Inauguration Meeting & Celebration of Lev Pitaevskii s 70 th Birthday. Bogoliubov excitations. with and without an optical lattice.

Transcription

1 Inauguration Meeting & Celebration of Lev Pitaevskii s 7 th Birthday Bogoliubov excitations with and without an optical lattice Chiara Menotti

2 OUTLINE OF THE TALK Bogoliubov theory: uniform system harmonic trap 1D optical lattice How to reveal Bogoliubov excitations experimentally: collective oscillations sound propagation structure factor and Bragg spectroscopy

3 BOGOLIUBOV THEORY Ĥ µ ˆN = dr ˆΨ [ h2 2 ] 2m + V ˆΨ + g 2 dr ˆΨ ˆΨ ˆΨ ˆΨ µ dr ˆΨ ˆΨ ˆΨ = Nϕ + δψ Diagonalization of Ĥ µ ˆN: [ h2 1 st 2 ] order: 2m + V (r) + gn ϕ(r) 2 µ ϕ(r) = GPE 2 nd order: δψ = ˆbj = u j (r)â j + vj (r)â j Bogoliubov eqs. j j E = E + j hω jˆb jˆb j

4 LINEARIZED GROSS-PITAEVSKII EQUATION ϕ(r, t) = e iµt/ h [ ϕ(r) + u(r)e iωt + v (r)e iωt] ϕ(r, t) i h t = µϕ(r, t) [ h2 2 2m [ h2 2 2m ] + V (r) µ + 2gN ϕ(r) 2 u(r) + gnϕ 2 (r) v = hω u(r) ] + V (r) µ + 2gN ϕ(r) 2 v(r) + gnϕ 2 (r) u(r) = hω v(r) dr [u i (r)u j (r) v i (r)v j (r)] = δ ij

5 BOGOLIUBOV SPECTRUM IN THE UNIFORM SYSTEM hω(q) = [ ] q 2 q 2 2m 2m + 2gn ( q (cq) m ) 2 sound velocity gn c = m healing length h ξ = 4mgn hω for q : phononic regime for q : free particle regime ξ 1 q

6 HARMONIC TRAP Classification with the harmonic oscillator quantum numbers Na/a ho > 1: collisionless hydrodynamics collective oscillations sound waves single particle λ R λ R (i) λ < ξ (ii) λ < d ω ω ho ω ho < ω < µ (i) ω > µ (ii) ω > µ(a ho /R) 4/3

7 MEASUREMENT OF THE COLLECTIVE OSCILLATIONS Dipole oscillation: ω D = ω z Quadrupole oscillation: (ω z ω ) ω Q = 5 2 ω z MIT Group

8 OBSERVATION OF SOUND PROPAGATION Speed of Sound (mm/s) Density (1 14 cm -3 ) M. R. Andrews, D. M. Kurn, H.-J. Miesner, D. S. Durfee, C. G. Townsend, S. Inouye, and W. Ketterle (1997)

9 1D OPTICAL LATTICE Geometry and parameters ( πz ) V (z) = s E R sin 2 d d : lattice spacing q B = hπ/d : Bragg momentum E R = qb 2 /2m : recoil energy x / d Parameters of the problem: s : optical lattice depth gn : interaction parameter where g = 4π h 2 a/m and n is the 3D average density

10 BOGOLIUBOV SPECTRUM IN PRESENCE OF THE LATTICE Bloch ansatz for the Bogoliubov amplitudes: u jq (z) = e iqz/ h ũ jq (z) v jq (z) = e iqz/ h ṽ jq (z) (q = quasi momentum) (j = band index) = ω j(q) 1 hω / E R IMPORTANT!!! The Bogoliubov spectrum is different from the Bloch energy bands, defined as the energy per particle of a condensate moving as a whole with quasi-momentum q q / q B

11 COLLECTIVE OSCILLATIONS IN A 1D OPTICAL LATTICE F. Cataliotti, S. Burger, C. Fort, P. Maddaloni, F. Minardi, A. Trombettoni, A. Smerzi, M. Inguscio, Science (21) M. Krämer, L. Pitaevskii, and S. Stringari, PRL (22) ψ (z) 2 V(z) z / d Dipole oscillation : ω D = Quadrupole oscillation : ω Q = m m ω z m 5 m 2 ω z!! optical lattice + interactions might give rise to dynamical instabilities!! A. Smerzi, A. Trombettoni, A.R. Bishop and P. Kevrekidis, PRL (22)

12 SOUND PROPAGATION IN A 1D OPTICAL LATTICE F. Dalfovo, M. Krämer, C. Menotti, L. Pitaevskii, A. Smerzi and S. Stringari time-dependent GPE in the linear regime work in progress time x / d s = 2, τ = 1 higher bands excitations: hω / E R time q / q B x / d s = 2, τ = 1

13 OPEN QUESTIONS on-set of the non-linear regime behaviour in the non-linear regime dynamical instabilities in large amplitude sound waves?

14 DIRECT MEASUREMENT OF BOGOLIUBOV EXCITATIONS collective oscillations sound propagation structure factor and Bragg spectroscopy: N-photon Bragg scattering S(p, ω) in the uniform system and in a 1D optical lattice measurement of S(p, ω) of a BEC trapped in a harmonic trap under which conditions one measures S(p, ω)?

15 N-PHOTON STIMULATED SCATTERING Ω + ω ka θ p Ω kb p = N h k A k B = N 2 hk sin(θ/2) ω = N ω one measures ω π D.M. Stamper-Kurn, A.P. Chikkatur, A. Görlitz, S. Inouye, S. Gupta, D.E. Pritchard and W. Ketterle, PRL (1999)

16 S(p, ω) IN THE UNIFORM SYSTEM S(p, ω) = n n ˆρ p g 2 δ(ω ω ng ) = = U p + V p 2 δ(ω ω(p)) 1 S(p) = S(p) = p2 /2m = hω(p) p /2mc ; p 1 ; p ξ 1 p suppression at small p indication of the phononic regime

17 PHONONIC CORRELATIONS AND INTERFERENCE S(p) = 1 N g ˆρ(p)ˆρ (p) g ˆρ (p) = k â k+pâp in Bogoliubov theory: F transform of the atomic density operator â p = u pˆb p v pˆb p â p = u pˆbp v pˆb p ˆρ (p) g (â pâ + â â p) g = N(â p + â p ) g S(p) = g (â pâ p + â pâ p + â pâ p + â p â p ) g = (u p + v p ) 2 when p, v p u p = S(p) due to phonons when p, v p = S(p) 1

18 FIRST EXPERIMENTAL RESULTS π Bogoliubov spectrum ω π π ξ Static structure factor π ξ µ π ξ ω ω π µ µ J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson, PRL (22)

19 S(p, ω) OF A BEC IN A 1D OPTICAL LATTICE C. Menotti, M. Krämer, L. Pitaevskii and S. Stringari, cond-mat/ h ω / E j R S(p, ω) = Z j (p)δ(ω ω j (p)) j d/2 [ Z j (p) = u jq (z) + vjq(z) ] e ipz/ h ϕ(z)dz d/2 where q = p + 2lq B with l integer p / q B and q 1 st BZ (i) possibility of exciting low bands with high p (ii) possibility of exciting high excited states with low p (iii) phononic regime at every p/q B even

20 NUMERICAL RESULTS main features: S S Z p / q B Z 1 for p out of 1 st BZ suppression of Z 1 at large p Z 1 = for p = 2lq B q = effect of the lattice q =.8 q B effect of interactions: introduce phononic correlations ( Z 1 for s large) p/q B

21 DEPENDENCE OF Z 1 (p) on s 1 s = 1 s = 2 1 s = p/q B 1 s = p/q B 1 s = p/q B 1 s = p/q B p/q B p/q B

22 DEPENDENCE ON INTERACTIONS AND s Tight binding analytic solution: Z 1 (p) = ε(p) hω(p) e π2 σ 2 p 2 /2d 2 qb 2 ( ) pd ε(p) = 2 δ sin 2 2 h hω(p) = ε(p) [ε(p) + 2κ 1 ] Z p / q B.4 h ω(p) / E R.3 Z 1 (q B ) κδ/(κδ + 1).2.1 ε(p) / E R increasing s : δ Z 1 (q B ) p / q B increasing gn : κ Z 1 (q B ) non interacting (gn = ): κ 1 = Z 1 (q B ) 1

23 CONNECTION WITH EXPERIMENTS C. Tozzo and F. Dalfovo Local Density Approximation and beyond long time measurements and effect of radial excitations what does one really measure?

24 LOCAL DENSITY APPROXIMATION AND BEYOND J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo and F. Dalfovo, PRL (23) real systems are inhomogeneous and finite: Local Density Approximation τ < 2π ω time-dependent GPE with V (r, t) = V ho + θ(t)v B cos(kz ωt) τ > 2π ω

25 EFFECT OF RADIAL EXCITATIONS C.Tozzo and F. Dalfovo, cond-mat/3326 Even Odd Even Odd (1,) (2,) (1,1) (2,1) z 12 (3,) + (4,) + (3,1) (4,1) ω nk [ω ρ ] x A.A. Penckwitt and R.J. Ballagh (21) 1 5 η= k [a ρ ]

26 WHAT DOES ONE REALLY MEASURE IN BRAGG SCATTERING EXPERIMENTS? in a cylindrical condensate (ω z = ): S(p, ω) S( p, ω) = otherwise S(p, ω) S( p, ω) = if ω z << ω : 2 h π p V B 2 τ lim τ P z(p, ω, τ) 2 h π p V 2 B τ ω2 z P z (p, ω, τ) τ dτ P.B. Blakie, R.J. Ballagh and C.W. Gardiner, PRA (22) the cylindrical approximation is good for intermediate τ B the Bogoliubov branches are observable as distinguishable peaks in P z (t)

27 SUMMARY AND OUTLOOK Bogoliubov excitations in various geometries: theory and experiment in the harmonic trap and 1D optical lattices sound propagation in a 1D optical lattice addressing the different excitations through Bragg-scattering exps. calculation of S(p, ω) for a BEC in a 1D optical lattice deep understanding of Bragg experiments in elongated harmonic traps low-q phonon spectroscopy by revealing density fluctuations after expansion