Mark Acton (grad) Kathy-Anne Brickman (grad) Louis Deslauriers (grad) Patricia Lee (grad) Martin Madsen (grad) David Moehring (grad) Steve Olmschenk (grad) Daniel Stick (grad) US Advanced Research and Development Activity US Army Research Office US National Security Agency National Science Foundation FOCUS FOCUS Center Boris Blinov (postdoc) Paul Haljan (postdoc) Winfried Hensinger (postdoc) Chitra Rangan (postdoc/theory – to U. Windsor) Luming Duan (Prof., UM) Jim Rabchuk (Visiting Prof., West. Illinois Univ.) David Hucul (undergrad) Rudy Kohn (undergrad) Mark Yeo (undergrad) NSF

Trapped Atomic Ions I Quantum computing and motional quantum gates Christopher Monroe FOCUS Center & Department of Physics University of Michigan

“When we get to the very, very small world – say circuits of seven atoms - we have a lot of new things that would happen that represent completely new opportunities for design. Atoms on a small scale behave like nothing on a large scale, for they satisfy the laws of quantum mechanics…” “There's Plenty of Room at the Bottom” (1959 APS annual meeting) Richard Feynman

A quantum computer hosts quantum bits that can store superpositions of 0 and 1 classical bit: 0 or 1 quantum bit: |0 + |1 Benioff (1980) Feynman (1982) examples of “qubits”: N S N S h V H atoms particle spins photons

GOOD NEWS… quantum parallel processing on 2 N inputs Example: N=3 qubits  =a 0 |000  + a 1 |001  + a 2 |010  + a 3 |011  a 4 |100  + a 5 |101  + a 6 |110  + a 7 |111  f(x) …BAD NEWS… Measurement gives random result e.g.,   |101  f(x)

depends on all inputs quantum logic gates |0 |0  |0 |0 |0 |1  |0 |1 |1 |0  |1 |1 |1 |1  |1 |0 e.g., (|0 + |1)|0  |0|0 + |1|1 quantum XOR gate: superposition  entanglement |0  |0 + |1 |1  |1  |0 quantum NOT gate: …GOOD NEWS! quantum interference

Key resource: Quantum Entanglement not just a “choice of basis” e.g.   vs. |0,0 must be able to access subsystems individually (see Bell )  = ( + )( + ) Contrast = 2 || = 0.5 (  i )(  i )  = or (  i )(  i ) … Contrast = 2 || = 0.5  =  +  Contrast = 2 || = 1 not hard to qualify (entanglement thresholds) ideal:

 1 = | + |  2 = | + | + | + | + | + | very hard to quantify (esp. mixed states) Classical Information:S(AB)  S(A) + S(B) Quantum Information:S(AB) < S(A) + S(B) possible! Information Entropy

Quantum computer hardware requirements 1.Must make states like |000…0 + |111…1 2. Must measure state with high efficiency strong coupling to environment strong coupling between qubits weak coupling to environment x x + see E. Schrödinger (1935)

N qubits controlled coupling … to >99% accuracy * * provided things have been done right Quantum Information and Atomic Physics

0.3 mm 199 Hg + J. Bergquist, NIST Aarhus Boulder (NIST) Munich (MPQ) Hamburg Innsbruck Los Alamos McMaster Michigan Oxford Teddington (NPL) Ion Trap QC Groups: Trapped Atomic Ions J. Bergquist (NIST) | | qubit stored inside each trapped ion

2 Cd + ions

S P D |  |  Ca +, Sr +, Ba +, Yb + optical (10 15 Hz) 1 sec Energy Atomic Ion Internal Energy Levels (think: HYDROGEN) S P |  |  Be +, Mg +, Hg +, Cd +, Zn + microwave (10 10 Hz)  hyperfine qubit levels

State | NSNS NSNS Hyperfine Structure: States of relative electron/nuclear spin State | SNSN NSNS

111 Cd + atomic structure 1,1 1,0 1,-1 0,0 =215nm 2 S 1/2 2 P 3/2 2,2 2, GHz | | magnetic insensitive qubit (to 2 nd order)   (400 Hz/G 2 )·B·B   ( Hz/G)·B

1,1 1,0 1,-1 0,0 =215nm 2 S 1/2 2 P 3/2 2,2 2, GHz | | /2 = 50 MHz “bright” # photons collected in 100 s Probability 111 Cd + qubit measurement

1,1 1,0 1,-1 0,0 =215nm 2 S 1/2 2 P 3/2 2,2 2, GHz | | /2 = 50 MHz 99.7% detection efficiency “dark” Probability # photons collected in 100 s Cd + qubit measurement

1,1 1,0 1,-1 0,0 =215nm 2 S 1/2 2 P 3/2 2,2 2, GHz | | 111 Cd + qubit manipulation: microwaves microwaves coupling rate: g 

Time  (ms) Time (ms) ,1 1,0 1,-1 0,0 1,1 1,0 1,-1 0,0 Microwave Rabi Flopping Prob(10|00) Prob(11|00) prepare 00  waves measure fluorescence (bright or dark) :: sweep  g   10  100kHz

 (  s) prepare 00  waves measure fluorescence (bright or dark) :: increment  “Single shot” Rabi Flopping Prob(10|00)

Time (ms) Time (ms) ,1 1,0 1,-1 0,0 1,1 1,0 1,-1 0,0 Microwave Ramsey Inteferometry Prob( | ) prepare 00  waves measure fluorescence :: sweep  /2

1,1 1,0 1,-1 0,0 =215nm 2 S 1/2 2 P 3/2 2,2 2, GHz | | /2 = 50 MHz 111 Cd + qubit manipulation: optical Raman transitions /2  THz  coherent coupling rate (good): g R = g 1 g 2 /  direct coupling to P (bad): R dec =  g 1 g 2 /   want small  (but < FS !)

0.3 mm J. Bergquist, NIST

Thanks: R. Blatt, Univ. Innsbruck 40 Ca +

logical |0 m logical |1 m Another Qubit: The quantized motion of a single mode of oscillation  harmonic motion of a collective single mode described by quantum states |n  m = |0  m, |1  m, |2  m,..., where E = ħ  (n+½) PHONONS: FORMALLY EQUIVALENT TO PHOTONS  motional “data-bus” quantum bit spans|n  m = |0  m and |1  m 0 1 2

Coupling (internal) qubits to (external) bus qubit radiation tuned to  0  || || || || || || || || || || || || |1  m |0  m |1  m |0  m

S 1/2 P 3/2 | | excitation on 1 st lower (“red”) motional sideband (n=0)  ~ few MHz

S 1/2 P 3/2 | | excitation on 1 st lower (“red”) motional sideband (n=0)

Mapping: (  |  +  |  ) |0  m  |  (  |0  m +  |1  m ) S 1/2 P 3/2 |  |  S 1/2 P 3/2 |  | 

Mapping: (  |  +  |  ) |0  m  |  (  |0  m +  |1  m ) S 1/2 P 3/2 |  |  S 1/2 P 3/2 |  | 

Spin-motion coupling: some math interaction frame; “rotating wave approximation”  =  L  0 = detuning k = 2  = wavenumber x0x0 frequency of applied radiation

stationary terms arise in H at particular values of  “Lamb-Dicke Limit” ,n|H 0 | ,n  = ħg  = 0 “CARRIER” ,n  1|H 0 | ,n  = ħg  = +  “1 ST BLUE SIDEBAND” ,n  1|H +1 | ,n  = ħg  =  “1 ST RED SIDEBAND”

Doppler Cooling Raman spectrum of single 111 Cd + ion (start in |) | | n= PP “Red” Sideband |,n  |,n+1 “Blue” sideband |,n  |,n-1  (MHz) sideband strengths: thermal occupation distribution Thermometry: n  6

n | | Raman Sideband Laser-Cooling. n1n1. n1n1 | | n1n1 stimulated Raman ~-pulse on blue sideband spontaneous Raman recycling.. n=-1 n   recoil / trap << 1

Doppler Cooling Raman spectrum of single 111 Cd + ion (3.6 MHz trap) | | n=0 L. Deslauriers et al., Phys. Rev. A 70, (2004) Doppler + Raman Cooling | | n=0 PP n < PP “Red” Sideband |,n  |,n+1 “Blue” sideband |,n  |,n-1 n   (MHz) x 0 ~3 nm

Heating of a single Cd + ion from n0 Delay Time (msec) nn Trap Frequency (MHz) Heating rate dn/dt (quanta/msec) Quadrupole Trap (160  m to nearest electrode) Linear Trap (100  m to nearest electrode) Heating Rate dn/dt (quanta/msec) Decoherence of Trapped Ion Motion

40 Ca Hg Cd + Heating history in 3-6 MHz traps 9 Be + Distance to nearest trap electrode [mm] Ba + heating rate (quanta/msec) 137 Ba + IBM-Almaden (2002) 40 Ca + Innsbruck (1999) 199 Hg + NIST (1989) 9 Be + NIST (1995-) 111 Cd + Michigan (2003) Q. Turchette, et. al., Phys. Rev. A 61, (2000) L. Deslauriers et al., Phys. Rev. A 70, (2004)

Trap dimension [mm] S E (  )  (V/m) 2 /Hz 40 Ca Hg Cd Ba + 9 Be + 1/d 4 guide-to-eye Electric Field Noise History in 3-6 MHz traps ~ 1/d 4 Heating due to fluctuating patch potentials (?) d est. thermal noise

Quantum Gate Schemes for Trapped Ions 1. Cirac-Zoller 2. Mølmer-Sørensen 3. Fast Impulsive Gates

Universal Quantum Logic Gates with Trapped Ions Step 1 Laser cool collective motion to rest Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995) n=0

Universal Quantum Logic Gates with Trapped Ions laser j k Step 2 Map j th qubit to collective motion Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)

Universal Quantum Logic Gates with Trapped Ions laser j k Step 3 Flip k th qubit depending upon motion Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995) |  |  /2, 2, /2 sign flip |  |n=1  only ! 22 /2

Universal Quantum Logic Gates with Trapped Ions laser j k Step 4 Remap collective motion to j th qubit (reverse of Step 1) Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995) Net result: [| j + | j ] | k  | j | k + | j | k n=0

CNOT between motion and spin (1 ion): F=85% C.M., et. al., Phys. Rev. Lett. 75, 4714 (1995) CNOT between spins of 2 ions: F=71% F. Schmidt-Kaler, et. al., Nature 422, (2003). Demonstrations of Cirac-Zoller CNOT Gate

 =  m +  m During the gate (at some point), the state of an ion qubit and motional bus state is: Decoherence Kills the Cat

Direct coupling between | and | with bichromatic excitation ? uniform illumination |  + e i  |  22 |  |   g 2  +  = Rabi Freq= = 0 g 2 

Bichromatic coupling to sidebands uniform illumination | , |    |  |  n1n1 n n+1 n n  Mølmer/Sørensen Milburn/Schneider/James (1999)  kx 0 g  n+1) 2   kx 0 g  n) 2  +  = Rabi Freq= =  kx 0 g) 2   as long as kx 0  n+1<< 1: “Lamb-Dicke regime”) independent of motion !

Mølmer/Sørensen 2-ion entangling quantum gate – a “super” /2-pulse Big improvement – no focussing required no n=0 cooling required less sensitive to heating || |  |  | n1n1 n n+1 n n n1n1 n

Can scalable to arbitrary N! |  ···   |  ···  + |  ···  22 e.g., 6 ions |3,-3  = |  |3,3  = |  |3,-1  = |  + ··· |3,1  = |  + ··· | J,J z   Coupling: H =  J x 2 flips all pairs of spins  Entangling rate  N -1/2

Four-qubit quantum logic gate Sackett, et al., Nature 404, 256 (2000) |   |  + e i  | 

x p N=1 ion: Force = F 0 || (spin-dependent force) Same idea in a different basis     e i   (  enclosed area) laser N=2 ions     e i     e i     e.g., force on stretch mode only = /2: -phase gate NIST (2003): 97% Fidelity

Strong Field Impulsive Gates 2 S 1/2 2 P 1/2 | |  ++ 0,0 1,1 1,0 1,-1 0,0 1,1 1,0 1,-1 e.g. 111 Cd GHz strong coupling:  Rabi  >> and  Rabi ~ 1 (a)off-resonant laser pulse; differential AC Stark shift provides qubit-state-dependent impulse

|   |  = |  ’  ’  k = linear shift f = nonlinear shift = 2U dd t /ħ ++ d d ++ “dipole engineering”: U dd = m 1 m 2 /r 3 = (e d ) 2 /r 3 r |   e +i k -i f /2 |  = |  ’  ’  |   e -i k -i f /2 |  = |  ’  ’  |   |  = e i f |  ’  ’  quantum phase gate  (t) t sub s       Cirac & Zoller (2000)

Poyatos, Cirac, Blatt & Zoller, PRA 54, 1532 (1996) Garcia-Ripoll, Zoller, & Cirac, PRL 91, (2003)  p = 2ħk || ||  |e   U = || e 2i  a  a  † | | |e|e | | |e|e -pulse up -pulse down two sequential -pulses spin-dependent impulse (b) resonant ultrafast kicks

The trajectory of a normal motional mode of two ions in phase space under the influence of four photon kicks. Gray curve: free evolution. Black curve: four impulses kick the trajectory in phase space, with an ultimate return to the free trajectory after ~1.08 revolutions.

2 S 1/2 2 P 1/2 | | ++ 0,0 1,1 1,0 1,-1 =226.5 nm 10 psec no kick 2 P 3/2 1/(15 fsec) = FS splitting  e  3nsec |e|e Fast version of  z phase gate does not require Lamb-Dicke regime! e.g. 111 Cd + require  FS <<  pulse <<  e

Summary Trapped Ions satisfy all “DiVincenzo requirements” for quantum computing: 1. identifiable qubits 2. efficient initialization 3. efficient measurement 4. universal gates 5. small decoherence SO WHAT’S THE PROBLEM?!

ENIAC (1946)

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