Program on « QUANTUM DYNAMICS OUT OF EQUILIBRIUM » Institut Henri Poincaré 12 November - 14 December 2007 Pierre GASPARD Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium 1)

SEMICLASSICS AND PERIODIC-ORBIT QUANTIZATION OF CHAOTIC SCATTERING

2)

SLOWING DOWN OF QUANTUM DECAYS IN CLASSICALLY CHAOTIC SCATTERING

3)

DECAY OF QUANTUM STATISTICAL MIXTURES IN CLASSICALLY CHAOTIC SCATTERING

4)

NONEQUILIBRIUM TRANSIENTS AND TRANSPORT IN LARGE QUANTUM SYSTEMS

NONEQUILIBRIUM TRANSIENTS AND TRANSPORT IN LARGE QUANTUM SYSTEMS Pierre GASPARD Brussels, Belgium M. Nagaoka, Japan M. Esposito, Brussels & San Diego USA • REDUCED DESCRIPTION & MASTER EQUATION • LIOUVILLIAN RESONANCES • STOCHASTIC SCHRÖDINGER EQUATION

REDUCED DESCRIPTION IN MANY-BODY SYSTEMS quantum subsystem coupled to a heat reservoir: description in terms of a reduced density matrix ρˆ s (t) obeying some master equation. Ex: spin-boson model in the weak-limit coupling: Bloch-Redfield equations

ρˆ = ∑ ψ j P j ψ€ j

density matrix

j

Landau-von Neumann eq. observable of the subsystem:

€

statistical average:

€

1 ˆ ˆ [ H, ρˆ ] ≡ Lˆ ρˆ ih Aˆ = Aˆ ⊗ Iˆ

∂t ρˆ =

s

b

+iHˆ t/h ˆA(t) = tr Aˆ tr e−iHˆ t/h€ ˆ ρ0 e = trs Aˆ s ρˆ s (t) s s b

€ coarse-graining by tracing out the bath degrees of freedom

€

ˆ

ˆ

ρˆ t = e−iHt/h ρˆ 0 e +iHt/h

QUANTUM SUBSYSTEM WEAKLY COUPLED TO AN ENVIRONMENT Ex: spin-boson model in the weak-limit coupling Hamiltonian

Hˆ tot = Hˆ s + Hˆ b + λVˆ

Landau-von Neumann eq.

∂t ρˆ =

Vˆ = ∑ Sˆα Bˆ α

1 ˆ ˆ [ H tot , ρˆ ] ≡ Lˆ ρˆ ih

perturbative expansion:

€ ρˆ I (t) = ρˆ (0) + €

t

ˆˆ ∫ dt1LI (t1)ρˆ (0) + 0

tracing out the bath degrees of freedom ˆ Lˆs t

t

T

t1

t

∫ dt ∫ 1

0

€

initial density matrix: ˆ

e− βH b ρˆ (0) = ρˆ s (0) ⊗ Zb ˆˆ ˆˆ dt 2 LI (t1 )LI (t 2 ) ρˆ (0) + O(λ3 )

0

€ ˆ ˆˆ ˆˆ Lˆs T dτ LI (0)LI (−τ ) e ρˆ s (0) + O( λ3 )

ρˆ s (t) = e ρˆ s (0) + ∫ dT ∫ 0 t 0  dρˆ s (t) ˆˆ ˆˆ Lˆˆ0τ ˆˆ − Lˆˆ0τ = Ls + ∫ dτ LI e LI e dt  0

€

α

b

 + O(λ )ρˆ s (t) b  Redfield master equation: Markovian approximation t >> t b  R dρˆ sR (t)  ˆˆ ∞ ˆˆ Lˆˆ0τ ˆˆ − Lˆˆ0τ 3 = Ls + ∫ dτ LI e LI e + O(λ )ρˆ s (t) dt b   0 € € 3

SLIPPAGE OF INITIAL CONDITIONS autocorrelation function of bath coupling operators:

C(t) = Bˆ (t) Bˆ (0)

b

non-Markovian evolution: t

T

0

0

ˆ ˆˆ ˆˆ Lˆs T 3 ˆ ρˆ s (t) = e ρˆ s (0) + ∫ dT ∫ dτ L€ (0) L (− τ ) e ρ (0) + O( λ ) I I s ˆ Lˆs t

b

Markovian evolution: R s

ˆ Lˆs t

ρˆ (t) = e ρˆ s (0) +

€

t

∞

0

0

ˆ ˆˆ ˆˆ Lˆs T 3 ˆ dT d τ L (0) L (− τ ) e ρ (0) + O( λ ) ∫ ∫ I I s

slippage of initial conditions:

ρˆ s (t) ≈ e

€

€

ˆ LˆsR t

ˆˆ ˆ Sρ s (0)

b

€

SPIN-BOSON MODEL Hamiltonian

Δ ˆ H tot = − σˆ z + Hˆ b + λσˆ x Bˆ 2

1 ˆ H b = ∑ ( pˆ α2 + ωα2 qˆα2 ) 2 α

Bˆ = ∑ cα qˆα α

2 c spectral strength J(ω ) = ∑ α δ (ω − ωα ) = Kω s exp(−ω /ω c ) α 2ω α ∞ € ˆ (t) Bˆ (0) = ∫ dωJ(ω ) coth βhω cosωt − isin ωt  autocorrelation function: C(t) = B b   2 0

€ Bloch-Redfield Markovian master equation:

x =  y =  z =

σˆ x σˆ y σˆ z

€

 x˙ = Δy   y˙ = −(Δ + h ) x − gy  z˙ = − f − gz

Liouvillian resonances:

€

s = 0 invariant equilibrium state € population relaxation s = −g g  h s = − ± i Δ +  decoherence 2  2

€

∞  2  f = 4 λ ∫ sinΔt ImC(t) dt 0  ∞  2 g = 4 λ ∫ cosΔt ReC(t) dt  0 ∞  2 h = 4 λ ∫ sinΔt ReC(t) dt  0

QUANTUM MODEL OF DIFFUSION Hamiltonian: tight-binding electronic Hamiltonian coupled to a bath of bosons

cα2 J(ω ) = ∑ δ(ω − ωα ) = Kω exp(−ω /ω c ) α 2ω α

Ohmic spectral strength autocorrelation function:

C(t) = Bˆ (t) Bˆ (0)

b

€ Liouvillian resonances: diffusive branch:

= Qδ (t) tunneling amplitude of electrons: A 2

€

s = −2Qλ2 + 2 Q2 λ4 − (2Asinq /2) = −Dq 2 + O(q 4 )

diffusive coefficient:

A2 A2 D= 2 = Qλ πKλ2 k BT

conductivity:

€ €

€

e 2n σ= D kBT

Hamiltonian:

STOCHASTIC SCHRÖDINGER EQUATION Hˆ = Hˆ + Hˆ + λVˆ Vˆ = ∑ Sˆ Bˆ tot

s

b

α

α

α

Schrödinger eq.

ih ∂t Ψ(x s, x b ;t) = Hˆ tot Ψ(x s, x b ;t)

bath orthonormal basis:

€

Hˆ b χ n (x b ) = εn χ n (x b )

expansion of the total wave function:

€

Ψ(x s, x b ;t) = ∑ψ n (x s;t) χ n (x b ) n

a typical subsystem wave function ψ n (x s;t)obeys the stochastic Schrödinger equation:

€

i∂tψ (x s;t) = Hˆ sψ (x s;t) + λ ∑ηα (t) Sˆαψ (x s;t) − iλ

2

α

Gaussian noises:

€

t

−iHˆ s (t−τ ) ˆ ˆ d τ C (t − τ ) S e Sβψ (x s;t) + O(λ3 ) ∫ ∑ αβ α 0

αβ

η€α (t) = 0 ηα (t)ηβ (t') = 0

€

ηα* (t)ηβ (t') = Cαβ (t − t') This stochastic Schrödinger equation is associated with the non-Markovian master equation.

€

CONCLUSIONS & PERSPECTIVES The concept of Liouvillian resonances extends to many-body quantum systems where they describe the properties of relaxation and decoherence. The quantum Liouvillian resonances are given as the eigenvalues of the quantum master equations such as the Bloch-Redfield master equation. As the Pollicott-Ruelle resonances in classical dynamical systems, the quantum Liouvillian resonances give the dispersion relations of transport properties such as diffusion (as well as the associated decoherence). In the same way, a stochastic Langevin equation is associated to the Fokker-Planck master equation, it is possible to associate a stochastic Schrödinger equation to quantum master equations. In the present setting, they concern typical « subsystem wave functions », i.e., the coefficients of the expansion of the total wave function on an orthonormal basis of the bath. Such stochastic Schrödinger equations provide us with a framework to understand how stochasticity can manifest itself, e.g. in quantum measurement processes. In this sense, they are complementary to the master equations describing the process of decoherence. Dynamical randomness can be characterized by quantities such as the KS entropy per unit time.