Atomic spin chain

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A. Appendices

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 opti- cal lattices with different experimental techniques; however, we only take one- dimensional case as an example. As depicted inFigure 1, the atomic gases with F ¼1 are trapped in a one-dimensional optical lattice, which is formed by two π-polarized laser beams counter-propagating along they-axis. The light-induced lattice potential takes the form:

V_Lð Þ ¼r U_Lexp r⊥2=W_L²

cos²ðk_LyÞ (1) wherer⊥ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x²þz²

p , k_L ¼2π=λ_L is the wave number,W_L the width of the lattice beams, and the potential depthU_L ¼ℏj jΩ²=6ΔwithΩbeing the Rabi frequency andΔbeing laser detuning.

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

Figure 1.

Schematic diagram of spinor BECs confined in a one-dimensional optical lattice. A strong static magnetic field B is applied to polarize the ground-state spin orientations of the atomic chain along the quantizedz-axis.

is defined asΔ¼ω_L ω_a,ω_L being laser frequency andω_a atomic resonant frequency. We classify the optical lattice into two categories: red-detuned optical lattice (Δ,0) and blue-detuned oneΔ.0. For the former, the condensed atoms are trapped at the standing wave nodes where the laser intensity is approximately zero; the light-induced interactions are neglected in this case. In order to realize ferromagnetic phase transition, a strong static magnetic fieldBis applied to polarize the ground-state spin orientations of the atomic chain along the quantizedz-axis.

The Hamiltonian including the long-range MDDI between different lattice sites takes the following form [7–9]:

HM ¼∑

m

ð

dr ψ^^†_m

ℏ²

2M∇²þVL

^ ψ_m

þ ∑

m,m⁰,n,n⁰

ð dr

ð

dr⁰ψ^^†_mψ^^†_m0ψ^_n0ψ^_nV^col_mm0n⁰nðr;r⁰Þ

þ1 2 ð

dr ð

dr⁰ψ^^†_mψ^^†_m0ψ^_n0ψ^_nV₁ðr;r⁰ÞF^mnF^m⁰n⁰

1 2

ð dr

ð

dr⁰ψ^^†_mψ^^†_m0ψ^_n0ψ^_nV₂ðr;r⁰Þ F^mnðr r⁰ÞF^m⁰n⁰ ðr r⁰Þ

þ ∑

m,n

ð

drψ^^†_mð μBÞ_mnψ^_n

(2)

whereψ^_mðr;tÞðm ¼ 1;0Þare the hyperfine ground-state atomic field operators.

The total angular momentum operatorF^for the hyperfine spin of an atom has compo- nents represented by 33 matrices in the f ¼1;m_f ¼m

subspace. The final term in Eq. (2) represents the effect of polarizing magnetic field, andμis magnetic dipole moment. The interaction potential includes the two-body ground-state collisions and the magnetic dipole-dipole interactions, which can be described by the potentials:

V^col_mm0n⁰nðr;r⁰Þ ¼ λ_sδ_m0n⁰δ_mnþλ_aF^mnF^m⁰n⁰

δðr r⁰Þ, (3) V₁ðr;r⁰Þ ¼μ₀γ_B²

4π 1 r r⁰

j j³, (4)

V₂ðr;r⁰Þ ¼3μ₀γ_B² 4π

1 r r⁰

j j⁵: (5)

In collision interaction potential in Eq. (3),λ_s andλ_aare related to thes-wave scattering length for the spin-dependent interatomic collisions. In the magnetic dipole-dipole interaction potential Eqs. (4) and (5),μ₀ is the vacuum permeability;

the parameterγ_B¼ μ_Bg_F is the gyromagnetic ratio withμ_B being the Bohr mag- neton andg_Fthe Landég factor.

Under tight-binding condition and single-mode approximation, the spatial wave functionϕ_mð Þr of them-th condensate is then determined by the Gross-Pitaevskii (GP) equation [5]:

ℏ²

2M∇²þV_mð Þ þr λ_sðN_m 1Þjϕ_mð Þr j²

ϕ_mð Þ ¼r μ_mϕ_mð Þr , (6) with

V_mð Þ ¼r U_Lexp r_L²=W²

cos²ðk_LyþmπÞ,0,y,λ_L=2, (7)

andNmis the number of condensed atoms at them-th site;μ_m is the chemical potential. In this case, the individual condensate confined in each lattice behaves as spin magnets in the presence of external magnetic fields; such spin magnets form a 1D coherent spin chain with an effective Hamiltonian [5, 6]:

H_m ¼∑

m

λ_a⁰^S²_m γ_BS^mB ∑

n6¼m

J^mag_mn S^mS^n

!

, (8)

where we have defined collective spin operatorsS^mwith componentsS^^f_m^x;y;zg and λ⁰_a¼1

2λ_a ð

drjϕ_mð Þr j⁴: (9)

The parameterJ^mag_mn describes the strength of the site-to-site spin coupling induced by the static MDDI. It takes the form:

J^mag_mn ¼ μ₀γ_B² 16πℏ²

ð dr

ð

dr⁰ R1

r⁰

j j⁵jϕ_mð Þr j²jϕ_nðr r⁰Þj², (10) whereμ₀ is the vacuum permeability,R₁ ¼j jr^{0 2} 3y⁰². It will be seen from this that the MDDI is determined by many parameters; however, we need to choose a parameter which is easy controllable in the experiment. The realistic interaction strengths of the MDDI and LDDI coupling coefficients have been calculated in ref.

[4], in which we find the dependency of the transverse width.

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 quantizedz-axis) in the ground state of the ferromagnetic spin chain. Because of the Bose-enhancement effect, the spins are very large in these systems. As a result, at low temperature the spins can be treated approximately as classical vectors, and we can replaceS^n with its average valueS_n[1]. From the Hamiltonian (8), we can derive the Heisenberg equation of motion for operatorS_n ¼S_n^xiS_n^yi² ¼ 1 (normalized byℏ):

idS_n^þ

dt ¼γ_BB_zS^þ_n þ ∑

j6¼n

2J^mag_nj S_j^zS_n^þ S_n^zS_j^þ

: (11)

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 Bis strong enough, we have:

S_n^z¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S² S_n^þS_n q

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S² S_n^þS_n^þ q

: (12)

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 they-axis. This external laser field is properly

chosen so that a new dipole-dipole interaction will be introduced into the system, i.e., the so-called LDDI. Its Hamiltonian takes the form [10]:

H_opt ¼∑

m

∑

n6¼m

J^opt_mnS^_mS^^þ_n þS_m^þS_n

: (13)

Here, the coefficientJ^opt_mndescribes the site-to-site spin coupling induced by the LDDI, which has the concrete form:

J^opt_mn¼ Q 144ℏk³_L

ð dr

ð

dr⁰f_cð Þr⁰ exp R2

W²_L

cosðk_LyÞcos½k_Lðy y⁰ފeþ1W rð Þ ⁰ e 1jϕ_mð Þr j²jϕ_nðr r⁰Þj²:

(14)

In above integral we defineQ ¼γj jΩ1 ²=Δ²₁to describe the magnitude of the LDDI; hereγ is the spontaneous emission rate of the atoms, andΩ1 is the Rabi frequency;Δ1is the optical detuning for the induced atomic transition. A cutoff functionf_cð Þ ¼r expð r=L_cÞis introduced to describe the effective interaction range of the LDDI,Lcbeing the coherence length.R2 ¼r⊥2þjr⊥ r⊥0j² with the transverse coordinater_⊥ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x²þz²

p . The unit vectors have the form:

e0 ¼ez, (15)

e1¼∓exiey

, (16)

andeαðα¼x;y;zÞwhich are the usual geocentric Cartesian frame. The tensor W rð Þdescribes the spatial profile of the LDDI and has the form:

W rð Þ ¼3

4 1 3 cos²θ sinξ

ξ² þ cosξ ξ³

sin²θ cosξ ξ

, (17)

where we defineξ¼k_Ljr r⁰j^,andθis the angle between the dipole moment and the relative coordinater r⁰. Being different from MDDI in Eq. (8), it shows that the LDDI in this system leads to the spin coupling only in the transverse direction, i.e., the direction perpendicular to the external magnetic field, as shown in Eq. (13). The total Hamiltonian of the system can be written as [10].

Figure 2.

The spin coupling coefficientsJ^z_mn¼J^mag_mn andJ_ijas a function of the lattice site index for different transverse widths of the condensate in the MDDI and LDDI dominating optical lattice [4].

H¼HmþHopt: (18) Function (11) now has the form:

idϕ_n

dt ¼γ_BB_zϕ_nþ ∑

j¼n1

2SJ^mag_nj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ϕ_j ² r

ϕ_nþ ∑

j¼n1

4SJ_nj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 j jϕ_n ² q

ϕ_j: (19) The interaction coefficientJ_ijsatisfiesJ_ij ¼J^opt_ij þ¹₂J^mag_ij .

Figure 2shows the two spin coupling coefficients varying with the lattice sites and the transverse width of the condensate. It is clear that the spin coupling coeffi- cients are sensitive to the variation of the transverse widthW of the condensate and exhibit the similar long-range behavior.

3. Magnetic soliton in atomic spin chain