Spin–orbit coupling effects on localization and correlated tunneling for two interacting bosons in a double-well potential

We theoretically study the tunneling dynamics of two interacting spin–orbit-coupled (SOC) atoms trapped in a periodically perturbed double-well potential. We find that the phenomenon of coherent destruction of tunneling (CDT) can exist only for certain values of SOC, and two different mechanisms for the appearance of CDT are identified in this system. One is the conventional CDT resulting from quasi-energy degeneracy, while the other CDT originates from the dark Floquet state with zero quasi-energy for all values of the driving parameters. We discover that under double modulation combining the double-well potential shaking and a time-periodic Zeeman field, it is possible to realize spin-flipping single-atom Rabi tunneling and the CDT induced by the dark Floquet state at any value of SOC strength, which is not accessible with a single-drive field. Furthermore, we show that the detuning of Zeeman field with respect to the multiphoton energy is particularly significant in the case of the correlated two-particle tunneling mediated by SOC. We expect that these results will stimulate further exploration of the many-body dynamics in the interacting systems and expand the possibilities for manipulating the spin dynamics.


Introduction
Tunneling dynamics is arguably the most fundamental quantum phenomenon at the heart of many physical systems.It has been shown that time-periodic modulation can be used as a flexible knob to coherently control quantum dynamics.A seminal example is coherent destruction of tunneling (CDT) [1,2], a phenomenon discovered originally in a symmetric double-well potential perturbed by a monochromatic driving force, upon the occurrence of which the tunneling dynamics can be brought to a complete standstill.The CDT phenomenon has been shown to occur at the isolated degeneracy point of the quasienergies in the driven systems.Long studied theoretically, the seminal result has also been observed experimentally to date in many physical systems, for example, light tunneling inhibition in waveguide arrays [3], three-dimensional photonic lattices [4], a Bose-Einstein Condensate (BEC) in strongly shaken optical lattices [5], a single particle in a double-well potential [6], and chaotic microcavity [7].In recent years, the suppression of tunneling originating from a peculiar dark Floquet state with zero quasi-energy has been discovered [8], which provides a new tool for controlling the tunneling dynamics.In this regard, we would mention that the CDT can be realized as a consequence of the existence of the dark Floquet state by adjusting the amplitude of the periodic driving field applied to the right boundary site [9].It also has been demonstrated that the dark Floquet state can be used to suppress decay in fully governable open quantum systems [10].
Over the past two decades, there has been an intense and very broad range of research activities in the interacting systems.Due to the unprecedented level of controllability, tunability and cleanness, the many-body system realized with ultracold atoms is a perfect candidate for exploring the limit of strong correlations, where the atom-atom interactions can be experimentally tuned over a wide range, for example through Feshbach resonances [11][12][13].A distinctive feature in the strongly interacting system is the possibility of realizing long-lived bound pairs of interacting particles, also known as doublons, which have been observed in a large number of experiments [14][15][16][17].Recently, the two-atom systems with time-periodic modulation have attracted much attention, and many interesting phenomena have been reported, such as controlling transport of two particles in a bipartite lattice [18], second-order tunneling of two interacting bosons in a driven triple well [19], and simultaneously achieving CDT for both paired and unpaired states of two interacting bosons in a tight-binding lattice [20].The presented two-particle dynamics may provide intuitive access to the basic features of many-body systems.In the presence of periodic external fields, interacting systems may exhibit non-integrable properties, necessitating analytical studies using perturbation methods [21].It has been shown that the correlated pair tunnelling in the strongly interacting Floquet (periodically driven) systems can be understood via the second-order effect by means of a multiscale asymptotic analysis [19,20].
In recent years, spin-orbit coupling (SOC) in ultracold atomic physics has attracted a great deal of attention from both the experimental [22][23][24][25][26][27][28][29][30][31][32] and the theoretical [33][34][35][36][37][38][39][40][41][42][43][44][45][46] sides.In experiments, SOC can be created by coupling two hyperfine states of atoms via a pair of counter propagating Raman lasers [22], and the strength of the synthetic SOC exhibits excellent tunability [23].The mean-field dynamics of SO-coupled atoms in a double-well potential has been the subject of detailed study and some novel dynamical phenomena have been revealed, such as the Josephson effect [33], net atomic spin current [34], interaction induced spin localization [35], and so on.In addition, a number of studies have been carried out on controlling the spin tunneling of SOC atoms in a double-well potential driven by periodic forces.For instance, it has been shown that the range of modulation frequencies at which tunneling suppression occurs can be drastically increased by tuning the SOC strength for SO-coupled non-interacting BEC (or single cold atom) in a double-well trap subjected to periodic out-of-phase modulation [36].The coherent control of the spin dynamics of the SO-coupled atom by period drive field has been extended to non-Hermitian systems as well [37][38][39].The mean-field dynamics (or single-particle dynamics) of the SOC bosonic junction can typically be described by a four-mode Hamiltonian, so there are no dark Floquet states, since the dark Floquet state, in analogy to the well-known dark state in undriven systems, has been shown to exist only in the odd-mode model.There has also been limited research on the many-body dynamics of the SO-coupled BEC system in a double-well potential from the perspective of a two-mode Bose-Hubbard-like Hamiltonian [40][41][42].We note that the system of two interacting particles in the presence of SOC has recently been investigated in both two-dimensional [43] and one-dimensional [44] harmonic traps.The two-atom system is a bridge between the single-particle system and the many-particle systems, which may provide access to the crossover from few-body to many-body physics.However, much less is known so far about the dominant pair tunneling and localization mechanisms for the two-particle dynamics mediated by the SOC.
In this paper we study the tunneling dynamics of two interacting SO-coupled bosons in a double-well potential and our focus is on what new effects arise from the interplay between the periodic driving and the SO interaction.We begin with the simple single-drive field, i.e.only the double-well potential is periodically driven.In the multiphoton resonance regime, two different mechanisms for the appearance of CDT are identified and CDT is found to occur at particular values of SOC strength.When the SOC strength is an integer, there is a conventional CDT resulting from the quasi-energy degeneracy (actually a pseudo-degeneracy due to the finite-size effect).When the SOC strength takes on a half-integer value, the CDT mechanism originates from a peculiar dark Floquet state with zero quasi-energy, and the suppression of tunneling occurs over a wide range of driving parameters and is robust against the extension of the dynamical evolution time.We also reveal that SOC drastically changes the picture of correlated pair tunneling in the far-off-resonant regime.To achieve SOC tunability, we further consider the double modulation combining the double-well shaking and a time-periodic Zeeman field.We find that the CDT and spin-flipping single-atom Rabi oscillation can occur at arbitrary values of SOC strength, which is not accessible with a single-drive field.In this case, the CDT also arises from the dark Floquet state rather than the usual quasi-energy degeneracy, and the tunneling period of the single-atom Rabi oscillation can be tuned by adjusting the SOC strength.

Model equation
We consider two bosons with two pseudospin states (| ↑⟩ and | ↓⟩) hopping on a one-dimensional two-site optical lattice with synthetic SO coupling.The effective SO coupling can be implemented experimentally using a pair of counter propagating Raman lasers which generate a momentum-sensitive coupling between the two internal hyperfine states of the same atom.The single-particle Hamiltonian for a SO-coupled atom moving along the x direction in the absence of the external potential can be written in a gauge-field form [22]: ĥ = (px− Â) 2 2M + Ωσ z , where px is the atomic momentum operator, M is the atomic mass, Ω behaves as a Zeeman field, σx,y,z are the usual 2 × 2 Pauli matrices, and the effective SO coupling is embodied in a synthetic non-Abelian vector potential Â = −k R σy with the wave number of Raman laser k R characterizing the SO coupling strength.
2 cos(ωt) with driving frequency ω and driving amplitude f 0 .Employing the Fock basis represents n 1↑ spin-up (n 1↓ spin-down) particles occupying the lattice site 1, and n 2↑ spin-up (n 2↓ spin-down) particles occupying the lattice site 2, the state vector |ψ(t)⟩ of the two-atom system at any time t can be expanded as follows where c l (l = 1, 2, . . ., 10) are the time-dependent probability amplitudes of the two atoms being in the given state, and the corresponding probabilities are P l = |c l (t)| 2 (l = 1, 2, . . ., 10), obeying the normalization condition l P l = 1.The evolution equations for the probability amplitudes c l (t) can be obtained from the Schrödinger equation i∂ t |ψ(t)⟩ = Ĥ|ψ(t)⟩.For simplicity, h = 1 is assumed and the reference frequency ω 0 = 0.1E R is set so that the parameters ν, Ω, ω, f 0 are measured in units of ω 0 , and the time t is normalized to units of ω −1 0 .In experiment [22], the wavelength of the Raman laser is λ R = 804.1 nm, so the single-photon recoil energy is E R = k 2 R /2M = 22.5 kHz, with the Raman laser wave vector k R = 2π/λ R .

CDT induced by single-drive field in the multiphoton regime
In general, it is difficult to find analytical solutions to the driven Bose-Hubbard Hamiltonian (1).To gain some analytical insights, we will focus our investigation to the high-frequency regime (ω ≫ ν) and the multiphoton regime, where the static Zeeman field is an integer multiple of the frequency of the driving field, namely, Ω = Ω 0 = mω, (m = 1, 2, 3, . ..), and the atomic interaction strength is in resonance with the driving field (i.e.U = nω, n is an integer).In this case, the dynamics can be approximately described, at the first order of 1/ω, by an effective static Hamiltonian by applying the routinely used time-averaging (high-frequency approximation) method [46][47][48], where 2 cos ωtdt = f0 2ω sin ωt, and B(t) = ´t 0 Ωdt = Ω 0 t.Performing the integral in equation (3), we obtain the effective Hamiltonian: where σ± = 1 2 σx ± iσ y , χ = f0 ω , J γ (z) denotes the γ-order Bessel function of the variable z, the '+' in the plus-minus sign is associated with the conversion from the singly occupied site state to the doubly occupied site state (doublon), and '−' corresponds to the conversion from the doubly occupied site state (doublon) to the singly occupied site state.To study the system's time-evolution quantitatively, for convenience we term the minimum value of |⟨ψ(0)|ψ(t)⟩| 2 attained during the time evolution to be the localization, where we usually prepare the initial state as the Flock state |ψ(0)⟩ = |0, 0, 0, 2⟩.When the localization is not zero, tunneling is suppressed as the population can not transfer completely from the initial state to the other Fock states.If localization is 1, the tunneling effect will be completely suppressed, which we refer to as complete localization.
According to the time-dependent Schrödinger equation, we can numerically plot the localization of the state |0, 0, 0, 2⟩ as a function of the normalized driving parameters f 0 /ω and the effective SO coupling strength α, as shown in figure 1.In figure 1 we find that an almost perfect localization of |0, 0, 0, 2⟩ occurs when the SO coupling strength lies at α = 0 or α = 1/2.We then go on to analyze why complete localization of |0, 0, 0, 2⟩ occurs.
We first consider α = 0, from equation ( 4) we have the effective Hamiltonian: Equation ( 5) implies that quantum tunneling always involves spin conservation when α = 0. Consequently, the state |0, 0, 0, 2⟩ is only coupled to the state |0, 1, 0, 1⟩ and then indirectly to |0, 2, 0, 0⟩.Therefore we can treat the problem by using the three-mode model (TMM).From equation ( 5), in the Fock space spanned by the three Fock states, |0, 0, 0, 2⟩, |0, 1, 0, 1⟩, |0, 2, 0, 0⟩, Ĥeff becomes a matrix: From equation ( 6) we observe that when J 2n (χ) = 0, the effective tunneling between the three states |0, 0, 0, 2⟩, |0, 1, 0, 1⟩, |0, 2, 0, 0⟩ vanishes and thus the CDT occurs, which is the well-known result of the time-averaging approximation in the large frequency limit.Our direct numerical investigation of the original system (1) confirms the above results.Due to the time periodicity of the Hamiltonian (1), the energy spectrum of the undriven Bose-Hubbard Hamiltonian is replaced by the quasi-energy spectrum ε, which can be numerically calculated by diagonalizing the time evolution operator over one period.In figure 2(a) we plot the quasi-energies for the first photon resonance (n = 1) of the driven two-site Bose-Hubbard model (1) and compare them with the energies of the time-independent TTM model (6).For weak fields (χ < 2) the small deviations of the exact quasi-energies from the perturbative result are visible, but for higher field strengths the agreement is excellent.In figure 2(b) we plot the localization produced in this system with the initial state |0, 0, 0, 2⟩, and we can see that, as we have already seen in figure 1, the localization takes extremely low values, except for a series of very narrow peaks.These peaks are precisely aligned with the quasi-energy crossings at the zeros of J 2n (χ).This is the common explanation of CDT, which results from quasienergy degeneracy and happens only at the isolated parameter points.After enlarged view of the quasi-energies near J 2n (χ) = 0, we find that the collapse of quasi-energies is in fact not the exact crossing, but rather a fine structure of crossings and avoided crossings.This is the finite-size effect which has been conveniently exploited for selective CDT in a finite (truncated) lattice (or equivalent finite-level system) [49,50].This pseudocrossing of quasi-energies will lead to the instability of the CDT if we extend the evolution time, as shown in figure 2(b).This approximate degeneracy allows for a small but nonzero degree of tunneling between the two paired states |0, 0, 0, 2⟩ and |0, 2, 0, 0⟩, and thus the maximum value of the produced localization will become smaller and smaller as the evolution time is increased.Note that the peaks in figure 2(b) have a narrow range, which is due to the finite evolution time in numerical simulation.
To study this localization phenomenon qualitatively, we plot the quasi-energies for the first photon resonance (n = m = 1) of the two-atom system (1), as shown in figure 3(a).For brevity, we show only the three quasi-energies for the system confined in the space spanned by the three states {|0, 0, 0, 2⟩, |1, 0, 0, 1⟩, |2, 0, 0, 0⟩}.We notice that there is no degeneracy in the quasi-energy levels.This is in stark contrast to the case α = 0, where the CDT is rooted in quasi-energy degeneracy.Instead, we find that there is a Floquet state (which is called the dark Floquet state [8]) with zero quasi-energy for all values of f 0 /ω.In analogy to the well-known dark state, this dark Floquet state stands out not only for its zero quasi-energy but also for its unique population distribution among the Fock states.In figure 4, we display the time-averaged population probability ⟨P j ⟩ = ( ´T 0 dt|c j | 2 )/T for the given dark Floquet state.We can see that the dark Floquet state has a negligible population at the intermediate state |1, 0, 0, 1⟩, and as f 0 /ω is increased, ⟨P 6 ⟩ (the time-averaged population at state |0, 0, 0, 2⟩) exhibits a sequence of peaks centered on f 0 /ω = 2.4048, 5.5201, and 8.6537-the zeros of J 2n−2m (χ).
It is not difficult to see the suppression of the tunneling seen in figure 3 is linked to the existence of the dark Floquet state.We expand the initial state of |0, 0, 0, 2⟩ in terms of the Floquet states: During the dynamical evolution, the expansion coefficient b i evolves as b i e (−iε i t).We look at the case , where the dark Floquet state |ε 2 ⟩ has population one at state |0, 0, 0, 2⟩ while  it has zero population at |2, 0, 0, 0⟩ and |1, 0, 0, 1⟩.In this case, we have |b 2 | = 1 and b 1 = b 3 = 0, which corresponds to a complete suppression of the tunneling from |0, 0, 0, 2⟩ to |2, 0, 0, 0⟩ and |1, 0, 0, 1⟩.This suggests that the CDT observed in figure 3(b) has a different origin: it is the result of a dark Floquet state.Furthermore, we observe that the peak of ⟨P 6 ⟩ is rather broad, which implies that the dark Floquet is localized at the state |0, 0, 0, 2⟩ for a finite range of parameters.This explains the broadening of the suppression regime observed in figure 3(b), which is different from the case of α = 0, where tunneling suppression occurs only at isolated parameter points.Due to the different origin of the CDT, the peaks of localization for the case α = 1/2 are neither sharpened nor lowered as the evolution time is extended, as shown in figure 3(b).We further provide analytical insights for the broad peak of localization observed in figure 3(b).For the spin-flipping tunneling dynamics, the effective three-model model (8) has the eigenvalues given by with the corresponding three eigenvectors, The eigenstate |ε 2 ⟩ of zero energy, which can be represented in the Fock space as the coherent superposition is the well-known dark state, defined by zero coupling to the intermediate state |1, 0, 0, 1⟩.Apparently, the weights of the two wavefunction components of the dark state are adjusted by the Bessel functions J 2n−2m (χ) and J −2n−2m (χ) through Floquet control.For a wide range of driving parameters χ near resonance, the dark state is unbalanced, with the population on |0, 0, 0, 2⟩ being larger than on |2, 0, 0, 0⟩, that is, as confirmed in figure 4. At time t, the quantum state evolves according to By applying the initial condition |ψ(0)⟩ = |0, 0, 0, 2⟩ to equation (11), we obtain the occupation probability at the state |0, 0, 0, 2⟩ as From expression (12), we can immediately make two observations: which occurs over a wide range of χ), the minimum value of P |0,0,0,2⟩ (t) during the , which is a number greater than zero.The shortest evolution time to obtain the minimum value of P |0,0,0,2⟩ (t) is given by t min = 2π/δ, where δ = 2ε 1 represents the energy gap shown in figure 3 Recalling that the localization is defined by the minimum value of |⟨ψ( 0 , as the evolution time exceeds t min , the width of the peaks near resonance no longer decreases with increasing time.We note that a Floquet state with zero quasi-energy is also found in the spin-preserving system, as can be seen from the zero-energy flat band shown in figure 2(a).However, it is not related to the CDT discovered in figure 2(b) because this dark Floquet state is an equally weighted superposition of the two basis states |0, 0, 0, 2⟩ and |0, 2, 0, 0⟩, with zero coupling to the intermediate state |0, 1, 0, 1⟩.

SOC-related correlated pair tunnelling in the far-off-resonant regime
As is well known, the correlated pair tunnelling (also referred to as second-order tunnelling) is the dominating dynamical effect in the strongly interacting regime for two interacting ultracold atoms through a barrier in a double-well potential.And then a question naturally arises: how does the SO coupling modify the picture of the atom pair tunneling?In this section, we are trying to address these issues.To do this, we turn to the far-off-resonant regime with U/ω = µ (by 'far-off-resonant' we mean that the atomic interaction U should take values sufficiently far from any integer multiple of ω) and the Zeeman field is set as Ω = Ω 0 + ∆ = mω + ∆ (m = 1, 2, 3, . ..).As we will show later, the detuning ∆ of the Zeeman field with respect to the multiphoton energy plays a crucial role in the correlated pair tunnelling of the two SO-coupled atoms.
First we consider α = 1/2.In this case, the spin flipping accompanies the tunneling of the atom between the potential wells, and the quantum state of the SO-coupled two-atom system can be expanded as Inserting equation ( 14) into Schrödinger equation i∂ t |ψ(t)⟩ = Ĥ|ψ(t)⟩ , we obtain the following coupled equations: Making the transformation, c 2 cos ωtdt = f0 2ω sin ωt, τ = ωt, and ∆ = ϵ 2 ωβ, we can readily arrive at We seek for a solution to equation ( 16) as a power-series expansion in the smallness parameter ϵ: At the same time, we introduce multiple scales for time, τ 0 = τ , τ 1 = ϵτ , τ 2 = ϵ 2 τ , . .., and then replace the time derivatives by the expansion New J. Phys.26 (2024) 043020

H Wu et al
At the next order ϵ 2 , we have i ∂C (2) with 8 ,

M
(2) In order to avoid the occurrence of secularly growing terms in the solutions C (2) 8 the following solvability conditions must be satisfied: where we have set ln the derivation of equation ( 29), we have used the fomula e −iχ sin(ωt) = r J r (χ)e −irωt .Thus the evolution of the zeroth-order amplitudes A n up to the second-order long time scale is given by Substituting equations ( 17), (22), and (28) into equation (30) , and returning to the original time variable t , we obtain Equation ( 31) provides an accurate description of the dynamics of the original system with only spin-flipping coupling in the far-off-resonance regime, up to the second-order long time scale.From equation (31), we observe that the state |2, 0, 0, 0⟩ (with the probability amplitude A 2 ) is only coupled to |0, 0, 0, 2⟩ (with the probability amplitude A 6 ), while being fully decoupled from the state |1, 0, 0, 1⟩ (with the probability amplitude A 8 ).When the detuning ∆ is selected as , the diagonal energy difference between states |2, 0, 0, 0⟩ and |0, 0, 0, 2⟩ becomes zero, as seen by comparing the coefficients of A 2 and A 6 on the right-hand side of the first two equalities in equation (31).This allows the two paired states to undergo perfect Rabi oscillations with zero amplitude A 8 .To validate the analytical results, we compare the time evolution of the atomic probabilities of system (1) initialized in the state |ψ(0)⟩ = |0, 0, 0, 2⟩ for two cases: one without detuning between the Zeeman field and the drive field (∆ = 0), and the other with detuning (∆ = 0.002 924), with the system parameters set as α = 1/2, ω = 40, Ω = ω + ∆ = 40 + ∆, U = 1.4ω = 56, ν = 1, and f 0 /ω = 8.7.This comparison is shown in figure 6.When ∆ = 0 (that is Ω = Ω 0 = mω), where the energy gap Ω created by the spin flipping is bridged by the driving field, as shown in figure 6(a), in the large interaction regime U ≫ ν, it can be seen that both atoms essentially remain in the same well during the tunneling process, i.e. they tunnel as bound pairs, but they cannot fully transfer to the other well.In contrast, when ∆ = 0.002 924, complete pair tunneling with spin flipping is shown in figure 6(b), as the detuning ∆ is introduced to counteract the second-order energy difference between the two paired states |0, 0, 0, 2⟩ and |2, 0, 0, 0⟩, allowing the repulsively bound atom pairs to fully tunnel back and forth between the two potential wells.
We then consider the other case α = 0, where the tunneling process is accompanied by spin conservation.We write down C l = C (0) , and repeat the procedure as described above, and obtain the following equation where 10 , and Equation (32) gives a description of the conventional second-order tunneling without SOC between the paired states |0, 0, 0, 2⟩, and |0, 2, 0, 0⟩.Obviously, there is no need to introduce the additional detuning ∆ of the Zeeman field with respect to the multiphoton energy to realize the complete co-tunnelling between the states |0, 0, 0, 2⟩, and |0, 2, 0, 0⟩, as numerically demonstrated in figure 7.

Effect of combined modulations on the CDT phenomenon and single-atom Rabi oscillation
From the previous discussion, we know that the appearance of CDT occurs only at specific values of α in the context of the double-well potential subjected to periodic, out-of-phase modulation.In order to have SOC tunability, we consider the double modulation, which combines double-well potential shaking and modulated Zeeman field, that is, we assume that Zeeman term varies periodically in time as Ω = Ω(t) = Ω 0 + Ω 1 cos(ωt), with the same frequency ω of the driven potential.
As mentioned above, we set U = nω and Ω 0 = mω.From equation ( 1), for the high driving frequency ω ≫ ν, we employ the time-averaging method to derive the effective time-averaged Hamiltonian: where 2 cos ωtdt = f0 2ω sin ωt, and B(t) = ´t 0 Ω(t)dt = Ω 0 t + Ω1 ω sin ωt.Performing the integral in equation (34), we obtain the effective Hamiltonian: where σ± = 1 2 σx ± iσ y , χ = f0 ω and η = 2Ω1 ω , J γ (z) denotes the γ-order Bessel function of the variable z.In equation (35), the '+' in the plus-minus sign is associated with the conversion from the singly occupied site state to the doubly occupied site state (doublon), and '−' corresponds to the conversion from the doubly occupied site state (doublon) to the singly occupied site state.
In figure 8(b) we plot the localization of |0, 0, 0, 2⟩ as a function of η.We can see that the localization has a series of broad peaks centred on the zeros of J 2n−2m (χ + η).The numerical results establish again a firm link between the CDT and dark Floquet state.We examine the time-averaged populations of the dark Floquet state corresponding to ε 2 = 0 (figure 8(a)), and find that the localization peaks are exactly aligned with the points where the dark Floquet state has population one in the state |0, 0, 0, 2⟩, i.e. ⟨P 6 ⟩ = 1.As the dark Floquet state |ε 2 ⟩ is localized at state |0, 0, 0, 2⟩ over a wide range of parameters (figure 8(c)), the suppression of tunneling occurs accordingly over a wide range of parameters (figure 8(b)).As shown in figure 9, where we adjust the system parameters to match the condition (37), we see that the CDT can occur for arbitrary values of SOC strength.This phenomenon of CDT occurring for arbitrary values of SOC is unique to the doubly modulated system and is not possible for a single-drive field.
Finally, we discuss the single-atom Rabi oscillation with spin flipping by the application of combined modulation.Following the same line of reasoning, we can conclude from equation (36) that when J 2n (χ) = 0 and J −2n−2m (χ + η) = 0, the spin-flipping single-atom Rabi oscillation between the states |0, 0, 0, 2⟩ and |1, 0, 0, 1⟩ can be realized with the tunneling period T 2 = π/ω ′ 2 , ω ′ 2 = | √ 2ν sin(πα)J 2n−2m (χ + η)|.Obviously, the period of the Rabi oscillations can be modulated by adjusting the strength of the SOC, and when the strength of the SOC is an integer, the Rabi period becomes infinite and CDT appears.In figure 10, we plot the time evolution of the atomic probabilities for the system parameters satisfying the relations J 2n (χ) = 0 and J −2n−2m (χ + η) = 0, with different values of α, where we observe that an atom performs the spin-flipping Rabi oscillation between the potential wells, and the tunneling period is longer for α = 0.9 than for α = 0.6.When α = 1 (or 0), the system is frozen in its initial state |0, 0, 0, 2⟩ and CDT occurs.

Conclusions
In summary, we have theoretically studied the effects of SOC on the tunneling dynamics of the driven two-particle Bose-Hubbard model.We first carry out a theoretical analysis and numerical verification of the tunneling dynamics of the two interacting bosons mediated SOC in a periodically driven double-well potential for the cases of multiphoton resonance and far-off-resonance.For the resonant case, we find that there are two different mechanism for occurrence of CDT.When the tunneling process is exclusively spin-preserving, CDT occurs due to quasi-energy degeneracy, which only happens at isolated parameter values.When tunneling is exclusively spin flipping, CDT occurs due to the dark Floquet state, which occurs over a wide range of parameter.For the far-off-resonant case, where the strong interparticle interaction U is far from any integer multiple of the driving frequency, we find that complete correlated pair tunneling with spin flipping can be achieved by introducing a small detuning between the Zeeman field and the drive field to counteract the effective second-order energy offset between the paired states.We also find that double modulation has an advantage over single-drive field for coherent control of spin tunneling.When the double-well potential and Zeeman field are both periodically modulated, the CDT and spin-flipping single-atom Rabi oscillation can occur at arbitrary SOC strength values, which is not possible with a single-drive field.In this doubly modulated system, CDT is caused by the dark Floquet state and tunneling suppression occurs over a wide range of driving parameters.These features would underlie two-particle transfer phenomena in the realistic SO-coupled systems.

Figure 3 .
Figure 3. (a) Quasi-energies as a function of the normalized driving parameters f0/ω for α = 1/2.Solid lines (red) denote exact numerical results obtained from equation (1), which agree well with the high-frequency approximation solutions derived from equation (8) (black solid circles).(b) The localization as a function of the normalized driving parameters f0/ω for α = 1/2.The localization peaks are centred at f0/ω = 2.4048, 5.5201, 8.6537, . ..-the zeros of J 2n−2m ( f 0 /ω), where n = m = 1.Throughout the paper we consider the first photon resonance n = m = 1 unless otherwise stated.The initial condition and other parameters are the same as in figure 2.

Figure 4 .
Figure 4.The time-averaged population distributions for the dark Floquet state |ε2⟩ in figure 3(a).The other parameters are the same as in figure 3.

Figure 7 .
Figure 7. Time evolution of the occupation probabilities for the system (1) with the initial state |ψ(0)⟩ = |0, 0, 0, 2⟩ for the case of α = 0 with ∆ = 0.The blue and red lines denote the numerical results obtained from the original model(1), and the open circles denote the second-order perturbative from equation The other parameters are same as in figure