Quantum impurity systems out of equilibrium: real-time...

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 fur Theoretische Physik · Universitat Bremen

Dresden, August 21

Collaborators A. Schiller, R. Bulla, M. Vojta, S. Tornow,R. Peters, Th. Pruschke, S. Tautz, R.Temirov


Outline

1 IntroductionDecay due to an bosonic environmentSTM Spectra of a single molecular contactOccupation dynamics in pulse experiments

2 Theory of non-equilibrium dynamicsQuantum impurity systemsTime-dependent NRG

3 Results: spin and charge dynamicsBenchmark: decoherence of a QuBit: spin boson modelSpin- and charge dynamics

4 OutlookNEQ spectral functionsConclusion


Contents






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


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


STM Spectra of a single molecular contact

STM Spectra of a single molecular contact (Temirov et al 2007)

ultimate goal: dI/dV curve at finite bias!














Occupation dynamics in pulse experiments

Occupation dynamics measured in pulse experiments

J.M. Elzerman et al. Nature 430, 431-435

(2004)


Occupation dynamics in pulse experiments

Occupation dynamics measured in pulse experiments

J.M. Elzerman et al. Nature 430, 431-435

(2004)

measurement of

〈n↓(t)〉 =1

2〈ntot(t)〉 − 〈sz(t)〉


Contents






Real-time dynamics of an observable

〈O〉(t) = Tr[Oρ(t)

]Equilibrium: single condition ρ(t) = ρ0 = exp(−βHi )/Z

Nonequilibrium: two conditions: ρ0 and Hf

ρ(t) = e−iHf t ρ0eiHf t

Calculation of the trace using an energy eigenbasis of Hf

〈O〉(t) =∑n,m

〈En|O|Em〉〈Em|ρ0|En〉e−i(Em−En)t

Question

Can we obtain such an eigenbasis for interacting systems?








〈O〉(t) =∑n,m


Question









〈O〉(t) =∑n,m


Question









〈O〉(t) =∑n,m


Question









〈O〉(t) =∑n,m


Question



Quantum impurity systems

Solution of quantum impurity systems

Impurity Bath continuumξ 1

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




Impurity

ξ3

ξ2

ξ 1

321 N

ξΝ ∼Λ −Ν/2










Impurity

ξ3

ξ2

ξ 1

321 N

ξΝ ∼Λ −Ν/2










Impurity

ξ3

ξ2

ξ 1

321 N

ξΝ ∼Λ −Ν/2










Impurity

ξ3

ξ2

ξ 1

321 N

ξΝ ∼Λ −Ν/2










Impurity

ξ3

ξ2

ξ 1

321 N

ξΝ ∼Λ −Ν/2










Impurity

ξ3

ξ2

ξ 1

321 N

ξΝ ∼Λ −Ν/2









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 chainFBA and A Schiller, PRL 95, 196801 (2005), Phys. Rev. B 74, 245113 (2006)



The challenge


The problem


avoid overcounting


The solution




The challenge


The problem


avoid overcounting


The solution




The complete basis set of the NRG chain

Impurity

eEnvironment

321 N

|l,e,1>

|l,e,2>

|l,e,3>

|e>

complete basis: {|e〉} = {|αimp, α1, α2, α3, α4, · · · , αN〉}




Impurity

eEnvironmentξ 1

321 N

|l,e,1>

|l,e,2>

|l,e,3>

|e>

|k,e,1>

|k’,e,1>

complete basis: {|e〉} = {|k, e; 1〉}




Impurity

eEnvironmentξ2ξ 1

321 N

|l,e,2>

|l,e,2>

|l,e,3>

|e>

|k,e,2>

complete basis: {|e〉} = {|k, e; 2〉}+ {|l , e; 2〉}




Impurity

eξ2ξ 1

321 N

ξ3

|e>

|l,e,3>

|l,e,2>

|k,e,3>

complete basis: {|e〉} = {|k, e; 3〉}+∑3

m=2{|l , e;m〉}




Impurity

ξ2ξ 1

ξ3

321 N

ξΝ

|e>

|l,e,3>

|l,e,2>

|l,e,N>

complete basis: {|e〉} =∑N

m=2{|l , e;m〉}


Time-dependent NRG

Time-dependent NRG

〈O〉(t) =∑n,m


O: local operator, diagonal in e

reduced density matrix (Feynman 72, DMRG: White 92, NRG: Hofstetter 2000)

ρredll ′ (m) =

∑e

〈l , e;m|ρ0|l ′, e;m〉

RG upside down: discarded states contain the information onthe time evolution

FBA and A. Schiller PRL 95, 196801 (2005), PRB 74, 245113(2006)


Time-dependent NRG

Time-dependent NRG

〈O〉(t) =∑m

l or l ′ discarded∑l ,l

〈l |O|l ′〉e i(El−El′ )tρredl ′l (m)



ρredll ′ (m) =

∑e

〈l , e;m|ρ0|l ′, e;m〉




Time-dependent NRG

Time-dependent NRG

〈O〉(t) =∑m

l or l ′ discarded∑l ,l

〈l |O|l ′〉e i(El−El′ )tρredl ′l (m)



ρredll ′ (m) =

∑e

〈l , e;m|ρ0|l ′, e;m〉




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)


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)


Contents






Benchmark: decoherence of a QuBit: spin boson model

Spin-boson model: definition

ε

∆ H =ε

2σz −

∆

2σx

+∑q

ωqb†qbq

+σz

∑q

Mq(b†q + bq)

two-level system, Spin, Qubit

bosonic bath continuum: Hb =∑

q ωqb†qbq

interaction between spin and bath

J(ω) = π∑q

|Mq|2δ(ω − ωq) = 2παω1−sc ωs for 0 < ω ≤ wc

(Leggett et al. RMP 1987)




ε

∆ H =ε

2σz −

∆

2σx

+∑q

ωqb†qbq

+σz

∑q

Mq(b†q + bq)



q ωqb†qbq


J(ω) = π∑q






ε

∆ H =ε

2σz −

∆

2σx

+∑q

ωqb†qbq

+σz

∑q

Mq(b†q + bq)



q ωqb†qbq


J(ω) = π∑q





Spin-boson model: decoherence

Analytical solution for ∆ = ε = 0

pure state: |s〉 = (|0〉+ |1〉)/√

2

red. density matrix of the pure state ρloc = |s〉〈s|

ρloc

=1

2

(1 11 1

)=⇒ 1

2

(1 00 1

)

+ Decoherence: ρ01, ρ01 decays, thermodynamic state emerges

analytical solution (Unruh 95, Palma et al. 96 )

ρ10(t) = 〈1|TrBoson [ρ(t)] |0〉 = e−Γ(t)ρ10(0)

Γ(t) =1

π

∫ ∞

0dω J(ω) coth

( ω

2T

) 1− cos(ωt)

ω2





pure state: |s〉 = (|0〉+ |1〉)/√

2


ρloc

=1

2

(1 11 1

)=⇒ 1

2

(1 00 1

)




Γ(t) =1

π

∫ ∞

0dω J(ω) coth

( ω

2T

) 1− cos(ωt)

ω2





pure state: |s〉 = (|0〉+ |1〉)/√

2


ρloc

=1

2

(1 11 1

)=⇒ 1

2

(1 00 1

)




Γ(t) =1

π

∫ ∞

0dω J(ω) coth

( ω

2T

) 1− cos(ωt)

ω2





pure state: |s〉 = (|0〉+ |1〉)/√

2


ρloc

=1

2

(1 11 1

)=⇒ 1

2

(1 00 1

)




Γ(t) =1

π

∫ ∞

0dω J(ω) coth

( ω

2T

) 1− cos(ωt)

ω2




TD-NRG

0,001 0,01 0,1 1 10t*T

0

0,2

0,4

0,6

0,8

1

ρ 01(t

)/ρ 01

(0)

s=1.5s=1.0s=0.8s=0.6s=0.4s=0.2

analytical exact solution and TD-NRG: excellent agreement




TD-NRG plus analytic solution

0.001 0.01 0.1 1 10t*T

0

0.2

0.4

0.6

0.8

1

ρ 01(t

)/ρ 01

(0)

s=1.5s=1.0s=0.8s=0.6s=0.4s=0.2

analytical exact solution and TD-NRG: excellent agreement



TD-NRG: spin decay in the ohmic regime

0 20 40 60 80 100t*ωc

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

S z(t)

α=0.01α=0.1

∆1=0.2, s=1, T=3*10-8

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)




0 20 40 60 80 100t*ωc

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

S z(t)

α=0.01α=0.1α=0.5α=0.7

∆1=0.2, s=1, T=3*10-8








0 20 40 60 80 100t*ωc

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

S z(t)

α=0.01α=0.1α=0.5α=0.7

∆1=0.2, s=1, T=3*10-8








0 20 40 60 80 100t*ωc

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

S z(t)

α=0.01α=0.1α=0.5α=0.7α=1.2α=1.3α=1.4

∆1=0.2, s=1, T=3*10-8

103 104 105 106 107 108 109

-0.2

0

0.2

0.4

S z(t)

α=0.01α=0.1α=0.5α=0.7α=1.2α=1.3α=1.4







TD-NRG: surprise in the sub-ohmic regime s < 1

100

101

102

103

104

t*ωc

-0.4

-0.2

0

0.2

0.4S z(t

)α=0.001<α

c

αc=0.00825

α=0.012>αc

α=0.014>αc

s=0.1, T=10-5

QPC: αc ≈ s∆ + O(s2) (Vojta, Bulla, PRL 2005)

surprise: oscillatory solution even in the localized phaseα > αc (FBA, Vojta, Bulla, PRL 2007)



TD-NRG: surprise in the sub-ohmic regime s < 1

100

101

102

103

104

t*ωc

-0.4

-0.2

0

0.2

0.4S z(t

)α=0.001<α

c

αc=0.00825

α=0.012>αc

α=0.014>αc

s=0.1, T=10-5

QPC: αc ≈ s∆ + O(s2) (Vojta, Bulla, PRL 2005)

surprise: oscillatory solution even in the localized phaseα > αc (FBA, Vojta, Bulla, PRL 2007)


Spin- and charge dynamics

J.M. Elzerman et al. Nature 430, 431(2004)




TD-NRG:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2t*Γ

0

0.1

0.2

0.3

0.4

0.5

n dow

n(t)

5 10 15 20H/Γ

0.2

0.3

0.4

0.5

T1

〈n↓(t)〉 =1





TD-NRG:

0 0.5 1 1.5 2 2.5 3 3.5 4

0.6

0.8

1

n d(t)

H=5H=10H=15H=20

0 0.5 1 1.5 2 2.5 3 3.5 4t*Γ

0

0.1

0.2

0.3

0.4

0.5

S z(t)

〈n↓(t)〉 =1





J.M. Elzerman et al. Nature 430,

431(2004)

level scheme of the Q-dot

µ

Εd

H>0

t=0

H=0

time

at t = 0:

switching off H: H = 0 fort > 0

Ed = µBH/2 → Ed = U/2



Charge dynamics of a quantum dot

0.6

0.8

1n d(t

) U/Γ1=2

U/Γ1=4

U/Γ1=6

0.01 0.1 1 10 100t*Γ

1

0.6

0.8

1

n d(t)

U/Γ1=8

U/Γ1=10

U/Γ1=12

U/Γ1=18

(a) Γ0 = Γ1

(b) Γ0 = 0

time scale for charge relaxation

tch = 1/Γ1 = 1/(πρt2)



spin dynamics of a quantum dot

0

0.1

0.2s z(t

)U/Γ

1=8

U/Γ1=10

U/Γ1=12

U/Γ1=18

10-2

10-1

100

101

102

103

t*Γ1

0

0.1

0.2

s z(t)

U/Γ1=2U/Γ1=4U/Γ1=6

(c) Γ0 = Γ1

(d) Γ0 = 0

time scale for spin relaxation

tsp ∝ 1/TK ∝ exp(1/ρJ)



spin dynamics of a quantum dot

0

0.1

0.2s z(t)

U/Γ1=2U/Γ1=4U/Γ1=6U/Γ1=8U/Γ1=10U/Γ1=12U/Γ1=18

10-4 10-3 10-2 10-1 100 101 102

t/tK

0

0.1

0.2

s z(t)

(a) Γ0 = Γ1

(b) Γ0 = 0

time scale for spin relaxation

tsp ∝ 1/TK ∝ exp(1/ρJ)


Contents






TD-NRG: transient currents

Equilibrium:

µRµL

for t < 0:

bias µL − µR = 0


TD-NRG: transient currents

Equilibrium:

µRµL

for t < 0:

bias µL − µR = 0

Non-Equilibrium

µ

µL

R

for t > 0:

finite bias V = µL − µR


Quantum dot (SIAM) transient currents

0 50 100 150 200 250 300 350 400t*Γ

0

0.5

1

1.5

2

<I(

t)>

/V i

n (e

2 /h)

Ed=-2, U=4

Ed=-3, U=6

Ed=-4, U=8

Quantum Dot (SIAM)V=10

-3, T=2.5*10

-5, Λ=3

0 1 2 3 4 5 6 7 8 9 10t/t

curr

0

0.5

1

1.5

2

<I(

t)>

/V i

n (e

2 /h)

Ed=-2, U=4

Ed=-3, U=6

Ed=-4, U=8

transient time tcurr

grows with U

tcurr ∝ 1/TK


Quantum dot (SIAM) transient currents

0 50 100 150 200 250 300 350 400t*Γ

0

0.5

1

1.5

2

<I(

t)>

/V i

n (e

2 /h)

Ed=-2, U=4

Ed=-3, U=6

Ed=-4, U=8

Quantum Dot (SIAM)V=10

-3, T=2.5*10

-5, Λ=3

0 1 2 3 4 5 6 7 8 9 10t/t

curr

0

0.5

1

1.5

2

<I(

t)>

/V i

n (e

2 /h)

Ed=-2, U=4

Ed=-3, U=6

Ed=-4, U=8

4 5 6 7 8U

0.01

0.1

1/t cu

rr tcurr

TK

transient time tcurr

grows with U

tcurr ∝ 1/TK


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 NEQGreen function (see also Costi 97)

G rA,B(t) = lim

t′→∞−iTr

[ρ0(Hi )[A(t + t ′),B(t ′)]s

]Θ(t)

m,m′ contributions needed =⇒ two reduced density matrices

novel sum-rule conserving algorithm (FBA 2007)








G rA,B(t) = lim

t′→∞−iTr

[ρ0(Hi )[A(t + t ′),B(t ′)]s

]Θ(t)










G rA,B(t) = lim

t′→∞−iTr

[ρ0(Hi )[A(t + t ′),B(t ′)]s

]Θ(t)










G rA,B(t) = lim

t′→∞−iTr

[ρ0(Hi )[A(t + t ′),B(t ′)]s

]Θ(t)










G rA,B(t) = lim

t′→∞−iTr

[ρ0(Hi )[A(t + t ′),B(t ′)]s

]Θ(t)





Reconstruction of the equilibrium spectral function

-10 -8 -6 -4 -2 0 2 4 6 8 10ω/Γ

0

0.1

0.2

0.3

0.4

ρ(ω

)U=1U=2U=4U=6U=8U=10

0ω/Γ

0

0.1

0.2

0.3

0.4U=1U=2U=4U=6U=8U=10

Start with U = 0 and evolve to finite U using the TD-NRG

G rA,B(t,U) = lim

t′→∞−iTr

[ρ0(U = 0)[A(t + t ′),B(t ′)]s

]Θ(t)



Reconstruction of the equilibrium spectral function

-4 -3 -2 -1 0 1 2 3 4ω/Γ

0

0.1

0.2

0.3

0.4ρ(

ω)

Ed=-3,U=6Ed=-1,U=6Ed=-1,U=6Ed= 0,U=6


Conclusion

Conclusion

The time-dependent NRG

novel method to investigate real-time dynamics inquantum-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 inmore complex systems

scattering states NRG for steady-state currents?

application to charge transfer reaction in bio-molecules


Conclusion

Conclusion

The time-dependent NRG

novel method to investigate real-time dynamics inquantum-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 inmore complex systems

scattering states NRG for steady-state currents?

application to charge transfer reaction in bio-molecules

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