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

Nikolay A. Kostov ; Vladimir S. Gerdjikov ; Tihomir I. Valchev

Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

Résumé

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 ($k\to 0$) our solutions model a quasi-one dimensional quantum degenerate Bose–Fermi mixture trapped in optical lattice. In the limit $k\to 1$ 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.

Détail

Keywords: Bose–Fermi mixtures; one dimensional optical lattice.

Bibliographie

[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 | DOI

[2] Morsch O., Oberthaler M., “Dynamics of Bose–Einstein condensates in optical lattices”, Rev. Modern Phys., 78 (2006), 179–215 | DOI

[3] Modugno G., Roati G., Riboli F., Ferlaino F., Brecha R. J., Inguscio M., “Collapse of a degenerate Fermi gas”, Science, 297:5590 (2002), 2240–2243 | DOI

[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 pp., ages ; cond-mat/0011291 | DOI

[5] Tsurumi T., Wadati M., “Dynamics of magnetically trapped boson-fermion mixtures”, J. Phys. Soc. Japan, 69 (2000), 97–103 | DOI

[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 pp., ages ; cond-mat/0511206 | DOI

[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 pp., ages ; cond-mat/0404320 | DOI

[8] Salerno M., “Matter-wave quantum dots and antidots in ultracold atomic Bose–Fermi mixtures”, Phys. Rev. A, 72 (2005), 063602, 7 pp., ages ; cond-mat/0503097 | DOI

[9] Adhikari S. K., “Fermionic bright soliton in a boson-fermion mixture”, Phys. Rev. A, 72 (2005), 053608, 7 pp., ages ; cond-mat/0509257 | DOI

[10] Adhikari S. K., “Mixing-demixing in a trapped degenerate fermion-fermion mixture”, Phys. Rev. A, 73 (2006), 043619, 6 pp., ages ; cond-mat/0607119 | DOI

[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 | DOI | MR | Zbl

[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 pp., ages | DOI

[13] Roati G., Riboli F., Modugno G., Inguscio M., “Fermi–Bose quantum degenerate ${}^{87}$Rb-${}^{40}$K mixture with attractive interaction”, Phys. Rev. Lett., 89 (2002), 150403, 4 pp., ages ; cond-mat/0205015 | DOI

[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 ${}^{87}$Rb and ${}^{40}$K”, Phys. Rev. A, 70 (2004), 021601(R), 4 pp., ages ; cond-mat/0405419 | DOI

[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 | DOI

[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 pp., ages ; cond-mat/0007091 | DOI

[17] Busch Th., Anglin J. R., “Dark-bright solitons in inhomogeneous Bose–Einstein condensates”, Phys. Rev. Lett., 87 (2001), 010401, 4 pp., ages ; cond-mat/0012354 | DOI

[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 | DOI | MR | Zbl

[19] Sov. Phys. JETP, 38 (1974), 248–253 | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 pp., ages ; cond-mat/0007117 | DOI

[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 pp., ages ; cond-mat/0010099 | DOI

[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 pp., ages | DOI

[27] Abramowitz M., Stegun I. A. (ed.), Handbook of mathematical functions, Dover, New York, 1965