Doron Cohen Ben-Gurion University

Transcription

1 BEC dynamics in a few site systems Doron Cohen Ben-Gurion University Maya Chuchem (BGU/Phys) [,3,4] Tsampikos Kottos (Wesleyan) [3,4,5,6] Katrina Smith-Mannschott (Wesleyan) [3,4] Moritz Hiller (Gottingen) [3,4,5,6] Amichay Vardi (BGU/Chem) [,2,3] Erez Boukobza (BGU/Chem) [,2] $BSF, $DIP, $FOR76 [] Occupation dynamics & fluctuations (PRL 29) [4] Landau-Zener dynamics (PRL 29) [2] Occupation dynamics & fluctuations (PRA 29) [5] Quantum stirring (EPL 28) [3] Occupation dynamics & fluctuations (arxiv 2) [6] Quantum stirring (PRA 28)

2 H = i=,2 The Bose-Hubbard Hamiltonian (BHH) for a dimer [ E iˆn i + U ] 2 ˆn i(ˆn i ) K 2 (â 2â + â â2) N particles in a double well is like spin j = N/2 system H = EĴz + UĴ 2 z KĴx + const K = hopping U = interaction E = E 2 E = bias u NU K, E K Classical phase space H(θ, ϕ) = NK [ ] 2 2 u(cos θ)2 cos θ sin θ cos ϕ H(ˆn, ϕ) = (similar to Josephson/pendulum Hamiltonian) Ĵ z = (N/2) cos(θ) = ˆn = occupation difference Ĵ x (N/2) sin(θ) cos(ϕ), ϕ = relative phase Rabi regime: u < (no islands) Josephson regime: < u < N 2 (sea, islands, separatrix) Fock regime: u > N 2 (empty sea) Assuming u> and < c Sea, Islands, Separatrix c = ( u 2/3 ) 3/2 A c 4π ( u 2/3) 3/2

3 WKB quantization (Josephson regime) h = Planck cell area in steradians = 4π N+ ( ) A(E ν ) = 2 + ν h ν =,, 2, 3,... ω(e) de dν = [ h A (E)] ω K K = Rabi Frequency ω J NUK = u ω K ω + NU = u ω K ω x [ log ( )] N 2 ω J u

4 The preparations Eigenstates E ν are like strips along contour lines of H. Coherent state θϕ is like a minimal Gaussian wavepacket. Fock state n is like equi-latitude annulus. Fock n= preparation - exactly half of the particles in each site Fock coherent θ= preparation - all particles occupy the left site Coherent ϕ= preparation - all particles occupy the symmetric orbital Coherent ϕ=π preparation - all particles occupy the antisymmetric orbital

5 Wavepacket dynamics MeanField theory (GPE) = classical evolution of a point in phase space SemiClassical theory = classical evolution of a distribution in phase space Quantum theory = recurrences, fluctuations (WKB is very good).5.5 n/j n/j φ / π.5.5 φ / π Any operator  can be presented by the phase-space function A W(Ω)  = trace[ˆρ Â] = dω h ρ W(Ω)A W (Ω)

6 P (E) = the LDOS of the preparations.5 e n/j.5 π φ / π TwinFock preparation Zero preparation Pi preparation Edge preparation P(E ν ).2. P(E ν ).5.25 P(E ν ).2. P(E ν ) [ E (scaled) ( 2E NK ) 2 ] / E (scaled) BesselI[ E E - NU ] E (scaled) BesselK[ E E x NU ] 5 5 exp [ N E (scaled) ( E E ) ] 2 x ω J

7 M = [ ν M = the participation number P(E ν ) 2 ] = number of participating levels in the LDOS M / N /2.5.5 M / N / M N /2 vs u N for: Zero, Pi, Edge. u/n. u/n In the semiclassical analysis there is scaling with respect to (u/n) /2 which is [the width of the wavepacket] / [the width of the separatix] M [ ( N log u )] u [Pi] (quasi periodic large fluctuations) M [ ( N log u )] N [Edge] (easy to get the classical limit)

8 Recurrences and fluctuations S = J /(N/2) = (S x, S y, S z ) = Bloch vector OccupationDifference = (N/2) S z OneBodyPurity = (/2) [ + S x 2 + S y 2 + S z 2] FringeVisibility = [ S x 2 + S y 2] /2 Spectral analysis of the fluctuations: dependence on u and on N, various preparations. Zero prep Pi prep Edge prep S x.96 S x.4 S x t (scaled) t (scaled) t (scaled)

9 Temporal behavior Zero prep Pi prep Edge prep S x.96 S x.4 S x t (scaled) t (scaled) t (scaled) C x (ω).6.4 C x (ω).2 C x (ω) ω/ω J ω/ω J ω/ω J P(E ν ).5 P(E ν ).2 P(E ν ) E (scaled) E (scaled) 5 5 E (scaled)

10 The spectral content of S x w osc /w J. u/n ω osc 2ω J [Zero] ω osc [ log ( )] N 2ωJ [Pi] u ω osc 2 [ log ( )] N 2ωJ [Edge] u ) /2 2ωJ [u N] ω osc ( u N

11 Fluctuations of S x.5 Naive expectation: phase spreading diminishes coherence. In the Fock regime S x [Leggett s phase diffusion ] In the Josephson regime S x is determined by u/n. S x -.5 S x /3 [TwinFock] S x exp[ (u/n)] [Zero] S x 4/ log [ 32 (u/n)] [Pi] -. u/n RMS [ A t ] = [ M Ccl(ω)dω ] /2 N /4 * RMS[S x ].. RMS[S x ].. ~N -.9 ~N -.26 N.. u/n N =, N = 5, N =. RMS [S x (t)] N /4 [ Edge] RMS [S x (t)] (log(n)) /2 [ Pi] TwinFock: Self induced coherence leading to S x /3. Zero: Coherence maintained if u/n < (phase locking). Pi: Fluctuations are suppressed by u. Edge: Fluctuations are suppressed by N (classical limit).

12 The many body Landau-Zener transition 8 N=3, k=, U=.27 preparation adiabatic 6 4 E diabatic Energy c c Dynamical scenarios: adiabatic/diabatic/sudden sudden

13 Occupation Statistics PN; Var(n)x4 <n> adiabatic E n (t) PN Var(n) (a) sweep rate e-6.. sudden (b) (c) Y.2. u=.9, N=8 u=2.5, N= u=4.5, N=3 u=4.5, N=; SC u=4.5, N= u=4.5, N= X Sweep Rate (NK) 2 Adiabtic-diabatic (quantum) crossover Diabatic-sudden (semiclassical) crossover Sub-binomial scaling of Var(n) versus n D I A B A T I C 2 K eff/ N S U D D E N P R O C E S S NUK 2 K NUK Q u a n t u m A D I A B A T I C NK K K/N K/N K NK NK 2 U

14 Quantum Stirring in a 3 site system a k k 2 2 k c b k k 2 Control parameters: X = ( k 2 k ) X 2 = E (E =E 2 =) U = the inter-atomic interaction k c X = (X, X 2 ) Ĥ = 2 i= E i n i + U 2 2 i= ˆn i (ˆn i ) k c (ˆb ˆb 2 + ˆb 2ˆb ) k (ˆb ˆb + ˆb ˆb ) k 2 (ˆb ˆb 2 + ˆb 2ˆb ) The induced current: I = G E (G = G 2 ) The pumped particles: Q = Idt = G dx (per cycle)

15 Stirring of BEC a k k 2 b k k 2 strong attractive interaction: classical ball dynamics negligible interaction ( U κ/n): mega-crossing 2 k c k c weak repulsive interaction: gradual crossing strong repulsive interaction (U N κ): sequential crossing E n a 3 2 U= 3 b U= c U= d U= n i n n n G e+8 5e G G anal

16 Results for the geometric conductance G(R) = N (k 2 k2 2 )/2 [( ) 2 +2(k +k 2 ) 2 ] 3/2 G(J) [ k k 2 k +k 2 ] 3U G(F ) = ( k k 2 k +k 2 ) Nn= (δ n ) 2 [( n ) 2 +(2δ n ) 2 ] 3/2, mega crossing gradual crossing sequential crossing where: X2 ds X2 R = Rabi regime (U κ/n) J = Josephson regime (κ/n U Nκ) R B F = Fock regime (U Nκ) X X Observation: It is possible to pump Q N per cycle. En n=3 n=2 n= n= off plane section along X= X2

17 Summary Semiclassical and WKB analysis of the dynamics Occupation statistics in a time dependent scenario The adiabatic / diabatic / sudden crossovers Quantum stirring: mega / gradual / sequential crossings E preparation adiabatic diabatic c c sudden