Tuning the Quantum Phase Transition of Bosons in Optical Lattices

Transcription

1 Tuning the Quantum Phase Transition of Bosons in Optical Lattices Axel Pelster 1. Introduction 2. Spinor Bose Gases 3. Periodically Modulated Interaction 4. Kagome Superlattice 1

2 1.1 Quantum Phase Transition 2

3 1.2 Theoretical Description Bose-Hubbard Hamiltonian: Ĥ BH = t i,j â iâj + i [ ] U 2 ˆn i(ˆn i 1) µˆn i, ˆn i = â iâi 3

4 Quantum Phase Diagram: 1.3 Ginzburg-Landau Theory Santos and Pelster, PRA 79, (2009) Error bar: Extrapolated strong-coupling series Black line: Mean-field Blue line: 3rd strong-coupling order Red line: Landau theory Blue dots: Monte-Carlo data Extension to higher orders: Teichmann, Hinrichs, Holthaus, and Eckardt, PRB 79, (R) (2009) Hinrichs, Pelster, and Holthaus, Appl. Phys. B 113, 57 (2013) Excitation Spectra: Bradlyn, Santos, and Pelster, PRA 79, (2009) Graß, Santos, and Pelster, PRA 84, (2011) 4

5 Tuning the Quantum Phase Transition of Bosons in Optical Lattices Axel Pelster 1. Introduction 2. Spinor Bose Gases Mobarak and Pelster, LPL 10, (2013) 3. Periodically Modulated Interaction 4. Kagome Superlattice 5

6 2.1 Bose-Hubbard Model for Spin-1 Boson Ĥ BH = Ĥ (0) + Ĥ (1) = [ U0 2 ˆn i(ˆn i 1) + U ] 2 2 (Ŝ2 i 2ˆn i) µˆn i ηŝiz i = J ij â iαâjα ij α Ĥ (0) Ĥ (1) ˆn i = αˆn iα = αâ iαâiα ] [ ] [â iα,â jβ = â iα,â jβ = 0 [ ] â iα,â jβ = δ α,β δ i,j Ŝi = α,βâ iα F αβâ iβ > 0, Anti-ferromagnetic,e.g. 23 Na U 2 = < 0, Ferromagnetic, e.g. 87 Rb J, if i;j arenextneigbors J ij = 0, otherwise 6

7 2.2 Atomic Limit: Mott Phases 1 Η 0.2U 0 Anti ferro 0.5 Site-diagonal: Ĥ(0) = iĥ(0) i Ĥ (0) i S i,m i,n i = E (0) S i,m i,n i S i,m i,n i E (0) S i,m i,n i = µn i + U 0 2 n i (n i 1) U2 U0 0 1,1,1 0,0,2 2,2,2 1,1,3 3,3,3 Μ U 0 0,0,4 2,2,4 4,4,4 + U 2 2 [ Si (S i +1) 2n i ] ηmi 0.5 Ferro Μ Η U 0 Minimization of energy 1. η > 0 = m i = S i 2. Two competing effects: U 2 > 0 ˆ= minimize S i η > 0 ˆ= maximize S i 7

8 2.3 Quantum Phase Boundary (η = 0.2U 0 ) 8

9 2.4 Magnetic Superfluid Phases (η = 0.2U 0 ) 9

10 Tuning the Quantum Phase Transition of Bosons in Optical Lattices Axel Pelster 1. Introduction 2. Spinor Bose Gases 3. Periodically Modulated Interaction Wang, Zhang, Santos, Eggert, and Pelster, PRA 90, (2014) 4. Kagome Superlattice 10

11 3.1 Floquet Theory Ĥ(t) = Ĥ BH +Acos(ωt) i g(ˆn i ), ˆn i = â iâi Ĥ BH = i f(ˆn i ) t ij â iâj, f(ˆn i ) = U 2 (ˆn 2 i ˆn i ) µˆni U,t ω : time average over driving period Arimondo et al., Adv. Atom. Mol. Opt. Adv. 61, 515 (2012) Ĥ eff = i f(ˆn i ) t ij â i J 0(G(ˆn i,ˆn j ))â j G(ˆn i,ˆn j ) = g(ˆn j) g(ˆn j 1)+g(ˆn i ) g(ˆn i +1) ω 11

12 3.2 Examples Shaken lattice: Ĥ(t) = Ĥ BH +Acos(ωt) i ˆn i Modulated interaction: Ĥ(t) = ĤBH + A 2 cos(ωt) i (ˆn 2 i ˆn i ) Ĥ eff = i f(ˆn i ) tj 0 ( A ω ) ij â iâj Ĥ eff = i f(ˆn i ) t ij â i J 0 ( ) A ω (ˆn j ˆn i ) â j = renormalized hopping Eckardt et al., PRL 95, (2005) Lignier et al., PRL 99, (2007) Zenesini et al., PRL 102, (2009) = conditional hopping Rapp et al., PRL 109, (2012) Hubbard, PRSA 276, 238 (1963) Experimental verification: still open 12

13 t U strong-coupling method: 3.3 Second Order (2D) Μ U J 0 (A/ hω)=0.4 Freericks and Monien, PRB 53, 2691 (1996) Quantum Monte Carlo: Sandvik, PRB 59, R14157 (1999) second-order Landau theory: Wang, Zhang, Eggert, and Pelster, PRA 87, (2013) = error less than 6 % for 0 < A ω < x 2 2.4, J 0 (x 2 ) = 0 13

14 Ĥ eff = i 3.4 Effective Bose-Hubbard Model [ U 2 ˆn i(ˆn i 1) µˆn i ] t ij â i J 0 ( ) A ω (ˆn j ˆn i ) â j Approximation due to uniform renormalization of critical hopping with driving amplitude [ U ] Ĥ(x) = i 2 ˆn i(ˆn i 1) µˆn i tλ(x) <ij>â iâj λ(x) = 1 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x , x = A ω t U Ĥ eff Ĥ(x) Μ U ρ ρ s ρ eff ρ s eff µ/u 3D: A = ω QMC, 2D: t = 0.05U,A = 0.4 ω 14

15 3.5 Summary Bose-Hubbard model with periodic modulated interaction: { 1 } Ĥ eff = i 2 [U + Acos(ωt)] ˆn i(ˆn i 1) µˆn i t <ij>â iâj Approximation due to renormalization of hopping with driving amplitude Bose-Hubbard model with renormalized hopping: Ĥ(x) = i = same universality class = same critical exponents [ ] ( ) U A 2 ˆn i(ˆn i 1) µˆn i tλ ω Hinrichs, Pelster, and Holthaus, Appl. Phys. B 113, 57 (2013) <ij> â iâj 15

16 Tuning the Quantum Phase Transition of Bosons in Optical Lattices Axel Pelster 1. Introduction 2. Spinor Bose Gases 3. Periodically Modulated Interaction 4. Kagome Superlattice Zhang, Wang, Eggert, and Pelster, PRB 92, (2015) 16

17 4.1 Proposed Kagome Superlattice y (a) B A C x y x C A B NSW NSW NLW NLW (b) A B 6 8 µ/v Normal SW (NSW) Enhanced LW (ELW) 2(c) C B Optical potential with k = 3π/2λ LW,λ LW = 1064 nm,γ = V E /V 0 : V c /V 0 = γ 2 1+4γcos( 3kx)cos(ky)+2cos(2ky) 2cos(4ky) 4cos(2 3kx)cos(2ky) Bose-Hubbard model with µ = 4(γ 1)V 0 > 0: Ĥ = t (â iâj +â i â j )+ U ˆn i (ˆn i 1) µ ˆn i µ i 2 i,j i i i Aˆn 17

18 4.2 Quantum Phase Diagram QMC: stochastic cluster series expansion Sandvik, PRB 59, 14157(R) (1999) Landau theory Santos and Pelster, PRA 79, (2009) Wang, Zhang, Eggert, Pelster, PRA 87, (2013) SSD (striped solid) phase: fractional filling of 1/3 and 4/3 18

19 4.3 Insulating Phases QMC: βu = 300, L = 9, t/u = Total density: ρ = (ρ A +ρ B +ρ C )/3 Density difference: ρ = ρ A (ρ B +ρ C )/2 19

20 4.4 Anisotropic Superfluidity Superfluid density via winding number = Wx/y 2 /4βt ρ x/y s Pollock and Ceperley, PRB 36, 8343 (1987) Total superfluid density: ρ + s = (ρ x s +ρ y s)/2 }{{}}{{} Superfluid density difference: ρ s = ρ x s ρ y s ρ x s < ρ y s A preferred Effective square lattice ρ x s > ρ y s A full, B/C preferrred No supersolid due to artificial symmetry-breaking 20

21 4.5 Time-of-Flight Expansion Anisotropic parameter: I ± = ρx s ρ y s ρ x s +ρ y s Sign change indicates superfluid density is tensor Experimental detection by TOF absorption pictures 21