Li You, Georgia Tech (KITP Quantum Gases Conf 5/13/04) 1

Transcription

1 Li You D. L. Zhou B. Zeng, M. Zhang, Z. Xu (Tsinghua Univ.) C. P. Sun, (ITP) Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04)

2 i (,,, +,, e ϕ ) -GH Z =, i ( e ϕ ) = + W. M. Itano et al., Phs. Rev. A 4, 95 (990).!"#!#$ % & ' ( ) * +, - Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04)

3 . /00-. 4/005 (" 3$ A B Ψ ( r A, r B, ) = αiφi ( r A ) β jψ j ( r B ) i= j= = αφ ( ra ) + αφ ( ra ) + βψ ( rb ) + βψ ( rb ) +,, A B i, j, i A j B i, j, = A B ( r, r, ) φ ( r ) ψ ( r ) Ψ = ϒ '# Schmidt decomposition T ϒ = U DV * 6 Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 3

4 Φ = ( r, r, r, ) φ ( r ) φ ( r ) φ ( r ) 3 3 GP equation (coarse-grained mean-field) + + = M Vext r asc r r r ( ) φ ( ) φ ( ) µφ ( ) Ψ r, r Two-atom scattering state ( ) a r r r r sc umber fluctuations, Phase diffusions, Goldstone modes, Phs. Rev. Lett. 77, 3489 (996); 90, (003). 7 ' 7 % * 8 9 : Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 4

5 ; 9 / 6 ' * %! (mode) entanglement? ( a + b ) 0,0 +, (,0 +, + 0, +, ) 0,0, [ ϕ ( r ) ϕ ( r ) + ϕ ( r ) ϕ ( r )] a b a A b B b A a B L.-M. Duan, J. I. Cirac, and P. Zoller, Phs. Rev. A 65, (00) Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 5

6 "# F ωij fi f j i, j Ψ = 0 / = Z f f 0 k ' ' k k k John Schliemann, Daniel Loss, A. H. MacDonald, Phs. Rev. B 63, 0853 (00). John Schliemann, J. I. Cirac, M. Ku\'s, M. Lewenstein, and Daniel Loss, Phs. Rev. A 64, 0303 (00). "# Ψ B = βijbi bj 0 i, j ' ' Bk bk bk 0 k = = B,0,0, + B 0,,0, + B 0,0,, + 3 R. Pauskauskas and L. You, Phs. Rev. A 64, 0430 (00). Y.S. Li, B. Zeng, X.S. Liu, and G. L. Long, Phs. Rev. A 64, (00). Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 6

7 $ % # L. You D. L. Zhou B. Zeng, Z. Xu (Tsinghua Univ.) "% % 3 <" : i 0 + e j l 000 l 00 l 0 l 0 l &, : GHZ = ( ) W = ( ) 3!"# $%& '( Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 7

8 "% # " : e j i r s ( ) + t &, : GHZ W = z a a a + z a b bbb = ( ) ( a bb ba b bba ) + a b & L. You, Phs. Rev. Lett. 90, (003) ( ) -GHZ =,,, +,,, B. Zeng, D. L. Zhou, P. Zhang, Z. Xu, and L. You, Phs. Rev. A 68, 0436 (003). -GHZ ρ particle -GHZ > Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 8

9 a (k)a(k)a(k) a (k)a(k)a(k) Controlled entanglement of quantum degenerate atoms ' EPR, Phs. Rev. 47, 777 (935); J. S. Bell, Phsics (Long Island Cit, Y), 95 (964). A. Aspect, J. Dalibard, Roger, PRL 49, 804 (98) BELL ( ) = +! Bell Z. Y. Ou & L. Mandel, PRL. 6, 50 (988) Y. H. Shih & C. O. Alle, PRL. 6, 9 (988) Kwiat & Zeilinger et. al, PRL. 75, 4337 (995) ( )* Zeilinger A. Zeilinger et. al, PRL 8, 345 (999). 4-photons J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger Phs. Rev. Lett. 86, 4435 (00) Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 9

10 a (k)a(k)a(k) a (k)a(k)a(k) Controlled entanglement of quantum degenerate atoms + Wineland Monroe D. J. Wineland, and C. I. Monroe et. al, ATURE 404, 56 (000) five 9 Be + ions in linear trap )&* +,'--- Ω σ i< j σ ( i ) ( j ) R x x ( ) ( ) = Ω R + + i j J x "#$%,,,, +,,,, Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 0

11 % %, J = mode condensate Schwinger representation J x = ( b a + a b) i J = b a a b J z = ( b b a a ) ( ) J - H = uj z, ( ) ϕ ( ) 4 u= a + a a dr r aa bb ab a/ b Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04)

12 ./0 z x J, J, J, J x z z Can be used to generate an smmetric state "# Single atom Raman coupling H = ΩJ is readil implemented with external laser pulses Step 3 Step i π J iuj z t / i π J U ( t) e e e = e Step x iuj t / Ω u Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04)

13 "# D. S. Hall et al., Phs. Rev. Lett. 8, 453 (998) two-photon transition one-photon transition ikuni and Williams, cond-mat/ J, single atom tunneling rate of the nearest neighbor wells u, on-site collisional interaction Quantum Phase Transition u J z.6, M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, and I. Bloch, ature 43, 44 (00). D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phs. Rev. Lett. 8, 308 (998). Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 3

14 - # ( ) ϕ ( ) 4 u= a + a a dr r aa bb ab a/ b aaa abb aab ( π) ( ) ωtrap 30 khz ( ) 0( Hz ) -GHZ in 0 (milli-second u π )! "# $ % &% '( $ % )% '&** % +,-. E. A. Burt et. al., Phs. Rev. Lett. 79, 337 (997). 0 H = u S + Ω S z ( u t ) = ( k + ) π Ω ( u t ) + = m π u Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 4

15 +, 0()* =4 atoms 0 - = atoms. /!0 " 3 /! 4 " / Ω u, 0 0 f o r u p to 4 -a to m s / + "%. (,,,,,, ) + Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 5

16 * & &'% ( ) *' PRL, 9, (003). +*, Ψˆ + Ψ ˆ = Ψˆ 0 ˆ Ψ m f = c 4π 4 3 a a c π = + = M 3M a a ( ), ( ) Single mode approximation Ψ a ( t) ( r ) α α φ Phs. Rev. A 66, 060 (00). Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 6

17 a (k)a(k)a(k) Controlled entanglement of quantum degenerate atoms.%# ), a, a, a = a a + a a + a a ( 0 + ) + ( 0 0 ) J = a a + a a J + + = A = a a a 3 J = J z a + a + a a ( ) J = J + J + J = + 3 A A x z ).''/.!0). Law et al., PRL 8, 557 (998); Ho and Yip, PRL 84, 403 (000)../ S = 4T ( )( ) ( ) 3 + ε+ ε Y Y0 + G Y Phs. Rev. A 66, 0336 (00). Gell-Mann decomposition 3 ( ) 0 3 ( 0 ) 0 3 ( 0 ) ( ) 3 T = a a ; T = V = a a ; V = U = a a ; U = + Y = V 0 U + -- T ( ) G = V U + h. c. Y + + Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 7

18 * P. Facchi and S. Pascazio, Phs. Rev. Lett. 89, (00). H = H + KH K measurement Ω T 3 * Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 8

19 4()* =0 atoms 4 % 5 0 L. You D. L. Zhou B. Zeng, Z. Xu (Tsinghua Univ.) C. P. Sun (ITP) x 0 z Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 9

20 ! " ' #"# ( 5 æ q k q k ö = Õ cos 0 + sin ç è ø k = 5 H ( j) ( k) I µ å s s : J j, k =, j< k 6 5 p p qi + qi + i ( i) ( i) 0 x = cos 0 + sin p p qi + q + i i sin 0 i cos x = - + i ( i) ( i) Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 0

21 4()* æ p ö æ p ö æ ( j) p ö ( j) ( k) S = exp i ( ) exp i( ) s exp i s s ç è 4ø ç å 4 j ç å è ø è 4 j, k =, j < k ø = å i, j =, i< j ( i) ( j) ( i) ( j) ( I s s s s ) %89# 5 æ ö = + ç ø i i ç M Õ 0 x Õ x çè i= i= H. J. Briegel and R. Raussendorf, Phs. Rev. Lett. 86, 90 (00) % 5 æ q k k cos 0 sin q i ö = Õ + e f ç è ø k = ; " &<%89#5 æ ö æ ö = è ø è ø ( k) s ( k) k sk F å f S f çõ çõ { sk } k = k = : f f + - k k hk h = cos 0 + sin hk h = - sin 0 + cos ( k) k ( k) ( k) k ( k) Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04)

22 & ( k ) = ( + - sin k sin k ) ( + - sin sin ) p max h q f q f x 0 z 0.5 x z = 30 = 3. & =# 3% > # >#? J 7 ' = =#3% > & $, 6 < 3% > Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04)

23 % L. You D. L. Zhou, B. Zeng, Z. Xu (Tsinghua Univ.) C. P. Sun (ITP) an -bit error: E = e I + e X + e Z + e X Z i i0 i i i3 bit-flip: X : α 0 + β α + β 0 phase-error: Z : α 0 + β α 0 β Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 3

24 bit-flip code L L phase-flip code L L Initial state H = uj x ( ) ( ) ψ 0 = α 0 + β 0,,0 9 5% 8 - ancilla bits "% L L bit error rate p <, 3 ( bit ) error rate p + p ( < p) 3 3, error-detection (quantum measurement): P P P P no error bit flip on qubit one bit flip on qubit two bit flip on qubit three Recover: flip the flipped qubit again Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 4

25 "% % L L error-detection (quantum measurement): P 0 P P P no error phase flip on qubit one qubit two qubit three Recover: flip the flipped qubit again "%.% L 000 L 3 3 P. Shor, PRA 5, 493 (995). Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 5

26 7 % α 0 + β α 0,,0 + β,, + + Initial state ( ) ( ) ψ 0 = α 0 + β 0,,0 ancilla bits 3 ( α β ) 0 + 0,,0 σ ( ) ( ) z σ z H = uj x H = uj x Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 6

27 - ( α + β ) 0 0,,0 H = uj x &! ( α β ) 0 + 0,,0 = 8 α 0,0,0 0,0,0 0,0,0 + β,,,,,, 3 3 α ( 0,0,0,, ) + β ( 0,0,0 +,, ) 9 5% 8 - Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 7

28 -..$. * S ( ) = exp ( ga g a ) t ξ [ x, p x ] = i x p x. $ [ S i, S j ] = is k Sx S Sk 4 S Sstem of fixed particles S = S Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 8

29 ./0 % Coordinate rotation squeezes spin? z Squeezing: area preserving deformations S S M. Kitagawa and M. Ueda, Phs. Rev. A 47, 538 (993) D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phs. Rev. A. 50, 67 (994). random correlated M. Kitagawa and M. Ueda, PRA 47, 538 (993) Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 9

30 . $ x : = DS S ( ) ^ min )./0)',--' "!&&--(, $ ( S )( Sn ) ξ < ( Sn ) + ( Sn ) 3 n, n, n 3 Mutuall orthogonal unit vectors " ρ p ρ ρ ρ ρ () ( ) (3 ) ( ) k k k k k k A. Sorensen, L-M. Duan, J. I. Cirac, and P. Zoller, ature 409, 63 (00). Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 30

31 7 & # z U(t) = exp(-iφsz) φ U(t) = exp(-iχtsz ) S z φ = χts z z Coherent Minimum variance reduced from ½S to ½(S/3) /3 "# u ( ) ( ) ( µ λ ) Heff = a a aa + b b bb a a + b b a b + b a u = S z λ S A. Imamoglu, M. Lewenstein, and L. You, x Phs. Rev. Lett. 77, 5 (997) S S + = = a b S + S z = b b a a ( ) Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 3

32 4 :% $ G b ( t) a ( t) G exp u σ a/b( ) t τ c = u σ a / b "% % Science 9, 386 (00). ature 49, 5 (00). Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 3

33 "# & # Swirling Effect Cancels Out U(t) = exp[-χt(s + -S - )] x x x Coherent Squeezed Minimum variance reduced to ½ $ Li You K. Helmerson (IST) M. Zhang B. Deb (PRL, India) Kristian Helmerson and L. You, Phs. Rev. Lett. 87, 7040 (00). Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 33

34 - ' e Ω Ω g g For each atom: aborption of one photon emission of one photon :' e atomic excited state molecular excited state e g K K M g g K ( M) K Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 34

35 0 ' ( i) ( j) ( i) ( j) Ω R σ+ σ+ + σ σ For ever pair of atoms: absorption of one photon emission of one photon 4 H u J J R = ( x ) H S = uj x optimal spin squeezing snap shots Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 35

36 4 0; 4<== H = u J J H = uj R ( x ) S x 4 H = u J J R ( x ) 50% overlap with the maximall entangled state Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 36

37 ; < " $,= #3% > &!# Li You, Georgia Tech (KITP Quantum Gases Conf 5/3/04) 37