Optimization on a three-level heat engine working with two
noninteracting fermions in a one-dimensional box trap
Jianhui Wang,
a)
and Jizhou He
Department of Physics, Nanchang University, Nanchang 330031, China
(Received 16 September 2011; accepted 7 January 2012; published online 16 February 2012)
We setup a three-level heat engine model that works with two noninteracting fermions in a
one-dimensional box trap. Besides two quantum adiabatic processes, the quantum heat engine cycle
consists of two isoenergetic processes, along which the particles are coupled to energy baths at a
high constant energy E
H
and a low constant energy E
C
, respectively. Based on the assumption that
the potential wall moves at a very slow speed and there exists a heat leakage between two energy
baths, we derive the expressions of the power output and the efficiency, and then obtain the
optimization region for the heat engine cycle. Finally, we present a brief performance analysis of
a Carnot engine between a hot and a cold bath at temperatures T
H
and T
C
, respectively. We
demonstrate that under the same conditions, the efficiency g ¼ 1 E
C
=E
H
ðÞ
of the engine cycle is
bounded from above the Carnot efficiency g
c
¼ 1 T
C
=T
H
ðÞ.
V
C
2012 American Institute of
Physics. [doi:10.1063/1.3681295]
I. INTRODUCTION
The concept of quantum heat engine was introduced by
Scovil and Schultz-Dubois
1
and extended in many meaning-
ful later publications.
2–18
Various quantum systems can be
used as the working substance, such as spin systems,
7–12
har-
monic oscillator systems,
2
two-level or multilevel sys-
tems,
4,13,17,18
cavity quantum electrodynamics systems,
6,14
etc. The performance optimization of the quantum cycles
under constraints that determine the very path of the engine
evolution has attracted much interest.
2,7,11,12,15,18
Among
others, for the two-level engine model proposed by Bender
et al., the efficiency at the maximum value of power output
has been found to be
15
a universal constant, independent of
any parameter of the model.
In the present paper, we establish a multilevel engine
model, working with two noninteracting fermions confined in
a one-dimensional (1 D) box trap. Assuming that the potential
wall moves at a very slow but fixed average speed in one
cycle, we obtain the expressions for some important parame-
ters describing the performance characteristics of the cycle.
We determine the optimal ranges of the parameters to analyze
the general performance characteristics of the cycle model
with heat leakage. In order to compare this engine cycle with
the Carnot cycle, we describe the quantum Carnot cycle work-
ing with two noninteracting fermions in a 1 D box, under the
approximation that there are only three energy levels of the
particles are included. Comparison between the engine cycle
with the Carnot cycle shows that the value of the efficiency of
our engine model is bounded from above the Carnot value.
The plan of the paper is organized as follows: In Sec. II,
we briefly review the first law of thermodynamics in quantum
systems. In Sec. III we discuss the structure of a three-level
engine model that includes heat leakage. We study the optimi-
zation on power output, and we analyze the optimal range of
the efficiency and of the engine structure in Sec. IV.InSec.V
we discuss the relationship between the efficiency of the
engine model without heat leakage and that of the correspond-
ing quantum Carnot cycle. Sec. VI presents our conclusions.
II. THE FIRST LAW OF THERMODYNAMICS
Like the classical thermodynamics, the first law of ther-
modynamics in quantum-mechanical systems can be
expressed as a function of eigenenergies e
n
and probability
distributions p
n
,
dE ¼ dQþ dW; (1)
where
dQ¼
P
n
e
n
dp
n
(2)
and
dW ¼
P
n
p
n
de
n
(3)
stand for the heat exchange and work done, respectively,
during a thermodynamic process. As in a classical system in
which the generalized force Y
n
, conjugate to the generalized
coordinate y
n
, is written as Y
n
¼dW=dy
n
, one defines the
force for a quantum system as
F ¼
dW
dL
¼
P
n
p
n
de
n
dL
; (4)
where L is the generalized coordinate corresponding to the
force F.
III. A HEAT ENGINE MODEL OF TWO
NONINTERACTING FERMIONS IN A 1 D BOX TRAP
Schro¨dinger’s equation for a system can be given by
^
Hju
n
i¼e
n
ju
n
i,whereju
n
i and e
n
stand for the eigenstates
a)
Electronic mail: physwjh@gmail.com.
0021-8979/2012/111(4)/043505/6/$30.00
V
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2012 American Institute of Physics111, 043505-1
JOURNAL OF APPLIED PHYSICS 111, 043505 (2012)