Spin–orbit (SO) coupling is the interaction between spin and motion of a particle, which plays a crucial role in many important physical phenomena, e.g. spin-Hall effect [1], topological insulator [2], and the persistent spin helix [3]. For ultracold atoms, SO coupling can be created by utilizing the interaction between laser and atom, which yields Abelian or non-Abelian gauge fields for ultracold atoms in the dressed two hyperfine atomic internal states. As is well-known, the electrons are fermions in the material, but ultracold atoms may be bosons. Therefore, SO-coupled ultracold bosonic gases will lead to novel SO physics that has not been explored in solid materials.

Recently, SO coupling of ultracold atomic gases has been realized in experiments [4–9], which provides a brand new platform to investigate SO coupling physics, due to the unprecedented tunability of experimental parameters. A number of research works have focused on the interesting dynamics of SO-coupled ultracold atoms, for instance, quantum dynamics of SO-coupled Bose–Einstein condensates (BECs) in a double well [10–13], coherent control of an SO-coupled atom in a double-well potential [14], controlling stable spin tunneling in a non-Hermitian double-well system [15], Anderson localization of SO-coupled ultracold atomic gases in an optical lattice [16], Macroscopic Klein tunneling in SO-coupled BECs [17], Landau–Zener transition in an SO-coupled BEC [18], spin dynamics of SO-coupled BECs [19], nonequilibrium dynamics of two-component bosons in an optical lattice [20], controlling localization and directed motion of an SO-coupled single atom in a bipartite lattice [21], Bloch oscillation dynamics of an SO-coupled ultracold atomic gas in an optical lattice [22], quantum tunneling of an SO-coupled ultracold atom in an optical lattice with an impurity [23], controlling second-order tunneling of an SO-coupled atom in optical lattices [24], and so on.

As mentioned above, many works focus on the tunneling dynamical properties of SO-coupled ultracold atomic gases held in a double well or an optical lattice. To the best of our knowledge, the coherent control of spin tunneling for SO-coupled ultracold atoms confined in a triple well is rarely investigated. However, the triple-well model is an important one to study the coherent control of spin tunneling and is a bridge between the double-well and optical lattice models for a better understanding of the spin tunneling dynamics of SO-coupled ultracold atoms in the quantum wells. Thus, it motivates us to investigate the quantum spin tunneling in an SO-coupled bosonic triple-well system.

In this paper, we theoretically study the coherent control of spin tunneling for an SO-coupled boson confined in a driven triple well. In the high-frequency limit, the quasienergies of the SO-coupled ultracold atomic triple-well system are analytically obtained and the quasienergy spectra are shown. By adjusting the strength of the time-dependent driving field, we can manipulate the directed spin-flipping or spin-conserving tunneling of an SO-coupled boson along different pathways and in different directions. Further, we present an intriguing scheme of a quantum spin switch for transporting an SO-coupled boson accompanied with or without spin-flipping from well 1 to well 3. These results may be useful for the design of spintronic devices [25].

We consider an SO-coupled ultracold atom confined in a driven triple well and the Hamiltonian of this system reads [21, 24]where ${\hat{a}}_j^{†}=({\hat{a}}_{j\uparrow }^{†},{\hat{a}}_{j\downarrow }^{†})$ and ${\hat{a}}_j={({\hat{a}}_{j\uparrow },{\hat{a}}_{j\downarrow })}^{\mathrm{T}}$ (the superscript T denotes the transpose). ${\hat{a}}_{j\sigma }$ (${\hat{a}}_{j\sigma }^{†}$) is the annihilation (creation) operator of a pseudospin-σ (σ = ↑ , ↓ ) atom in the jth (j = 1, 2, 3) well. ν is the tunneling rate in the absence of SO coupling, γ denotes the strength of SO coupling, ${\hat{\sigma }}_y$ is the usual Pauli operator, the parameter Ω denotes the effective Zeeman-field strength, and ${\hat{n}}_{j\sigma }={\hat{a}}_{j\sigma }^{†}{\hat{a}}_{j\sigma }$ is the number operator. ${\epsilon }_1(t)={\epsilon }_1\cos (\omega t)$ and ${\epsilon }_2(t)={\epsilon }_2\cos (\omega t)$ are the periodic driving fields applied to well 1 and well 3 respectively [26, 27]. Here ϵ₁ and ϵ₂ are the driving amplitudes, and ω is the frequency of the driving field. Throughout this paper, ℏ = 1 and the parameters ν, ω, ϵ₁, ϵ₂, ω are normalized in units of the reference frequency ω₀ = 0.1E_r with $E_r=k_L^2/(2m)=22.5$ KHz being the single-photon recoil energy [4], and time t is measured in units of ${\omega }_0^{-1}$. In the experiment, the system parameters can be adjusted in a wide range as follows [13, 15, 21, 28]: ν ∼ ω₀, ϵ_1,2 ∼ ω ∈ [0, 100](ω₀), Ω ∼ ω.

We employ the Fock state basis ∣σ, 0, 0⟩ (or ∣0, σ, 0⟩ or∣0, 0, σ⟩) to denote the state of a pseudospin-σ atom occupying the well 1 (or well 2 or well 3) and no atom in the other two wells, the quantum state of the SO-coupled ultracold atomic system can be expanded aswhere a_k(t) (k = 1, 2,...,6) represents the probability amplitude of the boson being in the Fock state ∣σ, 0, 0⟩ or ∣0, σ, 0⟩ or ∣0, 0, σ⟩ (e.g. a₁(t) represents the probability amplitude of the boson being in Fock state ∣ ↑ , 0, 0⟩). The corresponding probability of the Fock state reads P_k(t) = ∣a_k(t)∣², which satisfies the normalization condition ${\sum }_{k=1}^6{P}_k(t)=1$. Inserting equations (1) and (2) into the time-dependent Schrödinger equation $\mathrm{i}\frac{\partial |\psi (t)⟩}{\partial t}=\hat{H}(t)|\psi (t)⟩$, one can obtain the coupled equations

Clearly, equation (3) is very hard to be solved exactly, because of the time-dependent coefficients. However, the spin tunneling dynamics of the SO-coupled ultracold atomic system can be studied analytically in the high-frequency limit [29, 30]. Therefore, we introduce the slowly varying function of time b_k(t)(k = 1, 2,…,6) through the transformation

In the high-frequency limit and by using of the Fourier expansion ${\mathrm{e}}^{\pm \mathrm{i}\int {\epsilon }_{1,2}(t)\mathrm{d}t}={\sum }_{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathcal{J}}_n(\frac{{\epsilon }_{1,2}}{\omega }){\mathrm{e}}^{\pm \mathrm{i}n\omega t}$ and ${\mathrm{e}}^{\pm \mathrm{i}\int [{\epsilon }_{1,2}(t)\pm \mathrm{\Omega }]\mathrm{d}t}={\sum }_{n^{\mathrm{\prime }}=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathcal{J}}_{n^{\mathrm{\prime }}}(\frac{{\epsilon }_{1,2}}{\omega }){\mathrm{e}}^{\pm \mathrm{i}(n^{\mathrm{\prime }}\pm \frac{\mathrm{\Omega }}{\omega })\omega t}$, the rapidly oscillating terms of the Fourier expansion with n ≠ 0 and $n^{\mathrm{\prime }}\pm \frac{\mathrm{\Omega }}{\omega }\ne 0$ can be neglected [31] and the equation (3) reduces to the formHere, $J_1=\nu \cos (\pi \gamma ){\mathcal{J}}_0(\frac{{\epsilon }_1}{\omega })$, $J_{2,3}=\nu \sin (\pi \gamma ){\mathcal{J}}_{\mp \frac{\mathrm{\Omega }}{\omega }}(\frac{{\epsilon }_1}{\omega })$, $J_4=\nu \cos (\pi \gamma ){\mathcal{J}}_0(\frac{{\epsilon }_2}{\omega })$, and $J_{5,6}=\nu \sin (\pi \gamma ){\mathcal{J}}_{\mp \frac{\mathrm{\Omega }}{\omega }}(\frac{{\epsilon }_2}{\omega })$ are the renormalized coupling constants and ${\mathcal{J}}_n(x)$ is the n-order Bessel function of x. It is worth noting that the spin tunneling dynamics of the original system (1) can be described effectively by equation (5), which is the basis of the following analysis.

Before moving on, we first briefly introduce the Floquet theorem [32, 33]. The basis of this theorem lies in the observation that for a time-dependent Hamiltonian with period τ, $\hat{H}(t)=\hat{H}(t+\tau )$, there exists a set of complete bases {∣$\mathrm{\Psi }$_p(t)⟩} which constitute the solutions of Schrödinger equation $\mathrm{i}\frac{\partial |\psi (t)⟩}{\partial t}=\hat{H}(t)|\psi (t)⟩$ in the formwhere the time-periodic functions ∣ φ _p (t) ⟩ are called Floquet states and E _p are quasienergies. For the system we are studying whose Hamiltonian is periodic in time with period τ = 2 π/ω, according to the Floquet theorem, the Floquet solution of this system can be constructed as ∣$\mathrm{\Psi }$ (t) ⟩ = ∣ φ (t) ⟩e^{− iEt}, where E is the Floquet quasienergy and $|\phi (t)⟩=A_1{\mathrm{e}}^{-\mathrm{i}\int [\frac{\mathrm{\Omega }}{2}+{\epsilon }_1\cos (\omega t)]\mathrm{d}t}|\uparrow ,0,0⟩+A_2{\mathrm{e}}^{-\mathrm{i}\int [-\frac{\mathrm{\Omega }}{2}+{\epsilon }_1\cos (\omega t)]\mathrm{d}t}|\downarrow ,0,0⟩$ + $A_3{\mathrm{e}}^{-\mathrm{i}\int \frac{\mathrm{\Omega }}{2}\mathrm{d}t}|0,\uparrow ,0⟩+A_4{\mathrm{e}}^{\mathrm{i}\int \frac{\mathrm{\Omega }}{2}\mathrm{d}t}|0,\downarrow ,0⟩+A_5{\mathrm{e}}^{-\mathrm{i}\int [\frac{\mathrm{\Omega }}{2}-{\epsilon }_2\cos (\omega t)]\mathrm{d}t}|0,0,\uparrow ⟩+A_6{\mathrm{e}}^{\mathrm{i}\int [\frac{\mathrm{\Omega }}{2}+{\epsilon }_2\cos (\omega t)]\mathrm{d}t}|0,0,\downarrow ⟩$ is the Floquet state. Based on the superposition principle of quantum mechanics, the non-Floquet state can be obtained by the linear superposition of the Floquet states [15, 34 –36], which implies the enhancement or suppression of quantum tunneling. We have introduced the stationary solution of equation ( 5 ) as b _k (t) = A _ke^−iEt with A_k being constant. Inserting such a form of b_k (t) into equation (5), the constant A _k and Floquet quasienergy E can be obtained. Here, we only give the Floquet quasienergies as with the constantsClearly, the quasienergies depend on the system parameters, except for the degenerate zero quasienergies E₁ and E₂.

From equation (7), it is surprisingly found that when the SO coupling strength γ is an integer or half integer, the constant β 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) popularly occurs at the degeneracy (crossing) point of the partial energy levels [34–36] and coherent destruction of tunneling (CDT) commonly happens at the collapse (crossing) point of all energy levels [37, 38]. As an example, we fix the parameters γ = 0.5, ω = 50, Ω = 100, ν = 1, and (a) ϵ₂ = 5.1356ω, (b) ϵ₂ = 2ω to plot the quasienergy spectra with quasienergy as a function of the driving parameter ϵ₁/ω, as shown in figures 1(a)–(b), respectively. Here, the circle points denote the analytical results based on equation (7) and the solid curves label the numerical results from the original model (1). The two sets of parameters lead to $E_3=E_5=-|{\mathcal{J}}_2(\frac{{\epsilon }_1}{\omega })|$ and $E_4=E_6=|{\mathcal{J}}_2(\frac{{\epsilon }_1}{\omega })|$, and $E_3=E_5=-\sqrt{{\mathcal{J}}_2{\left(\frac{{\epsilon }_1}{\omega }\right)}^2+{\mathcal{J}}_2{\left(2\right)}^2}$ and $E_4=E_6=\sqrt{{\mathcal{J}}_2{\left(\frac{{\epsilon }_1}{\omega }\right)}^2+{\mathcal{J}}_2{\left(2\right)}^2}$ in equation (7), respectively. In figure 1(a), we observe 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 ${\mathcal{J}}_2({\epsilon }_1/\omega )=0$ (i.e. ${\mathcal{J}}_2({\epsilon }_1/\omega )={\mathcal{J}}_2(5.1356)=0$). Not only that, the curves of quasienergies E₁ and E₂, E₃ and E₅, E₄ and E₆ are degenerate respectively. The perfect agreements between the numerical results (curves) and the analytical ones (circles) are shown. 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 E₄ or E₆. 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 E₁ and E₂, E₃ and E₅, E₄ and E₆), which have some small deviations from analytical results (circles).

Coherent manipulation of quantum tunneling is an interesting research subject in quantum information technologies [39, 40]. Generally, quantum tunneling of ultracold atomic systems relies intensively on the periodic driving external field. Thus, the quantum spin tunneling and transport of an SO-coupled ultracold boson held in a triple well can be manipulated by adjusting the driving parameters. In the section, we will focus on studying the directed tunneling and spin tunneling switch with or without spin-flipping.

For an SO-coupled ultracold atomic triple-well system, an attractive physical problem is how to control the directed spin-flipping or spin-conserving tunneling of an SO-coupled atom from the initial middle well to the left well or to the right well. Based on equation (5), it can be seen that when the SO-coupling strength γ is half integer or integer, quantum tunneling with or without spin-flipping can occur. Here, we will try to manipulate the directed 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 (5) that when the effective coupling constants satisfy J₁ = J₄ = J₅ = 0 and J₃ ≠ 0, b₃ is only coupled with b₂. It means that the SCDT occurs. We take the parameters γ = 0.5, ν = 1, ω = 50, Ω = 100, ϵ₁ = 2ω, ϵ₂ = 5.1356ω satisfying the conditions J₁ = J₄ = J₅ = 0 and J₃ ≠ 0 to plot the time evolutions of the probabilities in figure 2(a). Clearly, the spin-flipping tunneling channel between well 2 and well 3 is closed and the particle performs a Rabi oscillation of spin-flipping along this pathway between well 2 and well 1, namely, the directed spin-flipping tunneling takes place between initial state ∣0, ↑ , 0⟩ and state ∣ ↓ , 0, 0⟩ with tunneling time △t ≈ 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 in figure 2(a) except for ϵ₁ = 5.1356ω, which corresponds 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 (5) it can also be found when the effective coupling constants satisfy J₁ = J₃ = J₄ = 0 and J₅ ≠ 0, the probability function b₃ is only related to function b₆, which means the SCDT happens. We fix the same initial conditions and parameters as that in figure 2(a) except for ϵ₁ = 5.1356ω and ϵ₂ = 2ω to plot the time evolutions of the probabilities in figure 2(c). Clearly, the directed spin-flipping tunneling along another pathway between well 2 and well 3 occurs, in which the spin particle performs a spin-flipping Rabi oscillation between state ∣0, ↑ , 0⟩ and state ∣0, 0, ↓ ⟩ with tunneling time △t ≈ 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). The analytical results (dashed lines) based on equation (5) are in perfect agreement with the numerical results from the accurate model (3) in figure 2. Because the above-mentioned results are related to controlling 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.

From the above subsection, we find the time-dependent driving field affects dramatically spin tunneling dynamics of this system, thus we can propose a scheme of quantum spin tunneling switch with or without spin-flipping by means of sudden regulating of driving parameters to control the occurrence and suppression of quantum tunneling. The means have been performed in many research works [41–44]. Here, we present a scheme of quantum spin tunneling switch with spin-flipping by adjusting the driving strength as an example, as shown in figure 3. In figure 3(a), we take a spin-up boson initially occupied in well 1 and set the parameters γ = 0.5, ν = 1, ω = 50, Ω = 100, ϵ₁ = ϵ₂ = 5.1356ω corresponding to the collapse (crossing) point of all the quasienergies in figure 1(a). One can see that the spin-up boson is frozen in well 1, due to the CDT effect. At any given time t = t₁ = 2, the driving strength ϵ₁ is changed to ϵ₁ = 2ω and holds this value until the time is t = t₂ = 6.4. At the moment, the spin-up particle tunnels completely from state ∣ ↑ , 0, 0⟩ to state ∣0, ↓ , 0⟩. Then we immediately adjust the driving strength ϵ₁ to ϵ₁ = 5.1356ω, such that the state ∣0, ↓ , 0⟩ is kept which is attributed to the effect of CDT. At any given time t = t₃ = 8.4, we regulate the driving strength ϵ₂ to ϵ₂ = 2ω and preserve this value until t = t₄ = 12.8. At this time, the spin-down particle tunnels completely from state ∣0, ↓ , 0⟩ to state ∣0, 0, ↑ ⟩. Then, we return the driving strength ϵ₂ to ϵ₂ = 5.1356ω, so that the final state ∣0, 0, ↑ ⟩ is kept, due to the CDT effect. Therefore, the spin-up boson is successfully transported through two spin-flipping tunnels from well 1 to well 3 by adjusting the driving strength ϵ_i (i = 1, 2) of the driving external fields, namely, the quantum spin tunneling switch with spin-flipping is theoretically realized. From figure 3(a), it is also found that we can manipulate the spin-flipping tunneling of the spin particle from an arbitrarily initial occupied well to any other well. The corresponding spatial distributions of the spin particle at the tuning moments are shown in figure 3(b), where △t_i denotes transferring time between the different populations and the transferring time △t₁ = △t₂ = t₂ − t₁ = t₄ − t₃ = 4.4. Similarly, the quantum spin tunneling switch without spin-flipping can also be performed by adjusting the driving parameters.

In conclusion, we have investigated the coherent control of spin tunneling for an SO-coupled boson trapped in a driven triple well. In the high-frequency limit, we analytically obtain the Floquet quasienergies of the system and the fine quasienergy spectra are shown. By modulating the strength of the driving external field, we can control the directed spin-flipping or spin-conserving tunneling of an SO-coupled boson along different pathways and in different directions from the middle well to the left well or to the right well. The analytical results are demonstrated by numerical calculations and perfect agreements between both are shown. Further, based on the combined effect of CDT and SCDT, we propose a diverting scheme of quantum spin tunneling switch in which a spin particle is transported with or without spin-flipping from the initial well 1 to the final well 3. It is worth noting that we also can manipulate the spin-flipping or spin-conversing tunneling of a spin boson from an arbitrarily initial occupied well to any other well. These results may be useful in the design of spintronic devices and quantum information technology.