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