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Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates ∗ z 1 and Alexandru Nicolin 2 Antun Balaˇ 1 Scientific Computing Laboratory, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia 2 “Horia Hulubei” National Institute for Physics and Nuclear Engineering, P.O.B. MG-6, 077125 Bucharest, Romania ∗ Supported by: Serbian Ministry of Education and Science under Grant ON171017, CNCS-UEFISCDI under Grant PD122, Contract No. 35/28.07.2010, and European Commission through projects EGI-InSPIRE, PRACE-2IP, PRACE-3IP, and HP-SEE. LENCOS’12, Sevilla 11 July 2012

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Why ultracold quantum gases are interesting? Intensive progress in the field of ultracold atoms has been recognized by Nobel prize for physics in 2001 for experimental realization of Bose-Einstein condensation Cold alkali atoms: Rb, Na, Li, K . . . T ∼ 1 nK, ρ ∼ 10 14 cm − 3 Cold bosons, cold fermions Harmonic trap, optical lattice Short-range interactions, long-range dipolar interactions Spin-orbit-coupled BECs Tunable quantum systems concerning dimensionality, type and strength of interactions LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Faraday waves in BEC Two-component BEC Faraday waves in 2C BEC Conclusions and outlook Experimental observation of Faraday waves P. Engels, C. Atherton, M. A. Hoefer, PRL 98 , 095301 (2007) LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Two-component BEC systems Faraday waves in BEC Mean-field description of 2C BEC Two-component BEC Ground state - symbiotic pair Faraday waves in 2C BEC First excited - segregated state Conclusions and outlook Two-component BEC systems Experimentally realized with a broad variety of types of atoms and parameters of a system heterogeneous systems: different types of atoms homogeneous systems: same type of atoms, different internal (usually spin) states Rich dynamics and interplay of the parameters Several possible ground states A variety of possible dynamical evolutions We focus on the study of Faraday waves and patterns in cigar-shaped two-component 87 Rb BECs, with strong radial confinement, which is harmonically modulated We also study resonant waves, which appear for specific values of the frequency of radial modulation LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Two-component BEC systems Faraday waves in BEC Mean-field description of 2C BEC Two-component BEC Ground state - symbiotic pair Faraday waves in 2C BEC First excited - segregated state Conclusions and outlook Mean-field description of 2C BEC The system is described by a coupled system of GP equations: − � 2 » – i � ∂ Ψ 1 ( � r, t ) r, t ) | 2 + g 12 | Ψ 2 ( � r, t ) | 2 2 m 1 △ + V ( � r, t ) + g 11 | Ψ 1 ( � = Ψ 1 ( � r, t ) ∂t − � 2 » – i � ∂ Ψ 2 ( � r, t ) r, t ) | 2 + g 22 | Ψ 2 ( � r, t ) | 2 = 2 m 2 △ + V ( � r, t ) + g 21 | Ψ 1 ( � Ψ 2 ( � r, t ) ∂t where the couplings are given by: g 11 = 4 π � 2 a 1 , g 22 = 4 π � 2 a 2 , g 12 = g 21 = 2 π � 2 a int m 1 m 2 m eff Typical experimental values we consider for two hyperfine states of 87 Rb: N 1 = 2 . 5 · 10 5 , N 2 = 1 . 25 · 10 5 a 1 = 100 . 4 a 0 , a 2 = 98 . 98 a 0 , a int = 100 . 4 a 0 ω ρ ( t ) = ω ρ, 0 (1 + ǫ sin ω m t ) , ω ρ, 0 = 160 · 2 π Hz ω m = 250 · 2 π Hz , ǫ = 0 . 1 , ω z = 7 · 2 π Hz LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Two-component BEC systems Faraday waves in BEC Mean-field description of 2C BEC Two-component BEC Ground state - symbiotic pair Faraday waves in 2C BEC First excited - segregated state Conclusions and outlook Ground state - imaginary-time propagation 4.5 × 10 9 n 1 ( z ) 4.0 × 10 9 n 2 ( z ) 3.5 × 10 9 longitudinal density (m -1 ) 3.0 × 10 9 2.5 × 10 9 2.0 × 10 9 1.5 × 10 9 1.0 × 10 9 5.0 × 10 8 0.0 × 10 0 -80 -60 -40 -20 0 20 40 60 80 z ( µ m) Density profile of the converged eigenstate obtained by propagation in the imaginary time. Discretization parameters: N ρ = N z = 2000, ε = 10 − 4 /ω z . LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Two-component BEC systems Faraday waves in BEC Mean-field description of 2C BEC Two-component BEC Ground state - symbiotic pair Faraday waves in 2C BEC First excited - segregated state Conclusions and outlook Ground state - experimental realization K. M. Mertes, J. W. Merrill, R. Carretero-Gonz´ alez, D. J. Frantzeskakis, P. G. Kevrekidis, D. S. Hall, PRL 99 , 190402 (2007) LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Two-component BEC systems Faraday waves in BEC Mean-field description of 2C BEC Two-component BEC Ground state - symbiotic pair Faraday waves in 2C BEC First excited - segregated state Conclusions and outlook Segregated state - imaginary-time propagation 4.5 × 10 9 n 1 ( z ) 4.0 × 10 9 n 2 ( z ) 3.5 × 10 9 longitudinal density (m -1 ) 3.0 × 10 9 2.5 × 10 9 2.0 × 10 9 1.5 × 10 9 1.0 × 10 9 5.0 × 10 8 0.0 × 10 0 -80 -60 -40 -20 0 20 40 60 80 z ( µ m) First excited eigenstate obtained by imaginary-time propagation. Discretization parameters: N ρ = N z = 2000, ε = 10 − 4 /ω z . LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Two-component BEC systems Faraday waves in BEC Mean-field description of 2C BEC Two-component BEC Ground state - symbiotic pair Faraday waves in 2C BEC First excited - segregated state Conclusions and outlook First excited state - experimental realization C. Hamner, J. J. Chang, P. Engels, M. A. Hoefer, PRL 106 , 065302 (2011) LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Two-component BEC systems Faraday waves in BEC Mean-field description of 2C BEC Two-component BEC Ground state - symbiotic pair Faraday waves in 2C BEC First excited - segregated state Conclusions and outlook Energy: time dependence 316 segregated → symbiotic symbiotic 315 316.0 314 315.0 314.0 313 h ω z ) 0 10 20 30 40 50 E (units of - 312 308.262 308.260 311 308.258 0 10 20 30 40 50 310 309 308 0 50 100 150 200 250 300 350 400 450 imaginary time (s) Convergence of the total energy of the system during the imaginary-time propagation for the symbiotic pair and for the segregated state. LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Non-resonant Faraday waves Faraday waves in BEC Periods of Faraday waves Two-component BEC Resonant waves Faraday waves in 2C BEC Conclusions and outlook Symbiotic pair ground state - Faraday waves Emergence of Faraday waves as a result of real-time propagation. The radial frequency of the trap is modulated at the non-resonant frequency ω m = 250 · 2 π Hz, ǫ = 0 . 1. LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Non-resonant Faraday waves Faraday waves in BEC Periods of Faraday waves Two-component BEC Resonant waves Faraday waves in 2C BEC Conclusions and outlook Segregated state - Faraday waves Emergence of Faraday waves as a result of real-time propagation. The radial frequency of the trap is modulated at the non-resonant frequency ω m = 250 · 2 π Hz, ǫ = 0 . 1. LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012

Introduction Non-resonant Faraday waves Faraday waves in BEC Periods of Faraday waves Two-component BEC Resonant waves Faraday waves in 2C BEC Conclusions and outlook Periods of Faraday waves 100 100 FFT[ n 1 ] FFT[ n 1 ] FFT[ n 2 ] FFT[ n 2 ] 10 10 1 FFT amplitude FFT amplitude 1 0.1 0.01 0.1 0.001 k 1 k 2 k 3,1 k 3,2 k 1,2 k 1,1 k 2 k 3,2 k 3,1 0.01 0.0001 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 k ( µ m -1 ) k ( µ m -1 ) FFT of density profiles for the two condensates at t = 200 ms. For the symbiotic pair, the periods of waves are found to be 13 . 0 µ m and 12 . 5 µ m, while for the segregated state the periods are 11 . 6 µ m and 13 . 0 µ m. LENCOS’12, Sevilla A. Balaˇ z: Numerical study of Faraday waves in binary non-miscible Bose-Einstein condensates 11 July 2012