Real-time dynamics in Quantum Impurity Systems: A Time-dependent - PowerPoint PPT Presentation

Real-time dynamics in Quantum Impurity Systems: A Time-dependent Numerical Renormalization Group Approach Frithjof B Anders Institut für theoretische Physik, Universität Bremen Concepts in Electron Correlation, Hvar, 30. September 2005 Collaborator: A. Schiller, Hebrew University, Jerusalem, Israel R. Bulla, S. Tornow, University of Augsburg, Germany M. Vojta, University of Karlsruhe, Germany Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.1/18

What is a Quantum-Impurity System (QIS)? Quantum Impurity |α> |γ> (metallic) host bosonic bath quantum-impurity: embedded in a (metallic) host interacting with the environment of non-interacting particles (Bosons/Fermions) Problem: infrared divergence due to local degeneracy Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.2/18

What is a Quantum-Impurity System (QIS)? Quantum Impurity |α> |γ> (metallic) host bosonic bath Examples: transition metal ion Cu, Mn, Ce in a metal two-level system (Qubit) in a bosonic bath Quantum dot coupled to leads donor-acceptor centers of a large bio-molecule · · · Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.2/18

Goal of the Talk Our new Approach to Non-Equilibrium of QIS: based on the non-perturbative NRG uses the complete basis of the many body Fock space takes into account all energy scales describes short and long time scales does not accummulate an error ∝ t as the TD-DMRG ☞ breakthrough in the description of real time dynamics of non-equilibrium quantum systems: Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.3/18

Contents 1. Introduction Modelling of quantum dots Charge transfer in molecules (spin-boson model) 2. Non-equilibrium dynamics Time evolution of quantum systems New approach to quantum impurity problems 3. Results Dissipation and decoherence in a two level system Spin- and charge dynamics in ulta-small quantum dots AF-Kondo model spin precession 4. Summary and outlook Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.4/18

Modelling of a Quantum Dot � ǫ kσ c † H = kσ c kσ kσ � σ d σ + Un d ↑ n d [ E d ( t ) − σH ( t )] d † + ↓ σ � � � c † kσ d σ + d † + V ( t ) σ c kσ kσ Single Impurity Anderson Model (SIAM) charge fluctuation scale: Γ i = V 2 i πρ F infrared problem ☞ low temperature scale: T K ∝ exp( − πU/ 8Γ) Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.5/18

Spin-Boson Model qubit plus environment (Unruh) electron transfer in (bio)-molecules (Marcus, Schulten) | ↑� = | A � , | ↓� = | D � ǫσ z − ∆ � � � ω q b † b † � � H = 2 σ x + q b q + σ z M q q + b q q q � | M q | 2 δ ( ω − ω q ) ∝ 2 παω 1 − s ω s J ( ω ) = c q Leggett et. al. (RMP 1987), Xu and Schulten 1994, Bulla et. al. (2003) · · · Questions: influence of the bosonic spectrum J ( ω ) on the real time dynamics critical slowdown of the charge transfer process for large coupling Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.6/18

Where do we stand in the description of non-equilibrium, dissipation and decoherence in quantum systems? Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.7/18

Non-Equilibrium Dynamics of Quantum Systems quantum dynamics single quantum state: Schrödinger equation i � ∂ t | ψ > = H ( t ) | ψ > ensemble: density operator ρ ( t ) = e − iHt/ � ρ 0 e iHt/ � i � ∂ t ˆ ρ ( t ) = [ H, ˆ ρ ] ; finite size quantum system: only unitary dynamics, no dissipation dissipation and decoherence: infinitly large environment needed Subsystem Size of Subsystem − → 0 Environment Size of environment Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.8/18

NRG Approach to Quantum Impurity Problems H = H imp + H bath + H imp − bath 1. discretizing the bath Hamiltonian on a logarithmic energy mesh (Wilson 1975,Oliveira ) 0 −z −(z+1) −(z+1) −z 0 −Λ −Λ −Λ Λ Λ Λ 0 2. mapping onto a semi-infinite chain impurity t 1 t m−1 t m+1 t N−1 t 0 H m R m,N 3. diagonalizing the Hamiltonian H N +1 using the recursion √ � � � f † N +1 α f Nα + f † H N +1 = Λ H N + ξ Nα Nα f N +1 α α 4. truncate the basis set, go back to step 3 Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.9/18

Novel Many-Body Approach to NEQ of QIS impurity Subsystem t 1 t m−1 t m+1 t N−1 t 0 H m R m,N Environment use the NRG to generate a complete basis | l, e ; m � H m | l � = E m l | l � , l eliminated state e ∈ R m,N � � 1 = | l, e ; m �� l, e ; m | m l,e Puls at t = 0 : H ( t ) = H i Θ( − t ) + H f Θ( t ) operator ˆ O : property of the subsystem S Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.10/18

Novel Many-Body Approach to NEQ of QIS time-dependent NRG (TD-NRG) (FBA, A. Schiller, cond -mat/0505553, PRL 2005) Subsystem calculate ρ red NEQ ( t ) Environment � � � ˆ ρ ( t ) ˆ � α | ˆ � O | α ′ � ρ red O � ( t ) = Tr O = αα ′ ,m ( t ) m,αα ′ e − i ( E α − E α ′ ) t � ρ red � α, e ; m | ρ eq | α ′ , e ; m � αα ′ ,m ( t ) = e Feynman 1972, White 1992, Hofstetter 2000, · · · mimic bath contiuum: use Oliveira’s z -trick evolves towards the new steady state: [ H ( t > 0) , ρ ( ∞ )] = 0 Trace over the environment: dissipation and decoherence! Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.10/18

Spin-Boson Model H = ǫσ z − ∆ � � � ω q b † b † � � 2 σ x + q b q + σ z M q q + b q q q S x = 1 2 ( | ↑��↓ | + | ↓��↑ | ) Decoherence QuBit state 1/2 , α damp =0.1, α=0.1, ε=0, ∆ 0 =0.,ω c =1 N s =150, N z =16, N b =8, N iter =14, T=0.0078125, Λ=2 1 √ ( | ↑� + | ↓� ) s=1.5 2 0.5 s=1.5 (ana.) s=1.0 s=0.8 0.4 s=0.6 exact solution s=0.4 S x (t) s=0.2 0.3 0.2 P ( t ) = e − Γ( t ) 0.1 0 0.001 0.01 0.1 1 10 Leggett et al. , t*T Unruh, Mon Sep 19 11:46:59 2005 Palma et al. , Bulla et al. Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.11/18

Spin-Boson Model H = ǫσ z − ∆ � � � ω q b † b † � � 2 σ x + q b q + σ z M q q + b q q q J ( ω ) = 2 παω 1 − s ω s 0 < ω < ω c ; Ohmic case: s = 1 for c Fixed point: delocalized localized Toulouse Point: 1/2<α<α(∆) 0<α<1/2 c α(∆) < α c oszillatory overdamped α=1/2 Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.11/18

Spin-Boson Model H = ǫσ z − ∆ � � � ω q b † b † � � 2 σ x + q b q + σ z M q q + b q q q 1 =0.2, ε 1 =0, α=0.1, ω c =1, s=1, T=3*10 -8 N s =100, N b =8, N iter =25, N z =16, Λ=2, ∆ 0.5 2 παω 1 − s ω s J ( ω ) = c α=0.1 α=0.1 0.4 α=0.3 α=0.3 α=0.5 α=0.5 0.3 α=0.7 α=0.7 Ohmic Regime: s = 1 α=1.0 α=1.0 0.2 α=1.1 α=1.1 α=1.2 α=1.2 QPT at α c (∆) α=1.3 α=1.3 0.1 α=1.4 α=1.4 S z (t) Toulouse point 0 α = 1 / 2 -0.1 0.4 oszillatory α < 1 / 2 -0.2 0.2 S z (t) overdamped 0 -0.3 α c > α > 1 / 2 -0.2 -0.4 3 4 5 6 7 8 9 10 10 10 10 10 10 10 localize: α > α c -0.5 -1 0 1 2 3 4 5 10 10 10 10 10 10 10 t* ω c Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.11/18

Charge Fluctuation in a Small Quantum Dot H>0 H=0 � ǫ kσ c † H = kσ c kσ µ kσ � σ d σ + Un d ↑ n d [ E d ( t ) − σH ( t )] d † + ↓ Ε d σ � � � c † kσ d σ + d † + V ( t ) σ c kσ time kσ change of E d : change dynamics impurity levels change of mag. field H : spin dynamics change of V : route to new equilibrium Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.12/18

Charge Fluctuation in a Small Quantum Dot H>0 H=0 1 (a) Γ 0 = Γ 1 µ U/ Γ 1 =2 U/ Γ 1 =8 n d (t) 0.8 U/ Γ 1 =4 U/ Γ 1 =10 U/ Γ 1 =6 U/ Γ 1 =12 0.6 U/ Γ 1 =18 Ε d 1 (b) Γ 0 = 0 time n d (t) 0.8 impurity levels 0.6 0.01 0.1 1 10 100 t* Γ 1 Charge relaxation time scale : t ch = 1 / Γ 1 Frithjof Anders, Institut f¨ ur theoretische Physik · Universit¨ at Bremen Concepts in Electron Correlation, Hvar, 30/9/2005 – p.12/18