Neutral Atoms in Optical Lattices - from Quantum Simulators to Multiparticle Entanglement -

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1 Neutral Atoms n Optcal Lattces - from Quantum Smulators to Multpartcle Entanglement - Markus Grener, Olaf Mandel, Tm Rom, Artur Wdera, Smon Föllng T. W. Hänsch & I.B. LES HOUCHES 004 Introducton Durng the last 0 years a hgh degree of sophstcaton has been reached n the control of sngle quantum systems Atoms, Ions, Photons, Quantum Dots Ludwg-Maxmlans-Unversty, Munch, Max-Planck-Insttut for Quantum Optcs, Garchng Johannes-Gutenberg Unversty, Manz New challenges ahead: control, engneer and understand complex quantum system quantum computers, quantum smulators, novel (states of) quantum matter, advanced materals, mult-partcle entanglement Introducton Short Resume of Work wth the Mott State Control and engneer quantum matter at the ultmate lmt search for novel quantum phases solve longstandng open questons of condensed matter physcs Hgh precson spectroscopy Control nteracton propertes (mag. & opt. Feshbach resonances) producton of ground state moleculues realze Quantum smulators create large scale entanglement fast & robust qubts sngle ste addressng? Quantum Informaton applcatons The Mott state as a quantum regster Spn dependent potentals for ndvdual atoms O. Mandel et al., PRL 9, (003) Collaps and revval of the matter wave feld of a BEC M. Grener et al., Nature, 49, p. 5, 00. Controlled collsons massvely parallel quantum gate array controllable Isng nteracton O. Mandel et al. Nature, 45, p. 937, (003) Atomc/Molecular Physcs Entanglement Interferometry Precson measurement of atomc scatterng propertes A. Wdera et al., PRL 9, (004) State Selectve Producton of Molecules n Optcal Lattces T. Rom et al. (accepted for publcaton n PRL) Novel Quantum Phases Tonks-Grardeau Gas n an Optcal Lattce B. Paredes et al., Nature 49 (004)

2 Neutral Atoms n Optcal Lattces From a Classcal Gas to a Bose-Ensten Condensate Part Introducton Expermental Setup Loadng a BEC nto an Optcal Lattce From a Superflud to a Mott Insulator Collapse and Revval of a Macroscopc Quantum Feld Part Spn dependent lattce potentals a quantum atomc conveyor belt Neutral Atom Quantum Gate Arrays Part 3 Entanglement Interferometry Controlled Molecule Formaton n Lattces Concluson and Outlook Nobel prze 00! T >> T c Classcal Gas T > T c λ db = hmv T T < T c λ d db T = 0 Coherent Matter Wave Matter Waves at Work! How about the Partcles beneath the Waves? (MIT) Andrews et al., Scence (997)

3 How Can We Descrbe the Ground State of a Many-Body System n a Trap? From a Bose Gas wthout Interactons to a Strongly Correlated Bose System Many-Body State No Interactons Ψ ψ N External confnement Weak Interactons Ψ ψ nt N N partcle system Strongly Correlated System Ψ ψ nt N Neutral Atoms n Optcal Lattces Expermental Setup for Lattce Experments Part Introducton Expermental Setup Loadng a BEC nto an Optcal Lattce From a Superflud to a Mott Insulator Collapse and Revval of a Macroscopc Quantum Feld Part Spn dependent lattce potentals a quantum atomc conveyor belt Neutral Atom Quantum Gate Arrays Part 3 Entanglement Interferometry Controlled Molecule Formaton n Lattces Concluson and Outlook

4 Magnetc Transport of Cold Atoms Laser Coolng n Acton MOT BEC Magnetc Transport of Cold Atoms Plus a lot of Optcs and Electroncs (Table )!! M. Grener et al., PRA 63, 0340

5 Table How do we detect these quantum gases? Energy of a dpole n an electrc feld: Trappng Atoms n Lght Feld - Optcal Dpole Potentals Udp r r = d E Optcal Lattces 4 mm Polystyrol Partcles n Water r r An electrc feld nduces a dpole moment: d =αe Udp r α( ω) I( ) Red detunng: Atoms are trapped n the ntensty maxma Blue detunng: Atoms are repelled from the ntensty maxma beams 4 beams See R. Grmm et al., Adv. At. Mol. Opt. Phys. 4, (000). Poneerng work by Steven Chu see also work by: Hemmerch & Hänsch, Phllps, Grynberg, Cohen-Tannoudj, Salomon

6 Optcal Lattces D Lattce Potental Resultng potental conssts of an array of tghtly confng potental tubes BEC s splt nto up to 0,000 coupled D quantum gases (radal moton confned to zero pont oscllatons) V 0 up to 40 E recol ω r up to π 45 khz n 0-50 atoms per tube 3D Lattce Potental General Lattce Parameters Atomc Speces 87 Rb Wavelength 850 nm Wast (/e ) 5 µm Polarzaton Orthogonal between standng wave pars Intensty control All beams ntensty stablzed Resultng potental conssts of a smple cubc lattce BEC coherently populates more than 00,000 lattce stes V 0 up to 40 E recol ω r up to π 45 khz n -5 atoms on average per ste Lattce geometry Lattce spacng Smple cubc 45 nm

7 Neutral Atoms n Optcal Lattces Loadng the Atoms nto the Lattce Potental Part Introducton Expermental Setup Loadng a BEC nto an Optcal Lattce From a Superflud to a Mott Insulator Collapse and Revval of a Macroscopc Quantum Feld Part Spn dependent lattce potentals a quantum atomc conveyor belt Neutral Atom Quantum Gate Arrays Part 3 Entanglement Interferometry Controlled Molecule Formaton n Lattces Concluson and Outlook Here: Exponental ramp up n 80 ms wth a tmeconstant of 0 ms. Not very senstve to exact waveform, or ramp speed Start wth a pure condensate n a magnetc trap Turn on lattce potental adabatcally, so that the wave functon remans n the many body ground state of the system! Macroscopc Wave Functon of a BEC n an Optcal Lattce Detectng the Atoms n the Lattce Number of atoms on j th lattce ste φ ( x j ) ( j ) wx x j Ψ( x) = A x ( ) e j Phase of wave functon on j th lattce ste Spacng between neghborng lattce stes ( 45 nm) s too small to be detectable by optcal means! Localzed wave functon on J th lattce ste Swtch off the lattce lght Localzed wavefunctons expand and nterfere wth each other If there s a constant phase shft Df between lattce stes, the state s an egenstate (Bloch wavefuncton) of the lattce potental! Observe the multple matter wave nterference pattern! Quantum number characterzng these Bloch waves: h Crystal (Quas-) momentum q = φ λ (smulaton)

8 Momentum Dstrbutons D Matter Wave Interference Pattern of a BEC n an Optcal Lattce Momentum dstrbuton can be obtaned by Fourer transformaton of the macroscopc wave functon. Ψ( x) = A( x j ) wx ( x j ) e j φ ( x j ) Tme of flght measurement tme of flght ms 6 ms 0 ms 4 ms 8 ms Indvdual condensates n the lattce expand and nterfere wth each other, revealng the momentum dstrbuton of the atoms n the lattce. Interference Pattern of a 3D Lattce s-wave Scatterng (Experment) D Lattce experment theory s-wave scatterng sphere see also work from J. Walraven on d-wave scatterng!

9 Preparng Arbtrary Phase Dfferences Between Neghbourng Lattce Stes Mappng the Populaton of the Energy Bands onto the Brlloun Zones Phase dfference between neghborng lattce stes ( ) φ j = V' λ/ t/ h hω v Crystal momentum s conserved whle lowerng the lattce depth adabatcally! (cp. Bloch-Oscllatons) Crystal momentum But: dephasng f gradent s left on for long tmes! Populaton of n th band s mapped onto n th Brlloun zone! φ=0 φ=π A. Kastberg et al. PRL 74, 54 (995) M. Grener et al. PRL 87, (00) Free partcle momentum Expermental Results Pet Mondran Populatng Hgher Energy Bands Brlloun Zones n D Momentum dstrbuton of a dephased condensate after turnng off the lattce potental adabtcally Sngle lattce ste Energy bands D Stmulated Raman transtons between vbratonal levels are used to populate hgher energy bands. 3D Measured Momentum Dstrbuton!

10 Neutral Atoms n Optcal Lattces Part Introducton Expermental Setup Loadng a BEC nto an Optcal Lattce From a Superflud to a Mott Insulator Collapse and Revval of a Macroscopc Quantum Feld Part Spn dependent lattce potentals a quantum atomc conveyor belt Neutral Atom Quantum Gate Arrays Part 3 Entanglement Interferometry Controlled Molecule Formaton n Lattces Concluson and Outlook Lattce Potental s ncreased Observe at tme-offlght mages Rampng up the Lattce Potental Momentum Dstrbuton for Dfferent Potental Depths Can We Restore Coherence? 0 E recol Ramp down for dfferent tmes t and montor momemtum dstrbuton! Before rampng down 0. ms.4 ms 4 ms 4 ms E recol

11 How Long Does t Take to Restore Coherence? Bose-Hubbard Hamltonan Dephased Bose- Ensten condensate Expandng the feld operator n the Wanner bass of localzed wave functons on each lattce ste, yelds : ψ ˆ ( x) = aw ˆ ( x x ) Bose-Hubbard Hamltonan Tunnelng tme H = J n + U nˆ ( ˆ n ) aˆ ˆ ˆ aj + ε, j Tunnelmatrx element/hoppng element 3 h J = d xw( x x ) + V ( ) w( ) x m lat x x j Onste nteracton matrx element 4π h a U = m 3 d x w ( x) 4 0. ms.4 ms 4 ms 4 ms Coherence s restored on the order of a tunnelng tme! M.P.A. Fsher et al., PRB 40, 546 (989); D. Jaksch et al., PRL 8, 308 (998) Related experments n D: C. Orzel et al., Scence 9, 386 (00). Superflud Lmt Atomc Lmt of a Mott-Insulator H = J aˆ ˆ ˆ ( ˆ aj + U n n ), j Atoms are delocalzed over the entre lattce! Macroscopc wave functon descrbes ths state very well. M N Ψ ˆ SF a 0 = aˆ 0 H = J aˆ ˆ ˆ ( ˆ aj + U n n ), j Atoms are completely localzed to lattce stes! M n Ψ ( ˆ Mott a ) 0 a ˆ = 0 = Possonan atom number dstrbuton per lattce ste n= Atom number dstrbuton after a measurement Fock states wth a vanshng atom number fluctuaton are formed. n= Atom number dstrbuton after a measurement

12 The Smplest Possble Lattce : Atoms n a Double Well Quantum Phase Transton (QPT) from a Superflud to a Mott-Insulator Superflud State φ+φ l r φ+φ l r ( ) ( ) MI State φl φ r+ φr φl At the crtcal pont g c the system wll undergo a phase transton from a superflud to an nsulator! Average atom number per ste: Average onste Interacton per ste: 0.5 x x x <n> = <n> = <E nt > = ½U <E nt > = 0 Characterstc for a QPT Ths phase transton occurs even at T=0 and s drven by quantum fluctuatons! Exctaton spectrum s dramatcally modfed at the crtcal pont. U/J < g c (Superflud regme) Exctaton spectrum s gapless Crtcal rato for: U/J > g c (Mott-Insulator regme) Exctaton spectrum s gapped U/J = z 5.8 see Subr Sachdev, Quantum Phase Transtons, Cambrdge Unversty Press Superflud Mott-Insulator Phase Dagram Ground State of an Inhomogeneous System Jaksch et al. PRL 8, 308 (998) From Jaksch et al. PRL 8, 308 (998) From M. Nemeyer and H. Monen (prvate communcaton) For an nhomogeneous system an effectve local chemcal potental can be ntroduced µ =µ ε loc

13 Creatng Exctatons n the MI Phase Exctaton Probablty vs. Gradent Mott-Insulator wth n = atom per lattce ste 0 E recol t perturb = ms 3 E recol t perturb = 4 ms Wthout gradent potental Wth gradent potental Specal case: E j = U Energy Scales: hω ν 0 U U J S. Sachdev, K. Sengupta, S. M. Grvn, cond-mat/ E recol t perturb = 9 ms 0 E recol t perturb = 0 ms Resonance Gradent vs. Potental Depth Neutral Atoms n Optcal Lattces U theory Shaded green area denotes possble expermental values due to systematc uncertantes. Part Introducton Expermental Setup Loadng a BEC nto an Optcal Lattce From a Superflud to a Mott Insulator Collapse and Revval of a Macroscopc Quantum Feld Part Spn dependent lattce potentals a quantum atomc conveyor belt Neutral Atom Quantum Gate Arrays Part 3 Entanglement Interferometry Controlled Molecule Formaton n Lattces Concluson and Outlook

14 What Happens to the Relatve Phase of two Quantum Lquds over Tme? Dynamcal Evoluton of a Many Atom State due to Cold Collson Start wth a sngle Bose-Ensten condensate How do collsons affect the many body state n tme? Fundamental queston arses: BEC BEC Relatve phase φ Splt t nto two BECs wth a constant relatve phase φ Phase evoluton of the quantum state of two nteractng atoms: Collsonal phase () t = e U t / h What happens to the relatve phase between the two condensates over tme? What happens to the ndvdual wave functons of the two BECs over tme? M. Grener, O. Mandel, T. W. Hänsch and I. Bloch Nature, 49 (690), 00 Phase shft s coherent! Can be used for quantum computaton (see Jaksch, Bregel, Crac, Zoller schemes) Leads to dramatc effects beyond meanfeld theores! Collsonal phase of n- atoms n a trap: Et n / h= Un( n ) t/ h Tme Evoluton of a Coherent State due to Cold Collsons Freezng Out Atom Number Fluctuatons Coherent state n each lattce ste! + B A Ramp up lattce fast from the superflud regme (A) to the MI regme (B), such that atoms do not have tme to tunnel! α / Ψ = e n α n n! n = + e U t/ h Atom number fluctuatons at (A) are frozen!. Here α = ampltude of the coherent state. Here α = average number of atoms per lattce ste + e 3 Ut/ h / Ψ ( 0) = e α n α n n! n

15 Collapse and Revval of the Matter Wave Feld due to Cold Collsons Quantum state n each lattce ste (e.g. for a coherent state) n ( ) / / α Un n t h t e e n n! α Ψ () = Matter wave feld on the th lattce ste n () ˆ Ψ t = Ψ() t a Ψ() t The dynamcal evoluton can be vsualzed through the Q-functon Dynamcal Evoluton of a Coherent State due to Cold Collsons Q = αψ () t Characterzes overlap of our nput state wth an arbtrary coherent state α π. Matter wave feld collapses but revves after tmes multple tmes of h/u!. Collapse tme depends on the varance σ N of the atom number dstrbuton! Im(a) Yurke & Stoler, 986, F. Sols 994; Wrght et al. 997; Imamoglu, Lewensten & You et al. 997, Castn & Dalbard 997, E. Altman & A. Auerbach 00 Smlar to Collapse and Revval of Rab-Oscllatons n Cavty QED! Re(a) Dynamcal Evoluton of a Coherent State due to Cold Collsons Dynamcal Evoluton of the Interference Pattern tme t=50 ms t=50 ms t=00 ms t=300 ms Im(α) Re(α) t=400 ms t=450 ms t=600 ms G.J. Mlburn & C.A. Holmes PRL 56, 37 (986); B. Yurke & D. Stoler PRL 57, 3 (986) After a potental jump from V A =8E r to V B =E r.

16 Collapse and Revval N coh /N tot Revval Frequency vs. Lattce Potental Depth Oscllatons after lattce potental jump from 8 E recol to E recol h/u from theory Up to 5 revvals are vsble! Influence of the Atom Number Statstcs on the Collapse Tme t c /t rev for Dfferent Intal Potental Depths Fnal potental depth V B =E r Atom Number Statstcs n=, U/J 0 Atom Number Statstcs n=, U/J=7 V A =E r V A =4E r Independent proof of sub-possonan atom number statstcs for fnte U/J!

17 Controlled Collsons D. Jaksch et al., PRL 8,975(999), G. Brennen et al., PRL 8, 060 (999) A. Sorensen & K. Molmer, PRL 83, 74 (999) Movng the Lattce Potentals State Selectve Lattce Potentals 87 Rb Fne- structure Hyperfne structure I = I kx +θ/) I + 0 sn ( = I θ 0 sn ( kx / ) > 0> s - s + V ( x, ) V ( x, ) θ = θ 3 V0 ( x, θ ) = V ( x, θ) + V+ ( x, θ) 4 4

18 Delocalzaton by Hand Sngle Partcle State after Delocalzaton TOF π/ mcrowave pulse Shft π/ mcrowave pulse Intal state 0> More general, the sngle partcle state n > wll be: Ψ + Ψ e α + sngle partcle phase Sngle partcle phase depends on: m Accumulated knetc phase xt () dt h & Accumulated potental energy phase shft ) optcal potentals not constant durng transport ) magnetc feld fluctuatons can be removed by photon echo π/-π-π/ sequence If α s constant for all atoms, we can observe an nterference pattern wth absorpton magng after a tme of flght perod! α=0 α=π Shftng s Coherent! Temporal Phase Control Localzed Delocalzed stes V=70% Delocalzaton over two lattce stes ( x = 400 nm), shft tme 50 µs Varyng phase α of the last mcrowave pulse α = 0 +e ϕ + n ( ) 3 stes V=50% α = 80 4 stes We have been able to delocalze atoms over up to 7 lattce stes! O. Mandel, M. Grener etal.,phys. Rev. Lett. 9, (003)

19 Movng Atoms n Harmonc Potentals Mappng the Populaton of the Energy Bands onto the Brlloun Zones x [nm] 00 hω v Crystal momentum s conserved whle lowerng the lattce depth adabatcally! How fast can we move, n order to avod exctatons? 0 t t shft [µs] For a constant shft velocty v, frst order perturbaton theory yelds: v c () t = sn ( ωt/) ( a0ω) Crystal momentum Populaton of n th band s mapped onto n th Brlloun zone! A. Kastberg et al. PRL 74, 54 (995) M. Grener et al. PRL 87, (00) Free partcle momentum ω=π 30 khz, x = 00 nm Populatng Hgher Energy Bands Measurng the Exctaton Probablty vs. Shft Velocty Sngle lattce ste Stmulated Raman transtons between vbratonal levels are used to populate hgher energy bands. Energy bands Populaton of hgher vbratonal states (energy bands) can be mapped onto the correspondng Brlloun zones by adabatcally decreasng the lattce potental! A. Kastberg et al. PRL (995) M. Grener et al. PRL (00) Start wth ground state atoms Measured Momentum Dstrbuton! Constant Velocty Stop; measure remanng atoms n ground state Atoms moved over a dstance of approx. 00 nm

20 Complete Sequence used n the Experment Frst Trapped Atom Interferometer! Measure Atoms n State > Well suted for explorng Quantum Random Walks! 0 = +e ϕ ( ) Ramsey Frnge See also work by W. Ketterle, PRL 004 Controlled Collsons Buldng a Quantum Gate s acqured. D. Jaksch et al., PRL 8,975 (999) 4π h a 3 U = w0 ( x) w ( x ) d x m E = U n n In tme t Hold a phase factor of e 0 ϕ Et Hold / h = e D. Jaksch et al., PRL 8,975 (999) Input state Fnal state e φ Fundamental quantum gate for neutral atoms

21 Engneerng Entanglement Engneerng Entanglement ( + )( 0 + ) Engneerng Entanglement Engneerng Entanglement ( ) ( ) e ϕ ( + )( 0 + ) ( + )( 0 + )

22 Engneerng Entanglement Engneerng Entanglement ( e ϕ 0 + ) ( e ) Bell + ( + e ) ϕ ϕ + ( ) e ϕ ( ) e ϕ ( + )( 0 + ) ( + )( 0 + ) More General Entanglement Collapse and Revval of the Ramsey frnge Entangled State p/-pulse One atom per ste Shft trap Collsonal Phase Shft Shft trap p/-pulse Wth N atoms one obtans maxmally entangled cluster states from such a sequence! U( φ) = exp φ j + σ σ ( j) ( j+ ) z z D. Jaksch et al., PRL 8,975 (999), H.-J. Bregel et al., J.Mod.Opt. 47, 5 (000) H.-J. Bregel & R. Raussendorf PRL 86, 90 (00) & PRL 86, 588 (00). D. Jaksch, PhD-Thess, Innsbruck

23 Collapse and Revval of the Ramsey frnge Collapse and Revval of the Ramsey frnge One atom per ste φ = π ψ = BELL BELL { 0 ( 0 ) ( 0 + )} = c + One atom per ste φ = π ψ = BELL BELL { 0 ( 0 ) ( 0 + )} = c + Measured D. Jaksch, PhD-Thess, Innsbruck D. Jaksch, PhD-Thess, Innsbruck Entanglement Dynamcs I Condtonal Double-Slt α c β The gate operaton does not cause a loss of vsblty!

24 Condtonal Double-Slt Condtonal Double-Slt Condtonal Double-Slt Before π/-puls: ( e ϕ 0 + ) After π/-puls, detecton n 0 : Condtonal Double-Slt { 0 ( ) + 0 ( e 0 0 )} ϕ

25 Entanglement Dynamcs II Entanglement Dynamcs II 47% 30 µs 60 µs 90 µs 0 µs 50 µs 35% 36% 3% 6% 80 µs 0 µs 40 µs 70 µs 300 µs 7.6 % 6.5 % 5.3 % 4.7 % 4.7 % 330 µs 360 µs 390 µs 40 µs 450 µs Vsblty for φ=n π decreases for ncreasng n due to dephasng / decoherence. Vsblty for φ= (n+) π never falls below 4 %; best data for frst collapse and maxmum entanglement: V= 4.5 %. Experments here: Straghtforward extenson: Outlook 3600 copes of one dmensonal arrays of 60 atoms D or 3D entangled cluster states wth up to trapped atoms!!! H.-J. Bregel & R. Raussendorf PRL 86, 90 (00) & PRL 86, 588 (00). Neutral Atoms n Optcal Lattces Part Introducton Expermental Setup Loadng a BEC nto an Optcal Lattce From a Superflud to a Mott Insulator Collapse and Revval of a Macroscopc Quantum Feld Part Spn dependent lattce potentals a quantum atomc conveyor belt Neutral Atom Quantum Gate Arrays Part 3 Entanglement Interferometry Controlled Molecule Formaton n Lattces Concluson and Outlook

26 Controllng the effectve nteracton Controllng the effectve nteracton Two atoms n a coherent superposton of the nternal states: φ00 φ0 φ ( 00 0 ) 0 φ e + e + e 0 + e 4π h a, j 3 φ, j= t d x w( x) wj( x) m Two atoms n a coherent superposton of the nternal states: φ00 φ0 φ ( 00 0 ) 0 φ e + e + e 0 + e 4π h a, j 3 φ, j= t d x w( x) wj( x) m φ + φ φ 0 Entanglement evoluton for 00 0 χ ( U + U U ) Cf.: A. Sørensen et al., Nature 409, 63 (00), see also further work of D. Jaksch and L. You Hyperfne Feshbach resonance for 87 Rb State preparaton F =, m =+ + F =, m = : F F Startng from Mott-nsulatng phase wth one and two atoms per ste Extracted from: E.G.M. van Kempen et. al, PRL 88 #9, 0930 (00) M. Erhard et al., PRA 69, (004)

27 State preparaton Feshbach resonance I Loss channel B π/ π/ τ tme Measurement n D lattce confguraton Measured poston 9.(9) G 0 Magnetc feld values calbrated by mcrowave transtons Cf. Work of E. Cornell (JILA) Theoretcal predcton: E.G.M. van Kempen et. al, PRL 88 #9, 0930 (00) Expermental detecton of loss channel: M. Erhard et al., Phys. Rev. A 69, (004) Elastc channel: Entanglement dynamcs Two atoms per ste B Varable Phase α 0 0 τ Use of spn echo technques optonal π/ π/ tme Ramsey frnge: State selectve detecton of relatve atom number versus mcrowave phase α Phase α

28 Two atoms per ste 0 0 π/ ( 0 + ) ( 0 + Two atoms per ste ) 0 0 π/ ( 0 + ) ( 0 + ) ( φ00 e eφ0 0 + eφ0 0 + eφ Two atoms per ste 0 0 π/ ( 0 + ) ( 0 + ( φ0 = φ0 φ00 φ φ φ00 φ0 ( Global phase Two atoms per ste ) φ00 e eφ0 0 + eφ0 0 + eφ φ00 e e φ 0 + e φ 0 + ) 0 0 π/ ( 0 + ) ( 0 + ) ) φ00 e eφ0 0 + eφ0 0 + eφ ( ) e φ 0 + e φ 0 + ( π/ (Phase α) ( ) ( ) e φ + e φ ) )

29 Entanglement dynamcs φ φ ( e ) ( e ) Entanglement dynamcs φ φ ( e ) ( e ) For one specal α: φ = 0 (π,...): 0 0 For one specal α: φ = 0 (π,...): 0 0 Sgnal of Ramsey experment wth two atoms per ste φ = π/ (3π/,...): (( + ) ( ) ) φ = π/ (3π/,...): (( + ) ( ) ) φ = π (3π,...): φ = π (3π,...): α Entanglement evoluton Entanglement evoluton Scetch of optcal lattce: Scetch of optcal lattce: φ = 0 φ = π /

30 Entanglement evoluton Entanglement dynamcs N x Isolated atoms + N x Atom pars = Total sgnal (rescaled) Scetch of optcal lattce: φ=0 φ = π φ=π/ φ=π Lattce stes wth one and two atoms separated wthout destructon Increasng nteracton phase corresponds to longer nteracton tme Expermental results Measurement of elastc channel (a) B = 9.08 G (a) Measured data; Not senstve to sgn of scatterng length (b) Ftresults: Poston: 9.8(9) G Wdth: 5(4) mg (c) (b) Ft to dspersve profle; sgn of scatterng length ncluded π Measured n experment Can be extracted A. Wdera et al., PRL 9, (004) (c) Can be measured usng methods of: Grener et al., Nature 49, 5 (00) a = a + a a ( ) s, χ 00 0 A. Wdera et al., PRL 9, (004)

31 Neutral Atoms n Optcal Lattces Part Expermental Setup Loadng a BEC nto an Optcal Lattce From a Superflud to a Mott Insulator Collapse and Revval of a Macroscopc Quantum Feld Part Spn dependent lattce potentals a quantum atomc conveyor belt Neutral Atom Quantum Gate Arrays Part 3 Entanglement Interferometry Controlled Molecule Formaton n Lattces Concluson and Outlook Mcroarray of Identcal Two-Partcle Systems Great envronment to form molecules! Ultracold molecules va Feshbach resonances: C. Weman, D. Jn, Ch. Salomon, R. Grmm, G. Rempe, W. Ketterle PA n Optcal Lattces Advantages of Photoassocaton n Optcal Lattces Startng from a Mott nsulator phase wth exactly two atoms per lattce ste. The moton s descrbed by : h r r r r H = ( + ) + mω ( r + r ) + V( r r ) m The moton ur can be separated n center-ofmass ( R ) moton and relatve coordnate ( r r ) moton: h ur Hcm.. = + Mω R R M ur h r r Hrel. = r + µω r + V( r ) r µ Possblty to create a Mottnsulatorwth exactly two atom s perlattce ste.a com plete mcroscopc understandng of the two-atom dynam cs and PA s possble. Isolated sngle molecules HgherFranck-Condon factorthanks to a bound-bound transton nstead ofa free-bound transton. Hgh atom c denstes wthout-and 3-body decay No nelastc molecule-molecule co lsons 3/4 Ω Ω ωtrap Com plete controlofnternaland externaldegrees of freedom No mean-feld shfts and broadenng oflnewdth

32 Creatng a Molecular BEC by Meltng a Mott Insulator The Expermental Setup Proposal:J.I.Crac,P.Zoleretal.PRL Vol.89, (00) Soluton:. An atomc BEC s loaded nto an optcal lattce. The lattce depth s ncreased to create a MI wth exactly two partcles per ste 3. A molecular MI s produced by twocolor PA under tght trappng condtons 4. By decreasng the lattce depth the MI s melted and a molecular BEC s created n a quantum phase transton LA LB phase locked Usng a sequence of Raman lasers to reach the vbratonal ground state wth hgh effcency The Expermental Setup Moble Raman laser system : Measurement: -Color PA Transton nto the molecular excted electronc state. The molecule producton s detected va the atom loss.

33 Trap Loss as a Probe of the Occupaton Numbers Molecule Formaton va Two-Photon Photoassocaton - Controllng the Internal Degrees of Freedom - 3D lattce D lattce A selecton of groups performng PA experments: D. Henzen, P. Pllet, J. Wener, M. Leduc & C. Cohen-Tannoudj, W.D. Phllps, R. Hulet Ultracold molecules can also be created through Feshbach resonances: D. Jn, Ch. Salomon, R. Grmm, G. Rempe, W. Ketterle Resolvng the External Quantum States Full Control over all Internal and External Quantum Degrees of a Chemcal Reacton Should be Possble The Raman spectrum addtonally resolves the quantzed vbratonal levels of the trapped molecules. relatve moton h r r H rel. = r + µω r + V( r ) r µ center-of-mass moton h ur Hcm.. = + Mω R R M ur Ground State Molecules Molecules n st excted state By resolvng the vbratonal quantum state of the center of mass coordnate, one should be able to control the external degrees of freedom as well. Molecules n nd excted state Fnal Quantum State s Now Completely Controllable!

34 Photoassocaton to the last bound vbratonal state No Mean Feld Broadenng! Hgh Resoluton Spectroscopy at Hgh Denstes: Peak densty *0 5 cm -3 Compare e.g. Dan Henzen experments : Peak densty *0 4 cm -3 Lnewdth. khz Autler-Townes Splttng Lnewdth of the Raman Transton and Lfetme of the Ground State Molecules Autler-Townes splttng nduced on bound-bound transtons allows us to measure couplng strengths! Lfetme lmtng effects:. Spontaneous Ram an scatterng. Colsonaldecay 3. Resonantlght(ASE,lattce lasers) Ω Ω Ω eff. = Lne broadenng: Ifthe polarzabltyof the molecules s nottwce the atom c one,the nhom ogeneous external trappng potentalleads to a dfferentalenergy shft. see also work by C. Zmmermann

35 Concluson & Outlook Control quantum matter at the ultmate lmt Engneer novel many partcle quantum states wth fundamental and practcal applcatons (e.g. precson metrology) Understand and explore complex quantum matter (use ths enhanced knowledge to buld new materals) Quantum computers (need to control approx qubts, when???) Quantum smulators to solve tough problems n condensed matter physcs e.g. hgh-tc-superconductvty (seem to be more wthn reach over the next few years) Engneer complex matter (molecules) n a bottom-upapproach n precsely defned quantum states