Scaling behavior in ramps of the Bose Hubbard Model

1. Scaling behavior in ramps of the Bose Hubbard Model D. Pekker Caltech Without tilt: B. Wunsch, E. Manousakis, T. Kitagawa, E.A. Demler With tilt: K. Sengupta, B. K. Clark, M. Kolodrubetz

2. Ultracold Atoms for Quantum Simulator • R.P.Feynman Int. J. Theor. Phys. 21, 467 (1982). • use quantum simulator for (computationally) hard many-body systems major current effort to realize • Access to new many body phenomena • Long intrinsic time scales • interaction energy and bandwidth ~ kHz • system parameters easily tunable on timescales • Decoupling from environment • Long coherence times • Can achieve highly non-equilibrium quantum many body states • Ultracold atom toolbox • Optical lattices • Quantum gas microscope • Traps • Feshbach resonances • dipolar interactions • artificial gauge fields • artificial disorder “quantum Lego”

3. Optical Lattices • Retro-reflected laser – standing wave • AC-Stark shift – atoms attracted to maxima E2 (or minima depending on detuning) • Multiple lasers to make a 2D and 3D lattices Atoms in optical lattice Laser Mirror

4. Similarities: CM and Cold Atoms 400 300 200 temperature (K) Atoms in optical lattice 100 0 Antiferromagnetism and pairing at sub-micro Kelvin temperatures doping Antiferromagnetic and superconducting Tc of the order of 100 K Same model: http://www.wmi.badw-muenchen.de/FG538/projects/P4_crystal_growth/index.htm

5. Motivation: Quantum gas microscope Bakr et al., science 2010 Sherson et. al. Nature 2010

6. Spin systems • AFM phase of Hubbard model (with Fermions) • Not yet: difficult to quench spin entropy • Today: alternative approach initial final

7. Mission of this work: • Ultimate Goal: Understanding of Dynamics near QCP • Parametric tuning • Near (quantum) phase transitions • Universal character of dynamics • This talk • Try the program in “artificial spin” system ? • Methods for studying dynamics • How to observe scaling experimentally ? • time scales, finite size effects, trap inhomogeneity • criticality in dynamics is easier to see than in equilibrium !

8. Outline Part I: Introduction • Why universal scaling? Part II: Strongly tilted Bose Hubbard model • Mapping to Spin model • Methods: ED and tMPS • Dynamics: finite-size scaling crossover from Universal to Landau-Zener Part III: Superfluid-to-Mott ramps • Methods: ED, MF, CMF, MF+G, TWA • Static results • Dynamics: Fast (non-universal) &Slow (universal scaling) regimes E U

9. Passing through QCP: Universal Scaling Quantum Kibble-Zurek: non-adiabaticity of individual quasi-particle modes QCP tuning parameter rate of ramp dynamic exp x exp. *Usual assumption: defect production dominated by long wavelength low energy modes see, e.g. De Grandi, Polkovnikov

10. Scaling of energy and #qp Scaling of observables: measure properties of excited qp’s Number of modes excited (Fidelity) Excess Energy

11. Outline Part I: Introduction • Why universal scaling? Part II: Strongly tilted Bose Hubbard model • Map to Spin model • Methods: ED and tMPS • Dynamics: finite-size scaling crossover from Universal to Landau-Zener Part III: Superfluid-to-Mott ramps • Methods: ED, MF, CMF, MF+G, TWA • Static results • Dynamics: Fast (non-universal) &Slow (universal scaling) regimes E U

12. Strongly Tilted BH Model Resonant manifold: Ising-like quantum phase transition U E Paramagnet Anti-Ferromagnet Sachdev, Sengupta, Girvin PRB (2002)

13. Experimental Realization • Initial realization • Greiner, Mandel, Esslinger, Haensch, Bloch, Nature (2002) • detect gap U • Single site resolution • Simon, Bakr, Ma, Tai, Preiss, Greiner, Nature (2011) • tilted 1D chains • transition from PM to AFM

14. Tilted BH: mapping to spin model • Map BH to spin model Boson has not moved E Boson has moved U Forbidden configuration Hamiltonian: Phase Diagram: Ising universality class PM AFM • Due to constraint, not-integrable * we use units where J=1 Sachdev, Sengupta, Girvin PRB (2002)

15. Plan • Integrable vs. non-Integrable • Numerical Methods: ED & t-MPS • Theory of finite size crossover scaling • Numerical Results • Experimental observables

16. Integrable vs. non-integrable • QP interactions lead to relaxation in non-integrable models • What happens to power laws --- anomalous scaling exponents ? single q-p energies

17. Time evolution PM to QCP ramp: • Protocol • start deep in PM • evolve to the QCP • Exact Diag. • initial ground state • evolve with gap AFM PM t

18. t-MPS aka t-DMRG • Trial wave function approach • Pictorial representation • Systematic way to increase accuracy • increase bond dimension c Tr …

19. time evolution in t-MPS • step 1: apply the time evolution operator • step 2: project out forbidden configurations • step 3: reduce bond dimension • converge time step & bond dimension

20. Finite size effects Universal Scaling regimes • Fast Ramp • Non-universal: excite all q-p modes • Slow Ramp • KZ-like scaling: excite only long wavelength modes • Very Slow Ramp • LZ scaling: excite only longest wavelength mode (set by system size) single q-p energies const. v1/2 tuning prameter log nex v2 log v

21. Landau Zener • Where did power law come from? stop on QCP stop after QCP

22. Finite-size scaling function Universal Scaling regimes • Length scale • Dimensionless parameter • Modification to the scaling functions: • 1D Ising const. v1/2 log nex v2 log v correlation length exponent dynamic exponent

23. Most universal protocol: PM to QCP Observables: Residual energy Log-Fidelity Recover power-laws predicted for integrable models

24. Ramps PM to AFM v1/2 Residual energy Log-Fidelity • Why change in Residual energy power-law? v1/2 adiabatic non-adiabatic universal adiabatic non-universal excitations-> sites n=1 QCP

25. Protocol • For ramps that stop just beyond QCP, there can be a crossover of power laws • Stopping on QCP minimizes oscillations that obscure scaling • Most universal ramps: stop on QCP

26. Experimental observables: PM to QCP Missing even parity sites Spin-Spin correlations • Other observables: • Order parameter • Full distribution function

27. Conclusions: tilted bosons • Universal dynamics • First demonstration in non-integrable system • Finite sized systems • Universal crossover function from LZ to KZ scaling • Protocol is important • Scaling in smaller systems & shorter timescales • Experimentally feasible length and timescales • Easier to observe criticality than in equilibrium systems, no need to equilibrate! • Application: quantum emulators Thank: A. Polkovnikov MK, DP, BKC, KS, arXiv:1106.4031 C. De Grandi, A. Polkovnikov, A. W. Sandvik, arXiv:1106.4078

28. Outline Part I: Introduction • Why universal scaling? Part II: Strongly tilted Bose Hubbard model • Mapping to Spin model • Methods: ED and tMPS • Dynamics: finite-size scaling crossover from Universal to Landau-Zener Part III: Superfluid-to-Mott ramps • Methods: ED, MF, CMF, MF+G, TWA • Static results • Dynamics: Fast (non-universal) &Slow (universal scaling) regimes E U

29. Parametric ramp of 2D bosons (no tilt) trap tuning of optical lattice intensity Parametrically ramp from SF to MI at rate v Main Questions: timescales for “defect” production “Defect”: p-h symmetric point site with even # Bakr et. al. Science 2010

30. Spin-1 Model Truncated Hilbertspace Effective spin Hamiltonian Advantage: properties similar to BH model, but easier to analyze Defect density Bose-Hubbard Spin-1 • Same phase transitions • No p-h asymmetry • smaller Hilbert space • spin wave analysis Huber, Altman 2007

31. Methods we tried • Exact Diagonalization • small system sizes • no phase transitions • Mean Field • no low energy excitations • Cluster Mean Field • like ED, except self-consistent neighbohrs • some “low” energy excitations • Mean Field + Gaussian fluctuations • long wavelength modes: can capture scaling • modes non-interacting • Truncated Wigner • Similar to MFT+G, can capture instabilities

32. Mean field + fluctuations ba MF: b0 bf We need two vectors perpendicular to : & mean field quadratic fluctuations Dynamics: step 1 step 2 dynamics of quadratic modes Huber, Altman 2007

33. Plan • Test methods in equilibrium • phase boundary (test against QMC) • defect density • Run Dynamics • fast (compared to 1/J) • slow (compared to 1/J)

34. Validation: phase boundary using CMF Bose Hubbard Model Spin-1 Model • MF, CMF, MF+G: phase boundary • MF tends to favor ordered phase – too much SF • larger clusters more MI • qualitative agreement with QMC

35. Defect Density • Methods converge for large system/cluster size • Biggest discrepancies near phase transition • Both ED and CMF qualitatively OK for “fast” dynamics

36. Rapid ramping Eigenvalues: 3x3 Bose Hubbard • Describe short wavelength states • Exact digitalization of 3x3 system with PBC • Quasi-particles • Deep in SF: phase and amplitude • Deep in Mott: doubles and holes • Persistent gap ~ U • Fast ramp time scale ~1/U • Shift relative to experiment • Missing long wavelength modes • Inhomogeneity due to trap & disorder Defect production in ramp 1/U 1/J

37. Comparison for rapid ramps (CMF) Planck constant theory vs. experiment Short times: similar dynamics Higgs like oscillations – see Sat. talk Longer times: divergence

38. Slow ramping: MF+G (ms)-1 Each k: 2 parametrically driven SHO amplitude & phase Crossover into scaling regime tramp ~ 10/J Defect density saturates for shallow ramps (ms)-1

39. Protocol: Start from QCP Ramp deep into Mott Insulator (ms)-1 (ms)-1 (ms)-1 (ms)-1

40. Ramping time scales Fast dynamics Scaling with MFT exponents Scaling with RGexponents • Fast ramps: excite all modes (few site physics) • Slow ramps: excite long wavelength modes • Very slow ramps: excite very long wavelength modes – finite size effects Still missing: effects of the trap

41. CMF for inhomogeneous systems Time evolve each 2x2 plaquette [consistently] in the mean-field of its neighbors and m(r) from trap (Total: 30x30 plaquettes) Fitting parameter: size of Mott Shells Slow mass flow: hard to remove defects from center Final Density (after adiabatic ramp) chemical potential Initial Density 1/U 1/J

42. Truncated Wigner evolution (in progress) • Symmetry breaking in MI-SF • Configurations as product forms • Initial configuration from Wigner distribution • Dynamics: Schrodinger evolution color –

43. Conclusions: Mott-SF transition • High energy modes play an important role in fast ramps • Time scale 1/U appears • Critical scaling only for slow ramps and large systems • Optimized protocol useful for observing scaling • Cluster Mean Field is an effective tool for analyzing dynamics in inhomogeneous systems • Mass flow important: hard to remove defects from center