Quantum magnetism of ultracold atoms $$ NSF, AFOSR MURI, DARPA Harvard-MIT Theory collaborators: Robert Cherng, Adilet Imambekov, Vladimir Gritsev, Takuya Kitagawa, Mikhail Lukin, Susanne Pielawa, Joerg Schmiedmayer Experiments: Bloch et al., Schmiedmayer et al., Stamper-Kurn et al. Eugene Demler (Harvard)

Stoner instability. Double exchange Ferromagnetism in itinerant systems Antiferromagnetism Frustrated magnetic systems ? Magnetism in condensed matter systems

Quantum magnetism of ultracold atoms Familiar models, New questions Spin dynamics in 1d systems Luttinger model and nonequilibrium dynamics. New characterization: full distribution functions Ferromagnetic F=1 spinor condensates Quantum Hall ferromagnets in disguise. Skyrmion crystal phases

Spin dynamics in 1d systems: Ramsey interference experiments T. Kitagawa, S. Pielawa, A. Imambekov, J.Schmiedmayer, V. Gritsev, E. Demler arXiv:

Working with N atoms improves the precision by. Ramsey interference t 0 1 Atomic clocks and Ramsey interference:

Ramsey Interference with BEC Single mode approximation time Amplitude of Ramsey fringes Interactions should lead to collapse and revival of Ramsey fringes

1d systems in microchips Treutlein et.al, PRL 2004, also Schmiedmayer, Van Druten Two component BEC in microchip Ramsey Interference with 1d BEC 1d systems in optical lattices Ramsey interference in 1d tubes: A.Widera et al., B. PRL 100: (2008)

Ramsey interference in 1d condensates Collapse but no revivals A. Widera, et al, PRL 2008

Ramsey interference in 1d condensates A. Widera, et al, PRL 2008 Only partial revival after spin echo! Spin echo experiments Expect full revival of fringes

Spin echo experiments in 1d tubes Single mode approximation does not apply. Need to analyze the full model

Ramsey interference in 1d Time evolution Technical noise could also lead to the absence of echo Need “smoking gun” signatures of many-body decoherece Luttinger liquid provides good agreement with experiments. A. Widera et al., PRL Theory: V. Gritsev

Distribution Probing spin dynamics using distribution functions Distribution contains information about all the moments → It can probe the system Hamiltonian Joint distribution function can also be obtained!

Distribution function of fringe contrast as a probe of many-body dynamics Short segments Long segments Radius = Amplitude Angle = Phase

Distribution function of fringe contrast as a probe of many-body dynamics Preliminary results by J. Schmiedmayer’s group Splitting one condensate into two.

Short segments Long segments l =20 mm l =110 mm ExptTheory Data: Schmiedmayer et al., unpublished

Skyrmion crystals in ferromagnetic F=1 spinor condensates R. Cherng, Ph.D. Thesis

Spinor condensates. F=1 Three component order parameter: m F =-1,0,+1 Contact interaction depends on relative spin orientation When g 2 >0 interaction is antiferromagnetic. Example 23 Na When g 2 <0 interaction is ferromagnetic. Example 87 Rb Favors condensation into m F =0 state (or its rotation) Favors condensation into m F =1 state (or its rotation)

Spin textures in ferromagnetic Rb condensates m F =-1 m F =0 m F =+1 m F =-1 m F =0 m F =+1 Imbalanced (non-equilibrium) Initial populations Equal (equilibrium) Initial populations Vengalattore et al., PRL (2008)

Spin textures: checkerboard pattern Spectrum in Momentum Space Equal populations Transverse Longitudinal Vengalattore et al., PRL (2008)

Magnetic dipolar interactions in spinor condensates Comparison of contact and dipolar interactions. Typical value a=100a B q For 87 Rb m = m B and e =0.007 Spin dependent interactions in 87 Rb are small a 2 -a 0 = a B A. Widera, I. Bloch et al., New J. Phys. 8:152 (2006) Interaction of F=1 atoms

Energy scales Quadratic Zeeman (1 Hz) Spin dependent S-wave scattering (g s n=8 Hz) Dipolar interaction (g d n = 1 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 Fast Larmor precession strongly modifies effective dipolar interactions Fourier components of effective interaction (in-plane field)

Instabilities of ferromagnetic F=1 Rb condensate due to dipolar interactions Theory: unstable modes in the regime corresponding to Berkeley experiments. Cherng, Demler, PRL (2009) Experiments. Vengalattore et al. PRL (2008)

From microscopic Hamiltonian to effective low energy theory Dipolar and quadratic Zeeman A.Lamacraft, PRA (2008) Fixed densityMaximally polarized Magnetization Condensate phase Superfluid velocity Low energy manifold

Mermin-Ho relation Divergence flow Mermin-Ho Skyrmion density MagnetizationSkyrmion densitySuperfluid velocity

Non-linear sigma model Low-energy LagrangianSuperfluid flow related to skyrmion density Spin StiffnessSkyrmion interaction (Log) Superfluid kinetic energy

Magnetic crystals in spinor condensates

Effective Hamiltonian Spin dependent interactions Skyrmion interaction Interaction strengths

Minimal energy spin texture

Find all spin groups consistent with constraints Intrinsic constraints a)Zero net skyrmion charge b)Maximally polarized magnetization c)Explicit symmetry breaking via external field D 2 point group SG = p2mm, p2mg, p2gg Phenomenological constraints d)Rectangular lattice e)No easy-axis or easy plane f)Zero net magnetization

Minimal energy spin texture

Understanding spin textures

Skyrmions in ferromagnets Single skyrmion solution Spin spaceReal space Radial coordinate Azimuthal coordinate Ordinary ferromagnets. Equations of motion Spinor ferromagnets. Equations of motion ~

Exact solutions for spinor condensates Spin spaceReal space Stereographic coordinates Holomorphic coordinates Separation of variables static solution ansatz

Single skyrmion solutions Ordinary ferromagnetSpinor condensate ferromagnet

Lattice of skyrmions Ordinary ferromagnetSpinor condensate ferromagnet

Spin textures: skyrmion lattice Equal populations Transverse Longitudinal Skyrmion lattice solution without dipolar interactions

Spin textures Equal populations Transverse Longitudinal Skyrmion lattice solution with dipolar interactions

Quantum magnetism of ultracold atoms New questions, interesting physics Spin dynamics in 1d systems Luttinger model and nonequilibrium dynamics. New characterization: full distribution functions Ferromagnetic F=1 spinor condensates Quantum Hall ferromagnets in disguise. Skyrmion crystal phases Harvard-MIT