Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Mathematical Models of Markovian Dephasing Franco Fagnola Politecnico di Milano (joint work with J. E. Gough, H. I. Nurdin, L. Viola) 51 Symposium on Mathematical Physics Toru´ n, June 16–18, 2019 Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook John E . Gough Lorenza Viola Hendra I . Nurdin Aberystwyth Dartmouth College NSW Sydney Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook 1 Dephasing 2 Markovian models 3 Can dephasing be ascribed to classical noise? 4 Purely Brownian dilations 5 Outlook Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Dephasing Dephasing Decay of coherences in a preferred basis, usually eigenstates of the system Hamiltonian, without energy transitions (pure dephasing) Decoherence Any process that may cause loss of coherence Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Dephasing System. Hilbert space h, Hamiltonian H S = � n ǫ n P n State. Density matrix ρ Evolution. ρ → ρ t := T ∗ t ( ρ ), T QMS, GKSL generator L Dephasing. Decay of coherences: P m ρ t P n → t →∞ 0 for all m � = n without energy transitions : tr( ρ t P n ) constant Example: phase damping. T QMS on M 2 ( C ) generated by L ( x ) = γ ( σ z x σ z − x ) γ > 0 � x 11 x 12 e − γ t � � � x 12 x 11 T t = x 21 e − γ t x 21 x 22 x 22 Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Definition p and q two T t -invariant orthogonal projections mutually orthogonal to each other dephase under a QMS T if t →∞ T t ( p x q ) = 0 lim for all x ∈ B (h) . T is maximally dephasing if ∃ rank-one orthogonal projections ( p n ) n with � n p n = 1 l , T t ( p n ) = p n for all t , n , p n , p m dephasing for all n � = m . Schr¨ odinger picture (Baumgartner & Narnhofer, J. Phys. A (2008)) t →∞ p m T ∗ t ( ρ ) p n = 0 , lim whenever n � = m Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Maximally dephasing QMSs 1. T t ( p n ) = p n for all t ≥ 0, for all n 2. GKSL generator L ( x ) = i [ H , x ] − 1 � ( L ∗ ℓ L ℓ x − 2 L ∗ ℓ xL ℓ + xL ∗ ℓ L ℓ ) 2 ℓ H = H ∗ , 1 l , L 1 , L 2 , . . . linearly independent ∀ n 3. p n L ℓ = L ℓ p n , p n H = Hp n 4. basis ( e n ) n s.t. p n = | e n �� e n | � � L ℓ = λ ℓ, n | e n �� e n | , H = ǫ n | e n �� e n | n n λ ℓ, n ∈ C , ǫ n ∈ R Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Maximally dephasing QMSs i [ H , x ] − 1 � ( L ∗ ℓ L ℓ x − 2 L ∗ ℓ xL ℓ + xL ∗ L ( x ) = ℓ L ℓ ) 2 ℓ � � L ℓ = λ ℓ, n | e n �� e n | , H = ǫ n | e n �� e n | n n | λℓ, m | 2 | λℓ, n | 2 �� � � � − − + λ ℓ, m λ ℓ, n + i ( ǫ m − ǫ n ) t | e m �� e n | ℓ 2 2 T t ( | e m �� e n | ) = e = e ( − 1 2 | λ • , m − λ • , n | 2 + i ℑ� λ • , m ,λ • , n � + i ( ǫ m − ǫ n ) ) t | e m �� e n | Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Maximally dephasing QMSs i [ H , x ] − 1 � ( L ∗ ℓ L ℓ x − 2 L ∗ ℓ xL ℓ + xL ∗ L ( x ) = ℓ L ℓ ) 2 ℓ � � L ℓ = λ ℓ, n | e n �� e n | , H = ǫ n | e n �� e n | n n | λℓ, m | 2 | λℓ, n | 2 �� � � � − − + λ ℓ, m λ ℓ, n + i ( ǫ m − ǫ n ) t ℓ 2 2 T t ( | e m �� e n | ) = e | e m �� e n | = e ( − 1 2 | λ • , m − λ • , n | 2 + i ℑ� λ • , m ,λ • , n � + i ( ǫ m − ǫ n ) ) t | e m �� e n | decay Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Maximally dephasing QMSs i [ H , x ] − 1 � ( L ∗ ℓ L ℓ x − 2 L ∗ ℓ xL ℓ + xL ∗ L ( x ) = ℓ L ℓ ) 2 ℓ � � L ℓ = λ ℓ, n | e n �� e n | , H = ǫ n | e n �� e n | n n | λℓ, m | 2 | λℓ, n | 2 �� � � � − − + λ ℓ, m λ ℓ, n + i ( ǫ m − ǫ n ) t ℓ 2 2 T t ( | e m �� e n | ) = e | e m �� e n | = e ( − 1 2 | λ • , m − λ • , n | 2 + i ℑ� λ • , m ,λ • , n � + i ( ǫ m − ǫ n ) ) t | e m �� e n | phase Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Another definition J. E. Avron, M. Fraas, G. M. Graf, ... Commun. Math. Phys. (2012) call L dephasing iff { H } ′ ⊆ ker ( L ) ker ( [ H , · ] ) ⊆ ker ( L ) i.e. p projection. L ( p ) = 0 iff and only if [ L ℓ , p ] = 0 ∀ ℓ and [ H , p ] = 0. If H has simple spectrum, then { H } ′ = { H } ′′ and both definitions are equivalent. Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Open Syst. Inf. Dyn. 24 (2017) Aim. Characterize those QMSs that cannot be described as a unitary dilation using only classical, commutative noise processes and need the full Hudson-Parthasarathy theory. Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Dilations H E � H S ← → e i t H tot � U t H tot = H S ⊗ 1 l E + 1 l S ⊗ H E + interaction , T t ( x ) = Tr E ( U ∗ t ( x ⊗ 1 l E ) U t ) Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Essentially commutative (classical) dilation Maassen & K¨ ummerer, Commun. Math. Phys. (1987) Definition A dilation is essentially commutative if the algebra generated by U ∗ t ( x ⊗ 1 l E ) U t with x ∈ B (h) is isomorphic to B (h) ⊗ C with C commutative. i.e. the environment is commutative. Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook QMSs with essentially commutative dilations Theorem (K¨ ummerer & Maassen, Comm. Math. Phys. 1987) A QMS T on M m ( C ) generated by L ( x ) = G ∗ x + � ℓ L ∗ ℓ xL ℓ + x G admits an essentially commutative † if and only if i [ H , x ] − 1 � L 2 ℓ x − 2 L ℓ xL ℓ + xL 2 L ( x ) � � = ℓ 2 ℓ � V ∗ � � j xV j − x + κ j j where H = H ∗ , L ℓ = L ∗ ℓ , κ j > 0 and V j unitaries. Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook QMSs with essentially commutative dilations Theorem (K¨ ummerer & Maassen, Comm. Math. Phys. 1987) A QMS T on M m ( C ) generated by L ( x ) = G ∗ x + � ℓ L ∗ ℓ xL ℓ + x G admits an essentially commutative † if and only if i [ H , x ] − 1 � L 2 ℓ x − 2 L ℓ xL ℓ + xL 2 L ( x ) � � = ℓ 2 ℓ � V ∗ � � j xV j − x + κ j j where H = H ∗ , L ℓ = L ∗ ℓ , κ j > 0 and V j unitaries. † HP dilation with Brownian and Poisson noises only Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Hudson-Parthasarathy dilations H E = Γ( L 2 ( R d ; C )) symmetric (Boson) Fock space on L 2 ( R d ; C ) � � dU t = ( S jk − δ jk 1 l ) d Λ jk ( t ) jk � � � i H + 1 L j dA j ( t ) † − � � � L ∗ L ∗ + j S jk dA k ( t ) − k L k dt U t , 2 j jk k H , L j from GKSL, ( S jk ) jk unitary matrix of operators on h Mathematical Models of Markovian Dephasing
Dephasing Markovian models Can dephasing be ascribed to classical noise? Purely Brownian dilations Outlook Hudson-Parthasarathy dilations H E = Γ( L 2 ( R d ; C )) symmetric (Boson) Fock space on L 2 ( R d ; C ) � � dU t = ( S jk − δ jk 1 l ) d Λ jk ( t ) jk � � � i H + 1 L j dA j ( t ) † − � � � L ∗ L ∗ + j S jk dA k ( t ) − k L k dt U t , 2 j jk k H , L j from GKSL, ( S jk ) jk unitary matrix of operators on h ( A k ( t ) † + A k ( t )) t ≥ 0 , i ( A k ( t ) † − A k ( t )) t ≥ 0 Brownian motions Mathematical Models of Markovian Dephasing
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