# Electrical spin manipulation in double SrTiO3/LaAlO3 quantum dots

B. Szafran<EMAIL_ADDRESS>AGH University of Krakow, Faculty of Physics
and Applied Computer Science, Al. Mickiewicza 30, 30-059 Krakow, Poland  M.
Zegrodnik  AGH University of Krakow, Academic Centre for Materials and
Nanotechnology, Al. Mickiewicza 30, 30-059 Krakow, Poland  M. P. Nowak  AGH
University of Krakow, Academic Centre for Materials and Nanotechnology, Al.
Mickiewicza 30, 30-059 Krakow, Poland  R. Citro  CNR-SPIN, Via Giovanni Paolo
II, 132, Fisciano (Sa), Italy  P. Wójcik<EMAIL_ADDRESS>AGH
University of Krakow, Faculty of Physics and Applied Computer Science, Al.
Mickiewicza 30, 30-059 Krakow, Poland

###### Abstract

The spin dynamics in two electron double quantum dots embedded in two
dimensional electron gas at the interface between SrTiO3 and LaAlO3 is studied
by an exact numerical solution of the time-dependent Schrödinger equation, in
the context of the electric dipole spin resonance experiment. Based on the
three band model of $3d$-electrons localized at Ti ions on the square lattice
we analyze in details the singlet-triplet transition induced by the AC
electric field, in the magnetic field range close to the avoided crossing
which appears as a result of the spin-orbit coupling. Our calculations show
that for symmetric double quantum dots the single photon spin-flip transitions
is prohibited due to the parity symmetry and the transition can occur only by
the higher order two-photon processes. For a weakly asymmetric system, when
the first order singlet-triplet transitions are released due to the parity
symmetry breaking, the spin-flip transition has a character of the Rabi
oscillations for a low electric field amplitude. As the amplitude is increased
the frequency of the transition is blueshifted (redshifted) for the magnetic
field below (above) the single-triplet avoided crossing. Interestingly, for a
sufficiently high magnetic field and high AC field amplitude the electric
field drives the system across the avoided crossing inducing the spin-flip by
the Landau-Zener-Stueckelberg-Majorana transitions with 100% spin flip
probability for a slow sweep. Finally, the optimization of the geometrical
parameters of the system with respect to the time of spin-flip of its fidelity
is also presented.

##  I introduction

The ability to control the spin of electrons confined in quantum dots (QDs)
has attracted growing interest in recent years, as it lies at the heart of
developments towards a scalable spin-based quantum computer Burkard  _et al._
(2023); Nadj-Perge  _et al._ (2010a). The effiency of spin manipulation is
entirely contingent upon the properties of the material platform, which
dictate the strength of the spin-orbit coupling (SOC) Giglberger  _et al._
(2007); Rashba and Efros (2003); Bychkov and Rashba (1984) or the hyperfine
interaction with the nuclear spin bath. The latter is a source of spin
decoherence Khaetskii  _et al._ (2002) limiting the usability of a specific
material platform in accordance with one of the Loss di Vincenzo criteria Loss
and DiVincenzo (1998), whereby the quantum system must remain coherent for
times much longer than the duration of elementary logic gates. For this
reason, silicon-based QDs have been intensively studied over the last years
due to their prolonged spin coherence time Mills  _et al._ (2022).

With respect to that, another promising but still unexplored platform for QD-
based spin qubits technology are transition metal oxide heterostructures, such
as SrTiO3/LaAlO3 (LAO/STO) interfaces Joshua  _et al._ (2012); Maniv  _et al._
(2015); Monteiro  _et al._ (2019); Pavlenko  _et al._ (2012); Biscaras  _et
al._ (2012). The electronic properties of the two-dimensional electron gas
(2DEG) formed at the LAO/STO interface is determined by the 3$d$-orbitals Diez
_et al._ (2015); Khalsa  _et al._ (2013); van Heeringen  _et al._ (2013);
Wójcik  _et al._ (2021); Zegrodnik and Wójcik (2020), implying that direct and
indirect decoherence stemming from interaction with the nuclear bath can be
significantly mitigated, as it is proportional to the square of the wave
function at the nuclei position Khaetskii  _et al._ (2002) \- this value
diminishes to zero for $3$d-electrons. This property, combined with the strong
spin-orbit interaction Caviglia  _et al._ (2010); Yin  _et al._ (2020), high
mobility Ohtomo and Hwang (2004), and susceptibility to electrostatic gates at
a level comparable to semiconducting materials Bjørlig  _et al._ (2020);
Thierschmann  _et al._ (2018); Guenevere  _et al._ (2017), initiates the
currently ongoing research on LAO/STO-based quantum dots as potential spin-
qubits with inherent scalability of 2D systems. The first experimental
realization of electrostatically defined LAO/STO QDs has been already
reported, yielding a Coulomb blockade diamond characteristic for a well-
defined confinement Bjørlig  _et al._ (2020); Thierschmann  _et al._ (2018).

The demonstration of electrical spin manipulation in these systems constitutes
the next milestone in spin-qubit realization based on the transition metal
oxide heterostructures. Significant progress in this field has been recently
achieved in electrostatically defined semiconducting QDs Schroer  _et al._
(2011), where the microwave magnetic field, previously used to control spin
dynamics Koppens  _et al._ (2006, 2008), has been replaced by the ac gate
voltage Nadj-Perge  _et al._ (2010a, b, 2012); Golovach  _et al._ (2006). In
this technique, called electric dipole spin resonance (EDSR), the microwave
potential applied to one of the gates generates a time-dependent potential
which, due to the momentum-dependent spin-orbit interaction Giglberger  _et
al._ (2007); Rashba and Efros (2003); Bychkov and Rashba (1984), hybridizes
the spin-up and spin-down states, leading to Rabi oscillations between the
Zeeman-split electronic states Koppens  _et al._ (2006). For two electrons,
the SOC hybridization induces level repulsion between the spin singlet and
triplet states, which is often used to determine the strength and direction of
the effective SOC field Stepanenko  _et al._ (2012). Recently, EDSR technique
applied to double quantum dots (DQDs) allowed for a demonstration of Pauli
spin blockade Nowack  _et al._ (2007); Nadj-Perge  _et al._ (2010a). In this
regime, the system initially set to the spin triplet state $T_{-}(1,1)$, with
tunneling from the left to the right QDs prohibited by selection rules,
transmits to the spin singlet state $(0,2)$, which is possible due to the SOC
generated by the ac field applied to one of the gates Nowak  _et al._ (2012).

Although spin manipulation in LAO/STO QDs has not yet been demonstrated
experimentally, theoretical studies allow us to gain a first insight into the
spin dynamics in these multiorbital systems. In our recent paper Szafran  _et
al._ (2023), we have discussed in detail the manipulation of electron spin in
a single QD in the context of EDSR experiment. We have shown that for a single
electron, the spin-flip in the ground state has the character of a Rabi
resonance, while for two electrons, the singlet-triplet transition is
forbidden by parity symmetry. The latter one is possible via a second-order,
two-photon process, which exhibits a two-state Rabi character for low ac field
amplitude.

Here we extend our study to the case of two-electrons in double quantum dots.
Based on the time dependent scheme we simulate the singlet-triplet transition
induced by the AC electric field. We consider the magnetic field range in the
nearest of the avoided crossing which appears as result of the SO coupling.
The calculated transition between singlet-triplet states, its characteristics,
i.e. duration time and fidelity, are discussed with respect to the system
asymmetry which is needed to induce the single photon transition by the parity
symmetry breaking, as well as the coupling strength between the dots. Finally,
the full symmetric system is also analyzed with respect to the higher order
singlet-triplet transitions.

The manuscript is organized as follows: in Sec. II we present a theoretical
model of double quantum dots in 2DEG at the (001) LAO/STO interface as well as
the scheme used for time-dependent two electron calculations. Sec. III
contains the analysis of electronic spectrum of a two-electron double quantum
dots as well as results of time dependent simulations (EDSR), finally summary
and conclusions are included in Sec. IV.

##  II Theoretical model

###  II.1 The single-electron problem

We consider 2DEG in the quantum well created at the (001) LAO/STO interface
Popović  _et al._ (2008); Pavlenko  _et al._ (2012). The system is
characterized by a significant atomic SO coupling and a strong Rashba
interaction due to the asymmetry of the vertical electric field at the
interface Diez  _et al._ (2015); van Heeringen  _et al._ (2013); Khalsa  _et
al._ (2013). We use the real space representation Szafran  _et al._ (2023) of
the tight-binding Hamiltonian Diez  _et al._ (2015) spanned by $3d$ orbitals
of Ti ions that are arranged in a square lattice. The Hamiltonian of the sytem
is given by

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\sum_{\mu,\nu}\hat{C}^{\dagger}_{\mu,\nu}(\hat{H}^{0}+\hat{H}_{SO}+\hat{H}_{B})\hat{C}_{\mu,\nu}+$
$\displaystyle\sum_{\mu,\nu}\hat{C}^{\dagger}_{\mu+1,\nu}\hat{H}^{x}\hat{C}_{\mu,\nu}+\sum_{\mu,\nu}\hat{C}^{\dagger}_{\mu,\nu+1}\hat{H}^{y}\hat{C}_{\mu,\nu}+$
$\displaystyle\sum_{\mu,\nu}\hat{C}^{\dagger}_{\mu+1,\nu-1}\hat{H}_{mix}\hat{C}_{\mu,\nu}-$
$\displaystyle\sum_{\mu,\nu}\hat{C}^{\dagger}_{\mu+1,\nu+1}\hat{H}_{mix}\hat{C}_{\mu,\nu}+h.c.,$

where
$\hat{C}_{\mu,\nu}=(\hat{c}_{\mu,\nu,xy}^{\uparrow},\hat{c}_{\mu,\nu,xy}^{\downarrow},\hat{c}_{\mu,\nu,xz}^{\uparrow},\hat{c}_{\mu,\nu,xz}^{\downarrow},\hat{c}_{\mu,\nu,yz}^{\uparrow},\hat{c}_{\mu,\nu,yz}^{\downarrow})^{T}$
is the vector of electron anihilation operators corresponding to states with
the spin $\sigma=\uparrow,\downarrow$ on the orbital $d_{xy},d_{xz},d_{yz}$ at
the position $(\mu,\nu)$. In Eq. (II.1), $\hat{H}^{0}$ accounts for the
splitting of the $3d$ orbitals degeneracy and the in-plane external potential
$V(\mathbf{r})$ defining the quantum dot confinement

$\displaystyle\hat{H}^{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}4t_{l}-\Delta_{E}&0&0\\\
0&2t_{l}+2t_{h}&0\\\
0&0&2t_{l}+2t_{h}\end{array}\right)\otimes\hat{\sigma}_{0}$ (5)
$\displaystyle+$ $\displaystyle\left(\begin{array}[]{ccc}V_{\mu,\nu}&0&0\\\
0&V_{\mu,\nu}&0\\\ 0&0&V_{\mu,\nu}\end{array}\right)\otimes\hat{\sigma}_{0}.$
(9)

The atomic spin-orbit coupling is defined by $\hat{H}_{SO}$ and is given by

$\hat{H}_{SO}=\frac{\Delta_{SO}}{3}\left(\begin{array}[]{ccc}0&i\sigma_{x}&-i\sigma_{y}\\\
-i\sigma_{x}&0&i\sigma_{z}\\\ i\sigma_{y}&-i\sigma_{z}&0\end{array}\right)\;,$
(10)

where the matrix corresponds to ${\bf L\cdot S}$ with the orbital angular
momentum represented in the $t_{2g}$-orbitals basis. Finally, the coupling of
the external magnetic field to the spin and atomic orbital momentum of
electrons is taken into account by the Hamiltonian

$\hat{H}_{B}=\mu_{B}(\mathbf{L}\otimes\sigma_{0}+g\mathbf{1}_{3\times
3}\otimes\mathbf{S})\cdot\mathbf{B}/\hbar,$ (11)

where $\mu_{B}$ is the Bohr magneton, $g$ is the Landé factor,
$\mathbf{S}=\hbar\boldsymbol{\sigma}/2$ with
$\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ and
$\mathbf{L}=(L_{x},L_{y},L_{z})$ with

$\begin{split}L_{x}&=\left(\begin{array}[]{ccc}0&i&0\\\ -i&0&0\\\
0&0&0\end{array}\right),L_{y}=\left(\begin{array}[]{ccc}0&0&-i\\\ 0&0&0\\\
i&0&0\end{array}\right),L_{z}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&i\\\
0&-i&0\end{array}\right).\end{split}$ (12)

The hopping elements in Eq. (II.1) consist of the kinetic term (spin-
conserving intersite hopping) and the Rashba SOC (mixing between the $d_{xy}$
and the two other orbitals) and take the form

$\hat{H}^{x}=\left(\begin{array}[]{ccc}-t_{l}&0&0\\\ 0&-t_{l}&0\\\
0&0&-t_{h}\end{array}\right)\otimes\hat{\sigma}_{0}+\frac{\Delta_{RSO}}{2}\left(\begin{array}[]{ccc}0&0&-1\\\
0&0&0\\\ 1&0&0\end{array}\right)\otimes\hat{\sigma}_{0}\;,$ (13)

and

$\hat{H}^{y}=\left(\begin{array}[]{ccc}-t_{l}&0&0\\\ 0&-t_{h}&0\\\
0&0&-t_{l}\end{array}\right)\otimes\hat{\sigma}_{0}+\frac{\Delta_{RSO}}{2}\left(\begin{array}[]{ccc}0&-1&0\\\
1&0&0\\\ 0&0&0\end{array}\right)\otimes\hat{\sigma}_{0}.$ (14)

Finally, the hybridization between $d_{xz}$, $d_{yz}$ orbitals are taken into
account by

$\hat{H}_{mix}=\frac{t_{d}}{2}\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right)\otimes\hat{\sigma}_{0}.$ (15)

In our simulations, we take on the tight-binding parameters $t_{l}=875\;$meV,
$t_{h}=40\;$meV, $t_{d}=40\;$meV, $\Delta_{E}=47\;$meV, after Ref. Zegrodnik
and Wójcik (2020); Diez  _et al._ (2015), the Landé factor $g=3$ Ruhman  _et
al._ (2014); Bjørlig  _et al._ (2020) and the SO coupling parameters
$\Delta_{SO}=10$ meV, $\Delta_{RSO}=20$ meV Caviglia  _et al._ (2010); Yin
_et al._ (2020).

The external magnetic field induces the Aharonov-Bohm phase shifts for the
electrons moving along the interface, that we account for using the Peierls
approach, i.e. multiplying the elements of the Hamiltonian corresponding to
the hopping between ions placed at points ${\bf
r}_{a}=(x_{\mu_{a}},y_{\nu_{a}})$ and ${\bf r}_{b}=(x_{\mu_{b}},y_{\nu_{b}})$
by a factor
$e^{{i\frac{e}{\hbar}\int_{\bf{r}_{a}}^{\bf{r}_{b}}}\vec{A}\cdot\vec{dl}}$,
where ${\vec{A}}$ is the vector potential taken in the symmetric form
${\vec{A}}=(-y/2,-x/2,0)B$ for $\vec{B}=(0,0,B)$.

The in-plane external potential that models the double quantum dot system is
assumed as a superposition of two Gaussian functions

$\displaystyle V(x,y)$ $\displaystyle=$ $\displaystyle-
V_{0}\exp\left(-\frac{1}{2}\left((x+\frac{s_{x}}{2})^{2}+y^{2}\right)/R^{2}\right)$
(16) $\displaystyle-$ $\displaystyle
V_{1}\exp\left(-\frac{1}{2}\left((x-\frac{s_{x}}{2})^{2}+y^{2}\right)/R^{2}\right),$

where $s_{x}$ is the distance between the minima of the separate functions
(centers of the left and right QDs), $V_{0}$ and $V_{1}$ define the depth of
each potential minimum and $R$ is the size parameter of the separate
Gaussians.

###  II.2 The two-electron problem

The two-electron problem is defined by the Hamiltonian

$\hat{H}_{2}=\hat{H}(1)+\hat{H}(2)+\frac{e^{2}}{4\pi\epsilon_{0}\epsilon
r_{12}},$ (17)

with the dielectric constant $\epsilon=100$ Gariglio  _et al._ (2015). The
problem is solved with the exact diagonalization method with the basis spanned
by the atomic orbitals

$\displaystyle\Psi_{q}(x,y,\sigma)$ $\displaystyle=$
$\displaystyle\sum_{j}a_{j}^{q}d_{j}(x,y,\sigma)$ (18) $\displaystyle=$
$\displaystyle\sum_{r_{j},o_{j},s_{j}}a_{j}^{q}d_{r_{j},o_{j}}(x,y)S_{s_{j}}(\sigma),$

where the summation runs over the position of ions $r_{j}$, orbitals $o_{j}$
on the ion and the $z$-component of the spin indexed by $s_{j}$ while $S$ is
the spin-up or spin-down eigenstate. In the sums $j$ is equivalent to the
triple of indeces $(r_{j},o_{j},s_{j})$ and $d_{r_{j},o_{j}}$ is one of the
$3d$ orbitals localized on the ion position $r_{j}$. Evaluation of the Coulomb
integrals has been performed in real space using the two-center approximation
Guerrero-Becerra and Rontani (2014) and integration over the $d$ orbitals
after Ref. Szafran  _et al._ (2023). In the calculations, we use up to 50
lowest-energy single-electron states that produce 1225 Slater determinants as
a basis for the two-electron problem.

###  II.3 Time stepping

We study the spin flips induced as singlet-triplet transitions between the
eigenstates of the two-electron Hamiltonian with an external periodic
perturbation of the potential $V_{AC}(x,t)=-eFx\sin(\nu t)$. The time
evolution is determined by integration of the time-dependent Hamiltonian
$\hat{H_{2}}(t)=\hat{H_{2}}+V_{AC}(x_{1},t)+V_{AC}(x_{2},t)$. The solution is
obtained in the basis of time-independent Hamiltonian eigenstates,
$\Psi(t)=\sum_{m}c_{m}(t)\exp\left(-iE_{m}t/\hbar\right)|m\rangle$ with
$\hat{H_{2}}|m\rangle=E_{m}|m\rangle$. Substituting this form of the wave
function to the Schrödinger equation and projecting the resulting equation on
the eigenstates $\langle n|$ one obtains a system of equations for $c_{n}(t)$,

$\displaystyle i\hbar c^{\prime}_{n}(t)=-eF\sum_{m}$ $\displaystyle
c_{m}(t)\exp\left[i\left(E_{n}-E_{m}\right)t/\hbar\right]$ (19)
$\displaystyle\times\sin(\nu t)\langle n|x|m\rangle,$

that we solved using the Crank-Nicolson time stepping with the two-electron
ground-state as the initial condition, i.e. $c_{n}(t=0)=\delta_{n,1}$.

(a) |  (b)
---|---
(c) |  (d)
Figure 1:  The charge density for the two-electron singlet (a,c) and two-
electron triplet (b,d) for $B=0$. Results for a symmetric system with
$V_{0}=V_{1}=50$ meV and the spacing parameter $s_{x}=29.64$ nm (a,b) and
$s_{x}=31.2$ nm (c,d).

##  III Results

###  III.1 The two-electron eigenstates

In Fig. 1 we plotted the charge density of the lowest-energy spin singlet
(a,c) and the lowest-energy spin triplet (b,d) for a symmetric quantum dots
with spacing between the Gaussian minima of $s_{x}=29.64$ nm (a,b) and
$s_{x}=31.2$ nm (c,d). The results for a stronger coupling [Fig. 1(a,b)] and
in particular the form of the densities in the barrier region indicate that
the singlet (triplet) charge density has a bonding (anti-bonding) character.
For weaker interdot coupling [Fig. 1(c,d)] the charge densities in both spin
states are more similar and the bonding (anti-bonding) character of the states
is less pronounced.

(a) |  (b)
---|---
(c) |  (d)
Figure 2:  The low-energy spectrum for the electron pair in a symmetric
quantum dot system $V_{0}=V_{1}=50$ meV for the spacing parameter
$s_{x}=29.64$ nm (a,b) and $s_{x}=31.2$ nm (c,d). The color of the line
indicates the z-component of the total spin. The singlet-triplet avoided
crossing is enlarged in (b) and (d).

(a)(b)(c)

Figure 3: (a) The exchange energy, i.e. the energy difference between the
spin-triplet and singlet states for $B=0$ and $s_{x}=29.64$ nm (black line) as
well as $s_{x}=31.2$ nm (red line). The charge localization in the left (red
line) and right dot (blue line) in the singlet (solid line) and the triplet
state (dotted line) for $s_{x}=29.64$nm and $s_{x}=31.2$ nm are displayed in
(b) and (c), respectively.

The two-electron energy spectra for the considered DQDs are displayed in Fig.
2(a,b) for $s_{x}=29.64$ nm and in Fig. 2(c,d) for $s_{x}=31.2$ nm. At $B=0$
the ground-state is spin-singlet for any interdot distance $s_{x}$. The
singlet-triplet energy splitting (the exchange energy) at $B=0$ varies
strongly with $s_{x}$, e.g. the interdot tunnel coupling, and for
$s_{x}=29.64$ nm it is equal to 185.5 $\mu$eV, decreasing to only 33.1 $\mu$eV
for $s_{x}=31.2$ nm. The applied external magnetic field promotes the spin-
polarized triplet for the ground-state. The singlet-triplet ground-state
transformation occurs via an avoided crossing [Fig. 2(b,d)] that is opened by
the spin-orbit coupling. The position of the avoided crossing at the magnetic
field scale is determined by the exchange energy at $B=0$. For $s_{x}=29.64$
nm ($s_{x}=31.2$ nm) the center of the avoided crossing occurs at 1.05 T (0.19
T) with the width (e.g. minimal energy spacing between the anticrossing energy
levels) equal to 12.3 $\mu$eV for the stronger coupling and only 2.68 $\mu$eV
for the weaker coupling.

To account for an intrinsic or intentional asymmetry in the confinement
potential of the double quantum dot system we introduce unequal depths of the
Gaussian quantum wells forming the artificial molecule. In Fig. 3(a) we plot
the exchange energy at $B=0$ as a function of the right quantum dot potential
calculated for the left potential set at $V_{0}=50$ meV and $s_{x}=29.64$nm
(black line) as well as $s_{x}=31.2$ nm (red line). The exchange energy is
minimal for the symmetric system. For an asymmetric double dot in the spin-
singlet state both the electrons tend to occupy the deeper quantum dot [Fig.
3(b,c)] which lowers the singlet energy with respect to the spin triplet for
which the double occupancy of the lowest orbital is forbidden by the Pauli
exclusion principle. In contrast, the reaction of the charge localization to
the asymmetry in the triplet state is weak [Fig. 3(b,c)]. Note, moreover, that
a small asymmetry $V_{1}\neq V_{0}$ does not introduce a significant charge
redistribution to the charge in the spin-singlet state which is a result of
the electron-electron interaction. Only for $V_{1}\leq 49$nm or $V_{1}\geq 51$
nm the electrons in the singlet start to occupy the deeper dot. For the weaker
interdot coupling the transition to the deeper dot is more abrupt [cf. Fig.
3(b) and (c)].

---
Figure 4: Same as Fig. 2(a) with the color of the lines indicates the $\Pi$
parity – the negative one is plotted in blue, and the positive one in red.

For a symmetric potential the single-electron Hamiltonian eigenstates are also
eigenstates parity operator $\Pi=\mathrm{diag}[P,-P,-P,P,-P,P]$, where $P$ is
the scalar parity operator $P\psi(r)=\psi(-r)$. The lowest-energy singlet and
the lowest spin-polarized triplet that participate in the avoided crossing
have the same – negative – $\Pi$ parity – see Fig. 4. The same symmetry of the
lowest singlet and triplet states has a direct consequence for the spin
transitions driven by AC electric field, namely the dipole transition matrix
elements between the two lowest-energy levels vanish so the first-order
(single-photon) transitions is forbidden and only the second-order processes
(two-photon) can occur. Similar behavior has been observed for two electrons
in single quantum dot and discussed in details in our previous paper Szafran
_et al._ (2023).

###  III.2 EDSR in weakly asymmetric system

Let us first study the case of weakly asymmetric system with $V_{1}=51$ meV.
For small asymmetry the electron charge is still more or less evenly
distributed over the dots, but the asymmetry lifts the parity selection rule
that forbids the single-photon, first-order transitions.

(a)
---

(b)

Figure 5:  (a) Same as Fig. 2(b) only for the right quantum dot made deeper by
1 meV, e.g. $V_{1}=51$ meV. (b) The average z-component of the total spin –
with respect to the right vertical axis – in the ground-state (in blue) and
the first excited state (in red). The black line shows the dipole matrix
element between the two lowest-energy states (referred to the left vertical
axis).

Fig. 5(a) shows the spectrum for $s_{x}=29.64$ nm near the singlet-triplet
avoided crossing that is shifted to higher magnetic field by the asymmetry due
to the increased exchange energy [Fig. 3(a)] (to $B=1.22$ T from $1.05$ T in
the $V_{0}=V_{1}$ case). The width of the avoided crossing is also enlarged:
from 12.3$~{}\mu$eV (for $V_{1}=50$ meV) to 13.8 $\mu$eV (for $V_{1}=51$ meV).
The essential quantity for the first-order spin transitions is the dipole
matrix element $\langle\psi_{n}|x|\psi_{m}\rangle$ between the two-lowest
energy levels which is plotted by the black curve in Fig. 5(b). For an ideally
symmetric system $V_{1}=V_{0}$ a similar singlet-triplet avoided crossing of
energy levels is obtained with the interchange of the average values of the
spin between the two lowest-energy levels, but the transition dipole matrix
element is zero. For the system with lifted parity symmetry, the matrix
element reaches its maximum in the center of the avoided crossing which allows
the first order transition possible.

We studied the transitions from the ground-state to the first-excited state
near the single-triplet avoided crossing induced by the external AC sine
electric field, with the initial state set to the ground-state. The maximal
occupancy of the first-excited state for the simulation lasting $5$ ns is
given in Fig. 6 for $B=0.434$ T (a), $B=0.94$ T (b), $B=1.5$ T (c), and $B=2$
T (d) with the amplitude of the AC field increasing in steps of 5 $\mu$V/nm.

(a) |  (b)
---|---
(c) |  (d)
Figure 6:  The results of simulation of the time evolution for the weakly
asymmetric ($V_{1}=51$ meV) double quantum dot system with spacing of
$s_{x}=29.64$ nm in an AC external electric field $eFx\sin(\nu t)$. The
subsequent lines in the upper part of the plots correspond to the AC field
amplitude increased by 5 $\mu$V/nm offset by +1 each along the vertical axis.
The simulation lasts $5$ ns and starts from the ground-state. The lines show
the maximal occupancy of the first excited state. The panels correspond to
$B=0.434$ T (a), $B=0.94$ T (b), $B=1.5$ T (c) and $B=2$ T (d).

For the magnetic fields in Fig. 6(a,b) and Fig. 6(c,d) the ground-state is
singlet and triplet, respectively. At the amplitude of 5 $\mu$eV the largest
peak corresponds to the singlet-triplet transition that occurs at the driving
energy near the singlet-triplet energy difference. For $B=0.94$ T and $1.5$ T,
where the dipole matrix element is large, at the lowest-amplitude, we also
resolve the peaks of higher-order transitions at half and $1/3$rd of the
frequency for the single-photon transition processes. As the AC field
amplitude is increased the frequency of the spin-flip transition [Fig. 6(a,b)]
for the magnetic field below the single-triplet avoided crossing is
blueshifted at higher AC field amplitudes. On the other hand the frequency for
the spin transition for the magnetic field above the single-triplet avoided
crossing gets redshifted for high AC amplitudes [Fig. 6(c,d)]. In both cases,
in an intense AC field, we find an effective upbeat of the triplet energy with
respect to the singlet. In Fig. 6(c) we find that the regular transition
spectrum gets deteriorated at higher frequency with the disappearance of the
single-photon peak. For $B=0.94$ T at higher amplitude the AC field drives the
system across the singlet-triplet avoided crossing so that the process for the
spin-flip acquires the Landau-Zener-Stueckelberg-Majorana (LZSM) character
Ivakhnenko  _et al._ (2023).

(a)
---
(b)
(c)
Figure 7:  The lowest singlet and triplet energy levels for $B=0.94$ T (a),
$1.5$ T (b) and $2$ T (c) for the weakly asymmetric system and a constant
electric field $+eFx$. In the considered field range a singlet-triplet avoided
crossing opens for $B=1.5$ T and $2$ T that changes. In the AC simulation the
avoided crossing changes the spin-flip probability at low frequency from the
multiphoton transitions to the LZSM interference – see Fig. 6(c,d) and Fig. 8
at higher amplitude.  Figure 8: Same as Fig. 6(d) only as a function of the
inverse of the driving energy.

Fig. 7 shows the two lowest-energy states for the double dot system with a
time-independent electric field of $eFx$ added to the potential. The avoided
crossing for $B=1.5$ T [Fig. 7(b)] is reached at $|F|\simeq 20~{}\mu$V. Note,
that in AC field with the amplitude of $F\simeq 20~{}\mu$V/nm the maximal
occupancy of the singlet-state is no longer zero at low $h\nu$ limit [cf. Fig.
6(c)]. At higher amplitude the low-frequency limit in Fig. 6(c) corresponds to
an adiabatic sweep of the potential across the singlet-triplet avoided
crossing with 100% spin flip probability for a slow sweep. For nonzero AC
frequency the LZSM interference between the contributions from both the states
appear with a diabatic and an adiabatic components Ivakhnenko  _et al._ (2023)
generating the interference comb visible in Fig. 6(c) at higher amplitude and
low frequency. For $B=2$ T we reach a similar point at $F\simeq 30~{}\mu$V/nm
[cf. Fig. 6(d) and Fig. 7(c)]. The change of the character of the triplet-
singlet transition is better illustrated by the transfer probability plotted
as a inverse of the driving energy presented in Fig. 8 for $B=2$ T. As the
amplitude increases from 5$~{}\mu$V/nm to 25$~{}\mu$V/nm higher-order peaks
corresponding to multiphoton transitions appear but the corresponding peaks
fall below 100% at large $1/h\nu$. For $30~{}\mu$V/nm and above, the
transition probability peaks acquire the height of 100% that does not fall
below at large $1/h\nu$, and appear periodically on the $1/h\nu$ scale, which
are characteristics of the LZSM interference Ivakhnenko  _et al._ (2023).

(a) |  (b) |  (c)
---|---|---
Figure 9:  The results for weakly asymmetric double quantum dot with
$V_{1}=51$ meV and $s_{x}=29.64$ nm. (a) shows the position of the spin-flip
singlet-triplet resonance in the AC electric field as a function of its
amplitude for selected values of the external magnetic field. (b) The spin-
flip time as a function of the amplitude in the first-order transition. The
line for $1.5$ T is interrupted since the first order peak can not longer be
resolved at large $F$. (c) The maximal share of the states other than the two
lowest-energy ones in the time evolution of the wave function for the system
driven by the external AC field.

Figure 9(a) shows the position of the first order transition on the energy
scale $E_{r}=h\nu_{r}$ as a function of the AC amplitude. Note that, the
magnetic fields considered in Figs. 9(a) as well as 6 were chosen to produce
equal singlet-triplet energy splitting before and after the avoided crossings,
hence the coalescent lines at the limit of zero amplitude. The frequency
shifts before and after the singlet-triplet avoided crossing have opposite
signs but similar magnitudes. The line for $1.5$ T is interrupted above 35
$\mu$V/nm, since the resonance line can be hardly recognized in the spectrum
[cf. Fig. 6(c)].

A strong AC amplitude induces appearance of other states than the lowest-
energy couple in the time evolution. The maximal share of the higher energy
states as a function of the amplitude is given in Fig. 9(c) as the leakage of
the wave function outside the lowest-energy singlet and triplet. The
dependence was determined at the first-order resonant transition and as we can
see the leakage on the amplitude is similar for all the magnetic fields
considered. At the amplitude of $20~{}\mu$V/nm the maximal share of the higher
energy states is about 2.5%.

The dependence of the spin-flip time on the amplitude for the first-order
transition is plotted in Fig. 9(b). The minimal flip time of about 245 ps for
$B=1.5$ T is obtained at the amplitude of 25 $\mu$V/nm. For even stronger
amplitudes the flip time no longer decreases which is related to the potential
sweeps across the singlet-triplet avoided crossing and the leakage of the wave
function to higher energy states for which the evolution no longer have a two-
state Rabi resonance character Khomitsky  _et al._ (2020).

###  III.3 Optimization of the dipole matrix moment

In the Rabi oscillations regime the rate of the first-order spin-flip
transition is proportional to the value of the dipole matrix element. The
latter is strictly zero for an ideally symmetric double quantum dots. In the
precedent section we described the results obtained in a system with a weak
asymmetry but the value of the matrix element can be optimized with the system
parameters.

(a)
---
(b)
(c)
Figure 10:  Same as Fig. 5(b) but for $s_{x}=31.2$ nm and $V_{1}=51$ meV (a),
$V_{1}=54$ meV (b) and $60$ meV (c).

The dipole matrix element for $s_{x}=29.64$ nm is displayed in Fig. 5(b). In a
system with larger interdot barrier ($s_{x}=31.2$ nm), but still with the
small asymmetry $V_{1}=51$ meV as in Fig. 5(b), the singlet-triplet avoided
crossing occurs at lower magnetic field of $B=0.23$ T [cf. Fig. 10(a)] than
for the stronger interdot coupling due to a reduction of the exchange energy
[Fig. 3(a)]. The width of the singlet-triplet avoided crossing is 3.2 $\mu$eV
for $s_{x}=31.2$ nm. The maximal value of the dipole matrix element in the
center of the avoided crossing is about $0.323$ nm, e.g. about 3 times lower
than for $s_{x}=29.64$ nm [Fig. 5(b)].

The position of the singlet-triplet avoided crossing and the maximal value of
the dipole matrix element strongly depends on the asymmetry of the confinement
potential. In Fig. 10(b) and (c) we plotted the average spins and the dipole
matrix element near the singlet-triplet avoided-crossing for $s_{x}=31.2$ nm
with $V_{1}=54$ meV (b) and $V_{1}=60$ meV (c). For $V_{1}=54$ meV [Fig.
10(b)] the maximal value of the dipole matrix element is 40 times larger than
for the small asymmetry $V_{1}=51$ meV [Fig. 10(a)] and about 4 times larger
than for the large asymmetry $V_{1}=60$ meV [Fig.10(c)].

The solid lines in Fig. 11(a) show the absolute value of the dipole element at
the center of the singlet-triplet avoided crossing as a function of the right-
dot potential $V_{1}$. The asymmetry of the confinement potential pushes the
avoided crossing to higher values of the magnetic field (dashed lines referred
to the right axis). The dependence of the matrix element on the asymmetry is
non-monotonic which is related to the charge distribution in the singlet and
triplet states. Fig. 11(b) shows the charge in the left and right quantum dot
calculated for $B=10$ T – near the value of the magnetic field for which the
maximum of the dipole matrix element is achieved in Fig. 11(a) for $V_{1}=54$
meV. The gap in the lines of Fig. 11(b) corresponds to the region of the
singlet-triplet avoided crossing as a function of $V_{1}$ that we skipped for
the clarity of the plot. At the left (right) hand side of the avoided crossing
the ground-state for $B=10$ T is the spin triplet (singlet).

(a)
---
(b)
Figure 11:  (a) The solid lines show the dipole matrix element for the
transition between the two lowest-energy states at the center of the singlet-
triplet avoided crossing. The dashed lines show the position of the avoided
crossing (right vertical scale). The red and black lines correspond to the
$s_{x}$ parameter equal to $31.2$ nm and $29.64$ nm, respectively. (b) The
charge localized in the right and left quantum dots for $s_{x}=31.2$ nm in the
singlet (red lines) and triplet (blue lines) for $B=10$ T. The gap in the
lines was left for the avoided crossing of the singlet and triplet energy
levels for the clarity of the plot. The ground-state is singlet (triplet) at
the right(left)-hand side of the figure.

Comparing Fig. 11(a) and Fig. 11(b) we can conclude that the maximal value of
the dipole matrix element is achieved for $V_{1}$ which on the one hand
localizes the singlet entirely in the deeper dot but for which the triplet
charge distribution is still evenly distributed in both dots. In the region of
the avoided crossing the two-lowest energy states have hybridized wave
functions that due to the contribution of the triplet are delocalized over
both dots which produces a large value of the dipole matrix element. For large
$V_{1}$, on the other hand, both singlet and triplet states are confined
within a single deeper quantum dot with potential perturbed by the potential
of the left Gaussian. Due to this perturbation the dipole matrix element is
non-zero. Since the system is localized in a single dot the matrix element
does not achieve values as large as for the delocalized triplet state.
Comparing Fig. 3(b) and Fig. 11(a) we also notice that at higher $B$ the
triplet state is more easily localized in the deeper dot: for $V_{1}=60$meV
the charge in the right dot is $1.2e$ at B=0, but already $1.7e$ at $B=10$ T.

Figure 12: Singlet-triplet transition time for the double-dot system with
$s_{x}=31.2$ nm and $V_{1}=54$ meV driven by the AC field of $eFx\sin(\nu t)$
for three values of the magnetic field corresponding to different values of
the dipole moment, see Fig. 10(b).

For the studies of the transition times we chose $V_{1}=54$ meV – near the
maximum of the dipole matrix element [Fig. 11(a)] and three values of the
magnetic field after the singlet-triplet avoided crossing, where the ground-
state is the spin triplet – $7.9$ T, $8$ T and $9$ T – see Fig. 10(b). The
singlet-triplet transition times as functions of the AC amplitude is given in
Fig. 12. The transition times reach minima for lower values of the AC field
amplitude than in Fig. 9(b) and start to grow for the $F$ values that exceeds
the one for the LZSM transition regime. The LZSM regime is open for the
amplitudes of $F=1.95~{}\mu$V, 2.7$~{}\mu$V and 9.5 $\mu$V/nm for $B=7.89$ T,
$8$ T and $9$ T, respectively. Note, that for each value of the magnetic field
the minimum spin flip time of about 330 ps is achieved for some value of the
AC field amplitude. Since the amplitudes of the AC field near the minimal spin
flip time are smaller than in Fig. 9(b), so are the wave function leakage
values near the minimum which are only 0.5%, 0.02%, 0.0043% for
$F=17.5~{}\mu$V/nm, $5~{}\mu$V/nm, and $2.43~{}\mu$V/nm, corresponding to
$B=7.89$ T, $8$ T and $9$ T, respectively.

###  III.4 Spin flips for an ideally symmetric system

For completeness we present the results for an ideally symmetric system, for
which the direct spin-flip transition is forbidden by the selection rule. Fig.
13 shows the probability of the spin-flip-singlet for $s_{x}=29.64$ nm and the
external magnetic field set at $2$ T, for a simulation lasting $5$ ns and
starting from the ground-state triplet as the initial state. The subsequent
lines at the upper part of Fig. 13 correspond to a growing amplitude of the AC
field. The most pronounced peak that is observed in Fig. 13 at weak AC field
amplitude occurs at roughly half the singlet-triplet energy splitting that at
the field of $2$ T is equal to $\Delta E_{ST}=0.1663$ meV [cf. Fig. 2(b)]. The
spin-flip occurs for AC frequency set at this peak by a two-photon second-
order process. Also, no third-order (three-photon) peak at 1/3 of $\Delta
E_{ST}$ is observed. For higher amplitude the position of this peak gets red-
shifted and the peaks corresponding to even denominators, e.g. 1/4 and 1/6 of
$\Delta E_{ST}$ are formed.

---
Figure 13: Same as Fig. 6 only for a symmertic system with $V_{0}=V_{1}=50$
meV at $B=2$ T.

The study of the AC driven spin-flips in a symmetrix system with $s_{x}=29.64$
nm is summarized in Fig. 14 for two values of the external magnetic field
$B=1.5$ T and $B=2$ T, both corresponding to the ground-state triplet. The
position of the resonant second-order transition on the energy scale is
redshifted with the amplitude [Fig. 14(a)]. In the studied range of the AC
amplitudes – the spin-flip time is a decreasing function of the amplitude
[Fig. 14(b)]. For $1.5$ T the LZSM regime opens at 45 $\mu$eV/nm, and for $2$
T at still higher amplitudes which are outside the range studies in Fig. 14.

(a) |  (b)
---|---
Figure 14:  (a) The position of the second-order (two-photon) singlet-triplet
resonance in the AC field of $eFx\sin(\nu t)$ for two values of the magnetic
field. For both $B$ values the ground-state of the system is the spin triplet.
(b) The spin-flip transition time at the singlet-triplet resonance as a
function of the driving field amplitude. Results for a symmetric double
quantum dots with $s_{x}=29.64$ nm and $V_{0}=V_{1}=50$ meV.

##  IV Summary and conclusions

We have studied two electrons confined in a lateral double quantum dot defined
within the two-dimensional electron gas on the (001)-oriented LAO/STO surface.
Utilizing the three-band model of $3d$-electrons localized at Ti ions on a
square lattice, we have analyzed the energy spectrum of the system, paying
particular attention to the singlet-triplet avoided crossing induced by the SO
interaction. The calculated eigenstates have been employed as initial
conditions for time-dependent calculations. In the magnetic field range close
to the avoided crossing, the spin-flip singlet-triplet transition induced by
an AC electric field has been analyzed in details in the context of the
electric dipole spin resonance experiment.

We have demonstrated that for a fully symmetric system the first order
singlet-triplet transition are forbidden due to the parity symmetry of the
Hamiltonian which results in zeroing of the dipole matrix elements. To induce
the spin-flip transition we introduce the asymmetry in the confinement
potential in the form of unequal depths of the Gaussian quantum wells. Our
simulations for a weakly asymmetric system show that the spin-flip singlet-
triplet transition has the character of the Rabi oscillations for a low AC
field amplitude, with minimal flip time of about 245 ps. Interestingly, when
we increase the AC filed amplitude the frequency of the transition is
blueshifted (redshifted) for the magnetic field below (above) the single-
triplet avoided crossing. Moreover, we have noticed that for a magnetic field
above the anticrossing the regular transition spectrum is highly deteriorated
at a low frequency regime and the high AC field amplitude. This behavior has
been explained as resulting from the Landau-Zener-Stueckelberg-Majorana
transitions between the ground and first excited states which occur in the
system when the electric field is large enough to drive the system across the
avoided crossing.

The optimization of system geometrical parameters shows that the appropriate
potential asymmetry between the dots highly reduces the leakage of transition
to the higher states which can reach the value of about 0.05% with the spin
flip time at the order of $330$ ps. Finally our simulations for a fully
symmetric system provide insight into the possibility of the spin-flip
transition by the multiphoton processes. The singlet-triplet transition
studied here will be relevant for future EDSR experiments aimed at reading the
spin state of the dot and paving the way to a spin-orbit qubit at the oxides
interface.

##  V ACKNOWLEDGEMENT

This work is financed by the Horizon Europe EIC Pathfinder under the grant
IQARO number 101115190 titled "Spin-orbitronic quantum bits in reconfigurable
2D-oxides" and partly supported by program „Excellence initiative – research
university” for the AGH University. Computing infrastructure PLGrid (HPC
Centers: ACK Cyfronet AGH) within computational grant no. PLG/2023/016317 was
used.

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