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Einselection Without Pointer States Einselection without pointer states Einselection without pointer states - Decoherence under weak interaction Christian Gogolin Universit¨atW¨urzburg 2009-12-16 C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 1 / 26 Einselection without pointer states j Introduction New foundation for Statistical Mechanics X Thermodynamics X Statistical Mechanics Second Law ergodicity equal a priory probabilities Classical Mechanics [2, 3] C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 2 / 26 Einselection without pointer states j Introduction New foundation for Statistical Mechanics X Thermodynamics X Statistical Mechanics Second Law ergodicity ? equal a priory probabilities Classical Mechanics ? [2, 3] C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 2 / 26 Einselection without pointer states j Introduction New foundation for Statistical Mechanics X Thermodynamics X Statistical Mechanics [2, 3] C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 2 / 26 Einselection without pointer states j Introduction New foundation for Statistical Mechanics X Thermodynamics X Statistical Mechanics * Quantum Mechanics X [2, 3] C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 2 / 26 quantum mechanical orbitals discrete energy levels coherent superpositions hopping Einselection without pointer states j Introduction Why do electrons hop between energy eigenstates? C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 3 / 26 discrete energy levels hopping Einselection without pointer states j Introduction Why do electrons hop between energy eigenstates? quantum mechanical orbitals coherent superpositions C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 3 / 26 Einselection without pointer states j Introduction Why do electrons hop between energy eigenstates? quantum mechanical orbitals discrete energy levels coherent superpositions hopping C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 3 / 26 Einselection without pointer states j Introduction Table of contents 1 Technical introduction 2 Setup 3 Subsystem equilibration 4 Decoherence under weak interaction C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 4 / 26 Einselection without pointer states j Technical introduction Technical introduction C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 5 / 26 Include classical randomness ρ, 2 M(H) = j ih j X Tr[ρ] = hijρjii = 1 hAiρ = Tr[A ρ] i y − i H t ρt = Ut ρ0 Ut Ut = e mixtures: ρ = p 1 + (1 − p) 2 Einselection without pointer states j Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics j i 2 H A = Ay h j i = 1 hAi = h jAj i − i H t j ti = Ut j 0i Ut = e C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 6 / 26 X Tr[ρ] = hijρjii = 1 hAiρ = Tr[A ρ] i y − i H t ρt = Ut ρ0 Ut Ut = e mixtures: ρ = p 1 + (1 − p) 2 Einselection without pointer states j Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics j i 2 H A = Ay h j i = 1 hAi = h jAj i − i H t j ti = Ut j 0i Ut = e Include classical randomness ρ, 2 M(H) = j ih j C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 6 / 26 y − i H t ρt = Ut ρ0 Ut Ut = e mixtures: ρ = p 1 + (1 − p) 2 Einselection without pointer states j Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics j i 2 H A = Ay h j i = 1 hAi = h jAj i − i H t j ti = Ut j 0i Ut = e Include classical randomness ρ, 2 M(H) = j ih j X Tr[ρ] = hijρjii = 1 hAiρ = Tr[A ρ] i C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 6 / 26 mixtures: ρ = p 1 + (1 − p) 2 Einselection without pointer states j Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics j i 2 H A = Ay h j i = 1 hAi = h jAj i − i H t j ti = Ut j 0i Ut = e Include classical randomness ρ, 2 M(H) = j ih j X Tr[ρ] = hijρjii = 1 hAiρ = Tr[A ρ] i y − i H t ρt = Ut ρ0 Ut Ut = e C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 6 / 26 Einselection without pointer states j Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics j i 2 H A = Ay h j i = 1 hAi = h jAj i − i H t j ti = Ut j 0i Ut = e Include classical randomness ρ, 2 M(H) = j ih j X Tr[ρ] = hijρjii = 1 hAiρ = Tr[A ρ] i y − i H t ρt = Ut ρ0 Ut Ut = e mixtures: ρ = p 1 + (1 − p) 2 C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 6 / 26 1 deff ( ) = 1 deff ( ) = d d = max Tr[A ρ] − Tr[A σ] 0≤A≤1 Effective dimension 1 deff (ρ) = Tr(ρ2) Time average 1 Z T ! = hρtit = lim ρt dt T !1 T 0 Einselection without pointer states j Technical introduction Technical introduction Trace distance 1 D(ρ, σ) = kρ − σk 2 1 C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 7 / 26 1 deff ( ) = 1 deff ( ) = d d Effective dimension 1 deff (ρ) = Tr(ρ2) Time average 1 Z T ! = hρtit = lim ρt dt T !1 T 0 Einselection without pointer states j Technical introduction Technical introduction Trace distance 1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1 C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 7 / 26 1 deff ( ) = 1 deff ( ) = d d Time average 1 Z T ! = hρtit = lim ρt dt T !1 T 0 Einselection without pointer states j Technical introduction Technical introduction Trace distance 1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1 Effective dimension 1 deff (ρ) = Tr(ρ2) C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 7 / 26 1 deff ( ) = d d Time average 1 Z T ! = hρtit = lim ρt dt T !1 T 0 Einselection without pointer states j Technical introduction Technical introduction Trace distance 1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1 Effective dimension 1 deff (ρ) = deff ( ) = 1 Tr(ρ2) C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 7 / 26 Time average 1 Z T ! = hρtit = lim ρt dt T !1 T 0 Einselection without pointer states j Technical introduction Technical introduction Trace distance 1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1 Effective dimension 1 1 deff (ρ) = deff ( ) = 1 deff ( ) = d Tr(ρ2) d C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 7 / 26 Einselection without pointer states j Technical introduction Technical introduction Trace distance 1 D(ρ, σ) = kρ − σk 2 1 = max Tr[A ρ] − Tr[A σ] 0≤A≤1 Effective dimension 1 1 deff (ρ) = deff ( ) = 1 deff ( ) = d Tr(ρ2) d Time average 1 Z T ! = hρtit = lim ρt dt T !1 T 0 C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 7 / 26 Explicit construction P 1 expand in basis: j i = i ci jii hijji = δij 2 choose ci from a normal distribution P 2 3 normalize 1 = i jcij h j i = 1 Einselection without pointer states j Technical introduction Choosing random states Mathematical construction Haar measure on SU(n) −! \uniform" distribution µ(V ) = µ(UV ) U j0i = j i Prfj ig = PrfU j ig C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 8 / 26 Einselection without pointer states j Technical introduction Choosing random states Mathematical construction Haar measure on SU(n) −! \uniform" distribution µ(V ) = µ(UV ) U j0i = j i Prfj ig = PrfU j ig Explicit construction P 1 expand in basis: j i = i ci jii hijji = δij 2 choose ci from a normal distribution P 2 3 normalize 1 = i jcij h j i = 1 C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 8 / 26 Einselection without pointer states j Setup Setup C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 9 / 26 Einselection without pointer states j Setup Setup System, HS; H S Bath, HB; H B H SB S B ρt = TrB[ t] ρt = TrS[ t] 1 S Tr[(AS ⊗ B) t] = Tr[AS ρt ] reduced state ! locally observable C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 10 / 26 1 S Tr[(AS ⊗ B) t] = Tr[AS ρt ] reduced state ! locally observable Einselection without pointer states j Setup Setup System, HS; H S Bath, HB; H B H SB S B ρt = TrB[ t] ρt = TrS[ t] d t dt = i [ t; H ] C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 10 / 26 H 0 + H 0 / 1 Tr[H S] = Tr[H B] = Tr[H SB] = 0 Assumption A Hamiltonian has non{degenerate energy gaps iff: Ek − El = Em − En =) k = l ^ m = n or k = m ^ l = n Einselection without pointer states j Setup A very weak assumption on the Hamiltonian H = H S ⊗1 + 1 ⊗ H B + H SB C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 11 / 26 Assumption A Hamiltonian has non{degenerate energy gaps iff: Ek − El = Em − En =) k = l ^ m = n or k = m ^ l = n Einselection without pointer states j Setup A very weak assumption on the Hamiltonian H = H 0 + H S ⊗1 + 1 ⊗ H B + H SB H 0 / 1 Tr[H S] = Tr[H B] = Tr[H SB] = 0 C. Gogolin j Universit¨atW¨urzburg j 2009-12-16 11 / 26 Einselection without pointer states j Setup A very weak assumption on the Hamiltonian H = H 0 + H S ⊗1 + 1 ⊗ H B + H SB H 0 / 1 Tr[H S] = Tr[H B] = Tr[H SB] = 0 Assumption A Hamiltonian has non{degenerate energy gaps iff: Ek − El = Em − En =) k = l ^ m = n or k = m ^ l = n C.
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