# Photo-assisted spin transport in double quantum dots with spin-orbit
interaction

David Fernández-Fernández1, Jordi Picó-Cortés1,2, Sergio Vela Liñán3 and
Gloria Platero1 1 Instituto de Ciencia de Materiales de Madrid ICMM-CSIC,
28049 Madrid, Spain  2 Institute for Theoretical Physics, University of
Regensburg, 93040 Regensburg, Germany  3 Universidad Complutense de Madrid,
Facultad de Ciencias Físicas, 28040 Madrid, Spain

###### Abstract

We investigate the effect of spin-orbit interaction on the intra- and interdot
particle dynamics of a double quantum dot under ac electric fields. The former
is modeled as an effective ac magnetic field that produces electric-dipole
spin resonance transitions, while the latter is introduced via spin-flip
tunneling amplitudes. We observe the appearance of non-trivial spin-polarized
dark states, arising from an ac-induced interference between photo-assisted
spin-conserving and spin-flip tunneling processes. These dark states can be
employed to precisely measure the spin-orbit coupling in quantum dot systems.
Furthermore, we show that the interplay between photo-assisted transitions and
spin-flip tunneling allows the system to operate as a highly tunable spin
filter. Finally, we investigate the operation of the system as a resonant
flopping-mode qubit for arbitrary ac voltage amplitudes, allowing for high
tunability and enhanced qubit control possibilities.

  *

Keywords: semiconductor quantum dots, spin-orbit coupling, quantum transport,
spin qubits, dark states, ac-driving dynamics and transport, electron dipole
spin resonance

##  1 Introduction

Semiconductor spin qubits are among the most promising platforms for the
implementation of quantum computing [1, 2, 3, 4, 5, 6, 7, 8]. Some of their
main advantages are the long coherence times and the promise of high
scalability to achieve the large number of qubits needed for the realization
of quantum algorithms [9, 10, 11, 12, 13, 14]. In these systems, qubits are
encoded in the spin states of electrons or holes localized in quantum dots
(QDs). Manipulation of the qubit states can be performed employing electron
spin resonance (ESR) [15, 16, 17, 18] by applying an oscillating magnetic
field. However, the localization of the ac magnetic fields needed to address
individual dots is experimentally challenging [15]. To overcome this
limitation, an effective ac magnetic field is generated by electrically
driving the particle in a material with strong spin-orbit coupling (SOC) [19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], a magnetic field gradient [31,
32, 33], a spatially dependent hyperfine field [34, 35], or modulated
anisotropies of the effective $g$-factor [36, 37, 38]. All these alternatives
allow for electrical manipulation via the so-called electric-dipole spin
resonance (EDSR), which in recent years has been widely employed to obtain
high-fidelity one- and two-qubit gates [39, 40, 41, 42, 43, 44, 45, 46, 47].

In a double quantum dot (DQD), the particle (electron or hole) is delocalized
between the two sites, giving rise to molecular-like orbitals. The addition of
an ac electric or magnetic field results in unusual properties that arise from
quantum coherence and, in particular, from quantum interferences. Some of the
most relevant examples in QD systems are charge and spin dynamical
localization [48, 49, 50, 51], spin filtering [52, 53, 54], or long-range
transfer mediated by photo-assisted tunneling (PAT) [55, 56, 57]. These
effects can be analyzed using Floquet theory [58, 59, 60], which is an
excellent tool for addressing time-periodic Hamiltonians. Although all of
these phenomena have been widely studied both theoretically and
experimentally, the consequences of adding a strong SOC have only begun to be
extensively analyzed in recent years [61, 62, 63, 64]. In particular, the
transport signatures under both strong SOC and large ac voltage amplitudes
have not been widely investigated.

Transport through such a system is characterized by a spin-polarized current
due to the finite probability of spin-flip tunneling between states of
opposite spin in different QDs. In the presence of an ac electric field, these
transitions can occur with the absorption or emission of one or more photons.
We analyze the current and spin polarization through the system, including the
effect of excited states via an effective magnetic field. Furthermore, within
a certain parameter configuration, we find a set of non-trivial dark states
(DS) in which the interference between PAT spin-conserving and spin-flip
transitions occurs. These processes yield a complex and nonlinear current
output with remarkable features, including a potential read-out mechanism of
the SOC. Lastly, we consider the possibility of employing the setup as a
flopping-mode qubit, in which virtual transitions between the QDs in the
presence of a strong SOC allow for fully coherent manipulation of the spin.
The flopping-mode qubit has recently gained attention as a promising platform
for quantum computing in solid-state systems [65, 66, 67].

##  2 Theoretical framework

Figure 1: Schematic picture of a DQD in the x-y plane, connected to
unpolarized contacts. Particles can enter through the left lead and exit to
the right one with tunneling rates $\Gamma_{\mathrm{S,L}}$ and
$\Gamma_{\mathrm{D,R}}$, respectively. A uniform magnetic field is applied
pointing in the z-direction. The particle can tunnel between QDs following a
spin-conserving path $\tau_{0}$. Additionally, due to the SOC, a spin-flip
tunneling $\tau_{\mathrm{sf}}$ is also present. Finally, an ac electric field
is applied to the left gate $\epsilon_{\mathrm{L}}(t)$, which in combination
with the SOC gives rise to the OME term $\beta(t)$ (see text).

We consider the following Hamiltonian for the DQD ($\hbar=1$)

$\displaystyle\hat{H}\left(t\right)=$
$\displaystyle\hat{H}_{0}(t)+\hat{H}_{1}(t),$ (1a)
$\displaystyle\hat{H}_{0}(t)=$
$\displaystyle\sum_{\eta;\sigma}\epsilon_{\eta}(t)\hat{d}_{\eta,\sigma}^{\dagger}\hat{d}_{\eta,\sigma}+\sum_{\eta}\frac{E_{z}}{2}\hat{\sigma}_{z,\eta}$
$\displaystyle+\sum_{\eta\neq\eta^{\prime};\sigma,\sigma^{\prime}}\tau_{\eta,\sigma;\eta^{\prime}\sigma^{\prime}}\left(\hat{d}_{\eta,\sigma}^{\dagger}\hat{d}_{\eta^{\prime},\sigma^{\prime}}+h.c.\right),$
$\displaystyle\hat{H}_{1}(t)=$
$\displaystyle\sum_{\eta}\frac{\beta_{\eta}(t)}{2}\hat{\sigma}_{x,\eta},$ (1c)

where $\hat{d}_{\eta,\sigma}$ ($\hat{d}_{\eta,\sigma}^{\dagger}$) is the
annihilation (creation) operator at site
$\eta\in\left\\{\mathrm{L,R}\right\\}$, with spin
$\sigma\in\left\\{\uparrow,\downarrow\right\\}$. The first term in
$\hat{H}_{0}$ represents the QD energy levels $\epsilon_{\eta}(t)$, which can
be controlled by electric gates applied to individual dots. We consider a
sinusoidal time-dependent gate in the leftmost dot:
$\epsilon_{\eta}(t)=\epsilon_{\eta,0}+\epsilon_{\mathrm{ac}}\cos(\omega
t)\delta_{\eta,\mathrm{L}}$. The second term is the Zeeman splitting
$E_{z}=g\mu_{\mathrm{B}}B_{z}$, due to the external magnetic field $B$ applied
perpendicularly to the QDs plane. The third term represents the hopping
between dots. We consider two possible tunneling paths: a spin-conserving path
$\propto\delta_{\sigma,\sigma}$ of amplitude $\tau_{0}$ and a spin-flip path
with $\sigma\neq\sigma^{\prime}$ of amplitude $\tau_{\mathrm{sf}}$ that arises
due to the SOC. The direction of the SOC determines the form of the spin-flip
tunneling term. Using the SOC vector [68, 66] $\bm{\alpha}$, the spin-flip
term is $\propto\bm{\alpha}\cdot\hat{\bm{\sigma}}$. Here, without loss of
generality, we consider $\bm{\alpha}$ in the y-direction, such that
$\tau_{\mathrm{sf}}\in\mathds{R}$ (see A). Note that we have not included a
Coulomb interaction term as we will focus on the single-particle dynamics.

The Hamiltonian above does not explicitly include excited states of the QD
confining potential. However, in combination with the SOC and an ac electric
field, the intra-dot dynamics result in the appearance of an effective
magnetic field [69], as employed for the manipulation of spin qubits in EDSR
protocols. In the present model, this effective magnetic field is contained in
$\hat{H}_{1}(t)$. We derive this term in the context of a Schrieffer-Wolff
transformation (SWT) in A. Under SOC, inter- and intra-dot dynamics yield
distinct contributions to the spin motion [70]. Hence, we will refer to this
term as the orbital magneto-electric effect (OME) in order to distinguish it
from the similar effective magnetic field that arises due to the inter-dot
dynamics of the spin, which we refer to as the tunneling magneto-electric
effect (TME). This effect appears as a result of a finite spin-flip tunneling
amplitude, and we discuss it in more detail in section 5.

The OME is orthogonal to the SOC vector and therefore we consider it along the
x-direction. Moreover, for a homogeneous electric field along the axis of the
DQD, the OME term is the same in both dots
$\beta_{\mathrm{L}}(t)=\beta_{\mathrm{R}}(t)=\beta(t)$. Here, we consider
$\beta(t)=\beta_{\mathrm{SO}}\cos(\omega t)$, with
$\beta_{\mathrm{SO}}\in\mathds{R}$. Note that the frequency of the OME term is
the same as $\epsilon_{L}(t)$, as it arises from the same applied voltage. The
static contribution to the electric field produces only a negligible rotation
of the spin axis, and we disregard this effect in the following (see A).
Finally, we remark that the OME term appears only for $E_{z}\neq 0$, which is
needed to break time-reversal symmetry. In the case where $E_{z}$ is small,
the hyperfine interaction may produce an equivalent effect [69].

The total Hamiltonian, written on the basis
$\left\\{|L\uparrow\rangle,|L\downarrow\rangle,|R\uparrow\rangle,|R\downarrow\rangle\right\\}$,
reads

$\hat{H}=\left({\begin{array}[]{cccc}\epsilon_{L}(t)+E_{z}/2&\beta(t)/2&-\tau_{0}&-\tau_{\mathrm{sf}}\\\
\beta(t)/2&\epsilon_{L}(t)-E_{z}/2&\tau_{\mathrm{sf}}&-\tau_{0}\\\
-\tau_{0}&\tau_{\mathrm{sf}}&\epsilon_{R}+E_{z}/2&\beta(t)/2\\\
-\tau_{\mathrm{sf}}&-\tau_{0}&\beta(t)/2&\epsilon_{R}-E_{z}/2\end{array}}\right).$
(1b)

We consider the following parameter to characterize the relationship between
the spin-conserving and spin-flip tunneling amplitudes

$\chi\equiv\frac{1}{\tau_{0}/\tau_{\mathrm{sf}}+1},$ (1c)

so that $\chi=0$ corresponds to $\tau_{\mathrm{sf}}=0$, and $\chi=1$ to
$\tau_{0}=0$. We normalize the tunneling rates so that
$\tau_{0}+\tau_{\mathrm{sf}}=\tau$. Most systems exhibit a spin-flip
contribution to tunneling that is much smaller compared to the spin-conserving
one [71]. However, an applied external field can be employed to tune the SOC
and therefore the spin-flip contribution, e.g., in GaAs-based hole QDs [72,
73]. Spin polarization in the external leads can further produce synthetic SOC
in DQDs [74]. Therefore, we investigate arbitrary values of $\chi$ in the
following.

Current readout is performed by coupling the QD chain to a source (S) and
drain (D) leads. The coupling between the leads and the QD chain is given by

$\hat{H}_{\Gamma}=\sum_{l,\bm{k},\sigma,\eta}(\gamma_{l,\eta}\hat{c}_{l,\bm{k},\sigma}^{\dagger}\hat{d}_{\eta,\sigma}+h.c.),$
(1d)

with $\hat{c}_{l,\bm{k},\sigma}$ the annihilation operator for a particle in
the lead $l=\left\\{S,D\right\\}$, with spin $\sigma$ and momentum $\bm{k}$.
We consider the coupling between the leads and the QD chain to be spin-
conserving. We further consider the infinite bias limit, so that transport is
unidirectional from source to drain. Additionally, in this limit, all the
side-bands couple equally to the source lead. Thus, following the property of
the Bessel functions $\sum_{n}J_{n}(\epsilon_{\mathrm{ac}}/\omega)^{2}=1$, we
can neglect the effect of the renormalization of $\gamma_{l,\eta}$ due to the
ac electric field [55].

We consider the dynamics of the system via its reduced density matrix
$\hat{\rho}(t)$, which, under a Markovian approximation, satisfies the master
equation

$i\frac{d}{dt}\hat{\rho}(t)=[\hat{H}(t),\hat{\rho}(t)]+\mathcal{K}\hat{\rho}(t),$
(1e)

where

$\displaystyle\mathcal{K}\hat{\rho}=\sum_{\sigma}$
$\displaystyle\left[\Gamma_{\mathrm{S,L}}\left(\hat{d}_{L,\sigma}^{\dagger}\hat{\rho}\hat{d}_{L,\sigma}-\frac{1}{2}\left\\{\hat{d}_{L,\sigma}\hat{d}_{L,\sigma}^{\dagger},\hat{\rho}\right\\}\right)\right.$
(1f)
$\displaystyle\left.+\Gamma_{\mathrm{D,R}}\left(\hat{d}_{R,\sigma}\hat{\rho}\hat{d}_{R,\sigma}^{\dagger}-\frac{1}{2}\left\\{\hat{d}_{R,\sigma}^{\dagger}\hat{d}_{R,\sigma},\hat{\rho}\right\\}\right)\right],$

is the kernel superoperator for weak coupling in the infinite bias
approximation. Transition rates due to coupling with leads are defined as
$\Gamma_{l,\eta}=2\pi/|\gamma_{l,\eta}|^{2}D_{l}(\epsilon_{F})$, where
$D_{l}(\epsilon)$ is the density of states of the lead $l$ and $\epsilon_{F}$
is the Fermi energy. We consider the case of strongly interacting QDs, where
the energy difference between the single- and double-occupied states is much
larger than that of the rest of the energy scales of the system, allowing us
to investigate charge transport in the single-charge section of the stability
diagram.

The steady-state current can then be calculated as

$I^{\infty}=e\sum_{\sigma}\Gamma_{\mathrm{D,R}}\rho_{\mathrm{R}\sigma}^{\infty},$
(1g)

where

$\rho_{\mathrm{R}\sigma}^{\infty}=\lim_{t\rightarrow\infty}\frac{1}{T}\int_{t}^{t+T}ds\langle\mathrm{R}\sigma|\hat{\rho}(s)|\mathrm{R}\sigma\rangle,$
(1h)

is the occupation of the rightmost dot with spin $\sigma$ in the stationary
state averaged over one period $T=2\pi/\omega$ of the ac voltage. In the
following, we consider an identical coupling to both leads, so that
$\Gamma_{\mathrm{S,L}}=\Gamma_{\mathrm{D,R}}=\Gamma$.

##  3 Quantum transport in DQD with spin-orbit interaction

In the case where $E_{z}\sim 0$, quantum transport occurs only close to zero
detuning $\delta\equiv\epsilon_{R,0}-\epsilon_{L,0}=0$, where the levels of
the two dots are aligned. The presence of an ac field changes this picture by
allowing for PAT in which the particle can tunnel from one dot to the other
one by absorbing or emitting a certain number of photons. Then, an $n-$photon
resonance occurs for $\delta=n\omega,\;n\in\mathds{Z}$.

Figure 2: (a) Current through a DQD as a function of $\delta$ and $E_{z}$ for
$\chi=0.1$ and $\beta_{\mathrm{SO}}=E_{z}/2$. Highlighted in the figure, there
is a set of four characteristic processes numbered $1-4$. Process 1
corresponds to direct spin-conserving PAT through the two spin channels at
$\delta=0$. Process 2 corresponds to direct spin-flip PAT. Process 3
corresponds to spin-flip PAT involving one photon. Process 4 corresponds to
spin-conserving PAT involving one photon through the two spin channels at
$\delta=\omega$. All four processes are schematically represented on the left
side of the figure. (b) A set of cuts in panel (a) at different values of
$E_{z}$, with a small offset of the $y$-axis for clarity. (c) Spin
polarization of the current as a function of $\delta$ and $E_{z}$. (d) Spin
polarization (solid black, left axis) and total current (dashed red, right
axis) for $E_{z}=0.25\omega$ as a function of $\delta$, represented by the
horizontal cut in panel (c). In regions around $\delta=\pm 0.5\omega$, the
current can be switched directly from one polarization to the opposite one
with comparable intensity. Other parameters are $\omega=10\tau=100\Gamma$, and
$\epsilon_{\mathrm{ac}}=1.2\omega$.

For $E_{z}\neq 0$, the degeneracy of the spin doublets is broken, resulting in
a current that will generally be spin polarized. Then, we can distinguish
several processes, schematized in figure 2. First, spin-conserving resonances
with either no photons (process 1), also known as direct resonance, or with
the emission/absorption of a photon (process 4). As in the case without a
magnetic field, these occur at $\delta=n\omega$, in which the particle can
tunnel directly from one dot to the other. These resonances survive the
presence of a magnetic field, since the difference in energy between one
particle with the same spin in different dots is also $\delta$. Moreover,
since the levels are now split in spin by the magnetic field, there can also
be resonances when the states with opposite spin have the same energy, which
is enabled by the presence of a spin-flip component in the tunneling
amplitudes. This will occur whenever $\delta=\pm E_{z}$, as represented in
figure 2 (process 2). Moreover, in the presence of an ac bias, these
resonances are accompanied by a set of replicas due to PAT. This is process 3
in figure 2. As a result, when the current is represented as a function of
$\delta$ and $E_{z}$ as in figure 2(a), we find both the usual resonances at
$\delta=n\omega$ and a set of spin-flip resonances along the lines

$\delta+n\omega=\pm E_{z}.$ (1i)

In the $\delta-E_{z}$ representation of figure 2(a), these correspond mostly
to diagonal lines. Note that transport at these resonances is not blocked when
a particle tunnels from the source into the left dot with a direction that is
not energy-aligned with any state on the right dot (such as
$|L\downarrow\rangle$ in process 2 of figure 2). Due to the finite spin-flip
tunneling amplitude, the spin states always have a small component of the
opposite spin. Another way to see this is that the spin rotates as a result of
virtual tunneling to the dot via the aforementioned TME until it aligns with
the resonance.

However, there are several points where this simple picture breaks down. In
the vicinity of a PAT process with $n$ photons the states in the two dots are
strongly hybridized with energies

$E_{\pm}=\delta\pm\tau
J_{n}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right),$ (1j)

for the two molecular states, where $J_{n}(z)$ is the $n$th-order Bessel
function. There, the diagonal lines are no longer straight but curve at
$E_{z}\approx\pm\tau$, corresponding to the resonance conditions with the
energies given by equation 1j. This is clearly seen in figure 2(a) in the PAT
resonances near $E_{z}=0$. The overall shape of these resonances closely
follows hyperbolas in the $\delta-E_{z}$ representation shown there.

Another departure from this simple picture occurs near $E_{z}=\pm\omega$,
where the OME field induces resonant transitions between the two spin states.
There, the spin-flip resonances do not cross the spin-conserving PAT
resonances at $\delta=\pm\omega$. Instead, the spin-flip resonant lines are
repelled so that a current-free gap opens between the main resonances (1) and
the spin-flip resonances (2). Moreover, we see another effect on the current
at the main resonances at $\delta=0$. There, the spin-flip resonances cross
the main resonances at $E_{z}=\pm(\tau+\beta_{\mathrm{SO}})$ instead of simply
crossing at $E_{z}=\pm\tau$, as discussed above. The spin-flip resonances are
then still hyperbolas but vertically displaced from where they would be for
$\beta_{\mathrm{SO}}=0$. Finally, when photo-assisted spin-flip resonances
with zero or one photon meet, they do not get distorted (such as at
$\delta=\pm\omega/2$, $E_{z}=\pm\omega/2$ in figure 2(a)) as they correspond
to processes that transmit different spin polarizations.

The spin polarization of the current is defined as

$P_{\sigma}=\frac{I_{\uparrow}^{\infty}-I_{\downarrow}^{\infty}}{I_{\uparrow}^{\infty}+I_{\downarrow}^{\infty}},$
(1k)

where $I_{\sigma}^{\infty}=e\Gamma\rho_{\mathrm{R},\sigma}^{\infty}$. We have
represented $P_{\sigma}$ as a function of $\delta$ and $E_{z}$ in figure 2(c).
As expected, the spin polarization is non-zero at the spin-flip current
branches, provided that only one path is active (either
$|L\uparrow\rangle\rightarrow|R\downarrow\rangle$ or
$|L\downarrow\rangle\rightarrow|R\uparrow\rangle$). We are able to obtain spin
polarization close to 1, with the largest limitation in this setup being the
overlap of a spin-flip resonance with a spin-conserving PAT resonance. The
width of spin-conserving resonances can be tuned by reducing $\tau_{0}$, and
the separation between PATs can be increased by varying the ac voltage
frequency $\omega$.

This setup allows for several ways of controlling spin polarization. In figure
2(d), we show the polarization together with the total current. By varying
$\delta$ from $\delta\simeq 0.2\omega$ to $\delta\simeq 0.8\omega$, we are
able to shift from a direct PAT resonance with a strong spin polarization in
one direction to a one-photon PAT resonance with a strong spin polarization in
the opposite one. Hence, polarization can be inverted without crossing the
resonance at $\delta=0$. Since $\delta$ is often one of the easiest parameters
to control in experimental setups, this allows us to generate highly tunable
fully spin-polarized currents. This is remarkable, since the spin-flip
amplitude in this case has been considered very small ($\chi=0.1$). With a
correct tuning of $\chi$ and $\tau$ (reducing the width of the spin-conserving
resonances), the spin polarization can be adjusted to an even larger degree.

We note a slight asymmetry in the spin polarization in the two adjacent highly
polarized peaks at $\delta\simeq 0.2\omega$ and $\delta\simeq 0.8\omega$
resulting from the OME term. This can be seen in figure 2(c), where the
resonance lines have different widths depending on whether they cross the
$E_{z}=\pm\omega$, $\delta=\pm\omega$ points or not. For
$\beta_{\mathrm{SO}}\rightarrow 0$, the current associated with the two
resonances has the same absolute value of polarization (and is close to
$P_{\sigma}=\pm 1$, respectively).

Tunning the frequency of the ac voltage, the setup allows another mode of
generating spin-polarized currents. The DQD can be initialized in a
configuration in which direct resonances without photons are energetically
disfavored. Then, applying a voltage with the frequency tuned to either the
spin-conserving or the spin-flip resonance with absorption or emission or
photons, the resulting current will be spin-polarized in one direction or the
other, depending on the spin of the initial state. Both modes of operation
allow for fully electric control of the spin polarization of the current
without the need to modify the magnetic fields, which enables fast control of
the current under experimental conditions.

Next, we study the effect of varying the amplitude of the ac voltage. Due to
Landau-Zener-Stückelberg (LZS) interferences, if the QDs are in resonance
$\delta=n\omega$, photo-assisted transitions can be understood to occur with a
tunneling amplitude renormalized by a Bessel function as [58, 59, 75]

$\tau_{\zeta}\rightarrow\tau_{\zeta}J_{n}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right),\quad\zeta\in\\{0,\mathrm{sf}\\}.$
(1l)

Figure 3: (a) Current and (b) spin polarization through a DQD as a function of
$\delta$ and $\epsilon_{\mathrm{ac}}$. (a, bottom) Cuts along the horizontal
axis at $\epsilon_{\mathrm{ac}}=0$ (green), $\epsilon_{\mathrm{ac}}=3\omega$
(blue), and $\epsilon_{\mathrm{ac}}=6\omega$ (red), showing the appearance of
satellite peaks close to the main resonances for $\epsilon_{\mathrm{ac}}\neq
0$. (a, left) Cut the along the vertical axis (dashed lines) at
$\delta=n\omega$ with $n=0$ (pink), $n=1$ (lime), and $n=2$ (cyan), showing
the appearance of coherent destruction of tunneling, where the current drops
to zero. The horizontal solid lines denote the zeros of the Bessel function
$J_{n}(\epsilon_{\mathrm{ac}}/\omega)$. The parameters used are
$\omega=10\tau=100\Gamma$, $E_{z}=0.3\omega$, $\beta_{\mathrm{SO}}=0$, and
$\chi=0.2$.

Crucially, both the spin-conserving and spin-flip amplitudes are equally
modified by the presence of the ac field 111Note that this was already taken
into account in equation 1j.. This well-known expression holds except at the
points where the spin levels are in resonance as well ($E_{z}=m\omega$), as we
will see in the next sections. At the zeros of any given Bessel function,
destructive interference occurs, and both spin-conserving and spin-flip
tunneling rates are suppressed. This effect is known as coherent destruction
of tunneling (CDT) [49]. The resulting pattern is given in figure 3(a), where
we have represented the current as a function of both $\delta$ and
$\epsilon_{\mathrm{ac}}$, for a range that includes the first five photo-
assisted resonances in each direction of $\delta$, plus the resonance at
$\delta=0$. Spin-flip resonances are visible as thinner parallel lines along
the $\epsilon_{\mathrm{ac}}$ axis, two to each side of the main resonances. We
have also represented in figure 3(a) (bottom panel) a set of cuts at
$\epsilon_{\mathrm{ac}}=0$ (green), $\epsilon_{\mathrm{ac}}=3\omega$ (blue)
and $\epsilon_{\mathrm{ac}}=6\omega$ (red), showing the resonance pattern
(similarly to figure 2). Furthermore, figure 3(a) (left panel) shows the
interference pattern that leads to CDT. Here, we have represented a set of
cuts at values of $\delta=n\omega$ for $n=0$ (pink), $n=1$ (lime) and $n=2$
(cyan). We also include dashed lines to denote the first zeros of
$J_{n}(\epsilon_{\mathrm{ac}}/\omega)$, which coincide precisely with the dips
in the current at $\delta=n\omega$. Finally, in figure 3(b) we plot the spin
polarization for the LZS interferometry. The main spin-conserving resonances
have an unpolarized spin current $P_{\sigma}\sim 0$, since both the spin-up
and spin-down channels are present with the same probability. However, spin-
flip resonances at $\delta=n\omega\pm E_{z}$ are highly spin polarized
$P_{\sigma}\sim\pm 1$, as already mentioned above. These resonances, similar
to the spin-conserving ones, also exhibit a CDT effect where
$J_{n}(\epsilon_{\mathrm{ac}}/\omega)=0$.

###  3.1 Effect of the OME

In this section, we consider in detail the effect of the OME term. This will
be, in general, a weak effect because the amplitude of the SOC-induced OME
field is smaller than the electric voltage amplitude. However, as anticipated
above, the OME has a strong effect near the spin resonances at $E_{z}\simeq
n\omega$. At these points, the effective ac magnetic field can induce resonant
transitions between the two spin states on the same QD. To focus on this
regime, let us consider the Hamiltonian at any such resonance. We perform the
following unitary transformation

$\hat{U}(t)=\exp\left(\frac{-in\omega
t\hat{\sigma}_{z}}{2}\right)\exp\left(\frac{-i\epsilon_{\mathrm{ac}}\sin(\omega
t)(1+\hat{\tau}_{z})}{2\omega}\right),$ (1m)

where $\hat{\tau}_{i}$ are Pauli matrices associated to the charge (left/right
dot) degree of freedom. The leftmost operator transforms the system into the
rotating frame at the spin resonance. The rightmost operator transforms into
the interacting picture with respect to the ac voltage, removing the
corresponding term from the Hamiltonian at the cost of turning the tunneling
time-dependent. Its purpose is to allow for arbitrary ac voltage amplitudes
within a rotating-wave approximation (RWA) [60], under which we will work.

When the transformation of equation 1m is performed and the Jacobi-Anger
expansion is employed, the terms in the Hamiltonian are changed as

$\displaystyle\langle\eta\sigma|\hat{H}_{0}|\eta\sigma\rangle=E_{z}\to
E_{z}-n\omega,$ (1na)
$\displaystyle\langle\mathrm{R}\sigma|\hat{H}_{0}|\mathrm{L}\sigma\rangle=\tau_{0}\to\tau_{0}\sum_{k=-\infty}^{\infty}J_{k}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)e^{ik\omega
t},$ (1nb)
$\displaystyle\langle\mathrm{R}\uparrow(\downarrow)|\hat{H}_{0}|\mathrm{L}\downarrow(\uparrow)\rangle=\pm\tau_{\mathrm{sf}}\to\pm\tau_{\mathrm{sf}}\sum_{k=-\infty}^{\infty}J_{k}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)e^{i\left(k\mp
n\right)\omega t},$ (1nc)
$\displaystyle\langle\eta\uparrow|\hat{H}_{1}|\eta\downarrow\rangle=\beta\left(t\right)\to\frac{\beta_{\mathrm{SO}}}{2}\left(e^{i\left(n+1\right)\omega
t}+e^{i\left(n-1\right)\omega t}\right).$ (1nd)

Other terms are obtained by imposing hermiticity on the effective Hamiltonian.
Let us now consider the avoided crossing at $\delta\approx 0$, $n=1$, visible
in figure 2 (the resonance $n=-1$ follows a similar behavior). We apply now
the RWA, neglecting the time-dependent terms. This yields

$\hat{H}_{\mathrm{RWA}}=\left(\begin{array}[]{cccc}-\delta/2&\beta_{\mathrm{SO}}/4&-\tau_{\mathrm{0}}^{\mathrm{RWA}}&-\tau_{\mathrm{sf}}^{\mathrm{RWA}}\\\
\beta_{\mathrm{SO}}/4&-\delta/2&-\tau_{\mathrm{sf}}^{\mathrm{RWA}}&-\tau_{\mathrm{0}}^{\mathrm{RWA}}\\\
-\tau_{\mathrm{0}}^{\mathrm{RWA}}&-\tau_{\mathrm{sf}}^{\mathrm{RWA}}&\delta/2&\beta_{\mathrm{SO}}/4\\\
-\tau_{\mathrm{sf}}^{\mathrm{RWA}}&-\tau_{\mathrm{0}}^{\mathrm{RWA}}&\beta_{\mathrm{SO}}/4&\delta/2\end{array}\right),$
(1o)

where
$\tau_{\mathrm{0}}^{\mathrm{RWA}}=\tau_{0}J_{0}(\epsilon_{\mathrm{ac}}/\omega)$
and
$\tau_{\mathrm{sf}}^{\mathrm{RWA}}=\tau_{\mathrm{sf}}J_{1}(\epsilon_{\mathrm{ac}}/\omega)$.
Note that at the resonance between the two spin levels, the spin-conserving
tunneling amplitude is renormalized by $J_{0}(z)$ while the spin-flip
amplitude is renormalized by $J_{1}(z)$, with
$z=\epsilon_{\mathrm{ac}}/\omega$.

In this frame, the OME term acts as a constant magnetic field, resulting in a
splitting $\beta_{\mathrm{SO}}/2$ of the spins in the x-direction. The avoided
crossing between the spin-conserving and spin-flip resonances discussed above
can be understood in this picture as the OME term producing a splitting
between the spins that is only strong near the EDSR. Therefore, resonant
transport occurs now when $\delta=\pm\beta_{\mathrm{SO}}/2$. These new
resonances are visible in figure 4(a) as side peaks next to the main resonance
at $\delta=0$. Note that the position of the resonances shows a linear
relationship between detuning and $\beta_{\mathrm{SO}}$, as expected. However,
around $\beta_{\mathrm{SO}}\lesssim\tau$ the hybridization between states is
large enough to distort this simple picture (in the same way as for equation
1j).

Figure 4: (a) Current through a DQD as a function of $\delta$ and
$\beta_{\mathrm{SO}}$. A cut along the horizontal axis at
$\beta_{\mathrm{SO}}=0.4\omega$ is shown in panel (b). The black solid line is
obtained with the original Hamiltonian (equation 1b), while the red dashed
line is obtained by the effective Hamiltonian (equation 1qx). Two satellite
peaks are located at $\delta\simeq\pm\beta_{\mathrm{SO}}/2$, marked with the
blue square and red diamond symbols. The energy levels in these situations,
for the effective Hamiltonian, are shown schematically on the right side of
the figure. Here, the width of the arrows denotes the tunneling amplitudes for
each possible path, while the gray arrows represent tunneling paths that are
less favorable. In all panels, $E_{z}=\omega$,
$\epsilon_{\mathrm{ac}}=1.2\omega$, $\omega=10\tau=100\Gamma$, and $\chi=0.2$.

The previous results are valid provided that $\beta_{\mathrm{SO}}/\omega\ll
1$, as required by the RWA. We can further relax this condition by performing
the transformation

$\hat{U}^{\prime}(t)=\exp\left(\frac{-i\beta_{\mathrm{SO}}\sin\left(\omega
t\right)\hat{\sigma}_{x}}{2\omega}\right)$ (1p)

to the original Hamiltonian in equation 1b, and then apply $\hat{U}(t)$ given
in equation 1m. This transformation removes $\hat{H}_{1}(t)$ (the OME term) at
the cost of transforming the spin-flip and Zeeman splitting terms as

$\displaystyle\frac{E_{z}}{2}\hat{\sigma}_{z}\to$
$\displaystyle\frac{E_{z}}{2}\cos\left[\frac{\beta_{\mathrm{SO}}}{\omega}\sin\left(\omega
t\right)\right]\hat{\sigma}_{z}$ (1qa)
$\displaystyle+\frac{E_{z}}{2}\sin\left[\frac{\beta_{\mathrm{SO}}}{\omega}\sin\left(\omega
t\right)\right]\hat{\sigma}_{y},$
$\displaystyle\tau_{\mathrm{sf}}\hat{\tau}_{y}\hat{\sigma}_{y}\to$
$\displaystyle\tau_{\mathrm{sf}}\hat{\tau}_{y}\cos\left[\frac{\beta_{\mathrm{SO}}}{\omega}\sin\left(\omega
t\right)\right]\hat{\sigma}_{y}$
$\displaystyle-\tau_{\mathrm{sf}}\hat{\tau}_{y}\sin\left[\frac{\beta_{\mathrm{SO}}}{\omega}\sin\left(\omega
t\right)\right]\hat{\sigma}_{z}.$ (1qb)

The first term on the right-hand side of the expression for $E_{z}$ reflects
the renormalization effect due to the ac magnetic field. Compared to equation
1o, this term renormalizes the Zeeman splitting by a factor
$J_{0}(\beta_{\mathrm{SO}}/\omega)$. At destructive interferences where
$J_{0}\left(\beta_{\mathrm{SO}}/\omega\right)=0$, the Zeeman splitting is
completely suppressed in a process known as spin locking (SL) [51].
Nonetheless, this suppression occurs for values of $\beta_{\mathrm{SO}}$
beyond those discussed in A. In general, the main effect of this
renormalization is that EDSR now occurs at
$E_{z}J_{0}(\beta_{\mathrm{SO}}/\omega)=n\omega$. The second term on the
expression for $E_{z}$ reduces for $\beta_{\mathrm{SO}}/\omega\ll 1$ to the
(phase-shifted) OME term. At resonance $n=\pm 1$, this corresponds to the
substitution

$\frac{\beta_{\mathrm{SO}}}{2}\to
E_{z}J_{1}\left(\frac{\beta_{\mathrm{SO}}}{\omega}\right),$ (1qr)

e.g., in equation 1o. In the limit $\beta_{\mathrm{SO}}/\omega\ll 1$ we
recover the result of the previous section, while for
$J_{1}\left(\beta_{\mathrm{SO}}/\omega\right)=0$ the effective magnetic field
is suppressed in a similar way to the case discussed above where spin locking
takes place. Again, this effect occurs for parameters beyond the scope of the
approximations of A and we do not consider it any further.

Regarding the transformed spin-flip term, given in equation (1qb), it may seem
that the first term would also produce a similar Bessel function
renormalization. However, the spin-flip amplitude at an arbitrary
$\beta_{\mathrm{SO}}$ allows sidebands to be generated from both the ac
voltage and the OME term. The combination of these sidebands yields a more
complicated expression. After applying the transformation of equation 1m and
expanding the first term of equation (1qb) using the Jacobi-Anger expansion,
we find the following:

$\pm\tau_{\mathrm{sf}}\to\frac{\pm\tau_{\mathrm{sf}}}{2}\sum_{k=-\infty}^{\infty}\sum_{k^{\prime}=-\infty}^{\infty}J_{k}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{2k^{\prime}}\left(\frac{\beta_{\mathrm{SO}}}{\omega}\right)\left(e^{i(-k\mp
n+2k^{\prime})\omega t}+e^{i(-k\mp n-2k^{\prime})\omega t}\right),$ (1qs)

where $\pm$ corresponds to the sign of the two spin channels, as in equation
1b. In the RWA, we find

$\displaystyle\pm\tau_{\mathrm{sf}}\to$ $\displaystyle(\mp
1)^{n+1}\tau_{\mathrm{sf}}^{\mathrm{RWA}},$ (1qt)
$\displaystyle\tau_{\mathrm{sf}}^{\mathrm{RWA}}=$
$\displaystyle\tau_{\mathrm{sf}}\sum_{k=-\infty}^{\infty}J_{2k+n}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{2k}\left(\frac{\beta_{\mathrm{SO}}}{\omega}\right).$
(1qu)

Finally, consider the last term on equation (1qb). After applying the
transformation of equation 1m, we find a term $\propto|L\rangle\langle
R|\hat{\sigma}_{z}$ with amplitude

$\tau_{\mathrm{sf}}\sum_{k=-\infty}^{\infty}\sum_{k^{\prime}=0}^{\infty}J_{k}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{2k^{\prime}+1}\left(\frac{\beta_{\mathrm{SO}}}{\omega}\right)\left(e^{i\left(k+2k^{\prime}+1\right)\omega
t}-e^{i\left(k-2k^{\prime}-1\right)\omega t}\right).$ (1qv)

In the RWA, only the terms with $k=\pm(2k^{\prime}+1)$ contribute and we
obtain the real tunneling rate

$\tau_{\mathrm{si}}^{\mathrm{RWA}}=\tau_{\mathrm{sf}}\sum_{k=-\infty}^{\infty}J_{2k+1}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{2k+1}\left(\frac{\beta_{\mathrm{SO}}}{\omega}\right).$
(1qw)

This produces spin-conserving amplitudes of different absolute values
depending on the direction of the spin, i.e., the spin-conserving tunneling of
a spin-down particle occurs with a tunneling amplitude different from that of
a spin-up particle. Note that this term arises only because of the combination
of sidebands from both the ac voltage and the OME term, resulting in a finite
static component that contributes to the Hamiltonian even in the RWA.
Remarkably, both electric and magnetic sidebands arise from the same applied
voltage. In that sense, this is a constructive self-interference effect, in a
similar way to virtual tunneling processes that involve two PAT transitions
[55, 57].

Let us now compare the results for arbitrary $\beta_{\mathrm{SO}}$ with the
Hamiltonian in the $\beta_{\mathrm{SO}}\ll\omega$ limit, equation 1o. Again,
let us focus on $n=1$, $\delta\approx 0$. We diagonalize the spin subspace by
performing a rotation around the y-axis by an angle
$\theta\equiv\arctan(E_{z}^{\mathrm{RWA}}/\beta_{\mathrm{SO}}^{\mathrm{RWA}})$,
where $E_{z}^{\mathrm{RWA}}\equiv
E_{z}J_{0}\left(\beta_{\mathrm{SO}}/\omega\right)-\omega$ and
$\beta_{\mathrm{SO}}^{\mathrm{RWA}}\equiv
E_{z}J_{1}(\beta_{\mathrm{SO}}/\omega)$. Finally, the effective Hamiltonian in
its matrix form reads

$\hat{\widetilde{H}}=\left(\begin{array}[]{cccc}(\widetilde{E}_{z}-\delta)/2&0&-\widetilde{\tau}_{\uparrow}&-\widetilde{\tau}_{\mathrm{sf}}\\\
0&(-\widetilde{E}_{z}-\delta)/2&-\widetilde{\tau}_{\mathrm{sf}}&-\widetilde{\tau}_{\downarrow}\\\
-\widetilde{\tau}_{\uparrow}&-\widetilde{\tau}_{\mathrm{sf}}&(\widetilde{E}_{z}+\delta)/2&0\\\
-\widetilde{\tau}_{\mathrm{sf}}&-\widetilde{\tau}_{\downarrow}&0&(-\widetilde{E}_{z}+\delta)/2\end{array}\right),$
(1qx)

where we have defined the effective magnitudes as

$\displaystyle\widetilde{E}_{z}$
$\displaystyle\equiv\sqrt{(E_{z}^{\mathrm{RWA}})^{2}+(\beta_{\mathrm{SO}}^{\mathrm{RWA}})^{2}},$
(1qy) $\displaystyle\widetilde{\tau}_{\uparrow,\downarrow}$
$\displaystyle\equiv\tau_{0}^{\mathrm{RWA}}\mp\left[\sin(\theta)\tau_{\mathrm{sf}}^{\mathrm{RWA}}-\cos(\theta)\tau_{\mathrm{si}}^{\mathrm{RWA}}\right],$
(1qz) $\displaystyle\widetilde{\tau}_{\mathrm{sf}}$
$\displaystyle\equiv\cos(\theta)\tau_{\mathrm{sf}}^{\mathrm{RWA}}+\sin(\theta)\tau_{\mathrm{si}}^{\mathrm{RWA}}.$
(1qaa)

The effective model represents a DQD system in the presence of a homogeneous
Zeeman splitting $\widetilde{E}_{z}$, a spin-conserving tunneling rate which
is different in general for the spin-up and spin-down channels
$\widetilde{\tau}_{\uparrow,\downarrow}$, and a spin-flip tunneling rate
$\widetilde{\tau}_{\mathrm{sf}}$. The magnetic field can be simplified for
$\beta_{\mathrm{SO}}<\omega$ as
$\widetilde{E}_{z}=\beta_{\mathrm{SO}}/2+\mathcal{O}\left((\beta_{\mathrm{SO}}/\omega)^{5}\right)$,
recovering the linear relation between $\delta$ and $\beta_{\mathrm{SO}}$ for
the resonant condition explained above. Note that the effective model is
strictly only valid for small detuning $\delta\ll\omega$. The different spin-
conserving tunneling rates $\widetilde{\tau}_{\uparrow,\downarrow}$ contribute
in a characteristic fashion to the current. In figure 4(a), we observe a
significant asymmetry between the two secondary resonances depending on the
sign of $\delta$. As we shall see next, this asymmetry arises fundamentally
due to the fact that
$|\widetilde{\tau}_{\uparrow}|\neq|\widetilde{\tau}_{\downarrow}|$.

The two satellite peaks shown in figure 4(b) occur when the energies of two
levels with opposite spin are in resonance 222Note that the spin-up and spin-
down levels in the effective model are a non-trivial combination of the spin-
up and spin-down levels of our original model. However, the population of the
left and right dots in the effective model of equation 1qx still resembles the
same populations as in the original system of equation 1b.. This is the
situation schematized in figure 4. If $\delta<0$, the spin-down state of the
left dot is in resonance with the spin-up state of the right dot and the most
likely tunneling path is given by $\widetilde{\tau}_{\mathrm{sf}}$, while
$\widetilde{\tau}_{\downarrow}$ can be neglected at first order. Meanwhile,
the spin-up energy level on the left dot is out of resonance with an energy
difference given by $\widetilde{E}_{z}$. However, due to
$\widetilde{\tau}_{\uparrow}$, there is still a small but finite probability
of tunneling to the right dot [76]. In the case of $\delta>0$, we have a
similar situation, except that the non-resonant level (spin-down in this case)
tunnels with an amplitude given by $\widetilde{\tau}_{\downarrow}$. Using
equations (1qu, 1qw) with $k=-2,-1,\ldots,2$, and the parameters used in
figure 4(b), we obtain $\widetilde{\tau}_{\uparrow}\simeq 0.06\omega$ and
$\widetilde{\tau}_{\downarrow}\simeq 0.04\omega$, and
$\widetilde{\tau}_{\mathrm{sf}}\simeq 0.002\omega$. Since
$\widetilde{\tau}_{\uparrow}>\widetilde{\tau}_{\downarrow}$ the effective
model predicts a higher current for resonance at $\delta<0$, which is in
agreement with the numerical results obtained from the original Hamiltonian.
In figure 4(b), we compare the current obtained both with the total
Hamiltonian (equation 1b) and with the effective Hamiltonian (equation 1qx),
showing how the effective Hamiltonian predicts the presence of satellite peaks
close to the main resonance and the asymmetry in $\delta$.

The fact that the asymmetry in $\delta$ is a consequence of
$|\widetilde{\tau}_{\uparrow}|\neq|\widetilde{\tau}_{\downarrow}|$ can be
further seen by calculating the effective magnetic field due to the TME, which
is explicitly asymmetric in $\delta$. This is shown in more detail in C.

##  4 Dark state formation

Dark states are a well-known feature in open quantum systems, first described
in the context of the optical response of electrons to laser pumping. In
mesoscopic transport, a dark state refers to a steady state in which coherent
interference results in a current blockade even in a situation where it would
be expected to flow [77, 78, 79, 80]. In this particular setup, dark states
correspond to a particle confined to the left dot, as the right dot will be
immediately emptied by tunneling to the drain.

In the following, we focus on the main resonances $\delta=m\omega$,
$m\in\mathbb{Z}$, where a non-zero current is expected to flow. In this case,
as mentioned above, current blockade occurs due to the LZS interference
pattern at the points where
$J_{m}\left(\epsilon_{\mathrm{ac}}/\omega\right)=0$, yielding the
aforementioned CDT. Taking into account the resonance $\delta=0$, the current
through the DQD can be suppressed if the driving amplitude verifies
$J_{0}(\epsilon_{\mathrm{ac}}/\omega)=0$, which occurs close to
$\epsilon_{\mathrm{ac}}\simeq 2.4\omega$. As described above, this
renormalization is the same for both the spin-conserving and spin-flip
tunneling amplitudes. However, if the system is simultaneously in a
configuration in which the spin states are also in resonance with the ac
voltage frequency $E_{z}=n\omega$, the renormalization is not necessarily the
same, as seen in the previous section. We focus on this case in the following.

Let us consider the Hamiltonian in the rotating frame, as in equation 1o, for
an $n-$photon resonance. First, consider $\beta_{\mathrm{SO}}=0$. We analyze
below how the dark states are affected when including a non-zero OME term. In
the case of a spin-up particle tunneling from the left to the right dot,
emitting $n$ photons, the spin-flip tunneling rate is renormalized as
$\tau_{\mathrm{sf}}\rightarrow\tau_{\mathrm{sf}}J_{-n}(\epsilon_{\mathrm{ac}}/\omega)$.
On the other hand, if the particle tunnels from the left dot with spin down,
it will absorb $n$ photons, so that
$\tau_{\mathrm{sf}}\rightarrow\tau_{\mathrm{sf}}J_{n}(\epsilon_{\mathrm{ac}}/\omega)$.
This situation is shown schematically in figure 5(a). The effective
Hamiltonian under an RWA then reads

$\hat{H}_{\mathrm{RWA}}^{(n)}=\left({\begin{array}[]{cccc}0&0&-\tau_{0}J_{0}(\epsilon_{\mathrm{ac}}/\omega)&-\tau_{\mathrm{sf}}J_{n}(\epsilon_{\mathrm{ac}}/\omega)\\\
0&0&\tau_{\mathrm{sf}}J_{-n}(\epsilon_{\mathrm{ac}}/\omega)&-\tau_{0}J_{0}(\epsilon_{\mathrm{ac}}/\omega)\\\
-\tau_{0}J_{0}(\epsilon_{\mathrm{ac}}/\omega)&\tau_{\mathrm{sf}}J_{-n}(\epsilon_{\mathrm{ac}}/\omega)&0&0\\\
-\tau_{\mathrm{sf}}J_{n}(\epsilon_{\mathrm{ac}}/\omega)&-\tau_{0}J_{0}(\epsilon_{\mathrm{ac}}/\omega)&0&0\end{array}}\right).$
(1qab)

Since $J_{-n}(z)=(-1)^{n}J(z)$, if the number of photons is odd, then the
spin-flip term has the same sign for both spin directions. As discussed in A,
time-reversal symmetry imposes the requirement that the spin-flip term must be
$\propto\hat{\tau}_{y}$, in terms of the charge Pauli matrices introduced
above. This form of tunneling prohibits destructive interference between the
spin-conserving and spin-flip tunneling channels. However, if the number of
photons is odd, the RWA Hamiltonian in equation 1qab has a spin-flip term of
the form $\hat{\tau}_{x}\hat{\sigma}_{x}$, which does not exhibit this
protection. Instead, the combination of $E_{z}\neq 0$ (breaking time-reversal
symmetry) and $\epsilon_{\mathrm{ac}}\neq 0$ (resulting in PAT) allows the
spin-conserving and spin-flip paths to interfere destructively.

By exact diagonalization of the above Hamiltonian, we look for DS of the form
$|\mathrm{DS}\rangle=(|L\uparrow\rangle+e^{i\varphi}|L\downarrow\rangle)/\sqrt{2}$
for an arbitrary relative phase $\varphi$, which we find occurs when the ratio
$\chi$ between the spin-conserving and spin-flip amplitudes satisfies

$\widetilde{\chi}^{(n)}=\frac{J_{0}(\epsilon_{\mathrm{ac}}/\omega)}{J_{0}(\epsilon_{\mathrm{ac}}/\omega)\pm
i^{n+1}J_{n}(\epsilon_{\mathrm{ac}}/\omega)},$ (1qac)

where the superindex denotes the dependence of the dark state on the resonance
with the driving $E_{z}=n\omega$.

Since we must impose $\widetilde{\chi}^{(n)}\in\mathds{R}$ (see equation 1c),
the dark state appears only when an odd number of photons are absorbed or
emitted, i.e., $n=2k+1$ for $k\in\mathds{Z}$. In particular, $E_{z}=0$
corresponds to taking $n=0$, and equation 1qac has no real solution showcasing
the need to break the time-reversal symmetry to observe dark-state formation.
The dependence on the number of photons involved in the process is known as
the odd-even effect, and has already been observed in DQDs, both theoretically
and experimentally [68, 81, 82, 62, 63]. This type of process is generally
intrinsic to multilevel systems, in which multiple pathways destructively
interfere with each other [83].

Figure 5: (a) Scheme of a DQD, with $E_{z}=n\omega$, and renormalized spin-
conserving $\tau_{0}\rightarrow\tau_{0}J_{0}(\epsilon_{\mathrm{ac}}/\omega)$,
and spin-flip $\tau_{\mathrm{sf}}\rightarrow\tau_{\mathrm{sf}}J_{\pm
n}(\epsilon_{\mathrm{ac}}/\omega)$ tunneling rates. (b-c) Current through a
DQD as a function of $\chi$ and $\epsilon_{\mathrm{ac}}$. (b) $E_{z}=\omega$
and (c) $E_{z}=2\omega$. The dashed line represents the analytical prediction
for a dark state $\widetilde{\chi}^{(n)}$ (see equation 1qac), which has no
real solution for $n=2$. In both cases $\delta=0$, with
$\omega=10\tau=100\Gamma$ and $\beta_{\mathrm{SO}}=0$.

In figure 5(b-c) we show the current through a DQD in the cases of $n=1$ and
$n=2$, respectively. In the former case, we plot the analytical prediction for
the existence of a dark state, equation 1qac, which is in agreement with the
numerical results. If the spin-conserving tunneling path is not available,
i.e., $\tau_{0}=0$, or equivalently $\chi=1$, then the only possible tunneling
channel requires spin-flip, which is highly suppressed because of the large
energy gap between states with opposite spin produced by $E_{z}$. This process
also results in current blockade (unrelated to interference effects) and is
visible in the upper left corner of this figure. For $n=2$ we see that the
only two situations where the current is suppressed coincide with $\chi\simeq
1$, $\epsilon_{\mathrm{ac}}=0$, and with $\chi=0$,
$\epsilon_{\mathrm{ac}}\simeq 2.4\omega$, as expected from the even-odd
effect. In an experimental situation, in which the driving amplitude can be
precisely tuned, the appearance of a sharp decrease in current at a given
$\epsilon_{\mathrm{ac}}$ can be used to determine the value of $\chi$ and
hence the SOC strength. To minimize the uncertainty of the measurement,
several odd resonances for $E_{z}=(2k+1)\omega$ can be studied, providing a
precise way to characterize the SOC present in a given device.

Next, we fix the driving amplitude at $\epsilon_{\mathrm{ac}}=2\omega$ and
consider arbitrary values of $E_{z}$. The corresponding results are presented
in figure 6(a). The DS at the exact resonant condition $E_{z}=\omega$ extends
to non-resonant values of $E_{z}$, forming a parabola in the $E_{z}-\chi$
plane. Again, we perform an exact diagonalization of the RWA Hamiltonian given
by equation 1qab and look for dark states. Up to second order in
$(E_{z}-\omega)$, these are found for the condition

$\displaystyle\widetilde{\chi}^{(1)}=$
$\displaystyle\frac{J_{0}(\epsilon_{\mathrm{ac}}/\omega)^{2}-|J_{0}(\epsilon_{\mathrm{ac}}/\omega)J_{1}(\epsilon_{\mathrm{ac}}/\omega)|}{J_{0}(\epsilon_{\mathrm{ac}}/\omega)^{2}-J_{1}(\epsilon_{\mathrm{ac}}/\omega)^{2}}$
(1qad)
$\displaystyle-\frac{(E_{z}-\omega)^{2}}{\tau^{2}|J_{0}(\epsilon_{\mathrm{ac}}/\omega)J_{1}(\epsilon_{\mathrm{ac}}/\omega)|}+\mathcal{O}\left((E_{z}-\omega)^{4}\right).$

The prediction is in accordance with the numerical results (see the inset of
figure 6(a)). Far from the resonance, the RWA is no longer valid, so the above
equation does not apply. Once again, even if the dot levels are in resonance
at $\delta=0$, a trivial DS at $\chi=1$ appears. In that situation, $E_{z}$ is
no longer in resonance with the driving, so the particle cannot absorb or emit
a photon, and will be blocked in the left QD, as explained above.

Next, we consider the case for an arbitrary amplitude of the OME term. Here,
we find a more complex situation. The OME term induces fast spin rotations
inside the QDs, which prevents current blockade in situations where,
otherwise, the particle would remain stuck in the left dot. This effect is
most clearly seen in figure 6 for $\chi\simeq 1$, where the current is blocked
for $\beta_{\mathrm{SO}}=0$ (figure 6(a)) if $E_{z}\neq\omega$, while for
$\beta_{\mathrm{SO}}\neq 0$ (figure 6(b)) a non-zero current can flow. Dark
states, on the other hand, can still be found, now extending to all values of
$E_{z}$. Interestingly, the spin projection of the steady state is highly
polarized, depending on whether $0<E_{z}<\omega$, for which the particle
remains in the left QD with spin down, or $\omega<E_{z}<2\omega$ so that the
final spin projection is inverted. It is also worth mentioning that the DS
obtained far from the resonance $E_{z}\neq\omega$ are highly pure with
$\Tr(\rho^{2})\simeq 1$. These DSs could be used to store quantum information
inside a QD without the requirement to modify the tunneling from the leads or
between dots.

Due to the complexity of our system, no analytical results could be obtained
for the formation of DSs for $\beta_{\mathrm{SO}}\neq 0$. Nonetheless, we can
still gain an understanding of their appearance by employing the Floquet
formalism. For a $T$-periodic Hamiltonian, the time evolution of the wave
function can be written as
$|\Psi_{\alpha}(t)\rangle=e^{-i\varepsilon_{\alpha}t}|\phi_{\alpha}(t)\rangle$,
where $\varepsilon_{\alpha}$ are the quasienergies and $|\phi_{\alpha}\rangle$
are the Floquet states, obtained by diagonalizing the Floquet Hamiltonian
$\hat{\mathcal{H}}(t)\equiv\hat{H}(t)-i\partial_{t}$. According to the Von
Neumann-Wigner theorem [84], crossings between distinct energy levels occur
only if the corresponding states belong to different representations of the
symmetry group of the system. Quasienergy crossings follow the same behavior,
often signaling that the ac drive has reinstated a symmetry of the system by
suppressing the relevant terms in the Hamiltonian [51]. At these crossings,
the time evolution operator depends exponentially on the difference between
the quasienergies. In resonance, the unitary time evolution matrix becomes the
identity operator, freezing the long-term dynamics.

The quasienergies for $\beta_{\mathrm{SO}}\neq 0$ have been obtained by means
of numerical methods [85], and their crossings perfectly agree with the
location of the DS found by evolving the Lindblad master equation (see figure
6(b)). The Floquet quasienergies for $E_{z}=0.7\omega$ are shown in figure
6(c). The crossing is located at $\chi\simeq 0.6$, which coincides with the
numerical result for the open system.

Figure 6: Current through a DQD as a function of $\chi$ and $E_{z}$ for (a)
$\beta_{\mathrm{SO}}=0$, and (b) $\beta_{\mathrm{SO}}=0.2E_{z}$. Darker colors
represent the appearance of DSs. The inset in panel (a) represents a zoom into
the DS around $E_{z}\simeq\omega$, while the dashed black line denotes the
analytical prediction in equation (1qad). The dashed black line in panel (b)
represents the DS predicted by the crossing of Floquet quasienergies. (c)
Floquet quasienergies for $E_{z}=0.7\omega$ (dash-doted line, panel (b)), with
the energy crossing highlighted with a circle. $\delta=0$,
$\omega=10\tau=100\Gamma$, $\epsilon_{\mathrm{ac}}=2\omega$.

##  5 Flopping mode qubit operation

We next focus on the situation in which the spin motion is confined to one of
the dots. Neglecting the OME term for simplicity, the spin dynamics in that
case is governed by the interplay between the Zeeman splitting and the TME
first discussed in section 2. In this section, we consider this regime as a
possible implementation of a solid-state qubit.

Let us first consider the case where we have an electric field that is
constant in time. We can perform a SWT to obtain an effective Hamiltonian for
an isolated spin in one of the dots. This transformation allows us to obtain
the dynamics in one energy subspace that is weakly coupled and well separated
in energy from another. In this case, we consider the states for each QD as
each of the energy subspaces, so the effective Hamiltonian after the SWT is
valid provided that we are far from the resonances (both spin-conserving and
spin-flip ones), i.e., $|\delta|\gg\tau_{0},|\delta\pm
E_{z}|\gg\tau_{\mathrm{sf}}$. Under these conditions, we obtain an effective
Hamiltonian that is second order in the tunneling and contains all
contributions due to virtual tunneling to the other dot, given by

$\displaystyle\hat{H}_{\mathrm{eff}}$
$\displaystyle=\hat{H}_{\mathrm{eff}}^{(0)}+\hat{H}_{\mathrm{eff}}^{(2)},$
(1qae) $\displaystyle\hat{H}_{\mathrm{eff}}^{(0)}$
$\displaystyle=\frac{\delta}{2}\hat{\tau}_{z}+\frac{E_{z}}{2}\hat{\sigma}_{z},$
(1qaf) $\displaystyle\hat{H}_{\mathrm{eff}}^{\left(2\right)}$
$\displaystyle=\frac{\hat{\tau}_{z}}{2}(-\delta^{\left(2\right)}+b_{z}^{\left(2\right)}\hat{\sigma}_{z}+b_{\perp}^{\left(2\right)}\hat{\sigma}_{x}),$
(1qag)

where

$\displaystyle\delta^{\left(2\right)}$
$\displaystyle=\frac{2\tau_{0}^{2}}{\delta}+\tau_{\mathrm{sf}}^{2}\left(\frac{1}{\delta+E_{z}}+\frac{1}{\delta-
E_{z}}\right),$ (1qaha) $\displaystyle b_{z}^{\left(2\right)}$
$\displaystyle=\tau_{\mathrm{sf}}^{2}\left(\frac{1}{\delta+E_{z}}-\frac{1}{\delta-
E_{z}}\right),$ (1qahb) $\displaystyle b_{\perp}^{\left(2\right)}$
$\displaystyle=\tau_{0}\tau_{\mathrm{sf}}\left(\frac{1}{\delta+E_{z}}-\frac{1}{\delta-
E_{z}}\right).$ (1qahc)

The first term in equation (1qag) consists of renormalization of the detuning
$\delta$ due to virtual tunneling. Similarly, the second term is a
renormalization of $E_{z}$ which arises from spin-flip tunneling to the right
dot and back. The third term is an effective magnetic field in the direction
perpendicular to the external field that gives rise to $E_{z}$. It arises from
a combination of spin-conserving and spin-flip tunneling. In this process, the
spin virtually tunnels to the adjacent dot, flipping its spin, and then
tunnels back to the original dot through the spin-conserving path, producing
an effective spin rotation. This is the TME discussed in section 2 and, like
the OME, arises from the motion of the particle under the SOC (the inter-dot
dynamics, in this case). Note that, in the same manner as the OME, the TME
requires $E_{z}\neq 0$ to break the time-reversal symmetry.

The effective magnetic field induced by TME can be used in a similar manner to
that of the flopping-mode qubit [65, 67]. However, the qubit cannot be
manipulated via detuning alone due to the lack of two-axis control, unless the
spin-flip and tunneling amplitudes can be manipulated independently, i.e.,
unless $\chi$ can itself be tuned, e.g., by manipulating the overlap between
the particle wave functions centered at each dot [66], rapidly enough to avoid
dephasing. Nonetheless, for a time-dependent detuning, as discussed here, the
TME allows for resonant manipulation of the dot. Let us focus on the case
where $\epsilon_{\mathrm{ac}}\ll\omega$. Then, in the rotating frame and after
applying the RWA, we obtain an effective spin model for the resonance
$E_{z}=\omega$. Applying the ac gate with a phase $\phi$ allows two-axis
control of the qubit

$\hat{H}_{\mathrm{RWA}}=\frac{b_{z}^{\left(2\right)}}{2}\hat{\sigma}_{z}-\frac{\tilde{b}_{1,\perp}}{2}\left(\cos\phi\,\hat{\sigma}_{x}+\sin\phi\,\hat{\sigma}_{y}\right),$
(1qahai)

where

$\tilde{b}_{1,\perp}=\frac{4\epsilon_{\mathrm{ac}}\tau_{0}\tau_{\mathrm{sf}}E_{z}}{\delta\left(\delta^{2}-E_{z}^{2}\right)}.$
(1qahaj)

Otherwise, the frequency of the ac gate can be matched with the renormalized
splitting of the two levels of the qubit $E_{z}+b_{z}^{(2)}$. The correction
to $\tilde{b}_{1,\perp}$ in this case can be given as

$\tilde{b}_{1,\perp}\to\tilde{b}_{1,\perp}+\frac{2\epsilon_{\mathrm{ac}}\tau_{0}\tau_{\mathrm{sf}}b_{z}^{\left(2\right)}}{\left(\delta^{2}-E_{z}^{2}\right)^{2}},$
(1qahak)

which is small in the context of the previous approximations. In figure 7(a),
we show the dynamics of the flopping-mode qubit using the effective
Hamiltonian described above. For comparison, we also show the result obtained
by integrating the equation of motion of a closed system under the original
Hamiltonian given by equation 1b. Since both dot energy levels are out of
resonance, with $\delta=\omega/2$, the population of the right dot is small
and the particle remains in the left dot. The results obtained show how the
effective model correctly reproduces the dynamics of the system.

Figure 7: (a) Rabi oscillations in a closed system governed by the full
Hamiltonian (symbols) given by equation 1b. The dynamics given by the RWA
Hamiltonian (solid lines) given by equation 1qahai is plotted for comparison.
Rabi frequency is given by $\nu_{\mathrm{R}}$, shown in the figure. The
parameters used are $\delta=\omega/2$ and $\tau=0.1\omega$. (b) Rabi frequency
as a function of detuning for different total tunneling rates. The results are
obtained with the full Hamiltonian (symbols) and the effective RWA Hamiltonian
(solid lines). The total tunneling rate is $\tau=0.05\omega$ (yellow,
triangles), $\tau=0.1\omega$ (orange, squares), and $\tau=0.2\omega$ (brown,
pentagons). Other parameters, common for both panels, are
$\epsilon_{\mathrm{ac}}=0.2\omega$, $E_{z}=\omega$, $\chi=0.2$, and
$\beta_{\mathrm{SO}}=0$.

The Rabi oscillation frequency of the effective model is given by

$\nu_{\mathrm{R}}=\frac{1}{2\pi}\sqrt{\left(b_{z}^{(2)}\right)^{2}+\left(\tilde{b}_{1,\perp}\right)^{2}}.$
(1qahal)

We compare the Rabi frequency of the effective RWA Hamiltonian with that
obtained from the original Hamiltonian; see figure 7(b). The results agree for
$|\delta-n\omega|\gg\tau$, as expected.

It should be noted that this method of manipulation couples the spin with
electric fluctuations due to the dependence of $b_{z}^{(2)}$ and
$b_{\perp}^{(2)}$ on $\delta$. In purified silicon, where the magnetic noise
caused by the atomic nuclei is heavily suppressed, this may be the main source
of decoherence for the qubit [86, 87]. In the low ac gate amplitude described
here, the flopping-mode qubit lacks a natural sweetspot [88, 89] in which the
system is insensitive to electric noise to first order in the coupling to the
bath. In particular, electric noise that enters through $b_{z}^{(2)}$ could
only be efficiently suppressed at $\delta=0$, where the first derivative with
respect to $\delta$, i.e., the noise susceptibility, vanishes. However, at
this value, the qubit cannot be operated as the direct spin-conserving
transition is resonant. However, the perpendicular component of the effective
magnetic field of equation 1qahaj can be made insensitive to noise to first
order at $\delta=\pm E_{z}/\sqrt{3}$ which can improve qubit operation. This
dynamical sweetspot [57, 90] is observed as a minimum of the Rabi frequency at
$\delta\simeq 0.58\omega$ in figure 7(b).

In B we give expressions for the TME at arbitrary ac detuning amplitudes.
Unlike the simple limit shown in equation 1qahaj, these expressions are quite
complex, highlighting the fact that the effective magnetic field arises from
second-order virtual tunneling involving two PAT processes. In the arbitrary-
amplitude limit, as given in B, we can achieve a fine degree of spin
manipulation. Furthermore, we also obtain dynamical sweetspots for large ac
gate amplitudes.

##  6 Conclusions

We have studied the effect of SOC on spin transport in a periodically driven
DQD. Due to the presence of a spin-flip tunneling path, a highly spin-
polarized current can be achieved by tuning the onsite energy difference
between the dots. In this direction, we propose several mechanisms to obtain
on-demand highly polarized currents in both directions with fully electric
control, allowing fast switching of the polarization under experimental
conditions. Furthermore, the combination of the orbital dynamics inside the
dots and the ac voltage results in the appearance of an effective magnetic
field, powering the electric dipole spin resonances in the dots. In transport,
the effect of this field can be most clearly observed in the appearance of
characteristic avoided crossings of the tunneling resonances near the EDSR
condition $E_{z}\simeq n\omega$. Tunneling under this condition results in
striking novel phenomena, such as spin-dependent tunneling amplitudes due to
the constructive self-interference of the sidebands coming from the electric
and (effective) magnetic fields. Notably, both of these processes arise from
the same ac voltage, with the electric field being the direct result of this
voltage and the magnetic field effectively incorporating the dynamics of the
excited states of the potential. We also investigate the appearance of new
dark states when the system is driven under the EDSR condition, resulting from
the interference between photo-assisted spin-conserving and spin-flip
processes with different numbers of photons involved. Without a Zeeman
splitting, time-reversal symmetry protects against destructive interference
between these two processes. However, in the presence of a magnetic field,
together with an ac voltage, these two paths can interfere and create dark
states. These states exhibit a characteristic even-odd effect, appearing only
for odd sideband transitions. Moreover, since the current drop is very sharp,
its location can be used to characterize the SOC present in the system. These
dark states could be useful for quantum information storage, as they are both
highly pure and spin-polarized. Finally, we study the viability of flopping-
mode qubit operations when the particle is localized in one of the dots. We
provide expressions for the effective magnetic field arising from the interdot
motion of the particle under SOC for arbitrary ac-gate amplitudes and
particularize the experimentally (more easily) accessible case of small gate
amplitudes. In this situation, we identify a dynamical sweetspot induced by
the ac voltage, where the Rabi frequency is insensitive to charge noise to
first order in detuning. This point of operation may be important for novel
solid-state quantum computing platforms, such as isotopically purified
silicon, where decoherence due to the nuclear magnetic field can be
suppressed, and electric noise may be the most relevant source of decoherence.

G.P. and D.F.F. are supported by Spain’s MINECO through Grant No.
PID2020-117787GB-I00 and by the CSIC Research Platform PTI-001. D.F.F.
acknowledges support from FPU Program No. FPU20/04762. J.P.C. acknowledges DFG
funding through project B04 of SFB 1277 Emerging Relativistic Phenomena in
Condensed Matter. D. F. F. and J. P. C. have contributed equally to this work.

##  Appendix A Effective model

In this section, we briefly discuss the origin of the OME term
$\hat{H}_{1}(t)$ in the Hamiltonian of equation (1a). We follow a similar
derivation to [69, 91], employing a Schrieffer-Wolff transformation (SWT) to
obtain the effective Hamiltonian on the basis employed in the main text. We
consider the Hamiltonian for a single particle in a linear DQD (in the
x-direction) under both electric and magnetic fields and in the presence of
SOC modeled by the spin-orbit vector [92]
$\bm{\alpha}=(\alpha_{x},\alpha_{y},0)$ compatible with 2DEGs grown along the
[001] direction [93], i.e., the z-direction in our model. We consider an
electric field in the x-direction and a Zeeman splitting $E_{z}$ in the
direction normal to the QD plane (in the z-direction). All this together
yields the Hamiltonian

$\displaystyle\hat{H}\left(x,t\right)=\hat{H}_{k}+V\left(x\right)+\hat{H}_{e}\left(x,t\right)+\hat{H}_{z}+\hat{H}_{\mathrm{SO}},$
(1qaham) $\displaystyle\hat{H}_{k}=k^{2}/2m,$ (1qahan)
$\displaystyle\hat{H}_{e}\left(x,t\right)=exE\left(t\right),$ (1qahao)
$\displaystyle\hat{H}_{z}=E_{z}\hat{\sigma}_{z}/2,$ (1qahap)
$\displaystyle\hat{H}_{\mathrm{SO}}=\bm{\alpha}\cdot\hat{\bm{\sigma}}k.$
(1qahaq)

The scalar potential $V\left(x\right)$ for the DQD exhibits two minima at
$\pm\ell$, near which the potential can be chosen as harmonic
$V_{\mathrm{osc}}\left(x\right)=\left(1/2\right)m\omega_{0}^{2}x^{2}$. In the
tight-binding approximation, we first evaluate the local Hamiltonians in each
dot in the eigenfunctions of the individual harmonic potentials

$|\psi_{\eta,\nu,\sigma}\rangle=|\eta,\nu\rangle|\sigma\rangle,$ (1qahar)

with $\nu\in\mathbb{N}$ labeling the eigenstates of the harmonic potential and
$\sigma\in\left\\{\uparrow,\downarrow\right\\}$ the spin projection along the
z-axis. The orbital part $|\eta,\nu\rangle$ of these eigenfunctions can be
obtained by diagonalizing the Hamiltonian
$\hat{H}_{k}+V_{\mathrm{osc}}\left(x-\eta\ell\right)$, with $\eta=\pm$
corresponding to the left and right dots, as appropriate, and corresponds to a
Fock-Darwin function with shifted centers. Following this, around $\eta\ell$
we can write the terms of the Hamiltonian as

$\displaystyle\hat{H}_{k}+V_{\mathrm{osc}}\left(x-\eta\ell\right)=\omega_{0}\left(\hat{a}^{\dagger}\hat{a}+1\right),$
(1qahas)
$\displaystyle\hat{H}_{e}\left(\eta\ell,t\right)=el_{0}E\left(t\right)\left(\hat{a}^{\dagger}+\hat{a}\right)+e\eta
E\left(t\right)\ell,$ (1qahat)
$\displaystyle\hat{H}_{\mathrm{SO}}=\frac{i}{2l_{0}}\bm{\alpha}\cdot\hat{\bm{\sigma}}\left(\hat{a}^{\dagger}-\hat{a}\right),$
(1qahau)

with $\hat{a}^{\dagger},\hat{a}$ the Fock operators of the oscillator and the
Zeeman term unchanged. Moreover, we employ the characteristic oscillator
length $l_{0}=\sqrt{\omega_{0}/2m}$. We can separate this Hamiltonian into a
part that is static in the orbital dynamics

$\hat{H}_{\eta}^{\left(0\right)}=\omega_{0}\left(\hat{a}^{\dagger}\hat{a}+1\right)+eE\left(t\right)\eta\ell+\left(E_{z}/2\right)\hat{\sigma}_{z},$
(1qahav)

and a dynamic part

$\displaystyle\hat{H}_{\eta}^{\left(1\right)}$
$\displaystyle=el_{0}E\left(t\right)\left(\hat{a}^{\dagger}+\hat{a}\right)+(i/2l_{0})\bm{\alpha}\cdot\hat{\bm{\sigma}}\left(\hat{a}^{\dagger}-\hat{a}\right).$
(1qahaw)

When projected into the ground state, the first part reduces to the usual
description of single-state QDs. However, if we project the second part as
well, we will obtain an effective Hamiltonian that incorporates the action of
the SOC into the ground state. We do this by first performing a SWT
$\hat{H}_{\eta}^{\prime}\left(t\right)=e^{\hat{\Upsilon}}\hat{H}_{\eta}\left(t\right)e^{-\hat{\Upsilon}}$.
We further consider the adiabatic approximation with respect to the electric
field, valid provided that the driving frequency is much lower than the
oscillator frequency, i.e., $\omega\ll\omega_{0}$. Then the anti-Hermitian
operator $\hat{\Upsilon}$ is given by

$\left[\hat{H}_{\eta}^{\left(0\right)},\hat{\Upsilon}\right]=\hat{H}_{\eta}^{\left(1\right)},$
(1qahax)

resulting in

$\hat{\Upsilon}=\left(f+\bm{d}\cdot\hat{\bm{\sigma}}\right)\hat{a}^{\dagger}-\left(f^{*}+\bm{d}^{*}\cdot\hat{\bm{\sigma}}\right)\hat{a},$
(1qahay)

where

$\displaystyle f$ $\displaystyle=el_{0}E_{\eta}\left(t\right)/\omega_{0},$
(1qahaz) $\displaystyle d_{x}$
$\displaystyle=\frac{1}{2l_{0}}\frac{i\omega_{0}\alpha_{x}+\alpha_{y}E_{z}}{\omega_{0}^{2}-E_{z}^{2}},$
(1qahba) $\displaystyle d_{y}$
$\displaystyle=\frac{1}{2l_{0}}\frac{i\omega_{0}\alpha_{y}-\alpha_{x}E_{z}}{\omega_{0}^{2}-E_{z}^{2}},$
(1qahbb) $\displaystyle d_{z}$ $\displaystyle=0.$ (1qahbc)

The effective action of the SOC term in the ground state is described by the
Hamiltonian

$\displaystyle\hat{H}_{\eta}^{\left(2\right)}=\frac{1}{2}\left[\hat{\Upsilon},\hat{H}_{\eta}^{\left(1\right)}\right]=\frac{1}{2}$
$\displaystyle\left\\{\left[\left(f+\bm{d}\cdot\hat{\bm{\sigma}}\right),\left(el_{0}E\left(t\right)+(i/2l_{0})\bm{\alpha}\cdot\hat{\bm{\sigma}}\right)\right]\hat{a}^{\dagger}\hat{a}^{\dagger}\right.$
(1qahbd)
$\displaystyle-\left[\left(f^{*}+\bm{d}^{*}\cdot\hat{\bm{\sigma}}\right),\left(el_{0}E\left(t\right)-(i/2l_{0})\bm{\alpha}\cdot\hat{\bm{\sigma}}\right)\right]\hat{a}\hat{a}$
$\displaystyle+\left[\left(f+\bm{d}\cdot\hat{\bm{\sigma}}\right),\left(el_{0}E\left(t\right)-(i/2l_{0})\bm{\alpha}\cdot\hat{\bm{\sigma}}\right)\right]\hat{a}^{\dagger}\hat{a}$
$\displaystyle+\left(el_{0}E\left(t\right)-(i/2l_{0})\bm{\alpha}\cdot\hat{\bm{\sigma}}\right)\left(f+\bm{d}\cdot\hat{\bm{\sigma}}\right)$
$\displaystyle+\left[\left(el_{0}E\left(t\right)+(i/2l_{0})\bm{\alpha}\cdot\hat{\bm{\sigma}}\right),\left(f^{*}+\bm{d}^{*}\cdot\hat{\bm{\sigma}}\right)\right]\hat{a}^{\dagger}\hat{a}$
$\displaystyle\left.+\left(f^{*}+\bm{d}^{*}\cdot\hat{\bm{\sigma}}\right)\left(el_{0}E\left(t\right)+(i/2l_{0})\bm{\alpha}\cdot\hat{\bm{\sigma}}\right)\right\\}.$

The two-pair excitation processes
$\left(\propto\hat{a}\hat{a},\hat{a}^{\dagger}\hat{a}^{\dagger}\right)$
connect states that are separated in energy by $2\omega_{0}$ and therefore can
be neglected in our effective Hamiltonian approximation, while the terms
$\propto\hat{a}^{\dagger}\hat{a}$ do not contribute to the energy of the
ground state. Hence, we find the following

$\hat{H}_{\eta}^{\left(2\right)}=\frac{-E_{z}||\bm{\alpha}||^{2}}{2l_{0}^{2}\left(\omega_{0}^{2}-E_{z}^{2}\right)}\hat{\sigma}_{z}+\frac{E_{z}eE(t)}{\omega_{0}^{2}-E_{z}^{2}}\bm{\alpha}^{\perp}\cdot\hat{\bm{\sigma}}.$
(1qahbe)

where $\bm{\alpha}^{\perp}=(\alpha_{y},-\alpha_{x},0$). The first term shifts
the Zeeman splitting to

$\widetilde{E}_{z}=E_{z}\left(1-\frac{||\bm{\alpha}||^{2}}{2l_{0}^{2}\left(\omega_{0}^{2}-E_{z}^{2}\right)}\right),$
(1qahbf)

while the second term is the OME field that we sought. Crucially, it is
oriented perpendicular to the direction of the SOC field, i.e., to
$\bm{\alpha}$. Note that both terms require $E_{z}\neq 0$ to break the time-
reversal symmetry. Moreover, the constant part of the electric field $E(t)$ in
the second term will rotate the spin quantization axis. However, this rotation
is produced around the direction determined by $\bm{\alpha}$. As shown in the
following, the spin-flip amplitude is aligned in this direction and is
unaffected by this rotation. The Zeeman splitting along the new quantization
axis is given by

$\widetilde{E}_{z}\to\widetilde{E}_{z}\sqrt{1+\frac{e^{2}E_{0}^{2}||\bm{\alpha}||^{4}}{(\omega_{0}^{2}-E_{z}^{2})^{2}}}\approx\widetilde{E}_{z}\left(1+\frac{e^{2}E_{0}^{2}||\bm{\alpha}||^{4}}{2(\omega_{0}^{2}-E_{z}^{2})^{2}}\right),$
(1qahbg)

with $E_{0}$ being the constant part of the electric field. This is a next-
order effect compared to the shift from $E_{z}$ to $\widetilde{E}_{z}$ and can
be ignored. The OME term is similarly rotated, but this is also a higher-order
effect, and we disregard it as well.

Regarding tunneling amplitudes, we consider orthonormal Wannier functions of
the ground state of each dot, defined as [66]

$|w_{\eta,\sigma}\rangle=\frac{1}{\sqrt{N}}(|\psi_{\eta,0,\sigma}\rangle+\gamma|\psi_{\bar{\eta},0,\sigma}\rangle),$
(1qahbh)

where $N\equiv 1-2\gamma S+\gamma^{2}$, $\gamma\equiv(1-\sqrt{1-S^{2}})/S$,
and $S\equiv\langle\psi_{L,0,\sigma}|\psi_{R,0,\sigma}\rangle$ is the overlap
between the dot wave functions. Regardless of the particularities of $V(x)$,
we can consider a standard real-valued tunneling matrix element $\tau_{0}$
without loss of generality. The spin-flip tunneling amplitude can be obtained
as

$\tau_{\mathrm{sf}}=\langle
w_{L\sigma}|\hat{H}_{\mathrm{SO}}|w_{R\sigma^{\prime}}\rangle=\frac{1-\gamma^{2}}{N}\langle\sigma|\bm{\alpha}\cdot\hat{\bm{\sigma}}|\sigma^{\prime}\rangle\langle
L,0|k|R,0\rangle,$ (1qahbi)

with $|\eta,0\rangle$ the ground state of the respective harmonic oscillator,
as defined above. The general form of the expected value $\langle
L,0|k|R,0\rangle$ can be determined by imposing time-reversal invariance of
the spin-orbit Hamiltonian, i.e.,
$\mathcal{T}\hat{H}_{\mathrm{SO}}\mathcal{T}^{-1}=\hat{H}_{\mathrm{SO}}$. In
the most general way, the SOC Hamiltonian, written on the basis of
$\left\\{|L\uparrow\rangle,|L\downarrow\rangle,|R\uparrow\rangle,|R\downarrow\rangle\right\\}$,
reads as follows

$\hat{H}_{\mathrm{SO}}=\left({\begin{array}[]{cccc}0&0&0&\tau_{\mathrm{sf}}\\\
0&0&-\tau_{\mathrm{sf}}^{*}&0\\\ 0&-\tau_{\mathrm{sf}}&0&0\\\
\tau_{\mathrm{sf}}^{*}&0&0&0\end{array}}\right).$ (1qahbj)

Taking $\bm{\alpha}$ along the y-direction, as considered throughout this
work, we recover a term $\propto\hat{\tau}_{y}\hat{\sigma}_{y}$, which yields
a real spin-flip matrix of the form

$\hat{H}_{\mathrm{SO}}=\left({\begin{array}[]{cccc}0&0&0&\tau_{\mathrm{sf}}\\\
0&0&-\tau_{\mathrm{sf}}&0\\\ 0&-\tau_{\mathrm{sf}}&0&0\\\
\tau_{\mathrm{sf}}&0&0&0\end{array}}\right).$ (1qahbk)

The fact that the spin-flip tunneling term is $\propto\hat{\tau}_{y}$ is
crucial. Otherwise, the spin-flip and spin-conserving tunneling terms can
interfere destructively. Consider, for instance, that the tunneling is of the
form $\propto\hat{\tau}_{x}\hat{\sigma}_{x}$. By performing a $\pi/2$ rotation
around the y-axis, we obtain

$\hat{H}_{\mathrm{T}}=\left({\begin{array}[]{cccc}0&0&-\tau_{0}+\tau_{\mathrm{sf}}&0\\\
0&0&0&-\tau_{0}-\tau_{\mathrm{sf}}\\\ -\tau_{0}+\tau_{\mathrm{sf}}&0&0&0\\\
0&-\tau_{0}-\tau_{\mathrm{sf}}&0&0\end{array}}\right),$ (1qahbl)

which can exhibit a dark state when $\tau_{0}=\tau_{\mathrm{sf}}$, i.e., when
$\chi=0.5$. However, if, as here, we have a term
$\propto\hat{\tau}_{y}\hat{\sigma}_{y}$, the rotation yields

$\hat{H}_{\mathrm{T}}=\left({\begin{array}[]{cccc}0&0&-\tau_{0}-i\tau_{\mathrm{sf}}&0\\\
0&0&0&-\tau_{0}+i\tau_{\mathrm{sf}}\\\ -\tau_{0}+i\tau_{\mathrm{sf}}&0&0&0\\\
0&-\tau_{0}-i\tau_{\mathrm{sf}}&0&0\end{array}}\right),$ (1qahbm)

preventing destructive interference.

##  Appendix B TME for arbitrary ac amplitudes

In this appendix, we give the expressions for the TME terms in the time-
dependent case with arbitrary ac amplitudes. After a time-dependent SWT [94],
we obtain an effective Hamiltonian up to second order in the tunnel couplings,
given by

$\hat{H}_{\mathrm{eff}}^{\left(2\right)}\left(t\right)=\frac{\hat{\tau}_{z}}{2}\\{-\delta^{\left(2\right)}\left(t\right)+b_{z}^{\left(2\right)}\left(t\right)\hat{\sigma}_{z}+[b_{x}^{\left(2\right)}\left(t\right)\hat{\sigma}_{x}+b_{y}^{\left(2\right)}\left(t\right)\hat{\sigma}_{y}]\\},$
(1qahbn)

where the time-dependent detuning and Zeeman splittings are given by

$\displaystyle\delta^{\left(2\right)}\left(t\right)$
$\displaystyle=\sum_{\mu,\nu}J_{\mu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{\nu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)\cos[(\mu-\nu)\omega
t]$ (1qahbo)
$\displaystyle\times\left[\frac{2\tau_{0}^{2}}{\delta-\nu\omega}+\frac{\tau_{\mathrm{sf}}^{2}}{\delta+E_{z}+\nu\omega}+\frac{\tau_{\mathrm{sf}}^{2}}{\delta-
E_{z}+\nu\omega}\right],$ $\displaystyle b_{z}^{\left(2\right)}\left(t\right)$
$\displaystyle=\sum_{\mu,\nu}J_{\mu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{\nu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)\cos[(\mu-\nu)\omega
t]$ (1qahbp)
$\displaystyle\times\left[\frac{\tau_{\mathrm{sf}}^{2}}{\delta+E_{z}+\nu\omega}-\frac{\tau_{\mathrm{sf}}^{2}}{\delta-
E_{z}+\nu\omega}\right].$

The magnetic field gradient in the perpendicular direction is given by

$\displaystyle b_{x}^{\left(2\right)}\left(t\right)$
$\displaystyle=\sum_{\mu,\nu}J_{\mu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{\nu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)\cos[(\mu-\nu)(\omega
t+\phi)]$ (1qahbq)
$\displaystyle\times\left[\frac{\tau_{0}\tau_{\mathrm{sf}}}{\delta+E_{z}-\nu\omega}-\frac{\tau_{0}\tau_{\mathrm{sf}}}{\delta-
E_{z}-\nu\omega}\right]$ $\displaystyle b_{y}^{\left(2\right)}\left(t\right)$
$\displaystyle=\sum_{\mu,\nu}J_{\mu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{\nu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)\sin[(\mu-\nu)(\omega
t+\phi)]$ (1qahbr)
$\displaystyle\times\left(\frac{2\tau_{0}\tau_{\mathrm{sf}}}{\delta-\nu\omega}-\frac{\tau_{0}\tau_{\mathrm{sf}}}{\delta+E_{z}-\nu\omega}-\frac{\tau_{0}\tau_{\mathrm{sf}}}{\delta-
E_{z}-\nu\omega}\right).$

Note that these expressions involve two different photon numbers $\mu$ and
$\nu$, as they are virtual second-order tunneling processes that involve two
photo-assisted transitions [55, 57].

In an $n-$photon resonance $E_{z}\simeq n\omega$, the Hamiltonian in the RWA
is given by

$\hat{H}_{n}^{\left(2\right)}\left(t\right)=\frac{\hat{\tau}_{z}}{2}\\{-\widetilde{\delta}^{\left(2\right)}+(E_{z}+\widetilde{b}_{z}^{\left(2\right)}-n\omega)\hat{\sigma}_{z}+\widetilde{b}_{n,\perp}^{\left(2\right)}[\cos(n\phi)\hat{\sigma}_{x}+\sin(n\phi)\hat{\sigma}_{y}]\\},$
(1qahbs)

where the diagonal terms are given by

$\displaystyle\widetilde{\delta}^{\left(2\right)}$
$\displaystyle=\sum_{\nu}J_{\nu}^{2}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)\left[\frac{2\tau_{0}^{2}}{\delta-\nu\omega}+\frac{\tau_{\mathrm{sf}}^{2}}{\delta+E_{z}+\nu\omega}+\frac{\tau_{\mathrm{sf}}^{2}}{\delta-
E_{z}+\nu\omega}\right],$ (1qahbt)
$\displaystyle\widetilde{b}_{z}^{\left(2\right)}$
$\displaystyle=\sum_{\nu}J_{\nu}^{2}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)\left(\frac{\tau_{\mathrm{sf}}^{2}}{\delta+E_{z}+\nu\omega}-\frac{\tau_{\mathrm{sf}}^{2}}{\delta-
E_{z}+\nu\omega}\right),$ (1qahbu)

and the off-diagonal term has an amplitude

$\displaystyle\widetilde{b}_{n,\perp}^{(2)}$
$\displaystyle=\sum_{\nu}J_{\nu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{\nu+n}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)\left(\frac{\tau_{0}\tau_{\mathrm{sf}}}{\delta-\nu\omega}-\frac{\tau_{0}\tau_{\mathrm{sf}}}{\delta-
E_{z}-\nu\omega}\right)$ (1qahbv)
$\displaystyle-\sum_{\nu}J_{\nu}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)J_{\nu-n}\left(\frac{\epsilon_{\mathrm{ac}}}{\omega}\right)\left(\frac{\tau_{0}\tau_{\mathrm{sf}}}{\delta-\nu\omega}-\frac{\tau_{0}\tau_{\mathrm{sf}}}{\delta+E_{z}-\nu\omega}\right).$

Figure 8: Rabi frequency as a function of driving amplitude for different
total tunneling rates. The results are obtained with the full Hamiltonian
(symbols), the low-amplitude effective RWA Hamiltonian (dashed lines) shown in
equation 1qahai of the main text, and the RWA Hamiltonian (solid lines) of
equation 1qahbs taking into account up to five terms in the infinite series.
The total tunneling rate is $\tau=0.05\omega$ (yellow, triangles),
$\tau=0.1\omega$ (orange, squares), and $\tau=0.2\omega$ (brown, pentagons).
Other parameters are $E_{z}=\omega$, $\delta=\omega/2$, $\chi=0.2$ and
$\beta_{\mathrm{SO}}=0$.

If figure 8, we compare the Rabi frequency of a flopping-mode qubit obtained
with the full Hamiltonian, with the results given by the effective RWA
Hamiltonian shown above. To obtain numerical results, we have truncated the
summations to $\nu=-2,-1,\ldots,2$. The agreement between both results is
notable, even when working with tunneling rates of $\tau=0.2\omega$ We also
compare the prediction given by the low-amplitude effective Hamiltonian shown
in equation 1qahai of the main text. In the limit of
$\epsilon_{\mathrm{ac}}<\omega/2$ all the results coincide.

##  Appendix C TME under the spin resonance condition: asymmetry in $\delta$

To gain insight into the asymmetry in $\delta$ discussed in section 3.1 of the
main text, let us study the TME under spin resonance. That is, we consider the
Hamiltonian of equation 1qx in a situation where direct tunneling between the
two dots is energetically disfavored. This can always be satisfied by lowering
$\tau$. Hence, since in this appendix we are concerned with qualitative
understanding of the asymmetry, we do not regard the precise conditions of
validity and always assume a value of $\tau$ that makes the SWT valid. Under
this condition, virtual tunneling processes are dominant and we can employ a
SWT to obtain an effective Hamiltonian with the leading term being of second
order in the tunneling amplitudes, as done above for the flopping-mode
operation in section 5 of the main text. We apply these transformations in the
frame discussed in section 3.1 leading to equation 1qx, i.e., after applying
equation 1p and equation 1m), and working in the RWA. This is different from
the treatment of B, where SWT was applied before RWA. This amounts to
neglecting photo-assisted virtual tunneling processes. A complete treatment of
the TME in the presence of the OME lies outside of the scope of this work.

In this rotating frame, as discussed above, the $\beta_{\mathrm{SO}}$ term has
taken a role analogous to the Zeeman splitting. Applying the transformation
yields the following effective spin model for the left dot

$\hat{H}_{\mathrm{eff}}=\frac{1}{2}\left(\widetilde{E}_{z}+\widetilde{b}_{z}^{\left(2\right)}\right)\hat{\sigma}_{z}+\frac{\tilde{b}_{\perp}^{(2)}}{2}\hat{\sigma}_{x},$
(1qahbw)

where the effective magnetic field is given by

$\displaystyle\tilde{b}_{z}^{(2)}=$
$\displaystyle\frac{\widetilde{\tau}_{\downarrow}^{2}-\widetilde{\tau}_{\uparrow}^{2}}{\delta}+\frac{2\widetilde{E}_{z}\widetilde{\tau}_{\mathrm{sf}}^{2}}{\widetilde{E}_{z}^{2}-\delta^{2}},$
(1qahbx) $\displaystyle\tilde{b}_{\perp}^{(2)}=$
$\displaystyle\widetilde{\tau}_{\mathrm{sf}}\left(\frac{\widetilde{\tau}_{\downarrow}}{\widetilde{E}_{z}-\delta}-\frac{\widetilde{\tau}_{\uparrow}}{\widetilde{E}_{z}+\delta}-\frac{\widetilde{\tau}_{\uparrow}+\widetilde{\tau}_{\downarrow}}{\delta}\right).$
(1qahby)

The term $\widetilde{b}_{z}^{(2)}$ is non-zero in the presence of spin-
dependent tunneling amplitudes
$|\widetilde{\tau}_{\downarrow}|\neq|\widetilde{\tau}_{\downarrow}|$, and a
finite spin-flip tunneling rate $\widetilde{\tau}_{\mathrm{sf}}$. The
effective Hamiltonian is not symmetric (nor antisymmetric) in $\delta$.
However, it is symmetric under reflection $\delta\to-\delta$ and
$\epsilon_{\mathrm{ac}}\to-\epsilon_{\mathrm{ac}}$. Under this reflection, the
effective tunneling rates transform as
$\widetilde{\tau}_{\uparrow}\rightleftharpoons\widetilde{\tau}_{\downarrow}$,
and $\widetilde{\tau}_{\mathrm{sf}}\to-\widetilde{\tau}_{\mathrm{sf}}$ for an
odd resonance $E_{z}=(2k+1)\omega,k\in\mathds{Z}$, as discussed in section
3.1.

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