in interference experiments Izabella Lovas EQP seminar 2014.12.01. - PowerPoint PPT Presentation

Full statistics of density ripples in interference experiments Izabella Lovas EQP seminar 2014.12.01.

References • Dissertation of Adilet Imambekov • Schumm et al., Nature Physics 1, 57 (2005) • Hadzibabic et al., Nature 441, 1118 (2006) • Gritsev et al., Nature Physics 2, 705 (2006) • Imambekov et al., Phys. Rev. A 77, 063606 (2008) • Kitagawa et al., New J. Phys. 13, 073018 (2011)

Content • Low dimensional gases • Interference experiment with paralell low dimensional condensates interference fringes • Distribution of fringe amplitude: - Strong and weak interaction limits - Numerical results • Particle number distribution in time of flight measurements

Introduction Low dimensional gases: enhanced phase fluctuations quasi-condensate regime: suppressed density fluctuations, large phase fluctuations Experimental realization: anisotropic harmonic trap Investigation of phase fluctuations: interference measurements

Experimental setup paralell condensates pattern integrated along direction x meandering structure because interference of phase fluctuations pattern

Experimental results

Origin of interference picture propagation for time t localized particle density measurement              x ,0 a x d / 2 a x d / 2 1 2    i /2 a a e complex coefficients 1,2 1,2 2 p    dp    i t     ipx ipd /2 ipd /2 , 2 m x t e e a e a e  1 2 2     m md   2   2  2        x t , a a 2 a a cos x  1 2 1 2     t t

Characteristics of interference fringes Experimentally measured density profiles fitted as initial separation     cos md            f x z ,  1 c x z x      t Gaussian envelope time of flight relative phase of condensates 2 1      L /2  Integrated pattern: x i x dxc x e 2  L L /2 x x reduction of contrast with increasing integration length BKT transition can be detected ladder operators of the condensates  Analogous quantity: a a 1 2   2       L 2  ˆ † Random variable: L dxa x a x 1 2 0

Calculation of momenta annihilation operators of the two condensates Ignoring shot noise terms: L L L L                   2 2   † †  † ˆ L dx dx a x a x a x a x dx dx a x a x 1 2 1 1 2 2 1 2 2 1 1 2 1 2 0 0 0 0 identical condensates Higher momenta: L L           2    2 n      ˆ † † L ... dx ... dx dx ... dx a x ... a x a x ... a x 1 n 1 n 1 n 1 n 0 0     ˆ fluctuating phase    ˆ i x Luttinger liquid theory: a x e homogeneous density       † a x a x           i j , i j  † † ... ... a x a x a x a x         1 n 1 n     † † a x a x a x a x   i j i j i j i j

Calculation of momenta 1/2 K    healing       ˆ ˆ    i x y   Correlations for OBC:  h length e     x y    2 n       ˆ 2 n 2 1/ K 2 1/ K L A Z , A L 0 2 n 0 h Luttinger 1/ K     1 1 u u v v parameter      i j i j i j i j Z ... du ... dv 2 1   n n u v 0 0 i j , i j            1 1 1     G u u , G v v , G u v ,  i j i j i j   K i j i j i j , ... du ... dv e 1 n 0 0     • OBC (as before): G x y , log x y finite T can be     1 treated as well      • PBC:   G x y , log sin x y   

Probability density function PDF   2  ˆ    L Z        n 2 n , d W   n 2  ˆ Z L 2 0 Z    Weak interaction limit: 2 n K , 1 n Z 2     W      narrow peak 1 1 1       1/ K  Tonks-Girardeau limit: K 1, Z dx dy x y 2 0 0 1 Z       n 2 n n ! d e n Z 2 0   e     exponential distribution W

Physical interpretation of limiting distributions Weak interactions: phase coherence length >> system size constant relative phase along the condensates, perfect interference pattern  Strong interactions: phase coherence only over length h L  independent pairs of condensates with random phase h Interference: mean displacement of a random walk random vectors of equal length exponential distribution

Hubbard-Stratonovich transformation 1             Eigenvalue equation: dyG x y , y G f x f f 0             Spectral decomposition: G x y , G f x y f f f         2    G f           n n 2 2 1 1             u v u v f i f i f i f i   f 2 K  i 1 i 1  Z ... du ... dv e 2 n 1 n 0 0  1    2    2 Identity: 2 i x x 2 e dxe  2        decoupling of terms with different u , v f i f j

Probability density function    2 2   t dt e f             n n f    Z g t g t     2 n f f   f 1 2          G f G f       2      1   t x x     f f f f 2 K K   g t dxe f 0              Negative eigenvalues of G f g t g t f f    2 2   t dt e f         2     f         W g t       f   f 1 2   Numerical evaluation: Monte Carlo method

Numerical results T  temperature 0 OBC PBC K  3 K  5 K  2

Distribution in weak interaction limit     W      narrow peak 1     K   eigenfunctions of G(x,y) Rescaled variable: x 1           2sin 2 , 2 cos 2 f x fx fx PDF of for PBC: x   1        K  2 2 1 0 1 t t 2  1, f 2, f f 1 2 Kf Gaussian random variables x random variable: Gumbel distribution distribution in extreme value statistics Rare phase fluctuations dominate the interference picture.

Outlook d  Time of flight measurement, dimensional BEC 1 distribution of particle number in a cell around R    Gaussian approximation:           2  2    † ˆ ˆ ˆ x R 2 R n R , R dxe x t , x t ,    dk      ˆ ˆ ik x x t , e c t  k 2       2 ˆ ˆ i k t 2 m Free evolution during expansion: 0 c t e c k k     t    † ˆ ˆ Luttinger liquid theory momenta of 0: x ,0 x ,0 1 2 Expression for PDF using Hubbard-Stratonovich transformation numerical evaluation: Monte Carlo method

Summary • Quasi-condensate phase in low dimensional gases • Interference experiment with low dimensional condensates distribution of interference fringes -Weak interactions: converges to a single Dirac-delta peak Gumbel distribution after rescaling - Tonks-Girardeau limit: exponential distribution - Some numerical results • Future calculation: particle number distribution in time of flight measurements