Upload
ngongoc
View
229
Download
0
Embed Size (px)
Citation preview
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