Van Hove Limit for Infinitely Extended Open Quantum Systems David Taj Physics Dept., University of Fribourg, Switzerland CPT - UMR 6207 and Universit´ e de Toulon, France david.taj@gmail.com 29th November 2010 David Taj david.taj@gmail.com
Outline The Markovian Approach and Davies generators Our Generator : Quantum Fokker-Planck Equation Example : a Quantum Particle in 3D Space Proposal : a Quantum Collisionless Boltzmann Equation Outlook on Quantum Brownian Motion Conclusions Collaborations Prof. F. Rossi (Dip. Fisica, Politecnico di Torino) Prof. H. Fujita Yashima (Dip. Mat., Universit´ a di Torino) Prof. C-A Pillet (CPT - UMR 6207 et Universit´ e de Toulon) Prof. V. Gritsev (D´ ep. Phys., Universit´ e de Fribourg) David Taj david.taj@gmail.com
Beam Foil Spectroscopy An experiment to start with! David Taj david.taj@gmail.com
Need of a Quantum Theory of Relaxation Phenomena Coherent superposition of system eigenstates ⇒ non-trivial interplay between Coherent Dynamics and Energy-Relaxation/Decoherence Contact with microscopic quantum description at large times The fundamental equations governing the basic laws of Physics are time reversible and not dissipative. Macroscopic irreversible equations obtained through averaging over microscopic degrees of freedom (stochasticity) energy-time scale separation ( µ eV versus meV, etc.) neglecting recollisions (Markovicity) David Taj david.taj@gmail.com
Van Hove Limit in Quantum Open Systems H = H S ⊗ H B H 0 = H S ⊗ 1 + 1 ⊗ H B H ′ = Q ⊗ Φ , H λ = H 0 + λ H ′ System observables in Heisenberg picture The state on H is ρ = ρ s ⊗ σ β At time t, O λ ( t ) = e iH λ t O S ⊗ 1 e − iH λ t We measure � O λ ( t ) � ρ = tr [ ρ O λ ( t )] = tr [ ρ S O λ S ( t )] where O λ S ( t ) = P 0 O λ ( t ) system observable at time t P 0 X ⊗ Y = tr ( σ β Y ) X ⊗ 1 partial trace projection Markovian Approximation in the Van Hove Limit Define W λ t O S := O λ S ( t ) system evolution superoperator Expect W λ 0 ≤ t ≤ λ − 2 τ, t ∼ exp { L λ t } , λ ∼ 0 [1] L. Van Hove, Physica 21 517 (1955) David Taj david.taj@gmail.com
Exact System Evolution : the Memory Kernel Formulation on operator spaces B = B 0 ⊕ B 1 Banach spaces B i = P i ( B ) , P 1 = 1 − P 0 Z O = i [ H 0 , O ] and A O = i [ H ′ , O ] Liouvillians W λ t = P 0 exp { ( Z + λ A ) t }| B 0 subsystem evolution The Nakajima-Zwanzig master equation � t � t 1 W λ t = X λ t + λ 2 dt 2 X λ t − t 1 A 01 U λ t 1 − t 2 A 10 W λ dt 1 t 2 0 0 A ij = P i AP j splitted interaction U λ X λ t = P 0 U λ t = exp { ( Z + λ A 00 + λ A 11 ) t } , t [1] Nakajima, S., Prog. Theor. Phys. 20(6) 948-959 (1958) [2] Zwanzig, R, J. Chem. Phys. 33 1338 (1960) David Taj david.taj@gmail.com
The Born-Markov approximation: Davies generator Markovian Hypotheses (Bounds on Dyson Expansion) � λ − 2 τ dx � A 01 U λ x A 10 � < C , | λ | < 1 0 � λ − 2 τ dx � A 01 ( U λ ∀ τ > 0 , lim x − U x ) A 10 � = 0 λ → 0 0 Davies Markovian Approximation Theorem (MAT) � W λ ∀ τ > 0 lim sup t − exp { L λ t }� = 0 λ → 0 0 ≤ t ≤ λ − 2 τ with L λ := Z 0 + λ A 00 + λ 2 K D , and � ∞ K D = 0 dr U − r A 01 U r A 10 Davies generator K D well defined for arbitrary H S spectra [1] E. B. Davies, Markovian Master Equations II , Math. Ann. 219 147-158 (1976) David Taj david.taj@gmail.com
Confined Systems : Davies averaged generator 1 Time averaging map ♮ � T 1 K ♮ = lim dq U q KU − q 2 T T → + ∞ − T Then L λ = Z 0 + λ 2 K ♮ D satisfies MAT iff K D does incorporates Pauli Master Equation as [ Z 0 , K ♮ D ] = 0 describes resonances of the Liouvillian (Fermi Golden Rule) 2 , 3 generates a Quantum Dynamical Semigroup but only if P 0 is a partial trace only when A 00 = 0 (no average forces on the system) K ♮ well defined only if Z 0 has discrete spectrum [1] E. B. Davies, Commun. Math. Phys. 39 91-110 (1974) [2] Jaksic V., Pillet C.-A., Ann. Inst. H. Poincare Phys. Theor. 67 425-445 (1997) [3] Derezinski J., Jaksic V., J. Stat. Phys. 116 411-423 (2004) David Taj david.taj@gmail.com
Infinitely Extended systems : K D is the only candidate [1] H. Sphon, Rev. Mod. Phys. 53 3 (1980) K D employed only under severe restrictions [2] E.B. Davies, Ann. Inst. Henri Poincar´ e 28 1 (1978) Why K D is so bad? It does not generate a proper QDS [4] D¨ umcke R and Spohn H, Z. Phys. B 34 419 (1979) is it really a big problem after all, or just some transient? David Taj david.taj@gmail.com
Failure of K D approximation at large times Case of a two-level quantum-dot system in a thermal bosonic environment One particle sector � � a | ρ | a � � � f a � � a | ρ | b � p ρ = = p ∗ � b | ρ | a � � b | ρ | b � f b Very small perturbation of thermal distribution at t = 0 Characteristic interlevel splitting: 30 meV Very high temperatures! Analytically solved: divergences don’t come from numerics! Totally unphysical results for large times/steady states [1] Taj D., Iotti R.C., Rossi F. , Eur. Phys. J. B 72 3 (2009) David Taj david.taj@gmail.com
The main idea : symmetry could recover probabilities Figure: S. Weinberg ”The Quantum Theory of fields”, vol 1, Cambridge University Press (1995) Probabilities must be positive! It could help in getting a good (unique?) evolution equation Hidden time symmetries in the memory kernel could imply positive probabilities! David Taj david.taj@gmail.com
The Van Hove Limit: a new approach The Nakajima-Zwanzig master equation � t � t 1 W λ t = X λ t + λ 2 dt 2 X λ t − t 1 A 01 U λ t 1 − t 2 A 10 W λ dt 1 t 2 0 0 Davies’ change of variable in the integral kernel � σ � � 0 � � t 1 � λ 2 linear homogeneous = r 1 − 1 t 2 λ 2 jacobian Our change of variable in the integral kernel � σ � λ 2 / 2 � λ 2 q � � � t 1 � � λ 2 / 2 = + for some q ∈ R r 1 − 1 t 2 0 (we will remove the q -asymmetry in a second step) David Taj david.taj@gmail.com
Dynamical Scattering Time T λ Time rescaled interaction picture: W λ, i = X λ − λ − 2 τ W λ τ λ − 2 τ �� W λ, i 2 + q W λ, i d σ dr X λ 2 − q A 01 U λ r A 10 X λ =1+ τ − λ − 2 σ − r λ − 2 σ − r σ + λ 2 ( r 2 + q ) D ( λ, q ) Let T λ ≈ | λ | − ξ , λ ∼ 0 , 0 < ξ < 2 � | λ |� P 0 A 2 P 0 � 1 / 2 � − 1 e.g. T λ = Dynamical Scattering Time Memory effects removal under Markovian Hypotheses �� � τ � ∞ 2 / T 2 0 dr e − ( r 2 ) ξ > 0 ⇒ d σ dr ≈ 0 d σ λ , λ ∼ 0 D ( λ, q ) W λ, i 2 + q ) ≈ W λ, i ξ < 2 ⇒ σ , λ ∼ 0 σ + λ 2 ( r David Taj david.taj@gmail.com
Averaging with Dynamical Scattering Time Averaging among our generators � ∞ dr e − ( r / 2)2 K ( q , T ) = U − r 2 + q A 01 U r A 10 U r 2 − q = U q K (0 , T ) U − q T 2 0 { K ( q , T ) } q ∈ R corresponds to K ♮ (0 , T ) : use gaussian with σ = T λ ! Our Dynamical Time averaged generator �� + ∞ � t 1 � K T = P 0 dt 1 Φ( t 1 ) dt 2 Φ( t 1 ) P 0 −∞ −∞ Φ( t ) = √ δ T ( t ) U − t ( A − A 00 ) U t , 2 π T e − t 2 1 δ T ( t ) = √ 2 T 2 Results under the same Markovian Hypotheses of Davies MAT for L λ = Z 0 + λ A 00 + λ 2 K T λ (0 ≤ t ≤ λ − 2 τ, λ ∼ 0) L λ always well defined ∀ λ � = 0, independently of Z 0 spectrum! If � P 0 � = 1 then exp { L λ t } is a contraction !!! lim τ → + ∞ K | λ | − 1 τ = K ♮ D when ∃ , recovering Davies David Taj david.taj@gmail.com
K T generates a QDS on Operator Algebras Let B , B 0 be Operator Algebras with identity Let P 0 : B → B 0 Conditional Expectation Let P 0 ( i [ H λ , · ]) generate automorphisms ( H λ = H 0 + λ H ′ ) ”The” Quantum Fokker-Planck Equation �� � d ω L † P 0 ( i [ H λ , X ]) + λ 2 i 2 πω P 0 ( � λω � ∂ t X = L λω ) , X − λ 2 2 { P 0 ( � L λ � L λ ) , X } + λ 2 P 0 ( � L λ X � L λ ) Dynamically Averaged Coupling � + ∞ � δ T λ ( t ) e i ω t U t ( H ′ ) � L λω = dt L λω := P 1 ( L λω ) −∞ David Taj david.taj@gmail.com
A Free Quantum Particle in 3D Euclidean Space Inelastically Coupled to a Fermionic Heath Bath H ′ = Q ⊗ Φ Limit Dynamics for H 0 = H S ⊗ 1 + 1 ⊗ H B , h ( t ) = tr [ σ β Φ U t (Φ)] − tr [ σ β Φ] 2 , first order corrected! � � δ T λ ( t ) e i ω t e iH S t Q e − iH S t dt A ω,λ = √ 2 π � d ω s ( ω ) [ A † = − 2 π i √ ω,λ A ω,λ , X ] K T ( λ ) X 2 π � � � � � d ω − 1 A † + A † ˆ +2 π √ h ( ω ) ω,λ A ω,λ , X ω,λ XA ω,λ 2 2 π � dt h ( t )(1 + | t | ǫ ) < ∞ Markovian Hypotheses verified if For H S = ε ( P ) = P 2 2 in 3D, � p | Q | p ′ � = q ( ε p , ε p ′ ), q ∈ S ( R 2 ), there ∃ L s.t. � T ( λ ) K T ( λ ) − L � → 0, and [ Z 0 , L ] = 0. Thermal distributions of observables affiliated to H S are stationary under L if furthermore ˆ h ∈ S ( R ). I found a Pauli Equation with FGR conditionally on ρ S = ρ β ... David Taj david.taj@gmail.com
Summary of this section [1] Taj D., Ann. Henri Poincar´ e Online First (2010) David Taj david.taj@gmail.com
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