Improving the Efficiency of an Ideal Heat Engine:
The Quantum Afterburner
Marlan O. Scully
Department of Physics and Department of Electrical Engineering, Texas A&M University, TX 77843
Max-Planck-Institut f¨ur Quantenoptik, D-85748 Garching, Germany
(May 28, 2001: 23: 13)
By using a laser and maser in tandem, it is possible to obtain laser action in the hot exhaust
gases involved in heat engine operation. Such a ”quantum afterburner” involves the internal quantum
states of working gas atoms or molecules as well as the techniques of cavity quantum electrodynamics
and is therefore in the domain of quantum thermodynamics. As an example, it is shown that Otto
cycle engine performance can be improved beyond that of the ”ideal” Otto heat engine.
The laws of thermodynamics [1] are very useful in
telling us how things work and what things will never
work. For example, The ideal heat engine is a paradigm
of modern science and technology. We read in the text-
books that:
”it might be supposed that the ideal cycle analysis is too
unrealistic to be useful. In fact, this is not so. Real
gas cycles are reasonably close to, although always less
efficient than, the ideal cycles.”
But as technology develops, it behooves us to reexam-
ine thermodynamic dogma. The purpose of the present
paper is to reconsider the operating limits of ideal heat
engines in light of recent developments in quantum optics
such as cavity QED [2], the micromaser [3], and quantum
coherence effects such as lasing without inversion (LWI)
[4] and cooling via coherent control [5]. In particular,
we shall show that by extracting coherent laser radia-
tion from the ”exhaust” gas of a heat engine [6], e.g.,
the Otto cycle idealization of the automobile engine. We
here show that it is indeed possible to improve on the ef-
ficiency of an ideal Otto cycle engine, operating between
two fixed temperature reservoirs, by adding a quantum
afterburner which extracts coherent energy from the hot
exhaust gases of the heat engine.
In such a quantum Otto engine (QOE) the laser en-
ergy is supplied by a thermal reservoir in accord with
the first law of thermodynamics, and entropy balance is
maintained as required by the second law [1].
In what follows we present a physical picture for and
thermodynamic analysis of the QOE. In the conclusion
we make contact with related previous work. In the next
section we present the quantum engine concept physically
as an extension of the conventional Otto cycle engine as
in Fig.(1). Then we analyze the QOE, calculating the
efficiency and entropy flow, etc. The proposed scheme
is simple enough to permit reasonably complete analy-
sis; but, hopefully, realistic enough to be convincing. In
the conclusion, we examine our results in the context of
previous research on the subject.
In order to present the physics behind the QOE, con-
sider Fig.(1) in which the working fluid passes through
the cycle 1234561. As mentioned earlier, we extend
the classical Otto engine to include a laser arrangement
which can extract coherent laser energy from the inter-
nal atomic degrees of freedom. As depicted in Figs.(1a-e),
the QOE operates in a closed cycle in the following steps:
a.(1 → 2) The hot gas expands isentropically doing use-
ful (”good”) work W
g
= C
v
(T
1
− T
2
)whereT
2
= T
1
R
−1
,
C
v
is the heat capacity, R =(V
1
/V
2
)
(γ−1)
,andγ is the
ratio of heat capacities at constant pressure to constant
volume.
b.(2 → 3) Heat Q
out
= C
v
(T
2
− T
3
) is extracted at con-
stant volume by a heat exchanger at temperature T
3
.
c.(3 → 4) Maser-laser cavities are added and energy is
extracted from the hot internal atomic degrees of free-
dom by cycling the gas from left to right to left through
the laser-maser system held at temperature T
3
,withan
entropy decrease ∆S
int
∼
=
Nkln 3, as discussed later.
d.(4 → 5) The gas is then compressed isentropically to
volume V
4
= V
1
, requiring waste work W
w
= C
v
(T
4
−
T
3
)=(R−1)T
3
.
e.(5 → 6) The gas is again put in contact with the
heat exchanger (at temperature T
1
) and the external-
transnational degrees of freedom are heated isochorically
to T
1
by heat energy Q
ext
.
f. (6 → 1) Maser-laser cavities are again added and in-
ternal states are heated by an amount q
in
extracted from
the hot cavities at temperature T
1
, completing the cycle.
As a useful simplifying assumption, we consider the ex-
ternal and internal degrees of freedom to be decoupled.
Only when the atoms are passing through the maser-
laser system do they change their internal state. That is,
the atomic states are chosen to be very long lived when
not in the cavities. But they are strongly coupled to
the radiation field in the maser and laser cavities due to
the increased density of states of the radiation inside the
cavity. Thus, when an atom in the |bi state is passed
into the maser cavity it quickly comes into equilibrium
with the thermal radiation in the cavity. For example in
step 3 → 4 , after passing through the cold maser, |bi
state population is determined by the Boltzmann factor
governed by temperature T
3
. And for small enough T
3
the |bi state is effectively depopulated thus providing a
population inversion between states |ai and |bi since the
1
population in state |ai is still determined by T
1
. This is
the basis for lasing off the thermal energy of the exhaust
gases.
Thus the maser serves as the incoherent (”heat”) en-
ergy removal mechanism,q
in
, which enables the coherent
(useful) energy, w
l
, to be emitted by the laser. To under-
stand the work w
l
vs heat q
m
aspect of the problem we
need only compare the photon statistics for the incoher-
ent thermal field in the maser cavity with the coherent
laser field. The maser field density matrix is given by [7]:
r
m
nn
=¯n
n
m
/(¯n
m
+1)
n+1
(1a)
where ¯n
m
=1/[exp(¯hν
m
/kT
2
)−1], ¯hν
m
is the energy per
quantum of the maser field.
The density matrix describing the laser field proceeds
from an initial thermal state which is largest for small ¯n
l
,
to the sharply peaked coherent distribution given by
r
(l)
n,n
= r
0,0
n
Y
l=1
A(l +1)
1+(A/B)(l +1)
− C(l +1)+
¯n
l
¯n
l
+1
(1b)
where A(C) is the linear hain (loss) and B is the non-
linear saturation parameter, ¯n
l
is the average number of
thermal photons in the laser cavity at temperature T
2
with no atoms present.
Having established the fact that only the laser radia-
tion contributes useful work, we write the efficiency of
the QOE as:
η
qo
=
W
g
− W
w
+ w
l
Q
in
+ q
in
and since q
in
= w
l
+ q
m
we find:
η
qo
= η
0
+
w
l
(1 − η
0
) − η
o
q
m
Q
in
+ w
l
+ q
m
. (2)
where the ideal classical Otto engine efficiency is defined
as η
0
=(W
g
− W
w
)/Q
in
. Taking W
g
, W
w
,andQ
in
as
given in the discussion of QOE operation we have the
alternative expressions η
0
=1− 1/R =1− T
2
/T
1
.
In order to determine whether η
q0
, Eq.(2), is an im-
provement over η
0
we now turn to the calculation of w
l
and q
m
. A rigorous calculation requires a quantum the-
ory of the laser/maser type analysis and this will be given
elsewhere. However it is sufficient for the present pur-
poses to apply microscopic energy balance calculations
to obtain good expressions for the important quantities.
After the atom makes one pass through the maser-laser
system, the internal density matrix is given by:
ρ
one
(3) =
1
2
(p
1
a
+ p
3
b
)(Λ
a
+Λ
b
)+(p
1
c
+ p
1
b
− p
3
b
)Λ
c
(3)
where Λ
a
= |αihα|,α = a, b, and c, and in the no-
tation of Fig.(1), p
i
α
is the Boltzmann factor given by
p
i
α
= Z
−1
i
exp(−β
i
α
)whereβ
i
=1/kT
i
; T
i
is the reser-
voir temperature T
1
or T
2
and Z
i
=
P
exp −(β
i
α
). But
an atom will bounce many times back and forth through
the maser/laser cavities in moving the gas adiabatically
from right to left. After many bounces the atom settles
down into the mixed state:
ρ
many
(3) = p
3
b
Λ
a
+ p
3
b
Λ
b
+(1− 2p
3
b
)Λ
c
(4)
We calculate q
m
by noting that (p
1
a
− p
3
b
)N atoms go
from a → b → c and (p
1
b
− p
3
b
)N atoms go from b → c,
and in both cases add energy
b
−
c
to the maser field.
Thus on making the b → c transition, the total incoher-
ent energy added to the maser field by all N atoms is:
q
m
=(
b
−
c
)N[p
1
a
− p
3
b
+ p
1
b
− p
3
b
](5)
Likewise w
l
is obtained by noting that the number of
atoms going from a to b with the coherent emission of
laser radiation is N(p
1
a
− p
3
b
) . Energy
a
−
b
is given up
by each atom, and the total coherent energy (i.e. useful
work) given to the laser field is:
W
l
=(
a
−
b
)N(p
1
a
− p
3
b
)(6)
We now use Eqs.(5,6) for q
m
and w
l
in a form which
allows us to determine the sign of the efficiency enhance-
ment factor in Eq.(2). That is, we wish to establish the
conditions for which:
(1 − η
0
)w
l
>η
0
q
m
(7a)
We use Eqs.(5,6) and introduce the notation
αβ
=
α
−
β
to write Eq.(7a) as:
1
η
0
− 1
ac
bc
− 1
> 1+
p
1
b
− p
3
b
p
1
a
− p
3
b
(7b)
As an example, we may take η
0
=1/4,
ac
/
bc
=11so
that the LHS of Eq.(7b) is 30; furthermore noting that for
high enough T
1
that p
1
α
' 1/3 and so long as p
3
b
< 1/3
the RHS of Eq.(7b) equals 2. Hence the RHS (maser)
factor is an order of magnitude less than the LHS (laser)
factor in (7a,b), which indeed shows that η
q0
>η
0
as
desired. Finally we note that the von Neumann entropy,
S = −kNT rρ ln ρ, added in the heating of the internal
states to temperature T
1
:
S
int
(6 → 1) = −kN
X
p
1
α
ln p
1
α
− 2p
3
b
ln p
3
b
− p
c
ln p
c
(8)
where p
c
=1−2p
3
a
, is equal and opposite to that removed
in the 4 → 4 maser-laser energy-entropy extraction pro-
cess. Hence when T
1
is high enough and T
3
is low enough
that p
c
∼
=
1, p
a
= p
b
= 0 then (8) takes the simple form
S(6 → 1)
∼
=
kN ln 3 as noted earlier. We now turn to the
relation of the present results to that of previous work.
The landmark paper by Ramsey [9] on negative tem-
peratures in thermodynamics and statistical mechanics is
2
a pillar of quantum thermodynamics and is directly rele-
vant to the present work. In his paper he shows that his
work is contrary to the Kelvin-Planck [10] statement of
the second law, which has now been revised to the Kelvin-
Planck- Ramsey statement. Furthermore in ref [9] it is
noted that: systems at negative temperatures have vari-
ous novel properties of which one of the most intriguing
is that a heat engine operating in a closed cycle can be
constructed that will produce no other effect than the
extraction of heat from a negative temperature reservoir
with the performance of the equivalent amount of work.
But it is also noted that at both positive and negative
temperatures, cyclic heat engines which produce work
have efficiencies less than unity, i.e. they absorb more
heat than they produce work.
The present study, on the other hand, does not in-
volve negative temperature reservoirs. But it is possible
to envision a negative temperature as being associated
with the a → b transition once inversion is produced by
the maser interaction, and the present work has much in
common with that of Ramsey.
The work of Ramsey led to the introduction of the
quantum heat engine concept by Scovil and Schultz-
Dubois. In their paper [11] entitled, Three level masers
as heat engines they conclude that the limiting efficiency
of their three level maser engine model is that of the
Carnot cycle. And that their work may be regarded as
another formulation of the second law of thermodynam-
ics. In the present paper the atomic states are not to be
viewed as the engine. We focus on the different problem
of improving the efficiency of an ideal heat engine which
has a laser-maser system integrated into an Otto cycle
engine as in Fig. (1).
The present results are an extension of the work by
the author and colleagues listed in ref [6] . In particular,
the paper presented at the Dec. 1999 Japanese-American
Conference on Coherent Control entitled Using External
Coherent Control Fields to Produce Laser Cooling With-
out Spontaneous Emission and Sharpen Thermo d ynamic
Dogma [12] , gave specific examples and direct calcula-
tion, based primarily on breaking emission symmetry as
in lasing without inversion. Thus demonstrating that
cooling of internal states by external coherent control
fields is possible. There we also showed that such co-
herent schemes allow us to reach absolute zero in a finite
number of steps, in contrast to usual third law of ther-
modynamics dogma.
It is interesting to compare this work with the paper
[13] of Kosloff, Geva, and Gordon entitled Quantum Re-
frigerators in the Quest of Absolute Zer o in which they
have independently arrived at similar conclusions using
a similar model. They have established a bound for the
maximum cooling rate in the low temperature limit where
quantum behavior dominates.
In conclusion: we have shown that it is possible, in
principal, to improve on the efficiency of an ideal Otto cy-
cle engine by extracting laser energy from exhaust gases;
this is summarized in Fig.(2). However, as will be pre-
sented elsewhere, when a similar lasing-off-exhaust-atoms
scheme is analyzed for a Carnot cycle engine, efficiency is
not improved. The present results are in complete accord
with the second law.
The author wishes to thank G. Agnolet, H. Bailey, G.
Basbas, J. Caton, T. Lalk, S. Lloyd, A. Matsko, F. Nar-
ducci, N. Nayak, H. Pilloff, N. Ramsey, Y. Rostovtsev,
S. Scully, and M. S. Zubairy for helpful discussions. The
support of the Office of Naval Research, the National Sci-
ence Foundation, and the Robert A. Welch Foundation
is also gratefully acknowledged.
[1] For an insightful discussion of the physics behind the
second law see: E. Lieb and J. Yngvason, Phys. Today,
P.32, April 2000, and for applications see: K. Wark, in
Advanced Thermodynamics for Engineers McGraw-Hill
(1995).
[2] S. Haroche and D. Kleppner, Phys. Today, 42, 24, (1989),
and S. Haroche and J. Raimond, in Atomic and Molecu-
lar Physics, 20, ed. D. Bates and B. Bederson (Academic,
1985), p.350.
[3] D. Meshede, H. Walther, and G. Miller, Phys. Rev. Lett.
54, 551 (1985); A review is given by G. Raithel, C. Wag-
ner, H. Walther, L. M. Narducci, and M. Scully, in Cavity
Quantum Electrodynamics, ed. P. R. Berman (Academic,
Boston, 1994), p.57. The first theory for the micromaser
is given by P. Filipowicz, J. Javanainen, and P. Meystre,
Phys. Rev. A 34, 3077, (1985).
[4] For reviews of such studies see O. Kocharovskaya, Phys.
Rep. 219, 175 (1992), M. O. Scully, Phys. Rep. 219, 191
(1992), and S. Harris, Phys. Today 36, June (1997) as
well as chapter 7 of ref [7].
[5] M. Scully, Y. Aharonov, D. J. Tannor, G. Sussmann and
H. Walther, Using External Cohere nt Control Fields to
Produce Laser Cooling Without Spontaneous Emission,
to be published. And, C. Roos, D. Leibfried, A. Mundt,
F. Schmidt-Kaler, J. Eschner, R. Blatt, LANL: Quant-
ph/0009034, (2000).
[6] See for example F. Sears, Thermodynamics Addison-
Wesley (1959).
[7] See for example M. Scully and S. Zubairy, Quantum Op-
tics Cambridge Press 1997
[8] See M. Scully and W. Lamb, Phys. Rev. Lett. 16, 853
(1966). The luminary work of M. Lax on the subject
found in W. Louisell, Quantum Statistical Properties
of Radiation, John Wiley (1973), and the famous Laser
Theory Handbook article of H. Haken, Springer (1972).
[9] N. Ramsey, Phys. Rev. 103, 20 (1956).
[10] G. Hatsopoulos and J. Keenan, Principles of General
Thermodynamics, John Wiley & Sons, Inc., Reprint Edi-
tion (1981).
3
[11] H. Scovil and E. Schulz-Dubois, Phys. Rev. Lett. 2 262
(1969).
[12] Workshop on Coherent Control, R. Gordon Ed., World
Scientific, (2001); See also ref. [5].
[13] R. Kosloff, E. Gava, and J. Gordon, J. App. Phys. 87,
8093 (2000).
Figure Captions
Fig. (1) illustrates the steps, 1234561, in cyclic operation
of the quantum Otto engine. The atomic internal popu-
lations are depicted for the three level atom (levels a, b,
c) at each stage of operation. A detailed description of
operating steps a → f isgiveninthetext.
Fig. (2) is a temperature (T) entropy (S) plot for the
quantum Otto cycle engine. Note that the entropy is
the sum of entropy for the external (kinetic) and internal
(quantum) degrees of freedom.
Fig. (3) depicts the evolution of internal atomic popula-
tions for the case in which the atom first passes through
the maser-laser system; and then (because everything is
adiabatic and involves long times and many bounces)
bounces back and forth through the cavities many times.
As discussed in the text, after, a large number of bounces
the atom settles down into a configuration wherein most
of the population is in state c.
q
m
Laser
Isoentropic Expand
1
!
2
Maser
Isochoric Cooling
S
1
00
00
00
00
11
11
11
11
V
1
00
00
00
00
11
11
11
11
00
00
00
00
11
11
11
11
(
d
)
(
e
)
(
f
)
T
3
00
00
00
00
11
11
11
11
T
1
2
!
3
3
!
4
Q
in
W
w
Isoentropic Compress
4
!
5
Isochoric Heating
5
!
6
Internal Heating
6
!
1
a
b
c
p
3
a
p
3
b
p
3
c
a
b
c
p
3
b
p
3
c
a
b
c
p
3
a
p
3
b
p
3
c
W
l
(
a
)
(
b
)
(
c
)
q
in
Q
out
00
00
00
00
11
11
11
11
S
2
00
00
00
00
11
11
11
11
W
g
V
2
p
3
a
a
c
b
p
1
a
p
1
b
p
1
c
a
c
b
p
1
a
p
1
b
p
1
c
a
c
b
p
1
a
p
1
b
p
1
c
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
000000
00000
0
00000
0
111111
11111
1
11111
1
FIG. 1.
4
5
3
2
1
6
W
g
Q
out
Q
in
q
m
W
l
W
w
q
in
4
T
S
ext
+
S
int
FIG. 2.
c
a
b
c
a
b
c
a
b
c
a
b
00000
00000
00000
00000
00000
00000
00000
11111
11111
11111
11111
11111
11111
11111
00000
00000
00000
00000
00000
00000
11111
11111
11111
11111
11111
11111
00000
00000
00000
00000
00000
00000
11111
11111
11111
11111
11111
11111
00000
00000
00000
00000
00000
00000
11111
11111
11111
11111
11111
11111
c
b
a
(
a
)
1
st
Pass
Laser
Maser
(
b
)
(
c
)
00000
00000
00000
00000
00000
00000
11111
11111
11111
11111
11111
11111
00000
00000
00000
00000
00000
00000
11111
11111
11111
11111
11111
11111
2
nd
Pass
m
th
Pass
p
1
a
1
2
0
@
p
1
a
+
p
3
b
1
A
1
2
0
@
p
1
a
+
p
3
b
1
A
p
3
b
1
2
2
6
6
6
4
1
2
0
@
p
1
a
+
p
3
b
1
A
+
p
3
b
3
7
7
7
5
1
2
2
6
6
6
4
1
2
0
@
p
1
a
+
p
3
b
1
A
+
p
3
b
3
7
7
7
5
p
3
b
p
3
b
c
b
a
p
1
a
p
1
b
p
1
c
c
a
b
1
2
0
@
p
1
a
+
p
3
b
1
A
p
3
b
FIG. 3.
5
