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From Atoms to Quantum Computers: the classical and quantum faces of nature Antonio H. Castro Neto Dartmouth College, November 2003
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Newton’s equation: m dx = F d t 2 2 Isaac Newton
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Particles Waves Continuous and Deterministic Universe
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Erwin Schrödinger Quantum mechanics: A discrete and probabilistic Universe
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i h d d t 1 2 1 2 1 2 2 1 2 2 2 * Interference
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UPDOWN LINEAR SUPERPOSITION
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Where do Classical and Quantum Mechanics meet? Schrödinger's cat Life) + (Death) (Life) (Death) Wavefunction Collapse
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Schrödinger's cat: molecular magnets
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Two-Level System Classical Particle Quantum Particle
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Harmonic Oscillator
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Courtesy of P.Mohanty BU Ultra small Oscillators: Nanowires Width ~ 10 human hair -6
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Dissipation Coupling to the environment Damped Harmonic Oscillator
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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 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 ! e D 0 -N Pure State Mixture
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Jun Kondo Electron moving in a crystal with Magnetic impurities
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Kondo effect Spin Flip Multiple Spin flips z
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Don Eigler IBM Scanning Tunneling Microscope
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Quantum Computation Classical Computer: deterministic and sequential Factorization of: x = x 0 2 0 + x 1 2 1 + …. = (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
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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
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Quantum Frustration AHCN, E.Novais,L.Borda,G.Zarand and I. Affleck PRL 91, 096401 (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
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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.
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