Quantum Information Processing with Ultra-Cold Atomic Qubits

Transcription

1 Quantum Information Processing with Ultra-Cold Atomic Qubits Ivan H. Deutsch University of New Mexico Information Physics Group

2 Quantum Information Processing Classical Input State Preparation y in QUANTUM WORLD Control y out Measurement Classical Output

3 Example: Rydberg atom

4 Hilbert space and physical resources The primary resource for quantum computation is Hilbert-space dimension. Hilbert spaces of the same dimension are fungible, but the available Hilbert-space dimension is a physical quantity that costs physical resources. Single degree of freedom Action

5 Hilbert space and physical resources Many degrees of freedom Number of degrees of freedom Hilbert-space dimension measured in qubit units. Identical degrees of freedom Scalable resource requirement Strictly scalable resource requirement qudits

6 Quantum computing in a single atom Characteristic scales are set by atomic units Length Momentum Action Energy Bohr Hilbert-space dimension up to n 3 degrees of freedom

7 Quantum computing in a single atom Characteristic scales are set by atomic units Length Momentum Action Energy Bohr Poor scaling in this physically unary quantum computer 5 times the diameter of the Sun

8 First Conclusions All Hilbert Spaces of the same dimension are mathematically isomorphic and fungible. The dimension of Hilbert is a resource. Physics determines the structure of Hilbert space. Systems with multiple physical degrees of freedom give Hilbert space a tensor-product structure. Arbitrary superpositions lead to entangled states. Control of a many-body system is a necessary condition to have an exponentially large Hilbert space with out using an exponential physical resource. R. Blume-Kohout, C. M. Caves, and I. H. Deutsch, Found. Phys. 32, 1641(2002).

9 QIP = Many-body Control n-body Hilbert Space H = h 1 ƒ h 2 ƒl ƒ h n (dimension d n ) subsystem = body (dimension d) Fundamental Theorem of QIP An arbitrary unitary map on H can be constructed from a tensor product of: A finite set of single-body unitaries. { (1) u } i Any chosen entangling two-body unitary. u (2) ij u (1) (1) i ƒ u j

10 Optical Lattices

11 Why Optical Lattices? State Preparation Initialization Entropy Dump Laser cooling State Manipulation Potentials/Traps Control Fields Particle Interactions Quantum Optics NMR State Readout POVM State Tomography Process Tomography Fluorescence

12 Two Qubit Interaction: Three dimensional picture Planes of atoms interact pairwise (parallel operations)

13 Qubit-Qubit Interactions Electric dipole-dipole interaction. Optical AC Stark d~(s/2)ea 0 DC Stark (Rydberg) d~n 2 ea 0 G. K. Brennen et al. PRL (1999) D. Jaksch et al. PRL (2000) Ground electronic collision. V = 1 4 V (r) + 3 S 4 V (r) + (V (r) -V (r))r T T S s 1 r s 2 Magnetic dipole-dipole interaction. V = r m 1 r m 2-3( r m 1 e r r )( r m 2 e r r ) r 3 Real photon exchange (cavity QED) D. Jaksch et al. PRL (1999) L. You, M. Chapman PRA (2001) T. Pellizzari et al. PRL (1995)

14 Example: S-wave collisions 2,2 F q=0 D. Jaksch et al. PRL (1999) 1,1 F Ø q=p/4 fi Ø fi e if Ø Ø fi Ø Ø Ø fi Ø Ø f ~ (k L a)w osc t O. Mandel et al quant-ph O. Mandel et al quant-ph q=p/2 q=3p/4 q=p

15 General Ground-Electronic Collisions Exchange Interactions: V ( r) 3 S u r 1 S g V BO = 1 4 V r S( ) V r T ( ) r ( ) + V T ( r) + V S ( r) s 1 r s 2 Problem: Interaction does not conserve atomic quantum numbers Solution: Collisions between trapped but separated atoms

16 Atoms in Separated Traps z 0 ) H = ) p 1 2 2m + ) p 2 2 2m + V Ê trap r 1 - Dr z ˆ Ê Á + V Ë 2 trap r 2 + Dr z ˆ Á + V ˆ Ë 2 int (r) Dz V V R Dz = 0 Dz > 0 r Separation in harmonic case: ) H CM = ) H rel = ) 2 p CM 2M mw 2 osc ) 2 p rel 2 r 2m mw osc r R 2 r - Dze r 2 z + V ˆ int (r) Dz r Very different scales: k L z 0 = 0.1 fi z 0 ~ 20 nm R ~ 1 A o

17 Self-Consistent Scattering Length Model Regularized d-function interaction V int ( r ) = 2ph2 m a d(3) r ( ) r r scattering length: a = -lim k Æ0 tand 0 ( E) k E ( ) Near resonance a(e) replace constant a with energy dependent a eff Energy E Bolda et. al, PRA 66, 2001 V int ( r ) = 2ph2 m a eff ( E) d (3) ( r ) r r Energy dependent a eff E scattering length ( ) = - tand 0( E) k( E)

18 Self-Consistent Eigenvalue Solution ) H rel = ) 2 p rel 2m mw osc 2 ( r - Dze r z ) 2 + 2ph2 m a eff (E)d (3) r ( ) r r ˆ H rel y = E y Solve for eigenvalues as a function of fixed scattering length: E(a eff ) Simultaneous solutions Solve for scattering length as a function of fixed scattering length: a eff (E)

19 Atoms in Separated Traps: Eigenspectrum a eff = -z 0 < 0 a eff = 0.5z 0 > 0 energy E energy E trap separation Dz /z 0 trap separation Dz /z 0 Why is there an Anti-crossing for positive scattering lengths?

20 Total potential V Atoms in Separated Traps: Shape Resonance trap eigenstate molecular bound state r energy E Localization of resonance trap separation Dz E bound + V trap = 3 2 hw fi Dz = 3+ z z 0 a scatt

21 Energy dependent d-function interaction: Bound States Bound state of actual potential Pole in the S-matrix E b = h2 2 k b 2m = - h2 2 k b 2m with k b = ik b s l ( k b ) = e 2id l Æ therefore id l Æ a eff ( ) = - itanh ( id 0) k b Reg. d - function bound state ik b Æ 1 k b E d = - h2 = - h2 2 k b 2 2m a eff 2m = E b Effective scattering length model contains information about all bound states self-consistently

22 Energy Gap Calculation Example: Collisions in 87 Rb singlet a s = 93 a 0 = 0.39 z 0 triplet a t = 102 a 0 = 0.42 z 0 ( l = 789 nm, h = k z 0 = 0.1) Æ DE 0.03 hw Example: Collisions in 133 Cs singlet a s = 280 a 0 = 1.2 z 0 triplet a t = 2400 a 0 = 10 z 0 ( l = 852 nm, h = 0.1) energy gap DE variational estimate numerical calculation scattering length a Æ DE hw Quantum State State Control via via a Trap-Induced Shape Shape Resonance in in Ultra-Cold Atomic Atomic Collisions, R. R. Stock, Stock, I. I. H. H. Deutsch, and and E. E. Bolda, Bolda, quant-ph/

23 Shape Resonance and Conditional Logic V V V molecular bound state molecular bound state trap eigenstate r r trap eigenstate r

24 I.H. Deutsch, Dept. Of Physics and Astronomy University of New Mexico Collaborators: Physical Resource Requirements for Scalable Q.C. Carl Caves (UNM), Robin Blume-Kohout (LANL) Quantum Logic via Dipole-Dipole Interactions Gavin Brennen (UNM/NIST), Poul Jessen (UA), Carl Williams (NIST) Quantum Logic via Ground-State Collisions René Stock (UNM), Eric Bolda (NIST)