Dynamics of open quantum systems via resonances Marco Merkli Deptartment of Mathematics and Statistics Memorial University CQIQC-V, August 14, 2013
Open quantum systems • System + Environment models Hamiltonian H = H S + H R + λ V – H S = diag ( E 1 , . . . , E N ) system Hamiltonian (finite-dimensional) – Environment a ‘heat bath’ of non-interacting Bosons (Fermions) at thermal equilibrium ( T = 1 /β > 0) w.r.t. Hamiltonian � ω k a † H R = k a k k ω k dispersion relation – Interaction constant λ , interaction operator � g k a † � � V = G ⊗ k + h . c . k G = G † acts on the system, g k ∈ C is a form factor .
• Schr¨ odinger dynamics ρ tot ( t ) = e − i tH ρ S ⊗ ρ R e i tH ρ S arbitrary system inital state, ρ R thermal reservoir state • Irreversible dynamical effects (in S or R) are visible in the limit of continuous bath modes ( e.g. thermodynamic limit: ∞ volume) Examples: convergence to a final state, decoherence, loss of entanglement, dissipation of energy into the bath • The limits of: continuous modes, large time, small coupling,.... are not independent • Our approach starts off with infinite-volume (true) reservoirs; first we perform continuous mode limit, then we consider t → ∞ , λ → 0,....
The coupled infinite system • Liouville representation (purification, GNS representation): view density matrix as a vector in ‘larger space’ (ancilla) √ p j ψ j ⊗ ψ j ρ = � → Ψ S = � ◦ system state j p j | ψ j �� ψ j | j ◦ ∞ -volume reservoir equilibrium state → Ψ R ◦ Initial system-reservoir state: Ψ 0 = Ψ S ⊗ Ψ R • Dynamics generated by self-adjoint Liouville (super-)operator L Ψ t = e − i tL Ψ 0 , with L = L 0 + λ V , L 0 = L S + L R
Dynamics and spectrum of L = L 0 + λ V • Guiding principle: spectral decomposition “ e − i tL = � j e − i te j P j ” spec ( L S ) spec ( L R ) λ V spec ( L 0 ) spec ( L ) 0 0 • Stationary states ← → Null space (of L 0 , L ) – Non-interacting dynamics: multiple stationary states | ϕ j �� ϕ j | ⊗ ρ R – Interacting dynamics: single stationary state Equilibrum of system + reservoir under coupled dynamics
Resonances Unstable eigenvalues become complex ‘energies’ = resonances = eigenvalues of a spectrally deformed Liouville operator . Spectral deformation: Transformation U ( θ ), θ ∈ C → L ( θ ) = U ( θ ) LU ( θ ) − 1 spec ( L 0 ( θ )) spec ( L 0 ) ∈ C Im θ > 0 0 λ V continuous spectrum spec ( L ( θ )) ε j ( λ ) γ ∼ Im θ 0 • ε 0 = 0 is simple eigenvalue, Im ε j ( λ ) ∝ λ 2 > 0
Resonance representation of dynamics • Spectral decomposition of L ( θ ) e i tL ( θ ) “ = ” � e i t ε j P j + O ( e − γ t ) j • Dynamics of system-reservoir observables A � e i t ε j C j ( A ) + O ( e − γ t ) � A � t = j • Remainder decays more quickly than main term ( γ > Im ε j ) • ε j and C j calculable by perturbation theory in λ • For observables A of system alone, remainder is O ( λ 2 e − γ t ) • Return to equilibrium . The coupled system approaches its joint equilibrium state: lim t →∞ � A � t = C 0 ( A ).
Example: reduced dynamics of system Two spins coupled to common and local reservoirs Spin Hamiltonians B 1 , 2 σ z 1 , 2 1 , 2 ⊗ � g k ( a † Interact.: energy exchange/dephasing: σ x 1 , 2 / σ z k + a k ) • Dynamics of two-spin reduced state ρ t – Thermalization (convergence of diagonal of ρ t ): rate depends on exchange interaction only – Decoherence (decay of off-diagonals): rates depend on local & collective, exchange & dephasing interact. in a correlated way – Entanglement : estimates on entanglement preservation and entanglement death times for class of initial ρ 0
Isolated v.s. overlapping resonances � Energy level spacing of system σ • System-reservoir coupling constant λ � Isolated resonances regime: σ > > λ 2 • < λ 2 Overlapping resonances regime: σ < ε j ( λ ) O ( λ 2 ) Isolated ε j (0) 0 O ( σ ) Starting point: σ fixed, λ = 0 – Stationary system states: ρ S diagonal in energy basis ( H S ) Perturbation: λ � = 0 small – Unique stationary system state: equilibrium ∝ e − β H S – All decay times ∝ 1 /λ 2
O ( σ ) ε j (0) ε j ( σ ) O ( λ 2 ) Overlapping O ( σ 2 /λ 2 ) 0 Starting point: λ fixed, σ = 0 – Stationary system states: ρ S diagonal in the interaction operator eigenbasis ( G ) Perturbation: σ � = 0 small – Unique stationary system state: equilibrium ∝ e − β H S – Emergence of two time-scales ◦ t 1 ∝ 1 /λ 2 : approach of quasi-stationary states ◦ t 2 ∝ λ 2 /σ 2 > > t 1 : quasi-stat. states decay into equilibrium
A donor-acceptor model E 0 σ E E 0 0 0 0 1 1 H = 0 E + σ/ 2 0 + H R + λ 1 0 0 ⊗ ϕ ( g ) 0 0 E − σ/ 2 1 0 0 k ω ( k ) a † 1 k ( g k a † • H R = � k a k and ϕ ( g ) = � k + h . c . ), reservoir √ 2 spatially infinitely extended and at thermal equilibrium. • Donor-acceptor transition induced by environment.
Degenerate acceptor, σ = 0 • Stationary system states are convex span of equilibrium state ρ 1 ∝ e − β H S + O ( λ 2 ) and of ρ 2 ∝ | 0 1 − 1 �� 0 1 − 1 | . • Asymptotic system state ( t → ∞ ) depends on initial state ρ (0) p 0 0 1 + O ( λ 2 ) , ρ ( ∞ ) = 0 2 (1 − p ) α ( p ) 1 0 α ( p ) 2 (1 − p ) where p depends on ρ (0) • Final state is approached on time-scale t 1 ∝ 1 /λ 2 , ρ ( t ) − ρ ( ∞ ) = O ( e − t / t 1 ) ,
< λ 2 Lifted acceptor degeneracy, 0 < σ < • The total system (donor-acceptor + environment) has single stationary state: the coupled equilibrium state. Reduced to donor-acceptor system, it is (modulo O ( λ 2 )) ρ β ∝ e − β H S • Final state is approached on time-scale t 2 ∝ λ 2 /σ 2 ( > > t 1 ) ρ ( t ) − ρ β = O ( e − t / t 2 ) , • Manifold of stationary states for σ = 0 becomes quasi-stationary (decays on time-scale t 2 )
• Arbitrary initial state ρ (0) approaches quasi-stationary manifold, then decays to the unique equilibrium ρ β . ρ β t 1 ρ (0) t 2 quasi-stationary manifold • Evolution of donor-probability, p D ( t ) = [ ρ ( t )] 11 p D ( t ) ( e β ∆ E + 1) − 1 1 thermal ( e β ∆ E + 2) − 1 2 ( e β ∆ E + 1) − 1 1 t t 1 t 2 0 1+ p D (0) • p D (0) ∈ [0 , 1], p D ( t 1 ) = 1 1 e β ∆ E +1 , p D ( t 2 ) = e β ∆ E +2 (equil.) 2
Based on collaborations with G. P. Berman (Los Alamos National Laboratory) F. Borgonovi (Brescia University) I. M. Sigal (University of Toronto) H. Song (Memorial University)
Recommend
More recommend
