Laboratoire de Physique et Modelisation ´ des Milieux Condenses ´ Univ. Grenoble Alpes & CNRS, Grenoble, France
Second law of thermodynamics for non-markovian quantum machines Robert S. Whitney manuscript in preparation
?? New Trends in Quantum Heat and Thermoelectrics, Trieste — 23 Aug 2016
OVERVIEW ♣ Fluctuation theorems : better than 2nd law ♣ Quantum machine: heat ⇒ electrical power
??
OVERVIEW ♣ Fluctuation theorems : better than 2nd law ♣ Quantum machine: heat ⇒ electrical power • non-markovian = strong coupling to reservoirs (cotunneling, Kondo, etc)
• interactions in system (Coulomb blockade) ... but non-interacting reservoirs
• real-time Keldysh theory: far from equilib.
??
OVERVIEW ♣ Fluctuation theorems : better than 2nd law ♣ Quantum machine: heat ⇒ electrical power • non-markovian = strong coupling to reservoirs (cotunneling, Kondo, etc)
• interactions in system (Coulomb blockade) ... but non-interacting reservoirs
• real-time Keldysh theory: far from equilib.
?? ♣ Conclusion:
Quantum fluctuation theorems & 2nd law = classical fluctuation theorems & 2nd law
CLASSICAL FLUCTUATION THEOREMS & 2 nd LAW
Throw all bricks in air! Pgood =
N◦ of “good” states Total N◦ states
Entropy:
Sgood = ln N◦ of “good” states Sbad = ln N◦ of “bad” states h i Pbad→good = Pgood→bad × exp − ∆Sgood→bad
CLASSICAL FLUCTUATION THEOREMS & 2 nd LAW Seifert (2005)
electrons
T
photons/phonons
Q
T
Q
Any large reservoir at thermal equilibrium
∆S =
∆Q kB T
T2 T1
T3
CLASSICAL FLUCTUATION THEOREMS & 2 nd LAW Seifert (2005)
electrons
T
photons/phonons
Q
T
Q
Any large reservoir at thermal equilibrium
∆S = Fluctuation theorems: • Under right conditions
∆Q kB T
Evans-Searles (1994), Crooks (1998)
P (−∆S) = P (∆S) exp − ∆S
T2 T1
⇒ 2nd law on average h∆Si ≥ 0
T3
CLASSICAL FLUCTUATION THEOREMS & 2 nd LAW Seifert (2005)
electrons
T
photons/phonons
T
Q
Q
Any large reservoir at thermal equilibrium
∆S = Fluctuation theorems: • Under right conditions
Evans-Searles (1994), Crooks (1998)
P (−∆S) = P (∆S) exp − ∆S • Universal : Kawasaki (1967), Seifert (2005)
exp − ∆S = 1 • Other relations:
∆Q kB T
Jarzynski (1997), etc
⇒ 2nd law on average h∆Si ≥ 0
T2 T1
T3
PROOF via CLASSICAL "STOCHASTIC TRAJECTORIES" Proof of fluctuation theorem & hence 2nd law
Reviews: Seifert (2012), van den Broeck (2013), Benenti-Casati-Saito-Whitney (2016)
INGREDIENTS: (i) a classical Markov rate equation (master equation) X d Pb (t) = Γba Pa (t) − Γab Pb (t) dt a where Pb = prob. system is in state b (i) & Γba = rate a→ b due to reservoir i
PROOF via CLASSICAL "STOCHASTIC TRAJECTORIES" Proof of fluctuation theorem & hence 2nd law
Reviews: Seifert (2012), van den Broeck (2013), Benenti-Casati-Saito-Whitney (2016)
INGREDIENTS: (i) a classical Markov rate equation (master equation) X d Pb (t) = Γba Pa (t) − Γab Pb (t) dt a where Pb = prob. system is in state b (i) & Γba = rate a→ b due to reservoir i (ii) local detailled balance (microreversibility) h i (i) (i) (i) (i) Γab = Γba exp −∆Sba where ∆Sba =entropy change in i due to a→ b
EXAMPLES: EXISTING NANOSCALE MACHINES
TWO reservoirs
THREE reservoirs 1 for heat & 2 for current Glattli group (2015)
Reddy group (2015)
2 reservoir theory (& older expts) = Molenkamp group, Mahan-Sofo, Linke group 3 reservoir theory = Entin-Wohlmann et al, Sanchez ´ & Buttiker ¨
EXAMPLES: EXISTING NANOSCALE MACHINES
TWO reservoirs
THREE reservoirs 1 for heat & 2 for current Glattli group (2015) Worschech group (2015)
Reddy group (2015)
2 reservoir theory (& older expts) = Molenkamp group, Mahan-Sofo, Linke group 3 reservoir theory = Entin-Wohlmann et al, Sanchez ´ & Buttiker ¨
EXAMPLES: EXISTING NANOSCALE MACHINES
TWO reservoirs
THREE reservoirs 1 for heat & 2 for current Glattli group (2015) Worschech group (2015) Molenkamp group (2015)
HOT
Left Reddy group (2015)
2 reservoir theory (& older expts) = Molenkamp group, Mahan-Sofo, Linke group 3 reservoir theory = Entin-Wohlmann et al, Sanchez ´ & Buttiker ¨
Right
CLASSICAL (stochastic) TRAJECTORIES TH
Heat source
(1,0)
e-
H
L R
Sanchez-Buttiker ¨ (2011)
Strasberg-Schaller-Brandes-Esposito (2013)
H R RVR H
RVR
R
T0
RVR R
T0
e-
(0,0)
(1,1)
R H
(0,1)
eRate(0 → 1) ∝ Fermi Rate(1 → 0) ∝ 1−Fermi
local
⇒ detailed balance
CLASSICAL (stochastic) TRAJECTORIES
Sanchez-Buttiker ¨ (2011)
Strasberg-Schaller-Brandes-Esposito (2013)
TH
Heat source
(1,0)
e-
H
L R
R
T0
T0
e-
(1,1)
R H
(0,0)
(0,1)
e-
Evolution with time: Trajectory ζ =
(1,1)
0
R (0,1) H
t1
t2
(0,0)
L (1,0)
t3
t
CLASSICAL (stochastic) TRAJECTORIES
Sanchez-Buttiker ¨ (2011)
Strasberg-Schaller-Brandes-Esposito (2013)
TH
Heat source
(1,0)
e-
H
L R
R
T0
T0
e-
(1,1)
R H
(0,0)
(0,1)
e-
Evolution with time: Trajectory ζ = time-reverse ζ =
(1,1)
0
R (0,1) H
t1 (1,0) L
0
t2 (0,0)
t-t3
(0,0)
t3 H (0,1) R
t-t2
i Prob. of ζ = Prob. of ζ × exp − ∆Sres (ζ) h
L (1,0)
t-t1
t
(1,1)
t ⇒ Fluctation theorem
completely
quantum SUPERPOSITIONS, ENTANGLEMENT, etc
& GENERAL : strong-coupling, time-dependence, etc
PREVIOUS PROOFS OF 2 nd LAW FOR QUANTUM MACHINES
timendent pe e d in
an vi
ko
ar
m
co we up ak lin g
g n- tin norac te in
interacting time-dependent strong-coupling non-Markovian
weak-coupling = sequential tunnelling approx. (neglecting cotunnelling, etc)
PREVIOUS PROOFS OF 2 nd LAW FOR QUANTUM MACHINES
timendent pe e d in
interacting time-dependent strong-coupling non-Markovian
an vi
ko
ar
m
co we up ak lin g
Landauer scattering Nenciu (2007), RW (2013)
g n- tin norac te in
Keldysh (non-interacting) Esposito, Ochoa, Galperin (2015)
Master equation Seifert (2005), van den Broeck (2013) Lindblad equation Alicki (1979), etc.
weak-coupling = sequential tunnelling approx. (neglecting cotunnelling, etc)
REAL-TIME KELDYSH APPROACH quantum + non-markov + interactions + far from equilibrium Schoeller-Schon ¨ (1994)
♣ big simplifications: • interactions in system but NOT in reservoirs =⇒ many-body eigenbasis for system =⇒ free-particle eigenbasis for reservoirs • infinite N◦ of reservoir modes k =⇒ coupling to lowest (2nd) order for each k
Example Hamiltonian:
X X ˆ = H ˆ sys dˆ†n , dˆn + H Vn,k dˆ†n cˆk + dˆn cˆ†k + Ek cˆ†k cˆk k
interacting system
k
coupling
electron reservoirs
REAL-TIME KELDYSH APPROACH quantum + non-markov + interactions + far from equilibrium Schoeller-Schon ¨ (1994)
♣ big simplifications: • interactions in system but NOT in reservoirs =⇒ many-body eigenbasis for system =⇒ free-particle eigenbasis for reservoirs • infinite N◦ of reservoir modes k =⇒ coupling to lowest (2nd) order for each k
Evolution as function of time :
t0
A A
+
B
C
k1
k2 B'
+
D k3 C +
C'
time
REAL-TIME KELDYSH APPROACH quantum + non-markov + interactions + far from equilibrium Schoeller-Schon ¨ (1994)
♣ big simplifications: • interactions in system but NOT in reservoirs =⇒ many-body eigenbasis for system =⇒ free-particle eigenbasis for reservoirs • infinite N◦ of reservoir modes k =⇒ coupling to lowest (2nd) order for each k
Evolution as function of time :
t0
A A
+
B
C
k1
k2 B'
+
D k3 C +
C'
+
+ +
time
Some examples of this rotation
ENTROPY CHANGE IN RESERVOIRS (rotating double-paths)
i0 j0
i j
j i
j0 i0
i0 j0
i j
j i
j0 i0
i j
j i
i j
j i
i0 j0 i0 j0
i0 j0
+
+
t1 t2 t1
t2
+
+ + +
i j
j i
t~2 t~1
~
t2
+ ~
t1
+
j0 i0 j0 i0
+ + +
+
j0 i0
Some examples of this rotation
ENTROPY CHANGE IN RESERVOIRS (rotating double-paths)
i0 j0
i j
j i
j0 i0
i0 j0
i j
j i
j0 i0
i j
j i
i j
j i
i0 j0 i0 j0
i0 j0
i0 j0 t0
i j time
t
- S ( e res
)
=
+
t1 t2 t1
+ +
+
+
j i t0
t2
+
j0 i0 time
t
i j
j i
t~2 t~1
~
t2
+ ~
t1
+
j0 i0 j0 i0
+ + +
+
j0 i0
ENTROPY CHANGE IN RESERVOIRS (rotating double-paths) • Initial system density-matrix ˆ † ⇐ diagonal p ˆ0 p ˆ (0) ˆ (0) W ρˆsys (t0 ) = W 0 • Final (reduced) system density-matrix ˆp ˆ† ˆ ˆW ρˆsys (t) = W ⇐ diagonal p
ENTROPY CHANGE IN RESERVOIRS (rotating double-paths) • Initial system density-matrix ˆ † ⇐ diagonal p ˆ0 p ˆ (0) ˆ (0) W ρˆsys (t0 ) = W 0 • Final (reduced) system density-matrix ˆp ˆ† ˆ ˆW ρˆsys (t) = W ⇐ diagonal p Define “probability-weight” of double path, ζ , ˆ (0) to state i of p ˆ from state i0 of p 0
P ζ
=
+ + +
i0
+ +
i
+
+ Complex Conj.
0
t0 Result : P ζ
t
time
= P ζ × exp − ∆Sreservoirs (ζ)
ENTROPY CHANGE IN SYSTEM ∆Ssys = Ssys (t) − Ssys (t0 )
with von Neumann h
i Ssys (τ ) = Tr ρˆsys (τ ) ln ρˆsys (τ )
Seifert (2005) ⇒ QUANTUM Assign entropy to each state in diagonal basis of system’s (reduced) density matrix, ρˆsys , at beginning (t0 ) and end (t)
ENTROPY CHANGE IN SYSTEM ∆Ssys = Ssys (t) − Ssys (t0 )
with von Neumann h
i Ssys (τ ) = Tr ρˆsys (τ ) ln ρˆsys (τ )
Seifert (2005) ⇒ QUANTUM Assign entropy to each state in diagonal basis of system’s (reduced) density matrix, ρˆsys , at beginning (t0 ) and end (t)
(0)
(0)
i0 →i ∆Ssys = pi ln pi − pi0 ln pi0 (0)
where pi0 is prob. initial state is i0 in diag. basis of ρˆsys (t0 ) & pi is prob. final state is i in diag. basis of ρˆsys (t)
QUANTUM FLUCTUATION THEOREM 0
+ + +
i0
+ +
quantizing Seifert
i
+
0
t0
t
time
i0 →i i0 →i i0 →i DEFINE ∆Stotal = ∆Sres + ∆Ssys
+ some algebra =⇒ ALL CLASSICAL FLUCTUATION THEOREMS Crook’s, Jarzynski, Kawasaki, etc
=⇒
2nd LAW
♣ neglected entropy of entanglement between system & reservoirs
PROOF or just RECIPE ?? WHY assume we can NEGLECT: ♣ Entropy of entanglement between system & reservoirs ♣ Non-zero off-diagonal trajectories
PROOF or just RECIPE ?? WHY assume we can NEGLECT: ♣ Entropy of entanglement between system & reservoirs ♣ Non-zero off-diagonal trajectories 0
+ + +
i0
i j
+ +
+
0
t0
t
time
has no time-reverse for j 6= i ...they “average” to zero
PROOF or just RECIPE ?? WHY assume we can NEGLECT: ♣ Entropy of entanglement between system & reservoirs ♣ Non-zero off-diagonal trajectories 0
+ + +
i0
i j
+ +
+
0
t0
t
time
ASSUMPTIONS seem reasonable if no Maxwell demons i.e. assume cannot use knowledge of microscopic state of reservoir to get extra work from system
has no time-reverse for j 6= i ...they “average” to zero
CONCLUSIONS warning: work only just finished
We find QUANTUM FLUCTUATION THEOREMS for arbitrary quantum machine
??
interacting, NON-MARKOVIAN, time dependent, etc
CONCLUSIONS warning: work only just finished
We find QUANTUM FLUCTUATION THEOREMS for arbitrary quantum machine
??
interacting, NON-MARKOVIAN, time dependent, etc
... identical to CLASSICAL FLUCTUATION THEOREMS ⇒ proof of
2 nd LAW
CONCLUSIONS warning: work only just finished
We find QUANTUM FLUCTUATION THEOREMS for arbitrary quantum machine
??
interacting, NON-MARKOVIAN, time dependent, etc
... identical to CLASSICAL FLUCTUATION THEOREMS ⇒ proof of
Assumption:
2 nd LAW
no Maxwell demons for specific definition of “Maxwell demon”
