Tilted Bose Hubbard model

Transcription

1 Tilted Bose Hubbard model Phase transition in 1D Institut für theoretische Physik III Supervisor: Alexander Krupp Physik Journal 12 (2013) Nr.1 p.27 27/05/2013

2 Outline 1 2 The Tilted Bose Hubbard Model (BHM) 3 4

3 Set of problems with no viable classical computing simulation Controllable environment for quantum simulations Quantum magnetism High temperature superconductivity Quantum computing

4 Optical lattices Spatially periodic structure of electric fields Usually generated by superposition of laser beams Jaksch, D. et al. (2005) Annals of Physics 315(1), 52 79

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6 The Tilted Bose Hubbard Model (BHM) H = t i,j (ˆb iˆb j + ˆb jˆb i ) + U 2 j ˆb jˆb jˆb jˆbj Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

7 The Tilted Bose Hubbard Model (BHM) H = t i,j E j (ˆb iˆb j + ˆb jˆb i ) + U 2 e r jˆb jˆb j j ˆb jˆb jˆb jˆbj Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

8 The Tilted Bose Hubbard Model (BHM) a) b) V 0 x t U 10 2 U [a.u.] ε V [a.u.] t [a.u.] -2-3 Jaksch, D. et al. (1998) Physical Review Letters pp. 5 8

9 The Tilted Bose Hubbard Model (BHM) Mott insulator Minimize H in limit U t Average site occupation n 0 Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

10 The Tilted Bose Hubbard Model (BHM) Mott insulator Tunneling generates new state Energy difference U E Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

11 The Tilted Bose Hubbard Model (BHM) Mott insulator Define dipole operator ˆd j = ˆbjˆb j+1 n0 (n 0 + 1) with the conditions ˆd j ˆd j 1 ˆd j ˆd j ˆd j+1 ˆd j+1 = 0 Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

12 The Tilted Bose Hubbard Model (BHM) O(t 2 ) Dipole of length 2 Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

13 The Tilted Bose Hubbard Model (BHM) Rewrite Hamiltonian H = t i,j E j with dipole operators (ˆb iˆb j + ˆb jˆb i ) + U 2 e r jˆb jˆb j j ˆb jˆb jˆb jˆbj H d = t n 0 (n 0 + 1) j ( ˆd j + ˆd j ) + (U E) j ˆd j ˆd j

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15 The Tilted Bose Hubbard Model (BHM) Mott Insulator (MI) Broken symmetry phase (BSP) Tilt increase Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

16 The Tilted Bose Hubbard Model (BHM) Eigenenergies of H d dependend on n 0 and λ = U E t E/t λ N = 8, n 0 = 1 Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

17 The Tilted Bose Hubbard Model (BHM) Scaling E N z = N 1 at λ c Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

18 No tunneling event spin Tunneling event spin S j z = 1 2 ˆd j ˆd j Sx j = 1 ( ˆd j 2 + ˆd ) j Sy j = i ( ˆd j 2 ˆd ) j Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

19 Mott Insulator (MI) Broken symmetry phase (BSP) Tilt increase Sachdev, S. et al. (2002) Physical Review B 8120, 1 17

20 Constrains ˆd j ˆd j 1 Fulfilled by definition ˆd ˆd j j ˆd ˆd j+1 j+1 = 0 Add term to H d J ˆd ˆd j+1 j+1 ˆd ˆd j j = J ( Sz j+1 ( ) /2) 1 S j z 1 /2

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22 1D Ising chain with longitudinal and transverse field equivalent to 2D H s = J j (S j zs j+1 z h x S j x h z S j z) No analytical solution Ovchinnikov, A. A. (2003) Physical Review B 68.

23 Phase transition h z = 1 /J, h x = 2 2 t /J h z Multicritical point Paramagnetic 0.25 Antiferromagnetic h x Simon, J. et al. (2011) Nature 472(7343),

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25 d CCD e g f b y a z c Bakr, Waseem S. et al. (2009) Nature 462(8482)

26 5 μm 640 nm Bakr, Waseem S. et al. (2009) Nature 462(8482)

27 Spin chain Atom position in tilted lattice Single site readout (odd/even) a Δ <0: paramagnet o o o o o b Δ 0 c Δ >0: antiferromagnet e e e e e d Spin = = = Simon, J. et al. (2011) Nature 472(7343),

28 a c 680 nm y x b PM E/U AF Simon, J. et al. (2011) Nature 472(7343),

29 S j z = 1 2 pj odd p odd E/U Time (ms) Simon, J. et al. (2011) Nature 472(7343),

30 1.0 p odd Paramagnet Antiferromagnet p odd E/U Time (ms) E/U Simon, J. et al. (2011) Nature 472(7343),

31 p odd Energy/J Time (ms) h z Simon, J. et al. (2011) Nature 472(7343),

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33 Summary MI phase Set of resonant states in 1D MI ground state BSP Mapping to results p odd Paramagnet Antiferromagnet E/U Simon, J. et al. (2011) Nature 472(7343),

34 and outlook results in agreement with theory Optical lattices as a promising quantum simulator Higher dimensions Different lattice geometries Wikimedia Commons Valence bond solid

35 Thank you for your attention