Exact solutions for a spin-orbit coupled bosonic double-well system

Exact solutions have played a central role in many branches of physicsbarnes109 , due to the fact that they not only can supplement numerical simulations, but also can deepen understanding of the underlying physicsZakrzewski32 . Therefore, since the birth of quantum mechanics, the search for exact solutions of quantum systems has been ongoingbarnes109 . Although widely studied over the past few decades, up to now many accurate solutions are obtained, which are based on two-level (or two-state) systemsbarnes109 ; Zakrzewski32 ; landau2 ; zener137 ; rosen40 ; rabi51 ; hioe32 ; hioe30 ; hai87 ; luoxb95 ; xie82 ; vitanov9 ; JC51 , such as the paradigmatic Landau-Zenerlandau2 ; zener137 and Rabirabi51 models, driven two-level systemsbarnes109 ; xie82 , complete population inversion by a phase jump for a two-state systemvitanov9 , the Jaynes-Cummings modelJC51 , and so on. The accurate analytic solution has proved to be very important in the contexts of qubit controlgre5 ; poem107 ; hai29 and has been extended to a series of precise controlsZakrzewski32 ; hioe30 ; hai87 ; luoxb95 ; xie82 ; vitanov9 ; bambini23 ; kyo71 ; gang82 . Not only that, the accurate solutions are extremely useful for the fundamental importance of quantum systems. However, for quantum systems with more levels, the exact solutions are more difficult to handle and are still lackingbere ; llorente .

In recent years, the study of spin-orbit (SO) coupling has been a hot topic, which has led to a number of interesting physical phenomenakato306 ; bernevig314 ; zutic76 ; step93 . It is found that SO coupling is specially appropriate for qubit control in the spintronic deviceszutic76 ; step93 ; flindt97 ; bednarek101 and plays a crucial role in topological quantum computingnayak80 . Synthetic SO coupling of both Bose and Fermi gases has been realized in recent experimentslin471 ; wang109 ; cheuk109 ; zhang109 ; huang12 ; wu354 ; zhang128 , which opens a completely new and ideal avenue for investigating quantum dynamics of SO-coupled cold atomic systems. There have been a great number of research works focusing on novel dynamics of SO-coupled cold atomic systemskart117 ; zhang628 ; tang121 ; wu99 ; ng92 ; zhang609 ; garcia89 ; citro224 ; tang ; cheng89 ; qu19 ; orso118 ; chen86 ; hu93 ; olson90 ; zcw108 ; yu90 ; ji99 ; luo93 ; luo52 ; luo22 ; gar114 ; abdu98 ; luo2021 ; luo39 ; luo74 , for instance, quantum spin dynamicstang121 ; wu99 , spin Josephson dynamicszhang609 ; garcia89 ; citro224 ; tang and localizationcheng89 ; qu19 , Anderson transition of SO-coupled cold atomsorso118 , collective dynamicszhang109 ; hu93 , Bloch oscillation dynamicsji99 , selective coherent spin transportationyu90 , spin-dependent dynamic localizationluo93 , controlling stable spin dynamicsluo22 , transparent manipulation of spin dynamicsluo39 , coherent control of spin tunnelingluo74 , and so on. To our knowledge, many works related to the quantum dynamics of SO-coupled cold atomic systems have been studied by applying approximate methodshu93 ; tang ; olson90 ; zcw108 ; ji99 ; cheng89 ; qu19 ; orso118 ; chen86 ; yu90 ; luo93 ; luo22 ; citro224 ; gar114 ; abdu98 ; luo52 ; luo2021 , for instance, numerical simulationhu93 ; olson90 ; zcw108 ; ji99 ; cheng89 ; qu19 ; orso118 , variational approximationhu93 ; cheng89 ; chen86 , mean-field approximationzcw108 , high-frequency approximation(or rotating-wave approximation)yu90 ; luo93 ; luo22 ; citro224 ; tang ; gar114 ; abdu98 ; luo39 ; luo74 , multiple-time-scale asymptotic analysisluo52 ; luo2021 , etc. However, the research on quantum spin dynamics of SO-coupled ultracold atomic systems based on exact solutions is still extremely rarehai29 .

In this paper, we present a simple method to construct the exact analytic solutions by applying combined modulations for an SO-coupled boson trapped in a double well potential (a four-level or four-state system). For the synchronous combined modulations case and the spin-conserving tunneling case, we obtain the analytical accurate solutions in general forms, respectively. For the spin-flipping case under asynchronous combined modulations, when the driving parameters are appropriately fixed, we get the simple particular exact solutions. Based on the obtained analytical exact solutions, some interesting quantum spin dynamical phenomena are revealed. Under synchronous combined modulations, the arbitrary population transfer (APT) with and/or without spin-flipping, the controlled coherent population conservation (CCPC), and the controlled coherent population inversion (CCPI) between left and right wells are shown. Under asynchronous combined modulations, for the spin-conserving and spin-flipping cases, we find new parametric conditions for performing the CCPC and CCPI, respectively. Our results may be useful for the preparation of precise quantum entangled states and the design of spintronic device.

We consider an SO-coupled boson trapped in a driven double well, which is governed by the Hamiltonian

	$\displaystyle\hat{H}(t)$	$\displaystyle=$	$\displaystyle-\upsilon(t)(\hat{a}_{l}^{{\dagger}}e^{-i\pi\gamma\hat{\sigma}_{y}}\hat{a}_{r}+H.c.)$		(1)
			$\displaystyle+\varepsilon(t)\sum_{j}(\hat{n}_{j\uparrow}-\hat{n}_{j\downarrow}).$

Here $\hat{a}_{j}^{{\dagger}}=(\hat{a}_{j\uparrow}^{{\dagger}},\hat{a}_{j\downarrow}^{{\dagger}})$ and $\hat{a}_{j}=(\hat{a}_{j\uparrow},\hat{a}_{j\downarrow})^{T}$ ($T$ denotes the transpose). $\hat{a}_{j\sigma}^{{\dagger}}$ and $\hat{a}_{j\sigma}$ are the creation and annihilation operators of a spin-$\sigma$ ($\sigma=\uparrow,\downarrow$) boson in the $j$th ($j=l,r$) well respectively. $\gamma$ is the effective strength of SO coupling and $\hat{\sigma}_{y}$ is the $y$ component of usual Pauli operator. $H.c.$ represents the Hermitian conjugate of the preceding term. $\hat{n}_{j\sigma}$=$\hat{a}_{j\sigma}^{{\dagger}}\hat{a}_{j\sigma}$ denotes the number operator for spin $\sigma$ in well $j$, $\upsilon(t)$ and $\varepsilon(t)$ are the time-dependent tunneling amplitude without SO coupling and Zeeman field, respectively. For simplicity, Eq. (1) has been treated as a dimensionless equation in which $\hbar=1$ and the reference frequency $\omega_{0}=0.1E_{r}$ ($E_{r}=k_{L}^{2}/2M=22.5$kHz is the single-photon recoil energy lin471 ) are set such that the parameters in $\upsilon(t)$ and $\varepsilon(t)$ are in units of $\omega_{0}$, and time $t$ is normalized in units of $\omega_{0}^{-1}$.

Employing the Fock basis $|0,\sigma\rangle(|\sigma,0\rangle)$ to denote the state of a spin-$\sigma$ boson occupying the right (left) well and no atom in the left (right) well, the quantum state of the SO-coupled bosonic system can be expanded as

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

(2)

where $a_{m}(t)(m=1,2,3,4)$ represents the time-dependent probability amplitude of the boson being in state $|0,\sigma\rangle$ or $|\sigma,0\rangle$, for example, $a_{1}(t)$ represents the time-dependent probability amplitude of the boson being in state $|0,\uparrow\rangle$. The occupation probability reads $P_{m}(t)=|a_{m}(t)|^{2}$, which satisfies the normalization condition $\sum_{m=1}^{4}P_{m}(t)=1$. For the convenience of studying dynamics, we define the population imbalance $Z_{sq}(t)=P_{s}(t)-P_{q}(t)$ ($s,q=L,R,1,2,3,4$, $s\neq q$) in which $P_{L}(t)=P_{3}(t)+P_{4}(t)$ and $P_{R}(t)=P_{1}(t)+P_{2}(t)$ are the total occupation populations of spin-$\sigma$ boson in left and right wells respectively. Inserting Eqs. (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

	$\displaystyle i\overset{\cdot}{a}_{1}(t)$	$\displaystyle=$	$\displaystyle\varepsilon(t)a_{1}(t)-\upsilon(t)[sin(\pi\gamma)a_{4}(t)+\cos(\pi\gamma)a_{3}(t)],$
	$\displaystyle i\overset{\cdot}{a}_{2}(t)$	$\displaystyle=$	$\displaystyle-\varepsilon(t)a_{2}(t)+\upsilon(t)[sin(\pi\gamma)a_{3}(t)-\cos(\pi\gamma)a_{4}(t)],$
	$\displaystyle i\overset{\cdot}{a}_{3}(t)$	$\displaystyle=$	$\displaystyle\varepsilon(t)a_{3}(t)+\upsilon(t)[sin(\pi\gamma)a_{2}(t)-\cos(\pi\gamma)a_{1}(t)],$
	$\displaystyle i\overset{\cdot}{a}_{4}(t)$	$\displaystyle=$	$\displaystyle-\varepsilon(t)a_{4}(t)-\upsilon(t)[sin(\pi\gamma)a_{1}(t)+\cos(\pi\gamma)a_{2}(t)].$

Generally, it is very difficult to get the exact solution of Eq. (3), because of the different forms of the time-dependent modulation functions. Here, we will attempt to obtain the exact solution of Eq. (3) under the synchronous and asynchronous modulations, respectively.

For the synchronous modulations, we mean that the time-dependent driving functions $\varepsilon(t)$ and $\upsilon(t)$ obey the relation $\varepsilon(t)=\beta\upsilon(t)$hai87 , in which $\beta$ is a proportional constant. In such a case, by introducing a new time variable $\tau=\tau(t)=\int\upsilon(t)dt$, Eq. (3) reduces to

	$\displaystyle i\frac{\overset{}{da}_{1}(t)}{d\tau}$	$\displaystyle=$	$\displaystyle\beta a_{1}(t)-sin(\pi\gamma)a_{4}(t)-\cos(\pi\gamma)a_{3}(t),$
	$\displaystyle i\frac{\overset{}{da}_{2}(t)}{d\tau}$	$\displaystyle=$	$\displaystyle-\beta a_{2}(t)+sin(\pi\gamma)a_{3}(t)-\cos(\pi\gamma)a_{4}(t),$
	$\displaystyle i\frac{\overset{}{da}_{3}(t)}{d\tau}$	$\displaystyle=$	$\displaystyle\beta a_{3}(t)+sin(\pi\gamma)a_{2}(t)-\cos(\pi\gamma)a_{1}(t),$
	$\displaystyle i\frac{\overset{}{da}_{4}(t)}{d\tau}$	$\displaystyle=$	$\displaystyle-\beta a_{4}(t)-sin(\pi\gamma)a_{1}(t)-\cos(\pi\gamma)a_{2}(t),$		(4)

which is the linear differential equations of $\tau$ with constant coefficients. In order to obtain the solutions of Eq. (4), we introduce the stationary solutions, $a_{1}(t)=Ae^{-i\lambda\tau}$, $a_{2}(t)=Be^{-i\lambda\tau}$, $a_{3}(t)=Ce^{-i\lambda\tau}$, $a_{4}(t)=De^{-i\lambda\tau}$, where $A$, $B$, $C$, and $D$ are constants obeying the normalization condition $|A|^{2}+|B|^{2}+|C|^{2}+|D|^{2}=1$, and $\lambda$ is the characteristic value. Inserting the stationary solutions into Eq. (4), we obtain the four characteristic values $\lambda_{m}$ and the corresponding constants $A_{m}$, $B_{m}$, $C_{m}$, $D_{m}$ for $m=1,2,3,4$ as

	$\displaystyle A_{1,2}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2+2\alpha_{\mp}^{2}}},B_{1,2}=-D_{1,2}=A_{1,2}\alpha_{\mp},C_{1,2}=A_{1,2},$
	$\displaystyle A_{3,4}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{2+2\eta_{\pm}^{2}}},B_{3,4}=D_{3,4}=A_{3,4}\eta_{\pm},C_{3,4}=-A_{3,4},$
	$\displaystyle\lambda_{1,2}$	$\displaystyle=$	$\displaystyle\mp\sqrt{1+\beta^{2}-2\beta\cos(\pi\gamma)},$
	$\displaystyle\lambda_{3,4}$	$\displaystyle=$	$\displaystyle\mp\sqrt{1+\beta^{2}+2\beta\cos(\pi\gamma)},$		(5)

with constants $\alpha_{\pm}=[-\beta+\cos(\pi\gamma)\pm\sqrt{1+\beta^{2}-2\beta\cos(\pi\gamma)}]\csc(\pi\gamma)$, $\eta_{\pm}=[\beta+\cos(\pi\gamma)\pm\sqrt{1+\beta^{2}+2\beta\cos(\pi\gamma)}]\csc(\pi\gamma)$. Based on Eq. (5), we immediately get the four accurate stationary-like solutions (states) of the system

	$\displaystyle|\psi_{m}\rangle$	$\displaystyle=$	$\displaystyle(A_{m}\mid 0,\uparrow\rangle+B_{m}\mid 0,\downarrow\rangle+C_{m}\mid\uparrow,0\rangle+D_{m}\mid\downarrow,0\rangle)$		(6)
			$\displaystyle\times e^{-i\lambda_{m}\tau},$

for $m=1,2,3,4$, which are the exact incoherent destruction of tunneling (IDT) states.

According to the superposition principle of quantum mechanics, we readily obtain the general exact solution of Eq. (4) by employing Eq. (6) to the linear superposition as

$\displaystyle|\psi(t)\rangle=\underset{m=1}{\overset{4}{\sum}}s_{m}|\psi_{m}(t)\rangle.$

(7)

Here, $s_{m}$ for $m=1,2,3,4$ is an arbitrary superposition coefficient determined by the initial condition and normalization. Compared Eq. (7) with Eq. (2), the probability amplitudes are renormalized as

	$\displaystyle a_{1}(t)$	$\displaystyle=$	$\displaystyle\underset{m=1}{\overset{4}{\sum}}s_{m}A_{m}e^{-i\lambda_{m}\tau},a_{2}(t)=\underset{m=1}{\overset{4}{\sum}}s_{m}B_{m}e^{-i\lambda_{m}\tau},$
	$\displaystyle a_{3}(t)$	$\displaystyle=$	$\displaystyle\underset{m=1}{\overset{4}{\sum}}s_{m}C_{m}e^{-i\lambda_{m}\tau},a_{4}(t)=\underset{m=1}{\overset{4}{\sum}}s_{m}D_{m}e^{-i\lambda_{m}\tau}$		(8)

which is the general exact solution of Eq. (3) under the synchronous modulations. Obviously, any of the synchronous combined modulation functions $\upsilon(t)$ and $\varepsilon(t)$ will generate an exact analytical solution to the Schrödinger equation, which thus enables us to produce an infinite variety of analytically exact solution (8) for the SO-coupled bosonic system. Here, we will select a square sech-shaped (a pulse-shaped) modulation function as an example to study the precise control of quantum spin dynamics by applying the exact solution under the synchronous modulations for the SO-coupled bosonic double-well system.

Let the modulation function $\upsilon(t)=V$sech²$(\Omega t)$, where $V$ and $\Omega$ are the driving strength and inverse pulse width respectivelyhai87 , such that the new time variable $\tau(t)=\frac{V}{\Omega}\tanh(\Omega t)$. Next, we will mainly discuss the APT with and/or without spin-flipping, the CCPC, and the CCPI between left and right wells based on the general exact solution (8).

In this section, by using the tanh-shaped modulation $\varepsilon(t)=\varepsilon\tanh(\chi t)$ and sech-shaped modulation $\upsilon(t)$=$\upsilon$ sech$(\chi t)$, our aim is to get the exact solutions of the asynchronous modulation system for the cases of spin-conserving and spin-flipping tunneling respectively. Then, applying those exact solutions to study the CCPC and CCPI with or without spin-flipping. From Eq. (3), it can be seen that when $\sin(\pi\gamma)=0$, $a_{1}$ is only coupled with $a_{3}$ and $a_{2}$ is only coupled with $a_{4}$, which mean only the quantum tunneling without spin-flipping can occur. In order to investigate the exact CCPC and CCPI without spin-flipping, without loss of generality, we set $\cos(\pi\gamma)=1$ (e.g., $\gamma=2$), such that Eq. (3) reduces to two set of decoupled subequations as

	$\displaystyle i\overset{\cdot}{a}_{1}(t)$	$\displaystyle=\varepsilon(t)a_{1}(t)-\upsilon(t)a_{3}(t),$
	$\displaystyle i\overset{\cdot}{a}_{3}(t)$	$\displaystyle=\varepsilon(t)a_{3}(t)-\upsilon(t)a_{1}(t),$		(11a)
	$\displaystyle i\overset{\cdot}{a}_{2}(t)$	$\displaystyle=-\varepsilon(t)a_{2}(t)-\upsilon(t)a_{4}(t),$
	$\displaystyle i\overset{\cdot}{a}_{4}(t)$	$\displaystyle=-\varepsilon(t)a_{4}(t)-\upsilon(t)a_{2}(t).$		(11b)

Here, we solve Eq. (11a) as an example. By introducing the transformation $a_{1}(t)=b_{1}(t)e^{-i\int\varepsilon(t)dt}$ and $a_{3}(t)=b_{3}(t)e^{-i\int\varepsilon(t)dt}$ and inserting them to Eq. (11a), such that the Eq. (11a) reduces to

	$\displaystyle i\overset{\cdot}{b}_{1}(t)$	$\displaystyle=-\upsilon(t)b_{3}(t),$
	$\displaystyle i\overset{\cdot}{b}_{3}(t)$	$\displaystyle=-\upsilon(t)b_{1}(t).$			(12)

By using a time scale, $\tau=\tau(t)=\int\upsilon(t)dt$, and utilizing the relation $\frac{db_{1}(t)}{dt}=\frac{db_{1}(t)}{d\tau}\frac{d\tau}{dt}=\frac{db_{1}(t)}{d\tau}\upsilon(t)$, the Eq. (12) can be simplified to the form

	$\displaystyle i\frac{db_{1}(t)}{d\tau}$	$\displaystyle=-b_{3}(t),$
	$\displaystyle i\frac{db_{3}(t)}{d\tau}$	$\displaystyle=-b_{1}(t).$			(13)

Obviously, the general solution of Eq. (13) can be easily obtained as

	$\displaystyle b_{1}(t)$	$\displaystyle=A_{+}e^{i\tau}+A_{-}e^{-i\tau},$
	$\displaystyle b_{3}(t)$	$\displaystyle=A_{+}e^{i\tau}-A_{-}e^{-i\tau}.$			(14)

Such that we can get the exact solution of Eq. (11a) as

	$\displaystyle a_{1}(t)$	$\displaystyle=A_{+}e^{i\int[\upsilon(t)-\varepsilon(t)]dt}+A_{-}e^{-i\int[\upsilon(t)+\varepsilon(t)]dt},$
	$\displaystyle a_{3}(t)$	$\displaystyle=A_{+}e^{i\int[\upsilon(t)-\varepsilon(t)]dt}-A_{-}e^{-i\int[\upsilon(t)+\varepsilon(t)]dt}.$			(15)

Similarly, we can obtain the precise solution of Eq. (11b) as

	$\displaystyle a_{2}(t)$	$\displaystyle=B_{+}e^{i\int[\upsilon(t)+\varepsilon(t)]dt}+B_{-}e^{-i\int[\upsilon(t)-\varepsilon(t)]dt},$
	$\displaystyle a_{4}(t)$	$\displaystyle=B_{+}e^{i\int[\upsilon(t)+\varepsilon(t)]dt}-B_{-}e^{-i\int[\upsilon(t)-\varepsilon(t)]dt}.$			(16)

Here, the complex superposition constants $A_{\pm}$ and $B_{\pm}$ are determined by the initial conditions and normalization respectively.

These solutions, Eqs. (15) and (16), are the exact ones corresponding for the case of spin-conserving tunneling for arbitrary modulation functions $\upsilon(t)$ and $\varepsilon(t)$. Here, we only consider the asynchronous modulations $\varepsilon(t)=\varepsilon\tanh(\chi t)$ and $\upsilon(t)$=$\upsilon$ sech$(\chi t)$. In such case, from Eq.(15), we surprisingly find when the parameters satisfy

$\displaystyle\sin(\pi\upsilon/\chi)=0,$

(17)

the populations $P_{3}(-\infty)=P_{3}(+\infty)$ and $P_{1}(-\infty)=P_{1}(+\infty)$ with the population imbalances $Z_{31}(-\infty)=Z_{31}(+\infty)=\pm 2(A_{+}A_{-}^{\ast}+A_{+}^{\ast}A_{-})$ (here, $\pm$ sign corresponds to $\cos(\pi\upsilon/\chi)=\mp 1$ respectively), which mean the CCPC occurs as time goes from $-\infty$ to $+\infty$. As an example, we take the parameters $\gamma=2$, $\varepsilon=1$, $\upsilon=1$, $\chi=1$ and set the different initial conditions to plot the time evolutions of the population imbalance $Z_{31}(t)$ in Fig. 3 (a). It can be seen that for different initial conditions the population imbalances remain unchanged as $Z_{31}(-\infty)=Z_{31}(+\infty)$, namely, the CCPC happens.

When the parameters satisfy

$\displaystyle\cos(\pi\upsilon/\chi)=0,$

(18)

the populations $P_{3}(-\infty)=P_{1}(+\infty)$ and $P_{1}(-\infty)=P_{3}(+\infty)$ with the population imbalances $Z_{31}(-\infty)=-Z_{31}(+\infty)=\pm 2i(A_{+}A_{-}^{\ast}-A_{+}^{\ast}A_{-})$ (here, $\pm$ sign corresponds to $\sin(\pi\upsilon/\chi)=\pm 1$ respectively), which mean the occurrence of CCPI without spin-flipping as time $t=-\infty\rightarrow+\infty$. As an example, we take the same parameters and initial conditions as those of Fig. 3(a), except for $\chi=2$, and plot the time evolutions of the population imbalance $Z_{31}(t)$ in Fig. 3 (b). It can be seen that the population imbalance $Z_{31}(-\infty)=-Z_{31}(+\infty)$ under the different initial conditions, which means the CCPI occurs.

It is worth noting that the time-evolution curves of the population imbalance $Z_{31}(t)$ have oscillation around $t=0$ and the number of oscillating peak (or valley) for each curve is equal to the value $\upsilon/\chi$ and $\upsilon/\chi-1/2$ in Figs. 3(a) and (b) respectively, e.g., $\upsilon/\chi=1$ in Fig. 3(a) and $\upsilon/\chi-1/2=0$ in Fig. 3(b). Specially, when the initial population imbalance $Z_{31}(-\infty)=0$, the population of particle will keep standstill which means the occurrence of CDT, see Figs. 3(a) and (b). The analytical results (the squares) from Eq. (15) are in complete agreement with the numerical results (the curves) from Eq. (11a). For the exact solution (16), we can obtain the similar results.

Next, we study the exact CCPI with spin-flipping and set $\cos(\pi\gamma)=0$ in Eq. (3) which means only quantum tunneling with spin-flipping can occur. Without loss of generally, we fix $\sin(\pi\gamma)=1$ (e.g., $\gamma=0.5$), such that Eq. (3) reduces to two set of decoupled subequations as

	$\displaystyle i\overset{\cdot}{a}_{1}(t)$	$\displaystyle=\varepsilon(t)a_{1}(t)-\upsilon(t)a_{4}(t),$
	$\displaystyle i\overset{\cdot}{a}_{4}(t)$	$\displaystyle=-\varepsilon(t)a_{4}(t)-\upsilon(t)a_{1}(t),$		(19a)
	$\displaystyle i\overset{\cdot}{a}_{2}(t)$	$\displaystyle=-\varepsilon(t)a_{2}(t)+\upsilon(t)a_{3}(t),$
	$\displaystyle i\overset{\cdot}{a}_{3}(t)$	$\displaystyle=\varepsilon(t)a_{3}(t)+\upsilon(t)a_{2}(t).$		(19b)

Note that the Eqs. (19a) and (19b) are analogous to Eq. (3) in Ref. hai87 , so we use the established method in Ref. hai87 to solve Eq. (19a) as an example to show the exact CCPI with spin-flipping. Let $a_{4}(t)=\sqrt{sech(\chi t)}a(t)$ and from Eq. (19a) we can obtain the second-order equation

	$\displaystyle\ddot{a}(t)+[\upsilon^{2}-\frac{1}{2}\chi^{2}-i\varepsilon\chi+(\frac{1}{4}\chi^{2}+\varepsilon^{2}-\upsilon^{2})$
	$\displaystyle\times\tanh(\chi t)^{2}]a(t)=0,$		(20)

which is exactly solvable by the use of hypergeometric functionshai78 . Here, in order to get a simple solution, we select the driving parameters to satisfy

$\displaystyle\frac{1}{4}\chi^{2}+\varepsilon^{2}-\upsilon^{2}=0,$

(21)

which is shown in Fig. 3(c). Based on Eq. (21), we can obtain the simple general solution of Eq. (20) as $a(t)=C_{+}e^{(i\varepsilon+\frac{1}{2}\chi)t}+C_{-}e^{-(i\varepsilon+\frac{1}{2}\chi)t}$ with constant $C_{\pm}$ determined by the initial condition. Such that the simple exact solution of Eq. (19a) is obtained as

	$\displaystyle a_{1}(t)$	$\displaystyle=$	$\displaystyle-\frac{i(\chi+2i\varepsilon)\sqrt{sech(\chi t)}}{2\upsilon}$
			$\displaystyle\times[C_{+}e^{-\frac{1}{2}(\chi-2i\varepsilon)t}-C_{-}e^{\frac{1}{2}(\chi-2i\varepsilon)t}],$
	$\displaystyle a_{4}(t)$	$\displaystyle=$	$\displaystyle\sqrt{sech(\chi t)}[C_{+}e^{(i\varepsilon+\frac{1}{2}\chi)t}+C_{-}e^{-(i\varepsilon+\frac{1}{2}\chi)t}].$

It is worth noting that Eq. (22) implies $P_{1}(-\infty)=P_{4}(+\infty)=2|C_{+}|^{2}$ and $P_{4}(-\infty)=P_{1}(+\infty)=2|C_{-}|^{2}$ with the population imbalance $Z_{41}(-\infty)=-Z_{41}(+\infty)$, which mean the exact CCPI with spin-flipping can be implemented. As an example, we take the SO coupling strength $\gamma=0.5$ and fix the parameters $\varepsilon=\sqrt{0.21}$, $\upsilon=0.5$ and $\chi=0.4$ obeying Eq. (21) to plot the time evolutions of the population imbalance $Z_{41}(t)$ for the different initial conditions as shown in Fig. 3(d). It is obviously seen that the population imbalances $Z_{41}(-\infty)=-Z_{41}(+\infty)$, which mean the CCPI with spin-flipping happens. Here, we note that the time-evolution curve for the initial population imbalance $Z_{41}(-\infty)=0$ has oscillation, which is different from the case of CCPI without spin-flipping (see Fig. 3(b)). The exact results (the squares) from Eq. (22) are in good agreement with the numerical results (the curves) from Eq. (19a).

By using the homologous method of solving Eq. (19a), we readily get the simple exact solution of Eq. (19b) as

	$\displaystyle a_{2}(t)$	$\displaystyle=$	$\displaystyle-\frac{i(\chi-2i\varepsilon)}{2\upsilon\sqrt{sech(\chi t)}}[D_{+}(\tanh(\chi t)+1)e^{-\frac{1}{2}(\chi-2i\varepsilon)t}$
			$\displaystyle+D_{-}(\tanh(\chi t)-1)e^{\frac{1}{2}(\chi-2i\varepsilon)t}],$
	$\displaystyle a_{3}(t)$	$\displaystyle=$	$\displaystyle\sqrt{sech(\chi t)}[D_{+}e^{(i\varepsilon-\frac{1}{2}\chi)t}+D_{-}e^{-(i\varepsilon-\frac{1}{2}\chi)t}],$		(23)

where the constant $D_{\pm}$ is determined by the initial condition and the driving parameters must also obey Eq. (21). Likewise, it is also noticed in Eq. (23) that $P_{3}(-\infty)=P_{2}(+\infty)=2|D_{+}|^{2}$ and $P_{2}(-\infty)=P_{3}(+\infty)=2|D_{-}|^{2}$ with the population imbalance $Z_{32}(-\infty)=-Z_{32}(+\infty)$(not shown here). Note that the above results obtained in the case of spin-flipping are similar to ones obtained in the two-level system without SO coupling (see Ref. hai87 ), which goes against common intuition.

In summary, we have proposed a simple method of combined modulations to obtain the analytical exact solutions for an SO-coupled boson trapped in a driven double-well potential. For the cases of synchronous combined modulations and the non-spin-flipping tunneling, we have obtained the general analytical precise solutions of the system respectively. For the spin-flipping tunneling case under asynchronous combined modulations, we have gotten the particular accurate solutions in simple form when the driving parameters are appropriately chosen. Based on these obtained accurate solutions, we have revealed some intriguing spin dynamical phenomena. Under synchronous combined modulations, the APT with and/or without spin-flipping, the CCPC, and the CCPI between left and right wells have been displayed. Under asynchronous combined modulations, we have also performed the CCPC and the CCPI with spin-conserving or with spin-flipping, respectively. Surprisingly, we have found that the CCPI exists in the cases of synchronous and asynchronous combined modulations in the SO-coupled cold atomic system (four-level system), which generalizes the result obtained in the case of asynchronous modulation in the two-level system without SO coupling (see Ref. hai87 ). The results may have potential applications in the qubit manipulation and the preparation of accurate quantum entangled states.

This work was supported by the Hunan Provincial Natural Science Foundation of China under Grants No. 2021JJ30435 and No. 2017JJ3208, the Scientific Research Foundation of Hunan Provincial Education Department under Grants No. 21B0063 and No. 18C0027, and the National Natural Science Foundation of China under Grant No. 11747034.