Design and realization of exotic quantum phases in atomic gases H.P. Büchler and P. Zoller Theoretische Physik, Universität Innsbruck, Austria Institut für Quantenoptik und Quanteninformation der Österreichischen Akademie der Wissenschaften, Innsbruck, Austria M. Hermele and M.P.A. Fisher KITP, Santa Barbara
Atomic quantum gases Bose-Einstein condensation - Gross-Pitaevskii equation - non-linear dynamics control and tunability Quantum degenerate dilute atomic gases of fermions and bosons Rotating condensates - vortices - fractional quantum Hall Molecules - Feshbach resonances - BCS-BEC crossover - dipolar gases Optical lattices - quantum information - Hubbard models - strong correlations - exotic phases
Atomic gases in an optical lattice Preparation - lattice loading schemes - controlled single particle manipulations (entanglement) - decoherence of qubits Ring exchange interaction Thermodynamics b 4 b 3 - Hubbard models - design of Hamiltonians - strongly correlated many-body systems b 1 b 2 Measurement - momentum distribution - structure factor - pairing gap -... Exotic phases?
Bose-Hubbard tool box
Optical lattices - AC Stark shift off-resonant laser ω g e 1D, 2D, and 3D Lattice structures - standing laser configuration laser laser V (x) = V 0 sin 2 k x +... - characteristic energies E r = 2 k 2 2m - high stability of the optical lattice 10kHz V 0 /E r 50 Internal states - spin dependent optical lattices - alkaline earth atoms
Control of interaction Interaction potential: - effective range r 3 0 n 1 - pseudo-potential approximation U(r) a s 10 2 a 0 r 0 n 1/3 Scattering properties - scattering amplitude: - bound state energy : a s > 0 1 f(k) = 1/a s + ik E M = 2 ma 2 s Tuning of scattering length - changing the first bound state energy via an external parameter c 6 r 6 r/a 0 - magnetic Feshbach resonance a s Bohr radius - optical Feshbach resonance 0 ν
Microscopic Hamiltonian H = dx ψ + (x) ( 2 ) +V (x) 2m ψ(x)+ g 2 dx ψ + (x)ψ + (x)ψ(x)ψ(x) optical lattice g = 4π 2 a s m : interaction strength V > E r - strong opitcal lattice - express the bosonic field operator in terms of Wannier functions - restriction to lowest Bloch band (Jaksch et al PRL 98) V/E r ω BS w(x) ψ(x) = i w(x x i )b i x i x
Bose-Hubbard Model Bose-Hubbard model (Fisher et al PRB 81) H BH = J i,j b + i b j + U/2 i b + i b+ i b i b i U E r a s /λ J E r e 2 V/E r t hopping energy interaction energy Phase diagram Mott insulator - fixed particle number - incompressible - excitation gap µ/u n=2 MI n = 1 superfluid Superfluid - long-range order - finite superfluid stiffness - linear excitation spectrum J/U
Long-range order: Experiments Disappearance of coherence for strong optical lattices (Greiner et al. 02) V E r > 13 (Greiner et al., 02) Structure factor (c) 3D FWHM [µm] 100 50 Appearance of well defined two particle excitations Potential Depth [E R ] (U / J) 8 (1.3) 10 (2.6) 12 (4.9) 14 (8.7) 16 (15) 18 (24) 0 1 2 3 4 5 6 Modulation Frequency [khz] (Esslinger et al., 04)
Ring exchange interaction
Ring exchange Ring exchange - bosons on a lattice H R E = K [ b + 1 b 2b + 3 b 4 + b 1 b + 2 b 3b + 4 ] b 4 b 3 Applications: Dimer models b 1 b 2 - spin liquids, VBS - phases - topological protected quantum memory 2D spin systems - Neel order versus VBS - deconfined quantum critical points Lattice gauge theories - U(1) lattice gauge fields - a model QED kinetic energy
Toy model: - bosons on a lattice - resonant coupling to a molecular state via a Raman transition - molecule is trapped by a different optical lattice Ring exchange b 4 m molecular state b 3 Effective coupling Hamilton b 1 b 2 detuning coupling (Rabi frequency) H = ν m + m + g i j c ij [ m + b i b j + m b + i b+ j ]
Ring exchange b 4 b 3 First internal state V G (x, y) - Bosonic atoms in the corners of the square - Bose-Hubbard model b 1 b 2 y x Raman transition V H (x, y) Second internal state a l - Trapped in the center of the square - quenched hopping - angular momentum l = 0, ±1, 2 - interaction allow for a molecular state y x
Ring exchange E C 2 2C 4 2σ v 2σ d Symmetries - Hamilton is invariant under operations of the C 4v - symmetries of single particle states a l A 1 (l =0) 1 1 1 1 1 z A 2 1 1 1-1 -1 I z b 1 b 3 + b 2 b 4 b 1 b 2 +b 2 b 3 +b 3 b 4 +b 4 b 1 B 1 1 1-1 1-1 x 2 y 2 b 1 b 2 b 2 b 3 +b 3 b 4 b 4 b 1 B 2 (l =2) E (l =1) 1 1-1 -1 1 xy m, b 1 b 3 b 2 b 4 2-2 0 0 0 (x, y) (b 1 b 2 b 3 b 4, b 2 b 3 b 4 b 1 ) Energy levels - design of optical lattice - tune with the Raman transtition close to a s-wave molecule in the d-wave vibrational state - d-wave symmetry for molecular state m + = ca + 2 a+ 0 + d [ a + 1 a+ 1 + a+ 1 a+ 1]... - integrate out single-particle states a l
Toy model: - bosons on a lattice - resonant coupling to a molecular state via a Raman transition - molecule is trapped by a different optical lattice Ring exchange b 4 d- wave molecular state m b 3 Effective coupling Hamilton detuning coupling (Rabi frequency) symmetry of the molecule b 1 b 2 H = ν m + m + g i j c ij [ m + b i b j + m b + i b+ j ] d-wave symmetry m + [b 1 b 3 b 2 b 4 ] + c.c.
Ring exchange Effective low energy Hamiltonian H = ν m + m+gm + [b 1 b 3 b 2 b 4 ]+gm [ b + 1 b+ 3 b+ 2 b+ 4 ] Relation to Ring exchange - integrating out the molecule H = K [ b + 1 b 2b + 3 b 4+b 1 b + 2 b 3b + 4 n 1n 3 n 2 n 4 ] - perturbation theory K = g2 ν
Ring exchange Hamiltonian on a lattice - add hopping for the atoms - half-filling for the bosons J H = J ij b + i b j +ν i m + i m i+g m + [b 1b 3 b 2 b 4 ]+m [ b + 1 b + 3 b+ 2 b+ 4 ] Superfluid J K - superfluid of bosonic atoms - long-ranger order E zj ν k π/a decreasing detuning ν - intermediate regime - quantum phase transition? - exotic phases? Molecules J K - formation of molecules - non-trivial structure due to d-wave symmetry E zj ν k π/a
Lattice gauge theory χ(m) 2D lattice gauge theory χ(n) i - atoms on links with ring exchange and quenched hopping - gauge transformation n red corner m blue corner j b ij b nm b nm e i[χ(n) χ(m)] - represents a 2D dimer model 3D lattice gauge theory b ij - adding an additional dimension - atoms on the links of the lattice - moleculs in the center of the faces χ(n) - pure U(1) lattice gauge theory exhibits a phase transition from the Coulomb phase to a confining phase - presence of a Coulomb phase in the present model? (M. Hermele et al, PRB 2004)