Quantum Information Processingwith Ultra-Cold Atomic Qubits

Ivan H. DeutschUniversity of New Mexico

http://info.phys.unm.edu

Information Physics Group

Classical Input

Classical Output

QUANTUM WORLDy in

y out

Quantum Information Processing

State Preparation

Control

Measurement

http://gomez.physics.lsa.umich.edu/~phil/qcomp.htmlExample: Rydberg atom

Hilbert space and physical resourcesThe primary resource for quantum computation is Hilbert-space dimension.

Hilbert spaces of the same dimension are fungible, but the availableHilbert-space dimension is a physical quantity that costs physical resources.

Single degree of freedom

Action

Hilbert space and physical resources

Many degrees of freedom

Hilbert-space dimensionmeasured in qubit units.

Identical degreesof freedom

Number of degreesof freedom

quditsStrictly scalable resource requirement

Scalable resource requirement

Quantum computing in a single atomCharacteristic scales are set by “atomic units”

Length Action EnergyMomentum

Bohr

Hilbert-space dimension up to n3 degrees of freedom

Quantum computing in a single atomCharacteristic scales are set by “atomic units”

Length Action EnergyMomentum

Bohr

5 times the diameter of the Sun

Poor scaling in this physically unary quantum computer

• All Hilbert Spaces of the same dimension aremathematically isomorphic and fungible.• The dimension of Hilbert is a resource.• Physics determines the structure of Hilbert space.• Systems with multiple physical degrees of freedomgive Hilbert space a tensor-product structure.• Arbitrary superpositions lead to entangled states.• Control of a many-body system is a necessarycondition to have an exponentially large Hilbert spacewith out using an exponential physical resource.

First Conclusions

R. Blume-Kohout, C. M. Caves, and I. H. Deutsch,Found. Phys. 32, 1641(2002).

QIP = Many-body Control

†

H = h1 ƒ h2 ƒL ƒ hn

n-body Hilbert Space

“subsystem” = “body” (dimension d)

(dimension dn)

Fundamental Theorem of QIPAn arbitrary unitary map on H can be constructedfrom a tensor product of:• A finite set of single-body unitaries .• Any chosen entangling two-body unitary.

†

ui(1){ }

†

uij(2) ≠ ui

(1) ƒ u j(1)

Optical Lattices

Why Optical Lattices?

State Preparation• Initialization• Entropy Dump

State Manipulation• Potentials/Traps • Control Fields• Particle Interactions

State Readout• POVM• State Tomography• Process Tomography

Fluorescence

Laser cooling Quantum OpticsNMR

Two Qubit Interaction:Three dimensional picture

Planes of atoms interact pairwise(parallel operations)

• Real photon exchange (cavity QED)

Qubit-Qubit Interactions

• Magnetic dipole-dipole interaction.

†

V =r m 1 ⋅

r m 2 - 3( r

m 1 ⋅r e r)(

r m 2 ⋅

r e r )r3

†

V =14

VS (r) +34

VT (r) + (VT (r) -VS (r))r s 1 ⋅r s 2

• Ground electronic collision.

• Electric dipole-dipole interaction.• Optical AC Stark d~(s/2)ea0• DC Stark (Rydberg) d~n2ea0

G. K. Brennen et al. PRL (1999)

D. Jaksch et al. PRL (1999)

D. Jaksch et al. PRL (2000)

L. You, M. Chapman PRA (2001)

T. Pellizzari et al. PRL (1995)

Example: S-wave collisions

q=0

q=p/4

q=p/2

q=3p/4

q=p

†

F↑

†

FØ1,1

2,2

†

↑ ↑ fi ↑ ↑

↑ Ø fi eif ↑ Ø

Ø ↑ fi Ø ↑

Ø Ø fi Ø Ø

†

f ~ (kLa)wosct

D. Jaksch et al. PRL (1999)

O. Mandel et al quant-ph 0301169O. Mandel et al quant-ph 0301169

General Ground-Electronic Collisions

• Exchange Interactions:

†

V r( )

†

r

†

1Sg

†

3 Su

†

VBO =14

VS r( ) +34

VT r( ) + VT r( ) + VS r( )( ) r s 1 ⋅r s 2

• Problem: Interaction does not conserve atomic quantum numbers

• Solution: Collisions between trapped but separated atoms

Atoms in Separated Traps

z0

†

Dz

†

) H =

) p 1

2

2m+

) p 22

2m+ Vtrap

r r 1 -D

r z 2

Ê

Ë Á

ˆ

¯ ˜ + Vtrap

r r 2 +D

r z 2

Ê

Ë Á

ˆ

¯ ˜ + ˆ V int (r)

Very different scales:

†

kLz0 = 0.1fi z0 ~ 20 nm

†

R ~ 1 Ao

†

) H CM =

) p CM

2

2M+

12

mwosc2

r R 2

) H rel =

) p rel2

2m+

12

mwosc2 r r - Dzr e z

2+ ˆ V int (r)†

Dz = 0

†

Dz > 0

r

r

V

V

Dz

Separation in harmonic case:

R

Self-Consistent Scattering Length Model

• Regularized d-function interaction

†

Vintr r ( ) =

2ph2

ma d(3) r r ( ) ∂

∂rr

†

a = -limk Æ0

tand0 E( )k E( )scattering length:

†

Vintr r ( ) =

2ph2

maeff E( ) d(3) r r ( ) ∂

∂rr

†

aeff E( ) = -tand0 E( )

k E( )Energy dependentscattering length

Energy E

a(E) replace constant a with

energy dependent aeff

Bolda et. al, PRA 66, 2001

• Near resonance

Self-Consistent Eigenvalue Solution

†

) H rel =

) p rel2

2m+

12

mwosc2 r r - Dzr e z( )2

+2ph2

maeff (E)d(3) r r ( ) ∂

∂rr

†

ˆ H rel y = E y

• Solve for eigenvalues as a function of fixedscattering length:

†

E(aeff )

• Solve for scattering length as a function offixed scattering length:

†

aeff (E)

Simultaneous solutions

Atoms in Separated Traps: Eigenspectrum

†

aeff = -z0 < 0

†

aeff = 0.5z0 > 0

Why is there an “Anti-crossing” for positive scattering lengths?

trap separationen

ergy

Etrap separation

ener

gy E

†

Dz /z0

†

Dz /z0

• Total potential

Atoms in Separated Traps: “ Shape Resonance”

†

Ebound + Vtrap =32

hw fiDzz0

= 3+z0

2

ascatt2

V

rmolecular boundstate

trap eigenstate

trap separation Dz

ener

gy E

• Localization of resonance

Energy dependent d-function interaction:Bound States

• Bound state of actual potential

†

Eb =h2kb

2

2m= -

h2kb2

2mwith kb = ikb

Effective scattering length model contains informationabout all bound states self-consistently

†

sl kb( ) = e2id l Æ • therefore idl Æ •

†

aeff kb( ) = -i tanh id0( )

ikb

Æ1

kb

• Pole in the S-matrix

†

Ed = -h2

2m aeff2 = -

h2kb2

2m= Eb

• Reg. d - function bound state

• Example: Collisions in 87Rbsinglet as = 93 a0 = 0.39 z0

triplet at = 102 a0 = 0.42 z0( l = 789 nm , h = k z0 = 0.1)

Æ DE £ 0.03 hw

• Example: Collisions in 133Cssinglet as = 280 a0 = 1.2 z0triplet at = 2400 a0 = 10 z0

( l = 852 nm, h = 0.1)

Æ DE £ 0.3 - 0.5 hw

Energy Gap Calculation

scattering length a

ener

gy g

ap D

E

variational estimatenumerical calculation

“Quantum State Control via a Trap-Induced Shape Resonance in Ultra-ColdAtomic Collisions”, R. Stock, I. H. Deutsch, and E. Bolda, quant-ph/0304093

“Quantum State Control via a Trap-Induced Shape Resonance in Ultra-ColdAtomic Collisions”, R. Stock, I. H. Deutsch, and E. Bolda, quant-ph/0304093

“ Shape Resonance” and Conditional Logic

V

r

molecular bound state

trap eigenstate

V

r

V

r

molecular bound state

trap eigenstate

Carl Caves (UNM), Robin Blume-Kohout (LANL)

http://info.phys.unm.edu/~deutschgroup

Gavin Brennen (UNM/NIST), Poul Jessen (UA),Carl Williams (NIST)

I.H. Deutsch, Dept. Of Physics and AstronomyUniversity of New Mexico

Collaborators:• Physical Resource Requirements for Scalable Q.C.

• Quantum Logic via Dipole-Dipole Interactions

René Stock (UNM), Eric Bolda (NIST)• Quantum Logic via Ground-State Collisions