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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 1 ! 1! V (ri − rj ) + W (ri , rj , rk ) + . . . Veff ({ri }) = 2 6 i!=j i!=j!=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) - string nets: degenerate Hilbertspace for loop gases (Fidowski, et al, ‘06) Route towards exotic and topological phases? 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 molecular ensembles (quantum memory) AMO- solid state interface - 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) ZZ XX 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 rotation of the molecule Low energy description - rigid rotor in an electric field : angular momentum dipole moment : dipole operator N =2 Accessible via microwave - anharmonic spectrum - electric dipole transition N =1 N =0 } - microwave transition frequencies - no spontaneous emission Interaction between polar molecules Hamiltonian kinetic energy Without external drive - van der Waals attraction Static electric field - internal Hamilton - finite averaged dipole moment trapping potential rigid rotor electric field interaction potential Dipole-dipole interaction Dipole-dipole interaction - anisotropic interaction - long-range repulsion attraction - Born-Oppenheimer valid for: attraction Instability in the many-body system - collaps of the system for increasing dipole interaction - roton softening - supersolids? (Goral et. al. ‘02, L. Santos et al. ‘03, Shlyapnikov ‘06) Stability: - strong interactions - confining into 2D by an optical lattice z a⊥ { confining potential oscillator wavefunction Stability via transverse confining Effective interaction +1.5 V(R,z)l!3/D - interaction potential with transverse trapping potential 2 ! " 1 z V (r) = D 3 − 3 5 + r r mωz2 2 2 z/l! z 1 0 - characteristic length scale 0 - potential barrier: larger than kinetic energy {1.5 0.5 1 0 1 attempt frequency Tunneling rate: - semi-classical rate (instanton techniques) - Euclidean action of the SE = h̄ instanton trajectory numerical factor: Dm h̄2 a⊥ "2/5 R/l! 3 kinetic energies Γ = A exp (−SE /h̄) ! 2 C bound states 4 Crystalline phase Hamiltonian interaction strength: - polar molecules confined into a two-dimensional plane Dm Eint = 2 rd = Ekin ! a rd = 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 rd ≈ 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 shifted away by external DC/AC fields Static electric field - along the z-axes - splitting the degeneracy of the first excited states degeneracy - induces finite dipole moments |e, 0! |e, 1! |e, −1" ∆ Ω dg = !g|dz |g" de = !e, 1|dz |e, 1" |g! Mircowave field - coupling the state |g! and ∆ Ω |e, 1! : 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 { { ! p2 ! ! (i) i stat ex H= + Vtrap (ri ) + H0 + Hint + Hint 2m i i i - external potentials: - dipole-Dipole interaction - restriction to the two internal states: - trapping potential - optical lattices |g!i |e, 1!i Two-level System - rotating wave approximation (i) H0 = 1 2 ! ∆ Ω Ω −∆ - two eigenstates " - two-level system in an effective magnetic field = hSi |+!i = α|g!i + β|e, 1!i |−"i = −β|g"i + α|e, 1"i and energies ! E± = ± Ω2 + ∆2 /2 Dipole-dipole interaction Microwave photon exchange D = |!e, 1|d|g"|2 ≈ d2 /3 - H ex int " + − # 1!D + − =− ν(ri − rj ) Si Sj + Sj Si 2 2 i!=j dipole-dipole interaction Induced dipole moments √ - ηd,g = de,g / D H stat int 1 − cos θ ν(r) = r3 Pi = |g!"g|i 1! Dν (ri − rj ) [ηg Pi + ηe Qi ] [ηg Pj + ηe Qj ] = 2 i!=j Qi = |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 ex stat E (1) ({ri }) = !G|Hint + Hint |G" ! - |G! = (α|g!i + β|e, 1!1 ) - i E (1) 1 ! Dν(ri − rj ) ({ri }) = λ1 2 i!=j dipole-dipole interaction: 1 − 3 cos θ Veff (r) = λ1 r3 Dimensionless coupling parameter - ! λ1 = α ηg + β ηe 2 2 "2 - tunable by the external electric field and the ratio Ω/∆ . − α2 β 2 dE/B - for a magic rabi frequency the dipole-dipole interaction vanishes λ1 = 0 Born-Oppenheimer potential Second order perturbation E (2) ({ri }) ! = k!=i!=j + ! i!=j 2 |M | √ D2 ν (ri − rk ) ν (rj − rk ) ∆ 2 + Ω2 2 |N | 2 √ [Dν (ri − rj )] ∆ 2 + Ω2 : repulsive two-body interaction Matrix elements - M = αβ !" # α ηg + β ηe (ηe − ηg ) + (β − α )/2 ! " 2 2 2 N = α β (ηe − ηg ) + 1 2 : three-body interaction 2 2 2 $ Effective Hamiltonian Effective interaction 1 ! 1! V (ri − rj ) + W (ri , rj , rk ) Veff ({ri }) = 2 6 i!=j i!=j!=k - two-body interaction 2 V (r) = λ1 D ν (r) + λ2 DR03 [ν (r)] - three-body interaction W (r1 , r2 , r3 ) = γ2 R03 D [ν(r12 )ν(r13 ) + ν(r12 )ν(r23 ) + ν(r13 )ν(r23 )] - validity is restricted to D √ = R03 " a3 ∆ 2 + Ω2 interparticle distance (i) transverse confining into 2D (ii) vanishing dipole-dipole interaction +1.5 V(R,z)l!3/D z/l! 1 0 0 {1.5 0.5 1 0 1 2 R/l! 3 4 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 Wijk = W0 ! 6 a |Ri − Rj |3 |Ri − Rk |3 " + perm Supersolids on a triangular lattice H= (1) ! ! 1 † Uijonnithe njtriangular lattice −J b bj + bosons Supersolidi hardcore 2 !ij" i#=j" Stefan Wessel(1) and Matthias Troyer(2) Institut für Theoretische Physik III, Universität Stuttgart, 70550 Stuttgart, Germany and (2) Theoretische Physik, ETH Zürich, CH-8093 Zürich, Switzerland (Dated: May 18, 2006) 1 Uij ∼ the phase diagram : static electric We determine of hardcore bosonsfield on a triangular lattice with nearest neighbor 3 − j| repulsion, paying |i special attention to the stability of the supersolid phase. Similar to the same model PACS numbers: Quantum Monte Carlo simulations 1 Next and to the widely Wessel Troyer, PRLobserved (2005) superfluid and Bosecondensed broken U (1) symmetry and “crysMelko et al.,phases PRB,with (2006) talline” density wave ordered phases with broken transsymmetry, the supersolid phase, breaking both - lational supersolid close to half filling the U (1) symmetry and translational symmetry has been strong nearest neighbor aand widely discussed phase that is hard to find both in interactions experiments and in theoretical models. Experimentally, evidence for 1/2 a possible supersolid phase in bulk 4 He has n= recently been presented [1], but the question of whether ! 10 a trueU/J supersolid has been observed is far from being set[2, 3], leavingnext-nearest the old question of supersolid behavior - tled stable under in translation invariant systems [4, 5] unsettled for now. neighbor interactions More precise statements for a supersolid phase can be made for bosons on regular lattices. It has been proposed that such bosonic lattice models can be realized 0.8 PS solid !=2/3 PS 0.6 supersolid ! 505298 v1 11 May 2005 on a square lattice we find that for densities ρ < 1/3 or ρ > 2/3 a supersolid phase is unstable and the transition between a commensurate solid and the superfluid is of first order. At intermediate fillings 1/3 < ρ < 2/3 we find an extended supersolid phase even at half filling ρ = 1/2. 0.4 superfluid PS solid !=1/3 PS 0.2 0 0 0.1 0.2 t/V 0.3 0.4 0.5 FIG. 1: Zero-temperature phase diagram of hardcore bosons 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?