Dynamics and thermodynamics of a drivenquantum system

Juzar Thingna5 July, 2017

University of Luxembourg, Luxembourg

Outline

• Mechanisms for relaxation– Dense reservoir: Red�eld regime– Sparse reservoir: Landau-Zener regime

• Thermodynamics– First law (Markovian and non-Markovian e�ects)– Second law

• Summary

1

Model

Free fermionic Hamiltonian

H (t ) = εtd†d +

L∑n=1

ϵnc†ncn +

γ

2

L∑n=1

(d†cn + H.c.

)≡ HS (t ) + HR +V

HS

Finite number ofreservoir levels

L

• Exactly solvable• Decoupled initial condition ρtot (0) = ϱ (0) ⊗ e−β (HR−µNR )

ZR

2

Red�eld-like kinetics: Dense reservoir

t

ϵ1

ϵL

Bare Reservoir

3

Red�eld-like kinetics: Dense reservoir

t

ϵ1

ϵL

Bare Reservoir

Bare System

3

Red�eld-like kinetics: Dense reservoir

γ

t

ϵ1

ϵL

Bare Reservoir

Bare System

Assumptions

• Dense reservoir and strong mixing Lγϵn+1−ϵn

� 1

• Weak system-reservoir coupling γ � 1

M. Esposito and P. Gaspard, Phys. Rev. E 68, 066112 (2003).

3

Red�eld-like kinetics: Dense reservoir

Time-dependent Red�eld master equation

dtϱnn =∑i

Liinnϱii

dtϱnm =∑i, j

Li jnmϱi j ∀n ,m, i , j

with

L1122 = −L

1111 = T

1221 (t ) +T

12∗21 (t ) L12

12 = −[T 1221 (t ) +T

21∗12 (t )

]+ iεt

L2211 = −L

2222 = T

2112 (t ) +T

21∗12 (t ) L21

21 = −[T 2112 (t ) +T

12∗21 (t )

]− iεt

Populations and coherence decouple

H. Zhou, J. Thingna, P. Hänggi, J.-S. Wang, and B. Li, Scienti�c Reports 5, 14870 (2015).

4

Red�eld-like kinetics: Dense reservoir

Non-Markovian Transition rates (Real + Imaginary parts)

T 2112 (t ) =

∫ t

0dt ′ei

∫ t ′

0 εt ′−τ dτ∫ ∞

−∞

dϵ

2πΓ[1 − f (ϵ )]e−iϵt

′

︸ ︷︷ ︸Bath Correlators

T 1221 (t ) =

∫ t

0dt ′e−i

∫ t ′

0 εt ′−τ dτ

︷ ︸︸ ︷∫ ∞

−∞

dϵ

2πΓ f (ϵ )eiϵt

′

5

Red�eld-like kinetics: Dense reservoir

Non-Markovian Transition rates (Real + Imaginary parts)

T 2112 (t ) =

∫ t

0dt ′ei

∫ t ′

0 εt ′−τ dτ∫ ∞

−∞

dϵ

2πΓ[1 − f (ϵ )]e−iϵt

′

︸ ︷︷ ︸Bath Correlators

T 1221 (t ) =

∫ t

0dt ′e−i

∫ t ′

0 εt ′−τ dτ

︷ ︸︸ ︷∫ ∞

−∞

dϵ

2πΓ f (ϵ )eiϵt

′

Occupied state population p (t ) = 〈d†d〉

dtp (t ) = T+ (t )[1 − p (t )] −T − (t )p (t )

with the transition rates T + (t ) = 2Re[T 1221 (t )] and

T − (t ) = 2Re[T 2112 (t )].

5

Red�eld-like kinetics: Dense reservoir

×102

0 10 20 30 40 50Time t

0

0.2

0.4

0.6

0.8

1

p(t)

0 20 40 60 80Time t

0 20 40 60 80 100Time t

0.6

0.7

0.8

0.9

1

p(t)

0 150 3000.6

0.7

0.8

0.9

1

p(t)

Strong mixing

Weak mixing

Autonomous

Driven

Autonomous

6

Landau-Zener kinetics: Sparse reservoir

ϵn

ϵn+1

ϵL

tn tn+1 · · · tL

µ Filled

Empty

γ

Assumptions (Linear driving εt = εt )

• Sequential crossing: ∆ϵn = ϵn+1 − ϵn � γ

• Landau-Zener validity time:

τc︷︸︸︷∆ϵnε�

τlz︷ ︸︸ ︷1√|ε |

max(1,

γ√|ε |

)7

Landau-Zener kinetics: Sparse reservoir

Landau-Zener Markov Chain

p (tn+1) =

System Gain︷ ︸︸ ︷R p (tn ) +

Reservoir Loss︷ ︸︸ ︷(1 − R) f (ϵn )︸︷︷︸

Fermi

Landau-Zener transition probability: R = exp[−πγ 2

2ε

]

8

Landau-Zener kinetics: Sparse reservoir

Landau-Zener Markov Chain

p (tn+1) =

System Gain︷ ︸︸ ︷R p (tn ) +

Reservoir Loss︷ ︸︸ ︷(1 − R) f (ϵn )︸︷︷︸

Fermi

Landau-Zener transition probability: R = exp[−πγ 2

2ε

]

Continuous time Landau-Zener master equationFast driving regime ε � γ 2 =⇒ R ≈ 1 − πγ 2

2ε and coarse graining intime

dtp (t ) = T+ (∞)[1 − p (t )] −T − (∞)p (t )

Markovian version of Red�eld-like equation

F. Barra and M. Esposito, Phys. Rev. E 93, 062118 (2016).

8

Landau-Zener kinetics: Sparse reservoir

×102

p(t) 0 20 40 60 80

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 800.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.80.992

0.994

0.996

0.998

1

0 20 40 60 800

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8Time t

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

L = 100, ϵL = 100, β = 0.1

∆ϵ � γ ; τc � τlz ∆ϵ � γ ; τc = τlz ∆ϵ � γ ; τc � τlz

∆ϵ = γ ; τc = τlz ∆ϵ = γ ; τc � τlz ∆ϵ � γ ; τc � τlz

γ 2 = 1ε = 1

γ 2 = 1ε = 100

γ 2 = 100ε = 100

γ 2 = 10−4

ε = 10−2γ 2 = 10−2

ε = 1γ 2 = 10−2

ε = 100

9

Take home message # 1

Driving a system can help it dissipate (via the Landau-Zenermechanism) compensating for the sparseness of the reservoir

Thermodynamics: Identities

First lawdtU = Q + W

Rate of change in Internal energydtU = Tr [dtρ (t ) {HS (t ) +V }] + Tr [ρ (t )dtHS (t )]

Heat currentQ = IE − µIN

Energy currentIE = −Tr [dtρ (t )HR]Particle currentIN = −Tr [dtρ (t )NR]

Rate of workW = Wmech + µIN

Rate of mechanical workWmech = Tr [dtHS (t )ρ (t )]

M. Esposito, K. Lindenberg and C. Van den Broeck, New J. Phys. 12, 013013 (2010).

11

Thermodynamics: Red�eld regime

Energy currentIE = T

+ (t )[1 − p (t )] − T − (t )p (t )Particle currentIN = T

+ (t )[1 − p (t )] −T − (t )p (t )

Rate of mechanical workWmech = εtp (t )

Modi�ed (non-Markovian) transition rates

T + (t ) = −2Re[∫ t

0dt ′e−i

∫ t ′

0 εt ′−τ dτ∫ ∞

−∞

dϵϵ

2πΓ f (ϵ )eiϵt

′

]

T − (t ) = 2Re[∫ t

0dt ′ei

∫ t ′

0 εt ′−τ dτ∫ ∞

−∞

dϵϵ

2πΓ[1 − f (ϵ )]e−iϵt

′

]

J. Thingna, J. L. García-Palacios, and J.-S. Wang, Phys. Rev. B 85, 195452 (2012).

12

Thermodynamics: Landau-Zener regime

Markov chainEnergy changeE (tn+1) = ϵn[p (tn+1) − p (tn )]Particle changeN (tn+1) = p (tn+1) − p (tn )

Mechanical workWmech (tn+1) = (ϵn+1 − ϵn )p (tn+1)

13

Thermodynamics: Landau-Zener regime

Markov chainEnergy changeE (tn+1) = ϵn[p (tn+1) − p (tn )]Particle changeN (tn+1) = p (tn+1) − p (tn )

Mechanical workWmech (tn+1) = (ϵn+1 − ϵn )p (tn+1)

Continuous time Landau-ZenerEnergy currentIE = εt [T + (∞)[1 − p (t )] −T − (∞)p (t )]Particle currentIN = T

+ (∞)[1 − p (t )] −T − (∞)p (t )

Rate of mechanical workWmech = εtp (t )

Identical to ‘Traditional’ thermodynamics with Markovian QMEsH. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press.

13

Thermodynamics

0 20 40 60 80Time t

−5

4

13

22

W

−4

−1.5

1

3.5

Q

−1

1.5

4

6.5

W

−0.5

1

2.5

4Q

0 20 40 60 80Time t

0

8

16

24

−12

−8

−4

0

−3

−1

1

3

−1.5

−0.5

0.5

1.5

0 0.1 0.2 0.3 0.4 0.5t

−6−3.5−11.54

0 0.1 0.2 0.3 0.4 0.5t

−3−1.5

01.53

Non-Markovian

Markovian

Red�eld Landau-Zener

14

Take home message # 2

Dynamics is insu�cient to capture thermodynamics

Periodic driving

Second lawEntropy production︷ ︸︸ ︷

∆iS = D[ρ (t ) | |ϱ (t ) ⊗ ρeqR ] =Shanon entropy︷︸︸︷

∆S − βQ

Fast driving Slow driving

Markovchain

LZQMEdt∆iS < 0

16

Periodic driving

Second lawEntropy production︷ ︸︸ ︷

∆iS = D[ρ (t ) | |ϱ (t ) ⊗ ρeqR ] =Shanon entropy︷︸︸︷

∆S − βQ

Fast driving Slow driving

Markovchain

LZQMEdt∆iS < 0

16

Multiple Reservoirs

HamiltonianHS (t ) = εtd

†d ; Hα =∑n∈α

ϵnc†ncn

V =γ

2

∑n∈α

(d†cn + H.c.

)ρ (0) = ϱd (0) ⊗ Π⊗

e−βα (Hα−µαNα )

Zα

ϵn

ϵn+1

...

ϵL

tn tn+1 · · · tL

µhµc

0 20 40 60 80Time t

0

0.2

0.4

0.6

0.8

1

p(t)

dtp (t ) =∑α

T +α (∞)[1 − p (t )] −T −α (∞)p (t )

∆TT =

∆µµ = 0.1 Linear response

∆TT =

∆µµ = 0.5

∆TT =

∆µµ = 0.9

17

Summary

• Strong mixing between the system and reservoir = Red�eldregime.

• Driving a system can relax the strong mixing condition =Landau-Zener regime.

• Thermodynamics requires more information than the dynamicswhich is most evident in the non-Markovian Red�eld regime.

• The Landau-Zener regime exists even for multiple reservoirsallowing us to explore transport due to �nite reservoirs.

18

Acknowledgments

Collaborators

• Assoc. Prof. Felipe Barra (University of Chile)

• Prof. Massimiliano Esposito (University of Luxembourg)

Questions