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Quantum Information Processing& Condensed Matter Physicswith Cold (Neutral) Atomsa tour cold atoms in optical lattices– AMO Hubbard-ology for beginners applications– quantum info and condensed matterPeter ZollerUNIVERSITY OFINNSBRUCKAUSTRIANACADEMY OFSCIENCESSFBCoherent Control ofQuantum Systems U networksIHP Zoller

a sneak preview:cold atoms in optical latticesIHP Zoller

Cold atoms in optical latticeslaser Loading cold bosonic or fermionicatoms into an optical lattice Atomic Hubbard models withcontrollable parameters– bose / fermi atoms– spin models optical lattice as a regulararray of microtraps for atomsOptical latticelasernonresonantlaserBloch bandAC Stark shifttunnelingoptical lattice as array of microtrapsIHP Zoller

Cold atoms in optical latticeslaser Loading cold bosonic or fermionicatoms into an optical lattice Atomic Hubbard models withcontrollable parameters– bose / fermi atoms– spin modelsoptical lattice as a regulararray of microtraps for atoms Example: Bose Hubbard modelH XJαβ b†α bβα6 β1 X † † Ubα bα bα bα2αkinetic energy:hopping b , b † interaction:onsite repulsionbosonsIHP Zoller

Cold atoms in optical latticeslaser Loading cold bosonic or fermionicatoms into an optical lattice Atomic Hubbard models withcontrollable parameters– bose / fermi atoms– spin models atomic physics: features– microscopic understanding ofHamiltonian– controlling & "engineering"interactions / Hamiltonian– "engineering Hubbard Hamiltonians"optical lattice as a regulararray of microtraps for atoms Example: Bose Hubbard modelH XJαβ b†α bβα6 β1 X † † Ubα bα bα bα2αkinetic energy:hopping b , b † interaction:onsite repulsionbosonsIHP Zoller

Cold atoms in optical latticeslaser Loading cold bosonic or fermionicatoms into an optical lattice Atomic Hubbard models withcontrollable parameters– bose / fermi atoms– spin models atomic physics: features– microscopic understanding ofHamiltonian– controlling & "engineering"interactions / Hamiltonian– adding internal degrees offreedom"engineering Hubbard Hamiltonians"optical lattice as a regulararray of microtraps for atoms filling the lattice with "qubits"IHP Zoller

Cold atoms in optical latticeslaser Loading cold bosonic or fermionicatoms into an optical lattice Atomic Hubbard models withcontrollable parameters– bose / fermi atoms– spin modelsoptical lattice as a regulararray of microtraps for atoms additional aspects / features– controlled lattice loading, e.g. unitfilling– measurement– – adding phonons – adding (controlled) dissipation IHP Zoller

Cold atoms in optical latticeslaser Loading cold bosonic or fermionicatoms into an optical lattice Atomic Hubbard models withcontrollable parameters– bose / fermi atoms– spin models atomic physics: features– microscopic understanding ofHamiltonian– controlling & "engineering"interactions / Hamiltonian– adding internal degrees offreedomoptical lattice as a regulararray of microtraps for atoms additional aspects / features– controlled lattice loading, e.g. unitfilling– measurement– – adding phonons – adding (controlled) dissipation "Atomic Hubbard toolbox"IHP Zoller

Why? condensed matter physics & quantum information condensed matter physics– strongly correlated systems: high Tc etc.– time dependent, e.g. quantum phasetransitions– – exotic quantum phases (?) quantum information processing– new quantum computingscenarios, e.g. "one way quantumcomputer""quantum simulator„ qubits on a lattice-analogue simulators- digital simulators entangling qubits via "Ising"(cluster state)experiments [Bloch et al. 2001, T. Esslinger et al., J. Denschlag et al.]– Superfluid-Mott Insulator Quantum Phase Transition– spin dependent lattice & entanglement– molecules IHP Zoller

1. “Analogue simulation” of cond mat Hubbard models We can build Hubbard models directly to simulate condensed mattermodels with controllable parametersExample:Thigh T c superconductivityH t (ci σ c jσ c jσ ciσ ) U ni ni i , j ,σspinliquidsiAFM SCdopingSolving open problems where theory does not give an answer:– ground state; strange quantum phase & properties– exotic excitationsIHP Zoller

2. and Lattice Spin Models: topological quantum computing exotic spin models Examples:KitaevDoucot, Feigelman, Ioffe et al.xxzzZZXX YYℓ 1ℓ 1z zx xH I spin i 1 j 1 J i,j i,j 1 cos i,j i 1,j protected quantum memory:degenerate ground states as qubitsH II spin J xj xk J x links Jz yj yky links zj zkz linksIHP Zoller

3. Digital Quantum Simulator – an example Idea: build a Hamiltonian as a time-averaged effective Hamiltonianfrom one and two qubit gateswe will show that we cando this “easily” with atomsExample: given Ising:H J2 b 〈a,b a z z we want to simulate the Heisenberg model:H J2 b a b a b 〈a,b a x x y y z zIsing HIsingle qubitrotations U(t)IHP Zoller

Raussendorf & Briegel, PRL 20014. Measurement based quantum computing Cluster state as a quantum resource: exploit quantum correlationsInformation processing via measurement of single qubits on a lattice2D cluster stateIHP Zoller

How to create a cluster state? prepare quantum memory in product state s x a C awith each qubit in state 1 0 z 1 z 2 switch on Ising interactionH int g t a,b 14 a z b z for a given time t 0 g t dt 2D cluster state c exp i t 10 4 b g t dt a,b a z z dt a C a This can be implemented efficiently with a spin-dependent optical lattice.IHP Zoller

Measurement based quantum computing Cluster state as a quantum resource: exploit quantum correlationsInformation processing via measurement of single qubits on a latticeIHP Zoller

Measurement based quantum computing Cluster state as a quantum resource: exploit quantum correlationsInformation processing via measurement of single qubits on a latticeflow of informationquantum gateIHP Zoller

Quantum Gates a a b a b Controlled-NOTtruth table CNOT:flow of information 0 0 0 0 0 1 0 1 of 1 1 Û1 1 0 rotationa single qubit 1 1 1 0 U1IHP Zoller

AMO Hubbard "toolbox" Optical lattices– basic ideas, properties & special topics Hubbard models– naive derivation & microscopic picture, spin models, validity Lattice loading & Measurements Time-dependent aspectsIHP Zoller

1. Optical lattices: basics AC Stark shift e nonresonantlaserAC Stark shiftΔE I eg g red detunedblue detuned I E 2 decoherence: spontaneous emissionΔE g Δ2good e laser1 24 Δ 1 iΓ spontaneousemissionΓ g bad E g g E g i 12 g Δ Γ 1-typical off-resonant lattice : γ sec 1In a blue detuned lattice this can bestrongly suppressedIHP Zoller

standing wave laser configurationlaser E E 0sin kze i t c. c.laseroptical potentialBloch bandtunnelingoptical lattice as array of microtrapsV x V 0 sin 2 kx k 2 Schrödinger equation for center of mass motion of atom i t x, t 2m 2 V x x, t 2IHP Zoller

1D, 2D and 3D lattice configurationslasersquare latticeRemarks:optical potentials generated by latticebeams with different frequencies addup incoherentlyinterferometric stability (!?)lasertriangular latticeIHP Zoller

harmonic background potential (e.g. laser focus, magnetic trap)undo by inverse harmonicpotential e.g. magnetic field superlattice random potentials– add more lasers from random direction or speckle pattern Single atom coherent dynamcis studied – Wannier-Bloch– quantum chaos: kicked systemsIHP Zoller

Jaksch et al 19982. Bose Hubbard in optical lattice: naïve derivation dilute bose gasH Zµ¶2h̄ψ† ( x) 2 VT ( x) ψ( x) d3 x2mZ1 g ψ† ( x)ψ† ( x)ψ( x)ψ( x) d3 x2collisions validity: dilute gas, a s a 0 /2 optical lattice4 a s 2mg ψ( x) Xαscattering lengthw( x xα ) bαWannierfunctionsIHP Zoller

Hubbard modelH J i,j b †i b j kinetic engery:hopping12U i b †i b †i b ib i i i b †i b iinteraction:onsite repulsionU g Z w( x) 4 d3 xfeature: (time dep) tunability from weakly to strongly interacting gasvalidity IHP Zoller

parameters approx. SF-Mott transition:recoil energy E R 2 k 2 /2m, V 0 9E RNa: E R 25 KHz, J 1 KHz, U 10 KHz,Rb: E R 3. 8 KHz . validity:a s a 0 /2, U Bloch and T 0 :kT J, U Bloch,density n 10 14 10 15 cm 3 (three particle loss)IHP Zoller

Hubbard model: microscopic picture HubbardUJ h̄ωtrapsolve in n 1,2,3, particle sectorconnect by tunneling (e.g in a tightbinding approx)n 2 atoms on one lattice site: moleculemolecular problem with added opticalpotential n 3 atoms on one lattice site: e.g. Efimov-type problem[n n {max} 3 killed by three body etc. loss]IHP Zoller

Julienne et al.two atoms on one site 2 2 2m 2R 2 2r 2U1212 2m 2 R 2 cm R E cm cm R 2 r 2 V r r E r h̄ωtrap12 2 r 2v 2 3vv 1v 0harmonicapproximation Born Oppenheimerpotentials including trapC6r6(schematicnot too scale)De 10 12 Hz 5 nmscattering length as a0trap size a0IHP Zoller

two atoms on one siteUh̄ωtrap12 2 r 2v 2 3vv 1v 0"2 atoms":onsite interactionU "shift"IHP Zoller

two atoms on one sitephotoassociationresonanceh̄ωtrapUΩ12 2 r 2v 2 3vv 1v 0"2 atoms":AC Starkshift opticalFeshbach resonanceUeff U bg Δ"molecule"1 24 ΔΔH U bg 1 24 Δ b †2 b 2IHP Zoller

Hubbard model including molecules HamiltonianH J b b †i b j 1 U b b †i b †i b ib i2 i,j i J m m†i mj 1 U m m†i m†i m imi Δm †i mi2 i,j atoms moleculeii 1 m †i b i b i h.c.2iRemarks:we have derived this only for sector:2 atoms or 1 moleculeinelastic collisions / loss for 2atoms and 1 molecules (?)IHP Zoller

Remark: quantum phases of "composite objects" molecular BEC via a quantum phase transitionatomic BECswitch onoptical latticemolecular BECquantummeltquantumfreezeMott insulator:pattern loadingphotoassociationconverts atoms100% to molecules molecular Mott insulatoroptical lattice:array of microtrapsIHP Zoller

Remarks:Straightforward generalization of these Hubbard derivations to bosons and / or fermions two-component mixtures of bosons / fermions dipolar gases / Hubbard models via heteronuclear molecules (longrange dipolar forces)Complaints: the time scales for tunneling are pretty long decoherence: spontaneous emission, laser / magnetic fieldfluctuations, Other ideas for interactions optical dipole-dipole interactions however loss Rydberg-Rydberg interactions in a static electric field (huge)μ1μ2V dip r IHP Zoller

e.g. Duan et al.Spin models optical lattice bosons in a Mott phase H J i,j , b †i b j 〈n i 〈n i 112U i, n i n i 1 U i n i n i H 〈i,j z zi zj xi xj yi yj J2Upretty small XXZ-model zi n i n i xi b † i b i b †i b i i i b † i b i b †i b i yideas for higher order H σ σ σ interactions IHP Zoller

Preparation of the lattice for quantum computing:one atom per lattice site superfluid – Mott insulator quantum phase transition in optical latticesIHP Zoller

Laser control: kinetic vs. potential energy shallow lattice : weak laserU small J largeweakly interacting system:J U(kinetic energy interactions)laser parameters(time dependent)deep lattice: intense laserstrongly interacting system:J U(kinetic energy interactions)U largeJ smallIHP Zoller

Quantum phase transition Hamiltonianexample: gasno commoneigenvectorskineticenergyinteractionenergyg 0: dilute gasg : strongly correlated systemIHP Zoller

Quantum phase transition Hamiltonian crossing the critical point (temperature T 0)no commoneigenvectorsEnergy eigenstate of H1 eigenstate of H0critical point why interesting?– fundamental interest– engineer interesting states . e.g entangled statesIHP Zoller

BH: Fischer et al. 1989optical lattice: Jaksch et al. 1998Superfluid – Mott insulator quantum phase transition N atoms in M lattice sites: (temperature T 0)α superfluid J U competing interactions Blochband2Jdelocalized atoms: BEC³ N³ N†††bk 0 vaci b1 . . . bM vacikIHP Zoller

Mott phase: J Ucommensurate filling N Mexcitations: gapped UUrobust!b†1 b†2 . . . b†M vaciincommensurate filling N / Mexcitations: J superfluidMott coreIHP Zoller

1D “exact” calculationsU/J 1Superfluid and Mottin 2D trapWe achieve loading with exactly1,2, atoms!n 2n 1C. KollathJ. von DelftU. SchollwöckU/J 10density:“wedding cake”U/J 17yJaksch et al. 1998xIHP Zoller