Bloch oscillations of spin-orbit-coupled cold atoms in an optical lattice and spin current generation

TL;DR

This paper investigates the dynamics of spin-orbit-coupled cold atoms in an optical lattice, revealing how inter- and intraladder couplings influence Bloch oscillations and exploring the potential for spin current generation.

Contribution

It provides a detailed analysis of Bloch oscillations in spin-orbit-coupled cold atoms, highlighting the effects of inter- and intraladder couplings and the breaking of Galilean invariance.

Findings

Intraladder coupling causes standard Bloch oscillations.

Interladder coupling results in high-frequency oscillations.

Spin-orbit interaction leads to out-of-phase spin component oscillations.

Abstract

We study the Bloch oscillation dynamics of a spin-orbit-coupled cold atomic gas trapped inside a one-dimensioanl optical lattice. The eigenspectra of the system is identified as two interpenetrating Wannier-Stark ladder. Based on that, we carefully analyzed the Bloch oscillation dynamics and found out that intraladder coupling between neighboring rungs of Wannier-Stark ladder give rise to ordinary Bloch oscillation while interladder coupling lead to small amplitude high frequency oscillation superimposed on it. Specifically spin-orbit interaction breaks Galilean invariance, which can be reflected by out-of-phase oscillation of the two spin components in the accelerated frame. The possibility of generating spin current in this system are also explored.

Equations

$$\hat{H}$$

$${\hat{H}}_{SO}$$

$$\hat{H}$$

$$+i\frac{J}{2}\sin \left(\pi \gamma \right)\left(|j,\uparrow ⟩⟨j+1,\uparrow |-|j,\downarrow ⟩⟨j+1,\downarrow |\right)$$

$$+\frac{\hslash \Omega }{2}|j,\uparrow ⟩⟨j,\downarrow |+H.c.]-Fd\sum _{\sigma }j|j,\sigma ⟩⟨j,\sigma |$$

$$+\frac{\hslash \delta }{2}\left(|j,\uparrow ⟩⟨j,\uparrow |-|j,\downarrow ⟩⟨j,\downarrow |\right)\}$$

$$J=-2\int dz{w}_{j+1}\left(z\right)\left[-\frac{d^2}{d{z}^2}+U_0{\sin }^2\left(k_{l}z\right)\right]w_j\left(z\right),$$

$$|q,\sigma ⟩=\sqrt{\frac{d}{2\pi }}\sum _{j=-\infty }^{\infty }|j,\sigma ⟩e^{iqjd}.$$

$$\hat{H}\left(q\right)=\left\langle q\left|\hat{H}\right|q\right\rangle=\left(\begin{array}[c]{cc}H_{d}^{+}&\hbar\Omega/2\\ \hbar\Omega/2&H_{d}^{-}\end{array}\right),$$

$$-iF\frac{\partial {\psi }_{\uparrow }\left(q\right)}{\partial q}-J\cos \left(qd-\pi \gamma \right){\psi }_{\uparrow }\left(q\right)+\frac{\hslash \delta }{2}{\psi }_{\uparrow }\left(q\right)$$

$$+\frac{\hslash \Omega }{2}{\psi }_{\downarrow }\left(q\right)=E{\psi }_{\uparrow }\left(q\right),$$

$$-iF\frac{\partial {\psi }_{\downarrow }\left(q\right)}{\partial q}-J\cos \left(qd+\pi \gamma \right){\psi }_{\downarrow }\left(q\right)-\frac{\hslash \delta }{2}{\psi }_{\downarrow }\left(q\right)$$

$$+\frac{\hslash \Omega }{2}{\psi }_{\uparrow }\left(q\right)=E{\psi }_{\downarrow }\left(q\right),$$

$${\psi }_{\uparrow }^{\nu }\left(q\right)$$

$${\psi }_{\downarrow }^{\nu }\left(q\right)$$

$$\frac{\hslash \Omega }{2}\sum _{m^{\prime }}{i}^{m-m^{\prime }}{J}_{m-m^{\prime }}\left(\frac{2J}{Fd}\sin \left(\pi \gamma \right)\right)B_{m^{\prime }}^{\nu }$$

$$+\left(mFd+\frac{\hslash \delta }{2}\right)A_m^{\nu }=E_{\nu }{A}_m^{\nu },$$

$$\frac{\hslash \Omega }{2}\sum _{m^{\prime }}{\left(-i\right)}^{m-m^{\prime }}{J}_{m-m^{\prime }}\left(\frac{2J}{Fd}\sin \left(\pi \gamma \right)\right)A_{m^{\prime }}^{\nu }$$

$$+\left(mFd-\frac{\hslash \delta }{2}\right)B_m^{\nu }=E_{\nu }{B}_m^{\nu },$$

$$W_{j,\uparrow }^{\nu }$$

$$W_{j,\downarrow }^{\nu }$$

$$\frac{dz}{dt}$$

$$+e^{i\pi \gamma }{\psi }_{j,\downarrow }^{\ast }\left(t\right){\psi }_{j+1,\downarrow }\left(t\right)].$$

$$\frac{dz}{dt}$$

$$+\sum _j{W}_{j,\downarrow }^{\nu \ast }{W}_{j+1,\downarrow }^{{\nu }^{\prime }}{e}^{i\pi \gamma }]e^{i\left(E_{\nu }-E_{{\nu }^{\prime }}\right)t/\hslash }\}.$$

$$z_{\uparrow \left(\downarrow \right)}\left(t\right)$$

$$\times \left[1-e^{i\left(E_{\nu }-E_{{\nu }^{\prime }}\right)t/\hslash }\right]\}+z_{\uparrow \left(\downarrow \right)}\left(0\right)$$

$${\psi }_j\left(t=0\right)={\left(a\sqrt{\pi }\right)}^{-1/2}{e}^{-{\left(j-j_0\right)}^2/2a^2+i{q}_{0}jd}\left(\genfrac{}{}{0ex}{}{1}{0}\right),$$

$${\hat{F}}_z=\left[{\hat{H}}_{SO},\left[\hat{\mathbf{z}},{\hat{H}}_{SO}\right]\right]=\frac{{\hslash }^3{k}_R\Omega }{m}{\hat{\sigma }}_y{\mathbf{e}}_z,$$

$${\hat{J}}_S^i\left(t\right)=\frac{d}{dt}\left({\hat{\sigma }}_i\hat{z}\right).$$

$$J_S^z\left(t\right)$$

$$-\frac{Jd}{\hslash }{e}^{i\pi \gamma }{\psi }_{j,\downarrow }^{\ast }\left(t\right){\psi }_{j+1,\downarrow }\left(t\right)+2\Omega dj{\psi }_{j,\uparrow }^{\ast }\left(t\right){\psi }_{j,\downarrow }\left(t\right)]$$

$$=\sum _{\nu ,{\nu }^{\prime }}\mathrm{Im}\{a_{\nu }^{\ast }{a}_{{\nu }^{\prime }}\sum _j[\frac{Jd}{\hslash }{W}_{j,\uparrow }^{\nu \ast }{W}_{j+1,\uparrow }^{{\nu }^{\prime }}{e}^{-i\pi \gamma }$$

$$-\frac{Jd}{\hslash }{W}_{j,\downarrow }^{\nu \ast }{W}_{j+1,\downarrow }^{{\nu }^{\prime }}{e}^{i\pi \gamma }+2\Omega dj{W}_{j,\uparrow }^{\nu \ast }{W}_{j,\downarrow }^{{\nu }^{\prime }}]e^{i\left(E_{\nu }-E_{{\nu }^{\prime }}\right)t/\hslash }\},$$

Full text

I introduction

Bloch oscillation describe that inside a lattice potential a particle will perform periodic oscillation instead of constant acceleration when subject to a constant external force. It was first proposed in electronic system blochoriginal , however have not been observed until the use of semiconductor superlattice blochexperiment due to the small lattice constant and imperfections in conventional crystal. The frequency of Bloch oscillation is propotional to the applied force $F$, which can have potential application in precision measurement. Besides that, the dynamics concerning particles moving in periodic structures is itself important due to that it is a pure quantum effect and reflects the properties of energy band such as the topology topology . These extends people’s interest in Bloch oscillation beyond the electronic system. Bloch oscillation have been experimentally observed in optical system opticalBloch and ultracold atoms trapped in an optical lattice castin1996 ; niuPRL1996 ; atomBlochreview . Recently it was demonstrated that impurity moving in quantum liquids can also display the behavior of Bloch oscillation impurity ; impuritytheory . Theoretically Bloch oscillation can be well-understood within adiabatical approximation in which the particles move in Bloch energy band under the action of the force castin1996 . The eigenstate of Bloch oscillation is also well-known as Wannier-Stark ladder (WSL) WSLreview .

On the other hand besides the external centre-of-mass motion, particles possess internal degree-of-freedom such as the electronic spin. Pseudospin can also be constructed from the atomic internal energy level structure. Through the mechanism of spin-orbit (SO) coupling particle’s orbital motion can be connected to its spin dynamics and lead to rich physics. Recently SO coupling have been successfully implemented in neutral atom soReview ; 1dso ; chenPRL2012 . Along with that, interesting physics have been predicted in SO-coupled atomic system such as dipole oscillation chenPRL2012 ; puPRA2012 , Zitterbewegung leblancNJP2013 ; quPRA2013R , spin-dependent pairing dongPRA2013 , SO-modulated Anderson localization zhouPRA2013 ; orsoPRL2017 ; shermanPRL2015 , SO-modulated atom optics zhouatomoptics and exotic dynamics lanPRA2014 ; EngelsPRL2015 ; flat band ; panPRA2016 ; engelsPRL2017 .

Then it is natural to ask how Bloch oscillation will be affected by SO interaction. In the present work we will investigate the Bloch oscillation dynamics of SO-coupled cold atoms in a one-dimensional optical lattice. An important motivation lies in the recent achievement of SO-coupled Bose-Einstein condendates (BEC) in a one-dimensional optical lattice EngelsPRL2015 , which guarantee that the results obtained here can be readily observed in experiment. In previous theoretical works, Larson and co-workers investigated Bloch oscillation of SO-coupled BEC in a two-dimensional optical lattice, in which transverse spin current and atomic Zitterbewegung are predicted LarsonPRA2010 . Bloch oscillation of a SO-coupled helicoidal molecule was studied by Caeteno in caetanoPRB2014 . Kartashov et al. studied Bloch oscillation in one-dimensional optical and Zeeman lattices in the presence of SO coupling, in which they give a detailed discussion on the amplitude and wavepacket width of Bloch oscillation KartashovPRL2016 . Although the WSL eigen-spectra have been given in KartashovPRL2016 , its relation with the oscillation dynamics was not clarified yet. Here we will solve the dynamics using the theory of WSL. We show that one can understand the properties of Bloch oscillation dynamics in the presence of SO coupling via analyzing the coupling of WSLs. Especially in the case with finite Zeeman detuning which was not considered in KartashovPRL2016 , the two spin components will display unusual out-of-phase oscillation. In addition we show how this can serve as an unambiguous proof of broken Galilean invariance caused by SO interaction. Since SO interaction can play a crucial role in generating and manipulating spin current SpinCurrentScience , we’ll also look into the possibility of generating spin current in the present one-dimensional system.

The article is organized as follows: In Sec. II we present our model and the dynamics are solved with WSL. Section III is devoted to the detailed discussion of Bloch oscillation. The possibility of generating spin current in the present system is explored in Sec. IV. Finally we conclude in Sec. V.

II model

As shown in Fig. 1, our model is based on the recent experiment EngelsPRL2015 with a 87Rb BEC prepared in a one-dimensional optical lattice along the $z$-direction, inside which the effective SO interaction is induced via coupling the $\left|1,-1\right\rangle$ ($\left|\downarrow\right\rangle$) and $\left|1,0\right\rangle$ ($\left|\uparrow\right\rangle$) hyperfine states with Raman lasers. In addition to that, here we consider that a constant external force $F$ is exerted on the atoms via tilting the optical lattice. The effective single-particle Hamiltonian reads

[TABLE]

in which the SO coupling is embodied in the effective vector potential $\hat{A}=-m\alpha\hat{\sigma}_{z}$ ($\alpha=\hbar k_{R}/m$ characterizes SO coupling strength with $k_{R}$ the Raman beam wavevector), $\Omega$ is the Raman coupling strength with $\delta$ the two-photon detuning. The periodic potential is characterized by the depth $U_{0}$ and period $d=\pi/k_{l}$.

By performing lowest energy band truncation and assuming tight binding approximation, Hamiltonian (1) can be expanded in the $\sigma$-Wannier basis $\left|j,\sigma\right\rangle$ (with $j$ the lattice site index) as

[TABLE]

in which the spin-dependent hopping matrix element $\hat{T}=J\exp\left(-i/\hbar\int\hat{A}dl\right)/2$ is obtained through Peierls substitution peierls , $J$ is the tunneling amplitude without SO coupling, $\gamma=k_{R}/k_{l}$. $J$ can be calculated as

[TABLE]

with $w_{j}\left(z\right)=w\left(z-z_{j}\right)$ is the Wannier state of the lowest energy band at the $j$-th site which can be obtained numerically lattice parameter . Here we consider the case of $U_{0}>0$ with $z_{j}=jd$.

In order to find out the eigenstates of Hamiltonian (2), it will be more convienent to transform it into the Bloch basis via the Fourier transformation HartmannNPJ2004

[TABLE]

One can then obtain

[TABLE]

with $H_{d}^{\pm}=-J\cos\left(qd\mp\pi\gamma\right)\pm\hbar\delta/2-iF\partial/\partial q$. The eigenvalue problem then resort to

[TABLE]

Consider that $\psi^{\nu}\left(q\right)=\left[\psi_{\uparrow}^{\nu}\left(q\right),\psi_{\downarrow}^{\nu}\left(q\right)\right]^{T}$ to be the $\nu$-th eigensolution of Eqns. (6) with the corresponding eigenvalue $E_{\nu}$, it can be solved via performing the Fourier expansion

[TABLE]

where $A_{m}^{\nu}$ and $B_{m}^{\nu}$ are expansion coefficients with the truncation number $M$. Through numerical calculation we found that $M=50$ to be a good approximation for the parameters considered in the present work. Substitute (7) into Eqns. (6), one can have

[TABLE]

with $J_{n}\left(z\right)$ the $n$th-order Bessel functions of the first kind. One can then numerically solve Eqns. (8) and obtain the coefficients $A_{m}^{\nu}$, $B_{m}^{\nu}$ and the corresponding eigenenergy $E_{\nu}$. The Wannier amplitudes of the corresponding eigenvector read

[TABLE]

In the case without SO coupling the eigenenergy of Eqs. (6) is known as WSL Wannier1960 , which consists of quantized energy levels with equal energy spacing $Fd$. In the presence of SO coupling WSL still exists, as can be seen from the Hamiltonian (1) with $\hat{H}\left(z\right)\psi\left(z+d\right)=\left(E+Fd\right)\psi\left(z+d\right)$. However the coupling between two pseudo-spin states will lead to two interpenetrating WSL which positioned symmetrically around [math] KartashovPRL2016 , with an intra-ladder separation $s$, as shown in Fig. 2(a). The inter-ladder spacing within the two WSL is still $Fd$. By considering that, we can label the WSL eigenenergy with $\nu_{1\left(2\right)}$ and $E_{\nu_{1\left(2\right)}}=\nu_{1\left(2\right)}Fd\mp s/2$. The intra-ladder spacing $s$ is a composite function of $\gamma$, $\Omega$ and $\delta$. As shown in Fig. 2(b), $s$ is a periodic function of $\gamma$. When $\delta=0$, $s=0$ for integer values of $\gamma$, the two WSL overlaps. This can be seen from that Eqs. 6(a) and (b) are the same by replacing $\psi_{\uparrow}\left(q\right)\rightarrow\psi_{\downarrow}\left(q\right)$ at $\delta=0$ and integer $\gamma$, signaling identical dynamics for the two spin components. Interestingly in addition to that, at some specific values of $\gamma$ maked by asterisks in Fig. 2(b) $s=Fd$, also indicating overlaping WSL. A nonzero $\delta$ separates the two ladder even at $\gamma=0$.

The relation between the WSL spectrum and dynamics can be understood from the mean velocity. The velocity operator can be defined as $d\hat{z}/dt=i\left[\hat{H},\hat{z}\right]/\hbar$, using the Hamiltonian (2) and assume the atomic wavefunction $\left|\psi\left(t\right)\right\rangle=\sum_{j,\sigma}\psi_{j,\sigma}\left(t\right)\left|j,\sigma\right\rangle$, one can calculate the mean velocity as

[TABLE]

Then one can take advantage of Wannier-Stark eigenstates by considering that $\psi_{j,\sigma}\left(t\right)=\sum_{\nu}a_{\nu}W_{j,\sigma}^{\nu}\exp\left(-iE_{\nu}t/\hbar\right)$ with $a_{\nu}=\sum_{j,\sigma}W_{j,\sigma}^{\nu\ast}\psi_{j,\sigma}\left(0\right)$, and the mean velocity can be expressed as

[TABLE]

The particle mean position $z\left(t\right)=z_{\uparrow}\left(t\right)+z_{\downarrow}\left(t\right)$ can then be derived via integrating Eq. (11) over time, in which

[TABLE]

symbol the mean position of spin-$\sigma$ component. Eq. (12) predict that the oscillation frequencies are ruled by the energy difference between two Wannier-Stark levels with the amplitude of each frequency inversely propotional to the energy distance of those Wannier-Stark states and propotional to the overlap of their wavefunctions.

In the absence of SO coupling it is well-known that the Wannier-Stark eigenstate $W_{j}^{\nu}$ have the form of Bessel function of the first kind ($J_{\nu+j}\left(z\right)$) with $W_{j+1}^{\nu}=W_{j}^{\nu+1}$ HartmannNPJ2004 , then $\sum_{j}W_{j}^{\nu\ast}W_{j+1}^{\nu^{\prime}}=\sum_{j}W_{j}^{\nu\ast}W_{j}^{\nu^{\prime}+1}$ take the value $1$ for $\nu=\nu^{\prime}+1$ and [math] otherwise. It indicates that in the oscillation dynamics each rung of the WSL is only coupled to its neighboring rung with the Bloch frequency $\omega_{B}=\left(E_{\nu}-E_{\nu^{\prime}}\right)/\hbar=Fd/\hbar=2\pi/T_{B}$. One can notice that in the presence of SO coupling the coupled equations (8) indicate two WSL in which any rung of the ladder is coupled to all the rungs of the other ladder, which will substantially modify the Bloch oscillation dynamics. This will be discussed in detail in the subsequent section.

III bloch oscillation dynamics

The Bloch oscillation dynamics have been studied in KartashovPRL2016 for the case of $\delta=0$. The results predicted there can be well understood under adiabatical theory. When $F$ is weak enough not to induce interband transitions the adiabatic approximation can be applied, under which the atoms move adiabatically along the energy band with the quasimomentum $q\left(t\right)=q\left(0\right)+Ft/\hbar$. One can predict that the frequency of Bloch oscillation is propotional to $Fd$ with the amplitude propotional to the bandwidth. The properties of Bloch oscillation can then be captured via further looking into the energy band structure, which can be obtained through diagonalizing the Hamiltonian (5) without the force ($F=0$). This result in a two-band structure with $\varepsilon_{\pm}\left(q\right)=-J\cos qd\cos\pi\gamma\pm\sqrt{J^{2}\sin^{2}qd\sin^{2}\pi\gamma-\hbar\delta J\sin qd\sin\pi\gamma+\hbar^{2}\delta^{2}/4+\hbar^{2}\Omega^{2}/4}$. Two major results are predicted in KartashovPRL2016 : (i) In analogue to increasing the potential depth $U_{0}$ of the optical lattice, SO interaction can take the same effect of band flattening flat band . In this case the bloch oscillation amplitude will be suppressed and thus make it difficult to measure. An example for this is given at $\gamma=0.5$ with the energy band shown in Fig. 3(a). (ii) Since that in the adiabatic approximation the mean velocity of the atom $v\left(q\right)=d\varepsilon\left(q\right)/\hbar dq$, the change in the band structure indicate that the atomic dynamics will subject to strong modification. As an example, for the band profile at $\gamma=0.8$ shown in Fig. 3(b), the initial atomic moving direction will be reversed.

These phenomena can also be explained using the theory of WSL. By considering that the eigenstate of the system consists of two interpenetrating WSL, one can group their contribution to the dynamics into two terms. Similar to the case without SO coupling, start from Eqs. (8) and (9) one can prove that within each ladder $W_{j+1,\sigma}^{\nu_{i}}=W_{j,\sigma}^{\nu_{i}+1}$ $\left(i=1,2\text{ label the two ladders}\right)$ still hold true, then according to Eq. (12) one can conclude that in the presence of SO interaction the Bloch oscillation dynamics in general are still dominated by intra-ladder coupling between neighboring rungs within each ladder, indicating the oscillation frequency $T_{B}$. At $\delta=0$, due to the symmetry between spin-$\uparrow$ and $\downarrow$ components, we have $\sum_{j}\left|W_{j,\uparrow}^{\nu}\right|^{2}=\sum_{j}\left|W_{j,\downarrow}^{\nu}\right|^{2}=1/2$, then according to Eqs. (11) and (12) one can predict that $z\left(t\right)=0$ at $\gamma=0.5$ and $dz/dt<0$ at $\gamma=0.8$ for initial small $t$, indicating that Bloch oscillation dynamics are substantially modified by SO interaction.

We assume that initially the atomic wavefunction

[TABLE]

to be a spin-polarized Gaussian wave-packet with width $a$, where $j_{0}$ is the center of the wave-packet while $q_{0}$ denotes the initial quasimomentum. In our calculations the parameters are chosen as $j_{0}=0$ and $q_{0}=0$. The dynamics are simulated using the method of eigenstate expansion and the results are demonstrated in Figs. 3(c)-(f), from which one can see that the results of numerical simulation are consistent with the above theoretical analysis.

Besides intraladder coupling, interladder coupling also contribute to the oscillation dynamics. We calculate the value of $\sum_{j}W_{j,\sigma}^{\nu_{1}\ast}W_{j+1,\sigma}^{\nu_{2}}$ and found out that for relatively large $\left|\nu_{1}-\nu_{2}\right|$ (approaching $100$) it really matters. This can be traced to the symmetry within WSL. Eq. (8) indicate that if $\left(A_{m},B_{m}\right)$ are eigensolutions with eigenvalue $E_{\nu}$, then $\left(-B_{-m}^{\ast},A_{-m}^{\ast}\right)$ are eigensolutions with eigenvalue $-E_{\nu}$. Due to the large energy difference of interladder coupling, it will superimpose small amplitude high frequency oscillation on the dynamics dominated by intraladder coupling.

An interesting case is that at $\gamma=0.5$, since the intraladder coupling are canceled out, then the dynamics deviating from $z=0$ is the result of interladder coupling, which is shown in Fig. 3(g). One can observe small amplitude high frequency oscillations, which become prominent around $t=nT_{B}/2$. Similar behavior can also be observed for $\gamma=0.8$ in Fig. 3(h), in which the small oscillations are superimposed on the traditional Bloch oscillation.

The Klein four-group KartashovPRL2016 or $\mathcal{CPT}$ symmetry symmetry is conserved by the Hamiltonian $\hat{H}_{SO}+U_{0}\sin^{2}\left(k_{l}z\right)$ at $\delta=0$, then in the corresponding energy band the eigenfunctions are symmetric for spin-$\uparrow$ and $\downarrow$ ($\psi_{\uparrow}\left(q\right)=\psi_{\downarrow}\left(-q\right)$) at the centre and edge of Brillouin zone, which can also be seen from Eqs. (6). Then within adiabatical theory one can predict that $\left\langle\hat{\sigma}_{z}\right\rangle=0$ when the atoms pass through the centre and edge of Brillouin zone. However this symmetry is broken at finite $\delta$. At finite $\delta$ the upper energy band and the lower one are shifted to opposite directions with respect to $q=0$, as shown in Fig. 4(a). Physically this band asymmetry can be captured through Bloch oscillation via exerting force in opposite directions. The numerical results are shown in Figs. 4(b) and (c), in which a force $F$ are considered to be exerted along the $+z$ and $-z$ direction, respectively. At $\delta=0$ one would expect that these two dynamics are identical, here the different dynamics signal the energy band asymmetry. Since the atomic initial state can be viewed as the superposition of the upper and lower eigenstate of the two bands, then in adiabatic limit they will subject to different dispersion under the action of the force. This cannot take place at $\delta=0$ where the energy band are always symmetric and the two bands possess almost identical dispersion. The combined effect will lead to different oscillation dynamics for the two spin components as we illustrated in Fig. 4(e), the dynamics become out-of-phase for the two spin components. One can also notice that in Fig. 4(d) the high frequency oscillations for the two components are out-of-phase, this is because $W_{j,\uparrow}^{\nu\ast}W_{j+1,\uparrow}^{\nu^{\prime}}=-W_{j,\downarrow}^{\nu\ast}W_{j+1,\downarrow}^{\nu^{\prime}}$ for interladder couplings. In the meanwhile, $\left\langle\hat{\sigma}_{z}\right\rangle$ deviate from [math] when the wavepacket passes through the centre and edge of the Brillouin zone, as shown in Figs. 4(f) and (g).

In the case without SO coupling, one can introduce a linearly time-dependent frequency difference $\Delta\nu\left(t\right)=-Ft/md$ between the two lattice beams castin1996 , the lattice potential becomes $U_{0}\sin^{2}\left[k_{l}z-\pi\int_{0}^{t}d\tau\Delta\nu\left(\tau\right)\right]$ and in an accelerated frame it is equivalent to exerting a constant inertial force $F$ on the atoms trapped in a stationary lattice. However this equivalence cannot be established in the presence of SO coupling. This is due to that the SO Hamiltonian $\hat{H}_{SO}$ breaks Galilean invariance as the physical momentum $p_{z}-\hat{A}$ does not commute with $\hat{H}_{SO}$. In this case going into a moving inertial frame will result in an additional time-dependent term $-\alpha Ft\hat{\sigma}_{z}$ in Hamiltonian (1), which play the role as a time-dependent effective detuning.

We calculate the oscillation dynamics in the stationary frame (lab frame) with the exerting force $F$ and that in the accelerated frame within which the atoms are subject to an effective force $F$ as well as an effective time-dependent detuning $-\alpha Ft\hat{\sigma}_{z}$, the results are shown in Fig. 5. The dynamics in the lab frame are simulated with eigenstate expansion while that in the accelerated frame are calculated by means of the Fourth-order Runge-Kutta method. Both the initial state are given by Eq. (13). In the numerical simulation we consider the recoil energy $E_{r}=\hbar^{2}k_{l}^{2}/2m=8.55J$ for a typical experimental value of $U_{0}=4E_{r}$. As one can expect, in the lab frame the oscillation dynamics for spin-$\uparrow$ and $\downarrow$ components are in phase, as shown in Figs. 5(c) and (e). However the dynamics shown in Figs. 5(d) and (f) indicate that they are out-of-phase (phase separated in the time domain) in the accelerated frame. This interesting dynamics can be readily captured in experiment and serve as a clear proof of broken Galilean invariance, which is also the mechanism underlying other unusual behaviors such as the deviation of dipole oscillation frequency in a harmonically trapped system chenPRL2012 ; puPRA2012 , the ambiguity in defining Landau critical velocity in SO coupled condensates wuEPL2012 , finite-momentum dimer bound state in a SO coupled Fermi gas dongPRA2013 and asymmetric expansion of SO coupled atomic Bose gas engelsPRL2017 . The effect of broken Galilean invariance can be signified via introducing a frequency difference between the two laser beams forming the optical lattice EngelsPRL2015 .

IV spin current generation

An interesting question is how to create a spin current with SO coupling SpinCurrentScience . Spin current have been experimentally generated in a SO-coupled BEC via spin Hall effect spielmanNature2013 and quenching SpinCurrentQuench . In theory, Larson et al. studied bloch oscillations of atomic BEC in a tilted two-dimensional (2D) optical lattice LarsonPRA2010 , in which the atoms are subject to a 2D SO interaction $\hat{H}_{SO}\propto\hat{p}_{x}\hat{\sigma}_{x}+\hat{p}_{y}\hat{\sigma}_{y}$ and in turn give rise to a spin-dependent effective force propotional to $\hat{\sigma}_{z}$. As a result an oscillating transverse spin current can be generated. For the present 1D system we have

[TABLE]

indicating an SO aroused effective force along $\mathbf{e}_{z}$-direction and propotional to $\hat{\sigma}_{y}$.

Here we would like to explore the possibility of generating spin current in the present 1D system with this effective force. As suggested by Shi et al. spin current , the spin current operator along the $z$-direction can be defined as

[TABLE]

Follow the very similar procedure as deducing Eqs. (10) and (11), make use of the WSL eigenstate, the mean-value of $\sigma_{z}$-component of spin current can be calculated as

[TABLE]

which predicts that in addition to the coupling between different rungs, the last term in Eq. (16) indicate that the coupling between spin-$\uparrow$ and $\downarrow$ components also contribute to the spin current, resulting from that the effective force $\hat{F}_{z}$ is propotional to $\hat{\sigma}_{y}$.

In order to illustrate the contribution of this term, one can choose $\gamma=1$ at which the major intraladder contribution from first two terms in Eq. (16) canceled out at $\delta=0$ due to the symmetry. Physically it is equivalent to that the two spin components are performing identical Bloch oscillation and in the meanwhile subject to on-site Raman coupling, as one can see from the Hamiltonian (2). In the case $\left\langle\hat{\sigma}_{z}\right\rangle=0$ for the Bloch eigenstate without the force. We then numerically calculate $J_{S}^{z}\left(t\right)$ and the results are shown in Fig. 6. One can expect that in the absence of Raman coupling no spin current can be generated since that spin-$\uparrow$ and spin-$\downarrow$ components both move adiabatically along the energy band and exhibit typical properties of Bloch oscillation, as can be seen from Fig. 6(a) and (b). The small amplitude high frequency oscillation is aroused by interladder coupling as we discussed in Sec. III. The time evolution of spin current $J_{S}^{z}$ exhibit the behavior of collapse and revival shown in Fig. 6(c), reminiscent of the Jaynes-Cummings model in quantum optics JC experiment . This collapse and revival behavior can be understood as a result of the complex interplay between the external force $F$ and the intrinsic force $F_{SO}=\hbar k_{R}\Omega\hat{\sigma}_{y}/2$ aroused by SO interaction. One can also understand this collapse and revival behavior the same as Zitterbewegung LarsonPRA2010 . Zitterbewegung results from coherent coupling between eigenstates of Dirac cone with different helicity Zitterbewegung and have been successfully observed in experiment with cold atoms leblancNJP2013 ; quPRA2013R , while here the trembling oscillation is aroused by spin swapping.

We also examined the case with finite Zeeman detuning. As one can expect, although the spin-$\uparrow$ and spin-$\downarrow$ components are performing different oscillation, it will be immersed in the dynamics aroused by Raman coupling and in general the spin current will exhibit the dynamics of collapse and revival. In order to achieve constant directional spin current, one can either adapt time-dependent SO coupling chienPRA2013 or unbiased external force spin ratchet .

V summary and outlook

Before concluding the paper, we need to note that in the presence of SO interaction one should be very careful while using the above lowest energy band truncation. As was pointed out by Zhou and Cui cuiPRB2015 , in this case tight-binding models have limitations in predicting the correct single-particle physics due to the missed high-band contributions. Physically the Raman lasers inducing SO interaction also inevitably couple atoms to high-lying bands which will significantly affect the single-particle physics panPRA2016 . Experimentally atomic BEC can also be prepared in excited bands of an optical lattice zhouxjPRA2013 . Ao and Rammer also pointed out that high-band contributions can substantially affect the Bloch oscillation dynamics pingao . Contributions from higher Bloch bands will be important and interesting in orbital optical lattices hemmerichreview . By considering that, we compare the results presented in this work with those obtained through numerical simulation of the corresponding Schrodinger equation and found good agreement in the case of large energy gap and small external force.

In summary, we have studied the Bloch oscillation dynamics of a SO-coupled cold atomic gas trapped inside a 1D optical lattice. The eigen-spectra of the system have been identified as two interpenetrating WSL. The Bloch oscillation dynamics in this system can be well-understood via analyzing the coupling between different rungs of the WSL. In the presence of finite Zeeman detuning, we show that the two spin components can display out-of-phase oscillation. This can also serve as an unambiguous proof of broken Galilean invariance aroused by SO coupling. In addition to that, we numerically explored the possibility of generating spin current in the present system. Since SO interaction have been implemented in BEC in a 1D optical lattice EngelsPRL2015 , our findings of the interesting dynamical phenomena should be within reach of present-day experiments. For BEC it will be interesting to study the impact of interparticle collisions on Bloch oscillation gaulPRA2011 and spin current generation, which can be investigated by the Gaussian variational approach smerzi ; chenPRA2014 . It will also be interesting to investigate Landau-Zener tunneling LeePRA2015 ; wuNPJ2003 . These will be left for further investigation.

Acknowledgements.

We thank Han Pu and Yongping Zhang for careful reading and many helpful comments on the manuscript. This work is supported by the National Natural Science Foundation of China (Grants No. 11374003, No. 11774093, No. 11574086, No. 91436211, No. 11654005), the National Key Research and Development Program of China (Grant No. 2016YFA0302001) the Shanghai Rising-Star Program (Grant No. 16QA1401600), and the Science and Technology Commission of Shanghai Municipality (Grants No. 16DZ2260200 and No. 16ZR1409800).