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Superfluid dynamics of BEC in a periodic potential Augusto Smerzi INFM-BEC & Department of Physics, Trento LANL, Theoretical Division, Los Alamos Collaboration with: Chiara Menotti INFM-BEC & Department of Physics, Universita` di Trento Andrea Trombettoni INFM & Department of physics, Universita` di Parma BEC trapped in a periodic potential (r , t ) 2 2 2 i Vext (r ) g 0 t 2m The interatomi c interactio n The trapping potential (s wave scattering length approximat ion) m Lattice field VL (r ) V0 sin 2 (k x) r2 ( y 2 z 2 ) 2 a 0 repulsive 4 2 a g 0 m 2 2 Driving field VD (r ) m r x a 0 attractive 2 BEC expanding in a 1D optical lattice No interaction --> a = 0 Density profile Trapping potential Momentum distribution d intersite spacing q B Bragg momentum BEC expanding in a 1D optical lattice a N0 A. Trombettoni and A. Smerzi, PRL 86, 2353 (2000) BEC expanding in a 1D optical lattice Interacting atoms --> a > 0 Density profile Trapping potential Momentum distribution Preliminary experimental evidences of self-trapping B. Eiermann, M. Albiez, M. Taglieber, M. Oberthaler University of Konstanz BEC expanding in a 1D optical lattice Interacting atoms --> a > 0 Dynamical variables n (t ) N n (t ) ei n (t ) Tunneling rate Array of weakly coupled BEC k n ( x) T V n 1 ( x) Josephson oscillations 1. Atoms are condensed in the optical and magnetic fields. 2. The harmonic confinement is instantaneously shifted along the x direction. N 105 Rd atoms x 2 10 Hz r 2 100 Hz V0 5 E R d 0.5 m Array of Josephson junctions driven by a harmonic external field Josephson oscillations center of mass position : n N n n relative phases : p n 1 n d (t ) 2 K sin p(t ) dt d 2 2 p(t ) m x ( ) (t ) dt 2 Oscillations of the three peaks of the interferogram. Blue circles: no periodic potential The array is governed by a pendulum equation F.S. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Science 293, 843 (2001) Small amplitude pendulum oscillations Jos xd 2m K Triangles: GPE; stars: variational calculation of K Circles: experimental results Relation between the oscillation frequency and the tunneling rate Breakdown of Josephson oscillations The interwell phase coherence breaks down for a large initial displacement of the BEC center of mass Questions: 1) Why the interaction can break the inter-well phase coherence of a condensate at rest confined in a periodic potential ? 2) Why a large velocity of the BEC center of mass can break the inter-well phase coherence of a condensate confined in a periodic potential and driven by a harmonic field ? Which are the transport properties of BEC in periodic potentials ? The discrete nonlinear equation (DNL) n n * * i VD ( n n 1 n n 1 c.c.) n U n n t k ( n n 1 ) n 1 k ( n n 1 ) n 1 2 2 2 2 Tunneling depends on the height of the interwell barriers Dynamical variables and on the interactio n kn (t )n ( x)NTn (tV) e n1n( x) i ( t ) g 0 3n ( x) n 1 ( x) a Newtonian Dynamics of a wave-packet n (t ) (t ) 2 exp ip[n (t )] i n (t ) C f [n (t )] 2 (t ) The collective coordinate s ξ(t), p(t),σ(t), δ(t) satisfy variationa l Euler Lagrange equations 1 v g mE sin p VD ( ) p mE1 1 2 n m ( N ) n E n n 2 n Bloch energies & effective masses n (t ) N 0 e i ( p n t ) are eigenstate s of the DNL 1 EE N 0 cos p mE 1 loc cos p m Effective masses depend on the height of loc the inter-well barriers and on the density 2 m ( N 0 ,V0 ) 2 2 p 1 1 2E m ( N 0 ,V0 ) N0 2 p 2 1 E p 0 In the limit p 0 V0 0, m mE m See also M. Kramer, C. Menotti, L. Pitaevskii and S. Stringari, unpublished Bogoliubov spectrum n (t ) N 0 ei ( p n t ) n (t ) n (t ) uq e i q t ei[( p q ) n t ] , N0 1 q p quasi momentum of the large amplitude traveling wave q quasi momentum of the perturbati on mode 1. Replace in the DNL 2. After linearization, retrieve the dispersion relation ( p, q, N 0 ) Bogoliubov spectrum sin p cos 2 p cos p 4 q 2 q B sin q 2 sin N 0 sin 2 m m 2 mE N 0 2 m 1 m 1 2 E O 1 m Free limit (periodic potential OFF ) p 1 q 4 1 2 B q N q 0 2 m m 4 m N 0 Sound-wave & energetic instability Energetic instabilit y B 0 sound velocity vs , B q ( q 0 ) mE vg c m 1 c N 0 cos p mE N 0 sin p vg mE 2 m c 2 E2 vg2 m Landau criteria for breakdown of superfluid ity in free space : c 2 vg2 Cfr. with B. Wu and Q. Niu, PRA64, 061603R (2001) Dynamical instability The system becomes modulation ally unstable when ωB becomes complex c2 1 N 0 cos p 0 mE N 0 In the free (V 0) limit the Bogoliubov spectrum The amplitude of the perturbation modes grows exponentially fast, dissipating the energy 1 q 4 of 1thelarge μ amplitude wave-packet ωB vg q m 2 4 m N 0 N0 q 2 is always real (when a 0) No dynamical instabilities 1 1 2 2 q 2 sin q cos p sin N 0 cos p 2 m breakdown2 ofmsuperfluidity New mechanism for the of a BEC E N 0 1 (q, p) in a periodic potential Comparison between analytical and numerical dispersion relation 1 p 4 Full line: analytical (DNL) Dots: numerical (GPE) 3 p 4 Dashed line : m mE Breakdown of superfluidity for a BEC driven by a harmonic field Density at t=0,20,40 ms as a function of the Quasi-momentum vs. time for three different initial displacements: 40, 80, 90 sites Position. Initial displacements: 50, 120 sites v g mE1 sin p p VD ( ) The system becomes dynamically unstable at the critical group velocity vg m1 A. Smerzi, A. Trombettoni, P.G. Kevrekidis, A.R. Bishop, PRL 89, 170402 (2002) The Frontier Technological applications • Interferometry at the Heisenberg limit • Quantum information Foundational problems • Quantum – classical correspondence principle • Schroedinger cats, entanglement Tools • Quantum many-body dynamical theory