Theory and Methodology

The dynamics of Bose-Einstein condensates (BECs) can be approximately described in the mean-field approximation by the Gross-Pitaevskii (G-P), or nonlinear Schrödinger, equation for the macroscopic condensate wave function. BECs in optical lattices have many special phenomena, which are created by the interference between the nonlinearity due to the atomic interactions and the exotic dispersion relations from the periodic potential produced by interference of laser beams. The chemical potential of atoms trapped in a period potential exhibits band structures containing band gaps. No linear eigenmodes exist within a band gap. Louis et al.[10] demonstrated numerically that there are nonlinear localized modes of the condensate existing in the band gap, which are matter-wave gap solitons. On the other hand, dispersion relations of photons in nonlinear photonic crystal also exhibit the band-gap structure. Nonlinear localized photon-modes, which are optical gap solitons [13], are shown inside the photonic gap [14, 15].

To understand the physical properties of the gap soliton, the one-band effective-mass theory of a nonlinear Schrödinger equation is derived and solved analytically to obtain gap soliton solutions. We start with G-P equation to describe the dynamics of BECs in optical lattices. The linear band structure of BECs in optical lattices is solved by the coupled mode theory. The effective mass of BECs is defined. The numerical simulation method is also given.

2-1 Bose-Einstein condensates in optical lattices

The dynamics of a Bose-Einstein condensate in an optical lattice can be described by the Gross-Pitaevskii (GP) [16] or the nonlinear Schrödinger equation for

the macroscopic condensate wave function Ψ( , )r t ,

is a one-dimensional periodic potential produced by the interference of laser beams, where is the lattice constant and is the potential depth.

⎦ is an optical trapping potential with frequencies ω_x and ω_⊥. Due to the high confinement in the y-z plane, the trap is elongated along the x direction (i.e.ω_x ω_⊥). Therefore, we express the wave function as Ψ( ; )r t =a y z( , ) ( , )ψ x t , where is described by a solution of the two-dimensional radially symmetric quantum harmonic-oscillator

problem, . By applying the

transformation one-dimensional G-P equation is derived as

2 2

It is more convenient to use dimensionless quantities by normalizing

, , , and

/ _o

T t T= X =x L/( / 2) ψ ϕ= / L^{1/ 2}₁ V_o =E_o/E , and choose T_o =mL²/ 4 ,

1 _s 2/ 2

L =ω_⊥ a mL , and . After these transformations, an effective one-dimensional G-P equation in dimensionless variables is derived as

4 2/

Schrödinger equation with a periodic potential. If σ  is positive (negative), atoms are resulting in repulsive (attractive) interaction. Eq.(2.3) possesses an integral of motion

N ^∞ ϕ2dX

=

∫

−∞ ^{, (2.4)}

which accounts for the conservation of the number N of atoms in the condensate.

2-1-1 Band Structure

The E-k band structure of a BEC in optical lattices determines basic properties of the matter waves under linear Schrödinger equation. To find the band structure we assume that the linear part of Eq. (2.3) admits stationary solutions of the form ( , )ϕ X T =φ( ) exp(X −iET), thus obtaining the following eigenvalue problem,

Eq. (2.5) possesses periodic solutions, known as Floquet-Bloch (FB) modes. By using plane-wave methods, the periodic potential can be expanded as

The Bloch functions can be expressed as

( )

Substituting Eq.(2.6) into Eq.(2.5), we have

(

^{2E V− o−}

(

^{k m+ π}

)

²

)

^am+V₂^oam−1+V₂^oam+1= . (2.8) ⁰ The accuracy of the method depends on the number of plane waves considered in the expansion, as well as on the form and the depth of the potential. We assume the potential is relatively shallow, and the Bloch modes between the first and the second band can be accurately described by keeping only two terms of the expansion, i.e.m= −0, 1. Eq.(2.6) is rewritten as

φ( , )X k =a e_o ^ikX +a e₋₁ ^ikXe^{−i Xπ} . (2.9) Considering the dominant terms in Eq.(2.8), we obtain the following coupled equations

We substitute Eq.(2.9) into Eq.(2.7) to satisfy the orthonormality of the Bloch functions, and then the coefficients a_oand a₋₁ are given by Substituting these coefficients into the Bloch functions in Eq.(2.9), we obtain the lower band Bloch waves at wave vector k

( )

And, the upper band Bloch waves at wave vector k

( )

We have a nontrivial solution in Eq.(2.10), when the determinant of coefficients vanishes. E-k band structure of BECs in one-dimensional optical lattices is then derived as

2 2 2 the extreme point of the first Brillouin zone i.e.

o 2.5

At the energy,E_a, the Bloch wave is an even function given by ( , ) 1 2cos The condensate wave function in the lowest energy of the first band at k=0 is

given by

The first band gap is explicitly ∆ =E E_c since is the lowest energy in the E-K spectrum.

Ec

Up to the present, we have derived the energy band structure and obtained three Bloch functions, named φ_a( )X , ( )φ_b X and ( )φ_c X at chosen points in the -E K structure under the linear regime of the nonlinear Schrödinger equation with periodic potentials.

In the following, we will introduce the effective-mass theory to the nonlinear Schrödinger equation with periodic potentials. The envelop wave function of BECs in a periodic potential can be described by an effective nonlinear Schrödinger equation, where the periodicity is absorbed into the effective mass.

2-2 The Effective Mass Theory

Effective mass theory is a well-known approximation in solid state physics for studying dynamics of an electron in semiconductor, described by nonlinear Schrödinger equation. The dynamics of a Bose-Einstein condensate loaded into an optical lattice are described by the Gross-Pitaevskii (G-P) equation. The two systems, electrons in semiconductor and BECs in optical lattices, are analogous.

Therefore, we can introduce the effective mass theory to study BECs in optical lattices.

2-2-1 Brief review of effective mass theory

We give a brief review of effective mass theory in solid state system [17].

Starting with a general form of linear Schrödinger equation

( ) ( ) ( )

^,

where is a time independent Hamiltonian that, in most instance, contains a periodic potential. represents an external influence; it is not periodic and may be time dependent. The eigenfunctions of the characteristic equation given by

Ho is the band index and is the lattice vector. The Bloch functions are orthonormal according to

Note that in this equation and everywhere else, the integral on r involves complete space. They also form a complete set that

( ) (

^{k r k r, , '}

)

³

(

^{r r}

n l

n

φ^∗ φ d k⁼δ ^{− '}

)

∑∫

^{. (2.25)}

It is sometimes desirable to expand the wave function simply related to the Bloch functions at a single k-point of a band structure. Therefore, we introduce a basis of the so-called Kohn-Luttinger functionsχn

( )

^{k r,} which are defined in terms of the Bloch functions at some conveniently chosen point in the first Brillouin zone ko as

( )

^{k r, i(k ko)r}

(

^{k r}_o^,

)

^{ik r}

(

^{k r}

n e n e

χ = ^{− i} φ = ^{i u}_{n o}^,

)

k

, (2.26) where is the cell periodic function. The Kohn-Luttinger functions obey the same

orthonormality and completeness relations as the Bloch functions. Thus, an arbitrary wave function can be expanded as

un are determined as the follows

Ho

( ) ( )

We can break up the integral over whole space into an integral over unit cells since is periodic if is periodic.

(

^,

with Ω being the volume of a unit cell. Thus, Eq.(2.28) yields

(

^o

) {

^{nl l}

(

^{2 2}

)(

^{2 2}

) ^{( ) ( )}

^nl

⁽ }

Consequently, the expansion coefficients An

( )

k^,t of Eq.(2.27) satisfy

( )( ) ^{( )}

This is the Schrödinger equation under the effective mass theory. For a stationary state of energy E, i.e.,An

( )

k^,t →An

( )

k , Eq.(2.32) can be represented as

After some algebra, the last term in Eq.(2.33) is represented as

The only terms of Eq.(2.33), which represent coupling between bands, are those involving the momentum matrix elements p_nl.

We will consider the simpler non-degenerate case in the following discussion.

By introducing a unitary transformation to the expansion coefficients, where is Hermitian and is in some sense “small”, Eq.(2.33) is then replaced by

C e A= ⁻iS

The first three terms in Eq.(2.35) are equivalent to the expression given in the Eq.(2.36) for the energy as a function of wave vector to second order in δk, so that we replace those terms by En

( )

^k

(

En

( )

^k −E C

)

n

^{( )}

^k +

∫

d q U³

⁽

^k−^q

^{) ( )}

Cn ^q =^{0 (2.37)} This equation resembles the Schrödinger equation in momentum space for one

particle in the potential U. There is a significant difference in that the effective mass tensor is involved, rather than the free electron mass . All of the effects of the periodic potential are incorporated in the effective mass.

m^∗ m_o

It is desirable to transform Eq.(2.37) to a differential equation in ordinary space. We

define a function Fn

( )

^r by

( )

^{r i k r}

( )

³

n n k

F =

∫

e^{δ i} C d k ^(2.38)

The integration in Eq.(2.38) includes only the Brillouin zone. Next, multiply Eq.(2.37) by ^exp

(

^{iδk ri}

)

and integrate over the Brillouin zone. Let us consider the

where α_ij is the reciprocal effective mass tensor, given implicitly in Eq.(2.35) for

(

^{m mn∗}

) ( )(

_αβ ^{= m 2 ∂2En ∂ ∂k kα β}

)

. By substituting Eq.(2.35) into Eq.(2.38) last term of Eq.(2.37) is transformed as

( ) ( )

If r is large compared to a lattice spacing, the errors from approximation are not important for slowly varying impurity potentials. Therefore, we are able to simplify Eq.(2.42) as

( )

This is the transformed effective mass equation and does not contain any terms coupling different bands.

Now, we introduce Wannier functions an

(

^r−^R_µ

)

, characterized by a band index and a lattice site vector R_µ, to express the wave function in terms of orthogonal localized functions.

After some algebra, the Bloch functions can be expressed as

( )

^,

( )

₃

(

first-order correction. To this order

An C_n

If we are interested in the wave function associated with a particular impurity level under the conduction band, for instance n=c, we have finally

( )

r _c

(

k ro^,

) ( )

F

ϕ =φ r (2.49)

The impurity function is then an oscillatory band wave function modulated by a slowly varying, but exponentially decreasing, envelope function ^F

( )

^{r .}

2-2-2 Effective mass theory in BECs

We, in turn, adopt the effective mass equation [Eq.(2.44)] to consider a Bose-Einstein condensate in an optical lattice, described by G-P equation. For the envelope functions vary not faster than on a scale of three lattice constants, the nonlinear terms should be added to Eq.(2.44) in a straightforward way

( ) ( ) ( ) ( )

²

( )

where σ is the strength of the nonlinear interatomic interaction. x and_i x_jare cartesian components of r . Finally, we have the transformed effective mass equation contains nonlinear interaction without coupling bands.

Now, we proceed to consider the nonlinear Schrödinger equation without external influence as in Eq.(2.3), the effective mass equation in Eq.(2.51) is rewritten as

chemical potential at the extreme point of the band. This is the effective-mass equation near the extreme points of the band, , by making a unitary transformation to diagonalize the Hermitian matrix of Eq.(2.3). In deriving Eq.(2.52), we have assumed that the band mixing is small. If the band mixing is not negligible, a two-band model is necessary to describe BECs in optical lattices. We believe that is usually ignored in the applications of the effective-mass theory [12], cannot be ignored and is important to describe BECs in optical lattices, qualitatively and quantitatively. The effective nonlinear interaction strength

ko in solid-state physics. With second-order differential of Eq.(2.14), effective mass

at is given by

The effective mass can be positive or negative depending on the band. With proper atomic interaction σ_n and the detuning chemical potential δ_n, solitons may emerge from band edge E k_n( )_o .

Eq.(2.52) is a time-independent nonlinear Schrödinger equation governing the

condensate envelope function F Xn

( )

, which is related to the condensate wave function by

( )

_n

(

_o^,

) ( )

_n

n

X k X F X

ϕ =

∑

φ from Eq.(2.49). If we are interested in the condensate wave function associated with a particular localized state developed from a certain band, from Eq.(2.49), we have

( ) (

^,

) ( )

n X n k X F Xo n

ϕ =φ (2.57)

where is band index marked as , , or in this thesis. n a b c

To give a brief summary, we have derived the condensate wave function of a time-independent nonlinear Schrödinger equation with a periodic potential to be an oscillatory band wave function modulated by a slowly varying envelope function

n

( )

F X . In next chapter, we introduce the bright and dark solitons arise from the BECs in optical lattices and discuss their properties.

2-3 Numerical Method

For obtaining the numerical results of one-dimensional G-P equation in Eq.(2.3) at steady state, we introduce numerical differentiation for approximating a second-order derivative, Newton-Raphson method for finding the solutions of nonlinear equations, and numerical integration for calculating the number of atoms in a BEC [Eq.(2.4)].

2-3-1 Numerical Differentiation

We use central-difference formula of order ^{O h}

( )

² as a second-order derivative

( )

Adding the two Eqs.(2.58)(2.59), we have

( ) ( ) ( ) ( )

^{2 (4)}

( )

^{4 (6)}

( )

If the series in Eq.(2.60) is truncated at the fourth derivative, there exists a valuecthat lies in

[

^{x h x h− , +}

]

^{so that}

This gives us the desired formula for approximating ^{f x′′}

( )

'' 1

2-3-2 Newton-Raphson theorem

[18]

We apply Newton-Raphson (or simply Newton’s) method for finding the solutions of nonlinear equations. Assume that ^{f ∈C a b2}

[ ]

^, and there exists a

will converge top for any initial approximationpo∈

[

p−δ^,p+δ

]

. The function

is called the Newton-Raphson iteration function. Since ^{f p}

( )

⁼⁰it is easy to see that . Thus, the Newton-Raphson iteration for finding the root of the equation is accomplished by finding a fixed point of the equation . Since Newton’s method rely on the continuity of

( )

⁰ enough compared to the sum of the first two terms. Hence, it can be neglected and we can use the approximation

( ) ( )( )

0≈ f p_o + f p′ _o p p− _o (2.67)

Solving forp in Eq.(2.67), we get^{p≈ p}_o^{− f p}

( ) ( )

_o ^{f p′} _o . This is used to define the next approximationp₁ to the root

( ) ( )

When p_k−1 is used in place of p_o in Eq.(2.68), the general rule Eq.(2.63) is established. For most applications this is all that needs to be understood. However, to fully comprehend what is happing we need to consider the fix-point iteration function. The key is in the analysis of ^{g x′}

( )

2-3-3 Numerical Integration

[19]

We adopt a three-point formula exact up to polynomials of degree two. This is true; moreover, by a cancellation of coefficients due to left-right symmetry of the formula, the three-point formula is exact for polynomials up to and including degree three, i.e. ^{f x}

( )

^{= x3}. We have the Simpson’s rule fourth derivative of the function f evaluated at an unknown place in the interval

and therefore the error is . Note that the formula gives the integral over an interval of size , so the coefficients add up to two.

h5

2h

在文檔中 K．P 理論於玻色-愛因斯坦凝聚態在光晶格之應用 (頁 14-32)