Conclusion and Perspective - K．P 理論於玻色-愛因斯坦凝聚態在光晶格之應用

4-1 Conclusion

We have applied the effective mass theory to study the dynamics of Bose-Einstein condensates (BECs) in optical lattices with either attractive or repulsive atom interactions. The macroscopic condensate wave function is describes by Gross-Pitaevskii (G-P) equation. We have derived the analytic soliton solution near band edge by including band edge energy as a parameter of solitons. The analytic soliton solution is found to be a Bloch function from the periodicity modulated by a soliton envelope function of the effective mass equation in which the periodic external potential appears in the form of an effective mass. The band edge energy is regarded as background condensate atoms at a specific wave vector to form solitons. We have demonstrated that the analytic soliton solutions can be either bright or dark solitons for both attractive and repulsive atom interactions since the energy band structure can change the dispersion of the BEC wavepackets dramatically. Both bright and dark solitons corresponding to energy in band gap and energy within band, respectively, can be categorized as Bragg reflection type solitons due to a standing wave of counter propagating Bragg reflected Bloch waves and internal reflection type solitons due to mainly localized by the attractive (repulsive) potential. The relation between the number of atoms to form solitons and energy has also been studied. As the energy goes deep inside the energy band (energy band gap), the number of atoms in dark (bright) soliton increases. Numerically solved the G-P equation, we confirmed the analytic soliton solutions agree reasonably well with simulations. The higher accuracy occurs at smaller detuning chemical potential and smaller depth of optical lattice. Eventually, we compared our bright soliton solutions with repulsive

interaction with the experimental results reported by Eiermann et al. [20]. Good agreement has been revealed. In conclusion, we have demonstrated that the band edge energy has physical significance to describe BECs in optical lattices and we have found that BECs in optical lattices can be described, qualitatively and quantitatively, by the effective-mass theory.

4-2 Perspective

We have applied the effective mass theory to study the stationary state BECs in optical lattices, i.e., the time-independent G-P equation throughout the Thesis and have showed that BECs in optical lattices can be described, qualitatively and quantitatively, by the effective-mass theory. In the future work, we can consider the time-dependent G-P equation where many phenomena of BECs in optical lattices can be experimentally demonstrated and/or theoretically investigated. The prediction of modulational instability (MI) [21] is one of the phenomena. Under the MI condition, the wavevector has imaginary term after adding a small perturbation wave to the Bloch wave and, consequently, the wave “grows up”. In other words, solitons can occur under such condition. Konotop and Salerno [22] have studied the modulational instability in BECs in optical lattices by means of multiple-scale expansion. With such analysis, they obtain the velocity and inverse effective mass of the energy band, and explain the relation between the existence of solitons and the scattering length a_s. We can instead proceed to study MI of BECs in optical lattices by applying the effective mass theory to the time-dependent G-P equation.

References

[1] C. Salomon, J. Dalibard, A. Aspect, H. Metcalf, and C. Cohen-Tannoudji, Phys.

Rev. Lett. 59, 1659 (1987).

[2] B.P. Anderson and M.A. Kasevich, Science 282, 1686 (1998).

[3] F.S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A . Smerzi, M. Inguscio, Science 293, 843 (2001).

[4] J. H. Denschlag, J. E. Simsarian, H. H¨affner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. L. Rolston, and W. D. Phillips, J. Phys. B: At. Mol. Opt. Phys.

35, 3095 (2002).

[5] M. Cristiani, O. Morsch, J. H. M¨uller, D. Ciampini, and E. Arimondo, Phys. Rev.

A 65 063612 (2002).

[6] D. I. Choi and Q. Niu, Phys. Rev. Lett. 82, 2022 (1999).

[7] D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).

[8] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. 78, 985 (1997).

[9] A. Trombettoni and A. Smerzi, Phys. Rev. Lett. 86, 2353 (2001).

[10] P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar, Phys. Rev. A 67, 013602 (2003).

[11] N. K. Efremidis and D. N. Christodoulides, Phys. Rev. A 67, 063608 (2003).

[12] H. Pu, L. O. Baksmaty, W. Zhang, N. P. Bigelow, and P. Meystre, Phys. Rev. A 67, 043605 (2003).

[13] S.F. Mingaleev and Yu.S. Kivshar, Phys. Rev. Lett. 86, 5474 (2001).

[14] R. Slusher and B. Eggleton, eds., Nonlinear Photonic Crystals (Springer-Verlag, Berlin, 2003).

[15] Yu.S. Kivshar and G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).

[16] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).

[17] Callaway, Quantum Theory of the Solid State (Academic Press, New York, 1991).

[18] J. H. Mathews, Numerical Methods for Mathematics, Science, and Engineering (Prentice-Hall, London, 1992).

[19] W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in C (Cambridge University, New York, 1997).

[20] B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M.K. Oberthaler, cond-mat 0402178 (2004).

[21] F. Kh. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M.

Salerno, Phys. Rev. A 64, 043606 (2001).

[22] V. V. Konotop, and M. Salerno, Phys. Rev. A 65, 021602 (2002).

-0.50 0 -0.25 0.00 0.25 0.50 1

2 3 4 5 6 7

c

b

( π )

E

k

a

2

^nd

band gap

1

^st

band gap

Fig. 1 Typical energy band spectrum E k− of BECs for depth of optical lattices

o 2.5

V = . Points “a” and “b” are the band edges of the first and second bands at 2

k=π and point “c” is the lowest band edge at k =0. The shaded regions are the first and second band gap, respectively.

-30 -20 -10 0 10 20 30 numerical result of G-P equation (d), (e), and (f). The dashed line represents optical lattices.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0

1 2 3 4 5 6

N

E c

a b

1

^st

band 2

^nd

band

Fig. 5 Number of atoms in bright soliton as a function of energy for , and . The solid and dashed curves are the analytic and numerical results, respectively. The shaded regions are the first and second band, respectively.

N E a-type

-type

b c-type

-30 -20 -10 0 10 20 30 numerical result of G-P equation (d), (e), and (f). The dashed line represents optical lattices.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.6

0.9 1.2 1.5 1.8 2.1 2.4

Nc

E

c a

b

2

^nd

band gap 1

^st

band gap

Fig. 9 Analytic results of deficit number of atoms in dark soliton N_c as a function of energy for , and . The shaded regions are the first and second band gap, respectively.

E a-type b-type c-type

6 8 10 12 14 16 18 20 22 0.5

1.0 1.5 2.0 2.5 3.0 3.5 4.0

x10 -3

Nx o

-1/meff

Fig. 10 Product of number of atoms in a-type bright soliton N and soliton width xo as a function of inverse effective mass 1 m_eff . Comparison of Eiermann et al. experimental results (solid circles) and our analytic results (open circles) for the depth of optical lattices V_o is 0.8972.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1

2 3 4 5 6

E Vo=1.5

a b

c

(a)

1

^st

band 2

^nd

band

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 1

2 3 4 5

6 Vo=3.5

E

a b

c

(b)

1

^st

band

Fig. 11 Number of atoms and energy for different depth of optical lattices (a) and (b)

N E

Vo V_o =1.5 V_o =3.5. The solid and dashed curves are the analytic and numerical results, respectively.

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 45-59)