Simulation Results and Discussion.

We have applied the effective mass theory to the Gross-Pitaevskii (G-P) equation, nonlinear Schrödinger equation with optical lattices, to obtained the effective mass equation [Eq.(2.52)]

( ) ( ) ( ) ( )

2 2

* 2

1

2 ⁿ _{n n n n n} 0

mn

F X F X F X F X

X δ σ

− ∂ − +

∂ =

The condensate wave function associated with a particular localized state developed from a certain band is shown in Eq.(2.57)

( ) (

^,

) ( )

n X n k X F Xo n

ϕ =φ

which is an oscillatory band wave function ^φ_n

(

^{k X}_o^,

)

modulated by a slowly varying, but exponentially decreasing, envelope function F Xn

( )

which provides a concrete picture to study the properties of BECs for certain energy in band structure.

If the energy is within the band gap, gap solitons occur. These solitons, characterized by localized wavepackets of the condensates, are also called bright solitons. On the other hand, for the energy is within the band, dark solitons occur and are characterized by localized dips on the condensate density background.

In the following, we apply the effective mass equation to obtain the analytic solutions of bright and dark solitons, respectively. Both numerical simulations and experimental results reported by Eiermann et al. [20] are applied to confirm that this envelope function approach indeed provides us with useful qualitative insight into the condensate dynamics.

3-1 Bright Solitons

For the energy is within band gap, bright soliton solutions can be described by the envelope functions which satisfy effective mass equation [Eq.(2.52)] and are given by

( )

^sech

( )

n n n

F X = A B X (3.1)

where B_n = 2δ_{n n}m^* and A_n =B_n m^*_{n n}σ . Bright solitons are developed from states of the upper (lower) band edge of the band gap, if atomic interactions are attractive (repulsive). The sign of detuning chemical potential, atomic interactions and effective mass determines whether bright solitons exist or not. For the energy is below the band edge δ_n < , the effective mass 0 m_n^∗ > , and attractive atomic 0 interactions 0σ_n < . For the energy is above the band edge δ_n > , the effective 0 mass , and repulsive atomic interactions m_n^∗<0 σ_n > . From Eq.(3.1), we know 0 that the gap soliton width is inversely proportional to B_n and depends upon the detuning chemical potential δ_n, which is the relative chemical potential to the band edges. The properties of a gap soliton are determined by δ_n, therefore, in order to understand BECs in optical lattices qualitatively and quantitatively, the chemical potential ^{E k}_n

( )

_o at band edge is significant. Near the band edge, where δ and _n Bn are small, the soliton is much extended (occupying many lattice sites) in the space and decay slowly to infinity. Deep inside the band gap, where δ and _n B_n are large, the soliton is much confined in the space. These analytic soliton properties are consistent with the numerical studies of BECs in optical lattices [8, 10].

We proceed to add oscillatory Bloch waves φn

(

k Xo^,

)

to envelope function

n

( )

F X to obtain the condensate wave functions. According to the states in the band from which solitons are developed, we classify gap solitons into categories that are Bragg-reflection type and internal-reflection type, respectively. Note that we kept the depth of the optical lattices V_o =2 5. in our numerical studies.

3-1-1 Bragg-Reflection Type

Solitons which arise from the energy E_a, the first band at k=π 2, or , the second band at

Eb

2

k =π , are classified in Bragg-Reflection Type. From Eq. (2.15) and Eq. (2.16), we have E_a =1.859 and E_b =3.109, respectively. For the solitons arising from E_a, we have the energy goes up to the second band gap which implies

a 0

δ > , the effective mass [Eq.(2.54)] is negative, and the formation of gap solitons is under repulsive atomic interactions

0.339 ma^∗ = −

σ >0. For convenience, we name gap solitons which arise from as gap solitons. By multiplying the Bloch wave of Eq.(3.1), the condensate wave function of gap solitons developed from is given by

We plot a series of the typical gap solitons which are developed from the first band edge for repulsive atomic interactions for different energy under . These plots are Fig. 2(a), Fig. 2(b) and Fig. 2(c) which correspond to the energy

, , and Bloch wave under soliton envelope. For comparison, we numerically solve time-independent G-P equation [Eq.(2.3)] and plot in Fig. 2(d), Fig. 2(e) and Fig. 2(f)

for the a-type gap solitons at the corresponding energies.

On the other hand, for the solitons arising from , we have the energy goes down to the second band gap which implies

Eb b 0

δ < , the effective mass

[Eq.(2.55)] is positive, and the formation of gap solitons is under attractive atomic interactions

* 0.202 mb =

σ <0. For convenience, we name gap solitons which arise from as gap solitons. By multiplying the Bloch wave

Eb wave function of gap solitons developed from E_b is given by

( )

^sech

( )

^{1 sin} We plot a series of the typical gap solitons which are developed from the

second band edge for attractive atomic interactions for different energy under . These plots are Fig. 3(a), Fig. 3(b) and Fig. 3(c) which correspond to the symmetric Bloch wave under soliton envelope. For comparison, we again numerically solve time-independent G-P equation [Eq.(2.3)] and plot in Fig. 3(d), Fig.

3(e) and Fig. 3(f) for the b-type gap solitons at the corresponding energies.

The results of and gap solitons reveal high agreement between the analytic and numerical solutions, especially, when detuning chemical potential

-type

a b-type

δ is small. The oscillations of gap solitons are due to the cosine and sine n

standing wave functions at the and as the Bragg reflection (BR) type solitons. They are the modulated standing Bloch waves, resulting from the interference of counter-propagating Bragg reflected waves, by the soliton envelope

Ea E_b

function.

To obtain the analytic particle number related to the detuning chemical potential

N

δa for the gap solitons, we substitute Eq.(3.2) into the conservation of number of atoms in Eq.(2.4)

-type

To obtain the analytic particle number N related to the detuning chemical potential

δb for the gap solitons, we substitute Eq.(3.3) into the conservation of number of atoms in Eq.(2.4)

-type solitons are plotted in Fig. 5. From Eq.(3.4) and Eq.(3.5), we found the atomic number is approximately proportional to

N a-type b-type

N δ and it is small close to the band

edges and becomes zero at band edges. These properties are not only consistent with the numerical simulations [10, 11] but also with our simulations as shown in Fig. 5.

Going deeper inside the band gap, the particle number becomes large and the nonlinear effect is strong (solitons being more localized).

N

3-1-2 Internal-Reflection Type

Solitons which arise from the energy E_c, the first band at k =0 are classified in Internal-Reflection Type. From Eq.(2.21), we have E_c =1.096. For the solitons arising from E_c, we have the energy goes down to the first band gap which implies

c 0

δ < , the effective mass [Eq.(2.56)] is positive, and the formation of gap solitons is under attractive atomic interactions

c 1 m^∗ =

σ <0. For convenience, we name gap solitons arise from as gap solitons. By multiplying the Bloch wave

Ec c-type

(

^X,0

) (

^b_o ^b 1cos ^X

)

φ = + ₋ π [Eq.(2.19)] to the envelope function of Eq.(3.1), the wave function of gap solitons developed from E_c is given by

( )

^sech

( ) (

₁

)

c X A BX bo b cos

ϕ = + ₋ πX (3.6)

We plot a series of the typical gap solitons as shown in Fig. 4(a), Fig. 4(b) and Fig. 4(c) which correspond to the energy

-type c

0 1371

E= . , , and

, respectively. For comparison, we again numerically solve time-independent G-P equation [Eq.(2.3)] and plot in Fig. 4(d), Fig. 4(e) and Fig. 4(f) for the gap solitons at the corresponding energies. Note that the Bloch wave function is no longer a standing wave due to no coupling with counter-propagating wave. The soliton is mainly localized by the attractive potential rather than by Bragg reflection in the periodic structure; in this band, it resembles an ordinary guided wave modulated by a periodic structure and acts as internal reflection (IR) wave which does not contain zeros. It is a fundamental eigenmode gap soliton with small ripples and it becomes pure profile for the soliton energy deeper in the gap or with lower soliton energy.

0 5482

To obtain the analytic particle number related to the detuning chemical potential

N

δc for the gap solitons, we substitute Eq.(3.6) into the conservation of number of atoms in Eq.(2.4)

-type

4 2

Number of atoms in condensates N is also approximately proportional to δ and does not exhibit atomic population cutoffs at band edge as shown in Fig. 5. It becomes large as the localized state goes deeper inside the band gap where the nonlinear effect is strong (solitons being more localized). However, we found in Fig.

5 the numerical particle numbers of and both slightly larger than those obtained from the analytic formula of Eq.(3.4) and Eq.(3.5), but that for

is smaller.

For the energy is on the band, dark soliton solutions can be described by the envelope functions which satisfy effective mass equation [Eq.(2.52)] and are given by

( )

^tanh

( )

n n n

F X = A B X (3.9)

where B_n = δ_{n n}m^* and A_n =B_n m^*_{n n}σ . Dark solitons are developed from states of the upper (lower) band edge of the band gap, if atomic interactions are repulsive (attractive). The sign of detuning chemical potential, atomic interactions and effective mass determines whether dark solitons exist or not. For the energy is above the band edge δ_n > , the effective mass 0 m_n^∗> , and repulsive atomic 0 interactions 0σ_n > . For the energy is below the band edge δ_n < , the effective 0 mass , and attractive atomic interactions m_n^∗<0 σ_n < . The atomic interaction of 0 dark solitons is contrary to that of bright solitons while sign of the effective mass remains the same. From Eq.(3.9), we know that width of dark soliton is inversely

proportional to B and the background condensate density is proportional to _n . These two properties both depend upon the detuning chemical potential

An

δn, which is the relative chemical potential to the band edges. Near the band edge, where δ , _n B and n are small, the width of the localized condensate density dips is extended (occupying many lattice sites) in the space and the background condensate density is low. Deep inside the band, where

An

δ , n B and _n are large, width of the dips is much confined and the background density is high. We find that he chemical potential

An

n

( )

o

E k at band edge plays an important role in order to study BECs in optical lattices qualitatively and quantitatively. To confirm these properties of dark solitons, we solve the condensate wave functions analytically and numerically.

The procedure to seek for the condensate wave functions of dark solitons is analogous to that of bright solitons and is to add oscillatory band wave function

to envelope function

(

^,

n k Xo

φ

)

F Xn

( )

. According to the states on the band from which solitons are developed, we classify gap solitons into categories that are the Bragg-reflection type and internal-reflection type, respectively. Note that we kept the depth of the optical lattices V_o =2 5. in the numerical studies.

3-2-1 Bragg-Reflection Type

Same as for dark solitons which arise from the energy E_a, the first band at 2

k=π , or E_b, the second band at k=π 2, are classified in Bragg-Reflection Type.

For dark solitons arising from the energy E_a =1.859, we have the detuning chemical potential 0δ_a < and negative effective mass m_a^∗ = −0.339 [Eq.(2.54)]. The

atomic interaction to form dark solitons is turned into attractive σ <0 while the effective mass leaves unchanged. For convenience, we call dark solitons arise from

as dark solitons. By multiplying the Bloch wave Ea a-type

(

( , 2) 1 cos 2

a ^{X X}

)

φ π = π π [Eq.(2.17)] to the envelope function of Eq.(3.9), the wave function of dark solitons developed from E_a is given by

( )

^tanh

( )

^{1 cos} and Fig. 6(c) which correspond to the energy

-type a

1 1917

E = . , ,

and , respectively. For comparison, we numerically solve time-independent G-P equation [Eq.(2.3)] and plot in Fig. 6(d), Fig. 6(e) and Fig. 6(f) for the dark solitons at the corresponding energies.

1 4776

For dark solitons arising from the energy E_b =3.109, we have the detuning chemical potential δ_b > and positive effective mass 0 m_b^∗ =0.202 [Eq.(2.55)].

The atomic interaction to form dark solitons is turned into repulsive σ >0 and the effective mass leaves unchanged. For convenience, we call dark solitons arise from

as dark solitons. By multiplying the Bloch wave Eb b-type

(

( , 2) 1 sin 2

b ^{X X}

)

φ π = π π [Eq.(2.18)] to the envelope function of Eq.(3.9), the wave function of dark solitons developed from E_b is given by

( )

^tanh

( )

^{1 sin} and Fig. 7(c) which correspond to the energy

-type b

3 204

E= . , , and

, respectively. For comparison, we numerically solve time-independent G-P equation [Eq.(2.3)] and plot in Fig. 7(d), Fig. 7(e) and Fig. 7(f) for the

dark solitons at the corresponding energies.

3 4898

The results of and dark solitons reveal high agreement between the analytic and numerical solutions. Also, we show that the smaller detuning chemical potential

-type

a b-type

δ , the higher accuracy of the analytic results. Lower accuracy n

of the analytic results occurs around the position X =0. The oscillations of

and dark solitons are due to the cosine and sine standing wave functions at the and as the Bragg reflection (BR) type solitons. They are the modulated standing Bloch waves, resulting from the interference of counter-propagating Bragg reflected waves, by the soliton envelope function.

-type a -type

b

Ea E_b

3-2-2 Internal-Reflection Type

For dark solitons arising from the energy E_c =1.096 at ^{k =0} are classified as Internal-Reflection Type. We have the detuning chemical potential δ_c > and 0 positive effective mass [Eq.(2.56)]. The atomic interaction to form dark solitons is turned into repulsive

c 1 m^∗=

σ >0 and still the effective mass leaves unchanged.

For convenience, we name dark solitons arise from as dark solitons.

By multiplying the Bloch wave

Ec c-type

(

X,0

) (

b_o b 1cos X

)

φ = + ₋ π [Eq.(2.19)] to the

envelope function of Eq.(3.9), the wave function of dark solitons developed from is given by and Fig. 8(c) which correspond to the energy

-type c

1 1917

E = . , ,

and , respectively. For comparison, we again numerically solve time-independent G-P equation [Eq.(2.3)] and plot in Fig. 8(d), Fig. 8(e) and Fig. 8(f)

1 4776 E= . 1 7634

E= .

for the c-type dark solitons at the corresponding energies.

Mechanism of the formation of dark solitons is quite different from bright solitons. Condensate waves can propagate in linear regime since the energy of dark soliton is in band. With the consideration of atomic interaction, i.e., under nonlinear regime, phase difference ∆θ between the parts left and right to the dark soliton (DS) plane, a plane of minimum condensate density, is π . This phase difference can be regarded as destructive interference of two waves, and hence there exist a localized dip in condensate density. In aspect of energy band spectrum, since the energy band structure shift slightly, there exist the nonlinear localized mode, dip on condensate density, i.e. dark solitons. Therefore, a dark soliton can propagate and its shape leave unchanged. Dark solitons are characterized by the dependence of a complementary norm N_c of the condensate wave function on the chemical potential

E to represent a notch on the condensate density. We have N_c

( ) ( )

2 2

background soliton

Nc =

∫

^⎡⎣ϕ X −ϕ ^{X ⎤}⎦^dX

where represents a deficit of the condensate atoms associated with the formation of a dark soliton notch in the Bloch wave background. We analytically calculate the number of deficient atoms in condensate, which are plotted in Fig. 9. The quantitative measure of this deficit depends on the width of the notch and the peak density of the background. As a rule, wider solitons form on lower density Bloch waves near the lower edges of spectral bands. The higher density nonlinear Bloch waves, corresponding to large chemical potentials, carry narrower dark states.

Nc

Nc

We have numerically and analytically demonstrated that bright (dark) solitons developed from both of band edges of the first and second bands at k =π 2 into the

second band gap (within the first band) are Bragg reflection type, a standing wave of counter propagating Bragg reflected Bloch waves due to optical lattices, whereas, bright (dark) soliton below the first band (within the first band) is mainly localized by the nonlinear attractive (repulsive) potential rather than by Bragg reflection in optical lattices. It resembles the ordinary guided waves modulated by a periodic structure and has internal reflection (IR) wave profile which does not contain zeros.

The first experimental observation of bright matter wave solitons with repulsive interaction for ⁸⁷Rb in optical lattices was reported by Eiermann et al. [20]. The experimental conditions are transverse and longitudinal trapping frequencies

2 85 Hz

ω_⊥ = π× and ω =2π×0.5 Hz, and a standing light wave of wavelength 783 nm

λ = . They deduced a soliton width of x_o =6 µm from the absorption

images and the inverse effective mass 1 m_a^∗ = −0.1 from the experimentally measured soliton period, and they found the number of atoms is around 300. We proceed to compare our analytic results for gap solitons with Eiermann’s experimental results. We can, on the contrary, obtain the analytic bright solitons solutions by deducing the depth of the optical lattice

-type a

0 8972

Vo = . from the effective mass 1 m_a^∗ = −0.1 and the detuning chemical potential from the soliton width

5.322 10 3

δ = × ⁻

o 6 µm

x = . Note that the soliton solutions are dimensionless and satisfy the effective one-dimensional G-P equation [Eq.(2.3)]. With parameter transformation discussed in Chapter 2, we can obtain solitons solutions satisfying one-dimensional G-P equation [Eq.(2.2)] instead and derive the number of atoms in bright soliton in dimensional variables

2 2 2 variables [Eq.(2.4)]. By substituting all parameters into Eq.(3.13), is found to be which well agree with Eiermann’s experimental results. Eiermann’s experiments show that only partial condensate atoms can form bright solitons and the rest condensate atoms are regarded as background. In aspect of our analytic soliton solutions, the detuning potential

N

N 287

δ represents number of condensate atoms to form bright solitons, whereas the band edge energy E kn

( )

o is regarded as background condensate atoms. For further discussion, we consider the product of atom number and soliton width as a function of the effective mass varied by adjusting the depth of the periodic potential. As the δ is small, the second term in Eq.(3.4) is negligible and we found that the product of soliton width and the number of atoms is proportional to the inverse effective mass. These results are shown in Fig.10 and are compared with Eiermann’s experimental results [22]. Good coincidence between analytic solutions and experimental results is also revealed in Fig. 10. However, the foregoing are under small δ and . To complete our discussion, we plot numerical and analytical for

Vo

-N E V_o =1.5 and V_o =3.5 shown in Fig. 11(a) and Fig. 11(b), respectively. For small δ , the numerical and analytical results are in good agreement for both two values of V_o; for large δ , it is found that

has higher disagreement than

o 3.5 V =

o 1.5

V = . For large depth of the optical lattice, , the atoms are much confined in optical lattices, and hence the Bloch functions are not adequate to describe condensate atoms near band edge.

Consequently, the inaccuracy of the analytic soliton solutions occurs. In summary, the effective mass theory can describe BECs in optical lattices qualitatively and quantitatively, especially adequate for small optical lattice and detuning potential.

o 3.5 V =

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 32-45)