# Engineering Majorana bound states in coupled quantum dots in a two-
dimensional electron gas

Sebastiaan L.D. ten Haaf  QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, Delft, 2600 GA, The Netherlands  These authors
contributed equally to this work.  Qingzhen Wang  QuTech and Kavli Institute
of Nanoscience, Delft University of Technology, Delft, 2600 GA, The
Netherlands  These authors contributed equally to this work.  A. Mert Bozkurt
QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
Delft, 2600 GA, The Netherlands  Chun-Xiao Liu  QuTech and Kavli Institute of
Nanoscience, Delft University of Technology, Delft, 2600 GA, The Netherlands
Ivan Kulesh  QuTech and Kavli Institute of Nanoscience, Delft University of
Technology, Delft, 2600 GA, The Netherlands  Philip Kim  QuTech and Kavli
Institute of Nanoscience, Delft University of Technology, Delft, 2600 GA, The
Netherlands  Di Xiao  Department of Physics and Astronomy, Purdue University,
West Lafayette, 47907, Indiana, USA  Candice Thomas  Department of Physics and
Astronomy, Purdue University, West Lafayette, 47907, Indiana, USA  Michael J.
Manfra  Department of Physics and Astronomy, Purdue University, West
Lafayette, 47907, Indiana, USA  Elmore School of Electrical and Computer
Engineering, Purdue University, West Lafayette, 47907, Indiana, USA  School of
Materials Engineering, Purdue University, West Lafayette, 47907, Indiana, USA
Tom Dvir  QuTech and Kavli Institute of Nanoscience, Delft University of
Technology, Delft, 2600 GA, The Netherlands  Michael Wimmer  QuTech and Kavli
Institute of Nanoscience, Delft University of Technology, Delft, 2600 GA, The
Netherlands  Srijit Goswami<EMAIL_ADDRESS>QuTech and Kavli Institute
of Nanoscience, Delft University of Technology, Delft, 2600 GA, The
Netherlands

###### Contents

  1. I Methods
    1. I.1 Fabrication and Measurements
    2. I.2 Numerical transport calculations
    3. I.3 Effective model in the strong coupling regime
  2. II Supplementary Figure S1 to S10

##  I Methods

###  I.1 Fabrication and Measurements

Two devices were used to obtain the data presented in the main text (device A
and device B). Both were fabricated using techniques described in detail in
[1]. A narrow aluminum strip is defined in an InSbAs-Al chip by wet etching,
followed by the deposition of two normal Ti/Pd contacts. After deposition of
$20\text{\,}\mathrm{n}\mathrm{m}$ AlOx via atomic layer deposition (ALD), two
Ti/Pd depletion gates are evaporated. Following a second ALD layer
($20\text{\,}\mathrm{n}\mathrm{m}$ AlOx), seven Ti/Pd finger gates are
evaporated in order to define the QDs and tune the ABSs energy. The two
depletion gates define a quasi-1D channel with a width of about
$160\text{\,}\mathrm{n}\mathrm{m}$, contacted on each side by a normal lead.
The aluminium strip induces superconductivity in the middle section of each
device, with an induced gap on the order of
$200\text{\,}\upmu\mathrm{e}\mathrm{V}$. The presence of extended ABSs is
confirmed through tunnelling spectroscopy. ABSs are found to be present over a
large range of $V_{\mathrm{ABS}}$, the voltage applied to the gate covering
the hybrid region. Finger gates on the left and right of the aluminium define
QDs with charging energies above $1\text{\,}\mathrm{m}\mathrm{V}$ (Fig.
S1a,b).

Measurements were performed in a dilution refrigerator with a base temperature
of $20\text{\,}\mathrm{m}\mathrm{K}$. Each investigated device showed strong
hybridization between the QDs, as presented here. Device A was used to study
the field evolution of Majorana sweet spots and to obtain the measurements
presented in Fig. 1, Fig. 3 and Fig. 4. Device B was used to demonstrate the
role of spin-orbit coupling on the interdot interactions, and the control over
interactions through magnetic field as presented in Fig. 2. Transport
measurements are performed in AC and DC using a three-terminal set-up, where
the aluminum is electrically grounded. Each Ohmic lead is connected to a
current-to-voltage converter and biased through a digital-to-analogue
converter that applies both DC and AC biases. Offsets of the applied voltage-
bias on each side are corrected via independently calibrating the Coulomb
peaks in the QDs on each side. The voltage outputs of the current meters are
recorded with two digital multimeters and two lock-in amplifiers. When
applying a DC voltage to the left Ohmic ($V_{\mathrm{L}}$) the right lead
($V_{\mathrm{R}}$) is kept grounded and vice versa. AC excitations are applied
on each side with amplitudes around $5\text{\,}\upmu\mathrm{V}$ RMS and
frequencies of $19\text{\,}\mathrm{H}\mathrm{z}$ (left) and
$29\text{\,}\mathrm{H}\mathrm{z}$ (right). In this way, we measure the full
conductance matrix $G$ by first measuring the response of $I_{\mathrm{L}}$ and
$I_{\mathrm{R}}$ to $V_{\mathrm{L}}$ and then to $V_{R}$. We account for the
voltage-divider effect by correcting the conductances using known fridge line
resistances $($3.6\text{\,}\mathrm{k}\Omega$$ in device A,
$3.3\text{\,}\mathrm{k}\Omega$ in device B), as detailed in [2]. This
correction was done for all presented spectroscopy data, except for the data
shown in Fig 3d-f. Magnetic fields were applied using a 3D vector magnet. The
alignment of the magnetic field of device A is expected to be accurate within
±10° and calibrated through performing tunneling spectroscopy of the hybrid
section as a function of field angle. The alignment of device B is expected to
be accurate within ±5°.

Due to device instabilities or charge jumps, electrostatics of the QDs
experience small drifts over the course of the measurements. Investigated
orbitals were tracked while collecting the presented datasets. For each
tunnelling spectroscopy measurements at a sweet spot, where $V_{\mathrm{QDL}}$
and/or $V_{\mathrm{QDR}}$ were detuned, a CSD was obtained directly before and
directly after to ensure that no drifts occurred during such a measurement. If
such a drift occurred, the measurement was discarded and repeated. Such drifts
are the cause of small discrepancies in gate voltages between highlighted
paths in Fig. 3b and the measurements shown in Fig. 3d,e. The highlighted
paths represents the corrected path taken w.r.t. the CSD shown in Fig. 3b,
based on the CSDs obtained before and after the measurements.

In order to compare energy scales between experiments and numerical
calculations, the gate voltages $V_{\mathrm{QDL}}$ and $V_{\mathrm{QDR}}$ are
converted to electro-chemical potential energies $\mu_{\mathrm{L}}$ and
$\mu_{\mathrm{R}}$. For this purpose we extract the dimensionless lever arms
$\alpha$. When forming sub-gap YSR states in the QDs, the effective lever arm
of each QD around a zero-bias charge degeneracy can differ from the lever arm
of the uncoupled QD, depending e.g. on hybridization with the hybrid region
(Fig. S4), extensively addressed in [3]. For analysis we therefore estimate
both the normal-state lever arm (denoted $\alpha_{\mathrm{N}}$) and the lever
arm of the sub-gap YSR-states (denoted $\alpha_{\mathrm{YSR}}$), at the
specific $V_{\mathrm{ABS}}$ regimes of interest. Device B was operated in a
regime without significant difference between $\alpha_{\mathrm{N}}$ and
$\alpha_{\mathrm{YSR}}$, such that the analysis in Fig. 2d uses the lever arms
extracted in Fig. S2. With the orbitals in Fig. 4, sweet spots were
investigated at magnetic field values between $0\text{\,}\mathrm{m}\mathrm{T}$
and $300\text{\,}\mathrm{m}\mathrm{T}$, where the energy of excited states
$E_{\mathrm{det}}$ was extracted at a fixed detuning of $V_{\mathrm{QDR}}$.
For this dataset $\alpha_{\mathrm{YSR}}$ was only obtained at
$0\text{\,}\mathrm{m}\mathrm{T}$ (Fig. S1), such that any change in
$\alpha_{\mathrm{YSR}}$ as a function of magnetic field can not be accounted
for in the analysis in Fig. 4l and Fig. 4. When extracting $E_{\mathrm{det}}$
at constant detuning of $V_{\mathrm{QDR}}$ at each field, $\mu_{\mathrm{R}}$
may not be constant but rather is expected to decrease (Fig. S4). We note that
the lack of this correction may lead to slightly underestimating the g-factor
in Fig. 4h, which in turn will lead to a lowered estimation of the Majorana
polarization. For the extractions of $E_{\mathrm{det}}$ and
$E_{\mathrm{gap}}$, $G_{\mathrm{LL}}$ and $G_{\mathrm{RR}}$ line-traces are
obtained at the sweet spot and at a detuning of $V_{\mathrm{QDR}}$ of
$-2\mathrm{mV}$, corresponding to a detuning of $\mu_{\mathrm{R}}\approx
100\upmu\mathrm{e}V$. From each line-trace, the separation between the ZBPs
and the first higher energy excitation is extracted by fitting Gaussian peaks
symmetrically around zero-bias. Error-bars are given by the uncertainty in
these fits.

###  I.2 Numerical transport calculations

For all presented numerical results, a description of the system incorporating
both the two QDs and the middle hybrid section was used, recently introduced
in [4]. We employ the same model and highlight key points and used numerical
parameter below. The model considers tunneling between two normal QDs (L,R)
and a central QD (M), which is in proximity to a superconductor. Each site $j$
has electrochemical potential energy $\mu_{j}$. A spin-conserving hopping $t$
allows transport between the outer QDs and central QD. The effect of spin-
orbit interactions is included through a spin-flip hopping term
$t_{\mathrm{SO}}$ between the outer QDs and central QD. The presence of the SC
is included by attributing a superconducting pairing term $\Delta_{i}$ in each
QD. To match the experimental geometry,
$\Delta_{\mathrm{L,R}},<\Delta_{\mathrm{M}}$ Lastly, the left and right sites
are assigned a large on-site charging energy $U$ and a Zeeman splitting
$E_{z}$ between the $\lvert\downarrow\rangle$ and $|\uparrow\rangle$
occupation. The Hamiltonian is constructed as follows:

$\displaystyle H=$
$\displaystyle\sum_{j,\sigma}{\mu_{j}n_{j\sigma}}+\sum_{j}U_{j}n_{j\uparrow}n_{j\downarrow}+\sum_{j}{E_{Z}}_{j}(n_{j\downarrow}-n_{j\uparrow})$
$\displaystyle+$
$\displaystyle\sum_{j\neq\mathrm{M},\sigma}[t_{j}d_{j\sigma}^{\dagger}d_{C\sigma}+h.c.]$
$\displaystyle+$
$\displaystyle\sum_{j\neq\mathrm{M}}[t_{j}^{\mathrm{SO}}d_{j\downarrow}^{\dagger}d_{C\uparrow}-t_{j}^{\mathrm{SO}}d_{j\uparrow}^{\dagger}d_{C\downarrow}+h.c.]$
$\displaystyle+$
$\displaystyle\sum_{j}[\Delta_{\mathrm{j}}d_{j\uparrow}^{\dagger}d_{j\downarrow}^{\dagger}+h.c.]$

where $d_{j\sigma},d_{j\sigma}^{\dagger}$ and $n_{j\sigma}$ are the
annihilation, creation, and number operators respectively for each site. The
sum $j$ runs over the sites (L,M,R) and $\sigma$ over the spin degree of
freedom ($\uparrow,\downarrow$). For simplicity, a left and right symmetry is
assumed, such that a Majorana sweet spot lies along
$\mu_{\mathrm{L}}=\mu_{\mathrm{R}}$. Sweet spots are obtained by scanning the
parameter space spanned by $\mu_{\mathrm{L}}=\mu_{\mathrm{R}}$ and
$\mu_{\mathrm{M}}$ and finding degeneracies between lowest odd and even
eigenstates. All transport calculations are obtained using the rate equation
detailed in [4]. Similar to [4], we calculate the quantity of Majorana
polarization (MP) (originally introduced in [5]). This quantity, denoted
$M_{j}$, is defined per site $j$ as:

$\displaystyle M_{j}$
$\displaystyle=\frac{\sum_{\sigma}(w_{\sigma}^{2}-z_{\sigma}^{2})}{\sum_{\sigma}(w_{\sigma}^{2}+z_{\sigma}^{2})}$
$\displaystyle w_{\sigma}$ $\displaystyle=\langle
O\lvert(d_{j\sigma}+d_{j\sigma}^{\dagger})\lvert E\rangle$ $\displaystyle
z_{\sigma}$ $\displaystyle=\langle
O\lvert(d_{j\sigma}-d_{j\sigma}^{\dagger})\lvert E\rangle$

where $\lvert O\rangle$ and $\lvert E\rangle$ are the lowest-energy odd and
even states respectively. Due to symmetrically chosen parameters $\lvert
M_{\mathrm{L}}\rvert=\lvert M_{\mathrm{R}}\rvert=|M|$, such that a single MP
value can be extracted for a specific set of parameters. To provide an
experimental observable that reflects a high or low $|M|$, we investigate the
behavior of two transition energies, denoted in the main text as
$E_{\mathrm{gap}}$ and $E_{\mathrm{det}}$. Numerically these are obtained from
the Hamiltonian as the energy difference between the lowest even state and the
second-lowest odd state, at specific values of $\mu_{\mathrm{L}}$ and
$\mu_{\mathrm{R}}$. $E_{\mathrm{gap}}$ is obtained with $\mu_{\mathrm{L}}$ and
$\mu_{\mathrm{R}}$ set to corresponding to their sweet spot value.
$E_{\mathrm{det}}$ is obtained with $\mu_{R}$ detuned by
$\approx\Delta_{\mathrm{M}}$ . In the absence of the Zeeman term, both
$E_{\mathrm{gap}}$ and $E_{\mathrm{det}}$ are zero by definition, due to the
degeneracy of the odd states in the presence of time reversal symmetry. The
dependence of $M$, $E_{\mathrm{gap}}$ and $E_{\mathrm{det}}$ on
$E_{\mathrm{z}}$ is demonstrated in Fig. S10.

In order to compare to experiments, parameters are selected to match realistic
values. We set $\Delta_{\mathrm{M}}$ = $100\text{\,}\mathrm{\upmu
e}\mathrm{V}$ and $\Delta_{\mathrm{L,R}}=0.5\Delta_{\mathrm{M}}$. The charging
energy $U$ is fixed in both dots to be 10$\Delta_{\mathrm{M}}$, except for
Fig. 1d where a value of 7$\Delta_{\mathrm{M}}$ is used to better highlight
the behavior of all four avoided crossings in the large CSD. For the analysis
in Fig. 4, tunneling terms $t$ and $t_{\mathrm{SO}}$ are fine-tuned such that
at large $E_{\mathrm{z}}$ the sweet spot gap of
$30\text{\,}\upmu\mathrm{e}\mathrm{V}$ is obtained, to match the experimental
result. This gives $t_{\mathrm{SO}}=0.4t=0.7\Delta_{\mathrm{M}}$. Similarly,
to compare to the experimental results in Fig. S5 and Fig. S10, $t$ and
$t_{\mathrm{SO}}$ are fine-tuned to obtain a sweet spot gap of
$50\text{\,}\upmu\mathrm{e}\mathrm{V}$ at large $E_{\mathrm{z}}$. This results
in $t_{\mathrm{SO}}=0.4t=0.85\Delta_{\mathrm{M}}$.

###  I.3 Effective model in the strong coupling regime

The above description is used for all presented calculations. To provide an
intuitive understanding of these results, we invoke a description of the
system through an effective model, introduced in detail in [6]. Here we
provide a brief summary of the relevant findings. When the energy
$E_{\mathrm{ABS}}$ of the sub-gap state in the middle site is large compared
to tunnel couplings between the QDs, the middle site can be integrated out.
This leaves a description of the system including only effective couplings
between YSR states in the left and right QD sites. Additionally, it is assumed
that $E_{\mathrm{z}}$ is sufficiently large such that only the
$\lvert\downarrow\rvert$ occupation of each QD partakes in transport. In this
description, a Hamiltonian can be constructed in a singlet-doublet basis. The
ground state of each QD is either the $\lvert\downarrow\rangle$ doublet or a
singlet $\lvert S\rangle$ of the form $\lvert S\rangle=u_{\mathrm{L,R}}\lvert
0\rangle-v_{\mathrm{L,R|}}\lvert\downarrow\uparrow\rangle$. Here the $u,v$
components depend on the chemical potential energies $\mu_{\mathrm{L}}$ and
$\mu_{\mathrm{R}}$ of the left and right dot:
($u_{\mathrm{L,R}}^{2}=1-v_{\mathrm{L,R}}^{2}=\frac{1}{2}+\frac{\mu_{L,R}+U}{\sqrt{U^{2}+E_{\mathrm{ABS}}^{2}}}$).
The effective Hamiltonian is obtained:

$H^{\text{eff}}_{\text{coupling}}=\sum_{\sigma,\eta=\uparrow,\downarrow}(t_{\sigma\eta}c^{\dagger}_{L\sigma}c_{R\eta}+\Delta_{\sigma\eta}c^{\dagger}_{L\sigma}c^{\dagger}_{R\eta})+h.c.,$
(1)

where $c_{\mathrm{L,R}}$ and $c_{\mathrm{L,R}}^{\dagger}$ are annihilation and
creation operators for sites in left and right. Further,
$t_{\mathrm{\sigma\eta}}$ specifies the amplitude of an electron with spin
$\sigma$ from left site tunneling to occupy a state with spin $\eta$ in the
right side, or vice versa. $\Delta_{\mathrm{\sigma\mu}}$ specifies the
amplitude of the creatiing or breaking of a Cooper pair through electrons with
spins $\sigma$ and $\eta$ in the left and right QDs
($\sigma,\eta\in[\uparrow,\downarrow]$). Note here due to the spin-orbit
interaction, two equal-spin electrons can also be coupled via the
$\Delta$-term. Importantly, this description is equivalent to the weakly
coupled model discussed in [7], when changing from a charge occupation basis
to a YSR-state basis. Now, two types of coupling arise between the total even-
parity states (denoted $\Gamma_{\mathrm{E}}$) and between total-odd parity
states ($\Gamma_{\mathrm{O}}$). Each are combination of ECT and CAR
amplitudes:

$\displaystyle\Gamma_{\mathrm{E}}$ $\displaystyle=\langle SS\lvert
H^{\text{eff}}_{\text{coupling}}\lvert\downarrow\downarrow\rangle$
$\displaystyle=-\Delta_{\uparrow\uparrow}v_{L}v_{R}+\Delta_{\downarrow\downarrow}u_{L}u_{R}+t_{\uparrow\downarrow}v_{L}u_{R}-t_{\downarrow\uparrow}u_{L}v_{R},$
$\displaystyle\Gamma_{\mathrm{O}}$ $\displaystyle=\langle S\downarrow\rvert
H^{\text{eff}}_{\text{coupling}}\lvert\downarrow S\rangle$
$\displaystyle=-t_{\uparrow\uparrow}v_{L}v_{R}+t_{\downarrow\downarrow}u_{L}u_{R}+\Delta_{\uparrow\downarrow}v_{L}u_{R}-\Delta_{\downarrow\uparrow}u_{L}v_{R},$

The coupling $\Gamma_{\mathrm{O}}$ results in bonding and anti-bonding states
of the form $u\lvert\mathrm{S},\downarrow\rangle\pm
v\lvert\downarrow,\mathrm{S}\rangle$. $\Gamma_{\mathrm{E}}$ forms bonding and
anti-bonding states of the form
$\alpha\lvert\mathrm{S},\mathrm{S}\rangle\pm\beta\lvert\downarrow,\downarrow\rangle$.
When $\Gamma_{\mathrm{E}}=\Gamma_{\mathrm{O}}$ the even and odd ground states
become degenerate at $\mu_{\mathrm{L}}=\mu_{\mathrm{R}}$, equivalent to the
$t=\Delta$ condition in the ”poor Man’s Majorana” description [7]. In Fig. 2d,
we extract the distance between avoided crossing in experimentally obtained
CSDs, after converting the gate voltages to the YSR energies of each QD. In
the effective description this distance corresponds to analysing where the
line $\mu_{L}=\mu_{R}$ or $\mu_{L}=-\mu_{R}$ intersects a degeneracy between
the even and odd ground state energies. This gives two points, separated by
$\sqrt{8|\Gamma_{\mathrm{O}}^{2}-\Gamma_{\mathrm{E}}^{2}|}$.

##  II Supplementary Figure S1 to S10

Figure S1: Characterization of QDs and ABS spectroscopy for device A. (a-b)
Coulomb diamond measurements of the left and right QDs, in the regime used for
measurements presented in Fig. 4.. Charging energies are estimated to be
$1.1\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ in each QD. The lever arm of the
left and right dot is estimated to be $\alpha\approx$ 0.11. (c-d) Due to a
large tunnel coupling between the QDs and the hybrid section, YSR-states form
in the sub-gap spectrum of the QDs. Close-up tunneling spectroscopy
measurements around the charge degeneracy points in (a-b) reveal clear sup-gap
features within the Coulomb diamonds, whose dependence on $V_{\mathrm{QDL}}$
and $V_{\mathrm{QDR}}$ is typical for YSR-states. (e) Crossed Andreev
reflection and elastic co-tunneling require the presence of extended ABSs.
Local $G_{\mathrm{l}}$ and non-local conductance $G_{\mathrm{nl}}$ of the
hybrid region are measured via tunnelling spectroscopy and their identical
energy dependence as a function of $V_{\mathrm{ABS}}$ highlights that ABSs
extend across the entire hybrid section. Comparable behavior was observed in a
wide $V_{\mathrm{ABS}}$ range from 0 to $-1\text{\,}\mathrm{V}$. The
measurement presented in Fig. 1 is taken at the $V_{\mathrm{ABS}}$ with the
eye-shaped crossing. (f) ABS spectroscopy as a function external magnetic
field at $V_{\mathrm{ABS}}$ = $-623\text{\,}\mathrm{m}\mathrm{V}$. The effect
of splitting of the doublet state can be observed at low fields. A $g$-factor
of 5.5 is extracted by linear fitting of the lowest sub-gap states (dashed
line) in Fig. S1d.

Figure S2: Characterization of device B. (a) Scanning electron micro-graph of
device B, used to obtain the measurements presented in Fig. 2. Scale bar shown
is $100\text{\,}\mathrm{n}\mathrm{m}$. (b-c) Coulomb diamond measurements of
the left and right QDs. Charging energies are extracted to be
$1.4\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$. The lever arm
$\alpha_{\mathrm{N}}$ is extracted to be about 0.11 for each QD. This lever
arm is used for the extraction in Fig. 2d. (d-e) To validate the direction of
$B_{\mathrm{SO}}$ and to show the connection between interactions in strongly
coupled QDs and underlying ECT and CAR processes, we first measure ECT and CAR
currents in the weakly coupled dots, as detailed in [8]. With $B\perp
B_{\mathrm{SO}}$, measurements of CAR (d) and ECT (e) show the typical
blockades for same-spin and opposite-spin charge configurations respectively.
In (f-g), with $B\parallel B_{\mathrm{SO}}$, the spin non-conserving ECT and
CAR processes are observed to be revived. (h) Measuring CAR and ECT rates as a
function of magnetic field angle $\theta$ shows the currents for the spin non-
conserving processes are indeed smoothly controlled and become suppressed when
$\theta=0$. This supports the interpretation that $B\parallel B_{\mathrm{SO}}$
when B is perpendicular to the 1-D channel. (i-j) Next, the QDs are operated
with higher tunnelling rates between the QDs and the SC, to enable strong
couplings. Similar to Fig. 1e-f, CSDs are obtained in the strongly interacting
regime, taken with the verified $B\parallel B_{\mathrm{SO}}$ and $B\perp
B_{\mathrm{SO}}$ respectively.

Figure S3: Extended dataset of Figure 2. (a) CSDs measured at various magnetic
field angles $\theta$ between $0\text{\,}\mathrm{\SIUnitSymbolDegree}$ and
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$, used to extract the data shown in
Fig. 2d. (b) Example of the extraction process. For each obtained CSD,
$V_{\mathrm{QDL}}$ and $V_{\mathrm{QDR}}$ are converted to energies
$\mu_{\mathrm{L}}$ and $\mu_{\mathrm{R}}$ using lever arms obtained in Fig.
S2. Next, the conductance is extracted along a
$\mu_{\mathrm{L}}=-\mu_{\mathrm{R}}$ or $\mu_{\mathrm{L}}=\mu_{\mathrm{R}}$
line. (c) Two Gaussian peaks are fitted to extract the separation between the
two avoided crossings, from which the quantity
$\sqrt{|\Gamma_{\mathrm{o}}-\Gamma_{\mathrm{e}}|}$ is obtained (plotted in
Fig. 2d).  Figure S4: Characterisation of YSR-states in QDs of Figure 3. To
complement the data in Fig. 3, Fig S5 and Fig. S10, we measure the sub-gap
states in QDL and QDR (see Fig S1). Using this, we obtain the lever arms of
$V_{\mathrm{QDL}}$ and $V_{\mathrm{QDR}}$ on the YSR-state energies (denoted
$\alpha_{\mathrm{YSR}}$) (see Methods). (a-b) Sub-gap spectroscopy of QDL and
QDR at $B_{\mathrm{z}}$=$0\text{\,}\mathrm{m}\mathrm{T}$. From the slopes of
the states upon crossing $V_{\mathrm{L}},V_{\mathrm{R}}=0$, we estimate
$\alpha_{\mathrm{YSR}}\approx$ 0.045. Applying an external magnetic field
lowers the energy of ABSs in the hybrid region, as a result of Zeeman
splitting, This in turn will affect the YSR-spectrum of the QDs, due to
increased hybridization between the QDs and the ABS. (c-d) Measuring sub-gap
spectroscopy of QDL and QDR at
$B_{\mathrm{z}}$=$225\text{\,}\mathrm{m}\mathrm{T}$ for the same settings as
in (a-b) shows indeed the effective lever arm here decreases to
$\alpha_{\mathrm{YSR}}\approx$ 0.028.

Figure S5: Full conductance spectra at the sweet spot upon detuning
$V_{\mathrm{QDL}}$ and $V_{\mathrm{QDR}}$. A comparison between numerically
calculated conductance and measured conductance in support of Fig 3, measured
at $B=$225\text{\,}\mathrm{m}\mathrm{T}$$. Presented results show the
evolution of $G_{\mathrm{l}}$ and $G_{\mathrm{nl}}$ for four different cases:
(a-b) detuning $V_{\mathrm{QDL}}$, (c-d) detuning $V_{\mathrm{QDR}}$, (e-f)
detuning both $V_{\mathrm{QDL}}$ and $V_{\mathrm{QDR}}$ simultaneously along a
diagonal path and (g-h) detuning both anti-diagonally. For each case, we find
the behavior of both $G_{\mathrm{l}}$ and $G_{\mathrm{nl}}$ is well described
by the numerical results.  Figure S6: Extended dataset for Figure 3. (a) Sets
of CSDs obtained while varying $V_{\mathrm{ABS}}$ in the range presented in
Fig 3. The range of $V_{\mathrm{QDL}}$ and $V_{\mathrm{QDR}}$ is constant for
each measurement. The slight drift of the avoided crossing upon varying
$V_{\mathrm{ABS}}$ is owed to cross-capacitance between $V_{\mathrm{ABS}}$ and
the potential of the QDs.  Figure S7: Energy diagrams detailing non-local
transport. In CSDs presented in Fig. S3, a clear sign change is observed when
changing from the $\Gamma_{\mathrm{O}}$$>\Gamma_{\mathrm{E}}$ regime to the
$\Gamma_{\mathrm{O}}$$<\Gamma_{\mathrm{E}}$ regime. This can be understood by
considering the possible transport cycles. (a) When
$\Gamma_{\mathrm{O}}$$>\Gamma_{\mathrm{E}}$, $G_{\mathrm{nl}}$ is observed to
be negative in the measured CSDs (see Fig. 3c). (c) When
$\Gamma_{\mathrm{O}}$$<\Gamma_{\mathrm{E}}$, the same measurements yield a
positive $G_{\mathrm{nl}}$ (see Fig. 3a). Horizontal and vertical dashed lines
indicate $\upmu_{\mathrm{R}}$ = 0 and $\upmu_{\mathrm{L}}$ = 0 respectively.
The state of the uncoupled system is labelled in each quadrant. (b,d) In such
CSD measurements, zero-bias transport can take place when the odd and even
ground states are degenerate. For non-local transport to occur, the system can
accept a hole/electron from one lead, and relax non-locally to its original
state by either (b) donating a hole/electron to the opposite lead, giving rise
to negative $G_{\mathrm{nl}}$, or (d) accept a hole/electron from the opposite
lead, giving rise to positive $G_{\mathrm{nl}}$. The preferred path is
dictated by the quadrant in $\mu_{\mathrm{L}}$, $\mu_{\mathrm{R}}$ space where
the odd-even degeneracy occurs. (e) When $\mu_{\mathrm{L}}$,
$\mu_{\mathrm{R}}>0$ or $\mu_{\mathrm{L}}$,$\mu_{\mathrm{R}}<0$, the former
path is expected to dominate and the resulting $G_{\mathrm{nl}}$ will be
negative. (f) When $\mu_{\mathrm{L}}>0$ and $\mu_{\mathrm{R}}<0$ or vice
versa, the latter path is expected to dominate and resulting $G_{\mathrm{nl}}$
will be positive.  Figure S8: Numerical analysis of $E_{\mathrm{gap}}$ and
$E_{\mathrm{det}}$ Numerical calculations supporting the results presented in
Figure 4. Through the procedure detailed in SI-Methods, Majorana sweet spots
are obtained and analysed for fields between $0\text{\,}\mathrm{m}\mathrm{T}$
and $300\text{\,}\mathrm{m}\mathrm{T}$. (a) Field evolution of
$G_{\mathrm{RR}}$ line-traces at each sweet spot, showing the excitations
above the ZBPs gradually increasing in energy and then saturating at $\pm
30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{e}\mathrm{V}$. (b) Field
evolution of $G_{\mathrm{RR}}$ line-traces when QDR detuned by
$3\Delta_{\mathrm{ind}}$, showing the excited states increase linearly in
energy. From calculations in (a) and (b), $E_{\mathrm{gap}}$ and
$E_{\mathrm{det}}$ are obtained, given by the energy between the lowest even-
parity state and second-lowest odd-parity state. (c) Extraction of
$E_{\mathrm{gap}}$ (solid) and the Majorana polarization (dashed), for
different values of the tunneling parameter $t$. In each case $t_{so}=0.4t$.
Larger tunnel coupling results in larger hybridization between ABSs, in turn
lowering the MP at a specific magnetic field. (d) Extraction of
$E_{\mathrm{det}}$ for various values of detuning $\upmu_{R}$. In each case
the slope at low fields corresponds to $2E_{\mathrm{z}}$ (dashed grey line).
The larger the detuning of $\upmu_{R}$, the longer this holds. The dashed
black line shows the energy of the ABS $E_{\mathrm{ABS}}$. At large detuning
$E_{\mathrm{det}}$ will increase linearly with $2E_{\mathrm{z}}$, until
becoming of comparable $E_{\mathrm{ABS}}$ becomes the lowest energy scale.
Figure S9: Raw datasets for figure 4h Obtained ’sweet spots’ at magnetic
fields between $0\text{\,}\mathrm{m}\mathrm{T}$ and
$300\text{\,}\mathrm{m}\mathrm{T}$. (a-i) CSDs and tunnelling spectroscopy are
measured at each sweet spot, where $V_{\mathrm{QDR}}$ is detuned. From these
measurements $E_{\mathrm{det}}$ and $E_{\mathrm{gap}}$ are extracted, as
described in the main text and in SI-Methods. Figure S10: Extended dataset
supporting figure 4h. Reproduction of the main results from Fig. 4, using the
orbitals shown in Fig. 3. Data was obtained at 6 different field values $B$
between 0 and $250\text{\,}\mathrm{m}\mathrm{T}$. At each field
$V_{\mathrm{ABS}}$ is adjusted to tune to the sweet spot. a Extraction of
$E_{\mathrm{det}}$ and $E_{\mathrm{gap}}$, similar to the analysis presented
in Fig. 4l. From a linear for of $E_{\mathrm{det}}$ a g-factor of 5.7 is
estimated. (b) Numerically obtained $E_{\mathrm{det}}$ and $E_{\mathrm{gap}}$,
using parameters tuned to compare to (a). At 250mT, an estimate of $M\approx
0.9$ is obtained. Extrapolation for comparison to Fig. 4l yields $M\approx
0.92$ at 300mT. (c) and (d) Waterfall plots highlighting the line-traces used
to extract the data in (a). (e-j) Raw datasets of CSDs and tunnelling
spectroscopy measurements, from which (a-d) is extracted. Datasets at
$150\text{\,}\mathrm{m}\mathrm{T}$ and $225\text{\,}\mathrm{m}\mathrm{T}$
datasets are repeated from Fig. 3 and Fig. S5 respectively.

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