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