Quantum Refrigeration Quantum Refrigeration & &

Absolute Zero TemperatureAbsolute Zero Temperature

Yair RezekYair Rezek

Tova FeldmannTova Feldmann

Ronnie KosloffRonnie Kosloff

The Third Law of Thermodynamics

Heat Theorem:“The entropy change of any process

becomes zero when the absolute zero temperature is approached”

Unattainability Principle:“It is impossible by any procedure, no matter

how idealized, to reduce any system to the absolute zero of temperature”

Walter Nernst

1864-1941

P.T. Landsberg, Rev. Mod. Phys. 28, p. 363, 1956.

J Phys A: Math. Gen. 22, p. 139, 1989

F. Belgiorno J. Phys. A: Math. Gen. 36, p. 8165, 2003.

The Brayton (Otto) Cycle

Cold Bath

(at Tc)

Hot Bath (at Th)

Isochoric Cooling(cold isochore)

Adiabatic Compression(cold-to-hot adiabat)

Isochoric Heating(hot isochore)

Adiabatic Expansion(hot-to-cold adiabat)

∆Wch

∆Whc

∆Qc

∆Qh

2

c cV 2

h hV

Entropies

[ ln( )]VNS Tr

ln( )E j jj

S P P

VN ES S

Von Neumann Entropy:

Shannon Entropy of Energy:

The von Neumann entropy is always lower than the Shannon energy entropy (or equal to in a thermal state)

where Pj is the probability to measure energy eigenvalue Ej

Yet Another Third Law

( ) ln( ( )) 1 ln(1) 0 VN E B n nn

S S k p E p E

0

1

0

ˆ

0

E

E

e

e

1 0

ˆ 0

0

1

,( )E V N

T S EE

The entropy of the system approaches zero as the absolute temperature approaches zero.

Outside of equilibrium, temperature may be defined as:

S 0Tc 0

The First Law

[ , ] ( )d

X X i H X Xdt t

DL

( )d

H H Hdt t

DL

( )E H H W Qt

DL

Heisenberg equation for Open Quantum System:

Applying it to the Hamiltonian:

leads to the time-explicit First Law

Quantum dynamical interpretation:

The Model

• Ideal gas in square (1D) piston

• Quantum particles in (1D) harmonic potential

• Contact with heat bath • Weak coupling to simple thermalizing environment

c

Adiabatic Compression

Adiabatic Expansion

c h

2 2 21 1 ˆˆ ˆ( ) ( )2 2

H t P m t Qm

/ adiabatic parameter

, ( )D

iX H X X

LEquations of motion on the isochores:

0 0

0 2 0

0 2 0

0 0 0 0

eqHH H

L Ld

C Cdt

I I

Equations of motion on the adiabats:

0 0

2 2 0

0 2 0

0 0 0 0

H H

L Ld

C Cdt

I I

( ),i

X H t X

Cooling Rate in Pictures

c

Adiabatic Compression

Adiabatic Expansion

c h

Unattainability & 2nd Law

0

c h

c h

Q Q

T T

c cQ T 1

Entropy production for a cyclic process is only on the interface.

Entropy Production:

As Tc 0, the heat exchange Qc must diminish to maintain the 2nd law.

Isentropic Cycle

c c cQ T

h hc c

c c

TT

T

1coth( ) coth( )

2 2 2

c c h h

c cQ F

2

( ) h c

adiabat

c

3 cc

QCR T

UnitarityThe von Neumann entropy remains constant under unitary evolution. Isentropy in this sense is guaranteed at all temperatures.

For sufficiently slow change of frequency on the adiabatic segment, adiabatic theorem holds.

0

( )( ) (0)

tH t H

Closing the cycle, one obtains:

In order to maintain cooling at low temperatures, the coth factors necessitate changing the frequency:

For a linear frequency change:

Cooling per Cycle

1c cQ T

Isentropic

c

Adiabatic Compression

Adiabatic Expansion

c h

hc

ch

hc

Isentropic II

2P

Dimensionless measure of adiabacity:

Compression adiabat is fast Expansion adiabat is slower, but

grows faster at low Tc

Unattainability

1 3 for linear cCR T

Conclusions:

The Brayton model shows that:

• The heat theorem does not hold.

• Unattainability principle maintained.

Dynamic treatment of the cold bath is required for a more robust analysis.

“Heat Theorem”

1

t

Linear

te

t

Exponential

Adiabatic

Isentropic III

i iP

Isochores are long

Second Law

Entropy change related to energy exchange:1

{ ( )}J Tr HT

DL/ BH k T

thZ e { ( ) ln }B thJ k Tr L

Completely Positive Maps and Entropy Inequalities, Goran Linblad, Commun. Math. Phys. 40, 147-151 (1975)

0 0( | ) { ln( )} { ln( )}S Tr Tr

0 0( | ) ( | )S S

0 0( | ) ( | )S S

0( ( )) ( ( ) | ) 0d

t S tdt

0( ( )) {( ( ) ln } { ( ) ln }B Bt k Tr k Tr L L

relative entropy:

Lindblad’s theorem:

Assume steady state: is a completely positive map with generator L( ) (0) (0) tt e L