# Coherent control of spin tunneling in a spin-orbit coupled bosonic triple
well

Yuxin Luo1, Jia Yi2, Wenjuan Li1, Xin Xie1, Xiaobing Luo3,4, and Yunrong
Luo1111Corresponding author: lyr<EMAIL_ADDRESS>1Key Laboratory for Matter
Microstructure and Function of Hunan Province, and Key Laboratory of Low-
dimensional Quantum Structures and Quantum Control of Ministry of Education,
School of Physics and Electronics, Hunan Normal University, Changsha 410081,
China
2School of Management, Hunan University of Information Technology, Changsha
410151, China
3Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China
4Department of Physics, Jinggangshan University, Ji’an 343009, China

###### Abstract

We study the coherent control of spin tunneling for a spin-orbit (SO) coupled
boson held in a driven triple well. Under high-frequency approximation, we
analytically obtain the quasienergies of the SO-coupled bosonic triple-well
system and fine energy band structure is displayed. By adjusting the driving
parameters, we reveal that the directed selective spin-flipping or spin-
conserving tunneling of a SO-coupled boson occurs along different pathways and
in different directions. The analytical results are numerically confirmed and
perfect agreements are found. Further, a scheme of quantum spin switch with or
without spin-flipping is presented. These results may be useful for quantum
information processing and the design of spintronic devices.

###### pacs:

03.65.Xp, 32.80.Qk, 68.65.Fg, 71.70.Ej

##  I Introduction

Spin-orbit (SO) coupling, the interaction between the spin and motion of a
quantum particle, plays a crucial role in many exotic condensed matter
phenomena and applications, such as spin-Hall effectkato306 , topological
insulatorbernevig314 , the persistent spin helixkoralek458 , spintronic
deviceszutic , etc. In cold atomic gases, SO coupling can be generated through
the interaction between laser and atom, which yields Abelian or non-Abelian
gauge fields for atoms in the dressed two atomic internal states. An important
difference between electrons and ultracold atoms is that electrons are
fermions and ultracold atoms may be bosons. Therefore, SO-coupled ultracold
atomic gases will lead to novel SO physics that has not been explored in solid
materials.

In recent experiments, artificial SO coupling of both bosonic and fermionic
ultracold atoms has been realizedlin471 ; wang109 ; cheuk109 ; zhang109 ;
huang12 ; wu354 , which provides an ideal platform to study novel SO coupling
physics, owing to their unprecedented level of controllability over all
available experimental parameters. There have been a number of research works
focusing on the intriguing dynamics of the SO-coupled cold atomic gases, such
as Josephson dynamics of SO-coupled Bose-Einstein condensates in a double-well
potentialzhang609 ; garcia89 ; citro224 , selective spin transportation in a
SO-coupled bosonic junctionyu90 , Anderson localization of cold atomic gases
with effective SO interaction in a quasiperiodic optical latticezhou87 , Klein
tunnelingzhang628 , collective dynamicszhang109 ; chen86 ; hu93 ,
nonequilibrium dynamics of SO-coupled lattice bosonsng92 , tunable Landau-
Zener transitions in SO-coupled atomic gasesolson90 , spin dynamics of SO-
coupled Bose-Einstein condensates in a random potentialmardonov115 ,
controlling spin-dependent localization and directed transport in a bipartite
latticeluo93 , Bloch oscillations of SO-coupled cold atoms in an optical
latticeji99 , dynamics of SO-coupled cold atomic gases in a Floquet lattice
with an impurityluo52 , controlling stable tunneling in a non-Hermitian SO-
coupled bosonic junctionluo22 , the superflow of SO-coupled Bose-Einstein
condensates in optical latticesluoxb103 , controlling directed atomic motion
and second-order tunneling of a SO-coupled atom in optical latticesluo103 ,
and so on.

As mentioned above, many works on the tunneling dynamical properties of SO-
coupled ultracold atomic gases, which have focused on systems with double-well
or optical lattice potentials. To the best of our knowledge, research on the
coherent control of spin tunneling for SO-coupled cold atomic gases held in a
triple well is extremely rare. However, the triple-well system is a typical
model to demonstrate the coherent control of spin tunneling dynamics and is an
important bridge between the double well and the optical lattice systems to
fully understand the spin tunneling and transport of SO-coupled cold atomic
gases in the quantum wells. Thus, it motivates us to study the spin tunneling
dynamics in a SO-coupled bosonic triple-well system.

In this paper, we theoretically investigate the coherent control of spin
tunneling for a SO-coupled boson held in a driven triple-well potential. In
high-frequency approximation, the quasienergies of the SO-coupled cold atomic
triple-well system are analytically obtained and the quasienergy spectrum is
shown. By adjusting the time-dependent driving field, we can manipulate the
directed selective spin-flipping or non-spin-flipping tunneling of a SO-
coupled boson along different pathways and in different directions. The
analytical results are confirmed by direct numerical simulations and good
agreements are displayed. Further, we present an interesting scheme of quantum
spin tunneling switch with or without spin-flipping for transporting a SO-
coupled boson from well 1 to well 3. The results may be useful in the design
of spintronic devices and can be observed with the current experimental
setups.

##  II Analytical solutions and quasienergy spectra in the high-frequency
approximation

We consider a single SO-coupled ultracold boson held in a driven triple-well
potential, in which the dynamics of the system is governed by a Hamiltonian
luo93 ; li91

$\displaystyle\hat{H}(t)$ $\displaystyle=$
$\displaystyle-\nu(\hat{a}^{{\dagger}}_{1}e^{-i\pi\gamma\hat{\sigma}_{y}}\hat{a}_{2}+\hat{a}^{{\dagger}}_{2}e^{-i\pi\gamma\hat{\sigma}_{y}}\hat{a}_{3}+H.c.)$
$\displaystyle+$
$\displaystyle\frac{\Omega}{2}\sum_{j}(\hat{n}_{j\uparrow}-\hat{n}_{j\downarrow})+\sum_{\sigma}[\varepsilon_{1}(t)\hat{n}_{1\sigma}-\varepsilon_{2}(t)\hat{n}_{3\sigma}].$

Here $\hat{a}_{j}=(\hat{a}_{j\uparrow},\hat{a}_{j\downarrow})^{T}$ (the
superscript $T$ stands for the matrix transpose) is the two-component vector
with elements being the annihilation operators of spin-up and spin-down atoms
in the $j$th $(j=1,2,3)$ well, and $\hat{a}_{j}^{{\dagger}}$ denotes its
Hermite conjugation with elements being the creation operators. $\nu$ denotes
the tunneling amplitude without SO coupling, $\gamma$ characterizes the SO
coupling strength, $\hat{\sigma}_{y}$ is the $y$ component of Pauli operator,
$\Omega$ is the effective Zeeman field intensity, and
$\hat{n}_{j\sigma}=\hat{a}_{j\sigma}^{{\dagger}}\hat{a}_{j\sigma}$ denotes the
number operator for spin $\sigma$ $(\sigma=\uparrow,\downarrow)$ in well $j$.
The function $\varepsilon_{1,2}(t)=\varepsilon_{1,2}\cos(\omega t)$ is the
periodic driving field, in which $\varepsilon_{1,2}$ and $\omega$ are the
driving amplitude and frequency respectively li91 . Throughout this paper,
$\hbar=1$ is adopted and the dimensionless parameters $\nu$, $\Omega$,
$\varepsilon_{1,2}$, $\omega$ are in units of the reference frequency
$\omega_{0}=0.1E_{r}$, with $E_{r}=k_{L}^{2}/(2m)$ being the single-photon
recoil energy, and time $t$ is normalized in units of $\omega_{0}^{-1}$. Note
that the single-photon recoil energy is $E_{r}=k_{L}^{2}/(2m)=22.5$ kHz and
the Zeeman field $\Omega$ is set as $-40\omega_{0}\sim 40\omega_{0}$ in the
experimentlin471 , and the experimentally achievable system parameters can be
tuned in a wide range as follows yu90 ; luo93 ; luo22 ; chen107 :
$\nu\sim\omega_{0}$, $\varepsilon_{1,2}\sim\omega\in[0,100](\omega_{0})$,
$\Omega\sim\omega$.

Using the Fock basis $|\sigma,0,0\rangle$ (or $|0,\sigma,0\rangle$ or
$|0,0,\sigma\rangle$) to represent the state of a spin $\sigma$ atom occupying
the well 1 (or well 2 or well 3) and no atom in the other two wells, we can
expand the quantum state of the SO-coupled system as

$\displaystyle|\psi(t)\rangle$ $\displaystyle=$ $\displaystyle
a_{1}(t)|\uparrow,0,0\rangle+a_{2}(t)|\downarrow,0,0\rangle$ (2)
$\displaystyle+$ $\displaystyle
a_{3}(t)|0,\uparrow,0\rangle+a_{4}(t)|0,\downarrow,0\rangle$ $\displaystyle+$
$\displaystyle a_{5}(t)|0,0,\uparrow\rangle+a_{6}(t)|0,0,\downarrow\rangle,$

where $a_{k}(t)$ ($k=1,2,...,6$) denotes the time-dependent probability
amplitude of the boson being in the corresponding Fock state
$|\sigma,0,0\rangle$ or $|0,\sigma,0\rangle$ or $|0,0,\sigma\rangle$(e.g.,
$a_{1}(t)$ denotes the time-dependent probability amplitude of the atom being
in state $|\uparrow,0,0\rangle$). The corresponding probability reads
$P_{k}(t)=|a_{k}(t)|^{2}$, which obeys the normalization condition
$\sum_{k=1}^{6}P_{k}(t)=1$. Inserting equations (1) and (2) into Schrödinger
equation $i\frac{\partial|\psi(t)\rangle}{\partial
t}=\hat{H}(t)|\psi(t)\rangle$ results in the coupled equations

$\displaystyle i\dot{a}_{1}(t)$ $\displaystyle=$
$\displaystyle[\frac{\Omega}{2}+\varepsilon_{1}\cos(\omega
t)]a_{1}(t)-\nu\cos(\pi\gamma)a_{3}(t)$ $\displaystyle+$
$\displaystyle\nu\sin(\pi\gamma)a_{4}(t),$ $\displaystyle i\dot{a}_{2}(t)$
$\displaystyle=$ $\displaystyle[-\frac{\Omega}{2}+\varepsilon_{1}\cos(\omega
t)]a_{2}(t)-\nu\sin(\pi\gamma)a_{3}(t)$ $\displaystyle+$
$\displaystyle\nu\cos(\pi\gamma)a_{4}(t),$ $\displaystyle i\dot{a}_{3}(t)$
$\displaystyle=$
$\displaystyle-\nu\cos(\pi\gamma)[a_{1}(t)+a_{5}(t)]-\nu\sin(\pi\gamma)$
$\displaystyle\times$
$\displaystyle[a_{2}(t)-a_{6}(t)]+\frac{\Omega}{2}a_{3}(t),$ $\displaystyle
i\dot{a}_{4}(t)$ $\displaystyle=$
$\displaystyle-\nu\sin(\pi\gamma)[a_{1}(t)+a_{5}(t)]-\nu\cos(\pi\gamma)$
$\displaystyle\times$
$\displaystyle[a_{2}(t)+a_{6}(t)]-\frac{\Omega}{2}a_{4}(t),$ $\displaystyle
i\dot{a}_{5}(t)$ $\displaystyle=$
$\displaystyle[\frac{\Omega}{2}-\varepsilon_{2}\cos(\omega
t)]a_{5}(t)-\nu\cos(\pi\gamma)a_{3}(t)$ $\displaystyle-$
$\displaystyle\nu\sin(\pi\gamma)a_{4}(t),$ $\displaystyle i\dot{a}_{6}(t)$
$\displaystyle=$ $\displaystyle[-\frac{\Omega}{2}-\varepsilon_{2}\cos(\omega
t)]a_{6}(t)+\nu\sin(\pi\gamma)a_{3}(t)$ (3) $\displaystyle-$
$\displaystyle\nu\cos(\pi\gamma)a_{4}(t).$

It is difficult to obtain the exact analytical solutions of equation (3),
because of the periodically varying coefficients. However, in the high-
frequency approximationblanes470 ; thimmel9 , it can become a set of linear
equations with constant coefficients, which is analytically solvable.
Therefore, we introduce the slowly varying function of time
$b_{k}(t)(k=1,2,...,6)$ through the transformation
$a_{1}(t)=b_{1}(t)e^{-i\int[\frac{\Omega}{2}+\varepsilon_{1}\cos(\omega
t)]dt}$,
$a_{2}(t)=b_{2}(t)e^{-i\int[-\frac{\Omega}{2}+\varepsilon_{1}\cos(\omega
t)]dt}$, $a_{3}(t)=b_{3}(t)e^{-i\int\frac{\Omega}{2}dt}$,
$a_{4}(t)=b_{4}(t)e^{i\int\frac{\Omega}{2}dt}$,
$a_{5}(t)=b_{5}(t)e^{-i\int[\frac{\Omega}{2}-\varepsilon_{2}\cos(\omega
t)]dt}$,
$a_{6}(t)=b_{6}(t)e^{i\int[\frac{\Omega}{2}+\varepsilon_{2}\cos(\omega
t)]dt}$. Under the high-frequency approximation and by using of the Fourier
expansion $e^{\pm
i\int\varepsilon_{1,2}(t)dt}=\sum_{n=-\infty}^{\infty}\mathcal{J}_{n}(\frac{\varepsilon_{1,2}}{\omega})e^{\pm
in\omega t}$ and $e^{\pm
i\int[\varepsilon_{1,2}(t)\pm\Omega]dt}=\sum_{n^{\prime}=-\infty}^{\infty}\mathcal{J}_{n^{\prime}}(\frac{\varepsilon_{1,2}}{\omega})e^{\pm
i(n^{\prime}\pm\frac{\Omega}{\omega})\omega t}$, we can neglect these rapidly
oscillating terms of the Fourier expansion with $n\neq 0$ and
$n^{\prime}\pm\frac{\Omega}{\omega}\neq 0$zou46  and the set of differential
equation (3) is transformed to the form

$\displaystyle i\dot{b}_{1}(t)$ $\displaystyle=$ $\displaystyle-
J_{1}b_{3}(t)+J_{2}b_{4}(t),$ $\displaystyle i\dot{b}_{2}(t)$ $\displaystyle=$
$\displaystyle-J_{3}b_{3}(t)-J_{1}b_{4}(t),$ $\displaystyle i\dot{b}_{3}(t)$
$\displaystyle=$ $\displaystyle-
J_{1}b_{1}(t)-J_{3}b_{2}(t)-J_{4}b_{5}(t)+J_{5}b_{6}(t),$ $\displaystyle
i\dot{b}_{4}(t)$ $\displaystyle=$ $\displaystyle
J_{2}b_{1}(t)-J_{1}b_{2}(t)-J_{6}b_{5}(t)-J_{4}b_{6}(t),$ $\displaystyle
i\dot{b}_{5}(t)$ $\displaystyle=$ $\displaystyle-J_{4}b_{3}(t)-J_{6}b_{4}(t),$
$\displaystyle i\dot{b}_{6}(t)$ $\displaystyle=$ $\displaystyle
J_{5}b_{3}(t)-J_{4}b_{4}(t),$ (4)

where the effective coupling constants are written as
$J_{1}=\nu\cos(\pi\gamma)\mathcal{J}_{0}(\frac{\varepsilon_{1}}{\omega})$,
$J_{2,3}=\nu\sin(\pi\gamma)\mathcal{J}_{\mp\frac{\Omega}{\omega}}(\frac{\varepsilon_{1}}{\omega})$,
$J_{4}=\nu\cos(\pi\gamma)\mathcal{J}_{0}(\frac{\varepsilon_{2}}{\omega})$, and
$J_{5,6}=\nu\sin(\pi\gamma)\mathcal{J}_{\mp\frac{\Omega}{\omega}}(\frac{\varepsilon_{2}}{\omega})$
with $\mathcal{J}_{n}(x)$ being the $n$-order Bessel function of $x$. Equation
(4) is effective in the description of the spin tunneling dynamics of the
original system, which is the basis of the following analysis.

Figure 1: (Color online) Quasienergy as a function of the driving parameter
$\varepsilon_{1}/\omega$ for $\gamma=0.5,\nu=1,\omega=50,\Omega=100$, and (a)
$\varepsilon_{2}=5.1356\omega$; (b) $\varepsilon_{2}=2\omega$. Here, circles
label the analytical results and solid curves denote the numerical
correspondences. Hereafter, any parameter adopted in the figures is
dimensionless.

Based on the Floquet theoremshirley138 , the analytical Floquet state, non-
Floquet state, and quasienergies can be obtained from equation (4). The
Floquet solution is constructed as $|\psi(t)\rangle=|\varphi(t)\rangle
e^{-iEt}$, where $E$ is the Floquet quasienergy and
$|\varphi(t)\rangle=A_{1}e^{-i\int[\frac{\Omega}{2}+\varepsilon_{1}\cos(\omega
t)]dt}|\uparrow,0,0\rangle+A_{2}e^{-i\int[-\frac{\Omega}{2}+\varepsilon_{1}\cos(\omega
t)]dt}|\downarrow,0,0\rangle+A_{3}e^{-i\int\frac{\Omega}{2}dt}|0,\uparrow,0\rangle+A_{4}e^{i\int\frac{\Omega}{2}dt}|0,\downarrow,0\rangle+A_{5}e^{-i\int[\frac{\Omega}{2}-\varepsilon_{2}\cos(\omega
t)]dt}|0,0,\uparrow\rangle+A_{6}e^{i\int[\frac{\Omega}{2}+\varepsilon_{2}\cos(\omega
t)]dt}|0,0,\downarrow\rangle$ is the Floquet state with the same period as the
Hamiltonian (1). According to the superposition principle of quantum
mechanics, the non-Floquet state can be constructed by the linear
superposition of the Floquet states, which implies a quantum interference
effect among the Floquet states with different quasienergyluo22 ; lu379 ;
luo48 ; luo7 . We have introduced the stationary solution of equation (4) as
$b_{k}(t)=A_{k}e^{-iEt}$ with $A_{k}$ being constant. Inserting such a form of
$b_{k}(t)$ into equation (4), the constant $A_{k}$ and Floquet quasienergy $E$
can be obtained. Here, we give the Floquet quasienergies as

$\displaystyle
E_{1,2}=0,E_{3,4}=\mp\frac{\sqrt{\alpha-\sqrt{\beta}}}{\sqrt{2}},E_{5,6}=\mp\frac{\sqrt{\alpha+\sqrt{\beta}}}{\sqrt{2}},$
(5)

with the constants
$\alpha=2J_{1}^{2}+J_{2}^{2}+J_{3}^{2}+2J_{4}^{2}+J_{5}^{2}+J_{6}^{2}$ and
$\beta=J_{2}^{4}+4J_{1}^{2}(J_{2}-J_{3})^{2}+J_{3}^{4}+2J_{3}^{2}J_{5}^{2}+4J_{4}^{2}J_{5}^{2}+J_{5}^{4}+8J_{1}J_{4}(J_{2}-J_{3})(J_{5}-J_{6})-8J_{4}^{2}J_{5}J_{6}+J_{6}^{4}-2J_{6}^{2}(J_{3}^{2}-2J_{4}^{2}+J_{5}^{2})-2J_{2}^{2}(J_{3}^{2}+J_{5}^{2}-J_{6}^{2})$
being adjusted by the system parameters, so that all the quasienergies are
determined for a set of fixed parameters except for the degenerate zero
quasienergies $E_{1}$ and $E_{2}$.

From equation (5), it is surprisingly found that when the SO coupling strength
$\gamma$ is integer or half integer, the constant $\beta$ will equal to zero.
This will result in the quasienergies $E_{3}=E_{5}=-\sqrt{\alpha/2}$ and
$E_{4}=E_{6}=\sqrt{\alpha/2}$, which means that the new quasienergy degeneracy
occurs. To our knowledge, the degeneracy of energy levels generally implies
quantum decoherence, so it will lead to the intriguing phenomenon of quantum
tunneling, for instance, the selective coherent destruction of tunneling
(SCDT) takes place at the degeneracy (crossing) point of the partial energy
levels lu379 ; luo48 ; luo7  and coherent destruction of tunneling (CDT)
occurs at the collapse (crossing) point of all energy levelsgrossmann ; liu377
. As an example, we set the parameters
$\gamma=0.5,\omega=50,\Omega=100,\nu=1$, and (a)
$\varepsilon_{2}=5.1356\omega$, (b) $\varepsilon_{2}=2\omega$ to plot the
quasienergy spectra with quasienergy as a function of the driving parameter
$\varepsilon_{1}/\omega$, as in figure 1(a)-(b), respectively. Here, the
circle points denote the analytical results based on equation (5) and the
curves label the numerical results from the exact model (3). The two sets of
parameters lead to
$E_{3}=E_{5}=-|\mathcal{J}_{2}(\frac{\varepsilon_{1}}{\omega})|$ and
$E_{4}=E_{6}=|\mathcal{J}_{2}(\frac{\varepsilon_{1}}{\omega})|$, and
$E_{3}=E_{5}=-\sqrt{\mathcal{J}_{2}(\frac{\varepsilon_{1}}{\omega})^{2}+\mathcal{J}_{2}(2)^{2}}$
and
$E_{4}=E_{6}=\sqrt{\mathcal{J}_{2}(\frac{\varepsilon_{1}}{\omega})^{2}+\mathcal{J}_{2}(2)^{2}}$
in equation (5), respectively. In figure 1 (a), we observe that the curves of
quasienergies $E_{1}$ and $E_{2}$, $E_{3}$ and $E_{5}$, $E_{4}$ and $E_{6}$
are degenerate respectively. The numerical results (curves) are in good
agreement with the analytical ones (circles). Not only that, there exists some
collapse (crossing) points of the quasienergy spectra (i.e., see the amplified
inset of figure 1(a)), which correspond to the roots of equation
$\mathcal{J}_{2}(\varepsilon_{1}/\omega)=0$ (i.e.,
$\mathcal{J}_{2}(\varepsilon_{1}/\omega)=\mathcal{J}_{2}(5.1356)=0$). Compared
to the single energy band structure shown in figure 1 (a), the quasienergy
curves are divided to three energy bands in figure 1 (b). The energy gap
between two adjacent energy bands is equal to the minimum value
$\mathcal{J}_{2}(2)\approx 0.3528$ of quasienergy $E_{4}$ or $E_{6}$.
Furthermore, from figure 1 (b) it can be seen that the two energy-level curves
of each band from the numerical results are approximately degenerate (e.g.,
quasienergy curves of $E_{1}$ and $E_{2}$, $E_{3}$ and $E_{5}$, $E_{4}$ and
$E_{6}$), which have some small deviations from analytical results (circles).

##  III Directed tunneling and spin switch with or without spin-flipping

Coherent control of quantum tunneling is an interesting and important research
subject and possesses many potential applications in quantum information
technologiesmonroe ; kral . Based on the fact that the spin tunneling dynamics
of a SO-coupled cold atom depends strongly on the periodic driving external
field in a triple well, the quantum spin tunneling and transport of a SO-
coupled atom can be manipulated via adjusting the driving parameters. In the
section, we will focus on studying the directed tunneling and spin switch with
or without spin-flipping.

###  III.1 Directed tunneling with or without spin-flipping

Figure 2: (Color online) Time evolutions of the probabilities $P_{k}(t)$ for
the system parameters $\gamma=0.5,\nu=1,\omega=50,\Omega=100$, and (a)
$\varepsilon_{1}=2\omega$, $\varepsilon_{2}=5.1356\omega$; (b)
$\varepsilon_{1}=\varepsilon_{2}=5.1356\omega$; (c)
$\varepsilon_{1}=5.1356\omega$, $\varepsilon_{2}=2\omega$, starting the system
with a spin-up particle in the well 2. Dashed lines label the analytical
results and solid curves denote the numerical correspondences.

Figure 3: (Color online) (a) Time evolutions of the probabilities $P_{k}(t)$
for the system parameters $\gamma=0.5,\nu=1,\omega=50,\Omega=100$, and
$\varepsilon_{1}=\varepsilon_{2}=5.1356\omega$ in the time intervals $0\leq
t<t_{1}=2$, $t_{2}=6.4\leq t<t_{3}=8.4$, $t\geq t_{4}=12.8$, and
$\varepsilon_{1}=2\omega$, $\varepsilon_{2}=5.1356\omega$ in the time interval
$t_{1}=2\leq t<t_{2}=6.4$, and $\varepsilon_{1}=5.1356\omega$,
$\varepsilon_{2}=2\omega$ in the time interval $t_{3}=8.4\leq t<t_{4}=12.8$,
starting the system with a spin-up particle in the well 1. (b) Schematic
diagram of the scheme for spin-flipping transporting a spin-up particle from
well 1 to well 3. The time intervals $\triangle t_{1}=\triangle
t_{2}=t_{2}-t_{1}=t_{4}-t_{3}=4.4$ denote the transferring times between the
different populations.

For a SO-coupled cold atomic triple-well system, an attractive physical
problem is how to control the directed selective spin-flipping or spin-
conserving tunneling of a SO-coupled atom from initial middle well to left
well or to right well. Based on equation (4), it can be seen that when the SO-
coupling strength $\gamma$ is half integer or integer, quantum tunneling with
or without spin-flipping can occur. Here, we will try to manipulate the
directed selective spin-flipping tunneling of a spin-up boson initially
located in the well 2 as an example. In order to realize the directed spin-
flipping tunneling of the spin-up particle from well 2 to well 1, it is easily
found from equation (4) that when the effective coupling constants satisfy
$J_{1}=J_{4}=J_{5}=0$ and $J_{3}\neq 0$, $b_{3}$ is only coupled with $b_{2}$.
It means that the SCDT occurs. We take the parameters
$\gamma=0.5,\nu=1,\omega=50,\Omega=100,\varepsilon_{1}=2\omega,\varepsilon_{2}=5.1356\omega$
satisfying the conditions $J_{1}=J_{4}=J_{5}=0$ and $J_{3}\neq 0$ to plot the
time evolutions of the probabilities in figure 2 (a). Clearly, the spin-
flipping tunneling passage between well 2 and well 3 is shut off and the
particle performs a spin-flipping Rabi oscillation along this pathway between
well 2 and well 1, namely, the directed selective spin-flipping tunneling
takes place between initial state $|0,\uparrow,0\rangle$ and state
$|\downarrow,0,0\rangle$ with tunneling time $\triangle t\approx 4.4$. It
means the occurrence of SCDT and the set of parameters corresponds to the
degeneracy position of the partial quasienergies in figure 1 (a). If we set
the same parameters as that in figure 2 (a) except for
$\varepsilon_{1}=5.1356\omega$, which correspond to the collapse (crossing)
point of all the quasienergies in figure 1 (a), the quantum tunneling of the
particle will be frozen. It means the CDT occurs as shown in figure 2 (b).

Further, from equation (4) it can also be found when the effective coupling
constants satisfy $J_{1}=J_{3}=J_{4}=0$ and $J_{5}\neq 0$, the probability
function $b_{3}$ is only related to function $b_{6}$, which means the SCDT
happens. We fix the same initial condition and parameters as that in figure 2
(a) except for $\varepsilon_{1}=5.1356\omega$ and $\varepsilon_{2}=2\omega$ to
plot the time evolutions of the probabilities in figure 2 (c). It is clearly
seen that the directed selective spin-flipping tunneling along another pathway
between well 2 and well 3 occurs, in which the spin particle performs a Rabi
oscillation with spin-flipping between state $|0,\uparrow,0\rangle$ and state
$|0,0,\downarrow\rangle$ with tunneling time $\triangle t\approx 4.4$. It
means the SCDT occurs and the set of parameters corresponds to the degeneracy
location of the partial quasienergies in figure 1 (b). Here, we note that the
analytical results (dashed lines) based on equation (4) are in good agreement
with the numerical ones from equation (1) in figure 2. Because the above-
mentioned results are related to control the directed spin-flipping tunneling
of a spin boson, it is similar to the case of manipulating the directed spin-
conserving tunneling of the particle.

###  III.2 spin switch with or without spin-flipping

From the above subsection, we find the time-dependent driving field affects
dramatically spin tunneling dynamics of this system, thus we can design the
quantum spin switch with or without spin-flipping by applying a sequence of
sudden modulation of driving strength to control opening or closing of quantum
tunneling. The scheme of sudden modulation of driving parameters has been
performed in many research worksweiss ; creffield ; hai82 ; luo84 . Here, we
present a scheme of quantum spin switch with spin-flipping by adjusting
driving strength as an example, as shown in figure 3. In figure 3 (a), let a
spin-up boson occupy the state $|\uparrow,0,0\rangle$ initially and fix the
parameters
$\gamma=0.5,\nu=1,\omega=50,\Omega=100,\varepsilon_{1}=\varepsilon_{2}=5.1356\omega$
corresponding to the collapse (crossing) point of all the quasienergies in
figure 1 (a). One can see that the initially occupied state is kept, due to
the CDT effect. At an arbitrarily given time $t=t_{1}=2$, we change the
driving strength $\varepsilon_{1}$ to $\varepsilon_{1}=2\omega$ and keep this
value until $t=t_{2}=6.4$ at which the spin-up particle transits completely
from state $|\uparrow,0,0\rangle$ to state $|0,\downarrow,0\rangle$. Then we
return the driving strength $\varepsilon_{1}$ to the initial value such that
the state $|0,\downarrow,0\rangle$ is frozen which is attributed to the effect
of CDT. At an arbitrarily given time $t=t_{3}=8.4$, we change the driving
strength $\varepsilon_{2}$ to $\varepsilon_{2}=2\omega$ and keep this value
until $t=t_{4}=12.8$. At this time, the spin-down particle transits completely
from state $|0,\downarrow,0\rangle$ to state $|0,0,\uparrow\rangle$. Then, we
return the driving strength $\varepsilon_{2}$ to
$\varepsilon_{2}=5.1356\omega$, such that the CDT effect makes the final state
$|0,0,\uparrow\rangle$ is kept. So the spin-up boson is successfully
transported through two spin-flipping tunnels from well 1 to well 3 by
adjusting the driving strength $\varepsilon_{i}$ ($i=1,2$) of the time-
dependent driving fields, namely, the quantum spin tunneling switch with spin-
flipping is theoretically realized. The spatial distributions of the spin
particle at the tuning moments are exhibited in figure 3 (b), where $\triangle
t_{i}$ denotes transferring time between the different populations and the
transferring times $\triangle t_{1}=\triangle
t_{2}=t_{2}-t_{1}=t_{4}-t_{3}=4.4$. Similarly, the quantum spin tunneling
switch without spin-flipping can also be performed by modulating the driving
fields.

##  IV conclusion and discussion

In summary, we have studied the coherent control of spin tunneling for a SO-
coupled boson trapped in a driven triple well. In the high-frequency regime,
we analytically obtain the Floquet quasienergies of the system and the fine
quasienergy spectrum is shown. By adjusting the strength of driving external
field, we can control the directed selective spin-flipping or non-spin-
flipping tunneling of a SO-coupled boson along different pathways and in
different directions from the middle well to one of the other two wells. The
analytical results are demonstrated by direct numerical simulations and good
agreements are shown. Further, based on the combined effect of CDT and SCDT,
we propose an interesting scheme of quantum spin switch that a spin-up
particle is transported with or without spin-flipping from the initial well 1
to the final well 3. These results may be useful for quantum information
processing and the design of quantum spintronic devices. We expect that the
future experiments along the lines sketched here can be considered, which may
help us further to gain new insight into the quantum spin tunneling dynamics
of SO-coupled ultracold gases in multi-well systems.

## ACKNOWLEDGMENTS

This work was supported by the Hunan Provincial Natural Science Foundation of
China under Grants No. 2021JJ30435 and No. 2017JJ3208, the National Natural
Science Foundation of China under Grant No. 11747034, and the Scientific
Research Fund of Hunan Provincial Education Department under Grant No.
18C0027. Xiaobing Luo was supported by the Scientific and Technological
Research Fund of Jiangxi Provincial Education Department (numbers GJJ180559),
and Open Research Fund Program of the State Key Laboratory of Low-Dimensional
Quantum Physics (KF201903).

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