Jyrki Piilo NON-MARKOVIAN OPEN QUANTUM SYSTEMS. Turku Centre for Quantum Physics Non-Markovian Processes and Complex Systems Group

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1 UNIVERSITY OF TURKU, FINLAND NON-MARKOVIAN OPEN QUANTUM SYSTEMS Jyrki Piilo Turku Centre for Quantum Physics Non-Markovian Processes and Complex Systems Group

2 Turku Centre for Quantum Physics, Finland Vasiliev Piilo Maniscalco Lahti Suominen Turku, Suomi - Hefei, Anhui, Kiina - Google Maps Voit näyttää kaikki ruudulla näkyvät tiedot kartan vieressä olevan Tulosta-linkin avulla. Non-Markovian Processes and Complex Systems Group TURKU

3 Contents Lecture 1 1. General framework: Open quantum systems 2. Local in time master equations Lecture II 3. Solving local in time master equations: Markovian and non-markovian quantum jumps GROUND AND EXCITED STATE PROBABILITIES excited state ground state TIME [β 1 ] Lecture III 4. Measures of non-markovianity 5. Applications of non-markovianity

4 1. General Framework: Open Quantum Systems Open vs closed systems Dynamical map Semigroup Lindblad equation Derivations Example

5 Literature H.-P. Breuer and F. Petruccione: The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002) C. W. Gardiner and P. Zoller: Quantum Noise (Springer, 2004) U. Weiss Quantum Dissipative Systems (World Scientific, 1999)

6 Open quantum systems Any realistic quantum system coupled to its environment The open system exchanges energy and information with its environment We are interested in: how does the interaction influence the open system, equation of motion? Not interested in the evolution of large environment E.g., radiation field - matter interaction, cavity-qed, quantum optics, dissipation of energy,...

7 Closed system dynamics Pure state, state vector: Schrödinger s Equation: Ψ i d Ψ = H Ψ dt Solution: Ψ(t) = e iht/ Ψ(0) = U(t) Ψ(0) Mixed state, density matrix: ρ = P i (t) Ψ Ψ i Solution: ρ(t) =U(t)ρ(0)U (t) Liouville - von Neumann equation: i dρ =[H, ρ] dt Deterministic, reversible time evolution

8 Open system dynamics Total system closed: ρ T Open system: ρ S =Tr E ρ T Environment: ρ E =Tr S ρ T No initial correlations: ρ T (0) = ρ S (0) ρ E (0) Total system Hamiltonian E S T H = H S I E + I S H E + H I Evolution of the open system and dynamical map ρ S (t) =Tr E [ U(t)ρS (0) ρ E (0)U (t) ] =Φ t ρ S (0) Note: partial trace: ρ S =Tr E ρ T = i E ϕ i ρ T ϕ i E

9 Dynamical map Linear dynamical map for open system Φ t : ρ S (0) ρ S (t) =Φ t ρ S (0) Trace preserving Positive (P) Completely positive (CP) Φ I n positive in extended space (Φ I n )ρ SA 0 E S A Specific properties and master equation...

10 Semigroup, Lindblad generator Φ t : ρ S (0) ρ S (t) =Φ t ρ S (0) Semigroup property: Φ t1 +t 2 =Φ t1 Φ t2 t 1 t 2 t 1 + t 2 It follows that: Dynamical map: Lindblad generator L Φ t = e Lt ) Lρ S = i[h, ρ S ]+ k γ k (A k ρ S A k 1 2 A k A kρ S 1 2 ρ SA k A k And the master equation...

11 Lindblad equation Lindblad-Gorini-Kossakowski-Sudarshan master equation (1975) unitary part dissipator (non-unitary part) decay rate jump operators Lindblad or jump operators: Decay rate constants: γ k 0 A k Guarantees physical validity of the solution (CP) (semigroup: master equation has to be of this form and validity guaranteed)

12 Microscopic derivation

13 Microscopic derivation Total system Hamiltonian (system S; environment or bath B, total system SB) H = H S I B + I S H B + H I Total system evolution in interaction picture (L-vN) i dρ SB dt Integral form =[H I,ρ SB ] ρ SB (t) =ρ SB (0) i t Plug this into L-vN (weak system-environment interaction) dρ SB (t) dt = i [H I(t),ρ SB (0)] 1 2 t 0 ds[h I (s),ρ SB (s)] 0 ds[h I (t), [H I (s),ρ SB (s)]] + O ( ) 1 3

14 dρ SB (t) dt Microscopic derivation = i [H I(t),ρ SB (0)] 1 2 t 0 ds[h I (t), [H I (s),ρ SB (s)]] + O ( ) 1 3 Factorized initial condition ϱ SB (0) = ϱ S (0) ϱ B (0) Trace over the bath gives dϱ S dt with (t) = t 0 Tr B [H I (t),ϱ SB (0)] = 0 dstr B {[H I (t), [H I (s),ϱ SB (s)]]} Stationary, macroscopic environment ϱ B (0) = ϱ B Born approximation (weak coupling) ϱ SB (t) ϱ S (t) ϱ B

15 Microscopic derivation Born approximation gives... t dϱ S (t) = dstr B {[H I (t), [H I (s),ϱ S (s) ϱ B ]]} dt 0 Markov 1: no dependence on previous states (short reservoir correlation/memory time τ B ) ϱ S (s) ϱ S (t) dϱ S (t) dt = t Redfield equation 0 dstr B {[H I (t), [H I (s),ϱ S (t) ϱ B ]]} Markov II: induced SB correlations decay fast, allows to extend the time integration to infinity...

16 Microscopic derivation dϱ S dt (t) = 0 Born-Markov equation dstr B {[H I (t), [H I (t s),ϱ S (t) ϱ B ]]} Born approximation (weak coupling) ϱ SB (t) ϱ S (t) ϱ B Markov approximation: time scales τ B τ S The contribution to the integral in the Redfield equation from short time interval during which the system state does not change very much Exact solution of Born-Markov not necessarily easy Not yet in Lindblad form...

17 Microscopic derivation Born-Markov equation dϱ S dt (t) = 0 dstr B {[H I (t), [H I (t s),ϱ S (t) ϱ B ]]} How to go from here to Lindblad form? Interaction Hamiltonian H I = α A α B α Defining the eigenoperators of the system A α (ω) = Π(ɛ)A α Π(ɛ ) ɛ ɛ=ω where Π(ɛ) projects to eigenspace of H S with eig. value ɛ allows to write the master equation as...

18 Microscopic derivation dϱ s dt (t) = ω,ω with Γ αβ (ω) α,β 0 e i(ω ω)t Γ αβ (ω)[a β (ω)ϱ S (t)a α(ω ) A α(ω )A β (ω)ϱ S (t)] + h.c. dse iωs B α(t)b β (t s) and reservoir correlation function B α(t)b β (t s) Tr B {B α(t)b β (t s)ϱ B } Real and imaginary parts Γ αβ (ω) = 1 2 γ αβ(ω)+is αβ (ω) For stationary reservoir, homogeneous in time B α(t)b β (t s) = B α(s)b β (0) Almost in the Lindblad form... One more approximation: fastly oscillating terms average out: secular approximation...

19 Microscopic derivation dϱ S dt (t) = i[h LS,ϱ S (t)] + Lϱ S (t) Lamb shift H LS = term (energy renormalization) S αβ (ω)a α(ω)a β (ω) ω α,β Dissipator Lϱ S = ω α,β γ αβ [A β (ω)ϱ S A α(ω) 1 ] 2 {A α(ω)a β (ω),ϱ S } By diagonalizing the dissipator, we finally obtain Lindblad form Lϱ S = ω γ α (ω) α [ A α (ω)ϱ S A α(ω) 1 2 {A α(ω)a α (ω),ϱ S } ]

20 So far: Time-evolution of the total system Tracing over the environment Born, Markov, and secular approximations Lindblad master equation (Markovian, semigroup)

21 Example: two-level atom in vacuum [ dϱ = i[h, ϱ]+γ σ ϱσ + 1 ] dt 2 {σ +σ,ϱ} System energy H = ω 0 σ z Lindblad operator σ = g e Decay rate Γ Exponential decay from excited state ρ ee(t) = e Γt ρ ee(0) ρ gg(t) = ρ gg (0)+(1 e Γt )ρ ee(0) ρ eg(t) = e Γt/2 ρ eg(0) e σ = g e g

22 II. Non-Markovian open systems: local in time master equations Projection operator techniques Nakazima-Zwanzig (memory kernel) TCL (time-convolutionless) Example

23 Open systems: Beyond semigroup dϱ S (t) dt = L S ϱ S (t) Semigroup iff generator s in Lindblad form L dϱ S (t) dt = L S (t)ϱ S (t) Time-dependent generator L S (t) dϱ S (t) dt = t o dsk S (t s)ϱ S (s) Memory kernel K S (t s)

24 Nakazima-Zwanzig projection operator technique Total system Hamiltonian H = H 0 + αh I Total system equation of motion (interaction picture) dϱ dt (t) = iα[h I(t),ϱ(t)] αl(t)ϱ(t) here L is the total system Liouville superoperator Basic idea: project to relevant and irrelevant parts of the total system Pϱ Tr B [ϱ] ϱ B Qϱ = ϱ Pϱ Relevant part Irrelevant part

25 Nakazima-Zwanzig projection operator technique Pϱ Tr B [ϱ] ϱ B Relevant part Qϱ = ϱ Pϱ Irrelevant part Here, ϱ B is a fixed environmental state (stationary env.) The projection superoperators have the properties: P + Q =I P 2 = P Q 2 = Q [P, Q] =0

26 Nakazima-Zwanzig projection operator technique The task: derive equation of motion for the relevant part, that is: for ϱ S (t) =Tr B ϱ(t) dϱ dt (t) = iα[h I(t),ϱ(t)] αl(t)ϱ(t) Pϱ = αplϱ t Qϱ = αqlϱ t or by inserting P + Q =Ibetween Liouvillean and density matrix Pϱ = αplpϱ + αplqϱ t Qϱ = αqlpϱ + αqlqϱ t

27 Nakazima-Zwanzig projection operator technique Pϱ = αplpϱ + αplqϱ t Qϱ = αqlpϱ + αqlqϱ t Coupled differential equations for the two parts The formal solution for the irrelevant part t Qϱ(t) =G(t, t 0 )Qϱ(t 0 )+α dsg(t, s)ql(s)pϱ(s) t 0 where the propagator [ G is t ] G(t, t 0 ) T exp α dsql(s) t 0 Inserting the solution to the equation of the relevant part...

28 Nakazima-Zwanzig projection operator technique...and using also (makes the first r.h.s. term to vanish) Tr B [H I (t 1 ) H I (t 2n+1 )ϱ B ]=0 P L(t 1 ) L(t 2n+1 )P =0 (odd moments of HI vanish, valid for thermal env. state)...finally gives the Nakazima-Zwanzig equation Pϱ t t (t) =αpl(t)g(t, t 0)Qϱ(t 0 )+α 2 dspl(t)g(t, s)ql(s)pϱ(s) t 0 Note: The 1st term on the r.h.s. contains initial correlations with the environment (vanish for initial product state) The 2nd term on the r.h.s. contain memory kernel K(t, s) =α 2 PL(t)G(t, s)ql(s)p Exact equation for the relevant part, challenging to solve...

29 Nakazima-Zwanzig projection operator technique...however, to 2nd order in the coupling constant α gives { ϱ t } S t (t) = α2 Tr B ds[h I (t), [H I (s),ϱ s (s) ϱ B ]] t 0 This is the same as the previous equation after Born approximation Question: Is is possible to eliminate the memory kernel in the Nakazima-Zwanzig equation Is it possible to have local in time non-markovian equation?

30 Open systems: Beyond semigroup dϱ S (t) dt = L S ϱ S (t) Semigroup iff generator s in Lindblad form L dϱ S (t) dt = L S (t)ϱ S (t) Time-dependent generator L S (t) dϱ S (t) dt = t o dsk S (t s)ϱ S (s) Memory kernel K S (t s)

31 Time convolutionless master equation (local in time, time dependent generator)

32 Time-convolutionless master equations How to construct local in time generator? (basic idea: introduce backward propagator) Start again from the formal solution for the irrelevant part Qϱ(t) =G(t, t 0 )Qϱ(t 0 )+α t t 0 dsg(t, s)ql(s)pϱ(s) Introduce backward propagator for the total system (s<t) ϱ(s) =Ḡ(t, s)(p + Q)ϱ(t) with (antichronological [ ordering) t ] Ḡ(t, s) T exp α ds L(s ) This gives for the irrelevant part Qϱ(t) =G(t, t 0 )Qϱ(t 0 )+α s t dsg(t, s)ql(s)pḡ(t, s)(p + Q)ϱ(t) t 0 Σ(t)

33 Time-convolutionless master equations Defining superoperator (depends on t only) t Σ(t) α dsg(t, s)ql(s)pḡ(t, s) t 0 We can write for irrelevant part as Qϱ(t) =G(t, t 0 )Qϱ(t 0 )+Σ(t)Pϱ(t)+Σ(t)Qϱ(t) [I Σ(t)]Qϱ(t) =G(t, t 0 )Qϱ(t 0 )+Σ(t)Pϱ(t) If the inverse of I Σ(t) exists (small enough coupling) Qϱ(t) =[I Σ(t)] 1 G(t, t 0 )Qϱ(t 0 )+[I Σ(t)] 1 Σ(t)Pϱ(t) Time evolution of Q depends on initial state and relevant part, also no dependence on previous point s. Plugging this in for the equation for the relevant part...

34 Time-convolutionless master equations Pϱ(t) = K(t)Pϱ(t)+I(t)Qϱ(t 0 ) t with time local generator and inhomogeneity K(t) =αpl(t)[i Σ(t)] 1 Σ(t)P I(t) =αpl(t)[i Σ(t)] 1 G(t, t 0 )Q Exact local in time equation Generally complicated Geometric series and series expansion in coupling constant [I Σ(t)] 1 = K(t) =α + n=1 + n=0 [Σ(t)] n PL(t)[Σ(t)] n P = + n=1 α n K n (t)

35 Time-convolutionless master equations Expanding also Σ(t) = Pϱ(t) t K(t) =α + k=1 gives, e.g., + = K(t)Pϱ(t)+I(t)Qϱ(t 0 ) PL(t)[Σ(t)] n P = n=1 n=1 α k Σ k (t) + α n K n (t) K 1 (t) =PL(t)P =0 K 2 (t) =... t 0 dt 1 PL(t)L(t 1 )P 1st order 2nd order TCL

36 Time-convolutionless master equations The second order TCL leads to the following equation dρ S (t) dt = α 2 t 0 dstr B {[H I (t), [H I (s),ρ S (t) ρ B ]]} Comparing to 2nd order Nakazima-Zwanzig (previously) dρ S (t) dt t = α 2 dstr B {[H I (t), [H I (s),ρ S (s) ρ B ]]} 0 Also similarity to Redfield equation...

37 Example: two-level atom in vacuum TCL2 dϱ dt Decay rates Markov Γ= 1 π TCL Γ(t) = 1 π = i[h, ϱ]+γ + 0 t 0 ds ds [ σ ϱσ + 1 ] 2 {σ +σ,ϱ} dωj(ω) cos[(ω ω 0 )s] = constant dωj(ω) cos[(ω ω 0 )s] Here, J is the spectral density of the Bosonic environment And the open system dynamics is Markov ϱ ee (t) =e Γt ϱ ee (0) TCL ϱ ee (t) =e R t 0 dsγ(s) ϱ ee (0)

38 Example: two-level atom in vacuum TCL2 dϱ dt Decay rates Markov Γ= 1 π TCL Γ(t) = 1 π = i[h, ϱ]+γ + 0 t 0 ds ds [ σ ϱσ + 1 ] 2 {σ +σ,ϱ} dωj(ω) cos[(ω ω 0 )s] = constant dωj(ω) cos[(ω ω 0 )s] Here, J is the spectral density of the Bosonic environment And the open system dynamics is Markov ϱ ee (t) =e Γt ϱ ee (0) TCL ϱ ee (t) =e R t 0 dsγ(s) ϱ ee (0)

39 Example: two-level atom in vacuum TCL2 DECAY RATE [ ] POPULATIONS (a) (b) ground NMQJ: aa NMQJ: bb analytical excited Decay rate oscillates having periods of negativity Excited state population oscillates COHERENCES (c) NMQJ: ab analytical Decoherence - re-coherence cycles TIME [1/ ] Compare to Markovian, semigroup evolution: exponential decay

40 End of lecture 1 1. General framework: Open quantum systems 2. Local in time master equations Next lecture: Solving local in time equations by Markovian and non-markovian quantum jumps