Spin-Boson Model. A simple Open Quantum System. M. Miller F. Tschirsich. Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012

Transcription

1 Spin-Boson Model A simple Open Quantum System M. Miller F. Tschirsich Quantum Mechanics on Macroscopic Scales Theory of Condensed Matter July 2012

2 Outline 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson Modell 4 Summary 2 / 45

3 Outline Bloch-Equations 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson Modell 4 Summary 3 / 45

4 Bloch-Equations Open Quantum System No ideal separation: system environment Environment: heat bath, statistics, random Decoherence: quantum interference classical mixture Dissipation: energy transfer to environment 4 / 45

5 Open Quantum System Bloch-Equations Rewiev: Density matrix formalism Example Pure = 0 1 ρ pure = = 1 ( ) Example Mixed 0 or 1 ρ mixed = 1 ( ) / 45

6 Spin Precession Bloch-Equations Two level System: Spin-1/2 External Magnetic field: B (0,0,B z ) + fluctuations δb(t) Semiclassical: no interplay spin field Fluctuations: irreversible dynamics, Decoherence 6 / 45

7 Spin precession Bloch-Equations Hamiltonian (no fluctuations) Ĥ 0 = B x 2 ˆσ x Classical Magnetization M M(t) S(t) := ˆσ ˆσ = (σ x,σ y,σ z ) spin-vector (Pauli operators). 7 / 45

8 Bloch-Equations Spin precession Rotating Frame: Heisenberg: ˆσ (t) = e iĥ 0t ˆσe iĥ 0 t dˆσ (t) dt Ehrenfest-Theorem ] = i [Ĥ0, ˆσ (t) = B ˆσ (t) M(t) = x ˆσ (t) x = M(0)cos(B x t) Rabi oscillations: reversible, unitary evolution 8 / 45

9 Fluctuations Bloch-Equations Hamiltonian (with noise) Ĥ = B x 2 ˆσ x δb(t) ˆσ 2 random noise δb(t) B x Density matrix Bloch-sphere ρ (t) = 1 (I + S(t) σ) 2 9 / 45

10 Fluctuations Bloch-Equations Hamiltonian (with noise) Ĥ = B x 2 ˆσ x δb(t) ˆσ 2 random noise δb(t) B x Density matrix Bloch-sphere ρ (t) = 1 (I + S(t) σ) 2 9 / 45

11 Fluctuations Bloch-Equations Rotating frame (noise dynamics only): Von Neumann eq.: ρ (t) = e iĥ0t ρ (t)e iĥ0t dρ dt [Ĥ ] = i (t),ρ (t) Integrate, 2nd order recursion: t ρ (t) = ρ [Ĥ ] (0) i ds (s),ρ (s) dρ dt dρ dt δb (1) = i [Ĥ (t),ρ (0) 0 t [Ĥ ds (t),[ĥ (s),ρ (t) 0 ] t [Ĥ ]] ds (t),[ĥ (s),ρ (s) 0 ]] δb (1) 10 / 45

12 Fluctuations Bloch-Equations Rotating frame (noise dynamics only): Von Neumann eq.: ρ (t) = e iĥ0t ρ (t)e iĥ0t dρ dt [Ĥ ] = i (t),ρ (t) Integrate, 2nd order recursion: t ρ (t) = ρ [Ĥ ] (0) i ds (s),ρ (s) dρ dt dρ dt δb (1) = i [Ĥ (t),ρ (0) 0 t [Ĥ ds (t),[ĥ (s),ρ (t) 0 ] t [Ĥ ]] ds (t),[ĥ (s),ρ (s) 0 ]] δb (1) 10 / 45

13 Fluctuations Bloch-Equations Rotating frame (noise dynamics only): Von Neumann eq.: ρ (t) = e iĥ0t ρ (t)e iĥ0t dρ dt [Ĥ ] = i (t),ρ (t) Integrate, 2nd order recursion: t ρ (t) = ρ [Ĥ ] (0) i ds (s),ρ (s) dρ dt dρ dt δb (1) = i [Ĥ (t),ρ (0) 0 t [Ĥ ds (t),[ĥ (s),ρ (t) 0 ] t [Ĥ ]] ds (t),[ĥ (s),ρ (s) 0 ]] δb (1) 10 / 45

14 Fluctuations Bloch-Equations Rotating frame (noise dynamics only): Von Neumann eq.: ρ (t) = e iĥ0t ρ (t)e iĥ0t dρ dt [Ĥ ] = i (t),ρ (t) Integrate, 2nd order recursion: t ρ (t) = ρ [Ĥ ] (0) i ds (s),ρ (s) dρ dt dρ dt δb (1) = i [Ĥ (t),ρ (0) 0 t [Ĥ ds (t),[ĥ (s),ρ (t) 0 ] t [Ĥ ]] ds (t),[ĥ (s),ρ (s) 0 ]] δb (1) 10 / 45

15 Longitudinal Noise Bloch-Equations Hamiltonian (longitudinal noise) Ĥ (t) = δb x 2 ˆσ x δb(t) = (δb x (t),0,0) along precession axis noisy Rabi frequency dρ t δbx (t)δb x (s) [ ds ˆσx, [ ˆσ x,ρ ( s t) ]] dt δb 0 4 δb ( = Γ 0 ρ 01 (t) ) 2 ρ 10 (t) 0 Environment correlation time: System timescale δb x (t)δb x (s) δbx peaked around s = t Exponential dephasing: ρ 01 (t) = ρ 01 (0)eΓ 2 t 11 / 45

16 Longitudinal Noise Bloch-Equations Hamiltonian (longitudinal noise) Ĥ (t) = δb x 2 ˆσ x δb(t) = (δb x (t),0,0) along precession axis noisy Rabi frequency dρ t δbx (t)δb x (s) [ ds ˆσx, [ ˆσ x,ρ ( s t) ]] dt δb 0 4 δb ( = Γ 0 ρ 01 (t) ) 2 ρ 10 (t) 0 Environment correlation time: System timescale δb x (t)δb x (s) δbx peaked around s = t Exponential dephasing: ρ 01 (t) = ρ 01 (0)eΓ 2 t 11 / 45

17 Longitudinal Noise Bloch-Equations Hamiltonian (longitudinal noise) Ĥ (t) = δb x 2 ˆσ x δb(t) = (δb x (t),0,0) along precession axis noisy Rabi frequency dρ t δbx (t)δb x (s) [ ds ˆσx, [ ˆσ x,ρ ( s t) ]] dt δb 0 4 δb ( = Γ 0 ρ 01 (t) ) 2 ρ 10 (t) 0 Environment correlation time: System timescale δb x (t)δb x (s) δbx peaked around s = t Exponential dephasing: ρ 01 (t) = ρ 01 (0)eΓ 2 t 11 / 45

18 Longitudinal Noise Bloch-Equations Animation (longitudinal noise) Averaging over random realizations of longitudinal δ B(t) 12 / 45

19 Transversal Noise Bloch-Equations Hamiltonian (transversal noise) where Ĥ (t) = δb z 2 ˆσ z (t) δb(t) = (0,0,δB z ) perpendicular; ˆσ z (t) = e B x ˆσ y t rotating frame: dρ t δbz (t)δb z (s) [ ds ˆσz (t), [ ˆσ z (s),ρ (t) ]] dt δb 0 4 δb RWA Γ ( 1 ρ 00 (t) ρ 11 (t) ρ 01 (t) ) 2 ρ 10 (t) ρ 11 (t) ρ 00 (t) Γ 1 = 1 2 t t ds e ib x s δb z (0)δB z (s) δbz Exponential dephasing rate Γ state gets mixed: ρ 00 (t) ρ 11 (t) = ( ρ 00 (0) ρ 11 (0) ) e Γ 1t 13 / 45

20 Transversal Noise Bloch-Equations Hamiltonian (transversal noise) where Ĥ (t) = δb z 2 ˆσ z (t) δb(t) = (0,0,δB z ) perpendicular; ˆσ z (t) = e B x ˆσ y t rotating frame: dρ t δbz (t)δb z (s) [ ds ˆσz (t), [ ˆσ z (s),ρ (t) ]] dt δb 0 4 δb RWA Γ ( 1 ρ 00 (t) ρ 11 (t) ρ 01 (t) ) 2 ρ 10 (t) ρ 11 (t) ρ 00 (t) Γ 1 = 1 2 t t ds e ib x s δb z (0)δB z (s) δbz Exponential dephasing rate Γ state gets mixed: ρ 00 (t) ρ 11 (t) = ( ρ 00 (0) ρ 11 (0) ) e Γ 1t 13 / 45

21 Transversal Noise Bloch-Equations Animation (transversal noise) Averaging over random realizations of transversal δ B(t) 14 / 45

22 Bloch Equations Bloch-Equations Summary: Rabi oscillations + fluctuations in Bloch-sphere representation ( ds dt = B }{{ S Γ } 2S z z Γ 2 + Γ ) 1 (S }{{} x x + S y y) 2 coherent mixing }{{} dephasing Bloch (1964): Phenomenological description of dissipation in NMR No prediction of the rates Γ 15 / 45

23 Outline Classical Dissipations 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson Modell 4 Summary 16 / 45

24 Classical Dissipations Langevin equation Include environment in our model Classical Hamiltonian Langevin equations 17 / 45

25 Langevin equation Classical Dissipations Hamiltonian H = p2 2M + V (q) + 1 }{{} 2 System N k=1 p2 k + m k ω 2 ( k xk x equ m k (q) ) 2 k }{{} coupling V(q) System: Particle (p,q) in Potential V (q) Environment: Bath of harmonic oscillators (p k,x k ) Weak Interaction: linear coupling from equilibrium positions x equ k (q) = c k m k ωk 2 q Interaction potential V I = q k x k λ k, weighted modes (λ k ) 18 / 45

26 Classical Dissipations Langevin Equation Coupled System - Environment Dynamics: ( M q + λ 2 k /m k ωk 2 k ) q + Vq (q) = λ k x k k m k ẍ k + m k ωk 2 x k = λ k q ( q = H p ṗ = H q ;ẋ k = H ṗ p k k = H ) q k Solve inhomogenous system Eliminate environment by resubstitution Initial conditions: Thermal equilibrium 19 / 45

27 Langevin Equation Classical Dissipations Langevin equation t M q (t) + M Memory friction kernel 0 ds γ (t s) q (s) + V q (q) = ξ (t) γ (t) = Θ(t)κ (t) ξ (t): fluctuating force κ (t) = 1 M k λ 2 k m k ω 2 k cos(ω k t) ξ (t) = 0, ξ (t)ξ (0) = Mk B T κ (t) 20 / 45

28 Classical Dissipations Langevin Equation Summary: Classical fluctuations: Dissipation (no decoherence). Correlators Temperature, mode coupling λ k Now: do it quantum mechanically! 21 / 45

29 Outline Spin-Boson Modell 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson Modell 4 Summary 22 / 45

30 Spin-Boson Modell A review of the spin model Any two state system (TSS) can be modeled by the spin formalism. e E(t) g 23 / 45

31 Spin-Boson Modell A review of the spin model Unperturbed TSS ( ) E1 0 H = 0 E 2 eigenv. { g, e } e g 24 / 45

32 Spin-Boson Modell A review of the spin model Unperturbed TSS ( ) E1 0 H = 0 E 2 eigenv. { g, e } e g Perturbed TSS - tunneling terms appear B(t) e g ( ) E1 W H = 12 eigenv. { +, } W 21 E 2 General Hamiltonian of a TSS H TSS = εσ z + σ x ε = 1 2 (E 1 E 2 ) = 1 2 (W 12 W 21 ) 25 / 45

33 Environment Spin-Boson Modell model environment as a bath of bosons behaving as oscillators Environment TSS Total Hamiltonian given by H = H TSS + H B + H int H B = k p 2 k 2m k m kω 2 k x 2 k = k hω k a k a k (drop zero point energy) 26 / 45

34 Spin-Boson Modell Interaction env. and TSS can be modeled as H int = σ z λ k x k (t) k x k (t) (a k + a k) is the position of the k-th harmonic oscillator λ k is the coupling strength between oscillator and spin can contain term proportional to σ z (describing spin s energy) or σ x (describing spin flips) The λ k s are given by the environment s spectral density J(ω) = i λ 2 i 2m i ω i δ(ω ω i ) (2) 27 / 45

35 Exact Solution Spin-Boson Modell Hamiltonian Ĥ = Ĥ S + Ĥ E + Ĥ I = ω ) ω k â kâk + σ z λ k (â k + â k k k Interaction picture evolution (unitary) Initial state: thermal equilibrium Û (t) = ˆT e i dsh I (s) = ϕ (t) ˆV (t) ρ (0) = ρ S (0) ρ E ρ E = 1 Z E e ĤE /k B T 28 / 45

36 Exact Solution Spin-Boson Modell Density matrix ρ (t) with Decoherence function ρ ij (t) = i tr B { V (t)ρ (0)V 1 (t) } j ρ 10 (t) = ρ 01 (t) =: ρ 10 (0)exp{Γ(t)} 4λk Γ(t) = lntr B {ρ 10 (t)} = 2 coth(ω k /2k B T )(1 cosω k t) k ω k Continous limit: Spectral density of environment (D (ω): density of modes) J (ω) = 4D (ω)λk 2 ω Γ(t) = 0 dω J (ω) ω 2 coth(ω/2k BT )(1 cosωt) 29 / 45

37 Exact Solution Spin-Boson Modell Density matrix ρ (t) with Decoherence function ρ ij (t) = i tr B { V (t)ρ (0)V 1 (t) } j ρ 10 (t) = ρ 01 (t) =: ρ 10 (0)exp{Γ(t)} 4λk Γ(t) = lntr B {ρ 10 (t)} = 2 coth(ω k /2k B T )(1 cosω k t) k ω k Continous limit: Spectral density of environment (D (ω): density of modes) J (ω) = 4D (ω)λk 2 ω Γ(t) = 0 dω J (ω) ω 2 coth(ω/2k BT )(1 cosωt) 29 / 45

38 Spin-Boson Modell Master equation for open quantum systems master equation" (for derivation Zurek or Petruccione) describes behavior of ρ = tr env ρ T master eq. in Lindblad form ρ(t) = ī h [H s,ρ] + γ k (A k ρa k 1 k 2 A k A kρ 1 2 ρa k A k) 30 / 45

39 Spin-Boson Modell Master equation for open quantum systems master equation" (for derivation Zurek or Petruccione) describes behavior of ρ = tr env ρ T master eq. in Lindblad form ρ(t) = ī h [H s,ρ] + γ k (A k ρa k 1 k 2 A k A kρ 1 2 ρa k A k) initially ρ T (0) = ρ(0) ρ env (0) Markovianity, i.e. env. is memoryless γ k 0 31 / 45

40 Master eq. and SBM Spin-Boson Modell e σ z g Consider SBM-Hamiltonian prepare initial state H = hω 0 2 σ z }{{} H s + k hω k a k a k + σ z λ k (a k + a k) } k {{ } H int ψ(0) = 1 2 ( g + e ) ρ(0) = 1 2 ( ) / 45

41 Spin-Boson Modell Master eq. and SBM ρ = ī h [H s,ρ] + γσ z ρσ z γ 2 σ z σ z ρ γ 2 ρσ z σ z Information about Spectrum contained in γ 0 Lindblad operators A = A = σ z σ 2 z = I and H s = ω 0 2 σ z master equation ρ = iω 0 s [σ z,ρ] + γσ z ρσ z γρ 33 / 45

42 Master eq. and SBM Spin-Boson Modell Evaluating i ρ j i,j {e,g} leads to this set of equations: ρ gg = 0 ρ ee = 0 ρ eg = ( iω 0 2γ) ρ eg ρ ge = (iω 0 2γ) ρ ge 34 / 45

43 Master eq. and SBM Spin-Boson Modell Evaluating i ρ j i,j {e,g} leads to this set of equations: ρ gg = 0 ρ ee = 0 ρ eg = ( iω 0 2γ) ρ eg ρ ge = (iω 0 2γ) ρ ge Leads to time evolution ρ(t) = 1 ( 1 e iω0t e 2γt ) 2 e iω0t e 2γt 1 Trace is preserved but interference terms decay! Interaction with Environment turns ρ(t) into statistical mixture, we observe decoherence! 35 / 45

44 Master eq. and SBM Spin-Boson Modell e σ x g Consider a diffierent interaction term H int = σ x k λ k q k = (σ + + σ ) λ k (a k + a k) k 36 / 45

45 Master eq. and SBM Spin-Boson Modell e σ x g Consider a diffierent interaction term H int = σ x k λ k q k = (σ + + σ ) λ k (a k + a k) k under rotating wave approximation (RWA) we neglect the terms σ + a k + σ a k and obtain H = hω 0 2 σ z + k hω k a k a k + λ k (σ a k + σ + a k ) (3) k 37 / 45

46 Master eq. and SBM Spin-Boson Modell We prepare the TSS in the exited state ψ(0) = e ρ(0) = Lindblad Operators are A = σ and A = σ + ( ) master equation for σ x coupling ρ = iω 0 s [σ z,ρ] + γσ ρσ + γ 2 σ + σ ρ γ 2 ρ σ + σ 38 / 45

47 Master eq. and SBM Spin-Boson Modell Evaluating g ρ g and e ρ e : ρ gg = γ ρ ee ρ ee = γ ρ ee (4) solving leads to ρ(t) = ( e γt ρ eg ) 1 e γt ρ ge Excited state decays to ground state over time through interaction with environment! We observe spontanious emission. 39 / 45

48 Outline Summary 1 Bloch-Equations 2 Classical Dissipations 3 Spin-Boson Modell 4 Summary 40 / 45

49 Summary Summary Open Quantum systems: Dissipation (classical) Decoherence Spin Boson Modell: Exactly solveable General approach: Master-equations 41 / 45

50 Summary Experiments with Spin Boson Model investigate a cooled 1D Coulomb crystal with N = 50 ions focus laser on a central ion assume linear spectral density J(ω) ω, high temperature regime Plot σ z (t), we observe effects quantum such as quantum revivals initial spin relaxation propagates along chain and reflects at boundary 42 / 45

51 Summary Experiments with Spin Boson Model Figure: Porras, Cirac et al / 45

52 Spin Boson Model Summary thank you for your attention 44 / 45

53 For Further Reading I Appendix For Further Reading H.-P. Breuer, F. Petruccione: The theory of open quantum systems. Oxford University Press, U. Weiss: Quantum Dissipative Systems World Scientific, F. Bloch: Nuclear Induction Physical Review 70, (1946) 45 / 45