Resonant and non-resonant modulated amplitude waves for binary Bose–Einstein condensates in optical lattices. (English) Zbl 1098.81881

Summary: We consider a system of two Gross–Pitaevskii (GP) equations, in the presence of an optical-lattice (OL) potential, coupled by both nonlinear and linear terms. This system describes a Bose–Einstein condensate (BEC) composed of two different spin states of the same atomic species, which interact linearly through a resonant electromagnetic field. In the absence of the OL, we find plane-wave solutions and examine their stability. In the presence of the OL, we derive a system of amplitude equations for spatially modulated states, which are coupled to the periodic potential through the lowest order subharmonic resonance. We determine this averaged systemś equilibria, which represent spatially periodic solutions, and subsequently examine the stability of the corresponding solutions with direct simulations of the coupled GP equations. We find that symmetric (equal-amplitude) and asymmetric (unequal-amplitude) dual-mode resonant states are, respectively, stable and unstable. The unstable states generate periodic oscillations between the two condensate components, which are possible only because of the linear coupling between them. We also find four-mode states, but they are always unstable. Finally, we briefly consider ternary (three-component) condensates.


81V80 Quantum optics
81Q50 Quantum chaos
Full Text: DOI arXiv


[1] Agrawal, G.P., Nonlinear fiber optics, (1995), Academic Press San Diego, CA
[2] Alfimov, G.L.; Kevrekidis, P.G.; Konotop, V.V.; Salerno, M., Wannier functions analysis of the nonlinear schrodinger equation with a periodic potential, Phys. rev. E, 66, 046608, (2002)
[3] Anderson, B.P.; Kasevich, M.A., Macroscopic quantum interference from atomic tunnel arrays, Science, 282, 5394, 1686-1689, (1998)
[4] Baizakov, B.B.; Konotop, V.V.; Salerno, M., Regular spatial structures in arrays of bose – einstein condensates induced by modulational instability, J. phys. B: at. mol. opt. phys., 35, 5394, 5105-5119, (2002)
[5] Ballagh, R.J.; Burnett, K.; Scott, T.F., Theory of an output coupler for bose – einstein condensed atoms, Phys. rev. lett., 78, 5394, 1607-1611, (1997)
[6] Band, Y.B.; Towers, I.; Malomed, B.A., Unified semiclassical approximation for bose – einstein condensates: application to a BEC in an optical potential, Phys. rev. A, 67, 5394, 023602, (2003)
[7] Berg-Sørensen, K.; Mølmer, K., Bose – einstein condensates in spatially periodic potentials, Phys. rev. A, 58, 2, 1480-1484, (1998)
[8] Bronski, J.C.; Carr, L.D.; Carretero-González, R.; Deconinck, B.; Kutz, J.N.; Promislow, K., Stability of attractive bose – einstein condensates in a periodic potential, Phys. rev. E, 64, 2, 056615, (2001)
[9] 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, 8, 1402-1405, (2001)
[10] 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, 8, 036612, (2001)
[11] Burger, S.; Cataliotti, F.S.; Fort, C.; Minardi, F.; Inguscio, M., Superfluid and dissipative dynamics of a bose – einstein condensate in a periodic optical potential, Phys. rev. lett., 86, 20, 4447-4450, (2001)
[12] Burnett, K.; Edwards, M.; Clark, C.W., The theory of bose – einstein condensation of dilute gases, Phys. today, 52, 12, 37-42, (1999)
[13] Busch, Th.; Cirac, J.I.; Pérez-García, V.M.; Zoller, P., Stability and collective excitations of a two-component bose – einstein condensed gas: a moment approach, Phys. rev. A, 56, 4, 2978-2983, (1997)
[14] Carretero-González, R.; Promislow, K., Localized breathing oscillations of bose – einstein condensates in periodic traps, Phys. rev. A, 66, 4, 033610, (2002)
[15] Choi, D.-I.; Niu, Q., Bose – einstein condensates in an optical lattice, Phys. rev. lett., 82, 10, 2022-2025, (1999)
[16] Dalfovo, F.; Giorgini, S.; Pitaevskii, L.P.; Stringari, S., Theory of bose – einstein condensation on trapped gases, Rev. modern phys., 71, 3, 463-512, (1999)
[17] Deconinck, B.; Frigyik, B.A.; Kutz, J.N., Dynamics and stability of bose – einstein condensates: the nonlinear Schrödinger equation with periodic potential, J. nonlinear sci., 12, 3, 169-205, (2002) · Zbl 1009.35078
[18] Deconinck, B.; Kutz, J.N.; Patterson, M.S.; Warner, B.W., Dynamics of periodic multi-component bose – einstein condensates, J. phys. A—math. gen., 36, 20, 5431-5447, (2003) · Zbl 1038.82056
[19] P. Engels, personal communication.
[20] Esry, B.D.; Greene, C.H.; Burke, J.P.; Bohn, J.L., Hartree – fock theory for double condensates, Phys. rev. lett., 78, 19, 3594-3597, (1997)
[21] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied mathematical sciences, (1983), Springer-Verlag New York, NY · Zbl 0515.34001
[22] Hagley, E.W.; Deng, L.; Kozuma, M.; Wen, J.; Helmerson, K.; Rolston, S.L.; Phillips, W.D., A well-collimated quasi-continuous atom laser, Science, 283, 5408, 1706-1709, (1999)
[23] D.S. Hall, personal communication.
[24] Ho, T.-L.; Shenoy, V.B., Binary mixtures of Bose condensates of alkali atoms, Phys. rev. lett., 77, 16, 3276-3279, (1996)
[25] Inouye, S.; Andrews, M.R.; Stenger, J.; Miesner, H.J.; Stamper-Kurn, D.M.; Ketterle, W., Observation of Feshbach resonances in a bose – einstein condensate, Nature, 392, 6672, 151-154, (1998)
[26] Ketterle, W., Experimental studies of bose – einstein condensates, Phys. today, 52, 12, 30-35, (1999)
[27] Köhler, T., Three-body problem in a dilute bose – einstein condensate, Phys. rev. lett., 89, 21, 210404, (2002)
[28] Kuklov, A.; Prokof’ev, N.; Svistunov, B., Detecting supercounterfluidity by Ramsey spectroscopy, Phys. rev. A, 69, 21, 025601, (2004)
[29] Louis, P.J.Y.; Ostrovskaya, E.A.; Savage, C.M.; Kivshar, Y.S., Bose – einstein condensates in optical lattices: band-gap structure and solitons, Phys. rev. A, 67, 21, 013602, (2003)
[30] Machholm, M.; Nicolin, A.; Pethick, C.J.; Smith, H., Spatial period-doubling in bose – einstein condensates in an optical lattice, Phys. rev. A, 69, 21, 043604, (2004), ArXiv:cond-mat/0307183
[31] Machholm, M.; Pethick, C.J.; Smith, H., Band structure, elementary excitations, and stability of a bose – einstein condensate in a periodic potential, Phys. rev. A, 67, 21, 053613, (2003)
[32] Malomed, B.A., Polarization dynamics and interactions of solitons in a birefringent optical fiber, Phys. rev. A, 43, 1, 410-423, (1991)
[33] Malomed, B.A.; Skinner, I.M.; Chu, P.L.; Peng, G.D., Symmetric and asymmetric solitons in twin-core nonlinear optical fibers, Phys. rev. E, 53, 4, 4084-4091, (1996)
[34] Malomed, B.A.; Wang, Z.H.; Chu, P.L.; Peng, G.D., Multichannel switchable system for spatial solitons, J. opt. soc. am. B, 16, 8, 1197-1203, (1999)
[35] Mueller, E.J., Superfluidity and Mean-field energy loops; hysteretic behavior in bose – einstein condensates, Phys. rev. A, 66, 8, 063603, (2002)
[36] Myatt, C.J.; Burt, E.A.; Ghrist, R.W.; Cornell, E.A.; Wieman, C.E., Production of two overlapping bose – einstein condensates by sympathetic cooling, Phys. rev. lett., 78, 8, 586-589, (1997)
[37] Pethick, C.J.; Smith, H., Bose – einstein condensation in dilute gases, (2002), Cambridge University Press Cambridge, UK
[38] Porter, M.A.; Cvitanović, P., Modulated amplitude waves in bose – einstein condensates, Phys. rev. E, 69, 8, 047201, (2004), ArXiv:nlin.CD/0307032
[39] M.A. Porter, P. Cvitanović, A Perturbative Analysis of Modulated Amplitude Waves in Bose-Einstein Condensates, Chaos, in press (ArXiv:nlin.CD/0308024).
[40] Pu, H.; Bigelow, N.P., Collective excitations, metastability, and nonlinear response of a trapped two-species bose – einstein condensate, Phys. rev. lett., 80, 6, 1134-1137, (1998)
[41] Pu, H.; Bigelow, N.P., Properties of two-species Bose condensates, Phys. rev. lett., 80, 6, 1130-1133, (1998)
[42] Rand, R.H., Topics in nonlinear dynamics with computer algebra, Computation in education: mathematics, science and engineering, (1994), Gordon and Breach Science Publishers USA · Zbl 0828.73080
[43] Rand, R.H., Dynamics of a nonlinear parametrically-excited PDE: 2-term truncation, Mech. res. commun., 23, 3, 283-289, (1996) · Zbl 0868.35007
[44] R.H. Rand, Lecture Notes on Nonlinear Vibrations, A Free Online Book. Available at http://www.tam.cornell.edu/randdocs/nlvibe45.pdf2003.
[45] Salasnich, L.; Parola, A.; Reatto, L., Periodic quantum tunnelling and parametric resonance with cigar-shaped bose – einstein condensates, J. phys. B: at. mol. opt. phys., 35, 14, 3205-3216, (2002)
[46] Son, D.T.; Stephanov, M.A., Domain walls of relative phase in two-component bose – einstein condensates, Phys. rev. A, 65, 14, 063621, (2002)
[47] Trippenbach, M.; Góral, K.; Rz\(a_¸\) żewski, K.; Malomed, B.; Band, Y.B., Structure of binary bose – einstein condensates, J. phys. B: at. mol. opt. phys., 33, 14, 4017-4031, (2000)
[48] Trombettoni, A.; Smerzi, A., Discrete solitons and breathers with dilute bose – einstein condensates, Phys. rev. lett., 86, 11, 2353-2356, (2001)
[49] Weideman, J.A.C.; Herbst, B.M., Split-step methods for the solution of the nonlinear Schrödinger equation, SIAM J. numerical anal., 23, 3, 485-507, (1986) · Zbl 0597.76012
[50] B. Wu, R.B. Diener, Q. Niu, Bloch waves and Bloch bands of Bose-Einstein condensates in optical lattices, Phys. Rev. A 65 (2002) 025601.
[51] B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, M. K. Oberthaler, Bright Bose-Einstein gap solitons of atoms with repulsive interaction, Phys. Rev. Lett. 92 (2004) 230401.
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