From Atoms to Quantum Computers: the classical and quantum faces of nature Antonio H. Castro Neto Dartmouth College, November 2003

Newton’s equation: m dx = F d t 2 2 Isaac Newton

Particles Waves Continuous and Deterministic Universe

Erwin Schrödinger Quantum mechanics: A discrete and probabilistic Universe

i h d   d t  1 2   * Interference

UPDOWN LINEAR SUPERPOSITION

Where do Classical and Quantum Mechanics meet? Schrödinger's cat  Life) +  (Death)  (Life)  (Death) Wavefunction Collapse

Schrödinger's cat: molecular magnets

Two-Level System Classical Particle Quantum Particle

Harmonic Oscillator

Courtesy of P.Mohanty BU Ultra small Oscillators: Nanowires Width ~ 10 human hair -6

Dissipation Coupling to the environment Damped Harmonic Oscillator

Decoherence Universe: system of interest + environment System of interest:  and  Environment:  n,m=  Decoupled at t=0:   After a time t=  :   1 2 n n m U 1 2 n U * * D U 1 n 2 m U n m 1 2 m n * * Classical Result !  e D 0 -N Pure State Mixture

Jun Kondo Electron moving in a crystal with Magnetic impurities

Kondo effect Spin Flip Multiple Spin flips z

Don Eigler IBM Scanning Tunneling Microscope

Quantum Computation Classical Computer: deterministic and sequential Factorization of: x = x x …. = (x 0,x 1,x 2,…x N ) Solution: Try all primes from 2 to √x → 2 N/2 =e N ln(2)/2 Quantum Computer: probabilistic and non-sequential Basis states:  x 0,x 1,x 2,…x N ) Arbitrary state:  y i }) = ∑ {x i } c {x i } ({y i })  x i }) Probability: | c {x i } ({y i }) | 2 Shor’s algorithm: N 3 Exponential explosion! Power law growth

Solid State Quantum Computers _Scalable: large number of qubits _States can be initiated with magnetic fields _Quantum gates: qubits must interact _Qubit specific acess Big challenge: How to make the qubits interact and have little decoherence? Use of low dimensional materials – E. Novais, AHCN cond-mat

Quantum Frustration AHCN, E.Novais,L.Borda,G.Zarand and I. Affleck PRL 91, (2003) Environment with large spin (classical) S=½ The energy is dissipated into two channels coupled to S x and S y. However: [S x,S y ] = i ћ S z

Conclusions _“There is a lot of room at the bottom” R.Feynman _There is a lot of beauty and basic phenomena. _ Experiments are probing the boarders between classical and quantum realities and also the frontiers of technology. _ New theoretical approaches and ideas are required.