quantum simulations with

Quantum Simulations with

Yb+ crystal

~5 mm

Trapped Atomic Ions

dc

rf

dc

dc

rf

dc

dc

rf

dc

dc

rf

dc

3-layer geometry:• single rf electrode• scalable to larger structures, natural for junctions

2S1/2(600 Hz/G @ 1 G)

wHF/2p = 12 642 812 118 + 311B2 Hz

| = |0,0

| = |1,0

171Yb+ hyperfine spin

2S1/2

2P1/2

369 nm

2.1 GHz

/2g p = 20 MHz

|

|

171Yb+ spin detection

(600 Hz/G @ 1 G)

wHF/2p = 12 642 812 118 + 311B2 Hz

# photons collected in 800 ms0 5 10 15 20 25

0

1

Pro

bab

ility

|z

2S1/2

2P1/2

369 nm

/2g p = 20 MHz

|

|

2.1 GHz

171Yb+ spin detection

>99% detectionefficiency

# photons collected in 500 ms0 5 10 15 20 25

0

1

Pro

bab

ility

|z |z

(600 Hz/G @ 1 G)

wHF/2p = 12 642 812 118 + 311B2 Hz

(600 Hz/G @ 1 G)

wHF/2p = 12 642 812 118 + 311B2 Hz2S1/2

2P1/2

|

|

171Yb+ spin manipulation

D = 33 THz

355 nm (10 psec @ 100 MHz)

2P3/2

/2g p = 20 MHz

National Ignition Facility: 351nm(Livermore National Laboratory)

Pavg ~ 5W at 355nm10 psec pulses, 120 MHz rep rate

0 10 20 30 pulse energy (nJ)

picosecondspin control

1

0

P(↑|↓)

See talk by Jonathan Mizrahi (Sunday)J. Mizrahi, et al., ArXiv 1307.0557 (2013)

Internal states of these ions entangled

Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)CM, et al., Phys. Rev. Lett. 74, 4714 (1995)

Q. Turchette, et al., Phys. Rev. Lett. 81, 3631 (1998)F. Schmidt-Kaler, et al., Nature 422, 408 (2003)

Trapped Ion Quantum Computer(Cirac-Zoller)

Cirac and Zoller, Phys. Rev. Lett. 74, 4091 (1995)CM, et al., Phys. Rev. Lett. 74, 4714 (1995)

Q. Turchette, et al., Phys. Rev. Lett. 81, 3631 (1998)F. Schmidt-Kaler, et al., Nature 422, 408 (2003)

Trapped Ion Quantum Computer(Cirac-Zoller)

Internal states of these ions entangled

Cirac-Zoller: number states of the QHO

• extreme cooling: requires a pure motional state

• not scalable: mode density problem

1

𝑘𝑥𝑟𝑚𝑠≪1𝑜𝑟 𝑛≪ℏ𝜔𝐸𝑅

=

Better: “spin-dependent displacements”

• only requires cooling to the Lamb-Dicke limit

• “virtual” coupling to phononsPossible

Mølmer & Sørensen (1999)Solano, de Matos Filho, Zagury (1999)Milburn, Schneider, James (2000)

F = F0|↑↑| - F0|↓↓|

global spin-dependent force

↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↓↑↓↑↓↑↓↓↑↓ ↑↓ ↑

↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑ ↓↑↓↑↓↑↓↓↑↓ ↑↓ ↑

|

|

ADD: Independent spin flips

B

F = F0|↑↑| - F0|↓↓|

global spin-dependent force

2S1/2

2P1/2

spin-dependent force (171Yb+)

|

|

1,-1 1,11,0

0,0

1,-11,11,0

0,0

B(x)

Magnetic field gradient

2S1/2

2P1/2

369 nm

spin-dependent force (171Yb+)

|

|

s+ s+

1,-1 1,11,0

0,0

1,-11,11,0

0,0

D

g

Position-dependent AC Stark shift

2S1/2

2P1/2

369 nm

spin-dependent force (171Yb+)

|

|

1,-1 1,11,0

0,0

1,-11,1

1,0

0,0

D

g

g

g

Red+blue sideband appliedsimultaneously

= Lamb-Dicke parameter

simultaneous sidebands

global spin-dependent oscillating force

i

ii

xi xkH )(̂

ki

tik

tik

ki

kixi

kk eaeabxk,

)()(0

)( ][ˆ †

normal mode transformation matrix: ion i, mode k

km2

1)()( 22

k

ki

i

ki bb

normal mode decomposition

Lamb-Dicke approximation:

1 rmskx

Aside: transverse Modes of an atom chain

transversemodes

frequency

. . .

transversemodes

frequency

S.-L. Zhu et al.,  Phys. Rev. Lett. 97, 050505 (2006)A. Serafini et al., New J. Phys. 11, 023007 (2009)

. . . . . .

axialmodes

0 1 2 3 4 5

fluorescence~ N()

Raman beatnote (MHz)

transverse x

transverse yaxial z

COM

COM

COMZigZag

ZigZag

(Dk nominally along x)

Raman spectrum of N=9 ions

Ramanbeatnotes:

wHF ± m

ki

tik

tik

ki

kixi

kk eaeabxkH,

)()(0

)( ][ˆ †

uppersidebands

frequencywHF+m

carrierlower

sidebands

wHF -m

global spin-dependent oscillating force

ki

tik

tik

ki

kixi

kk eaeabxkH,

)()(0

)( ][ˆ †

k

kkik

kii aa ])()([)(ˆ *

)sincos()(22

,

ki

k

ikiki ie

ik

†

phonon

s

k kk

k

kk

k

k

kjkijiji

bb

m

k

2

2sin

)(

)sin(

)(

)sin(

2

)()(

22

,,2

,

interaction between qubits (entangling gates etc..)

ji

jx

ixji

i

ixi iU

,

)()(,

)( )()(ˆexp)(

evolution operator

...)]](),([),([

6)](),([

2

1)(exp)(

232

0

1231

0

2

0

3

0

121

0

2

0

ttt

tHtHtHdtdtdti

tHtHdtdttdtHiU

0)sincos()(22

,

ki

k

ikiki ie

ik

How to avoid phonon creation?

(1) Pick detuning m and time t wisely “FAST MOLMER”

for all modes k

e.g.: m near single mode k only

→ ( -m wk)t = 2p m m=1,2,…

S.-L. Zhu, et al., Europhys Lett. 73 (4), 485 (2006).

Beatnote frequency

Ra

bi f

requ

enc

y

HF

x

p

“FAST MOLMER”

0)sincos()(22

,

ki

k

ikiki ie

ik

How to avoid phonon creation?

(1) Pick detuning m and time t wisely “FAST MOLMER”

for all modes k

e.g.: m near single mode k only

→ ( -m wk)t = 2p m m=1,2,…

S.-L. Zhu, et al., Europhys Lett. 73 (4), 485 (2006).

(2) “Adiabatically eliminate” phonons: | - m wk| >> hW0 “SLOW MOLMER”

1)sincos()( ,22

,

k

ikik

i

k

ikiki

iie

ik

x

p

Beatnote frequency

Ra

bi f

requ

enc

y

HF

“SLOW MOLMER”

)()(, ˆˆ j

xi

xji

jieff JH

k k

kj

kiji

ji

bb

m

kJ

22

2

, 2

)(

(2) “Adiabatically eliminate” phonons: | - m wk| >> hW0 “SLOW MOLMER”

How to avoid phonon creation?

1)sincos()( ,22

,

k

ikik

i

k

ikiki

iie

ik