## Abstract

In this chapter, we discuss the magnetic solitons achieved in atomic spinor Bose-Einstein condensates (BECs) confined within optical lattice. Spinor BECs at each lattice site behave like spin magnets and can interact with each other through the static magnetic dipole-dipole interaction (MDDI), due to which the magnetic soliton may exist in blue-detuned optical lattice. By imposing an external laser field into the lattice or loading atoms in a red-detuned optical lattice, the light-induced dipole-dipole interaction (LDDI) can produce new magnetic solitons. The long-range couplings induced by the MDDI and ODDI play a dominant role in the spin dynamics in an optical lattice. Compared with spin chain in solid material, the nearest-neighbor approximation, next-nearest-neighbor approximation, and long-range case are discussed, respectively.

### Keywords

- spinor Bose-Einstein condensates
- spin wave
- optical lattice
- magnetic soliton

## 1. Introduction

Soliton can be classified into different species, such as matter-wave soliton, magnetic soliton, optical soliton, and so on [1, 2]. The magnetic solitons, which describe the localized magnetization, are very important excitations in the Heisenberg spin chain in solid system in condensed matter [3]. Because of the defect and impurity in solid material, it is difficult to perform experimental observation and manipulation in these systems. In addition, magnetic solitons originate from the Heisenberg-like short-range exchange interaction between electrons in solid state systems, so the theoretical models and treatment are limited to only the approximation of nearest-neighbor interaction, it is disadvantageous to the study of long-range interaction-induced dynamics.

On the other hand, the spinor ultracold atoms in an optical lattice offer a pure and well-controlled platform to study spin dynamics. If the lattice trapping atoms is deep enough, the spinor atoms undergo a ferromagnetic-like phase transition that leads to a macroscopic magnetization of the condensates array; then, the individual sites become independent with each other. Spinor BECs at each lattice site behave like spin magnets and can interact with each other through the static MDDI, which can cause the ferromagnetic phase transition and the spin wave excitation. It is atomic spin chain in optical lattice [4, 5, 6, 7]. By tuning the system parameters, magnetic soliton can be produced; however a key difference of the atomic spin chain from solid-state one is that we are facing a completely new spin system in which the long-range nonlinear interactions play a dominant role. To demonstrate the high controllability, an external laser can be imposed into the lattice and induce light-induced dipole-dipole interaction (LDDI) to produce new magnetic solitons. In fact, the LDDI can be induced by loading the atoms in a red-detuned optical lattice as used in [5]. However, the method used here has good feasibility, and some novel physical phenomena can be observed.

In order to make comparison with the Heisenberg spin chain, we used the nearest-neighbor approximation and the next-nearest-neighbor approximation in our discussion. In addition, the two approximations are effective and reasonable with some appropriate parameters chosen in experiments.

In this chapter, we are going to be seeing how magnetic soliton in atomic spin chain is, which will be theoretically described. We first discuss the formation of atomic spin chain in optical lattice in Section 2. Then we shift our focus to the condition for magnetic soliton in Section 3. Section 4 presents the conclusion and outlook.

## 2. Atomic spin chain

### 2.1 Atomic spin chain with magnetic dipole: dipole interaction

The spinor BECs can be trapped within one-, two-, and three-dimensional optical lattices with different experimental techniques; however, we only take one-dimensional case as an example. As depicted in Figure 1, the atomic gases with

where

It has been theoretically and experimentally demonstrated that the condensate trapped in optical lattice undergoes a superfluid-Mott-insulator transition as the depth of the lattice wells is increased. If the lattice potential is deep enough, the condensate is divided into a set of separated small condensates equally located at each lattice site, which forms the atomic chain. The condensate localized in each lattice site is approximately in its electronic ground state, while the two lattice laser beams are detuned far from atomic resonance frequency, and the laser detuning is defined as

where

In collision interaction potential in Eq. (3),

Under tight-binding condition and single-mode approximation, the spatial wave function

with

and

where we have defined collective spin operators

The parameter

where

Here, we only consider the ferromagnetic condensates where the spins of atoms at each lattice site align up along the direction of the applied magnetic field (the quantized

Equation (11) determines the existence and the propagation of spin waves. From this equation, we can obtain the existence and dynamical evolving of the nonlinear magnetic soliton. In Sec.3, we attempt to show the long-range and controllable characteristics of MDDI and find how this long-range nonlinear dipole-dipole interaction affects the generation of the magnetic soliton. If the static magnetic field

### 2.2 Atomic spin chain with light-induced dipole-dipole interaction

In fact, the site-to-site interaction can also be tuned by laser field, and the exchange interaction induced by the laser fields also plays an important role in atomic spin chain in optical lattice. We consider that an external modulational laser field is imposed into the system along the

Here, the coefficient

In above integral we define

and

where we define _{,} and

Function (11) now has the form:

The interaction coefficient

Figure 2 shows the two spin coupling coefficients varying with the lattice sites and the transverse width of the condensate. It is clear that the spin coupling coefficients are sensitive to the variation of the transverse width

## 3. Magnetic soliton in atomic spin chain

### 3.1 Magnetic soliton under the MDDI

To confirm that the magnetic soliton can be formed in this system, we first consider the case that other interactions between spins are absent in this section. Namely if the solitons indeed exist, they can be generated by the MDDI alone

here

where

#### 3.1.1 Nearest-neighbor approximation with the MDDI

Here we firstly consider only the nearest-neighbor (NN) interactions. In other words, we assume that the on-site spin couples only to the spins at its two neighboring sites, and the longer range couplings are artificially cut off. In above equation, for small retortion from the ground state, we have:

Using the slow varying envelope approximation or long-wavelength limit, we have:

where

where

Since the spin coupling parameter

#### 3.1.2 Next-nearest-neighbor approximation with the MDDI

Even though the long-range effect of the dipole-dipole interaction is obvious, in this section, we consider a case slightly beyond the NN approximation, i.e., the next-nearest-neighbor (NNN) approximation, which is used in some solid-state systems. Under this approximation, each on-site spin is coupled to the four near-neighbor spins, two on its right side and the other two on its left side, respectively. In this case, we set:

Additionally, the second-order spatial derivative is introduced:

Then, we can easily get:

where

Because the LDDI is absent in this case, we have:

Then, Eq. (31) can be rewritten as [10]:

where

we have

#### 3.1.3 Long-range case with the MDDI

To study the effects of the long-range spin coupling on the nonlinear spin wave dynamics, we need consider the spin coupling at all sites through the whole optical lattice, though the MDDI decreases rapidly in some cases. We firstly use Eq. (22) to rewrite the nonlinear motion equation of

Then we set:

Considering the discreteness of the optical lattice, we can treat the symbolic terms as below:

After changing the sum to integration, Eq. (38) becomes:

Denoting

Taking above series into Eq. (44), the integration variable is changed, i.e.,

where we have used

The condition for Eq. (46) to have envelope soliton solution is

From the definitions of

This result is consistent with the instability condition of modulational instability of nonlinear coherent spin wave modes near the Brillouin zone boundary under the long-wavelength modulation [4]. There, the modulational instability criteria take the corresponding discrete form:

where

The function

### 3.2 Magnetic soliton under the LDDI

When we choose red-detuned optical lattice or an external modulational laser field is imposed into the system, both the static MDDI and the laser-induced LDDI begin to take effect on the dynamics of the soliton excitation, and the criterion for the existence of solitons could be changed. Here we also consider three interaction distances as doing in Section 3.1.

#### 3.2.1 Nearest-neighbor approximation with the LDDI

In this case, we can obtain the continuum limit NLSE for

The existence condition of magnetic soliton now has the form

Obviously, magnetic soliton can be observed in this case, which is different from the case where only the MDDI plays role.

#### 3.2.2 Next-nearest-neighbor approximation with the LDDI

In this case, the NLSE for

We assume that the intensity of the external laser is strong enough which corresponds to strong LDDI; then the terms like

#### 3.2.3 Long-range case with the LDDI

In fact, once the external laser is imposed into the lattice, the LDDI dominates the system rapidly; the NLSE now takes the form [10]:

where the coefficients are defined as below:

The value of

Similarly, Eq. (59) is consistent with the corresponding discrete form of MI condition of NCSW modes near the Brillouin zone boundary under the long-wavelength modulation [4]:

where

We plot the existence regions of magnetic solitons in atomic spin chain dominated by the LDDI in Figure 4. The longitudinal coordinates stand for the intensity of the modulated laser. It is clear that the magnetic soliton can be achieved by tuning the transverse width and the modulated laser, which demonstrates the high controllability of optical lattice.

## 4. Conclusion

In this chapter, we have shown the existence conditions of magnetic solitons that can occur in optical lattice. Compared to the more conventional solid-state magnetic materials, we discuss how the NN, the NNN, and the long-range interactions (the MDDI and LDDI) dominate the system. We also show that they can be tuned by the laser parameters and the shape of the condensate including the laser detuning, the laser intensity, and the transverse width of the condensate.

Besides studying the magnetic solitons, the atomic spin chain in optical lattice provides us with a useful tool to study the fundamental static and dynamic aspects of magnetism and lattice dynamics [12, 13, 14]. In experimental applications, the atomic spin chain has become potential candidates for multi-bit quantum computation due to their long coherence and controllability [13, 15, 16, 17]. The theoretical study of magnetic solitons in optical lattices will play a guiding role when the optical lattice is used in cold atomic physics, condensed matter physics, and quantum information.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant No. 11604086. Thanks to Ying-Ying Zhang for her contributions to physics discussion and English writing of the chapter.