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Strongly correlated many-body systems: from electronic materials to ultracold atoms to photons Eugene Demler Harvard University “Conventional” solid state materials Bloch theorem for non-interacting electrons in a periodic potential Consequences of the Bloch theorem B VH d Metals I EF EF Insulators and Semiconductors First semiconductor transistor “Conventional” solid state materials Electron-phonon and electron-electron interactions are irrelevant at low temperatures ky kx Landau Fermi liquid theory: when frequency and temperature are smaller than EF electron systems are equivalent to systems of non-interacting fermions kF Ag Ag Ag Strongly correlated electron systems Quantum Hall systems kinetic energy suppressed by magnetic field UCu3.5Pd1.5 CeCu2Si2 Heavy fermion materials many puzzling non-Fermi liquid properties High temperature superconductors Unusual “normal” state, Controversial mechanism of superconductivity, Several competing orders What is the connection between strongly correlated electron systems and ultracold atoms? Bose-Einstein condensation of weakly interacting atoms Density Typical distance between atoms Typical scattering length 1013 cm-1 300 nm 10 nm Scattering length is much smaller than characteristic interparticle distances. Interactions are weak New Era in Cold Atoms Research Focus on Systems with Strong Interactions • Feshbach resonances • Rotating systems • Low dimensional systems • Atoms in optical lattices • Systems with long range dipolar interactions Feshbach resonance and fermionic condensates Greiner et al., Nature (2003); Ketterle et al., (2003) Ketterle et al., Nature 435, 1047-1051 (2005) One dimensional systems 1D confinement in optical potential Weiss et al., Science (05); Bloch et al., Esslinger et al., One dimensional systems in microtraps. Thywissen et al., Eur. J. Phys. D. (99); Hansel et al., Nature (01); Folman et al., Adv. At. Mol. Opt. Phys. (02) Strongly interacting regime can be reached for low densities Atoms in optical lattices Theory: Jaksch et al. PRL (1998) Experiment: Kasevich et al., Science (2001); Greiner et al., Nature (2001); Phillips et al., J. Physics B (2002) Esslinger et al., PRL (2004); and many more … Strongly correlated systems Electrons in Solids Atoms in optical lattices Simple metals Perturbation theory in Coulomb interaction applies. Band structure methods wotk Strongly Correlated Electron Systems Band structure methods fail. Novel phenomena in strongly correlated electron systems: Quantum magnetism, phase separation, unconventional superconductivity, high temperature superconductivity, fractionalization of electrons … By studying strongly interacting systems of cold atoms we expect to get insights into the mysterious properties of novel quantum materials: Quantum Simulators BUT Strongly interacting systems of ultracold atoms and photons: are NOT direct analogues of condensed matter systems These are independent physical systems with their own “personalities”, physical properties, and theoretical challenges Strongly correlated systems of ultracold atoms should also be useful for applications in quantum information, high precision spectroscopy, metrology New Phenomena in quantum many-body systems of ultracold atoms Long intrinsic time scales - Interaction energy and bandwidth ~ 1kHz - System parameters can be changed over this time scale Decoupling from external environment - Long coherence times Can achieve highly non equilibrium quantum many-body states New detection methods Interference, higher order correlations Strongly correlated many-body systems of photons Linear geometrical optics Strongly correlated systems of photons Strongly interacting polaritons in coupled arrays of cavities M. Hartmann et al., Nature Physics (2006) Strong optical nonlinearities in nanoscale surface plasmons Akimov et al., Nature (2007) Crystallization (fermionization) of photons in one dimensional optical waveguides D. Chang et al., Nature Physics (2008) Outline of these lectures • • • • • Introduction. Systems of ultracold atoms. Bogoliubov theory. Spinor condensates. Cold atoms in optical lattices. Bose Hubbard model and extensions Bose mixtures in optical lattices Quantum magnetism of ultracold atoms. Current experiments: observation of superexchange • Fermions in optical lattices Magnetism and pairing in systems with repulsive interactions. Current experiments: Mott state • Detection of many-body phases using noise correlations • Experiments with low dimensional systems Interference experiments. Analysis of high order correlations • Non-equilibrium dynamics Emphasis of these lectures: • Detection of many-body phases • Dynamics Ultracold atoms Ultracold atoms Most common bosonic atoms: alkali 87Rb and 23Na Most common fermionic atoms: alkali 40K and 6Li Other systems: BEC of 133Cs (e.g. Grimm et al.) BEC of 52Cr (Pfau et al.) BEC of 84Sr (e.g. Grimm et al.), 87Sr and 88Sr (e.g. Ye et al.) BEC of 168Yb, 170Yb, 172Yb, 174Yb, 176Yb Quantum degenerate fermions 171Yb 173Yb (Takahashi et al.) Magnetic properties of individual alkali atoms Single valence electron in the s-orbital and Nuclear spin Hyperfine coupling mixes nuclear and electron spins Total angular momentum (hyperfine spin) Zero field splitting between states and For 23Na AHFS = 1.8 GHz and for 87Rb AHFS = 6.8 GHz Magnetic properties of individual alkali atoms Effect of magnetic field comes from electron spin gs=2 and mB=1.4 MHz/G When fields are not too large one can use (assuming field along z) The last term describes quadratic Zeeman effect q=h 390 Hz/G2 Magnetic trapping of alkali atoms Magnetic trapping of neutral atoms is due to the Zeeman effect. The energy of an atomic state depends on the magnetic field. In an inhomogeneous field an atom experiences a spatially varying potential. Example: Potential: Magnetic trapping is limited by the requirement that the trapped atoms remain in weak field seeking states. For 23Na and 87Rb there are three states Optical trapping of alkali atoms Based on AC Stark effect Dipolar moment induced by the electric field Typically optical frequencies. - polarizability Potential: Far-off-resonant optical trap confines atoms regardless of their hyperfine state Bogoliubov theory of weakly interacting BEC. Collective modes BEC of spinless bosons. Bogoliubov theory We consider a uniform system first - boson annihilation operator at momentum p, - strength of contact s-wave interaction - volume of the system - chemical potential For non-interacting atoms at T=0 all atoms are in k=0 state. Mean field equations Minimizing with respect to N0 we find BEC of spinless bosons. Bogoliubov theory We now expand around the nointeracting solution for small U0 . From the definition of Bose operators When N0 >>1 we can treat b0 as a c-number and replace by Mean-field Hamiltonian - means that p,-p pairs should be counted only once n0 = N0/V - density and m = n0 U0 Bogoliubov transformation Bosonic commutation relations are preserved when Bogoliubov transformation Mean-field Hamiltonian becomes Bogoliubov transformation Cancellation of non-diagonal terms requires To satisfy take Solution of these equations Mean-field Hamiltonian Bogoliubov modes Dispersion of collective modes Define healing length from Long wavelength limit, , sound dispersion Sound velocity Short wavelength limit, , free particles Probing the dispersion of BEC by off-resonant light scattering For details see cond-mat/0005001 Treat optical field as classical Excitation rate out of the ground state |g> Dynamical structure factor For small q we find PRL 83:2876 (1999) PRL 88:60402 (2002) Ω Rev. Mod. Phys. 77:187 (2005) Gross-Pitaevskii equation Gross-Pitaevskii equation Hamiltonian of interacting bosons Commutation relations Equations of motion This is operator equation. We can take classical limit by assuming that all atoms condense into the same state . The last equation becomes an equation on the wavefunction Gross-Pitaevskii equation Analysis of fluctuations on top of the mean-field GP equations leads to the Bogoliubov modes Two-component mixtures Two-component Bose mixture Consider mean-field (all particles in the condensate) Repulsive interactions. Miscible and immiscible regimes System with a finite density of both species is unstable to phase separation Bose mixture: dynamics of phase separation What happens if we prepare a mixture of two condensates in the immiscible regime? Two component GP equation Assume equal densities Analyze fluctuations Bose mixture: dynamics of phase separation Define density fluctuation phase fluctuation Equations of motion: charge conservation and Josephson relation Equation on collective modes Bose mixture: dynamics of phase separation Here When we get imaginary frequencies Imaginary frequencies indicate exponential growth of fluctuations, i.e. instability. The most unstable mode sets the length for pattern formation. Note that when . This is required by spin conservation. Most unstable mode PRL 82:2228 (1999) Spinor condensates F=1 Spinor condensates. F=1 Three component order parameter: mF=-1,0,+1 Contact interaction depends on relative spin orientation When g2>0 interaction is antiferromagnetic. Example 23Na When g2<0 interaction is antiferromagnetic. Example 87Rb F=1 spinor condensates. Hamiltonian - spin operators for F=1 Total Fz is conserved so linear Zeeman term should (usually) be understood as Lagrange multiplier that controls Fz. Quadratic Zeeman effect causes the energy of mF=0 state to be lower than the energy of mF=-1,+1 states. The antiferromagnetic interaction (g2>0) favors the nematic (polar) state (mF=0 and its rotations). The ferromagnetic interaction (g2<0) favors spin polarized state (mF=+1) and its rotations). Phase diagram of F=1 spinor condensates g2>0 Antiferromagnetic Shaded region: mixture of mF=-1,+1 states. This state lowers interaction energy. g2<0 Ferromagnetic Shaded region: mixture of all three states. There is XY component of the spin. Nature 396:345 (1998) Using magnetic field gradient to explore the phase diagram of F=1 atoms with AF interactions 23Na Nature 443:312 (2006) Magnetic dipolar interactions in ultracold atoms Magnetic dipolar interactions in spinor condensates q Comparison of contact and dipolar interactions. Typical value a=100aB For 87Rb m=mB and e=0.007 Bose condensation of 52Cr. T. Pfau et al. (2005) Review: Menotti et al., arXiv 0711.3422 For 52Cr m=6mB and e=0.16 Magnetic dipolar interactions in spinor condensates Interaction of F=1 atoms Ferromagnetic Interactions for 87Rb a2-a0= -1.07 aB A. Widera, I. Bloch et al., New J. Phys. 8:152 (2006) Spin-depenent part of the interaction is small. Dipolar interaction may be important (D. Stamper-Kurn) Spontaneously modulated textures in spinor condensates Vengalattore et al. PRL (2008) Fourier spectrum of the fragmented condensate Energy scales: importance of Larmor precession Magnetic Field •Larmor Precession (100 kHz) •Quadratic Zeeman (0-20 Hz) B F S-wave Scattering •Spin independent (g0n = kHz) •Spin dependent (gsn = 10 Hz) Dipolar Interaction •Anisotropic (gdn=10 Hz) •Long-ranged Reduced Dimensionality •Quasi-2D geometry d spin 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 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 XY components of the spins can lower the energy using modulation along z. X X Z components of the spins can lower the energy using modulation along x Strong instabilities of systems with dipolar interactions after averaging over precession Instabilities of F=1 Rb (ferromagnetic) 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) Berkeley Experiments: checkerboard phase Instabilities of magnetic dipolar interactions: general analysis a – angle between magnetic field and normal to the plane dn – layer thickness q measures the strength of quadratic Zeeman effect Instabilities of collective modes Magnetoroton softening Q measures the strength of quadratic Zeeman effect 87Rb condensate: magnetic supersolid ? Possible supersolid phase in 4He Phase diagram of 4He 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) T. Schneider and C.P. Enz (1971). Formation of the supersolid phase due to softening of roton excitations Resonant period as a function of T Interlayer coherence in bilayer quantum Hall systems at n=1 Hartree-Fock predicts roton softening and transition into a state with both interlayer coherence and stripe order. Transport experiments suggest first order transition into a compressible state. Eisenstein, Boebinger et al. (1994) L. Brey and H. Fertig (2000) Low energy effective model for ferromagnetic condensates Mass superflow current caused by spin texture Topological spin winding (Pontryagin index) Magnetic dipolar interaction. Favors spin skyrmions Requires net zero spin winding Magnetic crystal phases in F=1 87Rb ferromagnetic condensates Cherng, Demler, unpublished Experimental constraints: 1. Dihedral symmetry of the magnetic crystal 2. Rectangular lattice 3. Non-planar spin orientation Magnetic crystal lattices optimized within each class Optimal spin configuration. Magnetic crystal phases in F=1 87Rb ferromagnetic condensates Spin correlation functions for spin components parallel and perpendicular to the magnetic field Ultracold atoms in optical lattices. Band structure. Semiclasical dynamics. Optical lattice The simplest possible periodic optical potential is formed by overlapping two counter-propagating beams. This results in a standing wave Averaging over fast optical oscillations (AC Stark effect) gives Combining three perpendicular sets of standing waves we get a simple cubic lattice This potential allows separation of variables Optical lattice For each coordinate we have Matthieu equation Eigenvalues and eigenfunctions are known In the regime of deep lattice we get the tight-binding model - recoil energy - bandgap Lowest band Optical lattice Effective Hamiltonian for non-interacting atoms in the lowest Bloch band nearest neighbors Band structure Semiclassical dynamics in the lattice 1. Band index is constant 2. 3. Bloch oscillations Consider a uniform and constant force PRL 76:4508 (1996) State dependent optical lattices Fine structure for 23Na and 87Rb How to use selection rules for optical transitions to make different lattice potentials for different internal states. The right circularly polarized light couples to two excited levels P1/2 and P3/2. AC Stark effects have opposite signs and cancel each other for the appropriate frequency. At this frequency AC Stark effect for the state comes only from polarized light and gives the potential . Analogously state gives the potential Decomposing hyperfine states we find will only be affected by . , which State dependent lattice PRL 91:10407 (2003)