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