Download Quantum Refrigerator in the Quest for the Absolute Zero Temperature

Quantum Refrigerator in the Quest for the Absolute Zero
Temperature
Yair Rezeka and Ronnie Kosloffa
a The
Hebrew University, Jeruslaem, Israel
ABSTRACT
One of the formulations of the third laws of thermodynamics is that a processes become more isentropic as
one approaches the absolute zero temperatures. We examine this prediction by studying an operating model
of a quantum refrigerator pumping heat from a cold to a hot reservoir. The working medium consists of a
gas of noninteracting harmonic oscillators. The model can be solved in closed form in the quasi-static limit
or numerically for general conditions. It is found that the isentropic limit for Tc → 0 is approached only on
the expansion segment of the refrigeration cycle. The scaling of the cooling rate with temperature is shown to
be consistent with the second law of thermodynamics. This scaling is also consistent with the unattainability
principle which is an alternative formulation of the third law of thermodynamics.
Keywords: Refrigeration, Abolute Zero, Finite Time Thermodynamics, Quantum Heat Engine
1. INTRODUCTION
Walter Nernst stated the third law of thermodynamics as follows: “The entropy change of any process becomes
zero when the absolute zero temperature is approached.” This statement is known as the “Nernst Heat Theorem”.1 From a dynamical point of view this entropic formulation of the third law requires that a refrigeration
cycle will become increasingly isentropic when T → 0. We find that this condition is too restrictive. In cooling
cycles optimized for maximum cooling power, imposing a condition of isoentropicity increases the cycle time and
reduces the cooling power. The system compromises by approaching isentropic conditions on the part of the
cycle affected by the cold temperature, but has a rapid non isentropic evolution on the other cycle segment. The
resulting cycle is therefore not isentropic.
Walter Nernst stated an alternative, dynamic formulation of the third law of thermodynamics: “It is impossible by any procedure, no matter how idealized, to reduce any system to the absolute zero of temperature
in a finite number of operations”.2, 3 This statement has been termed the “Unattainability Principle”.1, 4–6 In
the present study this statement is viewed by the vanishing of the cooling rate Q̇c when pumping heat from a
cold bath whose temperature approaches absolute zero. The unattainability principle is demonstrated by the
decrease of the cooling rate with the temperature.
To study these issues, a reciprocating four stroke cooling cycle is employed where the working fluid consists
of a quantum medium. A quantum analog of the classical Brayton refrigeration cycle is established. The cooling
rate is replaced by the average refrigeration power per cycle Qc /τ where τ is the cycle period. The isentropic limit
of the refrigeration cycle is explored in Section 2. These results are compared to the performance determined by
numerical analysis in Section 3, which show a deviation from the isentropic limit.
2. THE QUANTUM BRAYTON CYCLE
A Brayton refrigeration cycle, pumping heat from a cold bath to a hot one, is composed of four segments (see Fig.
1). On the expansion and compression adiabats the system is thermally decoupled from the environment (hence
“adiabat”, which should not be confused with the limit of adiabatic i.e. slow change used below) and an external
field (the piston’s volume) is altered. On each isochore the external field (volume) is fixed, and the system is put
Send correspondence to Yair Rezek, E-mail: [email protected]
2
B
ho
e
th
so
ti
1.75
rm
1
τh
Cold isochore
m
her
1.25
sot
SE
di
τch
C
τc
Hot isochore
col
1.5
r
mp
Co
dia
na
io
ess
bat
τhc
Expansion adiabat
D
1
A
2
1.5
3
2.5
ω
Figure 1. A typical optimal cooling power cycle ADCB in the energy entropy SE and frequency ω plane. The isotherms
are indicated.
2
Tc
B
Th
1.8
S
1.6
C
1.4
1.2
1
0
B
A
D
1
2
3
4
T
5
6
7
8
Figure
P 2. The von Neumann entropy Svn = −kB tr{ρ̂ ln ρ̂} (solid line) and the energy Shannon entropy SE =
−kB j Pj ln Pj (dashed line) as a function of the internal temperature. The von Neumann entropy is seen to be
constant along the adiabatic segments, and is always lower than the energy entropy. The energy entropy changes weakly
in the expansion (bottom) adiabat and significantly on the compression (upper) adiabat. In an isoentropic cycle the energy
entropy would remain constant along the expansion adiabat; in a real cycle, such as presented here, it increases somewhat
and so reduces the entropy intake in the following (cold) isochore.
into contact with a thermal bath at a certain temperature. Within the context of finite-time thermodynamics,
each segment is carried out in a certain finite time; we will mark τh as the time spent in contact with the hot
bath, τhc as the expansion adiabat, τc as the cold isochore, and τch as the compression adiabat.
In a quantum Brayton cycle, the classical particles in the piston are replaced with a quantum medium in
a potential. For simplicity of analysis, the working medium consists of a gas of non-interacting particles in an
harmonic potential. The change of volume due to the action of the piston is translated to a change in curvature
of the harmonic potential k = mω 2 /2.
On the adiabatic segments the system is uncoupled to the environment, affected only by an external time2
1
P̂ +
dependent field. The dynamics are therefore governed by the time-dependent Hamiltonian Ĥ = 2m
1
2 2
2 mω(t) Q̂ ,
where the frequency ω(t) is the time-dependent field. The equation of motion for an operator
Ô of the working medium is the Heisenberg equation:
i
∂ Ô(t)
dÔ(t)
= [Ĥ(t), Ô(t)] +
.
dt
h̄
∂t
(1)
The external power of the compression/expansion segments is the opposite of the changes in the internal energy
of the working medium.7 In thermodynamic terms, since the system is thermally decoupled the only change in
energy is due to external work. Therefore inserting Ĥ for Ô in Eq. (1) leads to the definition of power
∂ Ĥ
dE
=P = h
i
dt
∂t
Rω
The work per segment can then be integrated W = ωif Pdt.
(2)
The dynamics on the adiabatic segments is unitary. Therefore, the von Neumann
entropy S vn = −kB tr{ρ̂ ln ρ̂}
P
is constant. In contrast, the energy entropy SE changes, where SE = −kB j Pj ln Pj and Pj = tr{|jihj|ρ̂} is
the probability of occupying the energy level j (Svn ≤ SE ). Isentropic conditions are defined as constant energy entropy SE and are obtained only under quasi-static (adiabatic) conditions. A faster change of frequency
increases SE , which is reflected in frictional losses.8, 9 These are due to friction-like phenomena related to the
non-commutativity of the Hamiltonian at different times.9, 10
In the Brayton cycle, heat is transfered to the working medium from the cold bath and ejected to the hot
bath under constant external field. Along these segments (the cold and hot isochores) the system is no longer
closed. For a system in contact with a typical large environment, the dynamics are generated by the completely
positive generator ddtÔ = L∗ (Ô) = h̄i [Ĥ, Ô] + L∗D (Ô).11 The dissipative term L∗D leading to thermal equilibrium
is:
1
1
L∗D (Ô) = k↓ ↠Ôâ − {↠â, Ô} + k↑ âÔ↠− {â↠, Ô}
.
(3)
2
2
The operators ↠(ω) , â(ω) are the raising/lowering operators of the working medium defined by [Ĥ, ↠] = +ω↠,
[Ĥ, â] = −ωâ, where ω is an internal frequency of the working medium. Thermal equilibrium is only reached
k
if k↑↓ = exp(− kh̄ω
). Considering an harmonic oscillator, one frequency ω suffices to characterize all the energy
BT
gaps at all temperatures. The heat flow from the cold/hot bath is associated with the change in energy of the
working medium due to dissipation:9, 10
Q̇ = hLD (Ĥ)i
(4)
This can be seen from considering the thermodynamic significance of the evolution along an isochore: since the
volume (frequency) is fixed no external work is exchanged, leading to the change in energy being solely due to
heat-exchange.
At thermal equilibrium the energy expectation value is sufficient to fully characterize the state of a system.
Our system will not be at equilibrium in general, however. It can be shown9 that there is a generalized Gibbs
state that describes the system and is characterized by three operators: the time dependent Hamiltonian Ĥ =
2
2
K(t) 2
K(t) 2
1
1
1
2m P̂ + 2 Q̂ , the Lagrangian L̂ = 2m P̂ − 2 Q̂ and the correlation Ĉ = ω(t) 2 (Q̂P̂+ P̂Q̂). The system will
maintain this generalized state during the cycle. Moreover starting from a different initial state it will reach such
a state as a limit cycle after the refrigerator operates for sufficient time. The system is hence fully characterized
by the expectation values of these three operators. The invariance is due to the set forming a closed Lie algebra,
which leads to a closed set of operators to the equation of motion on the adiabats as well as on the isochores. 9, 12
The state is a stable, attractor state. We will therefore be interested in the evolution of these three operators.
The dynamics of the operators on the adiabats is obtained by inserting the set to Eq. (1):



Ĥ
α
d 
L̂  (t) =  α
dt
0
Ĉ
−α
α
2ω


Ĥ
0
−2ω   L̂  (t) .
α
Ĉ
(5)
where α = ω̇
ω is the adiabatic parameter. The coefficients α and ω are all functions of ω(t) and are therefore time
dependent. The functional dependence of ω(t) on t is crucial for the performance of the refrigerator. Different
functional dependance on t lead to different scaling relations of the cooling.
On the isochores the energy displays an exponential approach to equilibrium:
dĤ
dt
=
− Γ(Ĥ − hĤieq Î)
(6)
where Γ = k↓ − k↑ is the heat conductance. k↑ /k↓ = e−h̄ω/kB T obeys detailed balance where ω = ωh/c and
T = Th/c are defined for the hot or cold bath respectively. hĤieq is the equilibrium expectation of the energy.
This evolution is simply part of an exponential approach to thermodynamic equilibrium with the bath the system
is coupled to. The heat transfer becomes: Q̇ = −Γ(hĤi − hĤieq ).
The operators L̂ and Ĉ display an oscillatory decay to an expectation value of zero at equilibrium:
d
L̂
−Γ −2ω
L̂
(t) =
(t)
2ω −Γ
dt
Ĉ
Ĉ
(7)
The equation of motion (5), (6) and (7) can be solved numerically for any combinations of control parameters
and time allocation on each segment. After a few cycles of operation the refrigerator settles to a periodic limit
cycle.13 The cooling power Q/τ as well as all other thermodynamical quantities are extracted from the state
evoltuion in the limit cycle.
2.1 The Isentropic Cycle
If the change in the external field ω is sufficiently slow along an adiabatic branch the adiabatic theorem can be
applied, resulting in fixed populations and constant energy entropy SE . Note that the functional form of ω(t) is
not important as long as it changes sufficiently slowly at all times. This is the isentropic limit. The dynamics of
the energy Ĥ are easily resolved by noting that the number operator N̂ = ↠â is a constant of the motion and
requiring a closed cycle, leading to the heat exchange along the cold isochore being
c
h
∆Qc = h̄ωc Neq
− Neq
(exc − 1)(exh − 1)
.
exc +xh − 1
(8)
i
where Neq
is the expectation value of the number operator at equilibrium with the hot or cold bath, and x i = Γi τi
is a dimensionless assessment of the time spent on the isochore obtained by multiplying the time spent on it (τ i )
with the relaxation rate induced by the bath (Γi ). This expression can be divided into two functions ∆Qc = F ·G,
where F is
(exc − 1)(exh − 1)
F =
.
(9)
exc +xh − 1
and G is
βc h̄ωc
βh h̄ωh
1
− coth
.
(10)
G = h̄ωc coth
2
2
2
To achieve refrigeration we require that we cool the cold bath, i.e. that the heat exchange along it will be
positive. In the isentropic limit we need G > 0 (as F > 0). This results in the condition βh h̄ωh > βc h̄ωc (the
exact opposite of the condition for operation as an Otto-cycle engine9 ). The compression ratio must exceed the
temperature ratio for us to have a (quasi-static) refrigerator C ≡ ωωhc > T ≡ TThc . Requiring the hot bath to be
hotter than the cold, βh < βc , this also implies ωh > ωc .
We are particularly interested at the limit where the cold baths temperature approaches zero. Keeping all
other parameters constant, this limit leads to a contradiction with the condition for cooling C > T above. A
more sensible limit is to lower the frequency as we lower the temperature so that ωTcc ≡ R remains constant. In
this limit we obtain a cooling per cycle of
∆QTc c →0 → Tc
h̄R
βh h̄ωh
h̄R
(coth(
) − coth(
))
2
2
2
(11)
This implies that the entropy drop in the cold bath per cycle is constant. The question now becomes the rate.
if a constant rate can be maintained the system can reach the zero entropy state and violate the third law.
The cooling rate (CR) becomes the cooling-per-cycle (∆Qc ) divided by the total cycle time (τtot = τh +
τhc + τc + τch ). The isentropic limit requires that the oscillator frequency changes very slowly along the adiabat.
Assuming (for simplicity) a linear change in the frequency along the adiabats (ω(t) = ω init +(ωf inal −ωinit )t/τadi ),
this is equivalent to (ωh − ωc )/ωc2 τadi , setting a minimal time to spend on the adiabatic branch (τadi ). Even
considering the limit case of negligible time on the isochores (so that the total cycle time will be minimized), we
arrive at the cooling rate
h̄R3
h̄R
βh h̄ωh
CRTc →0 ≤ Tc3
(coth(
) − coth(
))
(12)
4ωh
2
2
The cooling rate decreases as the cold baths temperature decreases. The colder it gets the harder it becomes to
cool, and we cannot cool all the way to zero temperature. This is a dynamical manifestation of the unattainability
principle: cooling slows down to a screeching halt as we approach zero temperature.
2.2 Other Qualities of the Isentropic Refrigerator
Another quantity of interest is the entropy production in a cycle. For a physical process, since the cycle is closed
the entropy created in the baths has to be positive (or zero). The expression for the entropy production
1
βh h̄ωh
βh h̄ωh
∆S = −βc ∆Qc − βh ∆Qh = = F · (βc h̄ωc − βh h̄ωh )(coth(
) − coth(
))
2
2
2
(13)
is always positive. Entropy is created in the universe by a quasi-static cycle, signifying an irreversible process.
The coefficient of performance for a refrigerator is the amount of cooling achieved divided by the work intake.
In the quasi-static limit, it amounts to
1
COP =
(14)
C−1
The COP is maximal when C = T , and in the Tc → 0 limit approaches zero. Again we are seeing that as we
approach zero temperature it becomes harder and harder to cool.
Finally, it is worthwhile to consider the first-order corrections to the isentropic limit. As we move away from
infinitely slow adiabats, internal friction becomes more dominant in the cycles operations. For an exponential
frequency along the adiabats ω(t) = ω0 eαt , and assuming sufficiently long isochores, the correction term for the
cooling per cycle is
2
h̄ωc
α
(f )
coth (βh h̄ωh )
(15)
∆Qc = −
2ωh
2
which is always negative: the friction always reduces the cooling.
The correction terms for the entropy production is precisely the same as for the engine, 9 for which we received
that friction always increases the entropy production as expected.
τhc
0.6
τ Fraction
0.5
0.4
0.3
τh
0.2
τc
0.1
τch
0
-0.1
0.4
0.6
0.8
1
Tc
1.2
1.4
1.6
Figure 3. The optimal time allocation fraction for each segment as a function of the cold baths temperature T c . The
fraction of time spent on the cold-to-hot adiabat (black dots) is seen to dominate as we approach zero temperature, at
the expense of the time spent on the hot (red boxes) and cold (blue triangles) segments. The cold-to-hot adiabat (green
triangles) is very fast, only a negligible amount of time is spent on it at all temperatures.
The correction for the COP is a bit complex, but amounts to lowering the COP due to friction:
COP (f ) = −
α2
ωh2 coth(βh h̄ωh ) + ωc2 coth(βh h̄ωc )
.
4ωc ωh (ωc − ωh )2 (coth(βh h̄ωc ) − coth(βh h̄ωh ))
(16)
Friction increases the produced entropy, and reduces the effectiveness and efficiency of the refrigerator. Since
it increases the energy entropy SE , it acts as a deterrent for a speedy expansion adiabat as that will reduce the
entropy (and therefore heat) exchange with the cold bath (D → C in Fig. 1). .
3. NUMERICAL INTEGRATION
It is possible to numerically integrate the equations of motion to obtain the limit cycle beyond the isentropic limit.
For each set of external parameters (ω(t),Tc ,Th ) and time allocations to the different segments, the refrigerator
will settle into a different limit cycle. A directed Monte-Carlo search was done to seek the time allocations that
yield the maximum cooling rate for each set of external parameters (see Fig. 3). Analyzing the optimal cycles as
Tc → 0, shows that generically the time allocated to the expansion adiabat (A → D) τ hc dominates (Cf. Fig. 2).
Short expansion times cause frictional heating of the working medium so that Tint ≥ Tc . Contrary to this, the
optimal time allocated to the compression adiabat (C → B) τch can be short. This is because there is sufficient
time allocated on the hot isochore (B → A) τh to dissipate the additional frictional heating to the hot bath.
These observations mean that as T → 0 the total cycle τ is of the order of τ hc . In addition sufficient time is
allocated to the isochores so that final points A and C are very close to the hot and cold isotherms.
Quantifying the unattainability principle requires finding a limiting scaling law between the rate of cooling
and the temperature Q̇c ∝ T δ . The second law of thermodynamics already imposes a restriction on δ since
it demands a positive entropy generation σ = Ṡ in a steady state refrigeration cycle. For cyclic processes the
entropy generation is concentrated on the hot and cold interfaces: σ = −Q̇c /Tc − Q̇h /Th . If Q̇h is finite then
the exponent of Q̇c ∝ T δ becomes restricted to δ > 1. Such an exponent has been realized in refrigerator
models14, 15 where the source of irreversibility is the heat transfer. The vanishing of Q̇c is also consistent with
2
π 2 kB
Tc 16
.
the vanishing of the quantum unit of heat transport 3h̄
Log(Qc/τtot)
0
ear
lin
-5
fit:
-2
9x
2.9
+
.85
-10
-15
-4
-3.5
-3
-2.5
-2
-1.5
Log(Tc)
-1
-0.5
0
0.5
Figure 4. The logarithm of the Cooling rate as a function of the logarithm of the cold baths temperature T c . The fitted
line’s slope indicates the power law, δ ≈ 3.
If sufficient time is allocated along an adiabat, the working medium maintains constant entropy conforming
with Nernst heat theorem. From the restriction on the nonadiabatic parameter α the cycle time is estimated
as τ ≈ τhc ≈ ωωh2 . As a result the scaling of the cooling power when Tc → 0 becomes Qc /τ ∝ Tc3 (cf. Eq.
c
12), which is consistent with the δ > 1 requirement. Such a scaling is found numerically for case of a linear
dependence of ω(t) on time. Fig. 4 shows the logarithm of the cooling power vs the logarithm of temperature
confirming the T 3 asymptotic scaling law for this case.
4. CONCLUSIONS
The studied refrigeration confirms Nerst unattenability principle. The cooling rate vanishes when T c → 0, at a
rate consistent with the second law of thermodynamics. The decrease in the linear frequency regime corresponds
to the expected performance in the isentropic refrigeration cycle. However, the cycles are not isentropic as the
compression adiabatic segment may be very short with little cost due to the exponential approach to equilibrium
on the following isochore. It is therefore preferable to decrease the time on this adiabat and increase the time spent
on the expansion adiabat, so as to reduce the internal friction on the expansion and improve the performance of
the cycle as a whole. The heat theorem is therefore maintained only in the expansion adiabat.
ACKNOWLEDGMENTS
This work was supported by the Israel Science Foundation, The Fritz Haber center is supported by the Minerva
Gesellschaft für die Forschung, GmbH München, Germany
REFERENCES
1.
2.
3.
4.
5.
P. T. Landsberg Rev. Mod.Phys. 28, p. 363, 1956.
W. Nernst Nachr. Kgl.Ges. Wiss. Gott. 1, p. 40, 1906.
W. Nernst er. Kgl. Pr.Akad. Wiss 52, p. 933, 1906.
P. T. Landsberg J. Phys A: Math.Gen. 22, p. 139, 1989.
F. Belgiorno J. Phys A: Math.Gen. 36, p. 8165, 2003.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
F. Belgiorno J. Phys A: Math.Gen. 36, p. 8195, 2003.
R. Kosloff J. Chem. Phys. 80, p. 1625, 1984.
T. Feldmann and R. Kosloff Phys. Rev. E 73, p. 025107(R), 2006.
R. K. Yair Rezek New J. Phys. 8, p. 83, 2006.
R. Kosloff and T. Feldmann Phys. Rev. E 65, p. 055102 1, 2002.
G. Lindblad Comm. Math. Phys. 48, p. 119, 1976.
T. Feldmann and R. Kosloff Phys. Rev. E 68, p. 016101, 2003.
T. Feldmann and R. Kosloff Phys. Rev. E 70, p. 046110, 2004.
T. Feldmann and R. Kosloff Phys. Rev. E 61, p. 4774, 2000.
E. G. Ronnie Kosloff and J. M. Gordon Applied Physics 87, p. 8093, 2000.
L. G. C. Rego and G. Kirczenow Phys.Rev.Lett. 81, p. 232, 1998.