Nonequilibrium dynamics of ultracold fermions Theoretical work: Mehrtash Babadi, David Pekker, Rajdeep Sensarma, Ehud Altman, Eugene Demler $$ NSF, MURI,

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

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

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

Outline

Relaxation of doublons in Hubbard modelExpts: Strohmaier et al., arXiv:0905.2963

Quench dynamics across Stoner instabilityExpts: Ketterle et al.,

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

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)

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 experiments

R. Joerdens et al., Nature 455:204 (2008)

Fermions in optical lattice.Decay of repulsively bound pairs

Experiments: N. Strohmaier 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 determined by many-body dynamics of strongly correlatedsystem of interacting fermions

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/6tDecay 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+

2

=

k1 k2

k = cos kx + cos ky + cos kz

Particle-hole emission is incoherent: Crossed diagrams unimportant

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

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

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

Doublon decay in a compressible state

Comparison of approximations Changes of density around 30%

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

Quench dynamics across Stoner instability

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

Magnetic domainscould not be resolved.Why?

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

Outline

Relaxation of doublons in Hubbard modelExpts: Strohmaier et al., arXiv:0905.2963

Quench dynamics across Stoner instabilityExpts: Ketterle et al.,

Quench dynamics across Stoner instability

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

Magnetic domainscould not be resolved.Why?

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

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 ”

Quench dynamics across Stoner instability

For MIT experiments domain

sizes of the order of a few F

D=3 D=2

When

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?

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