55footnotetext: All the authors contributed equally and are listed in
alphabetical order of their last names

# Machine learning unveils multiple Pauli blockades in the transport
spectroscopy of bilayer graphene double-quantum dots

Anuranan Das Machine learning unveils multiple Pauli blockades in the
transport spectroscopy of bilayer graphene double-quantum dots Department of
Electrical Engineering, Indian Institute of Technology Bombay, Powai,
Mumbai-400076, India  Adil Khan Machine learning unveils multiple Pauli
blockades in the transport spectroscopy of bilayer graphene double-quantum
dots Department of Electrical Engineering, Indian Institute of Technology
Bombay, Powai, Mumbai-400076, India  Ankan Mukherjee Machine learning unveils
multiple Pauli blockades in the transport spectroscopy of bilayer graphene
double-quantum dots Department of Physics, Indian Institute of Technology
Bombay, Powai, Mumbai-400076, India  Bhaskaran Muralidharan Machine learning
unveils multiple Pauli blockades in the transport spectroscopy of bilayer
graphene double-quantum dots<EMAIL_ADDRESS>Department of Electrical
Engineering, Indian Institute of Technology Bombay, Powai, Mumbai-400076,
India  Centre of Excellence in Quantum Information, Computation, Science and
Technology, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India

(August 28, 2024)

###### Abstract

Recent breakthroughs in the transport spectroscopy of 2-D material quantum-dot
platforms have engendered a fervent interest in spin-valley qubits. In this
context, Pauli blockades in double quantum dot structures form an important
basis for multi-qubit initialization and manipulation. Focusing on double
quantum dot structures, and the experimental results, we first build
theoretical models to capture the intricate interplay between externally fed
gate voltages and the physical properties of the 2-D system in such an
architecture, allowing us to effectively simulate Pauli blockades. Employing
the master equations for transport and considering extrinsic factors such as
electron-photon interactions, we thoroughly investigate all potential
occurrences of Pauli blockades. Notably, our research reveals two remarkable
phenomena: (i) the existence of multiple resonances within a bias triangle,
and (ii) the occurrence of multiple Pauli blockades. Leveraging our model to
train a machine learning algorithm, we successfully develop an automated
method for real-time detection of multiple Pauli blockade regimes. Through
numerical predictions and validations against test data, we identify where and
how many Pauli blockades are likely to occur. We propose that our model can
effectively detect the generic class of Pauli blockades in practical
experimental setups and hence serves as the foundation for future experiments
on qubits that utilize 2-D material platforms.

##  I Introduction

Quantum devices implemented on 2-D material platforms have garnered
significant attention due to their unique characteristics, including non-
equivalent valleys, strong spin-orbit coupling, electronically tunable
bandgaps, and reduced hyperfine interactions [1, 2, 3, 4, 5, 6, 7, 8]. This
makes materials like bi-layer graphene (BLG) and the transition metal
dichalcogenide family (TMDCs) [9] promising platforms for spin-valley qubits.
Recent experiments have explored the potential of utilizing the valley degree
of freedom for qubit initialization, storage, and readout. Achieving
confinement of electrons to quantum dots (QDs) on 2-D materials has, however,
presented several hurdles, such as defects, high effective electron masses,
and challenges in achieving precise gate voltage tunability using ohmic
contacts [10, 11, 12].
Initialization and read-out of qubits on any platform involves the ascertained
occurrence of Pauli-blockades [13, 14, 15, 16, 17, 18, 19] in the electronic
transport spectroscopy. Recent investigations have demonstrated that in 2-D
systems, the spin and the valley pseudospin are intertwined, leading to a
general class of Pauli blockades [20] that cannot be solely explained by
independently considering the spin or valley degrees of freedom. While the
theoretical understanding of the origins of multiple Pauli blockades has made
progress [21, 19], leveraging this knowledge to tune quantum dots by
identifying and characterizing multiple blockades still presents a significant
challenge.
Utilizing machine learning for the initialization and scaling of qubits in
quantum devices have highlighted the use of automation to detect and
characterize Pauli spin-blockades in semiconducting qubit systems [22, 23, 24,
25]. These algorithms can swiftly determine the occurrence of a spin-blockade,
enabling real-time readout of spin qubits. Given the rapid progress in spin-
valley qubit research, a fully automated algorithm for identifying multiple
spin-valley coupled Pauli blockades is certainly the need of the hour. In this
paper, we develop a comprehensive model to understand the transport mechanism
through quantum dots. We leverage this model to train a machine learning
algorithm, enabling us to predict blockade regimes in the 2-D material
platforms, as illustrated in Fig. 1d.
Our work builds upon a double quantum dot (DQD) transport setup [26, 27, 28,
29, 30], as depicted in Fig. 1a. The focus of our analysis is on Pauli
blockades, particularly in the regime where the total occupancy of the dots is
restricted to two electrons. To construct our models, we account for various
factors [31, 32, 33, 34] such as intrinsic spin-orbit (SO) coupling, spin-
Zeeman splitting, valley-Zeeman splitting, sequential spin (pseudo-spin)
conserving inter-dot tunneling, spin-flip tunneling, photon-assisted
tunneling, and Coulomb interactions. These elements are visualized in Fig. 1b
and Fig. 1c, and are the essential components of our model for understanding
the intricacies of the transport mechanism, and eventually making important
predictions.

(a)

(b)

(c)

(d)

Figure 1:  Device schematics and workflow. (a) The device structure under
consideration. The five finger gates and two split gates define the DQD
transport spectroscopy setup. Barrier gates $V_{B1}$ and $V_{B3}$ control the
coupling to the source and drain respectively, while $V_{B2}$ controls the
inter-dot tunneling. Plunger gates $V_{G1}$ and $V_{G2}$ determine the onsite
energies. (b) The DQD system is coupled to the left and right contacts by
coupling rates $\gamma_{L}$ and $\gamma_{R}$, respectively. Each dot is
described by the onsite energies $\epsilon_{i}$, the intra-dot interaction
$U_{ii}$, and the inter-dot interaction $U_{ij}$. The interdot tunneling is
descrbed by $t_{c}$, which is spin and valley conserving. Spin-flip tunneling
events are captured by $t_{flip}$. The dot tuning is controlled via gate
voltages $V_{G1}$ and $V_{G2}$. A finite source-drain bias of $V_{sd}$ is also
applied to facilitate current flow. The cross-capacitance between the gate
voltages is $C$. (c) Inelastic processes are crucial to realize bias triangles
in the high-bias regime. A basic mechanism for boson-assisted tunneling is
shown. Intra-dot transitions between $\epsilon_{b}$ and $\epsilon_{2r}$ is
blocked. An electron in level $\epsilon_{b}$ can loose energy via these
interactions to transit to level $\epsilon_{1r}$ and hence open conduction
channels involving states $\epsilon_{1r}$ and $\epsilon_{2r}$. (d) Flowchart
of the machine learning algorithm followed. The voltages and magnetic field
inputs (blue) set to the system determine the parameters in the Hamiltonian,
which, coupled with the I-V characteristics of the system (orange), are fed in
to extract bias triangles and determine Pauli blockade regimes (green) using a
machine learning algorithm.

By simulating the effects of spin-orbit coupling, spin-flip tunneling and
photon-assisted tunneling, we unveil two novel phenomena, which are not
present in spin-only systems: (i) multiple resonances and (ii) multiple Pauli
blockades. We demonstrate that these phenomena have distinct underlying
causes. Furthermore, we carefully train our machine learning model to not only
detect the presence of a Pauli blockade but also predict whether there are
multiple Pauli blockades or just multiple resonance peaks in the system. The
predictability of precise Pauli blockade regimes allows us to extract both
spin and valley information from the state using a single-shot readout,
effectively doubling the rate at which information is stored and read. We
firmly believe that our research lays the groundwork for future experiments
involving the initialisation and scaling of qubits on 2-D material platforms.
The paper is organised as follows. In Sec. II, we construct the effective
Hamiltonian for the DQD, solve for the relevant eigenstates, and obtain the
equations governing the current flow and thereby obtain the transport
spectroscopy. We then present the conditions under which Pauli blockades may
be realized, and under which they may be lifted. In Sec. III we perform
simulations on our model and generate a training data set using different
regimes and parameters. We also present the cases under which multiple
excitations and multiple blockades occur. We then use the generated data to
train our machine learning model and then test the same on different test
cases. Finally, we present our conclusions and scope for future work in Sec.
IV.

##  II Formalism and Methods

A schematic of the model under investigation is illustrated in Fig. 1a, and
Fig. 1b. The model involves a DQD setup [35, 36] with source and drain
electrodes shown in blue. Split gates (pink) are used to change the carrier
concentration, and finger gates (yellow) are used for fine-tuning the onsite
potentials on each quantum dot. Functionally, when a dc bias is applied across
this device, a current can be observed. The current flow can then also be
controlled by tweaking the gate voltages. We have a left (right) quantum dot
$LD(RD)$, whose onsite energy, $\varepsilon_{1}(\varepsilon_{2})$ can be
controlled using the lever arm voltage $V_{G1}(V_{G2})$. The model abstraction
for the DQD system is illustrated in Fig. 1a. The source drain bias voltage
$V_{SD}$ is applied across the electrodes labelled $V_{S}$ and $V_{D}$. The
voltage at gate $V_{B2}$ controls $t$, the inter-dot tunneling, which, in our
model, preserves the spin and the valley pseudo-spin. The coupling constants
between each lead and the channel, $\gamma_{1}$ and $\gamma_{2}$, are
controlled by barrier gate voltages $V_{B1}$ and $V_{B3}$. The backgate
voltage $V_{BB}$ can be tuned to control the overall degree of confinement.

###  II.1 Model Hamiltonian

We employ a modified version of the well-studied Hubbard Hamiltonian [37, 34,
20] to model the system. This model includes several components, such as
onsite energies and inter-dot tunneling. Additionally, there exists an onsite
Coulomb repulsion, denoted by $U_{1(2)}$, between pairs of electrons on each
dot, and an inter-dot Coulomb repulsion, denoted by $U_{12(21)}$. These
energies $U_{i}$ and $U_{ij}$ can be matrices indexed by $i(=1,2)$ and
$j(=1,2)$, allowing for a more flexible representation.
Each dot can host a conduction electron in one of the two valleys, either $K$
or $K^{\prime}$, and each valley can possess either spin $\uparrow$ or spin
$\downarrow$. Consequently, a single dot offers four available states:
$\ket{K\uparrow}$, $\ket{K\downarrow}$, $\ket{K^{\prime}\uparrow}$, and
$\ket{K^{\prime}\downarrow}$. In the absence of an external magnetic field,
these four energy states split into two Kramer pairs due to intrinsic spin-
orbit (SO) coupling [38, 39, 40, 41, 42]. However, when an external magnetic
field is introduced, the degeneracy of the states within each Kramer pair is
broken, affected by both the spin-Zeeman and the valley-Zeeman effects. The
energy shift due to spin-Zeeman splitting is given by $h_{S}=\sigma
g_{s}\mu_{B}B_{||}$, while the valley-Zeeman splitting contributes an energy
shift of $h_{V}=\tau g_{v}\mu_{B}B_{\perp}$. Here, $\sigma$ represents the
spin quantum number ($\sigma=+\frac{1}{2}$ for spin $\uparrow$ and
$\sigma=-\frac{1}{2}$ for spin $\downarrow$), and $\tau$ denotes the valley
pseudo-spin ($\tau=+\frac{1}{2}$ for valley $K$ and $\tau=-\frac{1}{2}$ for
valley $K^{\prime}$). The electron magnetic moment is denoted by
$\mu_{B}=5.79\times 10^{-5}$ eVT-1, $B_{||}$ and $B_{\perp}$ are the parallel
and perpendicular components of the external magnetic field, and $g_{s}$ and
$g_{v}$ represent the spin and valley g-factors, respectively.

Under such considerations, the Hamiltonian takes the form

$\displaystyle\hat{H}_{DQD}$
$\displaystyle=\underbrace{\varepsilon_{L}\hat{n}_{L}+\varepsilon_{R}\hat{n}_{R}}_{\text{Onsite
energy}}$ $\displaystyle\hskip
5.69054pt+\underbrace{\frac{U_{ii}}{2}\left(\hat{n}_{L}^{2}-\hat{n}_{L}+\hat{n}_{R}^{2}-\hat{n}_{R}\right)}_{\text{Onsite
repulsion}}+\underbrace{U_{ij}\hat{n}_{L}\hat{n}_{R}}_{\text{Interdot
replusion}}$ $\displaystyle\hskip
11.38109pt+\underbrace{\sum_{\tau,\sigma}t\hat{c}_{R\sigma\tau}^{\dagger}\hat{c}_{L\sigma\tau}+\text{h.c.}}_{\text{Interdot
sequential tunneling}}$ $\displaystyle\hskip
17.07164pt+\underbrace{\sum_{\alpha,\tau,}t_{SO}\hat{c}_{R\uparrow\tau}^{\dagger}\hat{c}_{L\downarrow\tau}+\hat{c}_{R\downarrow\tau}^{\dagger}\hat{c}_{L\uparrow\tau}+\text{h.c.}}_{\text{Interdot
spin-flip tunneling}}$ $\displaystyle\hskip
22.76219pt+\underbrace{\frac{\Delta_{SO}}{2}\sum_{\alpha,\tau,\sigma}\hat{c}_{\alpha\sigma\tau}^{\dagger}\left(\bm{\sigma}_{3}\right)_{\sigma\sigma}\left(\bm{\tau}_{3}\right)_{\tau\tau}\hat{c}_{\alpha\sigma\tau}}_{\text{Spin
orbit coupling}}$ $\displaystyle\hskip
28.45274pt+\underbrace{\left|h_{S}\right|\sum_{\alpha,\tau,\sigma}\hat{c}_{\alpha\sigma\tau}^{\dagger}\left(\bm{\sigma}_{3}\right)_{\sigma\sigma}\hat{c}_{\alpha\sigma\tau}}_{\text{Spin
Zeeman effect}}$ $\displaystyle\hskip
28.45274pt+\underbrace{\left|h_{V}\right|\sum_{\alpha,\tau,\sigma}\hat{c}_{\alpha\sigma\tau}^{\dagger}\left(\bm{\tau}_{3}\right)_{\tau\tau}\hat{c}_{\alpha\sigma\tau}}_{\text{Valley
Zeeman effect}},$ (1)

where the summations are defined over $\alpha\in\\{L,R\\}$,
$\sigma\in\\{\uparrow,\downarrow\\}$, $\tau\in\\{K,K^{\prime}\\}$. The terms
$\bm{\sigma}_{3}$ ($\bm{\tau}_{3}$) is the z-component of the Pauli matrix for
the spin (valley pseudo-spin), defined as

$\displaystyle\left(\bm{\sigma}_{3}\right)_{\uparrow\uparrow}$
$\displaystyle=1;\quad\left(\bm{\sigma}_{3}\right)_{\uparrow\downarrow}$
$\displaystyle=\left(\bm{\sigma}_{3}\right)_{\downarrow\uparrow}$
$\displaystyle=0;\quad\left(\bm{\sigma}_{3}\right)_{\downarrow\downarrow}$
$\displaystyle=-1$ (2a) $\displaystyle\left(\bm{\tau}_{3}\right)_{KK}$
$\displaystyle=1;\quad\left(\bm{\tau}_{3}\right)_{KK^{\prime}}$
$\displaystyle=\left(\bm{\tau}_{3}\right)_{K^{\prime}K}$
$\displaystyle=0;\quad\left(\bm{\tau}_{3}\right)_{K^{\prime}K^{\prime}}$
$\displaystyle=-1$ (2b)

The symbol $\hat{n}_{L}$ ($\hat{n}_{R}$) denotes the number operator for the
number of electrons on the left (right) dot, formulated as

$\displaystyle\hat{n}_{\alpha}=\sum_{\sigma,\tau}\hat{c}^{\dagger}_{\alpha\sigma\tau}\hat{c}_{\alpha\sigma\tau},$
(3)

where $\hat{c}^{(\dagger)}_{\alpha\sigma\tau}$ is the annihilation (creation)
operator for an electron on dot $\alpha$ with spin $\sigma$ in valley $\tau$.

In the absence of spin-flip tunneling and photon interactions, the above
Hamiltonian takes a block diagonal form and separates into nine sub-spaces,
each with an invariant total number of electrons $N=0,1,\cdots,8$. In
discussing the blockades in the two-electron occupancy regime, only the
subspaces $N=1$ and $N=2$ are relevant. We use
$L_{\zeta}\left(R_{\zeta}\right)$ to denote that there is an electron in the
state $\zeta$ on the left(right) quantum dot, where
$\zeta\in\\{K\uparrow,K\downarrow,K^{\prime}\uparrow,K^{\prime}\downarrow\\}$.
For instance, a system with two electrons: one in the left dot in state
$K\uparrow$ and the other in the right dot in state $K^{\prime}\downarrow$ is
represented as $\ket{L_{K\uparrow}R_{K^{\prime}\downarrow}}$. We also develop
the notation $(n_{L},n_{R})$ to represent a state with $n_{L}$ electrons on
the left dot and $n_{R}$ electrons on the right dot. The eigenstates are not
the $(n_{L},n_{R})$ states, but a superposition of $(n_{L},n_{R})$ states with
$n_{L}+n_{R}=N=\text{constant}$.

(a)

(b)

Figure 2:  Energy spectrum of the $N=2$ Fock space as is described in Sec.
II.2, as a function of the magnetic field applied. (a) The perpendicular
component of the magnetic field is held fixed at $B_{\perp}=0.8$ T, and the
parallel component is varied. The spin Zeeman effect splits the energy
spectrum according to $\Delta E=\mu_{B}g_{s}B_{||}$. (b) The parallel
component of the magnetic field is held fixed at $B_{||}=0.8$ mT, and the
perpendicular component is varied. The valley Zeeman effect splits the energy
spectrum according to $\Delta E=\mu_{B}g_{v}B_{\perp}$.

In addition to the above Hamiltonian, we also add a term for photon-assisted
tunneling (PAT) as described below. These additional effects often lift the
Pauli blockades by creating eigenstates that are superpositions of multiple
spin and valley states. The composite Hamiltonian of the system can be
modified to include the effects of boson-mediated scattering processes [43] as
shown in fig. 1c. The photon relaxation occurs in the sense that electrons in
the dot can change level by absorption and emission. Such an interaction
Hamiltonian is described as:

$H_{\mathrm{ph}}=H_{B}+H_{B-D}=\sum_{q}\omega_{q}d_{q}^{\dagger}d_{q}+\sum_{q\sigma
ij}g_{ph}\left(d_{q}^{\dagger}+d_{q}\right)c_{i\sigma}^{\dagger}c_{j\sigma},$
(4)

where the coupling amplitudes are independent of quantum numbers i, j and q.
Hence, $g_{q}^{ij}=g_{ph}$. Total bosonic coupling constant is thus
$\alpha_{\mathrm{ph}}(\omega)=2\pi
g_{\mathrm{ph}}^{2}\rho_{\mathrm{b}}(\omega)$, where
$\rho_{\mathrm{b}}(\omega)$ represents the photon density of states.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3:  Analysis of charge stability(CS) diagrams with emphasis on the
(1,1) to (0,2) transition and the effect of various physical parameter
assumptions. (a) CS diagrams with clearly defined bias triangles can be
obtained in the high source-drain bias regime($V_{sd}=-1.0mV$). (b) At
relatively lower biases($V_{sd}=-0.2mV$) however, the triangles are not
formed. For both the CS diagrams, the electron occupancy in the dots is
marked. (c) Bias triangles magnified from the encircled part in (a) for (1,1)
to (0,2) transition. For $V_{sd}=-1mV$ there is no blockade. (d) Reversing the
bias($V_{sd}=-1mV$) leads to blockade and a four-fold reduction in current is
seen. (e) On turning off PAT, further reduction in current(by 10 times) is
noticed. (f) Current vs detuning plot obtained by taking cross-section along
the white lines in figures (c), (d), and (e). The blockade mechanism is
explained in Sec. III.1.

###  II.2 Fock subspaces of the Hamiltonian

In the presence of sequential tunneling alone, the Hamiltonian is a block
diagonal matrix, each block representing the total number of electrons, $N$ in
the DQD. Exact solutions to the eigenstates have emerged recently [20]. The
$N=1$ sub-matrix of the Hamiltonian in (1) is an $8\times 8$ matrix and
consequently has eight distinct eigenstates. These eigenstates can be
categorized into two groups: the bonding states and the anti-bonding states,
as follows.

$\displaystyle\ket{B_{\zeta}}$
$\displaystyle=\xi\ket{L_{\zeta}}+\eta\ket{R_{\zeta}}$ (5a)
$\displaystyle\ket{AB_{\zeta}}$
$\displaystyle=\xi\ket{L_{\zeta}}-\eta\ket{R_{\zeta}}.$ (5b)

In the above formulation, $B$ represents the bonding states and are lower in
energy, while $AB$ represents the anti-bonding states and are higher in
energy.
$\zeta\in\\{K\uparrow,K\downarrow,K^{\prime}\uparrow,K^{\prime}\downarrow\\}$.
The $N=2$ sub-matrix of the Hamiltonian is a $28\times 28$ matrix with twenty
eight eigenstates. Based on their contribution towards the current through the
DQD, the eigenstates can be classified into three broad categories as follows.

$\displaystyle\ket{C_{\zeta_{1}\zeta_{2}}}$
$\displaystyle=\alpha\left(\ket{L_{\zeta_{1}}R_{\zeta_{2}}}-\ket{L_{\zeta_{2}}R_{\zeta_{1}}}\right)$
$\displaystyle\hskip
56.9055pt+\beta\ket{L_{\zeta_{1}}L_{\zeta_{2}}}+\kappa\ket{R_{\zeta_{1}}R_{\zeta_{2}}}$
(6a) $\displaystyle\ket{D_{\zeta_{1}\zeta_{2}}}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{L_{\zeta_{1}}R_{\zeta_{2}}}+\ket{L_{\zeta_{2}}R_{\zeta_{1}}}\right)$
(6b) $\displaystyle\ket{P_{\zeta}}$ $\displaystyle=\ket{L_{\zeta}R_{\zeta}},$
(6c)

where
$\zeta\in\\{K\uparrow,K\downarrow,K^{\prime}\uparrow,K^{\prime}\downarrow\\}$
and the combination $\zeta_{1}\zeta_{2}\in\\{K\uparrow K\downarrow,\ K\uparrow
K^{\prime}\uparrow,\ K\uparrow K^{\prime}\downarrow,\ K\downarrow
K^{\prime}\uparrow,\ K\downarrow K^{\prime}\downarrow,\ K^{\prime}\uparrow
K^{\prime}\downarrow\\}$ and $\zeta_{1}$ and $\zeta_{2}$ are taken over all
possible unordered combinations of $\zeta$. We therefore have six possible
combinations of $\zeta_{1}\zeta_{2}$. Each of the states $C$ occurs threefold,
with three different sets of values of $\alpha$, $\beta$, and $\kappa$. Thus,
in total, we have eighteen $C$ states, six $D$ states, and four $P$ states.
The state labels $C$ and $D$ stand for "conducting" and "dark" respectively,
according to their role in the equation for the current [20].
In the presence of spin-flip tunneling, the states belonging to the $N=1$
Fock-space are superpositions of two bonding (or anti-bonding) states. The
bonding superposition takes the form

$\displaystyle a\ket{B_{\zeta_{1}}}+b\ket{B_{\zeta_{1}}}$ $\displaystyle=\xi
a\ket{L_{\zeta_{1}}}+\eta a\ket{R_{\zeta_{1}}}$ $\displaystyle\hskip
17.07164pt+\xi b\ket{L_{\zeta_{2}}}+\eta b\ket{R_{\zeta_{2}}}$

and the antibonding state follows accordingly. In the $N=2$ subspace, we
obtain superpositions of two $C$ or $D$ states in a similar fashion. The
eigenvalue variation of various states in the $N=2$ manifold with respect to
$B_{\parallel}$ and $B_{\perp}$ is shown in Figs. 2a(a) and (b) respectively.
The existence of superposition increases the number of states that satisfy the
transport selection rules [29, 30, 20] and provides multiple transition
pathways to the same state, each having a different transition energy. This
manifests as multiple resonance peaks within a bias triangle, each
corresponding to a particular transition.

(a)

(b)

(c)

Figure 4:  Occurrence of multiple excitations and multiple blockades in a
single triangle. (a) On including spin-flip tunnelling arising out of spin-
orbit coupling ($t_{so}$) and applying a small parallel magnetic field
($B_{||}=80\mu T$), we observe new transitions, that were forbidden
previously. These excited transitions lead to multiple resonances that
manifest as multiple current peaks in the triangles. (b) In the absence of
spin-flip tunneling, the transitions happen between closely packed, almost
degenerate states, showing no signatures of multiple excitations. (c) A case
of multiple blockades with two excitation peaks. Figures (d), (e), and (f) are
cross-sections of the CS diagrams (a), (b), and (c), respectively.

###  II.3 Transport formulation

Despite having a solid grasp of the theory concerning current blockades in
single-degree-of-freedom DQDs [44, 45, 13, 46, 47], the behavior of the
current in DQDs with multiple degrees of freedom has remained a challenge,
until recently [20]. The overall current in such a system arises from the
intricate interplay between the probabilities of occupation for each
eigenstate and the transition rates between them. To address intricate
problem, we expand upon the existing master equation, which is well-known in
literature [45]. The eigenstates are labelled $\ket{N,i}$, where $N$ denotes
the total electron occupancy and $i$ denotes the $i^{\text{th}}$ state in the
corresponding Fock state subspace with total electron occupancy $N$. We define
the quantity $P^{N}_{i}$ to denote the probability of occupancy of the state
$\ket{N,i}$ and $R^{L(R)}_{(N_{1},i)\rightarrow(N_{2},j)}$ to denote the rate
of transition from the state $\ket{N_{1},i}$ to the state $\ket{N_{2},j}$ by
injection or removal of an electron from the source(drain). Henceforth, we
shall use the index $\alpha$ for the source ($\alpha=L$) or the drain
($\alpha=R$). The probabilities $P^{N}_{i}$ evolve over time as

$\displaystyle\dot{P}^{N}_{i}$ $\displaystyle=\sum_{j}\left[R_{(N\pm
1,j)\rightarrow(N,i)}P^{N\pm 1}_{j}-R_{(N,i)\rightarrow(N\pm
1,j)}P^{N}_{i}\right],$ (7)

where

$\displaystyle
R_{(N_{1},i)\rightarrow(N_{2},j)}=\sum_{\alpha\in\\{L,R\\}}R^{\alpha}_{(N_{1},i)\rightarrow(N_{2},j)}.$
(8)

###  II.4 Inelastic processes

Besides resonant tunneling, inelastic processes and co-tunneling often play a
pivotal role in determining the current-voltage characteristics. In fact, the
origin of the base current in the bias triangles is attributed to inelastic
processes[48, 49, 50]. To describe relaxation processes, an additional rate
term $R^{(ph)}$ is introduced to cater to the bosonic degrees of freedom. In a
similar spirit of weak tunneling consideration, we consider weak coupling to
the bosonic bath. Hence, the contributions from first order $\alpha_{ph}$ only
contributes. The transition rates are given by

$\displaystyle
R_{(N_{1},i)\rightarrow(N_{1},j)}=\sum_{\alpha\in\\{L,R\\}}R^{\alpha(ph)}_{(N_{1},i)\rightarrow(N_{1},j)}.$
(9)

We notice that the transitions take place among many-body states of the same
manifold, i.e., within states with same occupancy number. Individual bosonic
rates $R^{\alpha(ph)}_{(N_{1},i)\rightarrow(N_{1},j)}$ can be elaborated as

$\displaystyle
R^{\alpha(ph)}_{(N_{1},i)\rightarrow(N_{1},j)}=b(E)|\langle(N_{1},i)|c^{\dagger}c|(N_{1},j)\rangle|^{2},$
(10)

where $b(x)=sgn(x)\alpha_{ph}(x)n_{b}(x)$, with the Bose function
$n_{b}(x)=\frac{1}{(exp(x/k_{B}T)-1)}$ and $c^{\dagger}$ represents
corresponding creation operator.

###  II.5 Machine learning methodology

Pauli blockades are essential for qubit initialisation and readout. We
demonstrate an application of the above transport formalism to generate
simulated data and train a deep learning network for automated detection of
Pauli blockade. We follow an approach [51] using a residual convolutional
neural network which is 18 layers deep [52].
The training data set is generated by randomly varying multiple device
parameters in the transport simulations. The simulations generate charge-
stability plots (in terms of $I$ vs $V_{G1}$, $V_{G2}$) and a rectangular gate
voltage window enclosing a pair of bias triangles is then extracted. These
images are labelled as PB (Pauli blockade) and No PB (no Pauli blockade)
according to the nature of their respective charge transitions. Bias triangles
corresponding to positive and negative source-drain voltages are concatenated
together and presented as training data to the deep learning classifier. The
presence of Pauli blockade is indicated by the suppression of current in one
bias direction as compared to the other.
The machine learning model is implemented in PyTorch with CUDA. All the
extracted images are converted to a grayscale format to simplify the training
process and remove unnecessary colour information. A test-train split of 30%
is used and the simulated data is augmented by random stretching, rotation,
shearing, and contrast changes. Additional augmentations from the Torchvision
library including elastic transformation and sharpness adjustment are also
applied to some inputs. Training is run for 20 epochs or till sufficient
accuracy is reached, and a mini-batch size of 256 is used. The input data is
downsized to about 68x60 pixels to make it similar to the resolution of
typical current-voltage measurements. Softmax function is used to obtain
respective probabilities for the two classes, Pauli blockade (PB) and no Pauli
blockade (no PB). The sample is assigned to the class with higher predicted
probability of the two (greater than 50%).

Figure 5:  A plot of the energy difference between the $N=1$ and $N=2$
eigenstates for transitions allowed by selection rules. The plot is
corresponding to the multiple excitations as shown in FIG 4a. At a detuning of
around $1.78$ meV, we observe the start of the first blockade, this is
precisely where a $\mathcal{D}$ transition enters the source-drain bias window
in the presence of a corresponding $\mathcal{C}$ transition. The inset shows a
zoom in for detuning in the range of multiple peaks. There is another peak
observed when a new $\mathcal{C}$ transition enters the bias window, at around
$1.83$ meV.

##  III Results

The charge stability diagram is an important illustrative in understanding the
effect of transport dynamics in the system. It is essentially a current
(conductance) map plotted with respect to the gate voltages used to tune the
DQD potential barriers. We carry out a variety of analyses on the data that we
obtain from our physics based simulations. Some of the primary parameters that
play a crucial role in obtaining the conductance plots are (i) source-drain
bias ($V_{sd}$), (ii) photon coupling strength $\alpha_{ph}$, and (iii) the
tunneling coefficients $t_{c}$, $t_{so}$, and $t_{vo}$. The other tunable
parameters kept fixed are the temperature $T_{c}$ (= 0.9K) and cross-
capacitance C (= 0.2).
We first discuss the dependence of bias triangles on primary system parameters
as illustrated in Fig. 3. Next, we describe the occurrence of multiple peaks
and blockades in Fig. 4. Finally, we present some results from the use of deep
learning on our simulated data for detecting the occurrence of Pauli blockade.

###  III.1 Origin of the bias triangles

Figures 3a and 3b show the difference in charge stability diagrams for the
high and low bias regimes. In Fig. 3a, distinct triangles are visible when the
bias is set high ($V_{sd}=-1.0mV$). However, as depicted in Fig. 3b, we
observe that the distinct bias triangles are not visible for low bias
($V_{sd}=-0.2mV$). Initially, only a single transition (between ground states)
is possible in the low-bias regime. Multiple conduction channels open up for
electrons on increasing the bias, leading to conduction between $N-1$, $N$,
and $N+1$ states, and the corresponding excitations as permitted via the
transport selection rules [20].
We focus on transitions featuring the (0,2) and the (1, 1) configuration. At a
certain bias of $V_{sd}=-1.0mV$, Fig. 3c depicts the bias triangle with the
current. The triangle’s base corresponds to the ground state-ground state (GS-
GS) transition between two different charge (number) configurations, leading
to larger current values. On reversing the bias, as seen in Fig. 3d, it can be
noticed that there is about a three to four fold drop in the current values.
The PAT processes play the role of inelastic scattering crucial in forming the
triangle. If the boson coupling is turned off (Fig. 3e), we notice a further
decrement in the current(about 100 times smaller) than in the non-blockaded
case. A comparison of the blockaded and non-blockaded cases along a cut-line
is presented in Fig. 3f. In agreement with the theory, the GS-GS transitions
stay the same in both forward and reverse biases with only a drop in
magnitude.

###  III.2 Multiple peaks and blockades

The addition of spin-flip terms in the Hamiltonian leads to further splitting
of the states giving rise to more resonant lines in the triangles, with a
small spin-coupled magnetic field ($B_{||}=80\mu T$), and a finite tunneling
coefficient ($t_{c}=10^{-4}$). Figure 4a depicts a case with the spin-flip
present ($t_{so}=10^{-4}$), while Fig. 4b is the same triangle with the spin-
flip term absent ($t_{so}=0$). The contrast can be observed in that for the
former case, there are many minute peaks near the base of the triangle, while
for the latter, a single GS-GS transition is observed.
Under a certain set conditions of $B_{\perp}$ (=3.2 T), we report the
signature of multiple blockades. Figure 4c depicts one such case in which two
peaks sandwich a large dip across the white cut line. It can be inferred that
there are thus three multi-blockaded regions in the triangle. It is essential
to note that though the multiple excitation cases and the multi-blockade cases
discussed here seem similar in terms of the effect, the cause for their origin
varies.
Multiple Pauli blockades arise when specific conditions are met that lead to a
negative differential resistance (NDR) [29, 20]. Multiple blockades arise
because of the spin and valley Zeeman effects, wherein, the eigenergies in the
Fock space depend upon the eigenstate, i.e., the degeneracy in energy between
the $C$ and $D$ states is broken. The master equation then yields the relation
between the energies, and in turn the gate voltages, that will lead to
multiple Pauli blockades.
Multiple excitations occur in the presence of effects such as strong spin-
orbit coupling or photon-qubit interactions, which leads to multiple current
peaks (resonances). Multiple resonances arise due to a marked difference in
the transition energies between the eigenstates in the $N$ and $N+1$ occupancy
Fock-states. In the presence of sequential tunneling alone, the eigenstates
formed in the $N=2$ Fock space are of types $C$ or $D$ only. The coefficients
within the $C$ or the $D$ state govern the occurence of multiple blockades. In
contrast, spin-flip tunnelling results in the eigenstates being a
superposition of two $C$ states or $D$ states. In such cases, the coefficients
with which two $C$ (or $D$) states mix dictate where a resonance line can be
observed. This leads to multiple resonance lines within a single bias
traingle, unlike multiple bias triangles in case of pure sequential tunneling
case. Figure 5 demonstrates the relationship between the energy eigenspectrum
and the occurrence of blockades.

Figure 6:  Predicted labels for the simulated test data using the deep neural
network. The classifier takes in a concatenated pair of bias triangle images
at positive and negative source-drain voltage for the same $V_{G1}$ and
$V_{G2}$ range. The values below the pair of images indicate the probability
predicted by the classifier for Pauli Blockade and no Pauli Blockade occurring
in the given bias triangles.

###  III.3 Identification of Pauli blockades using machine learning

Figure 6 presents a few samples from the test data set along with labels and
probabilities as predicted by the deep learning classifier. It can be seen
that the lower image of the concatenated pair corresponds to a positive
$V_{SD}$ and the upper image to a negative $V_{SD}$. The overall test accuracy
obtained for our implementation is 96.15%. Images in the figure reflect the
actual input resolution to the classifier after augmenting and downsizing
them. For the purpose of assigning predicted labels and calculating test
accuracy, the sample is assigned to the class with higher probability, as
mentioned before.
As a way to benchmark the performance of our neural network trained on the the
simulated data set, we also perform the classification on a limited set of
data from experimental results which have appeared in the literature recently
[53, 15, 54]. An accuracy of 94.44% (17 out of 18 test cases predicted
correctly) was obtained on the experimental test data set, thus showing the
robustness of our neural network classifier and the simulated training data.
This also re-validates the importance of using simulations in addition to
experimental data for training, when the availability of experimental data is
scarce [51, 55, 56, 57].

Figure 7: Detection of multiple peaks using the machine learning classifier.
Predicted probability for the presence or absence of multiple peaks is given
below the corresponding images.

The machine learning procedure as mentioned above can also be used to detect
and classify other kinds of features in the bias triangles, such as the
occurrence of multiple peaks. This is depicted in Fig. 7, where the upper row
represents bias triangles showing multiple peaks, while the images in the
lower row only has a single GS-GS transition. The two classes are multiple
peaks present (Multi) and multiple peaks not present (No Multi). The training
procedure used in this case is similar to that for Pauli blockade detection
and the test accuracy obtained on simulated data is 98.36%.

##  IV Conclusion

Focusing on DQD structures in the BLG platform, and the experimental results
available in literature, we first built theoretical models to capture the
intricate interplay between externally fed gate voltages and the physical
properties of the DQD setup, allowing us to effectively simulate Pauli
blockades. Employing the master equations for transport and considering
extrinsic factors such as electron-photon interactions, we thoroughly
investigated all potential occurrences of Pauli blockades. Notably, our
research revealed two remarkable phenomena: (i) the existence of multiple
resonances within a bias triangle, and (ii) the occurrence of multiple Pauli
blockades. Leveraging our model to train a machine learning algorithm, we
successfully developed an automated method for real-time detection of multiple
Pauli blockade regimes. Through numerical predictions and validations against
test data, we identified where and how many Pauli blockades are likely to
occur. We propose that our model can effectively detect the generic class of
Pauli blockades in practical experimental setups and hence serves as the
foundation for future experiments on qubits that utilize 2-D material
platforms.

## Acknowledgements

The author BM acknowledges the support by the Science and Engineering Research
Board (SERB), Government of India, Grant No. CRG/2021/003102 and Grant No.
MTR/2021/000388.

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