Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Quantum impurity systems out of equilibrium:
real-time dynamics
Frithjof B. Anders
Institut für Theoretische Physik · Universität Bremen
Dresden, August 21
Collaborators
A. Schiller, R. Bulla, M. Vojta, S. Tornow,
R. Peters, Th. Pruschke, S. Tautz, R.
Temirov
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outline
1
Introduction
Decay due to an bosonic environment
STM Spectra of a single molecular contact
Occupation dynamics in pulse experiments
2
Theory of non-equilibrium dynamics
Quantum impurity systems
Time-dependent NRG
3
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin- and charge dynamics
4
Outlook
NEQ spectral functions
Conclusion
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Contents
1
Introduction
Decay due to an bosonic environment
STM Spectra of a single molecular contact
Occupation dynamics in pulse experiments
2
Theory of non-equilibrium dynamics
Quantum impurity systems
Time-dependent NRG
3
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin- and charge dynamics
4
Outlook
NEQ spectral functions
Conclusion
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Decay due to an bosonic environment
Decoherence of Qubits due to the environment
QuBits
Environment
Questions:
coherence time
influence of the type of bosonic bath
information loss ⇔ increase of qubit entropy
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Decay due to an bosonic environment
Charge transfer reaction: bosonic bath
Energy and charge transfer: D ∗ + A → D + A∗
Environment matters:
fluctuations determine relaxation rates
relaxation: coupling to molecular vibrations
single-particle versus two-particle transfer
collaborators: S. Tornow and R. Bulla
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
STM Spectra of a single molecular contact
STM Spectra of a single molecular contact
(Temirov et al 2007)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
STM Spectra of a single molecular contact
STM Spectra of a single molecular contact
(Temirov et al 2007)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
STM Spectra of a single molecular contact
STM Spectra of a single molecular contact
(Temirov et al 2007)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
STM Spectra of a single molecular contact
STM Spectra of a single molecular contact
ultimate goal: dI/dV curve at finite bias!
(Temirov et al 2007)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Occupation dynamics in pulse experiments
Occupation dynamics measured in pulse experiments
J.M. Elzerman et al. Nature 430, 431-435
(2004)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Occupation dynamics in pulse experiments
Occupation dynamics measured in pulse experiments
measurement of
J.M. Elzerman et al. Nature 430, 431-435
(2004)
hn̂↓ (t)i =
1
hn̂tot (t)i − hŝz (t)i
2
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Contents
1
Introduction
Decay due to an bosonic environment
STM Spectra of a single molecular contact
Occupation dynamics in pulse experiments
2
Theory of non-equilibrium dynamics
Quantum impurity systems
Time-dependent NRG
3
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin- and charge dynamics
4
Outlook
NEQ spectral functions
Conclusion
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Real-time dynamics of an observable
h
i
hÔi(t) = Tr Ô ρ̂(t)
Equilibrium: single condition ρ̂(t) = ρ̂0 = exp(−βHi )/Z
Nonequilibrium: two conditions: ρ̂0 and Hf
f
ρ̂(t) = e −iH t ρ̂0 e iH
f
t
Calculation of the trace using an energy eigenbasis of Hf
X
hÔi(t) =
hEn |Ô|Em ihEm |ρ̂0 |En ie −i(Em −En )t
n,m
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Real-time dynamics of an observable
h
i
hÔi(t) = Tr Ô ρ̂(t)
Equilibrium: single condition ρ̂(t) = ρ̂0 = exp(−βHi )/Z
Nonequilibrium: two conditions: ρ̂0 and Hf
f
ρ̂(t) = e −iH t ρ̂0 e iH
f
t
Calculation of the trace using an energy eigenbasis of Hf
X
hÔi(t) =
hEn |Ô|Em ihEm |ρ̂0 |En ie −i(Em −En )t
n,m
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Real-time dynamics of an observable
h
i
hÔi(t) = Tr Ô ρ̂(t)
Equilibrium: single condition ρ̂(t) = ρ̂0 = exp(−βHi )/Z
Nonequilibrium: two conditions: ρ̂0 and Hf
f
ρ̂(t) = e −iH t ρ̂0 e iH
f
t
Calculation of the trace using an energy eigenbasis of Hf
X
hÔi(t) =
hEn |Ô|Em ihEm |ρ̂0 |En ie −i(Em −En )t
n,m
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Real-time dynamics of an observable
h
i
hÔi(t) = Tr Ô ρ̂(t)
Equilibrium: single condition ρ̂(t) = ρ̂0 = exp(−βHi )/Z
Nonequilibrium: two conditions: ρ̂0 and Hf
f
ρ̂(t) = e −iH t ρ̂0 e iH
f
t
Calculation of the trace using an energy eigenbasis of Hf
X
hÔi(t) =
hEn |Ô|Em ihEm |ρ̂0 |En ie −i(Em −En )t
n,m
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Real-time dynamics of an observable
h
i
hÔi(t) = Tr Ô ρ̂(t)
Equilibrium: single condition ρ̂(t) = ρ̂0 = exp(−βHi )/Z
Nonequilibrium: two conditions: ρ̂0 and Hf
f
ρ̂(t) = e −iH t ρ̂0 e iH
f
t
Calculation of the trace using an energy eigenbasis of Hf
X
hÔi(t) =
hEn |Ô|Em ihEm |ρ̂0 |En ie −i(Em −En )t
n,m
Question
Can we obtain such an eigenbasis for interacting systems?
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
Solution of quantum impurity systems
Bath continuum
Impurity
ξ1
Wilson’s NRG (1975)
impurity: arbitrary complex local interactions
discretization of the bath continuum
switching on iteratively the couplings ξm ∝ Λ−m/2
temperature: Tm ∝ Λ−m/2
RG: elimination of the high energy states
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
Solution of quantum impurity systems
Impurity
1
ξ1
2
ξ
2
3
ξ
3
N
ξ
Ν
∼Λ
Wilson’s NRG (1975)
impurity: arbitrary complex local interactions
discretization of the bath continuum
switching on iteratively the couplings ξm ∝ Λ−m/2
temperature: Tm ∝ Λ−m/2
RG: elimination of the high energy states
−Ν/2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
Solution of quantum impurity systems
Impurity
1
ξ1
2
ξ
2
3
ξ
3
N
ξ
Ν
∼Λ
Wilson’s NRG (1975)
impurity: arbitrary complex local interactions
discretization of the bath continuum
switching on iteratively the couplings ξm ∝ Λ−m/2
temperature: Tm ∝ Λ−m/2
RG: elimination of the high energy states
−Ν/2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
Solution of quantum impurity systems
Impurity
1
ξ1
2
ξ
2
3
ξ
3
N
ξ
Ν
∼Λ
Wilson’s NRG (1975)
impurity: arbitrary complex local interactions
discretization of the bath continuum
switching on iteratively the couplings ξm ∝ Λ−m/2
temperature: Tm ∝ Λ−m/2
RG: elimination of the high energy states
−Ν/2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
Solution of quantum impurity systems
Impurity
1
ξ1
2
ξ
2
3
ξ
3
N
ξ
Ν
∼Λ
Wilson’s NRG (1975)
impurity: arbitrary complex local interactions
discretization of the bath continuum
switching on iteratively the couplings ξm ∝ Λ−m/2
temperature: Tm ∝ Λ−m/2
RG: elimination of the high energy states
−Ν/2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
Solution of quantum impurity systems
Impurity
1
ξ1
2
ξ
2
3
ξ
3
N
ξ
Ν
∼Λ
Wilson’s NRG (1975)
impurity: arbitrary complex local interactions
discretization of the bath continuum
switching on iteratively the couplings ξm ∝ Λ−m/2
temperature: Tm ∝ Λ−m/2
RG: elimination of the high energy states
−Ν/2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
Solution of quantum impurity systems
Impurity
1
ξ1
2
ξ
2
3
ξ
3
N
ξ
Ν
∼Λ
Wilson’s NRG (1975)
impurity: arbitrary complex local interactions
discretization of the bath continuum
switching on iteratively the couplings ξm ∝ Λ−m/2
temperature: Tm ∝ Λ−m/2
RG: elimination of the high energy states
−Ν/2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
The challenge
Non-equilibrium dynamics in quantum impurity systems
The problem
evaluation of all energy scales
avoid overcounting
NEQ-NRG spectral functions: Costi, PRB 55, 3003 (1997)
The solution
complete NRG basis set of the Wilson chain
FBA and A Schiller, PRL 95, 196801 (2005), Phys. Rev. B 74, 245113 (2006)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
The challenge
Non-equilibrium dynamics in quantum impurity systems
The problem
evaluation of all energy scales
avoid overcounting
NEQ-NRG spectral functions: Costi, PRB 55, 3003 (1997)
The solution
complete NRG basis set of the Wilson chain
FBA and A Schiller, PRL 95, 196801 (2005), Phys. Rev. B 74, 245113 (2006)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
The challenge
Non-equilibrium dynamics in quantum impurity systems
The problem
evaluation of all energy scales
avoid overcounting
NEQ-NRG spectral functions: Costi, PRB 55, 3003 (1997)
The solution
complete NRG basis set of the Wilson chain
FBA and A Schiller, PRL 95, 196801 (2005), Phys. Rev. B 74, 245113 (2006)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
The complete basis set of the NRG chain
Impurity
1
2
3
N
Environment e
|l,e,1>
|e>
|l,e,2>
|l,e,3>
complete basis: {|ei} = {|αimp , α1 , α2 , α3 , α4 , · · · , αN i}
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
The complete basis set of the NRG chain
Impurity
1
ξ1
2
3
N
Environment e
|k’,e,1>
|l,e,1>
|e>
|k,e,1>
|l,e,2>
|l,e,3>
complete basis: {|ei} = {|k, e; 1i}
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
The complete basis set of the NRG chain
Impurity
1
ξ1
2
ξ
2
3
N
Environment e
|l,e,2>
|e>
|k,e,2>
|l,e,2>
|l,e,3>
complete basis: {|ei} = {|k, e; 2i} + {|l, e; 2i}
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
The complete basis set of the NRG chain
Impurity
1
ξ1
2
ξ
2
3
ξ
3
N
e
|l,e,2>
|e>
|l,e,3>
|k,e,3>
complete basis: {|ei} = {|k, e; 3i} +
P3
m=2 {|l, e; mi}
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Quantum impurity systems
The complete basis set of the NRG chain
Impurity
1
ξ1
2
ξ
2
3
ξ
3
N
ξ
Ν
|l,e,2>
|e>
|l,e,3>
|l,e,N>
complete basis: {|ei} =
PN
m=2 {|l, e; mi}
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Time-dependent NRG
Time-dependent NRG
hÔi(t) =
X
hEn |Ô|Em ihEm |ρ̂0 |En ie −i(Em −En )t
n,m
Ô: local operator, diagonal in e
reduced density matrix
(Feynman 72, DMRG: White 92, NRG: Hofstetter 2000)
ρred
ll 0 (m) =
X
hl, e; m|ρ̂0 |l 0 , e; mi
e
RG upside down: discarded states contain the information on
the time evolution
FBA and A. Schiller PRL 95, 196801 (2005), PRB 74, 245113
(2006)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Time-dependent NRG
Time-dependent NRG
0
discarded
X l or l X
hl|Ô|l 0 ie i(El −El 0 )t ρred
hÔi(t) =
l 0 l (m)
m
l,l
Ô: local operator, diagonal in e
reduced density matrix
(Feynman 72, DMRG: White 92, NRG: Hofstetter 2000)
ρred
ll 0 (m) =
X
hl, e; m|ρ̂0 |l 0 , e; mi
e
RG upside down: discarded states contain the information on
the time evolution
FBA and A. Schiller PRL 95, 196801 (2005), PRB 74, 245113
(2006)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Time-dependent NRG
Time-dependent NRG
0
discarded
X l or l X
hl|Ô|l 0 ie i(El −El 0 )t ρred
hÔi(t) =
l 0 l (m)
m
l,l
Ô: local operator, diagonal in e
reduced density matrix
(Feynman 72, DMRG: White 92, NRG: Hofstetter 2000)
ρred
ll 0 (m) =
X
hl, e; m|ρ̂0 |l 0 , e; mi
e
RG upside down: discarded states contain the information on
the time evolution
FBA and A. Schiller PRL 95, 196801 (2005), PRB 74, 245113
(2006)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Time-dependent NRG
Discussion of the method
Advantage
resolving the contradiction: RG and including all energy scale
no accumulated error in time in contrary to the td-DMRG
exponentially long time scales accessible (up to t ∗ T ≈ 1)
Impact: sum rule conserving Green functions
Peters, Pruschke, FBA, Phys. Rev. B 74, 245114 (2006)
Weichselbaum, von Delft, PRL 99, 076402 (2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Time-dependent NRG
Discussion of the method
Advantage
resolving the contradiction: RG and including all energy scale
no accumulated error in time in contrary to the td-DMRG
exponentially long time scales accessible (up to t ∗ T ≈ 1)
Impact: sum rule conserving Green functions
Peters, Pruschke, FBA, Phys. Rev. B 74, 245114 (2006)
Weichselbaum, von Delft, PRL 99, 076402 (2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Contents
1
Introduction
Decay due to an bosonic environment
STM Spectra of a single molecular contact
Occupation dynamics in pulse experiments
2
Theory of non-equilibrium dynamics
Quantum impurity systems
Time-dependent NRG
3
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin- and charge dynamics
4
Outlook
NEQ spectral functions
Conclusion
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: definition
∆
ε
H =
∆
ε
σz − σx
2X 2
+
ωq bq† bq
q
+σz
X
Mq (bq† + bq )
q
two-level system, Spin, Qubit
P
bosonic bath continuum: Hb = q ωq bq† bq
interaction between spin and bath
X
J(ω) = π
|Mq |2 δ(ω − ωq ) = 2παωc1−s ω s for 0 < ω ≤ wc
q
(Leggett et al. RMP 1987)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: definition
∆
ε
H =
∆
ε
σz − σx
2X 2
+
ωq bq† bq
q
+σz
X
Mq (bq† + bq )
q
two-level system, Spin, Qubit
P
bosonic bath continuum: Hb = q ωq bq† bq
interaction between spin and bath
X
J(ω) = π
|Mq |2 δ(ω − ωq ) = 2παωc1−s ω s for 0 < ω ≤ wc
q
(Leggett et al. RMP 1987)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: definition
∆
ε
H =
∆
ε
σz − σx
2X 2
+
ωq bq† bq
q
+σz
X
Mq (bq† + bq )
q
two-level system, Spin, Qubit
P
bosonic bath continuum: Hb = q ωq bq† bq
interaction between spin and bath
X
J(ω) = π
|Mq |2 δ(ω − ωq ) = 2παωc1−s ω s for 0 < ω ≤ wc
q
(Leggett et al. RMP 1987)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: decoherence
Analytical solution for ∆ = ε = 0
√
pure state: |si = (|0i + |1i)/ 2
red. density matrix of the pure state ρloc = |sihs|
1 1 1
1 1 0
ρ =
=⇒
loc
2 1 1
2 0 1
+ Decoherence: ρ01 , ρ01 decays, thermodynamic state emerges
analytical solution (Unruh 95, Palma et al. 96 )
ρ10 (t) = h1|TrBoson [ρ(t)] |0i = e −Γ(t) ρ10 (0)
Z
ω 1 − cos(ωt)
1 ∞
Γ(t) =
dω J(ω) coth
π 0
2T
ω2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: decoherence
Analytical solution for ∆ = ε = 0
√
pure state: |si = (|0i + |1i)/ 2
red. density matrix of the pure state ρloc = |sihs|
1 1 1
1 1 0
ρ =
=⇒
loc
2 1 1
2 0 1
+ Decoherence: ρ01 , ρ01 decays, thermodynamic state emerges
analytical solution (Unruh 95, Palma et al. 96 )
ρ10 (t) = h1|TrBoson [ρ(t)] |0i = e −Γ(t) ρ10 (0)
Z
ω 1 − cos(ωt)
1 ∞
Γ(t) =
dω J(ω) coth
π 0
2T
ω2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: decoherence
Analytical solution for ∆ = ε = 0
√
pure state: |si = (|0i + |1i)/ 2
red. density matrix of the pure state ρloc = |sihs|
1 1 1
1 1 0
ρ =
=⇒
loc
2 1 1
2 0 1
+ Decoherence: ρ01 , ρ01 decays, thermodynamic state emerges
analytical solution (Unruh 95, Palma et al. 96 )
ρ10 (t) = h1|TrBoson [ρ(t)] |0i = e −Γ(t) ρ10 (0)
Z
ω 1 − cos(ωt)
1 ∞
Γ(t) =
dω J(ω) coth
π 0
2T
ω2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: decoherence
Analytical solution for ∆ = ε = 0
√
pure state: |si = (|0i + |1i)/ 2
red. density matrix of the pure state ρloc = |sihs|
1 1 1
1 1 0
ρ =
=⇒
loc
2 1 1
2 0 1
+ Decoherence: ρ01 , ρ01 decays, thermodynamic state emerges
analytical solution (Unruh 95, Palma et al. 96 )
ρ10 (t) = h1|TrBoson [ρ(t)] |0i = e −Γ(t) ρ10 (0)
Z
ω 1 − cos(ωt)
1 ∞
Γ(t) =
dω J(ω) coth
π 0
2T
ω2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: decoherence
TD-NRG
s=1.5
s=1.0
s=0.8
s=0.6
s=0.4
s=0.2
1
ρ01(t)/ρ01(0)
0,8
0,6
0,4
0,2
0
0,001
0,01
0,1
1
10
t*T
analytical exact solution and TD-NRG: excellent agreement
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin-boson model: decoherence
TD-NRG plus analytic solution
s=1.5
s=1.0
s=0.8
s=0.6
s=0.4
s=0.2
1
ρ01(t)/ρ01(0)
0.8
0.6
0.4
0.2
0
0.001
0.01
0.1
1
10
t*T
analytical exact solution and TD-NRG: excellent agreement
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Benchmark: decoherence of a QuBit: spin boson model
TD-NRG: spin decay in the ohmic regime
1
∆ =0.2, s=1, T=3*10
0.5
0.4
0.3
-8
α=0.01
α=0.1
0.2
Sz(t)
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
20
40
t*ωc
60
80
100
decaying, oscillatory solution α < 0.5
damped solution for 0.5 ≤ α < αc ≈ 1
delocalized phase α < αc , ∆r > 0
localized phase α ≥ αc
(Leggett et al. RMP 1987)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Benchmark: decoherence of a QuBit: spin boson model
TD-NRG: spin decay in the ohmic regime
1
∆ =0.2, s=1, T=3*10
0.5
0.4
0.3
-8
α=0.01
α=0.1
α=0.5
α=0.7
0.2
Sz(t)
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
20
40
t*ωc
60
80
100
decaying, oscillatory solution α < 0.5
damped solution for 0.5 ≤ α < αc ≈ 1
delocalized phase α < αc , ∆r > 0
localized phase α ≥ αc
(Leggett et al. RMP 1987)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Benchmark: decoherence of a QuBit: spin boson model
TD-NRG: spin decay in the ohmic regime
1
∆ =0.2, s=1, T=3*10
0.5
0.4
0.3
-8
α=0.01
α=0.1
α=0.5
α=0.7
0.2
Sz(t)
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
0
20
40
t*ωc
60
80
100
decaying, oscillatory solution α < 0.5
damped solution for 0.5 ≤ α < αc ≈ 1
delocalized phase α < αc , ∆r > 0
localized phase α ≥ αc
(Leggett et al. RMP 1987)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Benchmark: decoherence of a QuBit: spin boson model
TD-NRG: spin decay in the ohmic regime
1
∆ =0.2, s=1, T=3*10
0.5
0.4
0.3
0.2
Sz(t)
0.1
-8
α=0.01
α=0.1
α=0.5
α=0.7
α=1.2
α=1.3
α=1.4
0
-0.1
0.4
Sz(t)
-0.2
-0.3
0
-0.2
-0.4
-0.5
0
0.2
3
10
20
40
t*ωc
60
4
10
5
10
6
10
80
7
10
8
10
9
10
100
decaying, oscillatory solution α < 0.5
damped solution for 0.5 ≤ α < αc ≈ 1
delocalized phase α < αc , ∆r > 0
localized phase α ≥ αc
(Leggett et al. RMP 1987)
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
TD-NRG: surprise in the sub-ohmic regime s < 1
s=0.1, T=10
-5
α=0.001<αc
αc=0.00825
α=0.012>αc
α=0.014>αc
0.4
Sz(t)
0.2
0
-0.2
-0.4
0
10
1
10
2
10
t*ωc
3
10
4
10
QPC: αc ≈ s∆ + O(s 2 ) (Vojta, Bulla, PRL 2005)
surprise: oscillatory solution even in the localized phase
α > αc (FBA, Vojta, Bulla, PRL 2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
TD-NRG: surprise in the sub-ohmic regime s < 1
s=0.1, T=10
-5
α=0.001<αc
αc=0.00825
α=0.012>αc
α=0.014>αc
0.4
Sz(t)
0.2
0
-0.2
-0.4
0
10
1
10
2
10
t*ωc
3
10
4
10
QPC: αc ≈ s∆ + O(s 2 ) (Vojta, Bulla, PRL 2005)
surprise: oscillatory solution even in the localized phase
α > αc (FBA, Vojta, Bulla, PRL 2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Spin- and charge dynamics
J.M. Elzerman et al. Nature 430, 431(2004)
J.M. Elzerman et al. Nature 430, 431(2004)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Spin- and charge dynamics
TD-NRG:
0.5
0.5
T1
0.4
0.4
ndown(t)
0.3
0.3
0.2
10
5
20
15
H/Γ
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
t*Γ
hn̂↓ (t)i =
1
hn̂tot (t)i − hŝz (t)i
2
J.M. Elzerman et al. Nature 430, 431(2004)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Spin- and charge dynamics
TD-NRG:
nd(t)
1
0.8
H=5
H=10
H=15
H=20
0.6
0.5
0
0.5
1
1.5
0
0.5
1
1.5
2
2.5
3
3.5
4
2
2.5
3
3.5
4
Sz(t)
0.4
0.3
0.2
0.1
0
t*Γ
hn̂↓ (t)i =
1
hn̂tot (t)i − hŝz (t)i
2
J.M. Elzerman et al. Nature 430, 431(2004)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Spin- and charge dynamics
level scheme of the Q-dot
H>0
H=0
µ
Εd
t=0
time
at t = 0:
J.M. Elzerman et al. Nature 430,
431(2004)
switching off H: H = 0 for
t>0
Ed = µB H/2 → Ed = U/2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Spin- and charge dynamics
Charge dynamics of a quantum dot
nd(t)
1
(a) Γ0 = Γ1
U/Γ1=2
U/Γ1=4
U/Γ1=6
0.8
0.6
nd(t)
1
U/Γ1=8
U/Γ1=10
U/Γ1=12
U/Γ1=18
(b) Γ0 = 0
0.8
0.6
0.01
0.1
1
10
t*Γ1
time scale for charge relaxation
tch = 1/Γ1 = 1/(πρt 2 )
100
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Spin- and charge dynamics
spin dynamics of a quantum dot
U/Γ1=2
U/Γ1=4
U/Γ1=6
sz(t)
0.2
U/Γ1=8
U/Γ1=10
U/Γ1=12
U/Γ1=18
0.1
(c) Γ0 = Γ1
sz(t)
0
0.2
0.1
0 -2
10
(d) Γ0 = 0
10
-1
10
0
t*Γ1
10
1
10
time scale for spin relaxation
tsp ∝ 1/TK ∝ exp(1/ρJ)
2
10
3
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Spin- and charge dynamics
spin dynamics of a quantum dot
U/Γ1=2
U/Γ1=4
U/Γ1=6
U/Γ1=8
U/Γ1=10
U/Γ1=12
U/Γ1=18
sz(t)
0.2
0.1
(a) Γ0 = Γ1
sz(t)
0
0.2
0.1
0 -4
10
(b) Γ0 = 0
10
-3
10
-2
10
-1
t/tK
10
0
time scale for spin relaxation
tsp ∝ 1/TK ∝ exp(1/ρJ)
10
1
10
2
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Contents
1
Introduction
Decay due to an bosonic environment
STM Spectra of a single molecular contact
Occupation dynamics in pulse experiments
2
Theory of non-equilibrium dynamics
Quantum impurity systems
Time-dependent NRG
3
Results: spin and charge dynamics
Benchmark: decoherence of a QuBit: spin boson model
Spin- and charge dynamics
4
Outlook
NEQ spectral functions
Conclusion
Outlook
Introduction
Theory of non-equilibrium dynamics
TD-NRG: transient currents
Equilibrium:
µL
µR
for t < 0:
bias µL − µR = 0
Results: spin and charge dynamics
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
TD-NRG: transient currents
Equilibrium:
µL
Non-Equilibrium
µR
for t < 0:
bias µL − µR = 0
µL
µR
for t > 0:
finite bias V = µL − µR
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Quantum dot (SIAM) transient currents
Quantum Dot (SIAM)
-3
-5
V=10 , T=2.5*10 , Λ=3
Ed=-2, U=4
Ed=-3, U=6
Ed=-4, U=8
1.5
1
0.5
0
20
100
50
150
200
300
250
350
400
t*Γ
2
<I(t)>/V in (e /h)
2
<I(t)>/V in (e /h)
2
tcurr ∝ 1/TK
1.5
1
0.5
0
0
1
2
3
4
5
t/tcurr
transient time tcurr
grows with U
6
7
8
9
10
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Outlook
Quantum dot (SIAM) transient currents
Quantum Dot (SIAM)
-3
-5
V=10 , T=2.5*10 , Λ=3
Ed=-2, U=4
Ed=-3, U=6
Ed=-4, U=8
1.5
1
0.5
0
20
100
50
150
200
300
250
400
350
t*Γ
2
<I(t)>/V in (e /h)
2
<I(t)>/V in (e /h)
2
tcurr ∝ 1/TK
1.5
0.1
1/tcurr
1
0.5
0
tcurr
TK
0.01
0
1
2
3
4
5
t/tcurr
transient time tcurr
grows with U
4
7
6
U
5
6
7
8
8
9
10
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
NEQ spectral functions
Challenge: exact dI /dV curve
scattering states Bethe Ansatz?
scattering states NRG?
Keldysh Green function?
Warm-up problem: initial ρ0 plus Hf =⇒ steady state NEQ
Green function (see also Costi 97)
r
GA,B
(t) = 0lim −iTr ρ0 (Hi )[A(t + t 0 ), B(t 0 )]s Θ(t)
t →∞
m, m0 contributions needed =⇒ two reduced density matrices
novel sum-rule conserving algorithm (FBA 2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
NEQ spectral functions
Challenge: exact dI /dV curve
scattering states Bethe Ansatz?
scattering states NRG?
Keldysh Green function?
Warm-up problem: initial ρ0 plus Hf =⇒ steady state NEQ
Green function (see also Costi 97)
r
GA,B
(t) = 0lim −iTr ρ0 (Hi )[A(t + t 0 ), B(t 0 )]s Θ(t)
t →∞
m, m0 contributions needed =⇒ two reduced density matrices
novel sum-rule conserving algorithm (FBA 2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
NEQ spectral functions
Challenge: exact dI /dV curve
scattering states Bethe Ansatz?
scattering states NRG?
Keldysh Green function?
Warm-up problem: initial ρ0 plus Hf =⇒ steady state NEQ
Green function (see also Costi 97)
r
GA,B
(t) = 0lim −iTr ρ0 (Hi )[A(t + t 0 ), B(t 0 )]s Θ(t)
t →∞
m, m0 contributions needed =⇒ two reduced density matrices
novel sum-rule conserving algorithm (FBA 2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
NEQ spectral functions
Challenge: exact dI /dV curve
scattering states Bethe Ansatz?
scattering states NRG?
Keldysh Green function?
Warm-up problem: initial ρ0 plus Hf =⇒ steady state NEQ
Green function (see also Costi 97)
r
GA,B
(t) = 0lim −iTr ρ0 (Hi )[A(t + t 0 ), B(t 0 )]s Θ(t)
t →∞
m, m0 contributions needed =⇒ two reduced density matrices
novel sum-rule conserving algorithm (FBA 2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
NEQ spectral functions
Challenge: exact dI /dV curve
scattering states Bethe Ansatz?
scattering states NRG?
Keldysh Green function?
Warm-up problem: initial ρ0 plus Hf =⇒ steady state NEQ
Green function (see also Costi 97)
r
GA,B
(t) = 0lim −iTr ρ0 (Hi )[A(t + t 0 ), B(t 0 )]s Θ(t)
t →∞
m, m0 contributions needed =⇒ two reduced density matrices
novel sum-rule conserving algorithm (FBA 2007)
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
NEQ spectral functions
Reconstruction of the equilibrium spectral function
0.4
0.4
U=1
U=2
U=4
U=6
U=8
U=10
0.3 0.3
ρ(ω)
0.2
0.2
0.1
0.1
0
-10
0
0
ω/Γ
-8
-6
-4
-2
0
ω/Γ
2
4
6
8
10
Start with U = 0 and evolve to finite U using the TD-NRG
r
GA,B
(t, U) = 0lim −iTr ρ0 (U = 0)[A(t + t 0 ), B(t 0 )]s Θ(t)
t →∞
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
NEQ spectral functions
Reconstruction of the equilibrium spectral function
0.4
Ed=-3,U=6
Ed=-1,U=6
Ed=-1,U=6
Ed= 0,U=6
ρ(ω)
0.3
0.2
0.1
0
-4
-3
-2
-1
0
ω/Γ
1
2
3
4
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Conclusion
Conclusion
The time-dependent NRG
novel method to investigate real-time dynamics in
quantum-impurity systems
based on the NRG
complete many-body basis set
excellent agreement between analytic and numerics
Applicable to any quantum impurity problem
Question and Outlook
What can be learn about the spin and charge dynamics in
more complex systems
scattering states NRG for steady-state currents?
application to charge transfer reaction in bio-molecules
Outlook
Introduction
Theory of non-equilibrium dynamics
Results: spin and charge dynamics
Conclusion
Conclusion
The time-dependent NRG
novel method to investigate real-time dynamics in
quantum-impurity systems
based on the NRG
complete many-body basis set
excellent agreement between analytic and numerics
Applicable to any quantum impurity problem
Question and Outlook
What can be learn about the spin and charge dynamics in
more complex systems
scattering states NRG for steady-state currents?
application to charge transfer reaction in bio-molecules
Outlook