Quantum control of spin qubits in silicon

Size: px

Start display at page:

Download "Quantum control of spin qubits in silicon"

Gregory Ellis
5 years ago
Views:

1 Quantum control of spin qubits in silicon Belita Koiller Instituto de Física Universidade Federal do Rio de Janeiro Brazil II Quantum Information Workshop Paraty, 8-11 September 2009

2 Motivation B.E.Kane, Nature 393,133(1998) Qubits are the 31 P nuclear spins I=1/2. Patoms insi

3 Background Elementary gates for quantum computation Barenco et al PRA (1995) Combinations of two qubits exclusiveor + all single qubit operations may perform any unitary operation on arbitrarily many qubits, i.e., any operation in a QC. spin qubits exchange gate H S ( t) J ( t) S S 1 2 Loss & DiVincenzo PRA (1998) Kane Nature (1998) Transient Heisenberg coupling Evolution: ( t) U ( ) (0) exp{ t S t T i H S ( t' ) dt'} (0) 0 S Jtdt () J (mod2 ) U( ) U 0 0 S S S SWAP XOR-GATE: U XOR exp{ i( / 2) S }exp{ i( / 2) S } U ( ) exp{ i( / 2) S } U z z 1/ 2 z 1/ S S 1 S S ( )

Spin interactions in Si: 31 P R = e e n e n e R J A A B H R H 2 1 2 2 2 1 1 1 ) ( ) ( ) (

4 Spin interactions in Si: 31 P R = e e n e n e R J A A B H R H ) ( ) ( ) ( EXCHANGE 2-qubits operations n e n z n n e z B n e A B g B H 1-qubit operations HYPERFINE COUPLING

5 Why Si: 31 P? Lifetimes of P-bound electron and nuclear spin are extremely long in silicon*. Availability of the state-of-the-art crystal growth, processing, and isotope engineering technologies. Well understood physical properties. Possible integration with currently used Si devices. Si GaAs *Stable isotopes Nuclear spin Stable isotopes Nuclear spin 28 Si 92.2% 29 Si 4.7% 30 Si 3.1% 0 ½ 0 69 Ga 60.1% 71 Ga 39.9% 75 As 100 % 3/2 3/2 3/2

6 Experimental status (spin coherence) Coherence times of 60 ms for donor electron spin and of 65 ms to 1.75 s for donor nuclear spin have been measured for donors in Si. Tyryshkin et al. J. Phys.C 18 S783 (2006). Morton et al. Nature 455, 1085 (2008).

7 Experimental status (STM bottom-up) Towards the fabrication of P qubits for a Si quantum computer O Brien et al PRB 64, Atomically precise placement of single dopants in Si Schofield et al PRL 91, ~1 nm accuracy positioning of single P atoms in Si demonstrated

8 Top-down fabrication: ion implant As modeled by SRIM, a 14 kev 31 P + ion implanted into Si swith 5 nm SiO 2 surface layerd has a mean depth of 20 nm with lateral and longitudinal straggles of 8 and 11 nm, respectively. Review on P ion-implant for Si QC fabrication: Schenkel et al JAP 94, 7017 (2003)

9 Spin read-out Successful experimental implementation on ion-implanted samples reported in Silicon Qubit Workshop Berkeley August 24 to 26, 2009

10 Outline Bulk Si Shallow donors (P, As) in Si 2-qubit Exchange gate Doped photonic-crystal Silicon cavity

11 Bloch states in Si Lattice potential periodicity Translational symmetry: FCC lattice Diamond structure k 1 st Brillouin zone n ( k ) empty states 5.43 Å X Γ filled states k=0

12 Conduction-band edge Reciprocal space 6 equivalent minima k ( = 1,2,,6) = (0,0,k 0 ); (0, k 0,0); (k 0,0,0) k 0 = /a Eigenfunctions: 6 Bloch states: ( r ) exp[ ik ( r )] u ( r ) Planewave part (free electron-like) Periodic part (atomic-like)

superposition of degenerate Bloch states is also an eigenstate (not in Bloch s

13 Bloch states in Si conduction band-edge: k x Electronic probability density Bloch state at k k 1 u 2 2 k x x p x -like symmetry with lattice periodicity Any superposition of degenerate Bloch states is also an eigenstate (not in Bloch s form). Example: 1 6 k 1, 6 () r 2 () r ( r R0) Interference pattern: oscillatory incommensurate R 0

14 Outline Bulk Si Shallow donors (P, As) in Si 2-qubit Exchange gate Doped photonic-crystal Silicon cavity

15 Hydrogenic model for P in Si Si (IV) 14 e 14 p+ P (V) 15 e 15 p+ 2 2 * * [ /2 m U()] r () r Eb () r ~ + _ o 3 () r (1/ a*)exp( r/ a*), a* a0 ( m0 / m*) 30A Asymptotic exchange coupling of two hydrogen atoms (Herring&Flicker, 1964) donor pair exchange: 5 R 2 J ( R) ( ) a * U ( r) exp( 2R e r / a*) 2

16 Lowest energy levels for As in Si

17 Single substitutional donor at R 0 Kohn-Luttinger electronic ground state (A 1 symmetry) ) ( μ 0 1,6 μ μ μ 0 ) ( ) ( ) ( R r ik R e r u R r F r ) ( 0 2 R r e r V ENVELOPE FUNCTIONS (variational): deformed 1S hydrogenic orbitals centered at donor site PLANEWAVE PHASES PINNED AT DONOR SITE

Kohn-Luttinger ground state electronic charge distribution The oscillatory behavior of the donors wavefunctions in Si is well established.

18 Kohn-Luttinger ground state electronic charge distribution The oscillatory behavior of the donors wavefunctions in Si is well established. This behavior has no consequences for the conventional applications of n-doped Si. What is the impact of this behavior in the proposed Si-based quantum computer operations? Koiller, Capaz, Hu & Das Sarma PRB 70, (2004).

19 Outline Bulk Si Shallow donors (P) in Si 2-qubit exchange gate Doped photonic-crystal Silicon cavity

20 Donor pair exchange coupling Heitler-London approach 1 J( RA RB) J ( RA RB)cos ( k k ) ( RA RB) 36 Koiller, Hu & Das Sarma PRL 88, (2002) PRB 66, (2002) Strongly dependent on inter-donor distance No lattice periodicity Anisotropic Oscillatory behavior

21 Donor pair exchange coupling Si P X

22 Exchange anisotropy and oscillatory behavior Target * (R target,j target ) ---- For donors exactly aligned along the [100] crystal axis, the oscillatory behavior may be ignored in practice.

23 Inter-donor positioning uncertainties 1 st donor [100] R R uncertainty * 2 nd donor within a sphere with given uncertainty radius, centered at target point.* Distributions of exchange coupling R uncertainty * Small displacements form the tareget relative position. On the oder of atomic neighbor distances. Peaked at J~0 (very J target!!)

24 Inter-donor positioning uncertainties 1 st donor [100] R R uncertainty * 2 nd donor within a sphere with given uncertainty radius, centered at target point.* Distributions of exchange coupling R uncertainty * Exchange gates control: Nanofabrication challenge! Peaked at J~0 (very J target!!)

25 Can we build a large-scale quantum computer using semiconductor materials? scalable quantum computing in semiconductors may only be possible at the end of the road of Moore s Law Scaling: when devices are engineered and fabricated at the atomic level. B.E. Kane MRS BULLETIN, FEB 2005 or, new ideas are needed!

26 Outline Bulk Si Shallow donors (P) in Si 2-qubit Exchange gate Doped photonic-crystal silicon cavity

27 arxiv: Basic elements: Doped photonic-crystal Silicon cavity; Donor impurities placed at the antinodes of a cavity mode; Electrodes placed above donors, gated to produce electrical field at each donor position; Laser beams; Uniform magnetic field; Operating temperature 7K. Robust against small donor displacements.

28 Substitutional Donors in Si: Solid-state analogue of the H atom Relevant energy levels for Si:As Ramdas and S. Rodriguez, Rep. Prog. Phys. 44, 1297 (1981)

29 Generating well defined electron spin qubits from 1S(A1) Electronic spin state not well-defined due to hyperfine interaction. Solution: apply magnetic field strong enough to decouple nuclear and electronic spin states 400 MHz Qubits encoded in the subspace

30 Spin interactions in Si:As Optical cavity 1-qubit operations Spin-orbit coupling > > Raman-coupling of qubit states through off-resonant excitation of transitions involving two laser beams.

31 Spin interactions in Si:As Optical cavity 1-qubit operations 2-qubits operations δ i Raman-coupling of qubit states through off-resonant excitation of transitions involving two laser beams. Coupling between qubits mediated by cavity field.

32 2-qubits operations δ i δ i SWAP: δ i = δ i δ j j = all other N-2 donors except i and i.

33 Initialization and readout Initialization: opt. pumping Readout: radiative decay No spin-orbit coupling: dark > light > At T 7K only manifold 1S(A1) is populated. Optical pumping of state >. Readout by monitoring fluorescence light of a cycling transition.

36 Donor misplacements compatible with current nanofabrication capabilities present no problems here! = 100Å

37 Summary Theoretical investigations of the feasibility of 2-qubit operation based on donor-pair exchange coupling in Si, show fast oscillatory behavior of the coupling with interdonor position. Precisely controlling exchange gates for spin qubits in Si remains a nanofabrication challenge. We propose a scheme with donor-based electron spin qubits in Si THz cavities which combines the Si substitutional-donor quantum computing architecture with the optical initialization and manipulation processes (already demonstrated in ion traps and other atomic systems). 2-qubit operations are mediated by the vacuum field of the silicon material cavity, which couples to the donor states. The scheme is insensitive to small displacements of the donor impurities in the host.

38 Acknowledgments Theoretical investigations of the feasibility of 2-qubit operation based on donor-pair exchange coupling in Si, show fast oscillatory behavior of the coupling with interdonor position. Precisely controlling exchange gates for spin qubits in Si remains a nanofabrication challenge. Rodrigo Capaz UFRJ Xuedong Hu SUNY, Buffalo Sankar Das Sarma CMTC, U. Maryland

39 Acknowledgments We propose a scheme whith donor-based electron spin qubits in Si THz cavities which combines the Si substitutional-donor quantum computing architecture with the optical initialization and manipulation processes (already demonstrated in ion traps and other atomic systems). 2-qubit operations are mediated by the vacuum field of the silicon material cavity, which couples to the donor states. The scheme is insensitive to small displacements of the donor impurities in the host. M. Abanto, L. Davidovich, and R. L. de Matos Filho Inst. de Física Universidade Federal do Rio de Janeiro

40 Acknowledgments Support: Brazil CNPq FAPERJ Instituto do Milênio de Nanociências Instituto do Milênio de Informação Quântica USA CMTC (UMD)

arxiv:cond-mat/ v1 [cond-mat.mtrl-sci] 26 Feb 2004

arxiv:cond-mat/ v1 [cond-mat.mtrl-sci] 26 Feb 2004 Voltage Control of Exchange Coupling in Phosphorus Doped Silicon arxiv:cond-mat/42642v1 [cond-mat.mtrl-sci] 26 Feb 24 C.J. Wellard a, L.C.L. Hollenberg a, L.M. Kettle b and H.-S. Goan c Centre for Quantum