Atomic Dark-Bright Solitons: Theory and Experiments

Atomic Dark-Bright Solitons: Theory and Experiments D. J. Frantzeskakis* Department of Physics, University of Athens, Greece LENCOS 12, Seville, July 9-12, 2012 *In collaboration with: P. G. Kevrekidis, D. Yan (Amherst), R. Carretero-González (San Diego), M. Hoefer (North Carolina), P. Engels, J. Chang., C. Hamner (Washington), P. Schmelcher, S. Middelkamp, J. Stockhofe (Hamburg), J. Cuevas, A. Alvarez (Seville) V. Achilleos (Athens), V. Rothos (Thessaloniki) Outline Bοse-Einstein condensates (BECs) - Binary BECs - Gross-Pitaevskii mean-field description Dark-bright solitons in binary BECs - Single and multiple dark-bright solitons - SU(2) rotations: “beating” dark-dark solitons Dark-bright solitons at finite temperatures - Dissipative Gross-Pitaevskii equations - Different temperature-dependent damping regimes Conclusions and outlook Bose-Einstein condensates (BECs) BEC: State of matter in which a macroscopic number of particles share the same quantum state Theoretical prediction: Bose-Einstein (1925) Experimental observation: Cornell-Wieman-Ketterle-Hulet (1995) in ultracold atoms of 87Rb, 23Na and 7Li. Nobel Prize (2001) First experimental observation: magnetically-trapped spin states of 87Rb BEC (JILA group, PRL 1997) Binary Bose-Einstein condensates 2 2 222 2 112 )( 22 2 2 2 1 2 212 2 111 )( 11 2 2 1 2 2 ψψψψψ ψψψψψ        +++∇−=∂        +++∇−=∂ ggV m i ggV m i ext t ext t     Two coupled Gross-Pitaevskii equations (for different hyperfine states of the same atom species): intra-species atomic collisions inter-species atomic collisions Typically, e.g., for different spin states of 87Rb: 221211 ggg ≈≈ Binary BECs in the mean-field picture 2 2 2 2 12 2 2 1 2 2 2 11 2 1 )( 2 1 1)( 2 1 ψµψψψψ ψψψψψ      −+++∂−=∂      −+++∂−=∂ zVi zVi zt zt 1/ 2 1 )( 22 <<≡Ω Ω= ⊥ ωω z zzV Binary BECs in highly anisotropic (quasi-1D) harmonic traps Dark-bright solitons in homogeneous BECs tkDtDkztzzD ttz itz )]1(2[ 2 1 )(,tan)],([ })](exp{i[kzhsec),( ,sintanhcos),( 22 00 2 1 −+−===−= += += µθϕζ θζηψ ϕζϕψ  “Symbiotic” solitons The bright soliton (supported only by attractive interactions) exists only due to the interspecies interaction with the dark soliton bright soliton dark soliton dark soliton bright soliton z  Hamburg: Phase-imprinting of a dark soliton in state |1, 0> and filling the density dip with atoms in state |2, 0 >  Washington: Generation of dark-bright solitons by counterflow of two components in the |1, -1> and |2, -2> states Observations of dark-bright solitons Hamburg experiment S. Stellmer et al., Nat. Phys. 2008 Washington experiments C. Hammer et al., PRL 2011 S. Middelkamp et al., PLA 2011 Dark-bright solitons in the trap 2 2,1 ψ z Adiabatic dynamics of dark-bright solitons in the trap - steps to follow: Find an equation for the background wavefunction Find a perturbed NLS system for the dark-bright soliton wavefunction Assume: the soliton’s functional form remains the same but the soliton parameters are unknown functions of time Evolution of renormalized Hamiltonian  evolution of soliton parameters bright soliton dark soliton trapping potential background density ( )        ∆ −+= −−−++∂+∂= ∫ +∞ ∞− µ ϕ µ ψµψψψψ 223 2 2 22 2 2 1 2 2 2 1 sec 2 1 3 4 ||)1(2)1|||(||||| D N D dzE b zzDB Perturbation theory – Hamiltonian approach ( ) ( ) ( ) ( ) uVQuu z u t u i zdz dV VQu z t i b d 2 2 2 2 2 2 2 2 2 2 2 2 1 1 )(|| 2 1 12 2 1 )(1|| 2 1 υ µ υµυ υ υυ µ υυυ υυ −≡=−+− ∂ ∂ + ∂ ∂       ∂ ∂ +−≡=−+− ∂ ∂ + ∂ ∂ Approximate dark-bright soliton solution for Qd ≠ 0, Qb ≠ 0 as in the unperturbed case but with: )(tan)()(),(),( 0 ttDtztDDt ϕϕϕ →→→  Coupled GPEs as coupled perturbed NLSEs Use of the dark-bright soliton energy to find the evolution of the unknown time-dependent soliton parameters D, φ, z0 ),()exp()( 1 tztiz υµψ −Φ= )(||:ionapproximatFermiThomas 2 zV −=Φ− µ BEC wavefunction carrying a dark soliton: dark soliton Evolution of the dark-bright soliton energy: Oscillations of dark-bright solitons dz t Q t QDDNDD dt dE b b d db            ∂ ∂ + ∂ ∂ −=++= ∫ +∞ ∞− ∗∗ υυ ϕϕϕ Re2)tan(sec4 2   Equation of motion for the dark-bright soliton center: µ ωω b oscosc N r r r z dt z ≡         + − Ω ==+ ; )4/(14 1 2 ,0 d 2/1 2 0 2 2 0 2 22 / Ω osc ω µ / b N S. Middelkamp et al., Phys. Lett. A 2011 Busch-Anglin, PRL 2001 D N DDz b µ ϕϕ 2 cos,tan 22 0 −==  Dissipative dynamics at finite temperatures Coupled dissipative Gross-Pitaevskii equation (DGPEs) (Pitaevskii, Sov. Phys. JETP 1959) jj k kzjtj zVi ψµψψγ        −++∂−=∂− ∑ = 2 1 2 2 )( 2 1 )( )(1),(4;T~ µαµαγ α >>=<<= TkTk BBj Achilleos, Yan, Kevrekidis, Frantzeskakis, NJP 2012 ( ) ( ) ( ) ( )       ∂ ∂ +−≡=−+− ∂ ∂ + ∂ ∂         ∂ ∂ + ∂ ∂ +−≡=−+− ∂ ∂ + ∂ ∂ t u uVQuu z u t u i tzdz dV VQu z t i bb d d µγυ µ υµυ υ µ γ υ υυ µ υυυ υυ 2 2 2 2 2 2 2 2 2 2 2 2 1 1 )(|| 2 1 12 2 1 )(1|| 2 1 Coupled DGPEs as coupled perturbed NLSEs Temperature-induced antidamping of solitons 22 )2/1( V(z) z Ω= For adiabatic perturbation theory results in the following equation of motion (for sufficiently deep/slow solitons): 0 0 2 00 =+− zzaz osc ω  () osccrcrosc aaaasass ωω 2, 2 1 0 22 2,1 22 =−±=⇒=+− + ∈⇒> Rsaa cr 2,1 + ∈=⇒= Rssaa cr 21 Csaa cr ∈⇒< 2,1 • Super-critical case : • Critical case : • Sub-critical case : Cockburn, Nistazakis, Horikis, Kevrekidis, Proukakis, Frantzeskakis, PRL 2010; PRA2011 Achilleos, Yan, Kevrekidis, Frantzeskakis, NJP 2012 2/1 2 2 2 2 )4/(4 1 2 , 8 )4/(6 83 2         + − Ω =        +− + +        −= b b oscb b db b b b b d N NN N NN a µ ωγ µ γγ µ µ γ µ γµ Dark and dark-bright soliton trajectories Super-critical case :Sub-critical case : cr aa > cr aa < Temperature dark soliton bright soliton bright soliton dark soliton dark soliton dark-bright soliton The bifurcation diagram is “drifted” towards smaller values of γ ⇒ dark-bright solitons are more robust than dark solitons Conclusions Dark-bright solitons in binary BECs: - Singleand multiple-dark-bright solitons - SU(2) rotations: beating and regular dark-dark solitons - Dissipative dynamics of dark-bright solitons at finite temperatures - In all cases connection to experiments was provided Current and future work Vector solitons in multi-component BECs (including spinor BECs) - Scattering of vector solitons at narrow barriers - Dissipative dynamics of vector solitons in various settings - Vector soliton dynamics in higher-dimensional settings “Filled” (with the bright soliton) dark solitons are more robust than “bare” dark solitons in BECs against:  temperature-induced dissipation  transverse instabilities (dark-bright rings) Some relevant references: