. This would enable simultaneous tuning of both the nature of the in-

direct exciton and the inter-valley coupling strength, offering an additional degree of control over the excitonic landscape in such heterostructures.

Finally, this novel type-II quantum dot architecture exhibits a unique property: the electron can reside either within the dot or in one of the two AlAs barriers, with this spatial configuration being both controllable and deterministic. Meanwhile, the hole remains localized in the GaAs dot. This characteristic makes the system a strong candidate for implementing quantum protocols where the electron spin carries the quantum information, with an additional degree of freedom associated with its spatial position. This spatial degree of freedom could be optically manipulated via stimulated Raman adiabatic passage, enabling coherent population transfer between two spatially distinct states using a two-color optical pulse scheme, as proposed theoretically in Ref. 55.

## ACKNOWLEDGMENTS

This research is supported by the ANR (Agence Nationale de la Recherche) within the framework of the project Indiquate (ANR-22-CE47-0007), and the PEPR OQULUS (22-PETQ-0013). We also extend our gratitude to the Quantum Information Center at Sorbonne University for their financial support.

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Supplemental Material for:
Probing inter-valley coupling via exciton hybridization in a charge-tunable
GaAs/AlAs quantum dot

Arnault Raymond,* Paola Atkinson, Antoine Chapuis, Savvas Germanis,
Florent Margaillan, Mathieu Bernard, Valia Voliotis, and Benoit Eble†

Sorbonne Université, CNRS, Institut des Nanosciences de Paris, 4 Place Jussieu, 75252 Paris, France

CONTENTS

<table><tr><td>Supplemental Material</td><td>1</td></tr><tr><td>I. Observation of coexisting type-I and Type-II excitons in another GaAs/AlAs quantum dot</td><td>2</td></tr><tr><td>II. Magnetic field behaviour of the intervalley coupling</td><td>2</td></tr><tr><td>III. Numerical simulations methods and parameters</td><td>7</td></tr><tr><td>References</td><td>7</td></tr></table>

SUPPLEMENTAL MATERIAL

This supplemental material provides additional insights and data supporting the findings presented in the main text. It begins with further experimental results obtained from individual quantum dots, which confirm the reproducibility of the charge-state structure and the spectroscopic signatures discussed. It then outlines the theoretical approach used to describe the behavior of type-II excitons under both electric and magnetic field tuning. Finally, it details the numerical values and material parameters employed in the simulations used to interpret the experimental data and model inter-valley coupling mechanisms.

<footnote>* araymond@insp.jussieu.fr</footnote>

<footnote>† benoit.eble@insp.jussieu.fr</footnote>
# I. OBSERVATION OF COEXISTING TYPE-I AND TYPE-II EXCITONS IN ANOTHER GAAS/ALAS QUANTUM DOT

The spectroscopic measurements presented in the main article were performed on a single quantum dot. Similar experiments conducted on multiple other dots revealed a consistent charge-state structure. For instance, Fig. S1 shows characteristic features analogous to those reported in Fig. 1 of the main article. This provides strong evidence that the phenomena described in the main text are reproducible. These include the presence of several indirect excitonic transitions with both positive and negative Stark shift slopes, as well as the appearance of distinct anti-crossings. Specifically, the static dipole length falls within the same range as the dot studied in the main publication. These recurring signatures reinforce the robustness and generality of the observed inter-valley coupling and excitonic behavior.

<figure><img src="image_8.png" /></figure>

FIG. S1. Stark spectroscopy of GaAs/AlAs quantum dots demonstrates characteristics that are very similar to those presented in the main manuscript. This mapping was performed using a lower-resolution spectrometer, which did not allow for clear observation of anti-crossing for the neutral exciton. However, anti-crossing involving the negative trion is observed. Additionally, there is no anti-crossing with the hot positive trion. However, another anti-crossing is evident between the negative trion and the indirect negative Stark shift line.

# II. MAGNETIC FIELD BEHAVIOUR OF THE INTERVALLEY COUPLING

In this section, we present a simplified theoretical framework to describe the behavior of inter-valley coupling under an external magnetic field. The objective is to explain the differing behavior of the two orthogonal circular polarization anticrossings when the Zeeman splitting becomes comparable to or exceeds the inter-valley coupling strength.

Furthermore, the evaluation of the diamagnetic shift for the indirect transition remains imprecise in situations where the Zeeman-split linewidths exceed the diamagnetic shift across the magnetic field range, which is the case in the present study. Instead of relying exclusively on the $\sigma^+$ and $\sigma^-$ transition energies at a fixed electric field while tuning the magnetic field, a more accurate estimation is obtained by analyzing the full set of transition energies measured over the entire electric field range for each magnetic field value. This approach is essential for the precise determination of the anticrossing field, as it incorporates the maximum available information from the electric-field detuning spectra. In the following, we introduce a method to estimate the difference in diamagnetic shifts between direct and indirect excitons, based on the observed evolution of the anticrossing field as a function of the magnetic field in the Faraday configuration.

We restrict our theoretical analysis to the subspace spanned by the direct and indirect bright exciton states, denoted respectively as $|X_Γ^0\rangle$ for the direct exciton (electron in the $\Gamma$-valley of the GaAs QD) and $|X_X^0\rangle$ for the indirect exciton
(electron in the X-valley of the AlAs barrier), along with the two bright spin configurations $|+1\rangle$ and $|-1\rangle$. This yields a total basis set ($|X_F^0, +1\rangle, |X_X^0, +1\rangle, |X_F^0, -1\rangle, |X_X^0, -1\rangle$). We assume that we operate in a magnetic field regime where the anisotropic electron-hole exchange interaction can be neglected compared to the Zeeman splitting. The total Hamiltonian can then be decomposed into a Zeeman term, $H_Z$, and a coupling term, $H_C$, which is independent of the magnetic field. The latter includes the electric field interaction acting solely on the indirect states, as well as the inter-valley mixing described by two phenomenological parameters, $\alpha_+$ and $\alpha_-$.

In the given basis, we have

$$ \hat{H}_Z = \frac{\mu_B B_z}{2} \begin{pmatrix} g_{Dir} & 0 & 0 & 0 \\ 0 & g_{Ind} & 0 & 0 \\ 0 & 0 & -g_{Dir} & 0 \\ 0 & 0 & 0 & -g_{Ind} \end{pmatrix}, $$

where $g_{Dir}$ and $g_{Ind}$ are the effective Landé factor for the direct and indirect exciton respectively, and

$$ \hat{H}_C = \begin{pmatrix} E_{Dir}^0 & \alpha_+ & 0 & 0 \\ \alpha_+ & E_{Ind}^0 - \vec{F} \cdot \vec{p} & 0 & 0 \\ 0 & 0 & E_{Dir}^0 & \alpha_- \\ 0 & 0 & \alpha_- & E_{Ind}^0 - \vec{F} \cdot \vec{p} \end{pmatrix}, $$

where $E_i^0$ corresponds to the energy of eigenstate $|i\rangle$ in the absence of a magnetic field, $\vec{F}$ is the electric field, $\vec{p}$ the electric dipole moment for the indirect state, and $\alpha_+$ (resp $\alpha_-$) is the coherent coupling strength between the direct and indirect excitonic states in spin state $|+1\rangle$ (resp $|-1\rangle$), assumed to be spin-conserving, as experimentally evidenced in the main text by the lack of dark-bright mixing, and real (the generalization to the complex case is straightforward). The total Hamiltonian is therefore block diagonal because the holes are in an orthogonal spin states for the two bright spin configurations of the electron-hole pair, allowing to treat them separately. We thus can treat the case of the general 2-dimensional Hamiltonian

$$ \hat{H} = \begin{pmatrix} E_0 & \alpha \\ \alpha & E_1 \end{pmatrix}. $$

The eigenstate of this Hamiltonian can be written as

$$ \begin{cases} |+\rangle = \begin{pmatrix} \frac{\alpha}{\sqrt{\alpha^2+(E_+-E_0)^2}} \\ \frac{E_+-E_0}{\sqrt{\alpha^2+(E_+-E_0)^2}} \end{pmatrix} \\ |-\rangle = \begin{pmatrix} \frac{\alpha}{\sqrt{\alpha^2+(E_--E_0)^2}} \\ \frac{E_--E_0}{\sqrt{\alpha^2+(E_--E_0)^2}} \end{pmatrix} \end{cases} $$

with eigenenergies

$$ \begin{cases} E_+ = (E_0 + E_1)/2 + \sqrt{(E_1 - E_0)^2/4 + \alpha^2} \\ E_- = (E_0 + E_1)/2 - \sqrt{(E_1 - E_0)^2/4 + \alpha^2} \end{cases} $$

The anticrossing happens when energy $E_+$ and $E_-$ are closest together, that is when $E_0 = E_1$, giving an energy anticrossing gap $\Delta E = \alpha$.

At zero magnetic field, the $|+1\rangle$ and $|-1\rangle$ components have the same energetical behaviour (neglecting the anisotropic exchange interaction energy), yielding an anticrossing electric field governed by the equation (assuming the electric field is in the growth direction $z$ of the QD):

$$ \begin{align*} E_{Dir}^0 &= E_{Ind}^0 - \vec{F}^{AC} \cdot \vec{p} \\ \Leftrightarrow F_z^{AC,0} p_z &= E_{Ind}^0 - E_{Dir}^0 \\ \Leftrightarrow F_z^{AC,0} &= (E_{Ind}^0 - E_{Dir}^0) / p_z \end{align*} $$

where $p_z$ is the dipole moment of the indirect exciton in the growth direction $z$.
When a magnetic field is applied, the energy degeneracy between the $|+1\rangle$ and $|-1\rangle$ is lifted, and we have new energies for the direct and indirect components dependent on the spin configuration :

$$ \begin{cases} E_{Dir}^{\pm 1}(B_z) = E_{Dir}^{0} \pm \frac{\mu_B B_z g_{Dir}}{2} + \gamma_{Dir} B_z^2 \\ E_{Ind}^{\pm 1}(B_z) = E_{Ind}^{0} \pm \frac{\mu_B B_z g_{Ind}}{2} + \gamma_{Ind} B_z^2, \end{cases} $$

where $g_{Dir}$ (resp. $g_{Ind}$) is the effective Landé factor for the direct (resp. indirect) exciton, and $\gamma_{Dir}$ (resp. $\gamma_{Ind}$) its diamagnetic coefficient. We here assume that all the Landé factors and diamagnetic coefficients are constants and independent of the external fields for the sake of simplicity. If we assume $g_{Dir}$ and $g_{Ind}$ to be different, as demonstrated in section IV of the main article, we get new anti-crossing fields for the two bright spin configurations :

$$ \begin{aligned} F_z^{AC, \pm} &= \left( E_{Ind}^0 \pm \frac{\mu_B B_z g_{Ind}}{2} + \gamma_{Dir} B_z^2 - E_{Dir}^0 \mp \frac{\mu_B B_z g_{Dir}}{2} - \gamma_{Ind} B_z^2 \right) / p_z \\ &= F_z^{AC, 0} \pm \frac{\mu_B B_z}{2p_z} (g_{Ind} - g_{Dir}) + \frac{B_z^2}{p_z} (\gamma_{Ind} - \gamma_{Dir}). \end{aligned} $$

This analysis reveals that the primary effect of a magnetic field, in the presence of different effective g-factors and diamagnetic coefficients for the direct and indirect excitons, is to shift the anticrossing electric fields associated with the two bright spin configurations. The resulting field separation is given by $\Delta F_z = \frac{\mu_B B_z}{p_z} (g_{Ind} - g_{Dir})$ where $p_z$ denotes the indirect excitonic dipole moment, and the resulting mean anticrossing field shift by $(F_z^{AC,+} + F_z^{AC,-} - 2F_z^{AC,0})/2 = B_z^2 (\gamma_{Ind} - \gamma_{Dir})/p_z$.

The measurements of the separation of the anticrossing fields for the two circular polarizations and their mean shift compared to the $B=0T$ case are presented in figure S2. In order to determine the position of the anticrossing field for a given polarization, we measure the energy position of the lower branch of the anticrossing, and then fit the asymptotical behaviours for the direct (at high electric fields) and indirect excitons. The anticrossing field corresponds to the crossing of these asymptotes. The measurements indeed display both a separation of the anticrossing fields for the two circular polarizations, corresponding to the disparity in effective g-factors for the direct and indirect excitons, and a global shifting of the anticrossing towards higher electric fields, corresponding to a difference of the diamagnetic coefficients $\Delta\gamma = \gamma_{Ind} - \gamma_{Dir} > 0$. From this mean shift, we can estimate the difference to be $\Delta\gamma \simeq 19eV/T^2$ by fitting the to a parabolic function of the applied magnetic field, shown on figure S3.

Although a precise measurement of the diamagnetic shift for the direct exciton is lacking due to the limited number of magnetic field values used, we can rely on prior measurements performed on quantum dots with similar photoluminescence energies. These earlier results yield a typical value of $\gamma_{Dir} \sim 11\mu eV \cdot T^{-2}$ [1]. Based on this reference, we deduce a diamagnetic shift for the indirect exciton of approximately $\gamma_{Ind} \sim 30\mu eV \cdot T^{-2}$. This result is consistent with the general understanding that type-II excitons exhibit significantly larger lateral spatial extensions. This is particularly true in the present system, where the X-valley electron is well-described by an envelope wavefunction with ($n=0, l=2$) quantum numbers, extending laterally within the AlAs bottom barrier (i.e., in the plane perpendicular to the magnetic field).

This method also allows to estimate the difference between the g-factors to be of the order of $\Delta g \simeq 4.3$, corresponding within 5% to what was obtained by direct measurement of the energy difference.

A key implication of this result is that, due to their distinct magnetic responses, the degree of state mixing differs for each spin configuration at a given electric field. As a result, the energy splitting between the two anticrossed branches, defined as $E_{\pm}^{+1} - E_{\pm}^{-1}$, becomes electric-field dependent and can be calculated as:

$$ \begin{cases} \Delta E_{+}(F_z) = \frac{\mu_B B_z}{2} (g_{Ind} + g_{Dir}) + \sqrt{\left(E_{Dir}^0 - E_{Ind}^0 + F_z p_z + \frac{\mu_B B_z}{2} (g_{Dir} - g_{Ind}) + B_z^2 (\gamma_{Dir} - \gamma_{Ind})\right)^2 / 4 + \alpha^2} \\ - \sqrt{\left(E_{Dir}^0 - E_{Ind}^0 + F_z p_z - \frac{\mu_B B_z}{2} (g_{Dir} - g_{Ind}) + B_z^2 (\gamma_{Dir} - \gamma_{Ind})\right)^2 / 4 + \alpha^2} \\ \\ \Delta E_{-}(F_z) = \frac{\mu_B B_z}{2} (g_{Ind} + g_{Dir}) - \sqrt{\left(E_{Dir}^0 - E_{Ind}^0 + F_z p_z + \frac{\mu_B B_z}{2} (g_{Dir} - g_{Ind}) + B_z^2 (\gamma_{Dir} - \gamma_{Ind})\right)^2 / 4 + \alpha^2} \\ + \sqrt{\left(E_{Dir}^0 - E_{Ind}^0 + F_z p_z - \frac{\mu_B B_z}{2} (g_{Dir} - g_{Ind}) + B_z^2 (\gamma_{Dir} - \gamma_{Ind})\right)^2 / 4 + \alpha^2}. \end{cases} $$

To make the $F_z$ dependency more clear, let us look at a particular case where $p_z < 0$ and $g_{Ind} = -g_{Dir} = -g$, with $g > 0$, and $\gamma_{Dir} = \gamma_{Ind}$. In this case, we expect the energy splitting between the two spin components to have
<figure><img src="image_9.png" /><figcaption>FIG. S2. Anticrossing fields determined for a) a linear polarization at 0T, b) the $\sigma+$ and $\sigma-$ circular polarization at 2T, and c) the $\sigma+$ and $\sigma-$ circular polarization at 4T. The red dottel line represents the anticrossing field at 0T, and the blue (resp. green) dotted lines indicate the $\sigma+$ (resp. $\sigma-$) polarization anticrossing electric field in an applied magnetic field. The black dotted lines correspond to the fitted asymptotes at low electric field for the indirect exciton and high electric field for the direct exciton.</figcaption></figure>

opposite signs for the direct (positive) and indirect (negative) excitons. In this case, we have $\Delta E_-(F_z) = -\Delta E_+(F_z)$, and for the lower branch splitting we get :

$$ \Delta E_{-}(F_{z}) = \sqrt{\frac{(E_{Dir}^{0} - E_{Ind}^{0} + F_{z}p_{z} + \mu_{B}B_{z}g)^{2}}{4 + \alpha^{2}} - \frac{(E_{Dir}^{0} - E_{Ind}^{0} + F_{z}p_{z} - \mu_{B}B_{z}g)^{2}}{4 + \alpha^{2}}} $$
<figure><img src="image_10.png" /><figcaption>FIG. S3. Fit for the mean shift of the anticrossing field as a function of the magnetic field squared. The red band (resp. grey) represents the 95% (resp. 99.7%) confidence interval, estimated from the fit covariance parameters.</figcaption></figure>

In that case, when $F_z = F_z^{AC,+}$ or $F_z = F_z^{AC,-}$, we have

$$ \begin{cases} \Delta E_{-}(F_z^{AC,+}) = \alpha - \sqrt{(E_{Dir}^0 - E_{Ind}^0 + F_z p_z - \mu_B B_z g)^2 / 4 + \alpha^2} < 0 \\ \Delta E_{-}(F_z^{AC,-}) = \sqrt{(E_{Dir}^0 - E_{Ind}^0 + F_z p_z + \mu_B B_z g)^2 / 4 + \alpha^2 - \alpha} > 0 \end{cases} $$

showing that control of the electric field allows to tune the sign of the energetic splitting.

In general, if $g_{Ind}$ and $g_{Dir}$ are of opposite signs, there is always an electric field giving a zero energetic splitting for $\Delta E_{-}$ and a different electric field giving a zero energetic splitting for $\Delta E_{+}$.
### III. NUMERICAL SIMULATIONS METHODS AND PARAMETERS

Table I summarizes the material parameters used to solve Eq. (4) introduced in Section 5 of the main text. These parameters define the conduction and valence band profiles, effective masses, and dielectric properties of the GaAs/AlAs heterostructure system modeled in our calculations.

<table><caption>TABLE I. Numerical values of the parameters used in the calculations. These values are taken from references 2 and 3. However, it appears that only one value has been modified with respect to the literature. Specifically, the AlAs X-valley gap (indicated by ⋆) has been adjusted from 2.1 eV to 1.675 eV, which corresponds to a 20% discrepancy. This adjustment leads to a substantial agreement with the experimental outcome.</caption><thead><tr><th>Parameter</th><th>GaAs</th><th>Al<sub>x</sub>Ga<sub>1-x</sub>As (x > 0.3)</th><th>AlAs</th></tr></thead><tbody><tr><td colspan="4" style="text-align: center; font-weight: bold;">Valley Γ</td></tr><tr><td>Direct Gap E<sup>Γ</sup><sub>g</sub> (eV)</td><td>1.519</td><td>1.946 (x = 0.3), 2.096 (x = 0.4)</td><td>2.90</td></tr><tr><td>Band Offset Ratio</td><td>CB 67:VB 33</td><td>-</td><td>-</td></tr><tr><td>Luttinger Parameters γ<sub>1</sub>, γ<sub>2</sub></td><td>7.0, 2.1</td><td>-</td><td>3.76, 0.82</td></tr><tr><td>Reduced Electron Mass m<sub>e</sub>/m<sub>0</sub></td><td>0.066</td><td>0.086</td><td>0.13</td></tr><tr><td colspan="4" style="text-align: center; font-weight: bold;">Valley X<sub>z</sub></td></tr><tr><td>Indirect Gap E<sup>X<sub>z</sub></sup><sub>g</sub> (eV)</td><td>1.982</td><td>2.025 (x = 0.3), 2.051 (x = 0.4)</td><td>1.67*</td></tr><tr><td>Electron Mass Tensor m<sub>l</sub>/m<sub>0</sub>, m<sub>t</sub>/m<sub>0</sub></td><td>m<sub>l</sub> = 1.3, m<sub>t</sub> = 0.23</td><td>m<sub>l</sub> = 1.3 - 0.2x, m<sub>t</sub> = 0.23 - 0.04x</td><td>m<sub>l</sub> = 1.1, m<sub>t</sub> = 0.19</td></tr><tr><td colspan="4" style="text-align: center; font-weight: bold;">Dielectric Constant</td></tr><tr><td>Relative Permittivity ε<sub>r</sub></td><td colspan="3">11.3 (GaAs and AlAs average, no spatial dependence)</td></tr></tbody></table>

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