Nonequilibrium dynamics of ultracold atoms in optical lattices David Pekker, Rajdeep Sensarma, Takuya Kitagawa, Susanne Pielawa, Vladmir Gritsev, Mikhail Lukin Eugene 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 the CMB radiation. Jet production in particle decay. Heavy Ion collisions. Solid state devices

c Nonequilibrium quantum dynamics in “artificial” many-body systems Photons in strongly nonlinear medium Example: photon crystallization in nonlinear 1d waveguides Chang et al (2008) Strongly correlated systems of 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

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 absorbed by other degrees of freedom in the system Relaxation timescale is important for quantum simulations, adiabatic preparation

 Energy carried by spin excitations ~ J =4t 2 /U  Relaxation requires creation of ~U 2 /t 2 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 is converted to kinetic energy of single atoms Compressible state: Fermi liquid description Doublon can decay into a pair of quasiparticles with many particle-hole pairs U p-pp-p p-hp-h p-hp-h p-hp-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/t Decay probability

Doublon decay in a compressible state Doublon decay Doublon-fermion scattering Doublon Single fermion hopping Fermion-fermion scattering due to projected hopping

Fermi’s golden rule Neglect fermion-fermion scattering + other spin combinations Crossed diagram are not important + 2 G =  k1  k2  k = cos k x + cos k y + cos k z

Self-consistent diagrammatics Neglect fermion-fermion scattering Calculate doublon lifetime from Im S

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 presenttype 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 e 0 – characteristic energy of particle-hole pairs N p – number of diagrams included N – total number of diagrams

Self-consistent diagrammatics Including fermion-fermion scattering Correcting for missing diagrams Particle-hole self-energyDoublon life-time Typical energy transfer around 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] middle T low 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 contrast Hofferberth et al., 2008 Comparison of theory and experiments: no free parameters Higher 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 t 0 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: (2008)

Spin echo. Time reversal experiments Single mode approximation Predicts perfect spin echo The Hamiltonian can be reversed by changing a 12

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 oscillators can 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 fringes in one dimensional systems Fundamental limit on Ramsey interferometry Only q=0 mode shows complete spin echo Finite 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 analysis Kitagawa, Ueda, PRA 47:5138 (1993) Multimode analysis evolution 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: Compressibility measurements I. Bloch et al. arXiv:

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 experiments Probing 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 w matches Hubbard U

Lattice modulation experiments Probing dynamics of the Hubbard model R. Joerdens et al., arXiv:

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 BosonsFermions 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. A ij (t ’ ) replaced by probability of having a singlet Assume independent propagation of doublons and holes. Rate of doublon production A d(h) is the spectral function of a single doublon (holon)

Propogation of doublons and holes Hopping creates string of altered spins Retraceable Path Approximation Brinkmann & Rice, 1970 Consider the paths with no closed loops Spectral Fn. of single holeDoublon Production Rate Experiments

A d(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

Summary Fermions in optical lattice. Decay of repulsively bound pairs Ramsey inter- ferometry in 1d. Luttinger liquid approach to many-body decoherence Lattice modulation experiments as a probe of AF order T >> T N T << T N

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