Cold atoms in the presence of disorder and interactions
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1 Cold atoms in the presence of disorder and interactions Collaboration: A. Minguzzi, S. Skipetrov, B. van-tiggelen (Grenoble), P. Henseler (Bonn), J. Chalker (Oxford), L. Beilin, E. Gurevich (Technion). PRL 99, 66 (7); 1, (8). PRA 77, 3364 (8); 8, 1363 (9); 81, 3361 (1). Recent reviews: L. Fellani, C. Fort and M. Inguscio, Advances in Atomic, Molecular and Optical Physics 56, 119 (8). A. Aspect and M. Inguscio, Physics Today 6, 3 (9). L. Sanchez-Palencia and M. Lewenstein, Nature Physics 6, 87 (1). 1
2 Random Speckle Potential D z r r + Δr I() r Laser light Diffusive plate Observation plane Light intensity at point r in the observation plane ( Δr/ R ) J1 IrIr ( ) ( +Δ r) = Ir ( ) 1+ 4 Δr/ R R λz π D = - correlation radius of the random potential. / in the z direction, it is of order λ( z / D). The potential acting on the atoms V( r) I( r). δvr () δvr ( r) V ( Δr/ R ) 1 +Δ = Δr/ R J
3 V typical magnitude of the random potential. R correlation radius, ( /mr ) = Ε correlation energy. 1. V E E R V Consider a particle with energy E E. k = me R kr white noise limit. / 1/ 1 In the Born approximation, the scattering crossection (in 3D) on a typical barrier (or well) is σ The mean free path V E R l R σ m 3 1 V R 3
4 kl V 1 E V ε c E 3 μ chemical potential of the BEC. μ ε c μ ε c The critical density coherent weakly disordered BEC. condensate droplets, or localized bosons. n c m (a scattering length ) c a ε This picture is confirmed by an approach starting with the low density limit and tracing the interaction-induced delocalization. B. I. Shklovskii, Semiconductors (St. Petersburg), 4, 97 (8). G. Falco, T. Nattermann and V. Pokrovsky, PRB 8, (9).
5 . V E V R E E V For energy k / m V k R V / E 1 o On the classical level the problem reduces to that of percolation. E p percolation threshold. E > E " quantum percolation" threshold. ' p For p ' μ < E p isolated lakes of condensate. For ' μ > E p coherent BEC.
6 Free expansion of a BEC 1 Vtrap() r = mω r Ψ+ Vtr apψ + g Ψ Ψ= μψ. m μ V trp a μ r Ψ ( r) = n( r) = = (1 ). g g R μ R =, μ ω R a mω mω μ Ψ ( rt, = ) = nr (,) = (1 g r R ) V trap ( r) R μ r
7 At t= the condensate is released from the trap and it starts expanding according to: Ψ i = Ψ+ g Ψ Ψ t m Ψ ( rt, ) = nrt (, ) exp[ isrt (, )], v = S m n v 1 + divnv=, m + ( mv + gn) = t t Solution (self-similar): μ 1 r b nrt (, ) = (1 ), v( rt, ) = r 3 gb br b = /, () = 1, () =. 4 b ω b b b 1 At t = t ω t For Ψ( rt, ) nrt (, )exp( ir / a) Φ( r ). > t b=ωt v = r/ t -Linear expansion.
8 ε Expansion in the presence of disorder For t > t i V( r) Ψ = Φ ( k Ψ+ Ψ ) t m V trap Ψ(,) (,) (,,) μ V( r ) Ψ ( rt, ) =Φ( r) - the initial condition. Reset t back to. (, ) ( Ψ rt = GrRt,, ) Φ ( R ). 3 rt nrt dr dkpε ( k ) rrtw( k, R). (,, ) Pε ( k ) r R t - Quantum diffusion kernel. ε ( k ) = k / m W( k, R) - Wigner function corresponding to Φ( R ).
9 In the long time, large distance limit nrt k P rt ( π ) Φ ( k ) Fourier transform of Φ( R ). 3 dk (, ) Φ ( ) 3 ε ( k )(, ). 1 r /4Dεt For ε >> εc, Pε ( r, t) = e. 3/ (4 π Dt) 1 For ε < ε P r t = e 4π r r / ξ c, ε (, ). ε ξε ( ) ( ε ε) ν. c
10
11 nrt f ε μ N r c 1/ ν (, ) = ( ) ( ). 3 l r f x x x 3/ ( ) ( << 1), f( x) const ( x 1) Ν N loc ε μ for ε μ c 3/ ( ) ( c << ) S. Skipetrov, A. Minguzzi, B. van Tiggelen and B.S. PRL 1, (8).
12 Two comments about expansion of a Fermi gas, in a disordered potential. 1. The average density 3 (,, ) ( dr d k P, ) ε ( k ) r R t W k R 1 k mω R = Θ E ( ) d F π m < nrt ˆ(, ) > W The shape of cloud, in the long time large distance limit, is the same as for the BEC in the Gross-Pitaevskii approximation.. Even in the absence of disorder the density pattern, obtained in a single image, will look noisy and grainy. A. Legget, Rev. Mod. Phys. 73, 37 (1). * Atomic Speckles in the presence of disorder. E. Altman, E. Demler, M. Lukin, PRA 7, 1363 (4).
13 Free expansion of a BEC from a disordered trap Strongly anisotropic BEC (quasi 1D) Radial confinement V 1 ρ = ω ρ ω << μ ( ) m, Axial confinement is neglected but there is a potential V(z) Example: assume In equilibrium V ( z) = V cosk z ka << 1, a = μ / mω 1 1 n z V z m g ( ρ, ) = ( μ ( ) ω ρ ) At t= all potentials are switched off and the condensate expands according to: n v 1 + div( nv) =, m + ( mv + gn) = (v = Θ) t t m Initial conditions: n= n ( ρ, z ), v=.
14 The first stage of the expansion, t < t 1/ ω, is dominated by the nonlinearity: rapid radial expansion and, in addition, v ( zt, ) z 1 dv ( z) π = arctanω t Θ ( z) = V( z). mω dz ω Θ ( z) << 1 For -only small effects D. Clement, P. Bouyer, A. Aspect, L. Sanchez- Palencia PRA 77, (8). We consider Θ ( z) > 1 -large effects, atomic caustics The wave function factorizes as Ψ ( ρ, zt, ) =Φ( ρ, t) ψ( zt, ). ψ ( z, t) gives the density at point z, normalized by the radial factor Φ( zt, ) The function ψ(,) z t obeys: ψ ψ i =, with the initial condition t m z ψ (, z t ) = exp( iθ(). z )
15 m im m ψ (,) zt = dz'exp[ ( z z') + iθ (')] z = dze ' πit t πit iϕ ( z', z, t) In full analogy with optics: ϕ( z', z, t) = equation for the rays of atoms z ' ϕ( z', z, t) = z ' Example: condition forcaustics. Θ ( z) = Θ cos k z k Θ m z z' t = m 1 cos kz ' k Θ t = t * = m k Θ -characteristic time for caustic formation.
16 Density at caustics is controlled by the third derivative of the phase and is proportional to Θ 1/3. There is also a more singular caustic (cusp), with density proportional to 1/ Θ.
17 Y.P. Chen, J. Hitchcock, D. Dries, M. Junker, C. Welford, R. Hulet PRA 77, 3363 (8). μ Random potential with correlation length R = 15 μm. = 5.6 ω (c), (d) no disorder. V π =.3μ Θ = V.6 ω π V =.5μ Θ = V 4.4 ω (e), (f) - (g), (h) - (i), (j) - V = μ (beginning of fragmentation)
18 1D GAS, MANY BODY CORRELATIONS. μ ω >> na a >> So far we assumed the condition In the opposite case, na a < 1, the problem becomes strictly one-dimensional: In equilibrium all atoms reside in the ground state, χ ( ρ) of the harmonic oscillator. Many body correlations become important. When the gas is released from the trap, the radial expansion will be governed not by the interaction but by the zero point energy associated with radial motion. Phase imprinting can be done with the help of a short potential pulse: V(,) z t = Θ ( z), τ < t <. τ (the phase can be deterministic or random function of z) The radial part of the wave function, at t=, is χ ( ). ρ j Ψ ( z,... z ; t = ) = exp[ i Θ ( z )] Φ ( z,... z ), The axial part is 1 1 N j N j j 1 where Φ is the ground state, prior to the action of the pulse.
19 At time t=, just after the phase has been impressed, the trapping potential is switched off and the gas undergoes radial expansion. The z-dependent part of the manybody wavefunction evolves according to Ψ ( z,... z ; t) N 1 N i = z Ψ 1 zn t t j= 1 m zj (,... ; ), Ψ ( z,... z ; t = ) = exp[ i Θ ( z )] Φ ( z,... z ) with the initial condition 1 1 N j N j We are interested in the one-dimensional, z-dependent part of the particle density L N n1(,) z t = N Ψ ( z, z... zn; t) dz... dzn F( z, t). L In second quantized form L + Fzt (,) = Ψ ψ (,) zt ψ(,) zt Ψ. N
20 dp F z t = G z t n p π (,) p (,) ( ). L + ( ) ( ) ipz np= dz ψ zψ() e. N m im G z t = d z i ip π it t + Θ p (,) ζ exp[ ( ζ) ( ζ) ζ]. In the mean field approach np ( ) πδ ( p) = and Gp (,) z t reduces to ψ (,) zt considered previously (mean field). In the interacting gas the conditions for caustics are: Θ ξ / R 1, nr Θ 1 1/3 1/3 1 Weak interactions: ξ 1 n 1 Strong interactions (Tonks limit): ξ ξ = 1 αn 1 -healing length. and caustics can be formed. 1 n 1 No caustics.
As in my QM3 and QM4, it is useful to replace the phase by its gradient: In the present, nonlinear case the hydrodynamic equations take a form:
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