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 andDeterministicUniverse

Erwin Schrödinger

Quantum mechanics:A discrete and probabilisticUniverse

i h dd t

1 2

1 2

1 2 2 1

2 2 2

* *

Interference

UP DOWN

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

ClassicalParticle

QuantumParticle

<x(t)>

Harmonic Oscillator

Courtesy of P.Mohanty BU

Ultra smallOscillators:Nanowires

Width ~ 10 human hair -6

Dissipation

Couplingto theenvironment

Damped Harmonic Oscillator

DecoherenceUniverse: 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 1 2 1 2 2 1

2 2 2 * *

D U 1 n 2 m

U 1 2 1 2 n m 1 2 m n

2 2 2 * *Classical Result !

eD 0

-N

Pure State

Mixture

Jun Kondo

Electron movingin a crystal with Magnetic impurities

Kondo effect

Spin Flip

MultipleSpin flips

<S >z

Don EiglerIBM

ScanningTunnelingMicroscope

Quantum Computation

Classical Computer: deterministic and sequential

Factorization of: x = x0 20 + x1 21 + …. = (x0 ,x1 ,x2 ,…xN)Solution: Try all primes from 2 to √x → 2N/2 =eN ln(2)/2

Quantum Computer: probabilistic and non-sequential

Basis states: x0 ,x1 ,x2 ,…xN)Arbitrary state: yi}) = ∑{xi}

c{xi}({yi}) xi})

Probability: | c{xi}({yi}) |2

Shor’s algorithm: N3

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 thequbits interact and have littledecoherence?

Use of low dimensionalmaterials – E. Novais,AHCN cond-mat

Quantum Frustration AHCN, E.Novais,L.Borda,G.Zarandand I. Affleck PRL 91, 096401 (2003)

Environment withlarge spin (classical)

S=½

The energy is dissipated into two channelscoupled to Sx and Sy . However: [Sx ,Sy ] = i ћ Sz

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.