Scaling behavior in ramps of the Bose Hubbard Model

D. Pekker

Without tilt: B. Wunsch, E. Manousakis, T. Kitagawa, E.A. DemlerWith tilt: K. Sengupta, B. K. Clark, M. Kolodrubetz

Caltech

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”

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

LaserMirrorAtoms in optical lattice

Similarities: CM and Cold Atoms

Antiferromagnetic and superconducting Tc of the order of 100 K

Atoms in optical lattice

Antiferromagnetism and pairing at sub-micro Kelvin temperatures

Same model:

http://www.wmi.badw-muenchen.de/FG538/projects/P4_crystal_growth/index.htm

doping

tem

pera

ture

(K)

0

100

200

300

400

Motivation: Quantum gas microscope

Bakr et al., science 2010Sherson et. al. Nature 2010

Spin systems• AFM phase of Hubbard model (with Fermions)

– Not yet: difficult to quench spin entropy

• Today: alternative approach

initial

final

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 !

OutlinePart 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

EU

Passing through QCP: Universal Scaling

Quantum Kibble-Zurek: non-adiabaticity of individual quasi-particle

modes

*Usual assumption: defect production dominated by long wavelength low energy modes

rate of ramp

tuning parameter

see, e.g. De Grandi, Polkovnikov

x exp.dynamic exp

QCP

Scaling of energy and #qpScaling of observables:

measure properties of excited qp’s

Number of modes excited (Fidelity)

Excess Energy

OutlinePart 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

EU

Resonant manifold:

Ising-like quantum phase transition

Strongly Tilted BH Model

EU

Paramagnet Anti-Ferromagnet

Sachdev, Sengupta, Girvin PRB (2002)

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

Tilted BH: mapping to spin model• Map BH to spin model

EU

Boson has not moved

Boson has moved

Forbidden configuration

AFM PM

Due to constraint, not-integrable

Phase Diagram:Hamiltonian:Ising universality class

* we use units where J=1 Sachdev, Sengupta, Girvin PRB (2002)

Plan• Integrable vs. non-Integrable

• Numerical Methods: ED & t-MPS

• Theory of finite size crossover scaling

• Numerical Results

• Experimental observables

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

Time evolution• Protocol

– start deep in PM– evolve to the QCP

• Exact Diag.– initial ground state

– evolve with

AFM

PM

PM to QCP ramp: ga

p

t

t-MPS aka t-DMRG• Trial wave function approach

• Pictorial representation

• Systematic way to increase accuracy– increase bond dimension c

Tr …

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

Finite size effects

tuning prameter

single q-p energies

• 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)lo

g n e

x

log v

const.v1/2

v2

Universal Scaling regimes

Landau Zener• Where did power law come from?

stop after QCP

stop on QCP

Finite-size scaling function• Length scale

• Dimensionless parameter

• Modification to the scaling functions:

• 1D Ising

log

n ex

log v

const.v1/2

v2

Universal Scaling regimes

correlation length exponentdynamic

exponent

Most universal protocol: PM to QCPObservables:

Residual energy Log-Fidelity

Recover power-laws predicted for integrable models

Ramps PM to AFM

• Why change in Residual energy power-law?

Residual energy Log-Fidelity

adiabatic non-adiabaticuniversal

adiabaticnon-universal

QCP

excitations->sites n=1

v1/2

v1/2

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

Experimental observables: PM to QCP

• Other observables: – Order parameter– Full distribution function

Missing even parity sites Spin-Spin correlations

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. PolkovnikovMK, DP, BKC, KS, arXiv:1106.4031 C. De Grandi, A. Polkovnikov, A. W. Sandvik, arXiv:1106.4078

OutlinePart 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

EU

Parametric ramp of 2D bosons (no tilt)

tuning of optical lattice intensity

trap

Bakr et. al. Science 2010

Parametrically ramp from SF to MI at rate v

Main Questions: timescales for “defect” production

“Defect”:

site with even #

p-h symmetricpoint

Spin-1 Model

Advantage: properties similar to BH model, but easier to analyze

Huber, Altman 2007

Truncated Hilbert space

Effective spin Hamiltonian

Defect density

– smaller Hilbert space – spin wave analysis

– Same phase transitions

– No p-h asymmetry

Bose-Hubbard Spin-1

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

Mean field + fluctuations

mean field quadratic fluctuations

b0

ba

bf

MF:

We need two vectors perpendicular to : &

Dynamics: step 1 step 2dynamics of quadratic modes

Huber, Altman 2007

Plan• Test methods in equilibrium

– phase boundary (test against QMC)– defect density

• Run Dynamics– fast (compared to 1/J)– slow (compared to 1/J)

Validation: phase boundary using CMF

Spin-1 ModelBose Hubbard Model

• MF, CMF, MF+G: phase boundary• MF tends to favor ordered phase – too much SF• larger clusters more MI• qualitative agreement with QMC

Defect Density

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

Rapid ramping– 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

Eigenvalues: 3x3 Bose Hubbard

Defect production in ramp

1/U 1/J

Comparison for rapid ramps (CMF)Planck constant theory vs. experiment

Short times: similar dynamics

Higgs like oscillations – see Sat. talk

Longer times: divergence

Slow ramping: MF+G

Each k: 2 parametrically driven SHOamplitude & phase

Crossover into scaling regimetramp ~ 10/J

Defect density saturates for shallow ramps

(ms)-1

(ms)-1

Protocol:Ramp deep into Mott Insulator Start from QCP

(ms)-1 (ms)-1

(ms)-1 (ms)-1

Ramping time scales

• 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

Fastdynamics

Scaling with MFT exponents

Still missing: effects of the trap

Scaling with RG exponents

CMF for inhomogeneous systemsTime 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

1/U 1/J

chemical potential Initial Density Final Density (after adiabatic ramp)

Truncated Wigner evolution (in progress)

• Symmetry breaking in MI-SF• Configurations as product forms

• Initial configuration from Wigner distribution

• Dynamics: Schrodinger evolution

color –

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