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Quantum Jumps and Rate Operators in Open Quantum System Dynamics J. Piilo

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Quantum Jumps and Rate Operators in Open Quantum System Dynamics J. Piilo UNIVERSITY OF TURKU, FINLAND Quantum jumps and rate operators in open quantum system dynamics J. Piilo Turku Centre for Quantum Physics Non-Markovian Processes and Complex Systems Group Contents 1. Markovian: Monte Carlo Wave Function Dalibard, Castin, Molmer PRL 1992 2. Non-Markovian Quantum Jumps Piilo, Maniscalco, Härkönen, Suominen: PRL 2008 3. Unified framework: ROQJ - Rate operator quantum jumps Smirne, Caiaffa, Piilo PRL 2020 See also Chruscinski, Luoma, Piilo, Smirne arXiv:2009.11312 2 Preliminaries: problem setting S E ๏ Any realistic quantum system S is coupled to its environment E ๏ Master equation description: d⇥(t) 1 1 = i[H, ⇥ ]+ γ (t) A ⇥ (t)A† A† A ⇥ (t) ⇥ A† A dt − S k k S k − 2 k k S − 2 S k k ⇤k ⇥ ๏ Decomposition of the density matrix Stochastic ρ(t)= P (t) ⇥ (t) ⇥ (t) i | i ⇥ i | descriptions i 3 Simple classification of Monte Carlo/stochastic methods Markovian non-Markovian MCWF Fictitious modes (Imamoglu) Jump (Dalibard, Castin, Molmer) Pseudo modes (Garraway) Quantum Trajectories Doubled H-space (Breuer, Petruccione) methods: (Zoller, Carmichael) Triple H-space (Breuer) Non-Markovian Quantum Jump (Piilo et al) Non-Markovian QSD Diffusion QSD (Strunz, Diosi, Gisin, Yu) methods: (Diosi, Gisin, Percival...) Stochastic Schrödinger equations (Barchielli) Jump Diffusion 1 0.8 Plus: Wiseman, Gambetta, Budini, Gaspard, 0.6 0.4 Lacroix, Donvil and Muratore-Ginanneschi 0.2 excited state GROUND AND EXCITED STATE PROBABILITIES ground state (not comprehensive list, apologies for any 0 0 2 4 6 8 10 −1 TIME [β ] omissions) 4 Simple classification of Monte Carlo/stochastic methods This talk Markovian non-Markovian MCWF Fictitious modes (Imamoglu) Jump (Dalibard,Unified Castin, Molmer)frameworkPseudo modes missing (Garraway) Quantum Trajectories Doubled H-space (Breuer, Petruccione) methods: (Zoller, Carmichael) Triple H-space (Breuer) Non-Markovian Quantum Jump (Piilo et al) Non-Markovian QSD Diffusion QSD (Strunz, Diosi, Gisin, Yu) methods: (Diosi, Gisin, Percival...) Stochastic Schrödinger equations (Barchielli) Jump Diffusion 1 0.8 Plus: Wiseman, Gambetta, Budini, Gaspard, 0.6 0.4 Lacroix, Donvil and Muratore-Ginanneschi 0.2 excited state GROUND AND EXCITED STATE PROBABILITIES ground state (not comprehensive list, apologies for any 0 0 2 4 6 8 10 −1 TIME [β ] omissions) 4 Simple classification of Monte Carlo/stochastic methods This talk Markovian non-Markovian Connection missing Connection MCWF Fictitious modes (Imamoglu) Jump (Dalibard,Unified Castin, Molmer)frameworkPseudo modes missing (Garraway) Quantum Trajectories Doubled H-space (Breuer, Petruccione) methods: (Zoller, Carmichael) Triple H-space (Breuer) Non-Markovian Quantum Jump (Piilo et al) Non-Markovian QSD Diffusion QSD (Strunz, Diosi, Gisin, Yu) methods: (Diosi, Gisin, Percival...) Stochastic Schrödinger equations (Barchielli) Jump Diffusion 1 0.8 Plus: Wiseman, Gambetta, Budini, Gaspard, 0.6 0.4 Lacroix, Donvil and Muratore-Ginanneschi 0.2 excited state GROUND AND EXCITED STATE PROBABILITIES ground state (not comprehensive list, apologies for any 0 0 2 4 6 8 10 −1 TIME [β ] omissions) For a solution: see Luoma, Struntz, Piilo 4 PRL 2020 Markovian Monte Carlo Wave Function method 5 Density matrix and state vector ensemble Suppose now we want to solve the semigroup, Markovian GKSL equation d⇥(t) 1 1 = i[H, ⇥ ]+ γ A ⇥ A† A† A ⇥ ⇥ A† A dt − S k k S k − 2 k k S − 2 S k k ⇤k ⇥ Q: How to solve the master equation? ๏ Few exact models and analytical solutions ๏ Can we find the solution by evolving an ensemble of state vectors instead of directly solving the density matrix? Generally, we can decompose the density matrix as ρ(t)= P (t) ⇥ (t) ⇥ (t) i | i ⇥ i | i 6 Basics of stochastic state vector evolution Monte Carlo wave function method (Markovian) (Dalibard, Castin, Molmer, PRL 1992) ψ1(t0) ψ1(t1) ...... ψ1(tn) Ensemble of N state vectors ψ2(t0) ψ2(t1) ...... ψ2(tn) .... ψN (t0) ψN (t1) ...... ψN (tn) Time At each point of time, density matrix ρ as average of state vectors Ψi: 1 N ρ(t) = ⇥ (t) ⇥ (t) N | i ⇥ i | i=1 The time-evolution of each Ψi contains stochastic element due to random7 quantum jumps. Jump probability, example Time-evolution of state vector Ψi: At each point of time: decide if quantum jump happened. Pj: probability that a quantum jump occurs in a given time interval δt: Pj = δt Γ pe occupation probability time-step decay rate of excited state For example: 2-level atom E Probability for atom being transferred from the excited to the ground state and photon emitted. G 8 Example: driven 2-state system, Markovian Quantum jump: Discontinuous stochastic change of the state vector. Excited state probability P for a driven 2-level atom Excited state E coupling decay channel (deterministic) (random jump) G Ground state d⇥ 1 = i[H, ⇥]+Γ σ ⇥+ σ+σ , ⇥ dt − − − 2{ − } N ⇥ 1 ρ(t) = ⇥i(t) ⇥i(t) N | ⇥ | 9 i=1 Example: driven 2-state system, Markovian Quantum jump: Discontinuous stochastic change of the state vector. Excited state probability P for a driven 2-level atom Markovian Monte Carlo Excited state single realization ensemble average E coupling decay channel (deterministic) (random jump) P G Ground state Time Time d⇥ 1 damped Rabi oscillation = i[H, ⇥]+Γ σ ⇥+ σ+σ , ⇥ dt − − − 2{ − } of the atom N ⇥ 1 ρ(t) = ⇥i(t) ⇥i(t) N | ⇥ | 9 i=1 Markovian Monte Carlo wave function method Master equation to be solved: dρ(t) 1 † 1 † † = [HS ,ρ] + ∑ΓmCm ρCm − ∑Γm CmCm ρ + ρCmCm dt i 2 ( Master equation to )be solved h m m For each ensemble member : Solve the time dependent Schrödinger d ih Ψ(t) = H Ψ(t) equation. € dt Use non-Hermitian Hamiltonian H which H = Hs + Hdec includes the decay part Hdec. € Key for non-Hermitian Hamiltonian: i † H = − h Γ C C Jump operators C can be found from the dec 2 ∑ m m m m m dissipative part of the master equation. € † For each channel m the jump probability is δpm = δtΓm Ψ CmCm Ψ given by the time step size, decay rate, and € decaying state occupation probability. 10 Dalibard, Castin, Molmer: PRL 1992 € Markovian Monte Carlo wave function method Algorithm: 1. Time evolution over time step 2. Generate random number, did jump occur? No Yes 3. Renormalize before new time step 3. Apply jump operator before iHeffδt e− ⇥i(t) new time step ⇥i(t + δt) = | ⇥ Cj ⇥i(t) | ⇥ ⌅1 δp ⇥i(t + δt) = | − | Cj ⇥(t) || | || 4. Ensemble average over :s gives the density matrix and the expectation value of any operator A 1 A (t)= ψ (t) A ψ (t) ⇥ N i | | i ⇥ i 11 Markovian Monte Carlo wave function method Measurement scheme interpretation Two-level atom in vacuum Two-level atom MC evolution by Total system evolution Measurement scheme: C = Γ g e Jump operator continuous measurement of photons in the environment. ihΓ Hdec = − e e Non-Hermitian Hamiltonian 2 (cg g + ce e ) ⊗ 0 → 2 € Jump probability c' g + c' e ⊗ 0 + c g ⊗ 1 P = δtΓ ce ( g e ) ∑ λ λ λ € ๏Continuous measurement of the environmental state gives € € conditional pure state realizations for the open system ๏The open system evolution is average of these realizations 12 Markovian Monte Carlo wave function method Equivalence with the master equation: The state of the ensemble averaged over time step: (for simplicity here: initial pure state and one decay channel only) This gives comm. + anticomm. of m.e. This gives ”sandwich” term of the m.e. † φ(t + δt) φ(t + δt) C Ψ(t) Ψ(t) C σ(t + δt) = (1− P) + P † Average 1− P Ψ(t) C C Ψ(t) ”No-jump” path weight ”Jump” path weight t-evol. and normalization Jump and normalization Keeping in mind two things: a) the time-evolved state is (1st order in dt, before renormalization): € & iH δt Γδt † ) φ(t + δt) = (1 − s − C C+ Ψ(t) ' 2 * h b) the jump probability is: † P = δtΓ Ψ C C Ψ € it is relatively easy to see that the ensemble average corresponds to master equation d⇥(t) 1 1 = i[H, ⇥S]+ γk Ak⇥SAk† Ak† Ak⇥S ⇥SAk† Ak dt − 13− 2 − 2 € ⇤k ⇥ Questions: ๏What happens when the decay rates depend on time? ๏What happens when the decay rates turn temporarily negative? d⇥(t) 1 1 = i[H, ⇥ ]+ γ (t) A ⇥ (t)A† A† A ⇥ (t) ⇥ A† A dt − S k k S k − 2 k k S − 2 S k k ⇤k ⇥ 14 2. Non-Markovian Quantum Jumps Piilo, Maniscalco, Härkönen, Suominen: PRL 2008 15 Markovian vs. non-Markovian evolution (1) Markovian dynamics: Non-Markovian dynamics: Decay rate constant Decay rate depends on time, in time. obtains temporarily negative values. 16 Markovian vs. non-Markovian evolution (1) Markovian dynamics: Non-Markovian dynamics: Decay rate constant Decay rate depends on time, in time. obtains temporarily negative values. Example: 2-level atom in photonic band gap. Decay rate (exact) Time Pj = δt Γ pe < 0 Markovian description of quantum jumps fails, since gives negative jump probability. For example: negative probability that atom emits a photon. 16 Non-Markovian master equation Starting point: General non-Markovian master equation local-in-time: dρ(t) 1 † 1 † † = [HS,ρ] + Δ m (t)Cm ρCm − Δ m (t) CmCm ρ + ρCmCm dt i ∑ 2 ∑ ( ) h m m ๏ Jump operators Cm ๏ Time dependent decay rates Δm(t). € ๏ Decay rates have temporarily negative values. 17 Non-Markovian master equation Starting point: General non-Markovian master equation local-in-time: dρ(t) 1 † 1 † † = [HS,ρ] + Δ m (t)Cm ρCm − Δ m (t) CmCm ρ + ρCmCm dt i ∑ 2 ∑ ( ) h m m ๏ Jump operators Cm ๏ Time dependent decay rates Δm(t). € ๏ Decay rates have temporarily negative values. E Example: 2-level atom in photonic band gap.
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