Quantum Information Science Q with Trapped Ions

Quantum Information Science with Trapped Ions 2 Quantum Information Science EitlExperimental IlImplement ttiation with TdTrapped Ions

Rainer Blatt Institute of Exppy,y,erimental Physics, University of Innsbruck, Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences

Lectures:

1. GlGeneral aspec tdilttitts and implementation concepts quantum computation with trapped ions

2. Quantum information processing with Ca+ ions techniques, procedures, algorithms 1. Implementing quantum information processing

1.1 Generics of quantum information processing 1.2 Realization concepts 131.3 Ion trap quantum computer – the basics 1.4 Physics of ion traps 151.5 ItiftIon strings for quantum comput ttiation 1.6 Spectroscopy in ion traps 1.7 Laser cooling in ion traps 1.8 Addressing individual ions 1.9 Compensating AC Stark effects 2.0 Basic gate operations, composite pulses Level scheme of Ca+ qubit on narrow S - D quadrupole transition

P3/2  1s 854 nm

P1/2 866 nm D5/2 393 nm 397 nm D3/2

729 nm

S1/2 Spectroscopy with quantized fluorescence (quantum jumps)

P D absorption and emission cause fluorescence steps (digital quantum jump signal) monitor spectroscopy S

histogram of absorption events S orptions ss ab

D laser detuning State detection by quantized fluorescence

8 DidD state occupied S state occupied 7 ts detection efficiency: nn 6 5 99.85% ureme ss 4

3 f mea oo e # 2 g

0 0 20406080100120 count9ts per 9 ms Laser – ion interactions ...1

An ion trapped in a harmonic potential with frequency  interacting with the travelling wave of a single mode laser tuned close to a transition that forms an effective two -level system, is described by the Hamiltonian

k, l,  wavenumber, frequency and phase of laser radiation

m0: mass of the ion

for details see: D. LeibfCfried, C. Monroe, R. Blatt, D. Wineland Rev. Mod. Phys. 75, 281 (2003) Pauli matrices Laser – ion interactions ...2

With the Lamb-Dicke parameter we write (using RWA) in terms of creation and annihilation operators

In the interaction ppyicture defined by we obtain for the Hamiltonian

wihith coupling states with vibration quantum numbers Laser – ion interactions ...3 with laser tuned close to a resonance coupling to other resonances can be neglected ( ).

Time evolution of pairwise coupled states is then determined by the set of coupled equations

With the detuning and the Rabi frequency we obtain Rabi oscillations on resonance Laser – ion interactions ... Lamb-Dicke regime

For small ion oscillation amplitudes (Lamb-Dicke regime) we expand and obtain for the coupling

carrier excitation

detuning

red sideband excitation blue sideband excitation Laser – ion interactions ... unitary operators

For later use we already note that the action of a laser on the respective transitions can be written in the following form:

Operates on internal degrees Carrier pulse of length of freedom

Red sideband pulse of length Operate on internal and external degrees of freedom

Blue sideband pulse of length ENTANGLEMENT ! Spectroscopy of the S1/2 –D5/2 transition

Zeeman structure in non-zero magnetic field:

5/2 2-level-system: 3/2 1/2 D -1/2 5/2 -3/2 -5/2

S 1/2 1/2  -1/2

sideband quantum + vibrational degrees of freedom cooling state processing Quantized ion motion

coupled system 2-level-atom harmonic trap

excitation: various resonances spectroscopy: carrier and sidebands n = 0

D5/2 n = -1 n = 1

 S1/2 n = 0 1 2 Laser detuning Absorppption spectra: Exp erimental cy cle

P P P 3/2 3/2 3/2 854 nm P1/2 P1/2 866 nm P1/2 D D D 5/2 397 nm 5/2 397 nm 5/2 D D D 3/2 3/2 866 nm 3/2 729 nm S1/2 S1/2 S1/2

1st step: 2nd step:3rd step: probe with 729 nm detect fluorescence reset, Doppler cooling 5µs5 µs – 3ms3 ms 1 – 10 ms 1 – 10 ms

Average: Repeat experiment 100 - 1000 times, then step 729 nm laser to new detuning Absorption measurements: Pulse scheme

397nm 397nm + P 854nm 3/2 866nm P 397nm 1/2 854nm 866nm D5/2 729nm D3/2 detection 729nm time S preparation detection 1/2 excitation

- typically 5ms for each sequence - repeated typically 100 times - triggered by 50Hz line frequency - flexibly controlled by Labview Excitation spectrum of the S – D transition

- 32 - 12 12 32 52 12 12 12 12 12 excitation probability

0.5

-25 -20 -15 -10 -5 0 5 /MHz Laser detuning  at 729 nm Excitation spectrum of the S – D transition

red blue axial radial radial axial sidebands sidebands 0.5 carrier

0 -1710 -920 0 920 1710 kHz (spherical trap) 1. Implementing quantum information processing

e k v Laser atoms 

g momentttum transf er k abs ,  k em  p  nk abs   k em

k  0  p  n k abs

Doppler cooling limit:

k   v 1cm / s ED   m 2 for details see: Laser Cooling of Trapped Atoms • D. Wineland, W. Itano, Phys. Rev, A 20, 1521 (1979) • S. Stenholm, ATOM TRAP Rev. Mod. Phys. 58, 699 (1986) e Regimes:

weak confinement,       Doppler cooling   g ED   2, n  1 n 1,e n,e n 1,e    strong confinement, n 1, g sideband cooling n, g n 1, g 2 2 ES   4 1 2 n  1 cooling heating Cooling and heating of trapped atoms

 ne+ 1, n,e n 1,e 

n 1,g n,g n 1,g

strong confinement, sideband cooling, selective excitation of lower sideband,    optical pumping to ground state Cooling and heating of trapped atoms

 ne+ 1, n,e n 1,e 

n 1,g n,g n 1,g

weak confinement, Doppler cooling, simultaneous excitation of several n

Cooling cascade:

P3/2

D5/2 n 1 S 1/2 n n 1 Effective two-level system:

I. Marzoli et al., Phys. Rev. A 49, 2771 (1994) Absorption on quadrupole transition

(with motional sidebands)

 2 1 0 e

2 1 0 g Sideband absorption spectrum

99.9 % ground state population

0.8 0.8 after Doppler n 1.7 0.6 cooling z 0.6 D D P P 0.4 0.4

0.2 0.2 after sideband cooling 0 0 4.54 4.52 4.5 4.48 4.48 4.5 4.52 4.54 Detuning  (MHz) Detuning  (MHz)

Ch. Roos et al., Phys. Rev. Lett. 83, 4713 (1999) Cooling and heating

Ch. Roos et al., Phys. Rev. Lett. 83, 4713 (1999) cooling

1 ms

> 0.1

z cooling: 0.2 nn

< 0.01 phonon

0.001 0 1 2 3 4 5 6 7 CliTi(Cooling Time (ms ))

heating heating: 0.8 ms radial: 70 0.6 y phonon 0.4 0.2 z ms 0 axial: 190 0 10 20 30 40 50 60 phonon Delay Time (ms) Excitation spectrum of two ions Sideband cooling of two ions

RED sidebands BLUE sidebands

 z  3 y  y y 3 y z

0.6

0.5 tion aa

0.4

P0 P0 P0 popul 0.3 ee  98%  96%  95%

0.2 D-stat

0.1

0 -4.45 -4.4 -3.68 -3.63 -2.12 -2.07 2.07 2.12 3.63 3.68 4.4 4.45 Detuning at 729 nm (MHz) 1. Implementing quantum information processing

row ofbitif qubits in a linear Paul trap forms a quantum register

70 µm Addressing of individual ions

coherent maniltiipulation Paul trap of qubits 0.8

0.7

060.6 electrooptic 0.5 deflector 0.4 itation cc

0.3 Ex

0.2

010.1

0 -10 -8 -6 -4 -2 0 2 4 6 8 10 Deflector Voltage (V) dichroic beamsplitter  inter-ion distance: ~ 4 µm Fluorescence  addressing waist: ~25µm~ 2.5 µm detection CCD < 0.1% intensity on neighbouring ions Detection of 6 individual ions Coherent state manipulation

carrier D,1 (C) D,0

S,1 S,0 blue sideband carrier and sideband (BSB) Rabi oscillations with Rabi frequencies

Lamb-Dicke parameter Coherent state manipulation: carrier

carrier

Carrier transitions leave the motion unchanged Coherent state manipulation: sideband

Sideband transitions entangle motion and internal excitation

side ban d 1. Implementing quantum information processing

measurement of net P1/2 ac-Stark shift via Ramsey technique:

carrier AC – Stark carrier

pulse pulse pulse lation

3/2 uu D5/2 -1/2   T ...  -5/2

2 2 D - pop 729 nm

S1/ 2 ac-Stark shift is due to interaction on:

 carrier transition (S1/2 –D5/2)  D5/2 - P transitions  other S 1/2 -D5/2 Zeeman components AC – Stark shift of qubit transition 15 11 13 mmJ  J       22 22 22 qubit

sidebands

blue red Compensation of AC Stark shifts

Idea: Compensate Stark shifts by an additional compensating laser pulse

carrier and sidebands H. Häffner et al., compensating quant-ph/0212040 pulse

AC – Stark pulse and

comp. pulse n Ramsey settings for carrier carrier oo pulse pulse compensation of the T AC – Stark shifts   compensation ... populati

 --

2 blue sideband 2 D AC – Stark shift of qubit transition

H. Häffner et al., quant-ph/0212040

compensate

blue sideband on qubit transition Compensation of AC - Stark shift

AC – Stark 1 pulse and carrier comp. pulse carrier nn 0.9 pulse pulse T 0.8   opulatio compensation ... pp 0.7 

2 blue sideband 2 D - 0.6 citation xx 0.5 0.4 tate e ss 030.3 D 0.2 0.1 0 0 20 40 60 80 100 120 140 160 180 200 PlPulse leng th(th (µs ) HHäffH. Häffner e t al ., quant-ph/0212040 1. Implementing quantum information processing

D,2 D,1 D,0 transitions are described by unitary matrices:

S,2 carrier: S,1  S,0 Riee((),) ex p ii 2   computational red sideband: subspace Rieaea (,)  exp ii   † 1  2  

keep all blue sideband: populations in  subspace only ! Rieaea(,)    exp ii†  1  2  

A. M. Childs, I. L. Chuang, Phys. Rev. A 63, 012306 (2001) SWAP and composite SWAP operation

D,1 D,0 find simultaneously perfect SWAP operation with 4 rotation

S,1 on the DSSD,1 ,2 ,1 ,2  S,0 transition(s)  2  out of comppposite SWAP operation: perfect subspace if  SWAP |D,1> (|S,1>) RRcSWAP 1 2,0.732  populated operation  R1 22104622,1.046 

 R1  2,0.732

I. L. Chuang et al., Innsbruck (2002) |D,0> - |S,1> SWAP and composite SWAP operations

  RRcSWAP  11 2,0.732 R 2 2,1.046  R 1  2,0.732 

 RRSWAP  1  ,07320.732 RcSWAP single-step SWAP operation 3-step composite SWAP operation 3-step composite SWAP operation

I. Chuang et al., Innsbruck (2002)

1 1 3

 on DS,0 ,1 4 on DS,1 ,2 A phase gate with 4 pulses (2 rotation), blue sideband

  RR(,)      11 , 2 R 2,0 R 11 , 2 R 2,0

1 2 on SD,0 ,1

4 2 Population of |S,1> - |D,2> remains unaffected

 RR(,)     1111 2,2 RR ,0 2,2 R ,0

2 4

3 1 A single ion composite phase gate: Experiment state preparation S,0 , then application of phase gate pulse sequence

 2   2    00    2   00    22 1

0.9

0.8

0.7 nn 0.6

0.5 xcitatio

ee 040.4 -

5/2 0.3 D 0.2

0.1

0 0 20 40 60 80 100 120 140 160 180 Time (µs) A single ion composite CNOT gate: Experiment state preparation S,0 , then application of CNOT gate pulse sequence

p  Phase gate - 2 2 1

0.9 97. 8 (5) % 0.8

0.7 n oo 0.6

0.5 excitati

- 0.4 5/2

D 0.3

020.2

0.1

0 0 50 100 150 200 250 300 Time (µs) A single ion composite CNOT gate: Experiment

|D,1>   phase gate  prep. 2 2 1

ability 080.8 bb

n pro 0.6 oo

citati 0.4 xx

0.2 tate e ss -

5/2 0

D 0 50 100 150 200 250 300 Time (µs) Summary: Implementing quantum information processing