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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
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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”
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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
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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
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Motivation: Quantum gas microscope
Bakr et al., science 2010Sherson et. al. Nature 2010
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Spin systems• AFM phase of Hubbard model (with Fermions)
– Not yet: difficult to quench spin entropy
• Today: alternative approach
initial
final
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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 !
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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
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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
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Scaling of energy and #qpScaling of observables:
measure properties of excited qp’s
Number of modes excited (Fidelity)
Excess Energy
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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
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Resonant manifold:
Ising-like quantum phase transition
Strongly Tilted BH Model
EU
Paramagnet Anti-Ferromagnet
Sachdev, Sengupta, Girvin PRB (2002)
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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
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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)
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Plan• Integrable vs. non-Integrable
• Numerical Methods: ED & t-MPS
• Theory of finite size crossover scaling
• Numerical Results
• Experimental observables
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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
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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
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t-MPS aka t-DMRG• Trial wave function approach
• Pictorial representation
• Systematic way to increase accuracy– increase bond dimension c
Tr …
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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
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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
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Landau Zener• Where did power law come from?
stop after QCP
stop on QCP
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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
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Most universal protocol: PM to QCPObservables:
Residual energy Log-Fidelity
Recover power-laws predicted for integrable models
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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
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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
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Experimental observables: PM to QCP
• Other observables: – Order parameter– Full distribution function
Missing even parity sites Spin-Spin correlations
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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
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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
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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
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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
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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
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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
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Plan• Test methods in equilibrium
– phase boundary (test against QMC)– defect density
• Run Dynamics– fast (compared to 1/J)– slow (compared to 1/J)
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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
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Defect Density
• Methods converge for large system/cluster size• Biggest discrepancies near phase transition• Both ED and CMF qualitatively OK for “fast” dynamics
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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
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Comparison for rapid ramps (CMF)Planck constant theory vs. experiment
Short times: similar dynamics
Higgs like oscillations – see Sat. talk
Longer times: divergence
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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
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Protocol:Ramp deep into Mott Insulator Start from QCP
(ms)-1 (ms)-1
(ms)-1 (ms)-1
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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
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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)
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Truncated Wigner evolution (in progress)
• Symmetry breaking in MI-SF• Configurations as product forms
• Initial configuration from Wigner distribution
• Dynamics: Schrodinger evolution
color –
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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