Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 071, 14 pages      nlin.SI/0703057      doi:10.3842/SIGMA.2007.071

Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice

Nikolay A. Kostov a, Vladimir S. Gerdjikov b and Tihomir I. Valchev b
a) Institute of Electronics, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria
b) Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria

Received March 30, 2007, in final form May 17, 2007; Published online May 30, 2007

Abstract
We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k → 0) our solutions model a quasi-one dimensional quantum degenerate Bose-Fermi mixture trapped in optical lattice. In the limit k → 1 the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.

Key words: Bose-Fermi mixtures; one dimensional optical lattice.

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References

  1. Dalfovo F., Giorgini S., Pitaevskii L.P., Stringari S., Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys. 71 (1999), 463-512, cond-mat/9806038.
  2. Morsch O., Oberthaler M., Dynamics of Bose-Einstein condensates in optical lattices, Rev. Modern Phys. 78 (2006), 179-215.
  3. Modugno G., Roati G., Riboli F., Ferlaino F., Brecha R.J., Inguscio M., Collapse of a degenerate Fermi gas, Science 297 (2002), no. 5590, 2240-2243.
  4. Schreck F., Ferrari G., Corwin K.L., Cubizolles J., Khaykovich L., Mewes M.O., Salomon C., Sympathetic cooling of bosonic and fermionic lithium gases towards quantum degeneracy, Phys. Rev. A 64 (2001), 011402R, 4 pages, cond-mat/0011291.
  5. Tsurumi T., Wadati M., Dynamics of magnetically trapped boson-fermion mixtures, J. Phys. Soc. Japan 69 (2000), 97-103.
  6. Santhanam J., Kenkre V.M., Konotop V.V., Solitons of Bose-Fermi mixtures in a strongly elongated trap, Phys. Rev. A 73 (2006), 013612, 6 pages, cond-mat/0511206.
  7. Karpiuk T., Brewczyk M., Ospelkaus-Schwarzer S., Bongs K., Gajda M., Rzazewski K., Soliton trains in Bose-Fermi mixtures, Phys. Rev. Lett. 93 (2004), 100410, 4 pages, cond-mat/0404320.
  8. Salerno M., Matter-wave quantum dots and antidots in ultracold atomic Bose-Fermi mixtures, Phys. Rev. A 72 (2005), 063602, 7 pages, cond-mat/0503097.
  9. Adhikari S.K., Fermionic bright soliton in a boson-fermion mixture, Phys. Rev. A 72 (2005), 053608, 7 pages, cond-mat/0509257.
  10. Adhikari S.K., Mixing-demixing in a trapped degenerate fermion-fermion mixture, Phys. Rev. A 73 (2006), 043619, 6 pages, cond-mat/0607119.
  11. Belmonte-Beitia J., Perez-Garcia V.M., Vekslerchik V., Modulational instability, solitons and periodic waves in models of quantum degenerate boson-fermion mixtures, Chaos Solitons Fractals 32 (2007), 1268-1277, nlin.SI/0512020.
  12. Bludov Yu.V., Santhanam J., Kenkre V.M., Konotop V.V., Matter waves of Bose-Fermi mixtures in one-dimensional optical lattices, Phys. Rev. A 74 (2006) 043620, 14 pages.
  13. Roati G., Riboli F., Modugno G., Inguscio M., Fermi-Bose quantum degenerate 87Rb-40K mixture with attractive interaction, Phys. Rev. Lett. 89 (2002), 150403, 4 pages, cond-mat/0205015.
  14. Goldwin J., Inouye S., Olsen M.L., Newman B., DePaola B.D., Jin D.S., Measurement of the interaction strength in a Bose-Fermi mixture with 87Rb and 40K, Phys. Rev. A 70 (2004), 021601(R), 4 pages, cond-mat/0405419.
  15. Bronski J.C., Carr L.D., Deconinck B., Kutz J.N., Bose-Einstein condensates in standing waves: the cubic nonlinear Schrödinger equation with a periodic potential, Phys. Rev. Lett. 86 (2001), 1402-1405, cond-mat/0007174.
  16. Modugno M., Dalfovo F., Fort C., Maddaloni P., Minardi F., Dynamics of two colliding Bose-Einstein condensates in an elongated magnetostatic trap, Phys. Rev. A 62 (2000), 063607, 8 pages, cond-mat/0007091.
  17. Busch Th., Anglin J.R., Dark-bright solitons in inhomogeneous Bose-Einstein condensates, Phys. Rev. Lett. 87 (2001), 010401, 4 pages, cond-mat/0012354.
  18. Deconinck B., Kutz J.N., Patterson M.S., Warner B.W., Dynamics of periodic multi-components Bose-Einstein condensates, J. Phys. A: Math. Gen. 36 (2003), 5431-5447.
  19. Manakov S.V., On the theory of two-dimensional stationary self-focusing of electromagnetic waves, JETP 65 (1974), 505-516 (English transl.: Sov. Phys. JETP 38 (1974), 248-253).
  20. Belokolos E.D., Bobenko A.I., Enolskii V.Z., Its A.R., Matveev V.B., Algebro-geometric approach to nonlinear integrable equations, Springer, Berlin, 1994.
  21. Porubov A.V., Parker D.F., Some general periodic solutions to coupled nonlinear Schrödinger equations, Wave Motion 29 (1999), 97-109.
  22. Christiansen P.L., Eilbeck J.C., Enolskii V.Z., Kostov N.A., Quasiperiodic and periodic solutions for coupled nonlinear Schrödinger equations of Manakov type, Proc. R. Soc. Lond. Ser. A 456 (2000), 2263-2281.
  23. Eilbeck J.C., Enolskii V.Z., Kostov N.A., Quasiperiodic and periodic solutions for vector nonlinear Schrödinger equations, J. Math. Phys. 41 (2000), 8236-8248.
  24. Carr L.D., Kutz J.N., Reinhardt W.P., Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose-Einstein condensate, Phys. Rev. E 63 (2001), 066604, 9 pages, cond-mat/0007117.
  25. Bronski J.C., Carr L.D., Deconinck B., Kutz J.N., Promislow K., Stability of repulsive Bose-Einstein condensates in a periodic potential, Phys. Rev. E 63 (2001), 036612, 11 pages, cond-mat/0010099.
  26. Kostov N.A., Enol'skii V.Z., Gerdjikov V.S., Konotop V.V., Salerno M., Two-component Bose-Einstein condensates in periodic potential, Phys. Rev. E 70 (2004), 056617, 12 pages.
  27. Abramowitz M., Stegun I.A. (Editors), Handbook of mathematical functions, Dover, New York, 1965.

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