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.