Geometric Quantum Operations in Two-Level Atomic Systems

M. Tian, Z. W. Barber, J. A. Fischer, and Wm. R. Babbitt

Physics Department, Montana State University—Bozeman, MT 59717,Tian@physics.montana.edu

Quantum Information and Quantum Control, July 19~23, 2004, Toronto

Introduction

The geometric manipulation of the quantum states of qubitshas the potential to produce a full set of robust universal quantum operations for quantum computing. We have investigated the manipulation of the quantum states of two-level rare-earth atoms using laser-controlled geometric phases. A set of universal single qubit operations has been designed using resonant laser pulses. An operation equivalent to a phase gate has been demonstrated with thulium ions doped in yttrium aluminum garnet crystal.

Wave function & Bloch representationBloch representationWave function2-level atom

1

0

cos2

sin2

ii

ee

ϕ

β

βψ γ

−

⎛ ⎞⎜ ⎟

= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

1

3

2

β

ϕ

R

Single qubit operation U alters Bloch vector’s orientation

1

3

1

3

' Uψ ψ=

'R UR= 2

β

ϕ

R

2

'β

'ϕ

'RU

Universal single qubit operation

ϕ

β

1

2

3

R

3U

2U

3 3 2 2 3 1( ) ( ) ( )U U U Uδ δ δ=

Any arbitrary single qubit operationmade of two basic rotations: and 3U2U

/ 2

3 / 2

2

1( ) ,

1

cos( / 2) sin( / 2)( )

sin( / 2) cos( / 2)

i

i

eU

e

U

δ

δδ

δ δδ

δ δ

−

⎛ ⎞= ⎜ ⎟

⎝ ⎠⎛ ⎞

= ⎜ ⎟−⎝ ⎠

The basic Bloch rotations can be accomplished through controlledgeometric phase manipulation.

Geometric phase

After a cyclic evolution, the wave function gains geometric phase

βR

1

2

3

ϕ

α 2/γ α∆ = −

Geometric property:determined solely by the amount of the solid angle enclosed by the evolution path, independent of driving Hamiltonian, the quantum state of the system, and the shape of the path.

M.V. Berry, Proc. R. Soc, Ser A 392, 45 (1984), Y. Aharonov and J. Anandan . Phys. Rev. Lett. 58, 1593 (1987)

Geometric Phase in three-level atomic system, Phys. Rev. A 67. 011403 (R) (2003)Geometric manipulation of the quantum states of two-level atoms, Phys. Rev. A 69. 050301 (R) (2004)

Two-level atom driven by laser pulse

1

0

cos( )aE tω θ+aω tR Rd

d= Ω×

p

Rabi frequency: / , Pulse Area: A= .

Eµτ

Ω =Ω

:[ cos , sin , 0]θ θΩ −Ω − Ω

Optical Bloch equation:

1

2

3

R

Ω

Two-levelatom

Two basic rotations of an arbitrary Bloch vector for an arbitrary angle can be accomplished through pure geometric phase changes driven by laser pulse sequence satisfying along the evolution path. RΩ ⊥

Basic rotation 3( )U δ

1

0

1

2

3/ 2

3 / 2

1( )

1

i

i

eU

e

δ

δδ

−

⎛ ⎞= ⎜ ⎟

⎝ ⎠

3

cos2( )

sin2

iU

e ϕδ

β

β−

⎛ ⎞⎜ ⎟

=⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

/ 2

/ 2

cos2

sin2

i

i i

e

e e

δ

δ ϕ

β

β− −

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

Control pulses

0

1

10

0⎛ ⎞

= ⎜ ⎟⎝ ⎠

01

1⎛ ⎞

= ⎜ ⎟⎝ ⎠

/23( ) 0 0iU eδδ = /2

3( ) 1 1iU e δδ −=

Control pulse sequence (2 pulses): π pulse (θ =0), π pulse with θ = δ / 2

Operation on and3( )U δ 0 1

1

2

: pulse ( =0)

: pulse ( = / 2)

π θ

π δθ

Ω

Ω

3

1Ω

01

/ 2δ

2Ω

/ 20 0i ie π δ− +⇒Solid angle: 2π-δ

Geometric phase: -π+δ/22

1

2

3

1Ω2Ω

/ 2δ

1/ 21 1i ie π δ− −⇒

Solid angle: 2π+δ

Geometric phase: -π-δ/2

/23( ) 0 0iU eδδ = /2

3( ) 1 1iU e δδ −=

Basic rotation: 2 ( )U δ

1

0

1

3

1 11 1, 2 2

i ii i

⎛ ⎞ ⎛ ⎞+ = − =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

2

i− i+ 2 ( )U δ2

cos( / 2) sin( / 2)( )

sin( / 2) cos( / 2)U

δ δδ δ

δ⎛ ⎞

= ⎜ ⎟−⎝ ⎠

2/2( ) ii iU e δδ −+ = +

2/2( ) iiU ie δδ − = −

/2 pulse ( =0), pulse ( = + /2) ,and /2 pulse ( =0)π θ π θ π θδπThree-pulse control sequence

Control pulse sequence for 2 ( )U δ

1

2

3

: /2 pulse ( =0)

: pulse ( = + /2)

: /2 pulse ( =0)

π θ

π θ π δ

π θ

Ω

Ω

Ωi+

1

3

1,3Ω/ 2δ

2Ω

2

Solid angle: ,Geophase: - /2

δδ

/2ii ie δ−+ ⇒ +

1

2

3

1,3Ω/ 2δ

2Ω

i−Solid angle : 4 - ,Geophase: /2

π δδ

/2ii ie δ− ⇒ −

Observation of the rotation 3U

Photon echo processpulse 2

1

3

τpulse 1 τ

C1

2iEe θ−

3( )2U θ

timeInput pulses

Photonecho

pulse 1

timeInput pulses

pulse 2Photonecho

C2(θ)Add 2 π control pulses

Echo field

E2

1

3

2

2θ

Phase change vector rotation

Experiment

0 5 10 15 20Time (µs)

output

input P1 P2

C1 C2

R

PhotodiodeTi: S Laser

AOM Tm+3:YAG

Photo diodeFrequency stabilizationto spectral hole AWG

Flip mirror

Geometric phase shift and rotation angle

0.35

0.40

0.45

0.50

0.55

0.60

0.650

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 100Time (ns)

Inte

nsity

(a.u

.)

Experimental data calculated through FFT

0.0

0.5

1.0

1.5

2.0

0.0 0.2 0.4 0.6 0.8 1.0

Control phase (pi)R

otat

ion

angl

e (p

i)

Theory

Bloch vector rotation angled measured using photon echo with measurement limited standard deviation of 0.03π.

Phase shifted echo fields measured with heterodyne detection for phase shift varying from 0 ~ 2π, controlled by laser pulse of varying phase from 0 ~ π.

Summary

• Two basic Bloch vector rotations has been studied by means of laser controlled geometric phase changes.

• The control pulse sequences for the two basic geometric Bloch rotations have been designed.

• U3 rotation has been demonstrated in thulium doped yttrium aluminum garnet (YAG). • Geometric phase and rotation angles have been measured by heterodyned photon echo

up to the accuracy of the measurement limit.• U2 rotation can be demonstrated in a similar way.• Two-qubits CNOT gate be accomplished by pure geometric operation controlled by laser

pulse sequence.

•Krishna Rupavatharam, Spectrum Lab, Montana State U.

•Randy Reibel, Spectrum Lab, Montana State U.

•Stefan Kroll, Lund Institute of Technology, Sweden•Alexander Rebane, Montana State U.

•Cone/Sun group, Montana State U.

•Sci. Mat. Inc. Bozeman, MT

•Air Force Office of Scientific Research grant (F49620-98-1-0283)

Acknowledgments

Physical qubit based on Tm doped crystals

Properties of Tm:YAG at 4.2KFrequency selectivity: Γin~20GHz,

Γh<100kHz Coherence time: ~10µs for |e>,

~ms for |1>,Lifetime: ~800µs for |e>,

~10s for |1>,Oscillation strength: ~10-7 for 3H6 -3H4 transition

Zeeman splitting: ~60MHz/Tesla for 3H6

~20MHz/Tesla for 3H4

Transition wavelength: 793nm

N f( )

fInhomogeneous Broadening

Homogeneous BroadeningN f( )

ffInhomogeneous Broadening

Homogeneous Broadening

g∆

e

0

1

1υ0υ

34H

36H

A physical qubit is:• Represented by the atoms of identical energy level resonant at

a selected frequency,• Stored at hyperfine levels of the electronic ground state (1>

and 0> ) with coherence time of ~ms,• Addressed selectively by laser pulse for quantum state

initialization.• Manipulated individually with quantum operation <µsEnergy levels and Zeeman splitting

Multiple qubits and CNOT gate

Different qubits are labeled by their unique resonant frequencies, addressed individually by tuning excitation laser frequency to the channel of the selected qubits. Qubits can prepared by coherent and/or incoherent pummping through spectral hole burning process. Multiple frequency channels at one spatial spot favorites system scalability.Qubits are coupled by ion-ion interaction. CNOT gate can be realized by well-designed laser pulse sequence through either direct Rabi rotations or geometric rotations. The driving pulse sequence is independent of the qubits states.

Resonant frequency

num

ber o

f ato

ms

100MHz Frequency channelQubit A Qubit B

Random distributiondoping level

30MHz

10MHz

Spectral partition for qubits

g∆ ~

0 Aυ1Aυ 0 Bυ1Bυ