固态量子操纵和量子相变. Chang-Pu Sun ( 孙昌璞 ) Institute of Theoretical Physics, Chinese Academy of Sciences

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1 固态量子操纵和量子相变 Chang-Pu Sun ( 孙昌璞 ) Institute of Theoretical Physics, Chinese Academy of Sciences suncp@itp.ac.cn

2 Related Publications and References therein Lecture 3:. C. P. Sun, Y. Li, and X. F. Liu, Phys. Rev. Lett. 9, 4793 (3). S. Yang, Z. Song, and C. P. Sun, Phys. Rev. A 73, 37 (6) 3. X.-F. Qian, Y. Li, Y. Li, Z. Song, and C. P. Sun, Phys. Rev. A 7, 639 (5) 4. Z. Song, P. Zhang, T. Shi, and C.-P. Sun, Phys. Rev. B 7, 534 (5) 5. T. Shi, Y. Li, Z. Song, and C.-P. Sun,Phys. Rev. A 7, 339 (5) 6. Y. Li, T. Shi, B. Chen, Z. Song, and C.-P. Sun, Phys. Rev. A 7, 3 (5) 7. R. Xin, Z. Song and C.P. Sun, Physics Letters A, 34,, 3 (5) Review Article: Song Z, Sun CP,LOW TEMPERATURE PHYSICS 3 (8-9): 686 (5)

3 量子态存储

4 Revisit Quantum information (QI) Quantum bit (qubit) : A Robust and Controllable Two Level System Photon: Robust, but not Controllable Electronic Spin: Controllable, but not Robust Transfer the QI between difference qubits for different goals

5 Quantum Information Storage in QIP cn n QI Carrier Cn n QI Carrier en n QI transfer QI Processing QI transfer cn Mn( t) en Mn( t) en Mn( t) Quantum memory (QME)

6 Two kind of Quantum Information Storages Type I: Qubits is hard to have without construable interaction Three Level Atom in Optical Lattice (CI) and non-local, but QME is local and has (CI) Type II: L.V. Hau et al. Nature 397, 594 (999) C. Liu et al. Nature 49, 49 () QME has much longer decoherence time than that of Qubit. Sun, Li, Liu Taylor,et,Phys. al Rev., PRL Lett.., 9, 683 9, 4793 (3) 3 Song, Zhang, Sun, Qunt-phy/343

7 Atomic ensemble based Quantum Storage Electromagnetic Induced Transparency (EIT) L.V. Hau et al. Nature 397, 594 (999) C. Liu et al. Nature 49, 49 ()

8 Atomic responsibility in EIT media Ω p, ω p > > Ω c, ω c >

9 Dark State for Single Atom Dark state D : (quantum) g (classical) Ω HI D ( n) = D D = cos θ g n+ sinθ e n n n g n+ e n g n tan θ n( t) = Ω () t θ : π / ( n) D : e n g n+

10 Dark State of Many Atoms n a n a c n a a n n ac b N n c n k c n k n c n n! dn() θ sinθ cos θ = b c k! n! k= ( ) ( ) ( ) k k n k Nk k n k

11 Many Atom dark State With Few photons a b N, ( ) N d θ = b N b c a d c b c N ( θ ) = + c a ac, c c d c b c N ( θ ) = + c + c3 c N b

12 Adiabatic manipulation of QI of Photons Dark state polariton : n dn ( t) D = acosθ Csin θ, [ H, D] = θ n d n + n ( θ ) = D n! S() = ( cn n ) a C S( π /) = c a n n C S( θ) = cn dn( θ) θ : π / g n tan θ n( t) = Ω () t

13 An Experiment of Photon Storage on Atom Ensemble C. H. van der et al, JULY 3, VOL 3 SCIENCE

14 Quantum Memory =Collective Excitation in Atomic Ensemble Inhomogeneous homogeneous N HI = g + hc k = ( k ) ( kσ +..) Bosnic Collective Excitation N HI = g + hc a k = ( k ) ( σ +..) [ bb, + ] = b N [ k ] = J_ = σ N N k=

15 Quasi-Spin Wave based Quantum Storage Sun, Li, Liu, Phys. Rev. Lett., 9, Optical lattice Real crystal lattice Phonon induced decoherence N N ikab j ( j) ikac j ( j) I σab σac j= j= H = ga e +Ω e + hc..

16 Collective Excitation Quasi-spin wave Excitations a A N ab j j= ik r ( j) σ ba = e N c b C N ikcb rj j e σ bc N j= = N j= ikac rj ( j) ca T = e σ

17 Bosonization of Collective Modes Tow bosonic Excitons [ AA, + ] = [ CC, + ] = for N Hidden dynamic symmetry SU() h(3) [ CT, ] = A, [ A, T ] = C + [ T, T ] = T + [ T, T ] = ± T 3 3 ± ± [ SU(), h(3)] h(3) H () t = g NaA + +Ω () t T + hc.. +

18 Quantum Storage Based on Collective Excitation in Quantum Dots Lukin et al, PRL,4 Nuclei ensemble QME Song, Zhang, Shi, Sun, PRB 5, Spin-Boson Model

19 Quasi-Excitation Spin-Boson Model H z z N z j Iz j N j N g j I j h. c z j g j Iz j.5. B I j g j N i g i I i Energy spectrums (a) Models (b) B, B I z I N I N j Iz j g j g, I Ng [ BB, + ] = H = wb + B + ws z + g Bs B s- ( )

20 Decoherence by Non-Storage Modes C k = N [ k] ( i) i = i + I h I j h j. ωznk ρ R = exp( ) nk nk Z k T k n B k H + ( ω + gc C ) σ p z k k z k

21 Spin-Boson Model : Spin Wave N N z n = nμ n l Sl Sl+ l= l= H g B S J Ω s H = ω bb + σ + λ σ b + σ b N ( + ) z N N N N N N Electron Nuclei π k ωk = gnμnb + Js Jscos. N N i π kl N bk = e S N l= l, Universal Storage Unit

22 Decoherence by Non-Storage Modes N s V = λ ( σ χkbk + hc..) N + k = N λ s χ k s γ = π δ ωk λ. k= N N Ft () = Ψ Ψ R

23 量子态传输

24 Quantum State Transfer and Storage Storage transfer ( C + C ) M ( C + C ) M q Interaction Interaction M = M ' k k q ( C M + C M ) q' M ' ( C + C ) QST =Non-Local Quantum Storages

25 Quantum Information Meets Condensed Matter Physics? Or Condensed Matter Physics Beats Quantum Information?

26 Other Condensed Matter System seems to Beat Quantum Information? Finite Correlated Length vs. Decaying Entanglement A qubit N Data Bus L Correlation length L<N B qubit a long range order? ϕ ϕ or /N δ x x ' Ψ ( x) Ψ ( x ') exp( ), L ζ Δ L

27 Overcome Difficulty by Engineered Spin Chain With Always-On Interactions S. Bose, 3; M. Christandl, PRL, 9, (4) i = N + + θ i i i+ i i+ i= H = J ( S S + S S ) J = i( N i) i Engineered Couplings

28 Perfect Quantum State Transfer (QST) State transfer = Time evolution of state A B ψ + ( r r,) iht AB ψ( r,) ψ '( r, t) = e ψ( r,) ψ ( r, T) iht Ft ( ) = ψ( r+ r,) ψ'( rt, ) = ψ( r+ r,) e ψ( r,) AB Perfect F ( τ) =, Near perfect F ( τ) AB

29 QST by Difference of Known Transformation: A B ψ R ( r,) iht ψ( r,) ψ '( r, t) = e ψ( r,) ψ ( r, T) iht Ft ( ) = Rψ( r,) ψ'( rt, ) = Rψ( r,) e ψ( r,)

30 Quantum State Transfer by Flying Qubit Polarized photon A photon A photon B B Spin System without any dynamical controls Spin A Polarized particle(electron) A = a Vac, = a Vac ; A Spin A, A A, B B Magnon = = + SA...,.... A A

31 Wave Packet Spreading Time x Key point: Seeking Flying Qubit in solid-state system -x

32 Fidelity of polarization Fidelity of spatial WF Other option: Two orthogonal GWPs A α ( j N ) A ik j + e e a Vac, = A Ω j = A Ω j j, α ( j N ) A ik j + e e a Vac. j, Ft () = ψ( r,) ψ(, rt) B B Also Witness: detecting the property of solid-state systems

33 QST via Modulated XY Chain Spinless fermion, Tight binding model J n = n = J z = J, M J + M + i= N + = i i i+ + i= H J ( a a hc. ) J = in ( i) i i= N H = J ( S S + S S ) i= + + i i i+ i i+ H N = Jx, J =, E =±, ± 3,, J M=N/- subspace M. Christandl et al, PRL, 9,879(4)

34 Time Evolution = S(3) Rotation Ut e R t () = ij x t = x () Perfect QST = Quantum Rotation from J, J = J, J = N x z J, M = J + M + J, M' = J + M' + Y

35 Modulated XY chain: A general class for QST i= N H= J( S S + S S ) i= + + i i i+ i i+ J i i( N i) i = even = ( i + k)( N i + k) i = odd k =,,,, The spectrum E n N+ ( n k) n=,,, N/ = N + ( n + k) n = N/ +, N

36 Commensurate Spectrum Shi, Li, Song, and Sun, Phys. Rev. A 7, 339 (5) Our discovery : Ut ( ) = P Time evolution=parity Reflection

37 Rigorous result beyond Models Consider a system H, [ Pˆ, H] =, Pˆψ( r) = ψ( r) H φ ( r) = ε φ ( r), Pˆ φ ( r) = p φ ( r). n n n n n n If ε, p can be written in the form n n ε = E + C, n p n =± ( ) n, n is an arbitrary integer, any state evolve to ± ψ( r) after time π / E. n ψ( r) can Corollary n+ - n = odd number

38 Propagation of GWP: U= Simple networks: Transmission of GWP: Quantum revival: Fast transmission of GWP: Moving GWP H = J ( a a + hc..), j, σ k, σ + j, σ j+, σ H = J ε a a + k k, σ k, σ ring & chain fast & slow τ N, N full & fractional t= j-j j+ j+a t=

39 Propagation of GWP: U= ε k = J cos k, ε k k ψ A α π A ( j N ) i j + = e e aj, σ Vac Ω j Heff = vp, p = ka a k, σ + k, σ k, σ Ut ( ) = exp( ipvt) Tvt ( ), T ( x ) j = j + x σ σ Ring : τ = ( N + )/ J ; Chain : τ = N /( J )

40 Tight Binding Networks: Y-beam N l [ l ] + j l, j l, j+ j = H = ( t a a + H. c.). N, J A N, J A + + { a } { a } l, j R mi, N, J A b N, J A J nb J nc nb nc J + J = J A a (a) M, J A Realistic Space l R H + l Hjoint H m m H α, H = β (b) M, Virtual space J A

41 Tight Binding Networks Complicated networks: Chains + Nodes + Flux N l [ l ] + l l l j l, j l, j+ j = Chain : H = H ( N ) ( t a a + H. c. ). [ lm ] + Nodes : H ( t a a + H. c.) jo int ji l, j m, i { t, t,..., t N } [ l] [ l ] [ l] l Fux l : t i Φ + π +, j l j = te ; Φ, + = φ A d l j [ l ] j l j Our Aim : + + { } { a } a l, j R mi, N l + l l l l, j l, j+ j = H = H ( N ) t ( a a + H. c.).

42 TB Networks Decomposition at Critical Node A (a) Input t t n Output M N Real space t B B B m B m a (b) Input t M Virtual space Output N t a b b b m m Quantum Interference Effects t [ ABp ] N A = t = n t m H = H + H a m q = bq H α, H = β

43 Wave Packet Propagating along TB Networks A M t nb t nc Real space N t B C M Virtual space N t a b t t t + = ψτ ( ) = cos θψ ( N ) + sin θ ψ ( N ) nc nb B τ C τ Rigorous results, for arbitrary particle sectors S. Yang, Z. Song, and C.P. Sun, quant-ph/69

44 Reflection on the Node t nc The reflection factor R(t nc,t nb ) R(t nb,t nb ) α =.3 N A = N B = NC = 5. (a) t nb. (b) t nb Around the matching condition the reflection factor vanishes J = J + J = J = A nc nb B J C

45 Beam Splitter, Entangler The maximal concurrence C max ( J nc, J nb ) = max{ C( t)} as a function of and J nc J nb t nc C max (t nc,t nb ) C max (t nb,t nb ) (a) t nb. (b) t nb J = J = J nc nb /

46 Quantum Control By Flux A M t t nb nc φ N N B C n t φ =, tnb = tnc = : a a b n φ = + 4 t = t cos θ, t = t sin θ : nb nc Arbitrary φ, t = t cos θ, t = t sin θ : nb nc t a a = t nc t / μ = tcosφ nb = a b ~ t = t sinφ μ ~ t μ

47 Quantum Control By Flux t a = t nc t / μ = tcosφ nb = b ~ t = t sinφ μ ~ t μ A (a) t ψ A ( N ) μ ~ t μ φ A (τ ) φ B (τ ) M M μ = t cos Φ Real space~ t = t sin Φ initial t B final Transmission-reflection film: a = fa fa ( + ) a, j A, j B, j a = fa + fa ( A, j + B, j) b, j Φ Φ f ± = cos ± sin a b joint a (b) ψ N ) N ) a ( t φ a (τ ) φ (τ ) M b M Virtual space ψ b ( initial b final + H = H + H + H H jo int = t( a a, NabN, + H. c.).

48 Quantum Control By Flux 3 φ A t nab M N t nbd L D φ = n + : a b t nac N t ncd t = t = t cosθ nab t = t = t sinθ nac ncd nbd φ = n : a b a arbitrary φ : t = t = t = t = t/ nab ncd nac nbd a b ~ t ac ~ t bc i t = t( + e φ ) / ac c i t = t( e φ ) / bc

49 Lattice AB effect:, ψ( j, α) Q( j, φα, ) =. ψ( A, α) max max Δ = 5.55 ( α =.3) Δ = 6.65 ( α =.) Δ = ( α = )

50 Engineered Hubbard model t t t N 3 N N H = ( t c c + hc..) + U n n, j, σ + j j, σ j+, σ j j j t = j ( N j ) j σ =, Single-electron Bloch state Angular momentum states: J, M = J + M + J, M = J Mj+ J = ( N )/, M = J, J,, J +, J

51 TB Networks: U J, J J, J t t t N 3 N J, J + J, J (a) J, J + J, J J, J U U U U U (b) J, J + N J, J J, J N N Pseudo-spin operators: J = J ij = t c c ( ) ( ) ( ) + x y j j, j+, j J = J ij = t c c ( ) ( ) ( ) + x y j j+, j, j J = jc c ( σ ) + z jσ j, σ j, σ σ =, ( ) ( ) J = ( J ) σ σ + + [ J+, J ] = J z SO(3)

52 TB Networks: U Reduction H = J + J + V ( ) ( ) x x ; V = U J, MJ ;, M J, MJ ;, M M J, M; J, M' = J, M J, M' The intrinsic dynamic symmetry: SO (3) SO (3) The goal: D D = D [ J ] [ J ] J [ L] L= [ J ] D irreducible representation of SO(3)

53 Total angular momentum L, ;,, ;,, ) ( M J M J JM L JJ C LM M J M J =, ;,, ;,, ) (, M J M J C JM L JJ M L M M M M J M J LM = + = where are the Clebsch-Gordan coefficients. The interaction term is ',,, ;,, ;,, ;,, ;, ' L L C C U M J M J M J M J U V L M J M J L M J M J M LL M = =,,, ;,, ;, L L L M m M J m J LM m M J m J m C C = δ Due to the orthogonal relation of Clebsch-Gordan coefficients the on-site interaction can be reduced as the sum of the irreducible tensors, ) (, ) ( ] [ L JJ L JJ U W V L L L = = TB Networks: U

54 TB Networks: U N H = H L= ( L) ( L) ( L) [ L] [ L] H = H + W = Jx + W. t t t t U U U U (c) 3 t N N 3 N L ( L) + + = j j j+ + + j= L H ( t a a hc..) Ua a, t = ( L + j + )( L j) j Central potential barrier U

55 Numerical results F max peaks: (a) N=4 N=5 N=6 3 4 U Swapping fidelity iht F( U, t) = J, J; J, J e J, J; J, J The maxima of the fidelity F max ( U ) = max{ F( U, t), t } There indeed exist some U to get very high F max ( U )

56 Simple Mathematics ispt e x = x+ st ispt ψ () t = e ( c s ) s φ ψ () = ( c s ) s φ = ( c s ) e s isp φ ψ ( xt, ) = xψ ( t) = ispt c x e φ s s = cφ( x+ st) s s

57 量子态相变

58 Generalized Hepp-Coleman model: Ising model in a transverse field x z z ( e j j j+) H = J g e σ + σ σ j e ( ) H = H λ = H + V e e g e g H = J σ σ + z z g j j j V = λj σ. e j x j

59 Exact solution of Loschmidt echo k + ( λ ) = kε ( ) H A A / e e k k ikal e [ x] k [ + ] k [ ] A = σ u σ iv σ, ( ) k s e l e l l N s< l ε k ( λ) e Finite N k k e e ( ( )) ε =ε ( λ ) = J +λ λcos ka K=n/Na λ =λ c = Large N

60 Ground states H = J σσ λj σ z z x e j j j+ j j H = J σ σ + z z g j j j A G = k e G =... g g B G = k g ( ) ( )( ) B± k = cos αk A± k isin αk A k, + g + + ( k) ( k) k k G = icos α + sin α A A G k> e

61 Decoherence & Loschmidt echo Reduced Density matrix ( ) ( ) [ ρ t ] = c c D t s eg g e ( ) = ϕ ( ) ϕ ( ) D t t t g e α () t exp[ ihαt] G g ϕ = Loschmidt echo ( ) ( ) ( ) L( λ,t) = D t = ϕ t ϕ t g e ( k) ( e ) k L( λ, t) = [ sin α sin ε t ]. k>

62 动力学敏感性与量子混沌 Classical Chaos: Butterfly Effect( 蝴蝶效应 ): Slightly different initial conditions leads to exponential divergence of trajectories Same Dynamics Largely different Final State Slightly different initial condition Zurek,Nature,4,7(); PRL, 89,745 ()

63 量子混沌的基本概念 ϕ e () ϕ g () Unitary Transformation ϕ e() t ϕ g () t ϕ t ϕ t = ϕ U t U t ϕ = ϕ ϕ g( ) e( ) g() ( ) ( ) e() g() e() Peres, Conception of Quantum Mechanics,995 E () t Dynamic Sensitiveness E U e Loschmidt echo U g E () t ϕ ϕ ϕ ϕ g() t e() t = Ug () t Ue() t

64 Loschmidt echo via Local density of state L L t e dt G E E iωt ( ω) = ( ) = s δ( ω s), s L() t e γ t L( ω) ( ω Ω ) + γ

65 A heuristic analysis c K c L ( λ,t) F > L( λ,t), k> k K c S(, t) ln L ln F λ = > c k k [ ] ( ) sin α kλa /( λ) k k ε e J λ λ E(K ) ( λ) c S( λ, t) sin J[ λ]t ( ) E(K ) = 4π N (N + )(N + ) /(6N ) c c c c c ( ) L( λ,t) exp γt γ=4j E(K c)

66 Numerical results: far from the critical point Our result: Unpublished, but even posted in Arxive before the PRL paper Short time behavior c ( ) L( λ,t) exp γt

67 Universality of Loschmidt Echo Cucchietti, Fernandez-Vidal, Paz, quantph/6436 ( ) L ( λ, t) exp ut F(t) c Coupling independent u transverse field Ising model Boson Habburd model

68 Numerical results : near the critical point Small N Large N

69 量子相变环境导致退相干增强 near the critical point Far from critical point Small N Large N 由于联系了凝聚态物理中的量子相变非平衡统计物理中的量子混沌和量子信息中的量子退相干, 立即得到不同领域科学家的重视, 引发一系列后续的工作 ( 如由保真度描述量子相变和经典相变 ), 引用已经超过次

70 理论预言的实验证实之一 8 年德国 Suter 小组的核磁共振实验

71 理论预言的实验证实之二 9 年加拿大 Laflamme 小组的新实验

72 Implementation : Superconducting Quantum Network Circuit QED for Charge Qubit array

73 Model Hamiltonian H h B ( α) ( α) ( α ) = [ λ] ( λ σx + σz σ + z ) α α NEC : C / C Σ.5 m ( ) φx = η a+ a ( ω ) / η = ( S/ d) l / L ( ) HF = ωa a g a a+ aa σ ( α ) x α

74 Pseudo-Spin Representation: S S S = γγ + γ γ, zk k k k k = iγ γ + γ γ, xk k k k k = γ γ γ γ. yk k k k k G = Ok, O H k k> k> n = > k H ( k ) n k H = ε ( S cosα + S sin α ) ( k ) n nk zk nk xk nk k m ih t ih t Dmn e e k> k n = k

Z-Factory Physics. Physics for Modernized Z-Factory. Chao-Hsi Chang (Zhao-Xi Zhang) ITP, CAS

Z-Factory Physics. Physics for Modernized Z-Factory. Chao-Hsi Chang (Zhao-Xi Zhang) ITP, CAS Z-Factory Physics Physics for Modernized Z-Factory Chao-Hsi Chang (Zhao-Xi Zhang) ITP, CAS --- Working Group for Z-factory Physics --- Group Meeting @ WeiHai The topics for Z-factory Physics July 13-17,