Three-body Interactions in Cold Polar Molecules H.P. Büchler Theoretische Physik, Universität Innsbruck, Austria Institut für Quantenoptik und Quanteninformation der Österreichischen Akademie der Wissenschaften, Innsbruck, Austria In collaboration with: A. Micheli, G. Pupillo, P. Zoller
Three-body interactions Many-body interaction potential - Hamiltonians in condensed matter are effective Hamiltonains after integrating out high energy excitations V eff ({r i }) = 1 2 V (r i r j ) + 1 6 i j i j k W (r i, r j, r k ) +... two-particle interaction three-body interaction Application - Pfaffian wave function of fractional quantum Hall state (More and Read, 91) - exchange interactions in spin systems: microscopic models exotic phases (Moessner and Sondi, 01, Balents et al., 02, Moutrich and Senthil 02) Route towards exotic and topological phases? - string nets: degenerate Hilbertspace for loop gases (Fidowski, et al, 06)
Three-body interactions Extended Bose-Hubbard models - hardcore bosons H = J! b i bj!ij" 1! 1! Uij ni nj + Wijk ni nj nk. + 2 6 hopping energy i#=j i#=j#=k two-body interaction Goal - large interaction strengths - independent control of two- and three-body interaction three-body interaction 1D phase diagram µ/w n = 2/3 n = 1/2 Realizable with polar molecules n = 1/3 J/W
Polar molecules Hetronuclear Molecules + - electronic excitations 10 Hz 15 - vibrational excitations 1013 Hz - rotational excitations 10 Hz 10 - electron spin - nuclear spin Polar molecules in the electronic, vibrational, and rotational ground state - permanent dipole moment: d 1 9 Debye - polarizable with static electric field, and microwave fields dipole moment ea0 2.5Debye Strong dipole-dipole interactions tunable with external fields d1 d2 (d1 r) (d2 r) V (r) = 3 3 r r5
Interaction energies Particles in an optical lattice - lattice spacing a = λ/2 500nm - size of Wannier function ah.o 0.2a 2!2 π 2 - recoil energy Er = mλ2 Pseudo-potential - dominant interaction in atomic gases on-site interaction nearest-neighbor interaction U 0.5Er present but small U 0.5Er U1 10 3 Er Magnetic dipole moment - Chromium atoms with m 6µB Electric dipole moment - LiCs hetronuclear molecule d 6.5Debye 2 2 - increased by factor 1/α (137) U??? (a/ah.o )3 U1 U1 30Er
Polar molecules AMO- solid state interface molecular ensembles (quantum memory) - solid state quantum processor - molecular quantum memory (P. Rabl, D. DeMille, J. Doyle, M. Lukin, R. Schoelkopf and P. Zoller, PRL 2006) Cooper Pair Box (superconducting qubit) XX ZZ YY Spin toolbox - polar molecules with spin - realization of Kitaev model (A. Micheli, G. Brennen, P. Zoller, Nature Physics 2006)
Polar molecules Experimental status Raman laser / spontaneous emission - Polar molecules in the rotational and vibrational ground state - cooling and trapping techniques beeing developement: - cooling of polar molecules: e.g. stark decelerator D. DeMille, Yale J. Doyle, Harvard G. Rempe, Munich G. Meijer, Berlin J. Ye, JILA - photo association see J. Ye s talk (all cold atom labs) - bosonic molecules with closed electronic shell, e.g., SrO, RbCs, LiCs rotational and vibrational ground state
Polar molecules Crystalline phases - long range dipole-dipole interaction - interaction energy exceeds kinetic energy Three-body interaction - extended Hubbard models - tunable three-body interaction
Polar molecule Low energy description rotation of the molecule - rigid rotor in an electric field : angular momentum : dipole operator dipole moment N = 2 N = 1 N = 0 } Accessible via microwave - anharmonic spectrum - electric dipole transition - microwave transition frequencies - no spontaneous emission
Interaction between polar molecules Hamiltonian kinetic energy trapping potential rigid rotor electric field interaction potential Without external drive - van der Waals attraction Static electric field - internal Hamilton - finite averaged dipole moment
Dipole-dipole interaction Dipole-dipole interaction - anisotropic interaction - long-range attraction repulsion - Born-Oppenheimer valid for: attraction Instability in the many-body system Stability: z confining potential - collaps of the system for increasing dipole interaction - roton softening - supersolids? (Goral et. al. 02, L. Santos et al. 03, Shlyapnikov 06) - strong interactions - confining into 2D by an optical lattice a { oscillator wavefunction
Stability via transverse confining Effective interaction - interaction potential with transverse trapping potential V (r) = D [ 1 r 3 3z2 r 5 ] + mω2 z 2 z2 +1.5 z/ l 0 1 V( R, z) l 3 / D - characteristic length scale 0 0.5 - potential barrier: larger than kinetic energy {1.5 0 1 2 3 4 R l / 1 Tunneling rate: - semi-classical rate (instanton techniques) attempt frequency Γ = A exp ( S E / h) kinetic energies ( ) 2/5 - Euclidean action of the Dm S instanton trajectory E = h h 2 C a numerical factor: bound states
Crystalline phase Hamiltonian interaction strength: - polar molecules confined into a two-dimensional plane r d = E int = Dm E kin 2 a r d = 1 300 Kosterlitz-Thouless transition First order melting (Kalida 81) Quantum melting (H.P. Büchler, E. Demler, M. Lukin, A. Micheli, G. Pupillo, P. Zoller, PRL 2007) - indication of a first order transition - critical interaction strength r d 20
Three-body interactions 1! 1! V (ri rj ) + W (ri, rj, rk ) +... Veff ({ri }) = 2 6 i!=j i!=j!=k H.P. Büchler, A. Micheli, and P. Zoller cond-mat/0703688 (2007).
Single polar molecule Static electric field - along the z-axes - splitting the degeneracy of the first excited states degeneracy - induces finite dipole moments d g = g d z g d e = e, 1 d z e, 1 shifted away by external DC/AC fields e, 1 e, 0 Ω g e, 1 Mircowave field g e, 1 - coupling the state and Ω : detuning : rabi frequency - restrict to two states - ignore influence of e, 1 - rotating wave approximation - anharmonic spectrum - electric dipole transition - microwave transition frequencies - no spontaneous emission
Many-body Hamiltonian Many-body Hamiltonian H = i p 2 i 2m + i V trap (r i ) + i H (i) 0 + H stat int + H ex int { { Two-level System - external potentials: - trapping potential - optical lattices - dipole-dipole interaction - restriction to the two internal states: g i e, 1 i - rotating wave approximation H (i) 0 = 1 2 ( Ω Ω ) = hs i - two eigenstates + i = α g i + β e, 1 i i = β g i + α e, 1 i - two-level system in an effective magnetic field and energies E ± = ± Ω 2 + 2 /2
Dipole-dipole interaction Microwave photon exchange - D = e, 1 d g 2 d 2 /3 H ex int = 1 2 i j D 2 ν(r i r j ) [ S + i S j + S+ j S i ] Induced dipole moments dipole-dipole interaction ν(r) = 1 cos θ r 3 - η d,g = d e,g / D P i = g g i H stat int = 1 2 i j Dν (r i r j ) [η g P i + η e Q i ] [η g P j + η e Q j ] Q i = e, 1 e, 1 i
Born-Oppenheimer potentials Effective interaction (i) diagonalizing the internal Hamiltonian for fixed interparticle distance {ri }.! (i) stat ex H0 + Hint + Hint i (ii) The eigenenergies E({ri }) describe the Born-Oppenheimer potential a given state manifold. R0 (iii) Adiabatically connected to the groundstate G! = Πi +!i weak dipole interaction D = R03 " a3 2 + Ω2 interparticle distance ri rj
Born-Oppenheimer potential First order perturbation - - E (1) ({r i }) = G H ex int + H stat int G G = (α g i + β e, 1 1 ) i E (1) ({r i }) = 1 2 λ 1 i j Dν(r i r j ) dipole-dipole interaction: V eff (r) = λ 1 1 3 cos θ r 3 Dimensionless coupling parameter - λ 1 = ( α 2 η g + β 2 η e ) 2 α 2 β 2 - tunable by the external electric field and the ratio Ω/. de/b - for a magic rabi frequency the dipole-dipole interaction vanishes λ 1 = 0
Born-Oppenheimer potential Second order perturbation E (2) ({r i }) = k i j + i j M 2 2 + Ω 2 D2 ν (r i r k ) ν (r j r k ) N 2 2 + Ω 2 [Dν (r i r j )] 2 : three-body interaction : repulsive two-body interaction Matrix elements - M = αβ [( α 2 η g + β 2 ) η e (ηe η g ) + (β 2 α 2 )/2 ] [ ] N = α 2 β 2 (η e η g ) 2 + 1
Effective interaction V eff ({r i }) = 1 V (r i r j ) + 1 2 6 - two-body interaction Effective Hamiltonian i j V (r) = λ 1 D ν (r) + λ 2 DR 3 0 [ν (r)] 2 W (r i, r j, r k ) i j k - three-body interaction W (r 1, r 2, r 3 ) = γ 2 R 3 0D [ν(r 12 )ν(r 13 ) + ν(r 12 )ν(r 23 ) + ν(r 13 )ν(r 23 )] - validity is restricted to D 2 + Ω 2 = R3 0 a 3 interparticle distance +1.5 z/ l V( R, z) l 3 / D (i) transverse confining into 2D (ii) vanishing dipole-dipole interaction 0 1 0 0.5 {1.5 1 0 1 2 3 4 R l /
Bose-Hubbard model
Hubbard model Extended Bose-Hubbard models - hardcore bosons H = J! b i bj!ij" hopping energy - interaction parameters for strong optical lattices 1! 1! Uij ni nj + Wijk ni nj nk. + 2 6 i#=j two-body interaction Uij = V (Ri Rj ) i#=j#=k three-body interaction Wijk = W (Ri, Rj, Rk ) Polar molecule: LiCs: - dipole moment d 6Debye - hopping energy J/Er 0 0.5 - lattice spacing: - nearest neighbor interaction: λ 1000 nm Er 1.4 khz U/Er 30 3 W/Er 30 (R0 /al )
Hubbard model Three-body interaction - next-nearest neighbor terms W ijk = W 0 [ a 6 ] R i R j 3 R i R k 3 + perm
Supersolids on a triangular lattice H = J ij b i b j + 1 2 i j U ij n i n j U ij 1 i j 3 : static electric field Quantum Monte Carlo simulations Wessel and Troyer, PRL (2005) Melko et al., PRB, (2006) - supersolid close to half filling and strong nearest neighbor interactions n = 1/2 U/J 10 - stable under next-nearest neighbor interactions! 1 0.8 0.6 0.4 0.2 PS supersolid PS PS superfluid PS solid!=2/3 solid!=1/3 0 0 0.1 0.2 0.3 0.4 0.5 t/v
One-dimensional model H = J! b i bi+1 +W i! i ni 1 ni ni+1 µ! ni next-nearest neighbor interactions + Hn.n.n. i Bosonization Critical phase - hard-core bosons - instabilities for densities: n = 2/3 n = 1/2 n = 1/3 - quantum Monte Carlo simulations (in progress) - algebraic correlations - compressible - repulsive fermions µ/w Solid phases n = 2/3 - excitation gap - incompressible - density-density correlations n = 1/2! ni nj " - hopping correlations (1D VBS) n = 1/3 J/W!b i bi+1 b j bj+1 "
Conclusion and Outlook Polar molecular crystal - reduced three-body collisions - strong coupling to cavity QED - ideal quantum storage devices Lattice structure - alternative to optical lattices - tunable lattice parameters - strong phonon coupling: polarons Extended Hubbard models - strong nearest neighbor interaction - three-body interaction Novel quantum matter - supersolid phases - string nets?