Non-Markovian Quantum Dynamics of Open Systems: Foundations and Perspectives

Transcription

1 Non-Markovian Quantum Dynamics of Open Systems: Foundations and Perspectives Heinz-Peter Breuer Universität Freiburg QCCC Workshop, Aschau, October 2007

2 Contents Quantum Markov processes Non-Markovian dynamics: Correlated projection superoperators Non-Markovian generalization of the Lindblad equation Applications Perspectives

3 Quantum Markov processes ρ(0) = ρ S (0) ρ E tr E ρ S (0) unitary evolution [ ] ρ(t) = U t ρs (0) ρ E U t tr E dynamical map ρ S (t) = Φ t ρ S (0) Quantum dynamical map Φ t : ρ S (0) ρ S (t) = Φ t ρ S (0) = tr E { U t [ ρs (0) ρ E ] U t } superoperator, quantum operation, quantum channel CPT map

4 Quantum Markov processes Markov condition: Separation of time scales τ E τ R = dynamical semigroup: Φ t+s = Φ t Φ s = Φ t = e Lt = Markovian master equation in Lindblad form: Lρ S = i [H S, ρ S ] + λ d dt ρ S(t) = Lρ S (t) ( R λ ρ S R λ 1 2 {R λ R λ, ρ S } )

5 Non-Markovian dynamics Features of non-markovian dynamics: Environmental correlations do not decay rapidly: Markov condition τ E τ R violated Higher-order correlation functions are important, strong memory effects Finite revival times (finite reservoirs) Initial correlations/non-factorizing initial states: No dynamical map Φ t No master equation with Lindblad structure: d dt ρ S(t) L(t)ρ S (t)

6 Projection superoperator: Projection operator techniques total state ρ(t) = relevant part Pρ(t) Pρ(t) = tr E {ρ(t)} ρ E = ρ S ρ E = Nakajima-Zwanzig (NZ) equation: d dt Pρ(t) = t 0 dt 1 K(t, t 1 )Pρ(t 1 ) K(t, t 1 ) = memory kernel time integration over system s history

7 Time-convolutionless master equation Eliminating the memory kernel: ρ S (t) = Φ t ρ S (0) and d dt ρ S(t) = Φ t ρ S (0) = time-local master equation (TCL): d dt ρ S(t) = ( Φ t Φ 1 ) t ρ S (t) K(t)ρ S (t) Expansion of the TCL generator (ordered cumulants): K(t) = n=1 α n K n (t)

8 Structured reservoirs Example: Interaction with finite reservoir H = H S + H E + V γ/δɛ = Markov condition satisfied N 2 δǫ E V N 1 δǫ system environment H. P. B., J. Gemmer and M. Michel, Phys. Rev. E 73, (2006)

9 Correlated projection operators Standard projection: Pρ = (tr E ρ) ρ E = uncorrelated tensor product state Projections onto correlated states: 1. P is a linear, positive and trace-preserving projection: ρ Pρ linear ρ 0 = Pρ 0, P 2 = P tr{pρ} = trρ 2. P acts as a quantum channel on environment: P = I S Λ Λ = CPT map

10 Correlated projection operators Important physical implications: P maps product states to product states: P(A B) = A ΛB P maps separable states to separable states: ρ sep = i p i ρ i S ρi E Pρ sep = separable Projection completely determines reduced density matrix: ρ S tr E ρ = tr E {Pρ}

11 Representation theorem Pρ = i tr E {A i ρ} B i = i ρ i B i {A i }, {B i } = sets of linear independent observables Hilbert-Schmidt orthogonal: tr E {B i A j } = δ ij Normalization: Positivity: H. P. B., Phys. Rev. A 75, (2007) (tr E B i )A i = I E i A T i B i 0 i

12 Structured reservoir N 2 δǫ E V N 1 δǫ system environment Pρ = ρ 1 1 N 1 Π 1 + ρ 2 1 N 2 Π 2 d dt ρ 1 = γ 1 σ + ρ 2 σ γ 2 2 {σ +σ, ρ 1 } d dt ρ 2 = γ 2 σ ρ 1 σ + γ 1 2 {σ σ +, ρ 2 } ρ S (t) = ρ 1 (t) + ρ 2 (t)

13 Structured reservoir γ δɛ = d dt ρ 11(t) = (γ 1 + γ 2 )ρ 11 (t)+γ 1 ρ 11 (0) +γ 2 1 ρ 2 (0) 1 + γ 1 0 ρ 2 (0) 0 Not in Lindblad-Form, no semigroup, but CPT map Highly non-markovian: System never forgets initial data Initial correlations = no quantum dynamical map

14 Generalization of Lindblad equation Correlated projection superoperator: n Pρ(t) = ρ i (t) B i Dynamical transformation: i=1 {ρ i (0)} ρ S (0) = ρ i (0) i V t = {ρ i (t)} ρ S (t) = i ρ i (t) no dynamical map on reduced state space

15 Generalization of Lindblad equation Representation in extended state space H S C n : ρ ϱ(t) = 1 (t) 0 0 ρ i (t) i i = 0 ρ 2 (t) 0 i ρ n (t) Embedding into Lindblad dynamics on extended state space: ϱ(t) = e Lt ϱ(0) = Generalization of Lindblad equation: d ] dt ρ i = i [H i, ρ i + ( R ij λ ρ jr ij λ 1 2 jλ H. P. B., Phys. Rev. A 75, (2007) { R ji } ) λ Rji λ, ρ i

16 Correlated projection operators Conserved quantity: [H, C] = 0 Expectation values invariant under P: C tr{cρ} = tr{cpρ} P C = C = Master equation respects conservation law: d dt tr{cpρ(t)} = 0 Example: C = σ + σ + Π 2

17 Hyperfine interaction σ A k σ (k) H = ω 0 2 σ 3 + N k=1 spin star A k σ σ (k) Conserved quantity: 3-component of total spin J 3 = 1 2 σ σ (k) 2 3 = [H, J 3 ] = 0 H. P. B. and F. Petruccione, Phys. Rev. E 76, (2007) H. P. B., D. Burgarth and F. Petruccione, Phys. Rev. B 70, (2004) k

18 Hyperfine interaction = Construct projection that leaves invariant J 3 : P J 3 = J 3 Pρ = tr E {Π m ρ} 1 Π m ρ m 1 Π m m N m m N m 1 5 x Re ρ + 0 P At At J. Fischer and H. P. B., Phys. Rev. A, to appear (2007)

19 Transport in modular quantum systems H 0 = µ ˆV = µ ɛ i µ, i µ, i i c ij µ, i µ + 1, i + h.c. i,j

20 Pρ = q Projection onto Fourier modes tr [ Φ q ρ ] Φ q = F q (t)φ q q Φ q = C q Diffusive transport: F q (t) exp[ at] Ballistic transport: F q (t) exp[ bt 2 ] µ cos(qµ)π µ (b) slowest mode n N R. Steinigeweg, H. P. B. and J. Gemmer, Phys. Rev. Lett. 99, (2007) 100

21 Perspectives Projection onto entangled quantum states? Pρ = i tr E {A i ρ} B i = entangled state Methods of quantum information theory Entanglement criteria and measures Dynamical significance of entanglement

22 Perspectives Combination of TCL and NZ? TCL master equation: d Pρ(t) = K(t)Pρ(t) dt = exact on average for transport model NZ master equation: d t dt Pρ(t) = dt 1 K(t, t 1 )Pρ(t 1 ) 0 = exact for populations of spin star

23 Monte Carlo methods Stochastic representation of Lindblad dynamics: ρ S (t) = E ( ψ(t) ψ(t) ) ψ(t) = stochastic process in Hilbert space TCL master equation: ρ S (t) = E ( ψ 1 (t) ψ 2 (t) ) Generalized Lindblad equation: ρ S (t) = i E ( ψ i (t) ψ i (t) )

24 Exact Monte Carlo methods? First ansatz: ρ(t) = E ( Φ(t) Φ(t) ) Φ(t) = ψ(t) χ(t) = stochastic product state Does not work because of: ρ(t) = λ p λ Φ λ Φ λ = separable state = no representation of entangled states possible

25 Use a pair of stochastic states: Exact Monte Carlo methods? Φ i (t) = ψ i (t) χ i (t) i = 1, 2 Any state can be represented as: ρ(t) = E ( Φ 1 (t) Φ 2 (t) ) Bra and Ket evolve independently Reproduces the exakte non-markovian dynamics Problem: exponential growth of fluctuations H. P. B., Phys. Rev. A 69, (2004); Phys. Rev. A 70, (2004)

26 Summary Correlated projection operator techniques Highly non-markovian dynamics (infinite memory times) Correlations in initial state Non-perturbative Generalization of Lindblad theory Stochastic representations Further investigations: Projections onto entangled quantum states Time-dependent generators Combination of TCL and NZ: New expansion techniques