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Nonequilibrium dynamics of ultracold atoms in optical lattices
David Pekker, Rajdeep Sensarma, Takuya Kitagawa,Susanne Pielawa, Vladmir Gritsev, Mikhail LukinEugene Demler
$$ NSF, AFOSR, MURI, DARPA,
Collaboration with experimental groups of I. Bloch, T. Esslinger, J. Schmiedmayer
Harvard University
Nonequilibrium quantum dynamics of many-body systems
Big Bang and Inflation. Structure of the universe. From formation of galaxies to fluctuations in theCMB radiation.
Jet production in particle decay.Heavy Ion collisions.
Solid state devices
c
Nonequilibrium quantum dynamics in “artificial” many-body systems
Photons in stronglynonlinear medium
Example: photon crystallizationin nonlinear 1d waveguidesChang et al (2008)
Strongly correlated systemsof ultracold atoms
Outline
Fermions in optical lattice. Decay of repulsively bound pairs
Ramsey interferometry and many-body decoherence
Lattice modulation experiments
Fermions in optical lattice.Decay of repulsively bound pairs
Fermions in optical lattice.Decay of repulsively bound pairs
Experimets: T. Esslinger et. al.
Relaxation of repulsively bound pairs in the Fermionic Hubbard model
U >> t
For a repulsive bound pair to decay, energy U needs to be absorbedby other degrees of freedom in the system
Relaxation timescale is important for quantum simulations, adiabatic preparation
Energy carried by
spin excitations ~ J =4t2/U
Relaxation requires creation of ~U2/t2
spin excitations
Relaxation of doublon hole pairs in the Mott state
Relaxation rate
Very slow Relaxation
Energy U needs to be absorbed by spin excitations
Doublon decay in a compressible state
Excess energy U isconverted to kineticenergy of single atoms
Compressible state: Fermi liquid description
Doublon can decay into apair of quasiparticles with many particle-hole pairs
Up-p
p-h
p-h
p-h
Doublon decay in a compressible state
To calculate the rate: consider processes which maximize the number of particle-hole excitations
Perturbation theory to order n=U/tDecay probability
Doublon decay in a compressible state
Doublon decay
Doublon-fermion scattering
Doublon
Single fermion hopping
Fermion-fermion scattering due toprojected hopping
Fermi’s golden ruleNeglect fermion-fermion scattering
+ other spin combinations
Crossed diagram are not important
+
2
=
k1 k2
k = cos kx + cos ky + cos kz
Self-consistent diagrammatics Neglect fermion-fermion scattering
Calculate doublon lifetime from Im
Self-consistent diagrammatics Including fermion-fermion scattering
For fermions it is easy to include non-crossing diagrams
Diagrams not includedDiagrams included
Undercounting decay channels for doublons
No vertex functions to justify neglecting crossed diagrams
Correcting for missing diagrams
type present type missing
Self-consistent diagrammatics Including fermion-fermion scattering
Each diagram allows additional particle-hole pair production.Decay rate is determined by the number of particle-hole pairs.Correct the number of decay channels by counting the number of diagrams
0 – characteristic energy
of particle-hole pairs
Np – number of diagrams included
N – total number of diagrams
Self-consistent diagrammatics Including fermion-fermion scattering
Correcting for missing diagrams
Particle-hole self-energy Doublon life-time
Typical energy transferaround 8 t
Doublon decay in a compressible state
Doublon decay with generation of particle-hole pairs
Ramsey interferometry and many-body decoherence
Quantum noise as a probe of non-equilibrium dynamics
Interference between fluctuating condensates
1d: Luttinger liquid, Hofferberth et al., 2008
x
z
L [pixels]
0.4
0.2
00 10 20 30
middle Tlow T
high T
2d BKT transition: Hadzibabic et al, Claude et al
Time of flight
low T
high T
BKT
Distribution function of interference fringe contrastHofferberth et al., 2008
Comparison of theory and experiments: no free parametersHigher order correlation functions can be obtained
Quantum fluctuations dominate:asymetric Gumbel distribution(low temp. T or short length L)
Thermal fluctuations dominate:broad Poissonian distribution(high temp. T or long length L)
Intermediate regime:double peak structure
Can we use quantum noise as a probe of dynamics?
Working with N atoms improves the precision by .
Ramsey interference
t0
1
Atomic clocks and Ramsey interference:
Two component BEC. Single mode approximation
Interaction induced collapse of Ramsey fringes
time
Ramsey fringe visibility
Experiments in 1d tubes: A. Widera et al. PRL 100:140401 (2008)
Spin echo. Time reversal experiments
Single mode approximation
Predicts perfect spin echo
The Hamiltonian can be reversed by changing a12
Spin echo. Time reversal experiments
No revival?
Expts: A. Widera, I. Bloch et al.
Experiments done in array of tubes. Strong fluctuations in 1d systems.Single mode approximation does not apply.Need to analyze the full model
Interaction induced collapse of Ramsey fringes.Multimode analysis
Luttinger model
Changing the sign of the interaction reverses the interaction part of the Hamiltonian but not the kinetic energy
Time dependent harmonic oscillatorscan be analyzed exactly
Low energy effective theory: Luttinger liquid approach
Time-dependent harmonic oscillator
Explicit quantum mechanical wavefunction can be found
From the solution of classical problem
We solve this problem for each momentum component
See e.g. Lewis, Riesengeld (1969) Malkin, Man’ko (1970)
Interaction induced collapse of Ramsey fringesin one dimensional systems
Fundamental limit on Ramsey interferometry
Only q=0 mode shows complete spin echoFinite q modes continue decay
The net visibility is a result of competition between q=0 and other modes
Decoherence due to many-body dynamics of low dimensional systems
How to distinquish decoherence due to many-body dynamics?
Single mode analysisKitagawa, Ueda, PRA 47:5138 (1993)
Multimode analysisevolution of spin distribution functions
T. Kitagawa, S. Pielawa, A. Imambekov, et al.
Interaction induced collapse of Ramsey fringes
Fermions in optical lattice. Lattice modulation experiments as a probe of the Mott
state
Signatures of incompressible Mott state of fermions in optical lattice
Suppression of double occupancies T. Esslinger et al. arXiv:0804.4009
Compressibility measurementsI. Bloch et al. arXiv:0809.1464
Lattice modulation experiments with fermions in optical lattice.
Related theory work: Kollath et al., PRA 74:416049R (2006) Huber, Ruegg, arXiv:0808:2350
Probing the Mott state of fermions
Lattice modulation experimentsProbing dynamics of the Hubbard model
Measure number of doubly occupied sites
Main effect of shaking: modulation of tunneling
Modulate lattice potential
Doubly occupied sites created when frequency matches Hubbard U
Lattice modulation experimentsProbing dynamics of the Hubbard model
R. Joerdens et al., arXiv:0804.4009
Mott state
Regime of strong interactions U>>t.
Mott gap for the charge forms at
Antiferromagnetic ordering at
“High” temperature regime
“Low” temperature regime
All spin configurations are equally likely.Can neglect spin dynamics.
Spins are antiferromagnetically ordered or have strong correlations
Schwinger bosons and Slave Fermions
Bosons Fermions
Constraint :
Singlet Creation
Boson Hopping
Schwinger bosons and slave fermions
Fermion hopping
Doublon production due to lattice modulation perturbation
Second order perturbation theory. Number of doublons
Propagation of holes and doublons is coupled to spin excitations.Neglect spontaneous doublon production and relaxation.
d
h Assume independent propagation of hole and doublon (neglect vertex corrections)
= +
Self-consistent Born approximation Schmitt-Rink et al (1988), Kane et al. (1989)
Spectral function for hole or doublon
Sharp coherent part:dispersion set by J, weight by J/t
Incoherent part:dispersion
Propagation of holes and doublons strongly affected by interaction with spin waves
Schwinger bosons Bose condensed
“Low” Temperature
Propogation of doublons and holes
Spectral function: Oscillations reflect shake-off processes of spin waves
Hopping creates string of altered spins: bound states
Comparison of Born approximation and exact diagonalization: Dagotto et al.
“Low” Temperature
Rate of doublon production
• Low energy peak due to sharp quasiparticles
• Broad continuum due to incoherent part
• Spin wave shake-off peaks
“High” Temperature
Atomic limit. Neglect spin dynamics.All spin configurations are equally likely.
Aij (t’) replaced by probability of having a singlet
Assume independent propagation of doublons and holes.Rate of doublon production
Ad(h) is the spectral function of a single doublon (holon)
Propogation of doublons and holesHopping creates string of altered spins
Retraceable Path Approximation Brinkmann & Rice, 1970
Consider the paths with no closed loops
Spectral Fn. of single hole
Doublon Production Rate Experiments
Ad(h) is the spectral function of a single doublon (holon)
Sum Rule :
Experiments:
Most likely reason for sum rule violation:nonlinearity
The total weight does not scale quadratically with t
Lattice modulation experiments. Sum rule
SummaryFermions in optical lattice. Decay of repulsively bound pairs
Ramsey inter-ferometry in 1d.Luttinger liquidapproach to many-body decoherence
Lattice modulation experiments as aprobe of AF order
T >> TN T << TN
Harvard-MIT
Thanks to