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Atomic entangled states with BEC A. Sorensen L. M. Duan P. Zoller J.I.C. (Nature, February 2001) KIAS, November 2001. SFB Coherent Control €U TMR Entangled states of atoms j ª i6= j ' 1 i j '2 i :::j ' N i Motivation: • Fundamental. • Applications: - Secret communication - Computation - Atomic clocks Experiments: • NIST: 4 ions entangled. • ENS: 3 neutral atoms entangled. E E ' ' 4 3 This talk: Bose-Einstein condensate. E ' 103 Outline 1. Atomic clocks 2. Ramsey method 3. Spin squeezing 4. Spin squeezing with a BEC 5. Squeezing and atomic beams 6. Conclusions 1. Atomic clocks To measure time one needs a stable laser click The laser frequency must be the same for all clocks Innsbruck click Seoul click The laser frequency must be constant in time click Solution: use atoms to lock a laser !L !0 detector feed back In practice: Neutral atoms ions ! = ! + ± ! L 0 frequency fixed universal Independent atoms: Entangled atoms: 1 ± !=p p tn r e pN 1 ± ! = p e n t tn f ( N ) r e p • N is limited by the density (collisions). • t is limited by the experiment/decoherence. • We would like to decrease the number of repetitions (total time of the experiment). Figure of merit: • To achieve the same uncertainity: ± ! ± ! e n t= p »2 = (nrep )ent Tent N = = nr ep T f (N) 2 We want » ¿1 2. Ramsey method • Fast pulse: single atom 1 j 0 i! p ( j 0 i+ j 1 i ) 2 • Wait for a time T: single atom 1 ¡ i ( ! ¡ ! ) t 0 L !p ( j 0 i + e j 1 i ) 2 • Fast pulse: single atom · ¸· ¸ 1 1 ( ! ¡ ! ) t ( ! ¡ ! ) t s i n j 0 i + c o s j 1 i 0 L 0 L 2 2 • Measurement: # of atoms in |1> · ¸ 1 2 P = c o s ( ! ¡ ! ) t 1 0 L 2 Independent atoms Number of atoms in state |1> according to the binomial distribution: where If we obtain n, we can then estimate The error will be If we repeat the procedure we will have: Another way of looking at it Initial state: all atoms in |0> First Ramsey pulse: Jz Jz Jy Jy Jx Jx Free evolution: Measurement: Jz Jz Jy Jx Jx Jy In general 2 N ( ¢ J ) z » = 2 2 h J i+ h J i x y Jz 2 Jy where the J‘s are angular momentum operators Jx N X (k ) J j® ®= Remarks: k = 1 2 ¿ 1 • We want » 2 • Optimal: » ¸1 = N 2 • If » <1then the atoms are entangled. That is, X ½ =p 6 ½ ½ : : : ½ n 1 2 N n »2 measures the entanglement between the atoms 3. Spin squeezing • Product states: h J i = N = 2 x · -̧ N 1 p ( j 0 i+ j 1 i) 2 ¢ J = 0 x 2 N ( ¢ J ) z = 2 = 1 p » 2 h J i + h J i x y ¢ J = ¢ J = N = 2 h J i = h J i = 0 y z y z 2 No gain! Jz Jy Jx • Spin squeezed states: (Wineland et al,1991) h J i 'N = 2 x h J i = h J i = 0 y z p ¢ J < N = 2 z 2 N ( ¢ J ) z » = 2 < 1 2 h J i + h J i x y 2 These states give better precission in atomic clocks How to generate spin squeezed states? (Kitagawa and Ueda, 1993) 2 1) Hamiltonian: H =  J z ¡ i (  t J ) J z z U = e It is like a torsion »2 1 t= 0 »2m in » N ¡ 2=3  t' Ât min» 1=2N 2=3 Ât 1 2 = 3 2 N 2 » =1 2 » ' 1 2 = 3 N 22 =  ( J ¡ J ) 2) Hamiltonian: H z y t= 0 Ât' 1 N  t' 1 2 » =1 »2' 1 N 1 j ª i 'p ( j 0 ; : : : ; 0 i + j 1 ; : : : ; 1 i ) 2 »2 1 »2m in » N ¡ 1 Ât min» 1=2N Ât Explanation J 'N = 2 x " # J J Jx p y ;p z =i ' i N=2 N=2 N=2 J y X´ p N = 2 J z P´ p N = 2 2 ¡ x ª ( x ; 0 ) /e are like position and momentum operators Hamiltonian 1: t= 0 t> 0 N 2 2 H =  J = P z 2 Hamiltonian 2: ¢ ÂN ¡ 2 2 2 2 H = Â(Jz ¡ Jy ) = P ¡ X 2 t= 0 t> 0 4. Spin squeezinig with a BEC A. Sorensen, L.M Duan, J.I. Cirac and P. Zoller, Nature 409, 63 (2001) • Weakly interacting two component BEC 2 2 V T H d3 r r r jr j 2m ja,b laser 1 2 ja,b U ab d3 r r rar b r a b b trap U jj d 3rj rj rj rj r a + laser interactions Lit: JILA, ENS, MIT ... • Atomic configuration • optical trap AC Stark shift via laser: no collisions FORT as focused laser beam |0 | 1 | 1 ! aaa a bb a ab F 1 A toy model: two modes • we freeze the spatial wave function a x a x a b x b x b a x spatial mode function • Hamiltonian • Angular momentum representation b x H 1 U aa a 2 a 2 U ab a ab b 1 U bb b 2 b 2 2 2 a b ab H1 Uaa Ubb 2Uab J2z Jx 2 2 =  J ¡ J x z • Schwinger representation Jx 1 a b ab 2 Jy i a b ab 2 Jz 1 a a b b 2 A more quantitative model ... including the motion • Beyond mean field: (Castin and Sinatra '00) wave function for a two-component condensate N | c Na Nb |Na : a N a :t ; Nb : b Nb:t b Na 0 N bNN a a with xaNa : x, t d 3 x a Na ! Na xb Nb : x, t d 3x b N b! Nb |vac • Variational equations of motion • the variances now involve integrals over the spatial wave functions: decoherence • Particle loss Time evolution of spin squeezing • • Idealized vs. realistic model Effects of particle loss 21 1 10-1 10 2 loss -1 including motion -2 -2 10 10 -3 ideal -3 10 10 0 -4 10 -4 4 idealized model 8 12 t 16 20 10 0 4 8 t 20 % loss 12 16 20 X -4 10 Can one reach the Heisenberg limit? We have the Hamiltonian: 2 H =  J ¡ J x z 2 2 2 2 J + J + J = J = c o n s t a n t x y z | { z } H2 = Â(Jx2 ¡ Jz2) = Â(2Jx2 + Jy2 ¡ J 2) We would like to have: Idea: Use short laser pulses. short evolution short evolution short pulse short pulse 2 ¼ 2 ¼ 2 2 ¡ i 2 Jx ¡ i ±tJz i 2 Jx ¡ i Â2±tJx e e {z e }e ' 1¡ i±t(2Jx2 + Jy2) ' e¡ i±t(Jx¡ Jz) | 2 ¡i ± t J y e Conditions: -t = ¼ 2 t¿ ± t Stopping the evolution »2 1 »2m in » N ¡ 1 Ât min» 1=2N Ât Once this point is reached, we would like to supress the interaction The Hamiltonian is: 2 H =  J z Using short laser pulses, we have an effective Hamiltonian: 2 2 2 2 2 H =  J J + J + J = J = c o n s t a n t z x y z In practice: wait short pulse short pulses 5. Squeezing and entangled beams L.M Duan, A. Sorensen, I. Cirac and PZ, PRL '00 • • Atom laser atomic configuration Stark shift by laser: switch collisions on and off atoms • collisions Squeezed atomic beam |0 | 1 | 1 F 1 condensate pairs of atoms • Limiting cases squeezing sequential pairs collisional Hamiltonian 2 x x x 1 1 0 x x 20 x e i2t 1 1 condensate as classical driving field Equations ... • Hamiltonian: 1D model 2 xx H x Vx xdx i i 2m i1 x, t , x , tij x x i j g x, t x x e i2t h. c. dx, 1 1 • Heisenberg equations of motion: linear 2 e i2 t i x, t xx V x x, tg x, t x, t t 1 1 1 2m i x, t t 1 2 i2 t xx V x x, tg x, t x, t e 1 1 2m • Remark: analogous to Bogoliubov • Initial condition: all atoms in condensate Case 1: squeezed beams • Configuration g (x ,t) Â1 input: vaccum Â1 B output 1 B 1 condensate a 0 x • Bogoliubov transformation B 1 Â1 1 1 1 B 1  1Â1 1 1 • Squeezing parameter r | | | | tanh r |1| | 1| 1 • Exact solution in the steady state limit 1 Squeezing parameter r versus dimensionless detuning /g 0 and interaction coefficient g 0 t broadband two-mode squeezed state with the squeezing bandwidth g 0 . numbers: g 0 20kHz, a 3 m, v output flux of approx. 680 atoms/ms squeezing r 0 2 (large) 2 /m 9cm/s Case 2: sequential pairs • Situation analogous to parametric downconversion • Setup: collisions |0 | 1 | 1 F 1 symmetric potential • State vector in perturbation theory | t f x,y, t x ydx dy |vac 1 1 with wave function consisting of four pieces f x,y f LR x, y f RL x, y f LL x,y f RR x,y • After postselection "one atom left" and "one atom right" | eff f LR x,y x y x y dxdy|vac 1 1 1 1 | 1,1 LR |1, 1 LR 6. Conclusions • Entangled states may be useful in precission measurements. • Spin squeezed states can be generated with current technology. - Collisions between atoms build up the entanglement. - One can achieve strongly spin squeezed states. • The generation can be accelerated by using short pulses. • The entanglement is very robust. • Atoms can be outcoupled: squeezed atomic beams. Quantum repeaters with atomic ensembles L. M. Duan M. Lukin P. Zoller J.I.C. (Nature, November 2001) SFB Coherent Control €U TMR €U EQUIP (IST) Quantum communication: Classical communication: 1 Alice 101 Quantum communication: j Á i 0 0 1 Bob Alice j Á i j Á i Á i j Á i j Bob Quantum Mechanics provides a secure way of secret communication Classical communication: 1 1 0 1 Alice 0 01 Quantum communication: jÁ i Bob Alice Eve jÁ i jÁ i ½½ Bob Eve In practice: photons. laser y j0 i =a jv a c i vertical polarization 0 jÁi photons y j1 i =a jv a c i horizontal polarization 1 optical fiber Problem: decoherence. 1. Photons are absorbed: 2. States are distorted: _ Probability a photon arrives: P = e L =L 0 Quantum communication is limited to short distances (< 50 Km). ª i j Alice ½ We cannot know whether this is due to decoherence or to an eavesdropper. Bob Solution: Quantum repeaters. (Briegel et al, 1998). laser jª i repeater ½ jª i Questions: L = L 0 1. Number of repetitions < e 2. High fidelity: F = h ª j ½ j ª i '1 3. Secure against eavesdropping. Outline 1. Quantum repeaters: 2. Implementations: 3. 1. With trapped ions. 2. With atomic ensembles. Conclusions 1. Quantum repeaters The goal is to establish entangled pairs: (i) Over long distances. (ii) With high fidelity. (iii) With a small number of trials. Once one has entangled states, one can use the Ekert protocol for secret communication. (Ekert, 1991) Key ideas: 1. Entanglement creation: Establish pairs over a short distance 2. Connection: Connect repeaters Long distance 3. Pufication: Correct imperfections 4. Quantum communication: High fidelity Small number of trials 2. Implementation with trapped ions Entanglement creation: (Cabrillo et al, 1998) ion A laser Internal states ion A ion B jxi jxi ion B laser j0i j1i - Weak (short) laser pulse, so that the excitation probability is small. - If no detection, pump back and start again. - If detection, an entangled state is created. j0i j1i Description: Initial state: j 0 ; 0 i j v a c i ion A ion B jxi jxi After laser pulse: ( j 0 i + ² j x i ) ( j 0 i + ² j x i ) j v a c i A B £ ¤ j0; 0i + ²j0; xi + ²jx; 0i + o(x2) jvaci Evolution: j0; 0i jvaci + ²(bkj0; 1i j1ki + akj1; 0i j1ki ) + o(²2) Detection: bkj0; 1i § akj1; 0i ' j0; 1i § j1; 0i j0i j1i j0i j1i Repeater: Entanglement creation Gate operations: Connection Purification Entanglement creation 3 Implementation with atomic ensembles Atomic cell Internal states jxi Atomic cell j1i j0i - Weak (short) laser pulse, so that few atoms are excited. - If no detection, pump back and start again. - If detection, an entangled state is created. Description: n n 0 i j 0 i j v a c i Initial state: j n n After laser pulse: ( j 0 i + ² j x i ) ( j 0 i + ² j x i ) j v a c i Evolution: n n j 0 i j 0 i j v a c i + photons in several directions (but not towards the detectors) + 1 photon towards the detectors and others in several directions + 2 photon towards the detectors and others in several directions do not spoil the entanglement Detection: 1 photon towards the detectors and others in several directions + 2 photon towards the detectors and others in several directions negligible n Atomic 1 X i 2¼kj =n p e j1i An h0j „collective“ operators: n ay j = k= 1 n X y 1 p j a = 1 i h 0 j and similarly for b A n 0 n k = 1 Photons emitted in the forward direction are the ones that excite this atomic „mode“. Photons emitted in other directions excite other (independent) atomic „modes“. Entanglement creation: Sample A Apply operator y y ( a § b ) Sample B Measurement: Apply operator: a (A) Ideal scenareo A.1 Entanglement generation: Sample A After click: yy ( a + r ) j 0 ; 0 i (1) Sample R After click: (2) yy ( b + r ~ ) j 0 ; 0 i Sample B y y y y b + r ~ ) ( a + r ) j 0 ; 0 i Thus, we have the state: ( A.2 Connection: y y y y ( b + r ~ ) ( a + r ) j 0 ; 0 i j~ ri jri If we detect a click, we must apply the operator: ( r+ r ~ ) Otherwise, we discard it. yy We obtain the state: ( b + a ) j 0 ; 0 i A.3 Secret Communication: - Check that we have an entangled state: y y y y ~ ( b + a ~ ) ( b + a ) j 0 ; 0 i • Enconding a phase: y i ± y y y ~ ( b + e a ~ ) ( b + a ) j 0 ; 0 i • Measurement in A ( a + a ~ ) • Measurement in B: (b +~ b ) The probability of different outcomes +/- depends on ± One can use this method to send information. (B) Imperfections: - Spontaneous emission in other modes: No effect, since they are not measured. - Detector efficiency, photon absorption in the fiber, etc: More repetitions. - Dark counts: More repetitions - Systematic phaseshifts, etc: Are directly purified (C) Efficiency: Fix the final fidelity: F l o g N 2 Number of repetitions: r N Example: Detector efficiency: 50% Length L=100 L0 6 Time T=10 T0 43 (to be compared with T=10 T0 for direct communication) Advantages of atomic ensembles: 1. No need for trapping, cooling, high-Q cavities, etc. 2. More efficient than with single ions: the photons that change the collective mode go in the forward direction (this requires a high optical thickness). Photons connected to the collective mode. Photons connected to other modes. 3. Connection is built in. No need for gates. 4. Purification is built in. 4. Conclusions • Quantum repeaters allow to extend quantum communication over long distances. • They can be implemented with trapped ions or atomic ensembles. • The method proposed here is efficient and not too demanding: 1. 2. No trapping/cooling is required. 3. 4. Atomic collective effects make it more efficient. No (high-Q) cavity is required. No high efficiency detectors are required. Institute for Theoretical Physics Postdocs: - L.M. Duan (*) - P. Fedichev - D. Jaksch - C. Menotti (*) - B. Paredes - G. Vidal - T. Calarco Ph D: - W. Dur (*) - G. Giedke (*) - B. Kraus - K. Schulze P. Zoller J. I. Cirac FWF SFB F015: „Control and Measurement of Coherent Quantum Systems“ € EU networks: „Coherent Matter Waves“, „Quantum Information“ EU (IST): „EQUIP“ Austrian Industry: Institute for Quantum Information Ges.m.b.H.