Einselection without pointer states -

Einselection without pointer states Einselection without pointer states - Decoherence under weak interaction Christian Gogolin Universität Würzburg 2009-12-16 C. Gogolin Universität Würzburg 2009-12-16 1 / 26

Einselection without pointer states Introduction New foundation for Statistical Mechanics Thermodynamics Statistical Mechanics Second Law ergodicity equal a priory probabilities [2, 3] Classical Mechanics C. Gogolin Universität Würzburg 2009-12-16 2 / 26

Einselection without pointer states Introduction New foundation for Statistical Mechanics Thermodynamics Statistical Mechanics Second Law ergodicity equal a priory probabilities? [2, 3] Classical Mechanics? C. Gogolin Universität Würzburg 2009-12-16 2 / 26

Einselection without pointer states Introduction New foundation for Statistical Mechanics Thermodynamics Statistical Mechanics [2, 3] C. Gogolin Universität Würzburg 2009-12-16 2 / 26

Einselection without pointer states Introduction New foundation for Statistical Mechanics Thermodynamics Statistical Mechanics Quantum Mechanics [2, 3] C. Gogolin Universität Würzburg 2009-12-16 2 / 26

Einselection without pointer states Introduction Why do electrons hop between energy eigenstates? C. Gogolin Universität Würzburg 2009-12-16 3 / 26

Einselection without pointer states Introduction Why do electrons hop between energy eigenstates? quantum mechanical orbitals coherent superpositions C. Gogolin Universität Würzburg 2009-12-16 3 / 26

Einselection without pointer states Introduction Why do electrons hop between energy eigenstates? quantum mechanical orbitals discrete energy levels coherent superpositions hopping C. Gogolin Universität Würzburg 2009-12-16 3 / 26

Einselection without pointer states Introduction Table of contents 1 Technical introduction 2 Setup 3 Subsystem equilibration 4 Decoherence under weak interaction C. Gogolin Universität Würzburg 2009-12-16 4 / 26

Einselection without pointer states Technical introduction Technical introduction C. Gogolin Universität Würzburg 2009-12-16 5 / 26

Einselection without pointer states Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics ψ H A = A ψ ψ = 1 A ψ = ψ A ψ ψ t = U t ψ 0 U t = e i H t C. Gogolin Universität Würzburg 2009-12-16 6 / 26

Einselection without pointer states Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics ψ H A = A ψ ψ = 1 A ψ = ψ A ψ ψ t = U t ψ 0 U t = e i H t Include classical randomness ρ, ψ M(H) ψ = ψ ψ C. Gogolin Universität Würzburg 2009-12-16 6 / 26

Einselection without pointer states Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics ψ H A = A ψ ψ = 1 A ψ = ψ A ψ ψ t = U t ψ 0 U t = e i H t Include classical randomness ρ, ψ M(H) ψ = ψ ψ Tr[ρ] = i i ρ i = 1 A ρ = Tr[A ρ] C. Gogolin Universität Würzburg 2009-12-16 6 / 26

Einselection without pointer states Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics ψ H A = A ψ ψ = 1 A ψ = ψ A ψ ψ t = U t ψ 0 U t = e i H t Include classical randomness ρ, ψ M(H) ψ = ψ ψ Tr[ρ] = i i ρ i = 1 A ρ = Tr[A ρ] ρ t = U t ρ 0 U t U t = e i H t C. Gogolin Universität Würzburg 2009-12-16 6 / 26

Einselection without pointer states Technical introduction Quantum Mechanics on one slide Pure Quantum Mechanics ψ H A = A ψ ψ = 1 A ψ = ψ A ψ ψ t = U t ψ 0 U t = e i H t Include classical randomness ρ, ψ M(H) ψ = ψ ψ Tr[ρ] = i i ρ i = 1 A ρ = Tr[A ρ] ρ t = U t ρ 0 U t U t = e i H t mixtures: ρ = p ψ 1 + (1 p) ψ 2 C. Gogolin Universität Würzburg 2009-12-16 6 / 26

Einselection without pointer states Technical introduction Technical introduction Trace distance D(ρ, σ) = 1 2 ρ σ 1 C. Gogolin Universität Würzburg 2009-12-16 7 / 26

Einselection without pointer states Technical introduction Technical introduction Trace distance D(ρ, σ) = 1 2 ρ σ 1 = max Tr[A ρ] Tr[A σ] 0 A 1 C. Gogolin Universität Würzburg 2009-12-16 7 / 26

Einselection without pointer states Technical introduction Technical introduction Trace distance Effective dimension D(ρ, σ) = 1 2 ρ σ 1 = max Tr[A ρ] Tr[A σ] 0 A 1 d eff (ρ) = 1 Tr(ρ 2 ) C. Gogolin Universität Würzburg 2009-12-16 7 / 26

Einselection without pointer states Technical introduction Technical introduction Trace distance Effective dimension D(ρ, σ) = 1 2 ρ σ 1 = max Tr[A ρ] Tr[A σ] 0 A 1 d eff (ρ) = 1 Tr(ρ 2 ) d eff (ψ) = 1 C. Gogolin Universität Würzburg 2009-12-16 7 / 26

Einselection without pointer states Technical introduction Technical introduction Trace distance Effective dimension D(ρ, σ) = 1 2 ρ σ 1 = max Tr[A ρ] Tr[A σ] 0 A 1 d eff (ρ) = 1 Tr(ρ 2 ) d eff (ψ) = 1 d eff ( 1 d ) = d C. Gogolin Universität Würzburg 2009-12-16 7 / 26

Einselection without pointer states Technical introduction Technical introduction Trace distance Effective dimension D(ρ, σ) = 1 2 ρ σ 1 = max Tr[A ρ] Tr[A σ] 0 A 1 d eff (ρ) = 1 Tr(ρ 2 ) d eff (ψ) = 1 d eff ( 1 d ) = d Time average 1 T ω = ρ t t = lim ρ t dt T T 0 C. Gogolin Universität Würzburg 2009-12-16 7 / 26

Einselection without pointer states Technical introduction Choosing random states Mathematical construction Haar measure on SU(n) uniform distribution µ(v ) = µ(u V ) U 0 = ψ Pr{ ψ } = Pr{U ψ } Explicit construction 1 expand in basis: ψ = i c i i i j = δ ij 2 choose c i from a normal distribution 3 normalize 1 = i c i 2 ψ ψ = 1 C. Gogolin Universität Würzburg 2009-12-16 8 / 26

Einselection without pointer states Setup Setup C. Gogolin Universität Würzburg 2009-12-16 9 / 26

Einselection without pointer states Setup Setup System, H S, H S Bath, H B, H B H SB ρ S t = Tr B [ψ t ] ρ B t = Tr S [ψ t ] Tr[(A S 1 B )ψ t ] = Tr[A S ρ S t ] reduced state locally observable C. Gogolin Universität Würzburg 2009-12-16 10 / 26

Einselection without pointer states Setup Setup System, H S, H S Bath, H B, H B H SB ρ S t = Tr B [ψ t ] ρ B t = Tr S [ψ t ] dψ t dt = i [ψ t, H ] C. Gogolin Universität Würzburg 2009-12-16 10 / 26

Einselection without pointer states Setup A very weak assumption on the Hamiltonian H = H S 1 + 1 H B + H SB C. Gogolin Universität Würzburg 2009-12-16 11 / 26

Einselection without pointer states 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 Universität Würzburg 2009-12-16 11 / 26

Einselection without pointer states 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: E k E l = E m E n = k = l m = n or k = m l = n C. Gogolin Universität Würzburg 2009-12-16 11 / 26

Einselection without pointer states Subsystem equilibration Subsystem equilibration and fluctuations around equilibrium C. Gogolin Universität Würzburg 2009-12-16 12 / 26

Einselection without pointer states Subsystem equilibration Measure concentration in Hilbert space Theorem 1 For random ψ 0 P 1 (H) with d = dim(h) { Pr d eff (ω) < d } 2 e c d 4 d eff (ω) # energy eigenstates in ψ 0 [6] C. Gogolin Universität Würzburg 2009-12-16 13 / 26

Einselection without pointer states Subsystem equilibration Measure concentration in Hilbert space Theorem 1 For random ψ 0 P 1 (H) with d = dim(h) { Pr d eff (ω) < d } 2 e c d 4 d eff (ω) # energy eigenstates in ψ 0 = If d is large then d eff (ω) is large. [6] C. Gogolin Universität Würzburg 2009-12-16 13 / 26

Einselection without pointer states Subsystem equilibration Equilibration Theorem 2 For every ψ 0 P 1 (H) D(ρ S t, ω S ) t 1 d 2 S 2 d eff (ω) where ρ S t = Tr B ψ t ω S = ρ S t t ω = ψ t t [6] C. Gogolin Universität Würzburg 2009-12-16 14 / 26

Einselection without pointer states Subsystem equilibration Equilibration Theorem 2 For every ψ 0 P 1 (H) D(ρ S t, ω S ) t 1 d 2 S 2 d eff (ω) where ρ S t = Tr B ψ t ω S = ρ S t t ω = ψ t t = If d eff (ω) d 2 S then ρs t equilibrates. [6] C. Gogolin Universität Würzburg 2009-12-16 14 / 26

Einselection without pointer states Subsystem equilibration Speed of the fluctuations around equilibrium D(ρ S t, ρ S t+δt v S (t) = lim ) = 1 δt 0 δt 2 dρ S t dt 1 [5] C. Gogolin Universität Würzburg 2009-12-16 15 / 26

Einselection without pointer states Subsystem equilibration Speed of the fluctuations around equilibrium D(ρ S t, ρ S t+δt v S (t) = lim ) = 1 δt 0 δt 2 dρ S t dt 1 dρ S t dt = i Tr B [ψ t, H ] [5] C. Gogolin Universität Würzburg 2009-12-16 15 / 26

Einselection without pointer states Subsystem equilibration Speed of the fluctuations around equilibrium D(ρ S t, ρ S t+δt v S (t) = lim ) = 1 δt 0 δt 2 dρ S t dt 1 dρ S t dt = i Tr B [ψ t, H ] Theorem 3 For every ψ 0 P 1 (H) v S (t) t H S 1 + H SB d 3 S d eff (ω) [5] C. Gogolin Universität Würzburg 2009-12-16 15 / 26

Einselection without pointer states Subsystem equilibration Speed of the fluctuations around equilibrium D(ρ S t, ρ S t+δt v S (t) = lim ) = 1 δt 0 δt 2 dρ S t dt 1 dρ S t dt = i Tr B [ψ t, H ] Theorem 3 For every ψ 0 P 1 (H) v S (t) t H S 1 + H SB d 3 S d eff (ω) = If d eff (ω) d 3 S then ρs t is slow. [5] C. Gogolin Universität Würzburg 2009-12-16 15 / 26

Einselection without pointer states Subsystem equilibration Summary Typical states of large quantum systems have a high average effective dimension, their subsystems equilibrate and fluctuate slowly around the equilibrium state. C. Gogolin Universität Würzburg 2009-12-16 16 / 26

Einselection without pointer states Decoherence under weak interaction Decoherence under weak interaction C. Gogolin Universität Würzburg 2009-12-16 17 / 26

Einselection without pointer states Decoherence under weak interaction Approach 1: Effective dynamics standard QM: H S H B H SB dρ S t dt = i Tr B [ψ t, H ] C. Gogolin Universität Würzburg 2009-12-16 18 / 26

Einselection without pointer states Decoherence under weak interaction Approach 1: Effective dynamics standard QM: effective dynamics: H S H B H S H B H SB H SB L dρ S t dt = i Tr B [ψ t, H ] dρ S t dt = i [ρ S t, H S ] + i L(ρ S t ) C. Gogolin Universität Würzburg 2009-12-16 18 / 26

Einselection without pointer states Decoherence under weak interaction Approach 2: Decoherence à la Zurek Special Hamiltonian with pointer sates p : H = p p p H (p) Initial product state ψ 0 = ρ S 0 ψb 0 Einselection Off-diagonal elements in the pointer basis are suppressed: p ρ S t p = p ρ S 0 p ψ0 B U (p ) (p) t U t ψ0 B }{{} 1 [7] C. Gogolin Universität Würzburg 2009-12-16 19 / 26

Einselection without pointer states Decoherence under weak interaction Comparison Pros and cons unitary evolution general mechanism effective dynamics X einselection X C. Gogolin Universität Würzburg 2009-12-16 20 / 26

Einselection without pointer states Decoherence under weak interaction Comparison Pros and cons unitary evolution general mechanism effective dynamics X einselection X Can we find a more general mechanism based on standard Quantum Mechanics? C. Gogolin Universität Würzburg 2009-12-16 20 / 26

Einselection without pointer states Decoherence under weak interaction Yes we can! C. Gogolin Universität Würzburg 2009-12-16 21 / 26

Einselection without pointer states Decoherence under weak interaction Yes we can! Given the interaction is weak H SB H S, H S H B H SB C. Gogolin Universität Würzburg 2009-12-16 21 / 26

Einselection without pointer states Decoherence under weak interaction Yes we can! Given the interaction is weak H SB H S, when is the subsystem slow? v S (t) 1 2 = dρ S t dt 1 H S H SB H B C. Gogolin Universität Würzburg 2009-12-16 21 / 26

Einselection without pointer states Decoherence under weak interaction Yes we can! Given the interaction is weak H SB H S, when is the subsystem slow? v S (t) 1 2 = dρ S t dt 1 H S H SB H B dρ S t dt = i Tr B [ψ t, H ] C. Gogolin Universität Würzburg 2009-12-16 21 / 26

Einselection without pointer states Decoherence under weak interaction Yes we can! Given the interaction is weak H SB H S, when is the subsystem slow? v S (t) 1 2 = dρ S t dt 1 H S H SB H B dρ S t dt = i Tr B [ψ t, H ] = i Tr B [ψ t, H 0 + H S 1 + 1 H B + H SB ] C. Gogolin Universität Würzburg 2009-12-16 21 / 26

Einselection without pointer states Decoherence under weak interaction Yes we can! Given the interaction is weak H SB H S, when is the subsystem slow? v S (t) 1 2 = dρ S t dt 1 H S H SB H B dρ S t dt = i Tr B [ψ t, H ] = i Tr B [ψ t, H 0 + H S 1 + 1 H B + H SB ] [5] = = i Tr B [ψ t, H S 1 + H SB ] C. Gogolin Universität Würzburg 2009-12-16 21 / 26

Einselection without pointer states Decoherence under weak interaction Tow competing forces dρ S t dt = i Tr B [ψ t, H S 1 + H SB ] = i [ρ S t, H S ] + i Tr B [ψ t, H SB ] C. Gogolin Universität Würzburg 2009-12-16 22 / 26

Einselection without pointer states Decoherence under weak interaction Tow competing forces dρ S t dt = i Tr B [ψ t, H S 1 + H SB ] = i [ρ S t, H S ] + i Tr B [ψ t, H SB ] ρ S t H S C. Gogolin Universität Würzburg 2009-12-16 22 / 26

Einselection without pointer states Decoherence under weak interaction Tow competing forces dρ S t dt = i Tr B [ψ t, H S 1 + H SB ] = i [ρ S t, H S ] + i Tr B [ψ t, H SB ] ρ S t H SB H S C. Gogolin Universität Würzburg 2009-12-16 22 / 26

Einselection without pointer states Decoherence under weak interaction Decoherence through weak interaction Theorem 4 All reduced states ρ S t satisfy 2 Ek S ES l ρs kl 2 H SB + where max k l ρ S kl = ES k ρs t E S l dρ S t dt 1 [1] C. Gogolin Universität Würzburg 2009-12-16 23 / 26

Einselection without pointer states Decoherence under weak interaction Decoherence through weak interaction Theorem 4 All reduced states ρ S t satisfy 2 Ek S ES l ρs kl 2 H SB + where max k l ρ S kl = ES k ρs t E S l dρ S t dt 1 = If ρ S t is slow its off-diagonal elements are small. [1] C. Gogolin Universität Würzburg 2009-12-16 23 / 26

Einselection without pointer states Decoherence under weak interaction Conclusions Pros and cons unitary evolution general mechanism effective dynamics X einselection X our mechanism Applications electronic excitations of gases at moderate temperature radioactive decay environment assisted entanglement creation... C. Gogolin Universität Würzburg 2009-12-16 24 / 26

Einselection without pointer states Decoherence under weak interaction References [1] C. Gogolin, Einselection without pointer states, 0908.2921v2. [2] S. Lloyd, Black holes, demons and the loss of coherence: How complex systems get information, and what they do with it,. PhD thesis, Rockefeller University, April, 1991. [3] S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nature Physics 2 (2006) no. 11, 754. [4] N. Linden, S. Popescu, A. J. Short, and A. Winter, Quantum mechanical evolution towards thermal equilibrium, Physical Review E 79 (2009) no. 6, 061103, 0812.2385v1. [5] N. Linden, S. Popescu, A. J. Short, and A. Winter, On the speed of fluctuations around thermodynamic equilibrium, 0907.1267v1. [6] N. Linden, S. Popescu, A. J. Short, and A. Winter, Quantum mechanical evolution towards thermal equilibrium, 0812.2385v1. [7] W. H. Zurek, Environment-induced superselection rules, Phys. Rev. D (1982) no. 26, 1862 1880. C. Gogolin Universität Würzburg 2009-12-16 25 / 26

Einselection without pointer states Thank you for your attention! beamer slides: http://www.cgogolin.de C. Gogolin Universität Würzburg 2009-12-16 26 / 26