Quantum Computing with neutral atoms and artificial ions NIST, Gaithersburg: Carl Williams Paul Julienne T. C. Quantum Optics Group, Innsbruck: Peter Zoller Andrew Daley Uwe Dorner Peter Fedichev Peter Rabl
Outline [Quantum Computing based on] neutral atoms in optical lattices spin-dependent lattices high fidelity loading of large lattice arrays beyond Mott insulator Rabl et al. Entangling atoms two-atoms Feshbach & photoassociation gates atoms in pipeline structures Dorner et al. experiments by I. Bloch et al., LMU & NIST Gaithersburg All-optical quantum computing with quantum dots Suppressing decoherence via quantum-optical techniques
Background: atoms in spin-dependent optical lattices optical lattice: spatially varying AC Stark shifts by interfering laser beams trapping potential depends on the internal state internal states we can move one potential relative to the other, and thus transport the component in one internal state fine structure Alkali atom move via laser parameters
interactions by moving the lattice + colliding the atoms by hand theory: Jaksch et al. exp.: I. Bloch et al. atom atom 2 internal states e i φ e i φ e i φ e i φ move 0 0 0 0 0 collision by hand Ising type interaction: building block of the Universal Quantum Simulator H J 2 a,b z a z b nearest neighbor, next to nearest neighbor...
Correcting defects in optical lattices Theory: Zoller Cirac et al. (Ibk & MPQ) Preparation of qubits via a superfluid Mott insulator phase transition Mott insulator have still some defects... present LMU exp.: approx. out of 0 (not optimized) hole extra atom Questions: Even more regular loading? Can we heal defects? Self-healing? Rem: it seems difficult to do this in normal solids
Defect free optical crystals (for quantum computing) dressed energy levels bi 0.2 atom -0.2 δ U bb 2 atoms ai prepare a Mott insulator with n=2 atoms: 0.5 0 U aa defect: atom defect: 3 atoms 2 0 3 atoms After detuning sweep: exactly one atom in b -0.5 0 sweep detuning
Collision gates & speed validity of the Hubbard model we can tune to a resonance to have a (free space) scattering length U ν a 0 U 4 a s 2 m d 3 x w x 4 scattering length a s a 0
2 2 2m R 2 2 2m 2 R 2 cm R E cm cm R harmonic approximation 2 2 r 2 2 2 r 2 V r r E r photoassociation resonance 2 2 r 2 v v v 2 3 v 0 Born Oppenheimer potentials including trap C 6 r 6 (schematic not to scale) D e 0 2 Hz <5 nm
Coupling into molecular states via a Feshbach ramp switching speed ν trap
Feshbach switching in a spherical trap Start with the Feshbach resonance state 0 trap units above threshold Switch it suddenly close to threshold Wait ~ trap time (~ µs assuming MHz trap) Switch back A phase π is accumulated Fidelity: 0.9996 No state dependence required, however difficult atom separation with stateindependent potential State dependence: lattice displacement Non-adiabatic transport in optical lattice: simulation with realistic potential shape Fidelity 0.99999 in.5 trap times through optimal control theory
Feshbach ramp in an elongated trap 200 Assumptions: Cigar-shaped trap Transverse motion frozen Lower longitudinal density of states Bigger coupling to each level Magnetic field 50 00 0.2 0.4 0.6 0.8 Smaller non-adiabatic crossing Smoothly varying magnetic field Phase gate in trap time with fidelity 0.9996 Disadvantage: quasi-d requirement limits trap frequency & speed Phase 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 Idea: use this in a free-fall scheme where no state-dependent potential would be needed Ground-state population 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8
Alternative trap designs pattern loading, e.g for addressing single atoms laser NIST exp. group Microlens arrays Hannover
Atom chips magnetic traps issues: conservative potential surface effects single atom loading laser cooling linear ion trap loading from a BEC Mott insulator loading? reservoir (BEC) Atom (qubit) transport loading micro trap light for processing detector processing in arrays of micro traps Schmiedmayer Heidelberg, Munich, Harvard, Orsay control pad for selective addressing of each sub system
Optical-tweezers double trap for two single atoms Grangier A single atom is trapped in each site Vacuum chamber y x z Dipole trap Coils Resolution of the imaging system: one micron per pixel 4 µm N. Schlosser et al, Nature 4, 024 (200) Second trapping beam First trapping beam
Atoms in D pipelines beamsplitter atoms motivation: D experiments with optical (and magnetic) traps atoms beam splitter (Hannover)
Atoms in D lattices Beam splitter atom per site atoms in a longitudinal lattice are moved across a beam splitter transverse potential at a lattice site tunneling splitter increases spatial separation of the wells in an adiabatic way
Atoms in D lattices Beam splitter: attractive or repulsive interaction between adjacent atoms W nearest neighbor interaction product state attractive max entangled state repulsive Nearest neighbor interaction: cold collisions, dipole-dipole (Rydberg atoms) Jaksch et al. PRL 82, 975 (999), Jaksch et al. PRL 85, 2208 (2000) max entangled state
Atoms in D lattices Motivations: Interferometry (attractive interaction) very sensitive to (global) unbalance Store qubit in a protected quantum memory (repulsive interaction) unbalance: insensitive to (global) unbalance
Quantum Information Processing Identifying the ground states as qubits: J x = 0: Two degenerate ground states form a protected quantum memory Separated by a gap 2W from the higher excited states Insensitive to global fluctuations of the form (fluctuations in unbalance, fluctuations in tunneling barrier)
Two qubit gates Q Q 2 W Interaction W leads to state selective time evolution Truth table: Collectively enhanced phase:
All-optical spin quantum gates in quantum dots QIP: solid state implementation + scalable, fast + in line with present nanostructure developments - decoherence... coming from quantum optics quantum dots are like artificial atoms: engineering atomic structure spin-based optical quantum gates in semiconductor quantum dots ideas from quantum optics may help in suppressing decoherence
Coupling spin to charge via Pauli blocking /2i /2i /2i /2i 3/2i 3/2i 3/2i 3/2i x i c 0,/2 c 0, /2 d 0,3/2 vaci Idealized model: three-level system σ + α 0i + β i α 0i + β x i 0i c 0, /2 vaci i c 0,+/2 vaci
Selective phase via bi-excitonic interaction 0i a 0i b i a 0i b Laser addressing Exciton couples only to state > External electric field displaces electrons and holes Dipole-dipole interaction induces logical phase 0 0 0 0 0i a i b 0 0 0 0 i E e ab t i a i b E ab
Hole mixing problem Light holes couple to electron +/2 states via σ+ light Actual hole eigenstates comprise a certain admixture from light holes /2i σ + 3/2i /2i /2i 3/2i /2i Pauli blocking does not work perfectly A π + π pulse for the transition > - x> will leave behind some excitonic population x A different gate operation procedure is needed Model including hole mixing via effective weak coupling to state 0> Typical value for ε: ~0% εω 0 Ω
Hole-mixing tolerant laser excitation H = H 0 + δ ih + εω 2 0ihx + h.c. xi E/Ω 3 2 Start far from resonance Adiabatically change the detuning towards resonance Reach x> from > but not from 0> Adiabatically de-excite by returning to the initial situation No excitonic component 0i 0i -3-2 - 2 3 i Nonzero excitonic component - -2-3 xi α 0i + β i α 0i + β µcos θ2 i sin θ2 xi i /Ω
Adiabatically suppressing decoherence in gate operation pulse shapes Ω(t)=Ω 0 e (t/τ Ω) 2 (t)= h e (t/τ ) 2i coupling to phonons induces dephasing : spin-phonon model Ω 0 =3meV τ Ω =0ps E ab =2meV δ =0.5 mev = 3 mev τ =8.72 ps residual population in the unwanted excitonic states is smaller than -6 0 after gate operation X λjq (b jq + b jq ) xihx + ω j(q)b jq b jq phonon Hamiltonian in dressed-state basis " H ph = E + + X ω(q)b q b q +cos 2 θ 2 λ # q b q + b q +ih+ q + " E + X q sin θ 2 The same procedure avoids the effect of both hole mixing and phonon decoherence 0 K ω(q)b q b q sin 2 θ 2 λ # q b q + b q ih X λq b q + b q ( +i h + i h+ ) 5 K 5 K
Qubit read-out photon count simulation level scheme with decay error probability (~0.2% with 80% counting efficiency and 0% mixing) P ε = ε2 ( η) ε 2 + η " µ +ε 2# η +ε 2
Summary [Quantum Computing based on] neutral atoms in optical lattices spin-dependent lattices high fidelity loading of large lattice arrays beyond Mott insulator Entangling atoms two-atoms Feshbach & photoassociation gates atoms in pipeline structures All-optical quantum computing with quantum dots Suppressing decoherence via quantum-optical techniques