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)

http://iontrap.physics.lsa.umich.edu/

US Advanced Researchand Development Activity

US Army Research Office

US National Security Agency

National ScienceFoundation

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 IQuantum computing and motional quantum gates

Christopher MonroeFOCUS Center & Department of PhysicsUniversity 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

Sh

V

H

atomsparticlespins

photons

GOOD NEWS…quantum parallel processing on 2N 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

quantumlogic 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 quantumXOR gate:

superposition entanglement

|0 |0 + |1|1 |1 |0

quantumNOT 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

xx+

see E. Schrödinger (1935)

N qubits controlledcoupling

… to >99% accuracy*

* provided things have been done right

Quantum Information and Atomic Physics

N

i

jiN

jiij

ii tgtH

1

)()(

1,

)( ˆˆ)(ˆ)(2

1

0.3 mm

199Hg+

J. Bergquist, NIST

AarhusBoulder (NIST)Munich (MPQ)HamburgInnsbruck

Los AlamosMcMasterMichiganOxfordTeddington (NPL)

Ion Trap QC Groups:

Trapped Atomic Ions

J. Bergquist (NIST)

|

|

qubit storedinside eachtrapped ion

2 Cd+ ions

S

P

D

|

|

Ca+, Sr+, Ba+, Yb+

optical(1015 Hz)

1 sec

Energy

Atomic Ion Internal Energy Levels (think: HYDROGEN)

S

P

||

Be+, Mg+, Hg+, Cd+, Zn+

microwave(1010 Hz)

hyperfine qubit levels

State |

N

S

N

S

Hyperfine Structure: States of relative electron/nuclear spin

State |

S

N

N

S

111Cd+ atomic structure

1,11,01,-1

0,0

=215nm

2S1/2

2P3/22,22,1

14.53 GHz

|

|

magnetic insensitivequbit (to 2nd order) (400 Hz/G2)·B·B

(1400000 Hz/G)·B

1,11,01,-1

0,0

=215nm

2S1/2

2P3/22,22,1

14.53 GHz

|

|

/2 = 50 MHz

“bright”

# photons collected in 100 s0 5 10 15 20 25

0

1

Pro

babili

ty111Cd+ qubit measurement

1,11,01,-1

0,0

=215nm

2S1/2

2P3/22,22,1

14.53 GHz

|

|

/2 = 50 MHz

99.7% detectionefficiency

“dark”

0 5 10 15 20 25

Pro

babili

ty

# photons collected in 100 s

0

1

111Cd+ qubit measurement

1,11,01,-1

0,0

=215nm

2S1/2

2P3/22,22,1

14.53 GHz

|

|

111Cd+ qubit manipulation: microwaves

microwaves

coupling rate: g

Time (ms)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Time (ms)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1,11,01,-1

0,0

1,11,01,-1

0,0

Microwave Rabi FloppingP

rob(

10|0

0)P

rob(

11|0

0)

prepare00

wavesmeasure

fluorescence(bright or dark)

: :

sweep g 10100kHz

(s)

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300 350 400

prepare00

wavesmeasure

fluorescence(bright or dark)

: :increment

“Single shot” Rabi Flopping

Pro

b(10

|00)

Time (ms)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Time (ms)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1,11,01,-1

0,0

1,11,01,-1

0,0

Microwave Ramsey InteferometryP

rob(|)

Pro

b(|)

prepare00

waves measurefluorescence

: :

sweep

/2 /2

1,11,01,-1

0,0

=215nm

2S1/2

2P3/22,22,1

14.53 GHz

|

|

/2 = 50 MHz

111Cd+ qubit manipulation: optical Raman transitions

/2 0.1-1 THz

coherent coupling rate (good):gR = g1g2/

direct coupling to P (bad): Rdec = g1g2/

want small (but <FS!)

0.3 mm J. Bergquist, NIST

Thanks: R. Blatt, Univ. Innsbruck

40Ca+

logical |0m

logical |1m

Another Qubit: The quantized motion of a single mode of oscillation

harmonic motion of a collective single mode described byquantum states |nm = |0m, |1m, |2m,..., where E = ħ(n+½)PHONONS: FORMALLY EQUIVALENT TO PHOTONS

motional “data-bus” quantum bit spans |nm = |0m and |1m

•••

01

2

Coupling (internal) qubits to (external) bus qubit

radiation tuned to 0

| | | | |

| | | | |

|

|

|1m

|0m

|1m

|0m

•••

01

2

•••

01

2S1/2

P3/2

|

|

excitation on 1st lower (“red”) motional sideband (n=0)

~ few MHz

•••

01

2

•••

01

2S1/2

P3/2

|

|

excitation on 1st lower (“red”) motional sideband (n=0)

Mapping: (| + |) |0m | (|0m + |1m)

•••

01

2

•••

01

2S1/2

P3/2

|

|

•••

01

2

•••

01

2S1/2

P3/2

|

|

Mapping: (| + |) |0m | (|0m + |1m)

•••

01

2

•••

01

2S1/2

P3/2

|

|

•••

01

2

•••

01

2S1/2

P3/2

|

|

Spin-motion coupling: some math

)21( aa

)ˆ(ˆˆ2

1

2

ˆˆ 22

2

0 xExmm

pH z

interaction frame; “rotating wave approximation”

)ˆˆ( ˆˆ tixiktixik eegH

= L 0 = detuningk = 2 = wavenumber

)(ˆ 0titi eaaexx

mx

20

x0

))(ˆˆ(2

ˆˆ00

tixiktixik LL eeE

frequency ofapplied radiation

tieaaeikxtieaaeikx titititi

eegH

)()( 00 ˆˆ

stationary terms arise in H at particular values of

“Lamb-Dicke Limit”110 nkx

,n|H0|,n = ħg)ˆˆ(0 gH = 0

“CARRIER”

110 nkx

,n1|H0|,n = ħg n)ˆˆ)(( 01

aakxgH = +

“1ST BLUE SIDEBAND”

110 nkx

,n1|H+1|,n = ħg 1n)ˆˆ)(( 01 aakxgH

= “1ST RED SIDEBAND”

110 nkx

DopplerCooling

Raman spectrum of single 111Cd+ ion (start in |)

|

|

n=0

0.0

0.5

1.0

P

“Red”Sideband

|,n |,n+1

“Blue”sideband

|,n |,n-1

-3.6 +3.6 (MHz)

nnblue

nnred

ngtPI

ngtPI

)(sin

)1(sin

2

2

sideband strengths:

n

n n

nP

1

thermaloccupationdistribution

Thermometry: 1

n

n

I

I

red

blue

n 6

n

|

|

Raman Sideband Laser-Cooling

.

n1

. n1

|

|

n1

stimulated Raman ~-pulse on blue sideband

spontaneous Ramanrecycling

.

.

n=-1 n recoil/trap << 1

DopplerCooling

Raman spectrum of single 111Cd+ ion (3.6 MHz trap)

|

|

n=0

L. Deslauriers et al., Phys. Rev. A 70, 043408 (2004)

Doppler+ Raman Cooling

|

|

n=0

P

0.5

0.0

1.0

n < 0.05

0.0

0.5

1.0

P

“Red”Sideband

|,n |,n+1

“Blue”sideband

|,n |,n-1

n 6

-3.6 +3.6

-3.6 +3.6

(MHz)

(MHz)

x0 ~3 nm

Heating of asingle Cd+ ion from n0

Delay Time (msec)

0 10 20 30 40 500.0

0.5

1.0

1.5

n

Trap Frequency (MHz)

Heating rate dn/dt(quanta/msec)

1 2 3 4 5 60.01

0.1

1

10

Quadrupole Trap (160 m to nearest electrode)

Linear Trap (100 m to nearest electrode)

Heating Ratedn/dt

(quanta/msec)

Decoherence of Trapped Ion Motion

40Ca+

199Hg+

111Cd+

Heating history in 3-6 MHz traps

9Be+

Distance to nearest trap electrode [mm]

0.04 0.1 0.2 0.3 0.610-3

10-2

10-1

100

101

102

137Ba+

heating rate (quanta/msec)

137Ba+ IBM-Almaden (2002)

40Ca+ Innsbruck (1999)

199Hg+ NIST (1989)9Be+ NIST (1995-)

111Cd+ Michigan (2003)

Q. Turchette, et. al., Phys. Rev. A 61, 063418-8 (2000)L. Deslauriers et al., Phys. Rev. A 70, 043408 (2004)

Trap dimension [mm]

0.04 0.1 0.2 0.3 0.610-2

10-1

100

101

102

SE() 10-12 (V/m)2/Hz

40Ca+

199Hg+111Cd+

137Ba+9Be+

1/d4 guide-to-eye

Electric Field Noise History in 3-6 MHz traps

~ 1/d 4

Heating due tofluctuating patch potentials (?)

)(4

2

ES

m

q

d

est. thermal noise

Quantum Gate Schemes for Trapped Ions

1. Cirac-Zoller2. Mølmer-Sørensen3. Fast Impulsive Gates

Universal Quantum Logic Gateswith Trapped Ions

Step 1 Laser cool collective motion to rest

Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)

n=0

Universal Quantum Logic Gateswith Trapped Ions

laser

j k

Step 2 Map jth qubit to collective motion

Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)

Universal Quantum Logic Gateswith Trapped Ions

laser

j k

Step 3 Flip kth qubit depending upon motion

Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)

|

|10

10

10

10

/2, 2, /2

sign flip ||n=1 only !

2/2

Universal Quantum Logic Gateswith Trapped Ions

laser

j k

Step 4 Remap collective motion to jth 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, 408-411 (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 ?

uniformillumination

| + ei|2

|

|

g 2

+ = Rabi Freq =

= 0

g 2

Bichromatic coupling to sidebands

uniformillumination

|, |

|

|

n1n

n+1

n

n

Mølmer/SørensenMilburn/Schneider/James

(1999)

kx0gn+1) 2

kx0gn) 2

+ = Rabi Freq =

=kx0g) 2

as long as kx0n+1<< 1: “Lamb-Dicke regime”)

independentof motion !

1n n

1n n

2ˆ xH

i

i

e

eMS

MS

i

i

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

|

|

|

|

n1n

n+1

n n

n1n

n+1

Can scalable to arbitrary N!

|··· |··· + |···

2

e.g., 6 ions

|3,-3 = |

|3,3 = |

|3,-1 = | + ···

|3,1 = | + ···

| J,Jz

Coupling: H = Jx2

flips all pairs of spins

Entangling rate N-1/2

Four-qubit quantum logic gate

Sackett, et al., Nature 404, 256 (2000)

| | + ei|

x

p

N=1 ion: Force = F0|| (spin-dependent force)

Same idea in a different basis

ei

( enclosed area)

laser

N=2 ions ei ei

e.g., force on stretch mode only

= /2: -phase gateNIST (2003): 97% Fidelity

2ˆ zH

Strong Field Impulsive Gates

2S1/2

2P1/2

|

|

+

0,0

1,11,01,-1

0,01,1

1,0

1,-1

e.g. 111Cd+

14.5 GHz

strong coupling: Rabi>> and Rabi~ 1

(a) off-resonant laser pulse; differential AC Stark shift provides qubit-state-dependent impulse

| | = |’’= linear shift = nonlinear shift = 2Udd/ħ

++

++

“dipole engineering”: Udd = 12/r3 = (e)2/r3

r

| e+i-i/2 | = |’’| e-i-i/2 | = |’’| | = ei|’’

quantum phase gate

(t)

t

sub s

Cirac & Zoller (2000)

Poyatos, Cirac, Blatt & Zoller, PRA 54, 1532 (1996)Garcia-Ripoll, Zoller, & Cirac, PRL 91, 157901 (2003)

p = 2ħk

| ||e |e

U = ||e2iaa †

| |

|e

| |

|e

-pulseup

-pulsedown

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.

2S1/2

2P1/2

| |

+

0,0

1,11,01,-1

=226.5 nm10 psec no

kick

2P3/2

1/(15 fsec) = FS splittinge 3nsec|e

Fast version of z phase gate

does not require Lamb-Dicke regime!

e.g. 111Cd+

require FS << pulse << e

Summary

Trapped Ions satisfy all “DiVincenzo requirements” for quantum computing:

1. identifiable qubits2. efficient initialization3. efficient measurement4. universal gates5. small decoherence

SO WHAT’S THE PROBLEM?!

ENIAC(1946)

Next: Ion Traps and how to scale them!