Condensed Matter models for many-body systems of ultracold atoms Eugene Demler Harvard University Collaborators: Ehud Altman, Robert Cherng, Adilet Imambekov, Vladimir Gritsev, Takuya Kitagawa, Susanne Pielawa, David Pekker, Rajdeep Sensarma Experiments: Bloch et al., Esslinger et al., Schmiedmayer et al., Stamper-Kurn et al. Harvard- MIT

New Tricks for Old Dogs, Old Tricks for New Dogs Condensed Matter models for many-body systems of ultracold atoms

Dipolar interactions. Magnetoroton softening, spin textures, supersolid. New issues: averaging over Larmor precession, coupling of spin textures and vortices R. Cherng, V. Gritsev. In collaboration with D. Stamper-Kurn Luttinger liquid. Ramsey interferometry and many-body decoherence in 1d. New issues: nonequilibrium dynamics, analysis of quantum noise. V. Gritsev, T. Kitagawa, S. Pielawa. In collaboration with expt. groups of I. Bloch and J. Schmiedmayer Hubbard model. Fermions in optical lattice. Decay of repulsively bound pairs. New issues: nonequilibrium dynamics in strongly interacting regime. D. Pekker, R. Sensarma, E. Altman. In collaboration with expt. group of T. Esslinger Summary

Dipolar interactions in spinor condensates. Magnetoroton softening and spin textures R. Cherng, V. Gritsev. In collaboration with D. Stamper-Kurn

Roton minimum in 4 He Glyde, J. Low. Temp. Phys

Phase diagram of 4He Possible supersolid phase in 4 He A.F. Andreev and I.M. Lifshits (1969): Melting of vacancies in a crystal due to strong quantum fluctuations. Also G. Chester (1970); A.J. Leggett (1970) D. Kirzhnits, Y. Nepomnyashchii (1970), T. Schneider and C.P. Enz (1971). Formation of the supersolid phase due to softening of roton excitations

Roton spectrum in pancake polar condensates Santos, Shlyapnikov, Lewenstein (2000) Fischer (2006) Origin of roton softening Repulsion at long distances Attraction at short distances Stability of the supersolid phase is a subject of debate

Cold atoms: magnetic dipolar interactions xy x y y For 87 Rb m=m B and e=0.007 For 52 Cr m=6m B and e=0.16 Menotti et al., arxiv: r r r Short-ranged contact interactions Long-ranged, anisotropic dipolar interactions Contact vs. dipolar interactions

Magnetic dipolar interactions in spinor condensates Interaction of F=1 atoms Ferromagnetic Interactions for 87 Rb Spin-depenent part of the interaction is small. Dipolar interaction may be important (D. Stamper-Kurn) a 2 -a 0 = a B A. Widera, I. Bloch et al., New J. Phys. 8:152 (2006)

Spinor condensates at Berkeley M. Vengalattore et al., arXiv:

Spinor condensates at Berkeley

Competing energy scales Quadratic Zeeman (0-20 Hz) Spin dependent S-wave scattering (g s n=8 Hz) Dipolar interaction (g d n=0.8 Hz) Quasi-2D geometry B F Precession (115 kHz) Spin independent S-wave scattering (g s n=215 Hz) High energy scales Low energy scales

Dipolar interactions after averaging over Larmor precession

Dipolar interactions parallel to is preferred “Head to tail” component dominates Static interaction Averaging over Larmor precession z perpendicular to is preferred. “Head to tail” component is averaged with the “side by side”

Instabilities: qualitative picture

Stability of systems with static dipolar interactions Ferromagnetic configuration is robust against small perturbations. Any rotation of the spins conflicts with the “head to tail” arrangement Large fluctuation required to reach a lower energy configuration

XY components of the spins can lower the energy using modulation along z. Z components of the spins can lower the energy using modulation along x X Dipolar interaction averaged after precession “Head to tail” order of the transverse spin components is violated by precession. Only need to check whether spins are parallel Strong instabilities of systems with dipolar interactions after averaging over precession X

Instabilities: technical details

Quad ZeemanPrecession Spin dep. Dipolar Spin indep. Hamiltonian

Effective dipolar interaction: Spatial and time averaging Larmor precession comoving frame Gaussian profile Field Ansatz Time-averaged Quasi-2D Effective dipolar interaction

Effective dipolar interaction Time-averaged Quasi-2D Effective dipolar interaction dFdF dFdF

Collective Modes Spin Mode δf B – longitudinal magnetization δφ – transverse orientation Charge Mode δ n – 2D density δ η – global phase Mean Field Collective Fluctuations (Spin, Charge) δφδφ δfBδfB δηδη δnδn Ψ0Ψ0 Equations of Motion

Instabilities of collective modes Q measures the strength of quadratic Zeeman effect

Collective mode phase diagram R R0R0 C┴C┴ DBC┴DBC┴ CBCB D┴CBD┴CB D┴D┴

Berkeley Experiments: checkerboard phase M. Vengalattore, et. al, PRL 100: (2008)

Spin texture length scales M. Vengalattore et al., arXiv: Spin axis modulation ~30 μm Spin modulation ~10 μm Most unstable mode |k| 2 cost in kinetic energy |k| gain in dipolar energy l ~ 30 μm

Finding a stable ground state

Non-linear sigma model: Spin textures cause phase twists Spinor “vector potential”Energetic Constraints Equations of motion for η Effective kinetic energy

Non-linear sigma model Topological charge (net vorticity) Spin gradient Vortex interaction Dipolar interaction

Spin Textures Unit Cell Top. Charge QKinetic Energy Q<0 Q>0 Min KE Max KE

Spin Textures: Skyrmion Stripes Unit Cell Top. Charge QKinetic Energy Unit Cell Top. Charge QKinetic Energy

Spin Textures: Skyrmion Lattice Unit Cell Top. Charge QKinetic Energy Unit Cell Top. Charge QKinetic Energy

Quantum noise as a probe of non-equilibrium dynamics Ramsey interferometry and many-body decoherence T. Kitagawa, A. Imambekov, S. Pielawa, J. Schmeidmayer’s group. Continues earlier work with V. Gritsev, M. Lukin, I. Bloch’s group. Phys. Rev. Lett. 100: (2008)

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 et al., Phys. Rev. Lett. (2008) 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

Noise measurements using BEC on a chip Intereference of independent condensates Hofferberth et al., Nature Physics 2008 Average contrast Distribution function of fringe contrast

Distribution function of interference fringe contrast Hofferberth et al., Nature Physics 4:489 (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

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

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 with generation of particle-hole pairs

Dipolar interactions. Magnetoroton softening and spin textures in spinor condensates. Luttinger liquid. Ramsey interferometry and many-body decoherence in 1d. Hubbard model. Fermions in optical lattice. Decay of repulsively bound pairs. Outline

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