Time scales of diffusion and decoherence

Transcription

1 Time scales of diffusion and decoherence Janos Polonyi University of Strasbourg 1. Decoherence 2. OCTP: a QCCO formalism 3. Dynamical, static and instantaneous decoherence 4. Effective Lagrangian of a test particle interacting with an ideal gas 5. Static decoherence and diffusive time scales 6. Dynamical decoherence of harmonic systems 7. Summary

2 Decoherence Necessary condition of the classical limit System and its environment: H = H s H e Preferred system basis: { ψ n } Full system in pure, entangled state: Ψ = n c n ψ n χ n = Entanglement = mixed state: ρ = Ψ Ψ = )( ) c n c n ( ψ n χ n χ n ψ n nn ρ s = Tr e ρ = nn c n c n χ n χ n ψ n ψ n Decoherence: - suppression of interference terms (Schrödinger s cat) - diagonal ρ s in the preferred basis - additive expectation value in the preferred basis Ψ A Ψ = nn c n c n ψ n A ψ n χ n χ n n c n 2 a n - loss of information dissipation

3 Decoherence Environment = Open quantum systems = Effective theories = CTP Classical : Z = D[x]D[y]e S[x,y] = D[x]e S eff [x] e S eff [x] = D[y]e S[x,y] Quantum : 0 U 0 = D[x]D[y]e is[x,y] = D[x]e is eff [x] e S eff [x] = D[y]e is[x,y] 0 eff is not a pure state 0 eff ρ i Tr[Uρ i U ] = D[x + ]D[x ]D[y + ]D[y ]e is[x+,x + ] is [y,y ] = D[x + ]D[x ]e is eff [x +,x ] e S eff [x +,x ] = D[y + ]D[y ]e is[x+,y + ] is [x,y ] Mixed states, decoherence, dissipation x + x coupling

4 CTP formalism for closed systems (Schwinger 1961, Keldysh, 1964,...) Generator functional: e i W [ĵ] = TrT [e i dt(h(t) j + (t)x(t)) ]ρ i T [e i = D[ˆx]e i S[x+ ] i S[x ]+ i dtĵ(t)ˆx(t) = D[ˆx]e i S[ˆx]+ i dtĵ(t)ˆx(t) dt(h(t)+j (t)x(t)) ] Reduplication of the degrees of freedom: - ˆx = (x +, x ) - ψ A ψ x + x - p(x) = ψ (x)ψ(x) = independent x ±

5 CTP formalism for classical systems Polonyi 2012 Holonomic force: F (x, ẋ)δx = δx x U(x, ẋ) δẋ ẋu(x, ẋ) ( ) x + Semiholonomic force: x ˆx = - active: x + - passive: x (environment) Virtual work: x F (x, ẋ)δx = [δx x +U(ˆx, ˆx) + δẋ ẋ+u(ˆx, ˆx)] x + =x =x This extension enough to cover effective theories.

6 CTP formalism for classical systems Replaying the motion backward in time to recover semiholonomic forces ˆx(t) = flipping the time arrow { x( t) t i < t < t f, x( t) = x(2t f t) t f < t < 2t f t i, ( ) ( ) x + (t) x(t) x = (t) x(2t f t) Initial conditions: x ± (t i ) = x i ẋ ± (t i ) = v i Final condition : x + (t f ) = x (t f )

7 CTP formalism for classical systems Action, Green functions L(ˆx, ˆx) = L(x +, ẋ + ) L(x, ẋ ) + L spl (ˆx, ˆx) L spl (ˆx, ˆx) = i ɛ 2 (x+2 + x 2 ) Degeneracy for x + (t) = x (t) ψ ψ Time inversion: T : x ± x S[x +, x ] S [x, x + ] Generator functional: W [ĵ] = S[ˆx, ŷ] + ĵˆx, Green functions: 1 W [ĵ] = n! n=0 σ 1,...,σ n δs[ˆx, ŷ] δˆx = ĵ, δs[ˆx, ŷ] δŷ = 0 dt 1 dt n D σ1,...,σn (t 1,..., t n )j σ1 (t 1 ) j σn (t n )

8 CTP formalism for open quantum systems Decoherence x ± = x± xd 2 = x = x + +x 2 classical coordinate x d = x + x = O( ) quantum fluctuations QM: O( ) separation of the copies The same initial conditions for x + and x S eff [ˆx] = S 1 [x + ] S1[x ] + S 2 [ˆx] [ ] ] ] = S 1 x + xd S 1 [x xd + S 2 [x + xd 2 2 2, x xd 2 0 = δs 1[x] δx + δs 2[x +, x ] δx + x + =x =x semiholonomic forces Decoherence: suppression of the contributions to the density matrix by ImS eff 0 for large x d : ρ(x +, x ) = D[ˆx]e i S eff [ˆx] ˆx(t f )=ˆx

9 OTP formalism for the reduced density matrix CTP: Open Time Path (OTP): Trρ = TrT [e i dth(t) ]ρ i T [e i = D[ˆx]e i S[ˆx] x + (t f )=x (t f ) dt(h(t) ] Tr env ρ(x +, x ) = x + Tr env T [e i dth(t) ]ρ i T [e i dth(t) ] x = D[ˆx]e i S eff [ˆx] x ± (t f )=x ± Effective theory: an OCTP description, system : OTP environment: CTP

10 Free propagator Harmonic system: S 0 [ ˆφ] = ˆφ ˆD ˆφ Feynman Wightman ( ) ( ) i ˆD T [φx φ x,y = y ] φ y φ x D φ x φ y T [φ y φ x ] = i n + id i D f + id i D f + id i D n + id i ( 1 ˆD k = k 2 m 2 +iɛ 2πiδ(k 2 m 2 )Θ( k 0 ) ) 2πiδ(k 2 m 2 )Θ(k 0 1 ) k 2 m 2 iɛ Physical Green functions: D F = D n + id i, D ra = D n ± D f Off-shell: D n, on-shell: D f (φ + φ coupling), D i Tr env ρ(x +, x ) = D[ˆx]e i S eff [ˆx] x ± (t f )=x ± Radiation (far) field: environment excitations, dissipation

11 CTP formalism for open quantum systems Decoherence Equations of motion: S[ˆx] = 1 2 ˆx ˆD 1ˆx + S i [ˆx] + ˆxĵ = ˆx = ˆD[ δs δˆx ĵ] Iterative solution: sum of tree-graphs x ± = ( x ± ) xd 2 (, j± = ) jd 2 ± j j D ˆD j d = r j D a j d ( ) ( x 1 ) ˆD x d = 2 Dr x d 2D a x j: retarded effects, j d : advanced effects (boundary conditions) x: retarded response, x d : advanced response interactions: retarded and advanced effects mixed j d = 0: δs i[ˆx] δx d = O ( (x d ) 2n+1) = no x x d mixing = x d = 0 = retarded effects for observables j d 0: = advanced effects for x d 0 and decoherence

12 Decoherence by the Liouville space propagator Role of ImS eff in OTP ρ(x + f, x f ; t f ) = U(t)ρ(x + i, x i ; t i) (closed sys.) = D[x + ]D[x ]e i S[x+ ] i S[x ] ρ(x + (t i ), x (t i ); t i ) x ± (t f )=x ± f (open sys.) = D[x + ]D[x ]e i S eff [x +,x ] ρ(x + (t i ), x (t i ); t i ) x ± (t f )=x ± f Decoherence: Suppression by e 1 ImS eff [x,x d ]

13 Decoherence Dynamical, instantaneous, and static Dynamical Instantaneous Static Decay of Time dependent System dynamics Schrödinger s cat suppression of interference suppressed l dd, τ dd l id, τ id l sd, τ sd Scales: e 1 ImS eff [x,x d] e x d2 2l 2 (t), l = l( t τ ) τ sd τ diss (a collisional model Joos, Zeh, 1985,...) Are dissipation and decoherence really very different processes?

14 Test particle in an ideal gas Particle interacting with an ideal gas by the potential U(x): S[x, ψ, ψ] = S p [x] + S g [ψ, ψ] + S i [x, ψ, ψ] S p [x] = S g [ψ, ψ] = S i [x, ψ, ψ] = [ ] M dt 2 ẋ2 (t) V (x(t)) ] dtd 3 yψ (t, y) [i t + 2 2m + µ ψ(t, y) = ψ F 1 ψ dtd 3 yu(y x(t))ψ (t, y)ψ(t, y) = ψ Γ[x]ψ

15 Ideal gas environment Generating functional e i W [ĵ] = = D[ˆx]D[ ˆψ]D[ ˆψ ]e i Sp[ˆx]+ i ˆψ ( ˆF 1 +ˆΓ[ˆx]) ˆψ+ i D[ˆx]e i Sp[ˆx]+ i S infl[ˆx]+ i dt ĵ(t)ˆx(t) dt ĵ(t)ˆx(t) Influence functional: (Feynman, Vernon 1963) e i S infl[ˆx] = D[ ˆψ]D[ ˆψ ]e i ˆψ ( ˆF 1 +ˆΓ[ˆx]) ˆψ S infl [ˆx] = i Tr ln[ ˆF 1 + ˆΓ] = 1 σσ j σ G σσ j σ + O (ĵ3 ) 2 σσ =± Ĝ =, j σ (t, y) = U(y x σ (t))

16 Ideal gas environment Effective Lagrangian in O ( ), O ( x 2), O ( 2 t ) for an ideal fermi gas S infl [ˆx] = 1 2 σσ =± σσ j σ G σσ j σ + O (ĵ3 ) Ĝ =, j σ (t, y) = U(y x σ (t)) L eff = x d [ mẍ δmẍ kẋ + id 0 x d id 2 ẍ d] 1 δm = 48π 2 vf 2 dkuk 2 ixg 2 n 0k(x, y) mass ren. 1 k = 24π 2 dkkuk 2 ix G f 0k v (x, y) = m friction F τ d d 0 = 1 48π 2 dkk 2 Uk 2 G i 0k(x, y) decoherence 1 d 2 = 96π 2 vf 2 dkuk 2 ixg 2 i 0k(x, y) decoherence 0 = x d identity, m R ẍ = k ẋ δx d

17 Static decoherence Higher orders in x d L eff = x d [ mẍ kẋ δmẍ + iu d (x d ) id 2 ẍ d] U d (x d 1 ( sin x ) = 2π 2 dqq 2 d ) q q x d 1 Γ i 0q, 0 qualitative similarity with the collisional model of Joos & Zeh Static decoherence: e 1 IS ef [x,x d] = e τ sd (x d ) = t τ sd U d (x d ). x d τ diss τ sd (x d ) = F F (x, y) = 3 4 ( ) ɛf k B T, xd < 1 λ T 0 du 1 sin 4 πuy 4 πuy 1+e u x 0 du u 1+e u x

18 Limits of the static decoherence scenario Effective theory: perturbation expansion decoherence kinetic energy = τ sd k B T < 1 k B T T [K] < τ sd Collisional model: (i) Finite time step, t, in deriving the master equation: ρ < ρ during the build up of decoherence: τ coll = r 0 < τ sd v env m - Ideal fermi gas: < τ sd k 2 F - Air at room temperature: (ii) Multiple scatterings ignored: τ sd > τ dmin = 0 decoherence time at saturation cm 10 5 m/s = s < τ sd 1 dqν(q) q m σ tot(q) m = r 0 k F n g σ tot v F multiple scattering threshold σtot r 0

19 Dynamical decoherence Harmonic model Saddle point expansion of the Liouville space propagator the (exact): ρ(ˆx f ; t f ) = D[ˆx]e i S eff [ˆx] ρ(ˆx(t i ); t i ) ˆx(t f )=ˆx f Effective Lagrangian: E.O.M. N e i S eff (ˆx f,ˆx i,t ) L eff = x d [ mẍ kẋ mω 2 0x + id 0 x d id 2 ẍ d] δx d : mẍ = m 0 ω0x 2 kẋ + i(d 0 x d d 2 ẍ d ) δx : mẍ d = m 0 ω0x 2 d +kẋ d Exponentially increasing quantum fluctuations!

20 Dynamical decoherence Brownian motion mẍ = kẋ + i(d 0 x d d 2 ẍ d ) mẍ d = kẋ d k m Rex(t) Imx(t) x d (t) = 0.01 (solid line), 0.1 (dashed line) and 1 (dotted line) x d : relaxation backward in time - static decoherence scenario applies - τ id = τ diss

21 Dynamical decoherence Harmonic oscillator mẍ = m 0 ω 2 0x mνẋ + i(d 0 x d d 2 ẍ d ), ν = k m mẍ d = m 0 ω 2 0x d + mνẋ d Rex(t) Imx(t) x d (t) x d (t): Overshooting backward in time e 1 ImS eff e x d2 t l 2 dd (t) = e c(t)e τ dd 1 l 2 dd (t) = 1 ( ν 2ν (d 0 + d 2 ω0) 2 2 4ω 2 0 double exponential ) { 4e t(ν ν 2 4ω0 2) 4ω0 1 2 < ν 2 etν sin 2 ω νt 4ω0 2 > ν 2

22 Dynamical decoherence Anharmonic oscillator H.O.: Unstable quantum fluctuations = harmonic regime left quickly Relaxed state: - ballance between the linear and the non-linear terms of E.O.M. - non-perturbative quantum fluctuations

23 Summary 1. OCTP: to deal with QCCO problems 2. Decoherence: two set of characteristic scales Final state (instantaneous decoherence) Initial state (dynamical decoherence) 3. Ignoring the system dynamics: static decoherence 4. Time scales: no separation (no small parameter, as in QCD) τid = τ diss τsd τ diss τsd τ diss : only beyond the range of applicability 5. Qualitatively different cases: Brownian motion: static decoherence scenario applies Harmonic oscillator: Fast, double exponential decoherence Anharmonic oscillator: Non-perturbative asymptotic state