potential and its interference pattern Osaka-City-University, Japan - - PowerPoint PPT Presentation

Superfluid-insulator transition of a Bose-Einstein condensation in a periodic potential and its interference pattern

• Introduction
• Model: Gross-Pitaevskii equation
• Pure periodic potential
• Periodic and trapping potential

Osaka-City-University, Japan

• M. Kobayashi and M. Tsubota

1, Introduction

Superfluid-Mott insulator transition of trapped alkali atomic BEC in an optical lattice potential

Greiner et. al. Nature 415 39 (2002)

depth Potential V

0 =

V

Appearance of the interference Appearance of the interference pattern by the periodic potential pattern by the periodic potential Disappearance Disappearance

• f the pattern
• f the pattern

Disappearance of the long-range coherence by the deep periodic potential: Superfluid-Mott insulator transition

Summary of this work

• We discuss this system by using the Gross-Pitaevskii

We discuss this system by using the Gross-Pitaevskii (GP) equation with a periodic potential. (GP) equation with a periodic potential.

• Since the GP equation assumes the BEC, it is

Since the GP equation assumes the BEC, it is impossible to discuss the Mott insulator phase. impossible to discuss the Mott insulator phase.

• However the GP equation gives the detailed structure

However the GP equation gives the detailed structure

• f the amplitude and the phase of the BEC.
• f the amplitude and the phase of the BEC.
• Changing the potential depth, we investigate what

Changing the potential depth, we investigate what happens to the BEC. happens to the BEC.

2, Model: the GP equation

constant Coupling : potential Chemical : potential External : ) ( BEC

• f

function wave c Macroscopi : ) , ( ) , ( ) , ( ) ( 2 ) , ( i

2 2 2

g V t t t g V m t t µ Φ Φ         Φ + + µ − ∇ − = Φ ∂ ∂ x x x x x x  

Numerical calculation of this equation about two- Numerical calculation of this equation about two- dimensional system dimensional system

3,Pure periodic potential

We look for the ground state by introducing the We look for the ground state by introducing the dissipative term. dissipative term.

) ( cos ) ( cos ) (

2 2

Ky Kx V V − = x

) , ( ) , ( ) ( 2 ) , ( )

• (i

2 2 2

t t g V m t t x x x x Φ         Φ + + µ − ∇ − = Φ ∂ ∂ γ  

Ground state

) ( cos ) ( cos ) ( Potential

2 2

Ky Kx V V − = x

∫∫

= Φ = π π =

site

• 1

2 2 2 2 2 2

1 d ) ( 1 / x x

R R

E gK m K E 

2

) (x Φ

5 / =

R

E V 25 / =

R

E V 50 / =

R

E V 75 / =

R

E V

V Localization of the amplitude

The phase of the ground state

2

) (x Φ

V

) ( Phase x θ

[ ]

) ( i exp ) ( ) ( x x x θ Φ = Φ 5 / =

R

E V 25 / =

R

E V 50 / =

R

E V 75 / =

R

E V

Localization of the phase: breaking of the long-range correlation

Lowest excitation

Hartree-Fock-Bogoliubov equation

      ω =                   − µ + ∇ Φ − Φ + µ − ∇ − + = ϕ ϕ + Φ → Φ

∗ ω ∗ ω

) ( ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( 2 e ) ( e ) ( (x) ), ( ) ( ) (

2 2 2 2 2 2 i

• i

x v x u x v x u V m x g x g V m x v x u x

t t

   x x x x

Localization of the phase⇒Finite excitation energy: breaking superfluidity

2 4 6 8 10 50 100 150 200 ωlowest V0

Energy gap of Mott-insulator

A local interference pattern by the potential gradient

Greiner et. al. Nature 415 39 (2002)

2 4 6 8 10 50 100 150 200 ωlowest V0

difference Energy : E ∆

A energy gap is observed in Mott insulator phase Is there any relation ⇒ to the excitation energy gap given by Hartree-Fock-Bogoliubov equation?

4, Periodic and trapping potential

) ( ) ( cos ) ( cos ) (

2 2 2 2

y x Ky Kx V V

T

+ α + − = x

1 5 /

2 2

= α π =

R T R

E K E V

Ground state

5 / =

R

E V

2

) (x Φ

50 / =

R

E V 75 / =

R

E V

1 /

2 2 2 2 2

= π π =

R R

E gK m K E 

The phase of the ground state

The localization of the phase

) ( Phase x θ

2

) (x Φ

5 / =

R

E V 50 / =

R

E V 75 / =

R

E V

Removing only the trapping potential

1 /

2 2 2 2 2

= π π =

R R

E gK m K E 

50 / =

R

E V 75 / =

R

E V

2 4 6 8 10 12 0.5 1 1.5 2

V0/ER=30 V0/ER=50 V0/ER=100 V0/ER=200

center dx |Φ(x)|2

t

Even after removing the trapping potential, the localized wave function does not expand but oscillate.

Removing the combined potential

At the deep periodic potential, the interference pattern disappears.

1 /

2 2 2 2 2

= π π =

R R

E gK m K E 

50 / =

R

E V 75 / =

R

E V 120 / =

R

E V

Conclusions

• In the periodic potential, the phase of ground state

In the periodic potential, the phase of ground state localizes in each site and the energy gap appears in localizes in each site and the energy gap appears in the lowest excitation. the lowest excitation.

• After removing only the trapping potential, the

After removing only the trapping potential, the localized wave function does not expand but oscillate localized wave function does not expand but oscillate in each site. in each site.

• After removing the combined potential, the localized

After removing the combined potential, the localized wave function does not make interference pattern. wave function does not make interference pattern. Using the GP equation, we find the signals concerned Using the GP equation, we find the signals concerned with the superfluid–insulator transition. with the superfluid–insulator transition.