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.