arXiv:0808.2409v1 [quant-ph] 18 Aug 2008
Entangled quantum heat engines based on two two-spin systems
with Dzyaloshinski-Moriya anisotropic antisymmetric interaction
Department of physics, School of sciences,
Beijing University of Aeronautics and Astronautics,
Xueyuan Road No. 37, Beijing 100083, People’s Republic of China
We construct an entangled quantum heat engine (EQHE) based on two two-spin systems with
Dzyaloshinski-Moriya (DM) anisotropic antisymmetric interaction. By applying the explanations
of heat transferred and work performed at the quantum level in Kieus work [PRL, 93, 140403
(2004)], the basic thermodynamic quantities, i.e., heat transferred, net work done in a cycle and
efficiency of EQHE are investigated in terms of DM interaction and concurrence. The validity of
the second law of thermodynamics is confirmed in the entangled system. It is found that there
is a same efficiency for both antiferromagnetic and ferromagnetic cases, and the efficiency can be
controlled in two manners: 1. only by spin-spin interaction J and DM interaction D; 2. only
by the temperature T and concurrence C. In order to obtain a positive net work, we need not
entangle all qubits in two two-spin systems and we only require the entanglement between qubits
in a two-spin system not be zero. As the ratio of entanglement between qubits in two two-spin
systems increases, the efficiency will approach infinitely the classical Carnot one. An interesting
phenomenon is an abrupt transition of the efficiency when the entanglements between qubits in
two two-spin systems are equal.
PACS numbers: 03.67.Hk; 03.65.Ud; 75.10.Jm
The dominating feature of an industrial society is its ability to utilize, whether for wise
or unwise ends, sources of energy other than the muscles of men or animals. Except for
waterpower, where mechanical energy is directly available, most energy supplies are in the
form of potential energy of molecular or nuclear aggregations. In chemical or nuclear reac-
tions, some of this potential energy is released and converted to random molecular kinetic
energy. Heat can be withdrawn and utilized for heating buildings, for cooking, or for main-
taining a furnace at a high temperature in order to carry out other chemical or physical
processes. But to operate a machine, one of the problems of the mechanical engineers is to
withdraw heat from a high-temperature source and convert as large as a fraction as pos-
sible to mechanical energy. We can solve the problems by using heat engines which can
extract energy from its environment in the form of heat and do useful work. We consider
for simplicity a heat engine in which the so-called “working substance” is carried through a
cycle process, that is, a sequence of processes in which it eventually returns to its original
state. All the heat engines absorb heat from a source at a high temperature, perform some
mechanical work, and reject heat at a lower temperature. Consider a heat engine operating
in a cycle over and over again and let Qhand Qlstand for the heats absorbed and rejected
by the working substance per cycle. The net heat absorbed is Q = Qh+ Ql. The useful
output of the engine is the net work W done by the working substance, and from the first
law: W = Q = Qh+ Ql. The heat absorbed is usually obtained from the combustion fuel.
The heat rejected ordinarily has no economic value. The thermal efficiency of a cycle is
defined as the ratio of the useful work to the heat absorbed: ηT= W/Qh= (Qh+ Ql)/Qh.
Because of friction losses, the useful work delivered by an engine is less than the work W,
and the overall efficiency is less than the thermal efficiency. A Carnot cycle, in which all
the heat input is supplied at a single high temperature and all the heat output is rejected
at a single lower temperature, has an efficiency less than or equal to the Carnot efficiency
ηc= (Qh+Ql)/Qh= 1−Tl/Th, where Thand Tlare the temperature of the high-temperature
energy source and the low-temperature energy sink, respectively. This is supported by the
second law of thermodynamics and numerous experimental evidences.
Quantum heat engines (QHEs), in contrast, operated by passing quantum matter through
a closed series of quantum adiabatic processes and energy exchanges with heat baths, re-
right one, γ = 0, i.e., C1= 0, η can not be zero for a range of C2and the range will be
enlarged as the ratio kTh/kTlincreases. For a same γ, kTh/kTlmust be greater in order to
get a higher efficiency. This result is consistent with the second law of thermodynamics. In
other words, the second law of thermodynamics is proved to hold all the time even when
entanglement is indeed involved.
In conclusion, by quoting the quantum interpretations of heat and work from Ref.,
we construct an entangled quantum heat engine (EQHE) based on two two-spin systems
with Dzyaloshinski-Moriya (DM) anisotropic antisymmetric interaction. The basic thermo-
dynamic quantities, i.e., the heat transferred and the work done in a cycle, and the efficiency
of EQHE are investigated in terms of DM interaction and concurrence. The condition for a
positive work is given. Four features of main results can be found in this paper. First, the
efficiency can be controlled only by spin-spin interaction and DM interaction, it is indepen-
dent of the temperature. Second, we need not entangle qubits in two-spin system 1 and we
only require the entanglement in two-spin system 2 not be zero in order to obtain a positive
net work. The entanglement in two-spin system 1 will enhance the validity of the engine.
Third, for the same γ, the ratio kTh/kTlmust be greater in order to get a higher efficiency.
Fourth, an interesting phenomenon is an abrupt transition of the efficiency when C1= C2.
These results in the paper are consistent with the second law of thermodynamics. In other
words, the second law of thermodynamics is shown to be valid in this entangled system.
It is a pleasure to thank the reviewer and editor for their many fruitful discussions about
the topic. This work was supported by the National Natural Science Foundation of China
(Grant No. 10604053) and Beihang Lantian Project.
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