All optical quantum computation by engineering semiconductor macroatoms Irene D Amico Dept. of Physics, University of York (Institute for Scientific Interchange, Torino)
GaAs/AlAs, GaN/AlN Eliana Biolatti Fausto Rossi (Physics Dept, Politecnico di Torino GaAs/AlAs Paolo Zanardi (Institute for Scientific Interchange) GaN/AlN Sergio DeRinaldis Ross Rinaldi Roberto Cingolani (National Nanotechnology Laboratory, Lecce)
Outline Quantum information processing Semiconductor based implementations Quantum dots and quantum dot modeling GaAs and GaN based self-assembled quantum dots Exact diagonalization approach Engineering exciton-exciton coupling and quantum dot optical response Simulated experiments: C-NOT and state entanglement
Quantum information processing Basic motivation: to benefit from the natural parallelism of quantum mechanics (superposition ( principle) Solve complex mathematical problems (factorization of large integer numbers, database search) Simulation of complex systems (many-particles, quantum behavior) Quantum computers are explicitly governed by quantum mechanics, and they are based on two level quantum systems (qubits)
Quantum information processing Qubits can be in principle realized by any two-level quantum system: The polarization of a photon Two levels of a discrete energy spectrum Up or down spin state of an electron The state (dead or alive) of a Schroedinger s cat (maybe the cat would not be so happy though.)! bit, state: 0,1 qubit, state: a 0>+b 1>, a,b complex, SUPERPOSITION! Superposition: each qubit can store 0> and 1> simultaneously, i.e N qubits can store 2 numbers simultaneously and calculations can be performed simultaneously o each of these numbers QUANTUM PARALLELISM! N 1 0 0, 1 0 1 = =
Quantum information processing General computation scheme: 1. Preparation of the initial state 2. Its coherent propagation and manipulation 3. Its detection or measurement Problems: Decoherence: interaction with the environment &/or with non computational degrees of freedom (for example additional charges, phonons, additional energy levels) Scalability: necessity of building and addressing/controlling thousands of qubits Error correction Hardware Reference book: M.Nielsen and I. Chuang, Quantum computation and quantum information Cambridge Univ. Press, 2000
Semiconductor-based implementation Phosphorus nuclear spins in silicon (electrical manipulation)- B.E.Kane, Nature 393,133(1998) Electron spins in quantum dots (electrical manipulation)- D.Loss and D.P.DiVincenzo, PRA 57, 120 (1998) Electron spins in microcavity coupled quantum dot- Sherwin, Imamoglu, Montroy, PRA 60, 3508 (1999), M. Feng, I. D'Amico, P. Zanardi and F. Rossi, PRA 67, 014306 (2003) Excitons in quantum dots generated and coherently controlled by picosecond laser pulses- GaAs: Biolatti, Iotti, Zanardi and Rossi, PRL 85, 5647 (2000); E. Biolatti, I. D'Amico, P. Zanardi, and F. Rossi, PRB 65, 075306 (2002) GaN: DeRinaldis, I. D'Amico, E. Biolatti, R. Rinaldi, R. Cingolani, and F. Rossi PRB 65, 081309 (2002) Electron spins in quantum dots coupled by exciton-exciton interaction (Pauli-blocking + all-optical manipulation) E. Pazy, E. Biolatti, T. Calarco, I. D'Amico, P. Zanardi, F. Rossi, P. Zoller, Europhys. Lett. 62, 175 (2003)
Quantum dots Quasi-0-dimensional structure i.e. the confining lengths are DeBroglie wavelength of carriers (nanometers). This generates a This generates a discrete energy spectrum similar to the atomic spectrum (macroatoms macroatoms) Confinement: strong interactions among particles inside the dot Discrete spectrum: weak interaction with the environment by controlling dimensions and shape it is possible to engineer the electronic structure.
Quantum Dots as Hardware Top view InGaAs/GaAs QDs (from NNL-Lecce web page) Growth direction: stacked QDs Growth direction QDa QDb In-plane directions CB VB
Quantum Dots as Hardware QD: Quasi-0-dimensional discrete energy dimensional boxes: energy spectrum similar to the atomic spectrum. a b -- Growth direction CB 0> 1> exciton exciton + a b exciton VB quantum dots/qubits are coupled by exciton-exciton interactions = biexcitonic shift ε QDa QDb qubit a qubit b ε Two QD coupling two-qubit gate
Advantages with laser technology: possibility of generating and coherently controlling excitons on a subpicosecond time-scale slowly-varying (electrical or magnetical) external fields are avoided fully optical gating schemes are proposed scalability Disadvantages Short decoherence times (ps, ns) Perfect control of QDs growth (size and patterning) Exact knowledge of single dot spectrum (single dot addressing)
Exciton-exciton coupling Growth direction a b -- CB exciton + a b exciton VB quantum dots/qubits are coupled by exciton-exciton interactions = biexcitonic shift ε ε QDa QDb qubit a qubit b Two QD coupling two-qubit gate Pb: excitons are in general neutral objects!
Exciton-exciton coupling In the absence of an electric field excitons in different QDs interact very weakly (basically neutral objects) GaAs quantum dots zincblende structure static, in-plane, external electric field (75 kv/cm) to spatially separate electrons and holes and to create dipoles GaN quantum dots wurzite structure spontaneous polarization piezoelectric potential strong built-in electric field (MV/cm) in the z-direction creates intrinsic dipoles
dipole dipole GaAs GaN Parallel dipoles Stacked dipoles dipole dipole Exciton-exciton coupling (biexcitonic ( shift) ~ dipole-dipole interaction
GaAs /AlAs :-) tunable coupling :-) growth parameters to engineer electronic structure :-) well characterized material :-) biexcitonic shift ~3-4meV :-( external field can ionize trapped carriers :-( more complex circuits GaN/AlN :- coupling tunable by growth parameters only :-) growth parameters to engineer electronic structure :- not well characterized material :-) biexcitonic shift up to 8-9 mev :-) internal field does not ionize trapped carriers :-(stronger dephasing due to
Exact-diagonalization approach System Hamiltonian Hc Hcc Hcl single-particle contribution Coulomb interaction terms carrier-light coupling H = ( H c + H cc ) + H We perform a direct diagonalization of the manybody Hamiltonian H c + H cc for a given number N of excitons cl
Absorption spectra Given the exact many-body states and the N ε α corresponding energies : N α A 2 ( ) ( ) N H N 1 δ ε N ε 1 ω N 1 N ω N α β β cl α β α N =1 excitonic spectrum N = 2 biexcitonic spectrum
Optical response E. Biolatti, I. D Amico, P. Zanardi and F. Rossi, PRB 65, 75306 (2002) S. De Rinaldis, I. D Amico, E. Biolatti, R. Rinaldi, R. Cingolani and F. Rossi, PRB 65, R81309 (2002)
GaAs - System parameters Well defined QD, i.e. strong confinement regime large biexcitonic shift ε p p ~ a b Z 3 1 ε F + 1 2 2 2 m e ω e m h ω h i.e. large electric field but not too strong confinement optical response, i.e. not too large electric field I. D Amico and F. Rossi, APL 79, 1676 (2001) E. Biolatti, I. D Amico, P. Zanardi and F. Rossi, PRB 65, 75306 (2002) By an analytical model: 3.5 mev ε
GaN - System parameters p a p b 3 biexcitonic shift ε ~ 2 vs: Z barrier width QD height L QD base D d F~func(QD height, barrier width) S.DeRinaldis, I. D Amico and F. Rossi, Appl. Phys. Lett., 81, page 4236 (2002); Phys. Rev. B (to be published, 2004)
GaN: biexcitonic shift versus oscillator strength Optimal structure: QD height 2.5-3 nm S.DeRinaldis, I. D Amico and F. Rossi, Appl. Phys. Lett., 81, page 4236 (2002);
Computational subspace 0 1 l l absence of presence of ground -state exciton in QD ground -state exciton in QD Computational state-space The whole computational space is spanned by the basis: n = l n l n l ( ), = 0,1 l l Many-body effective Hamiltonian: ~ ε = ε + ε l l n l ' l ll ' l' 0 = ε ˆ ˆ ˆ l l n l + ε ll ' ll ' n l n 1 2 ~ H renormalized excitonic S. Lloyd, Science 261, 1569 (1993) transitions l '
Simulated experiments We perform a direct numerical solution of the Liouville-von Neumann equation ρ i [ ~, ] ρ = H ρ + t t ( ρ the density matrix), deco which contains the quantum mechanical unitary evolution plus decoherence processes. Decoherence processes are described by phenomenological dephasing times T1, T2.
ε a ε a 1 0 a 0 0 b a b Single-qubit operations ε a ε a 0 0 a b 1 0 a b Single color π - rotation sequence : 0 a 1 a
ε a ε b + ε ε b Conditional dynamics of excitons ε a ε b ε a ε b + ε
ε a Conditional two-qubit operations ε b + ε ε b (GaN) PRB 65, R81309 (2002) ε ε + ε a b C-NOT: 1 a b a b 0 1 1, 1 1 1 0 a b a b 0 0 0 0, 0 1 0 1 a b a b a b a b
Generation of entangled states GaAs GaN PRB 65, 75306 (2002) PRB 65, R81309 (2002) ( 0 + 1 ) 0 0 0 + 1 1 a a b a b a b
Conclusions All-optical implementation of quantum information processing with semiconductor quantum dots Systematic analysis of GaAs and GaN based quantum dots as QIC devices Detailed study on engineering quantum dot coupling by exciton-exciton interactions Sub-picosecond two-qubit gating based on exciton-exciton interactions and driven by multicolor sequences of ultrafast laser pulses synergic use of spin (memory) and charge (gating) degrees of freedom (E. Pazy, E. Biolatti, T. Calarco, I. D'Amico, P. Zanardi, F. Rossi, P. Zoller, Europhys. Lett. 62, 175 (2003)) coupling to microcavities (M. Feng, I. D'Amico, P. Zanardi and F. Rossi, Phys. Rev. A 67, 014306 (2003). ) study of pure dephasing in GaAs and GaN based QDs (B. Krummheuer, V.M. Axt, T. Kuhn, I. D'Amico, and F. Rossi, preprint 2004)