Nonequilibrium dynamics of ultracold atoms in optical lattices Nonequilibrium dynamics of ultracold atoms in optical lattices. Lattice modulation experiments and more David Pekker, Rajdeep Sensarma, Ehud Altman, Takuya Kitagawa, Susanne Pielawa, Adilet Imambekov, Mikhail Lukin, Eugene Demler Harvard University Collaboration with experimental groups of I. Bloch, T. Esslinger, J. Schmiedmayer $$ NSF, AFOSR, MURI, DARPA,

Outline Lattice modulation experiments Fermions in optical lattice. Probe of antiferromagnetic order Fermions in optical lattice. Decay of repulsively bound pairs. Nonequilibrium dynamics of Hubbard model Ramsey interferometry and many-body decoherence. Quantum noise as a probe of dynamics

Lattice modulation experiments with fermions in optical lattice. Probing the Mott state of fermions Sensarma, Pekker, Lukin, Demler, arXiv:0902.2586 Related theory work: Kollath et al., PRA 74:416049R (2006) Huber, Ruegg, arXiv:0808:2350

Same microscopic model 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

Probing many-body states Electrons in solids Fermions in optical lattice Thermodynamic probes i.e. specific heat System size, number of doublons as a function of entropy, U/t, w0 X-Ray and neutron scattering Bragg spectroscopy, TOF noise correlations Optical conductivity STM Lattice modulation experiments Beyond analogies with condensed matter systems: Far from equilibrium quantum many-body dynamics

Signatures of incompressible Mott state of fermions in optical lattice Suppression of double occupancies Jordens et al., Nature 455:204 (2008) Compressibility measurements Schneider et al., Science 5:1520 (2008)

Lattice modulation experiments Probing dynamics of the Hubbard model Modulate lattice potential Measure number of doubly occupied sites Main effect of shaking: modulation of tunneling Doubly occupied sites created when frequency w matches Hubbard U

Lattice modulation experiments Probing dynamics of the Hubbard model R. Joerdens et al., Nature 455:204 (2008)

Mott state Regime of strong interactions U>>t. Mott gap for the charge forms at Antiferromagnetic ordering at “High” temperature regime All spin configurations are equally likely. Can neglect spin dynamics. “Low” temperature regime Spins are antiferromagnetically ordered or have strong correlations

Schwinger bosons and Slave Fermions Constraint : Singlet Creation Boson Hopping

Schwinger bosons and slave fermions Fermion hopping Propagation of holes and doublons is coupled to spin excitations. Neglect spontaneous doublon production and relaxation. Doublon production due to lattice modulation perturbation Second order perturbation theory. Number of doublons

“Low” Temperature d h = + Self-consistent Born approximation Schwinger bosons Bose condensed d h Propagation of holes and doublons strongly affected by interaction with spin waves 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

Propogation of doublons and holes Spectral function: Oscillations reflect shake-off processes of spin waves Comparison of Born approximation and exact diagonalization: Dagotto et al. Hopping creates string of altered spins: bound states

“Low” Temperature Rate of doublon production Spin wave shake-off peaks Sharp absorption edge due to coherent 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 holes Hopping 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

Lattice modulation experiments. Sum rule 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

Fermions in optical lattice. Decay of repulsively bound pairs Experiment: ETH Zurich, Esslinger et al., Theory: Sensarma, Pekker, Altman, Demler

Fermions in optical lattice. Decay of repulsively bound pairs Experiments: 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

Relaxation of doublon hole pairs in the Mott state Energy U needs to be absorbed by spin excitations Relaxation requires creation of ~U2/t2 spin excitations Energy carried by spin excitations ~ J =4t2/U Relaxation rate Very slow Relaxation

Doublon decay in a compressible state Excess energy U is converted to kinetic energy of single atoms Compressible state: Fermi liquid description U p-p p-h Doublon can decay into a pair of quasiparticles with many particle-hole pairs

Doublon decay in a compressible state Perturbation theory to order n=U/t Decay probability To calculate the rate: consider processes which maximize the number of particle-hole excitations

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

Fermi’s golden rule G = Neglect fermion-fermion scattering gk1 gk2 2 + + other spin combinations Particle-hole emission is incoherent: Crossed diagrams unimportant gk1 gk2 gk = cos kx + cos ky + cos kz

Self-consistent diagrammatics Calculate doublon lifetime from Im S Neglect fermion-fermion scattering Emission of particle-hole pairs is incoherent: Crossed diagrams are not important Suppressed by vertex functions

Self-consistent diagrammatics Neglect fermion-fermion scattering Comparison of Fermi’s Golden rule and self-consistent diagrams Need to include 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

Including fermion-fermion scattering Correcting for missing diagrams type present type missing Assume all amplitudes for particle-hole pair production are the same. Assume constructive interference between 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 energy Higher order diagrams suppressed by additional powers of (t/U)2 For each energy count number of missing crossed diagrams R[n0(w)] is renormalization of the number of diagrams

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 high T L [pixels] 0.4 0.2 10 20 30 middle T low T high T BKT x z Time of flight low T 2d BKT transition: Hadzibabic et al, Claude et al 1d: Luttinger liquid, Hofferberth et al., 2008

Distribution function of interference fringe contrast Hofferberth, Gritsev, et al., Nature Physics 4:489 (2008) 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 First demonstration of quantum fluctuations for 1d ultracold atoms First measurements of distribution function of quantum noise in interacting many-body system

Can we use quantum noise as a probe of dynamics? Example: Interaction induced collapse of Ramsey fringes A. Widera, V. Gritsev, et al. PRL 100:140401 (2008), T. Kitagawa, S. Pielawa, A. Imambekov, J. Schmiedmayer, E. Demler

Ramsey interference Atomic clocks and Ramsey interference: 1 Ramsey interference is basically Larmor precession for spin ½ particles. If we take a state which is a superposition of spin up and spin down state, the two states will evolve with different energies. So the spin rotates in the plane with the frequency determined by the energy difference. Ramsey precession underlies atomic clocks and is done not with one atoms but with many. Atomic clocks and Ramsey interference: Working with N atoms improves the precision by .

Interaction induced collapse of Ramsey fringes Two component BEC. Single mode approximation. Kitagawa, Ueda, PRA 47:5138 (1993) Ramsey fringe visibility time Experiments in 1d tubes: A. Widera et al. PRL 100:140401 (2008) What is the role of interaction on Ramsey precession?

Spin echo. Time reversal experiments Single mode analysis The Hamiltonian can be reversed by changing a12 Predicts perfect spin echo

Spin echo. Time reversal experiments Expts: A. Widera et al., PRL (2008) Experiments done in array of tubes. Strong fluctuations in 1d systems. Single mode approximation does not apply. Need to analyze the full model No revival?

Interaction induced collapse of Ramsey fringes. Multimode analysis Low energy effective theory: Luttinger liquid approach 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

Ramsey interferometry Interaction induced collapse of Ramsey fringes in one dimensional systems 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 Fundamental limit on Ramsey interferometry How to distinquish decoherence due to many-body dynamics?

Interaction induced collapse of Ramsey fringes Single mode analysis Kitagawa, Ueda, PRA 47:5138 (1993) Multimode analysis evolution of spin distribution functions T. Kitagawa, S. Pielawa, A. Imambekov, J. Schmiedmayer, E. Demler

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

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