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

meandering structure because

• f phase fluctuations

paralell condensates interference pattern pattern integrated along direction x

Experimental results

Origin of interference picture

localized particle propagation for time t density measurement

 

2 2 2 1 2 1 2

, 2 cos m md x t a a a a x t t                   

     

1 2

,0 / 2 / 2 x a x d a x d       

complex coefficients

 

 

2

/2 /2 2 1 2

, 2

p i t ipx ipd ipd m

dp x t e e a e a e  

 

 



/2 1,2 1,2 i

a a e

 



Characteristics of interference fringes

     

, 1 cos md f x z c x z x t               

Experimentally measured density profiles fitted as

Gaussian envelope relative phase of condensates initial separation time of flight

Integrated pattern:

 

 

2 /2 2 /2

1

x x

L i x L x

dxc x e L

 



reduction of contrast with increasing integration length BKT transition can be detected Random variable:

     

2 2 † 1 2

ˆ

L

L dxa x a x   

ladder operators

• f the condensates

Analogous quantity:

1 2

a a



Calculation of momenta

Ignoring shot noise terms:

             

2 2 † † † 1 2 1 1 2 2 1 2 2 1 1 2 1 2

ˆ

L L L L

L dx dx a x a x a x a x dx dx a x a x   

   

annihilation operators of the two condensates identical condensates

Higher momenta:

         

2 2 † † 1 1 1 1

ˆ ... ... ... ... ...

L L n n n n n

L dx dx dx dx a x a x a x a x       

Luttinger liquid theory:

                   

† , † † 1 1 † †

... ...

i j i j n n i j i j i j i j

a x a x a x a x a x a x a x a x a x a x

 

        

 

 

ˆ

ˆ

i x

a x e



 

fluctuating phase homogeneous density

Calculation of momenta

Correlations for OBC:

   

 

1/2 ˆ ˆ K i x y h

e x y

 





         

healing length

 

2 2 2 1/ 2 1/ 2

ˆ ,

n n K K n h

L A Z A L   



 

     

 

,

1/ 1 1 2 1 , 1 1 1 , , , 1

... ... ... ...

i j i j i j i j i j i j

K i j i j i j i j n n i j i j G u u G v v G u v K n

u u v v Z du dv u v du dv e

 

   

          

   

• OBC (as before):

 

, log G x y x y  

• PBC:

 

 

1 , log sin G x y x y          

finite T can be treated as well Luttinger parameter

Probability density function

     

2 2 2 2

ˆ , ˆ

n n n

L Z d W Z L      



  

PDF

Weak interaction limit:

2 2

, 1

n n

Z K Z   

   

1 W     

narrow peak Tonks-Girardeau limit:

1 1 1/ 2

1,

K

K Z dx dy x y



   

 

1 2 2

!

n n n

Z n d e Z







 

 

W e  





exponential distribution

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



h

L 

independent pairs of condensates with random phase Interference:

mean displacement of a random walk random vectors of equal length

exponential distribution

Hubbard-Stratonovich transformation

Eigenvalue equation:

       

1

,

f f

dyG x y y G f x   



Spectral decomposition:

       

,

f f f

G x y G f x y   

     

 

 

   

 

2 2 2 1 1

1 1 2 2 1

... ...

n n f i f i f i f i f i i

G f u v u v K n n

Z du dv e

   

 

        

     

Identity:

2 2

2 2

1 2

i x x

e dxe

 



   





decoupling of terms with different

 

 

,

f i f j

u v  

Probability density function

 

 

 

 

2 2

2 1

2

f

t n n f n f f f

dt e Z g t g t 

    

           

 

 

 

       

2

1 2

f f f f

G f G f t x x K K f

g t dxe

          

  

Negative eigenvalues of

 

G f

 

 

 

 

f f

g t g t



 

 

 

 

2 2

2 1

2

f

t f f f

dt e W g t    

    

                 

 

Numerical evaluation: Monte Carlo method

Numerical results

T 

temperature OBC PBC

5 K  3 K  2 K 

Distribution in weak interaction limit

   

1 W     

narrow peak Rescaled variable:

 

1 x K    

PDF of for PBC:

x

     

2sin 2 , 2 cos 2

f x

fx fx    

1 K 

 

2 2 1, 2, 1

1 1 2 2

f f f

t t Kf 

 

   



Gaussian random variables eigenfunctions of G(x,y)

x random variable: Gumbel distribution

distribution in extreme value statistics Rare phase fluctuations dominate the interference picture.

Outlook

Time of flight measurement, dimensional BEC

1 d 

distribution of particle number in a cell around R Gaussian approximation: 



 

   

2 2

2 †

ˆ ˆ ˆ , , ,

x R R

n R R dxe x t x t  

    

  

   

ˆ ˆ , 2

ik x k

dk x t e c t    

Free evolution during expansion:

   

2

2

ˆ ˆ

i k t m k k

c t e c





0: t 

Luttinger liquid theory momenta of

   

† 1 2

ˆ ˆ ,0 ,0 x x  

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