Ming-Shien Chang. Institute of Atomic and Molecular Sciences Academia Sinica

Transcription

1 Ming-Shien Chang Institute of Atomic and Molecular Sciences Academia Sinica

2 Introduction to Quantum Information motivation preliminary & terminologies physical implementation Ion Trap QC/QS ion trap physics laser manipulation of ions implementation of a quantum entangling gate application: simulation of quantum magnetism Outlook current status and perspective 1

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5 Qubit = quantum two-level system = spin-1/ particle I 1 1 z 1 1 x 1 1 y i i Choices of qubit: any quantum two-level system conveniently accessible to us. e.g. Photons (flying qubit) Atoms (material qubit) i cos e sin Through out this talk, we mix-use the notations: {, 1} and {, }

6 QC ~ built upon a universal set of quantum logic gates Classical 1-bit NOT Quantum 1-qubit gate (rotation) 1 1 cos + e i sin 1 1 cos 1 e i sin -bit AND qubit XOR (controlled-not, or CNOT) superposition entanglement via CNOT [ + 1] + 1 1

7 Suppose we have a two-qubit phase gate CNOT PG One can realize a CNOT gate by the following steps: Rb ( x, ) R (, ).. b x b b PG b b a b a a a b

8 Superposition parallel processes N inputs f(x) Quantum interference enhances correct outcomes and suppresses erroneous outcomes. quantum logic gates depends on all inputs D. Deutsch (1985) Superposition and entanglement can speed up certain algorithms. n H n 1 H U Entangling Gate H n Solve for f :{,1} {,1} Deutsch-Jozsa (Cleve et al.) algorithm: deterministic and requires only one quiry

9 1. Scalable system of qubits: a b a b. A qubit-specific measurement capability 3. Initialization:,,, 4. A universal set of quantum gates 5. Decoherence times τ >> gate operation time a b 1 a b ; 1. Two additional criteria for quantum communication 6. Interconvert stationary and flying qubits 7. Transmit flying qubits between distant locations D. P. DiVincenzo, Fortschr. Phys , ()

10 Bottom-up Nuclear spin in QD Diamond NVC coherent scalable controllable measurable Trapped ions Atoms in OL Cavity QED molecule NMR Top-down Cooper pair box SQUID

11 Ion quantum computer Advantages Long storage time ( hours to months) Long coherence time ( seconds to minutes) Strong controlled interactions ( khz to MHz) State-of-the-art quantum entangling gates Cirac-Zoller gate Milburn-Schneider-James gate (σ z gate) Molmer-Sorensen gate (σ x gate) Ultrafast gate (multi-mode coherent control) Laserless gate (B field gradient) Perspectives Scalable to a large quantum system (currently < ions) Improve fidelity: toward fault-tolerant QC (infidelity < 1-3 ) fidelities of initialization, state detection, single- and two-qubit gates

12 What is an entangled state? E.g., two particles are entangled if their complete quantum state cannot be expressed as the product of the quantum states of each individual one. I.e., Product state: a, b a b Entangled states: 1 1, ab a b ( 1 ) ( 1 ), a ab b 1 1 a, b a b a b a, b a b a b a b a b Peris-Horodeck criterion, and O. Ghne and G. Toth, Phys Rep 474, 1 (9)

13 Entangled state generation Strategy I. delete some constituent states 1 1, ab a a b b ' a, b a b a b a b a b Strategy II. alter the phase (sign) of some constituent states a, b a b a b a b a b ' Uˆ a, b a b a b a b a b

14 Strategy I Strategy II photodetector beamsplitter photodetector ~ μm optical fibers U 1 i ( e 11) 171 Yb + 1 meter 171 Yb + i U( ) exp{ H}, ps 8.1 MHz H J z z. 1 Moehring et al., Nature 449, 68 (7)

15 Entanglement vs. entangling gate Strategy I. deletion Measurement based: probabilistic No outcome for 11 & Not a gate! Strategy II. spin-dependent phase shift i 1 1 i 1 Interaction based: deterministic A phase gate

16 measurement induced entanglement or 1 Hong-Ou-Mandel Interference - photon bunching for identical photons photon bunching i rr,, rr + e rr, bb,, bb + e bb, ab, AB, AB, i ab, AB, AB, destructive interference of paths, no outcome... unless two photons are not identical rb,, rb + e rb, i br, rb, e br, ab, A, B i a, b AB, A, B A, B A B coincidence photon detection a b Y.H. Shih and C.O. Alley, Proc. nd Int l Symp. Found. Quant. Mech, Tokyo (1986) Hong, Ou, and Mandel, Phys. Rev. Lett., 59, 44 (1987) Y.H. Shih and C.O. Alley, Phys. Rev. Lett. 61, 91 (1988)

17 Preservation of Entanglement Strategy I. deletion Measurement based: probabilistic Entanglement is destroyed upon measurement! Strategy II. spin-dependent phase shift i 1 1 i 1 Interaction based: deterministic Entanglement is preserved.

18 or 1 heralded entanglement photodetector beamsplitter photodetector = ( 1 blue red 1 ) ( blue + red ) ion state photon state optical fibers upon coincidence photon detection 1 Preserve entangled state for two atomic qubits Moehring et al., Nature 449, 68 (7) Olmschenk et al., Science 33, 486 (9) Maunz et al., PRL 1, 55 (9) Pironi et al., Nature 464, 11 (1) Hong, Ou, Mandel, PRL 59, 44 (1987) Y.H. Shih & C. O. Alley, PRL 61, 91 (1988) C. Simon and W. Irvine, PRL 91, 1145 (3) L.-M. Duan, et. al., Quant. Inf. Comp. 4, 165 (4)

19 Strategy I Strategy II beamsplitter photodetector photodetector ~ μm optical fibers U 1 i ( e 11) 171 Yb + 1 meter 171 Yb + i U( ) exp{ H}, ps 8.1 MHz H J z z. 1 Moehring et al., Nature 449, 68 (7)

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21 In this talk 171 Yb + is used.

22 Add micromotion discussion Electric Field Vectors E

23 photo ionization lasers cooling lasers E dc atomic beam rf ground static E rf

24 E dc rf ground static E rf

25 S 1/ F=1 37 nm 171 Yb + F= HF m HF 1.64 GHz 1 < few MHz 3-layer, 3-zone trap

26 P 1/ 1..8 S 1/ 1.6 GHz Probability State detection fidelity ~ 97% Photon number 5

27 γ = MHz P 1/ δ ~ few γ S 1/ 1.6 GHz

28 P 1/ S 1/

29 Coupling two ground hyperfine states (qubit states) with a Raman transition (two-photon transition) ~ THz P 1/ ~ MHz Spontaneous emission rate due to off-resonance excitation S 1/ ν =1.6 GHz 1 R 1 λ ~ 37 nm 1 We want spon 1 spon 1 R Δ Ω R

30 Coupling two ground hyperfine states (qubit states) with Raman transition (two-photon transition) P 1/ S 1/ ν =1.6 GHz R 1

31 Rabi oscillation 1..8 P( x1-3 Ramsey oscillation Time (ms) π/ π/ R ( ) Coherence time > 7 ms P() P( ) sin( ), : detuning : phase Time (ms) 5 6 7

32 cm Axial Modes zigzag Transverse Modes zigzag cm H pi 1 xv x i m i, j 1 k( aa k k ) k i ij j cm = center of mass mode transverse e.g. 5 axial 1 / 1 com com zig-zag m Axial cm zig-zag Transverse ion-ion spacing is ~m, while the vibration amplitude is ~ nm. James, APB 66, 181 (1998); Zhu, Monroe, and Duan, PRL 97, 555 (6).

33 cm Axial Modes zigzag Transverse Modes zigzag cm H pi 1 xv x i m i, j 1 k( aa k k ) k i ij j cm = center of mass mode transverse e.g. 5 axial 1 / 1 com com zig-zag m Axial cm zig-zag Transverse ion-ion spacing is ~m, while the vibration amplitude is ~ nm. James, APB 66, 181 (1998); Zhu, Monroe, and Duan, PRL 97, 555 (6).

34 Spin-motion coupling pˆ 1 H ˆ ˆ ˆ z m x E( xˆ) m ( a a 1 ) frequency of applied radiation E ( ˆ ˆ )( e ikxˆ i t L e ikxˆ i t L ) interaction frame; rotating wave approximation H g( ˆ e ikxˆ i t ˆ e ikxˆ i t ) x = L = detuning k = = wavenumber it it xˆ x( ae a e ) m x

35 H g ˆ e ˆ e it it it it ikx ( ae a e ) i t ikx ( ae a e ) it g{ ˆ e (1 ikx ( ae a e ) H. O.] HC..} it it it stationary terms arise in H at particular values of = H g( ˆ ˆ ) kx n 1 1,n H,n = ħg CARRIER = H g( kx )( a a ) 1 kx n 1 1,n1 H +1,n = ħg n 1 1 ST RED SIDEBAND = + H 1 g( kx)( ˆ a ˆ a ) kx n 1 1,n1 H,n = ħg n 1 ST BLUE SIDEBAND kx n 1 1 Lamb-Dicke Limit

36 Δk P 1/ ~ THz S 1/ ν ~1.6 GHz 1 1 Transition rate R ~ 1 MHz ~ few MHz carrier blue sideband red sideband Hc hc.. Hbsb a hc.. Hrsb a hc.. Lamb-Dicke Parameter x kx n 1

37 Δk S 1/ ν ~1.6 GHz 1 1 Transition rate R ~ 1 MHz ~ few MHz carrier blue sideband red sideband Hc hc.. Hbsb a hc.. Hrsb a hc.. Lamb-Dicke Parameter x kx n 1

38 Δk Two ions, transverse modes. Red Motional Sideband Tilt x, CM x, Tilt y, CM y Carrier Blue Motional Sideband CM y, Tilt y, CM x, Tilt x 1.5 Brightness Motional detuning (MHz)

39 1 st Step: Doppler Cooling P 1/ S 1/ ω T D n 1 k B P state linewidth vib. mode freq. I S (RS) C frequency I AS (BS) Thermometry: I I S AS n 1 n

40 P 1/ rsb -pulse n= -1 opt. pump n> ~ S 1/ after Raman Sideband cooling: Red sideband,n,n-1 n ~.5 Blue sideband,n,n+1 frequency frequency

41 ˆ r H ˆ ˆ rsb ahc.. 1 S 1/ 1

42 ˆ r H ˆ ˆ rsb ahc.. 1 S 1/ 1

43 ˆ r H ˆ ˆ rsb ahc S 1/ 1 S 1/ 1 State mapping: ( + ) m

44 ˆ r H ˆ ˆ rsb ahc S 1/ 1 S 1/ 1 State mapping: ( + ) m ( m + 1 m ) Cirac and Zoller, PRL 74, 491 (1995)

45 Scientific American, July 1, 8 laser cool to rest (n=), map j th qubit to phonon flip k th qubit if phonon present map phonon back to j th qubit entangled state! Cirac and Zoller, PRL 74, 491 (1995) Schmidt-Kaler et al., Nature 4, 48 (3)

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47 ˆ( ) A a * a Displacement Op.: ˆ e ( ) ( ) A D Coherent state: () t D( ) D t e D D iim( * ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) iim( D D D e ) * Here we have used Baker-Campbell-Hausdorff relation ee e A B C, where C AB [ AB, ] ([ A,[ AB, ]] [[ AB, ], B])

48 Ht ta ta * () [ () () ] n i Ut ( ) limexp{ Ht ( k ) t} exp{ i( t)} D( ( t)), n k 1 where t t/ n, t kt, and t () t dt' ('), t k t t' * ( t) Im dt' ( t') dt'' ( t'').

49 1 H aa f txa ftxa * ( ) () () H t f t x ae f t x a e * () () i t I () it f t i( ) t () Fe / F* x Fx it it HI () t ae a e i Ut ( ) exp{ ( Ht ( ') dt' dt' dt''[ Ht ( '), Ht ( '')] )} i t 1 t t ' Fx t it' () ' it t e dt (1 e ) Fx

50 Entangling Ions via Spin-Dependent Force Bichromatic Raman lasers create spin-dependent force: H Fxˆ( ) Fx ( aˆaˆ ) ˆ z x y (or, ) 1 imt imt z i m m im, Ht () [ ae ae ] rsb bsb Bichromatic Δk b r HF HF

51 Entangling Ions via Spin-Dependent Force Bichromatic Raman lasers create spin-dependent force: H Fxˆ( ) Fx ( aˆaˆ ) ˆ z x y (or, ) 1 i U( ) Texp{ H( t) dt} 1 i i exp{ H ( t) dt dt dt [ H ( t ), H ( t )] } ~ exp{ i } imt imt z i m m im, Ht () [ ae ae ] 1 1 z z 1 t Magnus expansion ( ) U() 11 i 1 i 1

52 z Fx H Fx (1 z) ~ z p -1 1 Right circularly polarized light : s Qubits subject to 1 st order Zeeman shift.

53 z Fx H Fx (1 z) ~ z p ~1 MHz THz s Don t work for clock states with dipole allowed transitions! ~ 1 GHz

54 x Fx H Fx (1 x) ~ x p ~1 MHz THz s Apply spin-dependent force and flip the spin simultaneously: σ z force in the x basis ~ 1 GHz

55 X Fx iti iti H x( ae a e ) x ~ Z + Z x ~ Z - Z X p X x Initial in = Z =( X - X ) motion= Apply bichromatic force: = X + X - = Z ( - -) + Z ( + -)

56 Spin dependent force time scan 1..8 (b) (c) without sb cooling with sb cooling P() time (ms)

57 σ x force on two ions: Molmer-Sorensen ( σ x σ x ) Gate Δk b HF r HF n+1 n n-1 n n n n n 1 n 1 t ( n1 n) t Molmer & Sorensen, PRL 8, 1971 (1999) & PRL 8, 1835 (1999)

58 σ x force on two ions: Molmer-Sorensen ( σ x σ x ) Gate Δk n b HF r HF n+1 n n-1 t T i T cos( ) e sin( ) n choose T, then 4 1 i 1 e ( n1 n) Molmer & Sorensen, PRL 8, 1971 (1999) & PRL 8, 1835 (1999) t

59 σ x force on two ions: Molmer-Sorensen ( σ x σ x ) Gate Δk n b HF r HF n+1 n n-1 t Generally t i t cos( ) e sin( ) t i t cos( ) e sin( ) t i ' t cos( ) e sin( ) t t i ' t cos( ) e sin( ) Molmer & Sorensen, PRL 8, 1971 (1999) & PRL 8, 1835 (1999) n

60 1 ( i e ) Brightness Parity C.M. Tilt.5. LP TP..1. Brightness.3.4 Time (ms) P( ) P( ) P( ) P( ).5.6 BP( ) P( ) P( ) Parity Normal node freq. (MHz) Gate fidelity: F~98%!

61 P( ) P( ) P( ) P( ) Parity oscillation 1. P()+P()-P()-P() e.g F~98% 3 4 phase (deg) Gate operation number 15 D F D 6 P()+P(() Parity Gate fidelity , with pure F ideal 1 i ideal e mix ideal P( ) P( ) F

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63 Ising spins in transverse B field: H J B i, j z z z x x ij i j i i XY model: x x y y H J ( ) i, j ij i j i j XXZ model : (~ Bose-Hubbard under Holstein-Primakoff transformation) x x y y z z z H Jij( i j i j ) Jij i j i, j Possible Observations Quantum phase transition Spin frustration Complex entangled states Provided tunable spin-spin interactions: Strength, Sign (ferro or anti-ferro), Range, Coupling graph (geometry).

64 1. Scalable system of qubits: a b 1 ; a b 1.. A qubit-specific measurement capability 3. Initialization:,,, 4. A Global universal and always-on set of quantum interactions gates (analogue version, no error accumulation) 5. Decoherence can serve as environment for the studied system. Easier, and closer to Feynman s original proposal for a QC.

65 H XY B i J ( i) () i ( j) y ij x x i j P() 1..5 Rabi oscillations R ( ) Ramsey oscillations π/ π/ Coherence time > 7 ms..5 (ms).1

66 (J ij ) i () i J () i ( j) y ij x x i j H B Axial com Axial Modes Axial zigzag Transverse Modes Transverse zigzag Ttransverse com Δk m 5 1 J m m i j i j m i, j, m m

67 Spin frustration in triangular lattice J (khz) 1-1 Frustration Frustration J 1 J Detuning s -1 = ( / com ) z

68 ground states vs. spin-spin couplings,,,,, J (khz) 1-1 Frustration Frustration J 1 J Detuning s -1 = ( / com ) z

69 Exact Ground State (Theory) Freericks and Duan ZIGZAG TILT COM P(Ferro)= P()+P() 1. B/J rms s COM z.

70 Quantum simulation: implementation Initializ ation Cooling Optical Pumping Ht () J Bt () R x (/) () i ( j) () i ij x x y i j i Ground state J B 1 Ground state R y (/) Detection () x x x Adiabatically following () T ˆ t t e H () dt () i Lloyd, Science 73, 173 (1996) and (8). Farhi et al., Science 9, 47 (1); ( ion simulator) Friedenauer et al., Nat. Phys., 4, 757 (8)

71 Phase diagram measurement Measurement Theory: given ramp Theory: GS Kim et al., Nature 465, 59 (1)

72 Universal phase diagram FM AFM

73 FM ground state:? or? AFM frustrated ground state: If we know the density matrix (8 x 8), we can know the underlying state. However, density matrix is hard to reconstruct.? J>? Luckily there is a short cut

74 FM ground state: AFM frustrated ground state: GHZ state Witness = (1) () (3) J z 4 5 ( J J ), W state Witness = 9 4 y z? J 1 i? () i J>? O. Ghne and G. Toth, Phys Rep 474, 1 (9). GHZ state witness Links frustration to ground state entanglement.

75 PM = P(all up) + P(all down) QS of transverse Ising model with all FM interactions time (ms) spins 3 spins 4 spins 5 spins 6 spins 7 spins 8 spins 9 spins Sharper phase transition as # of spins increases. Islam et al., Nature Communication (11)

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77 Harmonic external axial potential ( z ) μm 375 μm z r linear crystal: r z.73n.86 z 4 μm

78 Ion Trap Chips NIST-Boulder GaTech/MIT Maryland Ulm

79 Scaling a single crystal to >> 1 ions Uniformly-spaced ion crystal (spacing = s) r 7 (3) e 3 ms U() z log 1 z / R U 4 ~ z (quartic) R > > >

80 Raman transition with picosecond lasers I(f) f n =n- f = repetition rate=1/t = Comb offset from harmonics of = Phase slip b/t carrier & envelope each round trip Advantages: Built-in phase locked frequency comb for Raman transitions. E(t) t Requirements: 1. Bandwidth HF splitting. Lock carrier envelop phase r.t D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, Science 88, 635 ().

81 Impulsive (fast) spin-dependent kicks two sequential -pulses e e e e p = ħk no speed limit! no temperature limit! (only require harmonicity) eg: 4-pulse protocol Garcia-Ripoll, Zoller, & Cirac, PRL 91, (3)

82 A quantum computer with ~1 qubits G.-D. Lin, et al., EPL 86, 64 (9) p c Optimal control: finding the right pulse sequence to bring the atom back to its initial state, regardless of the details of that initial state. x c Garcia-Ripoll, Zoller, & Cirac, PRL 91, (3) PRL 14, 1451 (1), PRL 15, 95 (1)

83 Quantum Information Processing Trapped ion QC/QS o Driving coherent dynamics with lasers o Implementing a quantum entangling gate o Engineering spin-spin interactions o Quantum simulator of a three-spin network Outlook o Scalable to larger number of qubits (spins) o Entangling (fast) gates with a ultrafast pulse laser. 81

84 Grad Students Steven Olmschenk Jon Sterk Simcha Korenblit Dave Hayes Rajibul Islam Andrew Manning Jonathan Mizrahi Dave Hucul Crystal Senko Undergrads Brian Fields Kenny Lee Postdocs Dzmitry Matsukevich Kihwan Kim Peter Maunz Wes Campbell Le Luo Qudsia Quraishi Emily Edwards Theory Coworkers U of Michigan: Luming Duan Guin-Dar Lin Georgetown Univ. J. Freericks C.-C. J. Wang pfc@jqi