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 Osaka-City-University, Japan M. Kobayashi and M. Tsubota • Introduction • Model: Gross-Pitaevskii equation • Pure periodic potential • Periodic and trapping potential

1, Introduction Superfluid-Mott insulator transition of trapped alkali atomic BEC in an optical lattice potential Greiner et. al. Nature 415 39 (2002) Potential depth V 0 0 = V 0 Appearance of the interference Disappearance Appearance of the interference Disappearance pattern by the periodic potential of the pattern pattern by the periodic potential of 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 of the amplitude and the phase of the BEC. of 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 2   ∂  2 2 Φ = − ∇ − µ + + Φ Φ i  ( x , t ) V ( x ) g ( x , t ) ( x , t )   ∂ t 2 m     Φ ( x , t ) : Macroscopi c wave function of BEC V ( x ) : External potential µ : Chemical potential g : Coupling constant Numerical calculation of this equation about two- Numerical calculation of this equation about two- dimensional system dimensional system

3,Pure periodic potential 2 2 = − V ( x ) V cos ( Kx ) cos ( Ky ) 0 We look for the ground state by introducing the We look for the ground state by introducing the dissipative term. dissipative term. 2   ∂  2 2 (i - γ )  Φ ( x , t ) = − ∇ − µ + V ( x ) + g Φ ( x , t ) Φ ( x , t )   ∂ t 2 m    

Ground state 2 2  K Potential = E R 2 2 2 π m V ( x ) = − V cos ( Kx ) cos ( Ky ) 0 2 2 π = gK / E 1 R 2 ∫∫ Φ ( x ) d x = 1 1 - site V 0 2 Φ ( x ) = = = = V / E 5 V / E 25 V / E 50 V / E 75 0 R 0 R 0 R 0 R Localization of the amplitude

The phase of the ground state [ ] Φ ( x ) = Φ ( x ) exp i θ ( x ) V 0 2 Φ ( x ) Phase θ ( x ) = = = = V / E 5 V / E 25 V / E 50 V / E 75 0 R 0 R 0 R 0 R Localization of the phase: breaking of the long-range correlation

Lowest excitation Hartree-Fock-Bogoliubov equation -i ω t ∗ i ω t Φ ( x ) → Φ ( x ) + ϕ ( x ), ϕ (x) = u ( x ) e + v ( x ) e 2    2 2 − ∇ − µ + Φ V ( x ) g ( x )   u ( x ) u ( x )     2 m   =  ω     2 v ( x ) v ( x )        ∗ 2 2 − g Φ ( x ) ∇ + µ − V ( x )     2 m 10 8 Localization of the  ω lowest 6 phase ⇒ Finite excitation 4 energy: breaking 2 superfluidity 0 0 50 100 150 200 V 0

Energy gap of Mott-insulator A local interference pattern by the potential gradient Greiner et. al. Nature 415 39 (2002) 10 8  ω lowest 6 4 ∆ 2 E : Energy difference 0 0 50 100 150 200 V 0 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 2 2 2 2 = − + α + V ( x ) V cos ( Kx ) cos ( Ky ) ( x y ) 0 T 2 2  K E = R 2 π m = V / E 5 0 R 2 2 gK / π E = 1 R 2 π α T = 1 2 K E R Ground state 2 Φ ( x ) V / E = 5 V / E = 50 V / E = 75 0 R 0 R 0 R

The phase of the ground state The localization of the phase 2 Φ ( x ) Phase θ ( x ) V / E = 5 = V / E 50 V / E = 75 0 R 0 R 0 R

Removing only the 2 2  K E = R 2 π m trapping potential 2 2 π = gK / E 1 R 12 V 0 /E R =30 V 0 /E R =100 10 V 0 /E R =50 V 0 /E R =200 center d x | Φ ( x )| 2 8 6 4 2 0 V / E = 50 V / E = 75 0 0.5 1 1.5 2 0 R 0 R t Even after removing the trapping potential, the localized wave function does not expand but oscillate.

Removing the combined potential 2 2  K E = R 2 π m 2 2 π = gK / E 1 R = V / E = 50 V / E = 75 V / E 120 0 R 0 R 0 R At the deep periodic potential, the interference pattern disappears.

Conclusions 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. • 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.