Optimizing Cooling Processes A Dynamic View on the Third Law · 2012. 9. 3. · Karl Heinz...

Karl Heinz Hoffmann!Professur für Theoretische Physik, 

insbesondere Computerphysik!

A Dynamic View on the Third Law!IWNET2012, Roros!August 19-24, 2012!

Optimizing Cooling Processes  - 

A Dynamic View on the Third Law"

Karl Heinz Hoffmann!TU Chemnitz!

Germany!




The Third Law"

●  1906 Nernst   1911 Planck!

limT→0

S = 0




The Third Law"

●  magnetic refrigeration!

●  alternate!–  isotherms!–  adiabats!

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The Third Law"

●  consequence: absolute zero is unattainable!

●  infinite number of steps would be needed to reach absolute zero!

●  steps are thermodynamic equilibrium steps (infinitely slowly)!

What about non-equilibrium processes?!




Dynamic View on the Third Law"

●  Processes in finite time!

●  no longer thermodynamic equilibrium branches!

●  consider a refrigerator!

Finite-Time Thermodynamics  Endoreversible Thermodynamics!

Are there limits on the cooling rate?!




Dynamic View on the Third Law"

●  continuous refrigeration process: entropy production!

●  the Second Law:!

●  What exponent does apply in ?!δ

σ = −Q̇c/Tc + Q̇h/Th > 0

Rc = Q̇c <Q̇hTh

Tc

Rc ∝ T δc




A Quantum Refrigerator"

●  working medium:  an ensemble of quantum harmonic oscillators!

●  description by density operator!

●  the work parameter:  the frequency instead of the volume!

Ĥ =1

2mP̂

2+

m ω(t)2

2Q̂

2




A Quantum Refrigerator: Average Cooling Rate"

●  for a cyclic refrigerator the cooling rate is  the heat up-take per cycle time!

●  expectation:  

For given one needs to choose optimal!

Rc =Qcτ

=Qc

τhc + τc + τch + τh

0.5 1.0 1.5 2.0 2.5 3.0 3.50.5

1.0

1.5

2.0

2.5

3.0

3.5

Ω

n

τhc

τch

τc τh

τ( ωc( Tc ) )

Qc( ωc( Tc ) )

Tcωc, τhc, τc, τch, τh




The Dynamics of a Quantum Working Fluid"

●  Adiabats: externally driven Hamiltonian!

●  contact to heat bath, fixed frequency: dissipative Lindblad dynamics!

●  Lindblad term establishes thermal equilibrium to a bath!

Ĥ(ω(t))

dÔ(t)dt

=i

� [Ĥ(t), Ô(t)] +∂Ô(t)

∂t

dÔ(t)dt

= L∗(Ô) = i� [Ĥ, Ô] + L∗D(Ô)




Simplified Dynamics"

●  There exists a set of operators which is invariant under the dynamics!

●  These are the Hamiltonian, the Lagrangian, and the Correlation!

Ĥ, L̂, Ĉ

Ĥ =1

2mP̂

2+

m ω(t)2

2Q̂

2

L̂ =1

2mP̂

2− m ω(t)

2

2Q̂

2

Ĉ = ω(t)12(Q̂P̂ + P̂Q̂)




Entropies"

●  von Neumann entropy 

●  energy entropy 

with expectation values in energy eigen states!

●  in equilibrium!

pn = � n |ρ(Ĥ, L̂, Ĉ)| n �

SvN = SE

SvN = −kBTr(ρ̂ ln ρ̂)

SE = −kB�

n

pn ln pn




Adiabat Dynamics"

●  frequency changing linear in time [0,10] from 1 to 0.3!

0 2 4 6 8 100.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

t

Ω

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

t

CLE




Adiabat Dynamics"

●  frequency changing linear in time [0,10] from 1 to 0.3!

●  the mean occupation number n is not constant: quantum friction!

0 2 4 6 8 100.2

0.4

0.6

0.8

1.0

t

nE

0 2 4 6 8 100.94

0.95

0.96

0.97

0.98

0.99

t

SESvN




Heat Bath Contact"

●  fixed frequency and coupling to a bath with the contact temperature!! ! ! ! ! !W. Muschik, G. Brunk!

0

2

4

6

E

�1.0

�0.5

0.0

0.5

1.0

L

�1.0�0.5

0.0

0.5

1.0

C

0 2 4 6 8 10�0.50.00.51.01.52.02.53.0

t

CLE




Heat Bath Contact"

●  fixed frequency and coupling to a bath with the contact temperature!

0 2 4 6 8 102.02

2.04

2.06

2.08

2.10

t

SESvN




Heat Bath Contact"

●  fixed frequency and coupling to a bath with a different temperature!

0 10 20 30 40�0.50.00.51.01.52.02.53.0

t

CLE

0 10 20 30 401.0

1.2

1.4

1.6

1.8

2.0

t

SESvN




Heat Bath Contact"

●  fixed frequency and coupling to a bath with a different temperature!

0 1 2 3 4E

�1.0�0.5

0.00.5

1.0L

�1.0

�0.5

0.0

0.5

1.0

C

0 1 2 3 4E�1.0

�0.5

0.0

0.51.0

L

�1.0

�0.5

0.0

0.5

1.0

C




The Cooling Rate"

●  the cooling rate!

●  given the time needed for the adiabats:  optimize for relative time allocation on the isotherms (isofrequencies) optimize for time allocation between isotherms and adiabats!

0.5 1.0 1.5 2.0 2.5 3.0 3.50.5

1.0

1.5

2.0

2.5

3.0

3.5

Ω

n

τhc

τch

τc τh

neqCnC

nD

ωc

Rc =1

τhc + τc + τch + τh�ωc(nC − nD)

Roptc ≈�ωcneqcτhc

τhc = τch τh = τc




Adiabat Dynamics: the 1TURN Process"

●  special frequency protocol!

●  the optimal time from hot to cold!

0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

t

nE

0 5 10 15 20 250.94

0.96

0.98

1.00

1.02

t

SESvN

µ =ω̇

ω2= const

τ∗hc =12ω−1c




Time needed for IR Adiabats: the 1TURN Process"

●  for given choose proper speed and time !ωc, ωh

0.2 0.4 0.6 0.8 1.0E

0.0

0.1

0.2

L

0.00.10.2C

0.20.40.60.81.0E

0.0

0.1

0.2

L

0.0

0.1

0.2C

τ∗hc =12ω−1c Roptc < const. T 2c




Adiabat Dynamics: 3JUMP Process"

●  three frequency jumps between given frequencies!

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.71.0

1.2

1.4

1.6

1.8

2.0

t

Ω

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

�1

0

1

2

t

CLE

τ1

τ2




Adiabat Dynamics: 3JUMP Process"

●  three frequency jumps between given frequencies!

1.01.52.02.5E

�1.5 �1.0 �0.5 0.0 0.5L

0.0

0.5

1.0

C

0.00.51.0C

1.0

1.5

2.0

2.5

E

�1.5�1.0

�0.50.0

0.5

L




Shortest Time needed for Adiabats"

●  If frequency is confined between and  

●  proof on the basis of control theory, very technical 

3JUMP Processes are the adiabats  which lead in the shortest time  from equilibrium to equilibrium!

τ∗hc =1√

ωhω− 12c

ωhωc

Roptc < const. T32

c




Generalized Dynamics: Finite-Dimensional Lie Algebra"

●  There exists a set of operators which is invariant under the dynamics!

●  More general setting: there exists a set of operators  which form a Lie algebra with!

Ĥ, L̂, Ĉ

The Casimir Companion X

Frank Boldt1,∗ James D. Nulton2,† Bjarne Andresen3,‡ Peter Salamon2,§ and Karl Heinz Hoffmann1¶1 Institute of Physics, Chemnitz University of Technology – D-09107 Chemnitz, Germany

2 Department of Mathematical Science, San Diego State University – San Diego, California, 92182, USA3Niels Bohr Institute, University of Copenhagen,

Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark(Dated: July 13, 2012)

In this paper a new invariant of motion for Hamiltonian systems is introduced: the Casimircompanion. For systems with simple dynamical algebras (e.g., coupled spins, harmonic oscillators)our new invariant is useful in problems that consider adiabatically varying the parameters in theHamiltonian. In particular, it has proved useful in optimal control of changes in these parameters.The Casimir companion also allows simple calculation of the entropy of non-equilibrium ensembles.

PACS numbers: 05.70.Ln, 02.20.Sv, 05.70.-a, 02.30.YyKeywords: Nonequilibrium Processes, Entropy, Lie Algebra, Casimir Invariant, Optimal Control

INTRODUCTION

Classical mechanics as well as quantum mechanics hasto solve equations of motion. Finding invariants simpli-fies these equations. A useful tool to find these invariantsis the theory of Lie algebras. Given the Lie algebra asso-ciated to the equations of motion, straightforward invari-ants exist. These so-called Casimir invariants are familiarfrom quantum treatments of angular momentum.The Casimir operator was introduced by Hendrik

Casimir in 1931 [1]. It is a distinguished element of theenveloping algebra of a Lie algebra that commutes withevery element of the algebra. This feature makes it aconstant of the motion for any problem having the sym-metry of the Lie algebra. Traditionally, this means thatthe Hamiltonian of the problem commutes with each el-ement of the algebra.Recent work on the optimal control of quantum sys-

tems with low dimensional dynamical algebras [2–14] hasturned up a closely related invariant which we here dubthe Casimir companion X [15–18]. It appears to be asuseful for these problems as the Casimir operator is forproblems with traditional, static symmetries. X is cal-culated using the formula for the Casimir operator butinserting expectation values instead of operators.In the optimal control problems mentioned above

[15, 16], the constancy of X reduces the dimension ofthe problem by one – a feature which greatly simplifiesthe optimal control and facilitates its understanding andinterpretation [17, 18].A Hamiltonian problem with Hamiltonian H is said

to have a dynamical algebra [19] provided H can be ex-pressed as a linear combination of elements of the Lie al-gebra.

H =N∑

i=1

hi(t)Bi . (1)

It then follows that the dynamics of (say) the energy canbe reduced to the dynamics of a basis for the algebra

and this latter dynamics follows from the commutationrelations

[Bi,Bj] =N∑

k=1

c kij Bk , (2)

defining the algebra.Optimal control involving only the expectation values

of (say) the energy, are nicely handled in the Heisenbergrepresentation. The dynamics of any operatorA are then

dAdt

=ı

h̄[H,A] +

∂A∂t

, (3)

Taking the trace with the (constant) density matrix leadsto ordinary differential equations for the expectationvalue of A. To describe the dynamics of any operator ex-pressible in terms of the Lie algebra, the required numberof coupled differential equations is just the dimension ofthe algebra. The invariance of the Casimir companion Xfor these problems reduces the dimension by one. Fur-thermore the value of X is related to the entropy and theenergy of the system in a way that allows us to expressthe von Neumann entropy for nonequilibrium states interms of X .

THE CASIMIR COMPANION

Consider a system which is describable by a finite setof operators {Bi}Ni=1 forming a closed Lie algebra andwhere the Hamiltonian is a linear combination of theseoperators. The equations of motion (4) are easily seento be closed under this Lie algebra, provided the basis{Bi}Ni=1 is not explicitly time-dependent.

dBjdt

=∑

k

γ kj (t)Bk , (4)

Note that this applies even when parameters in theHamiltonian are varied. The no explicit time dependence







INTRODUCTION






H =N∑

i=1

hi(t)Bi . (1)



[Bi,Bj] =N∑

k=1

c kij Bk , (2)



dAdt

=ı

h̄[H,A] +

∂A∂t

, (3)




dBjdt

=∑

k

γ kj (t)Bk , (4)








INTRODUCTION






H =N∑

i=1

hi(t)Bi . (1)



[Bi,Bj] =N∑

k=1

c kij Bk , (2)



dAdt

=ı

h̄[H,A] +

∂A∂t

, (3)




dBjdt

=∑

k

γ kj (t)Bk , (4)





Generalized Dynamics: Finite-Dimensional Lie Algebra"

●  Known constant of the motion:  

The Casimir Operator!

●  There exists a second constant of the motion which is invariant under the dynamics:  

The Casimir Companion!

●  The von Neumann entropy – also in non-equilibrium – can always be expressed in terms of the Casimir Companion !

2

in the basis is not real a restriction because one can al-ways transform the time dependence from the basis tothe coefficients, leading to a time-dependent system ma-trix γ kj (t) and vice versa. This matrix can be calculated

directly, knowing the time-dependent coefficients hi(t) ofthe linear expansion in eq. (1) and the time-independentstructure constants c kij of the Lie algebra.From eq. (3-4) we get

γ kj (t) =N∑

i=1

ı

h̄hi(t)c kij . (5)

The metric g of the algebra can be calculated directlyfrom the structure constants:

gik =N∑

m,n=1

c min cn

km . (6)

Now the Casimir invariant C [20] is completely deter-mined

C =N∑

i,k=1

gikBkBi , (7)

with gik = [g−1]ki. In addition a second dimensionlessinvariant, the Casimir companion X , naturally exists:

X =N∑

i,k=1

gik〈Bk〉〈Bi〉 . (8)

Because of the linearity of equation (4), the expectationvalues Bi = 〈Bi〉 evolve according to the same equationsof motion as the operators Bi. The invariance of X fol-lows

Ẋ =N∑

i,k=1

gik(

〈Ḃk〉〈Bi〉+ 〈Bk〉〈Ḃi〉)

=N∑

i,k=1

2gik〈Ḃk〉〈Bi〉

=N∑

k,n,s=1

2ı

h̄hn(t)c snk 〈Bs〉〈Bk〉

=N∑

k,n,m=1

2ı

h̄hn(t)cnkm〈Bm〉〈Bk〉

=N∑

k,n,m=1

2ı

h̄hn(t)〈Bm〉〈Bk〉(cnkm + cnmk) = 0 ,

(9)

where we have used the symmetry of the metric, the an-tisymmetry of the structure constants, and the Jacobiidentity. The two invariants 〈C〉 and X are connected bythe system’s covariance matrix

X = 〈C〉 −N∑

i,k=1

gikCov(Bk,Bi) , (10)

with

Cov(Bi,Bj) =1

2〈BiBj + BjBi〉 − 〈Bi〉〈Bj〉 . (11)

As we shall see from the examples, this new invariantX isthe rescaled energy of the system in its equilibrium state.As the following examples show, its invariance allows usto quantify quantum friction stored in nonequilibriumstates. For sake of simplicity h̄ = 1 for the rest of thepaper.

EXAMPLE 1: THE QUANTUM HARMONICOSCILLATOR

The oscillator evolves according to the Hamiltonian:

H =P2

2m+

m

2ω(t)2Q2 , (12)

where P , Q, m and ω(t) are the momentum, position op-erators, mass of the particle and frequency of the oscil-lator, respectively. Adding the operators L and C closesthe finite Lie algebra induced by H

L =P2

2m−

m

2ω(t)2Q2 (13)

C =ω(t)

2(QP + PQ) . (14)

Using the commutation relations

[H,L] = −ı2ωC [L, C] = ı2ωH [C,H] = −ı2ωL ,(15)

the metric reads

[gik] = 8 ω(t)2

1 0 00 −1 00 0 −1

. (16)

This metric is explicitly time-dependent, because of theexplicitly time-dependent basis {H,L, C}. As remarkedabove, one could transform this basis into a static one,e.g., {Q2,P2, (QP + PQ)/2}. Using equation (10) withE = 〈H〉, the Casimir companion is

X =〈H〉2 − 〈L〉2 − 〈C〉2

8 ω2=

E2 − L2 − C2

8 ω2. (17)

At fixed ω, the equilibrated state satisfies 〈L〉 = 〈C〉 =0 [15, 21] and 〈H〉 achieves its minimum value 〈H〉eq..For this state, X becomes a scaled minimum energy:

X =〈H〉2eq.8 ω2

. (18)

All other states reachable with the dynamics havea higher rescaled energy albeit they have the samevon Neumann entropy which is a function only of X [15].The identity (10,11) after multiplying through by 8ω2

becomes(

〈H2〉 − 〈L2〉 − 〈C2〉)

−(

〈H〉2 − 〈L〉2 − 〈C〉2)

(19)

= Var(H)−Var(L)−Var(C) . (20)

2

in the basis is not real a restriction because one can al-ways transform the time dependence from the basis tothe coefficients, leading to a time-dependent system ma-trix γ kj (t) and vice versa. This matrix can be calculated

directly, knowing the time-dependent coefficients hi(t) ofthe linear expansion in eq. (1) and the time-independentstructure constants c kij of the Lie algebra.From eq. (3-4) we get

γ kj (t) =N∑

i=1

ı

h̄hi(t)c kij . (5)

The metric g of the algebra can be calculated directlyfrom the structure constants:

gik =N∑

m,n=1

c min cn

km . (6)

Now the Casimir invariant C [20] is completely deter-mined

C =N∑

i,k=1

gikBkBi , (7)

with gik = [g−1]ki. In addition a second dimensionlessinvariant, the Casimir companion X , naturally exists:

X =N∑

i,k=1

gik〈Bk〉〈Bi〉 . (8)

Because of the linearity of equation (4), the expectationvalues Bi = 〈Bi〉 evolve according to the same equationsof motion as the operators Bi. The invariance of X fol-lows

Ẋ =N∑

i,k=1

gik(

〈Ḃk〉〈Bi〉+ 〈Bk〉〈Ḃi〉)

=N∑

i,k=1

2gik〈Ḃk〉〈Bi〉

=N∑

k,n,s=1

2ı

h̄hn(t)c snk 〈Bs〉〈Bk〉

=N∑

k,n,m=1

2ı

h̄hn(t)cnkm〈Bm〉〈Bk〉

=N∑

k,n,m=1

2ı

h̄hn(t)〈Bm〉〈Bk〉(cnkm + cnmk) = 0 ,

(9)

where we have used the symmetry of the metric, the an-tisymmetry of the structure constants, and the Jacobiidentity. The two invariants 〈C〉 and X are connected bythe system’s covariance matrix

X = 〈C〉 −N∑

i,k=1

gikCov(Bk,Bi) , (10)

with

Cov(Bi,Bj) =1

2〈BiBj + BjBi〉 − 〈Bi〉〈Bj〉 . (11)

As we shall see from the examples, this new invariantX isthe rescaled energy of the system in its equilibrium state.As the following examples show, its invariance allows usto quantify quantum friction stored in nonequilibriumstates. For sake of simplicity h̄ = 1 for the rest of thepaper.

EXAMPLE 1: THE QUANTUM HARMONICOSCILLATOR

The oscillator evolves according to the Hamiltonian:

H =P2

2m+

m

2ω(t)2Q2 , (12)

where P , Q, m and ω(t) are the momentum, position op-erators, mass of the particle and frequency of the oscil-lator, respectively. Adding the operators L and C closesthe finite Lie algebra induced by H

L =P2

2m−

m

2ω(t)2Q2 (13)

C =ω(t)

2(QP + PQ) . (14)

Using the commutation relations

[H,L] = −ı2ωC [L, C] = ı2ωH [C,H] = −ı2ωL ,(15)

the metric reads

[gik] = 8 ω(t)2

1 0 00 −1 00 0 −1

. (16)

This metric is explicitly time-dependent, because of theexplicitly time-dependent basis {H,L, C}. As remarkedabove, one could transform this basis into a static one,e.g., {Q2,P2, (QP + PQ)/2}. Using equation (10) withE = 〈H〉, the Casimir companion is

X =〈H〉2 − 〈L〉2 − 〈C〉2

8 ω2=

E2 − L2 − C2

8 ω2. (17)

At fixed ω, the equilibrated state satisfies 〈L〉 = 〈C〉 =0 [15, 21] and 〈H〉 achieves its minimum value 〈H〉eq..For this state, X becomes a scaled minimum energy:

X =〈H〉2eq.8 ω2

. (18)

All other states reachable with the dynamics havea higher rescaled energy albeit they have the samevon Neumann entropy which is a function only of X [15].The identity (10,11) after multiplying through by 8ω2

becomes(

〈H2〉 − 〈L2〉 − 〈C2〉)

−(

〈H〉2 − 〈L〉2 − 〈C〉2)

(19)

= Var(H)−Var(L)−Var(C) . (20)




Entropies"

●  von Neumann entropy 

●  energy entropy!

●  one can define "contact temperature"!

X =E2 − L2 − C2

ω2

1Tcontact

=∂SE∂E

��ω

SvN = kB ln

��X − 1

4

�+√

X arcsinh

� √X

X − 14

�

SE = kB ln

��E2

ω2− 1

4

�+

�E2

ω2arcsinh

�E2

ω2

E2

ω2 −14

SE ≥ SvN




The Team"

Yair Rezek Ronnie Kosloff Peter Salamon Karl Heinz Hoffmann  

Fritz Haber Research Center San Diego Tech. Universität   Hebrew University Jerusalem State University Chemnitz!




Summary"

●  Dynamic view on the Third Law:  Study temperature dependence of the Cooling Rate!

●  "Quantum Refrigerator": parametric harmonic oscillators as working medium!

●  good indicators for non-equilibrium:  Von Neumann entropy, energy entropy!

●  Application: Maximum cooling rate of a quantum refrigerator!

●  There exist adiabatic processes in finite time !!!!

●  Generalization: Lie algebra dynamics and the Casimir Companion!




Selected References 1"

●  Optimal control in a quantum cooling problem  Salamon, Peter and Hoffmann, Karl Heinz and Tsirlin, Anatoly Applied Mathematics Letters (in press) (2012)!

●  Time-optimal controls for frictionless cooling in harmonic traps  Hoffmann, Karl Heinz and Salamon, Peter and Rezek, Yair and Kosloff, Ronnie Europhysics Letters 96(6): 60015-1--6 (2011)!

●  Optimal control of the parametric oscillator  Andresen, Bjarne and Hoffmann, Karl Heinz and Nulton, James and Tsirlin, Anatoly and Salamon, Peter  European Journal of Physics 32(3): 827--843 (2011) ; ISSN 0143-0807!

●  The quantum refrigerator: The quest for absolute zero  Rezek, Y. and Salamon, P. and Hoffmann, K. H. and Kosloff, R.  Europhysics Letters 85: 1--5 (2009)!

●  Maximum work in minimum time from a conservative quantum system  Salamon, P. and Hoffmann, K. H. and Rezek, Y. and Kosloff, R.  Physical Chemistry Chemical Physics11: 1027--1032 (2009)!




Selected References 2"

●  Optimal Process Paths for Endoreversible Systems  Hoffmann, K. H. and Burzler, J. and Fischer, A. and Schaller, M. and Schubert, S.  Journal of Non-Equilibrium Thermodynamics 28(3): 233--268 (2003)!

●  Recent Developments in Finite Time Thermodynamics  Hoffmann, K. H.  Technische Mechanik 22(1): 14--25 (2002)!

●  Endoreversible Thermodynamics  Hoffmann, K. H. and Burzler, J. M. and Schubert, S.  Journal of Non-Equilibrium Thermodynamics 22(4): 311--355 (1997)!

Optimizing Cooling Processes A Dynamic View on the Third Law · 2012. 9. 3. · Karl Heinz...

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