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Nonequilibrium dynamics of ultracold fermions
Theoretical work:Mehrtash Babadi, David Pekker, Rajdeep Sensarma, Ehud Altman, Eugene Demler
$$ NSF, MURI, DARPA, AFOSR
Experiments: T. Esslinger‘s group at ETHW. Ketterle’s group at MIT
Harvard-MIT
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Antiferromagnetic and superconducting Tc of the order of 100 K
Atoms in optical lattice
Antiferromagnetism and pairing at sub-micro Kelvin temperatures
Same microscopic model
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New Phenomena in quantum many-body systems of ultracold atoms
Long intrinsic time scales- Interaction energy and bandwidth ~ 1kHz- System parameters can be changed over this time scale
Decoupling from external environment- Long coherence times
Can achieve highly non equilibrium quantum many-body states
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Outline
Relaxation of doublons in Hubbard modelExpts: Strohmaier et al., arXiv:0905.2963
Quench dynamics across Stoner instabilityExpts: Ketterle et al.,
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Fermions in optical lattice.Decay of repulsively bound pairs
Ref: N. Strohmaier et al., arXiv:0905.2963Experiment: T. Esslinger’s group at ETHTheory: Sensarma, Pekker, Altman, Demler
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Signatures of incompressible Mott state of fermions in optical lattice
Suppression of double occupanciesJordens et al., Nature 455:204 (2008)
Compressibility measurementsSchneider et al., Science 5:1520 (2008)
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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
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Lattice modulation experiments
R. Joerdens et al., Nature 455:204 (2008)
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Fermions in optical lattice.Decay of repulsively bound pairs
Experiments: N. Strohmaier et. al.
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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 determined by many-body dynamics of strongly correlatedsystem of interacting fermions
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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
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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
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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/6tDecay probability
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Doublon decay in a compressible state
Doublon decay
Doublon-fermion scattering
Doublon
Single fermion hopping
Fermion-fermion scattering due toprojected hopping
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Fermi’s golden ruleNeglect fermion-fermion scattering
+ other spin combinations+
2
=
k1 k2
k = cos kx + cos ky + cos kz
Particle-hole emission is incoherent: Crossed diagrams unimportant
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Comparison of Fermi’s Golden rule and self-consistent diagrams
Need to include fermion-fermion scattering
Self-consistent diagrammatics Calculate doublon lifetime from Im Neglect fermion-fermion scattering
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Self-consistent diagrammatics Including fermion-fermion scattering
Treat emission of particle-hole pairs as incoherent include only non-crossing diagrams
Analyzing particle-hole emission as coherent process requires adding decay amplitudes and then calculating net decay rate. Additional diagrams in self-energy need to be included
No vertex functions to justify neglecting crossed diagrams
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Correcting for missing diagrams
type present type missing
Including fermion-fermion scattering
Assume all amplitudes for particle-hole pair production are the same. Assume constructive interferencebetween all decay amplitudes
For a given energy diagrams of a certain order dominate.Lower order diagrams do not have enough p-h pairs to absorb energyHigher order diagrams suppressed by additional powers of (t/U)2
For each energy count number of missing crossed diagrams
R[n0()] is renormalization
of the number of diagrams
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Doublon decay in a compressible state
Comparison of approximations Changes of density around 30%
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Why understanding doublon decay rate is important
Prototype of decay processes with emission of many interacting particles. Example: resonance in nuclear physics: (i.e. delta-isobar)
Analogy to pump and probe experiments in condensed matter systems
Response functions of strongly correlated systems at high frequencies. Important for numerical analysis.
Important for adiabatic preparation of strongly correlated systems in optical lattices
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Quench dynamics across Stoner instability
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Stoner model of ferromagnetismSpontaneous spin polarizationdecreases interaction energybut increases kinetic energy ofelectrons
Mean-field criterion
U N(0) = 1
U – interaction strengthN(0) – density of states at Fermi level
Does Stoner ferromagnetism really exist ?
Counterexample: 1d systems. Lieb-Mattis proof of singlet ground state
Kanamori’s argument: renormalization of U
then
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Magnetic domainscould not be resolved.Why?
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Stoner Instability
New feature of cold atoms systems: non-adiabatic crossing of Uc
Quench dynamics: change U instantaneously.Fermi liquid state for U>Uc. Unstable collective modes
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Outline
Relaxation of doublons in Hubbard modelExpts: Strohmaier et al., arXiv:0905.2963
Quench dynamics across Stoner instabilityExpts: Ketterle et al.,
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Quench dynamics across Stoner instability
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Stoner model of ferromagnetismSpontaneous spin polarizationdecreases interaction energybut increases kinetic energy ofelectrons
Mean-field criterion
U N(0) = 1
U – interaction strengthN(0) – density of states at Fermi level
Does Stoner ferromagnetism really exist ?
Counterexample: 1d systems. Lieb-Mattis proof of singlet ground state
Kanamori’s argument: renormalization of U
then
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Magnetic domainscould not be resolved.Why?
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Stoner Instability
New feature of cold atoms systems: non-adiabatic crossing of Uc
Quench dynamics: change U instantaneously.Fermi liquid state for U>Uc. Unstable collective modes
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Quench dynamics across Stoner instability
Find collective modes
Unstable modes determinecharacteristic lengthscale of magnetic domains
For U<Uc damped collective modes q =’- i ”For U>Uc unstable collective modes q = + i ”
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Quench dynamics across Stoner instability
For MIT experiments domain
sizes of the order of a few F
D=3 D=2
When
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Quench dynamics across Stoner instability
Open questions:
Interaction between modes. Ordering kinetics. Scaling?Classical ordering kinetics: Brey, Adv. Phys. 51:481
Stoner Instability in the Hubbard model?
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Relaxation of doublons in Hubbard modelExpts: Strohmaier et al., arXiv:0905.2963
Quench dynamics across Stoner instabilityExpts: Ketterle et al.,
Conclusions
Experiments with ultracold atoms open interesting questions of nonequilibrium many-body dynamics
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