^{1,a)}and Jizhou He

^{1}

### Abstract

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*, 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

_{C}*T*and

_{H}*T*, respectively. We demonstrate that under the same conditions, the efficiency of the engine cycle is bounded from above the Carnot efficiency .

_{C}This work was supported by the National Natural Science Foundation of China under Grant Nos. 11147200 and 11065008.

I. INTRODUCTION

II. THE FIRST LAW OF THERMODYNAMICS

III. A HEAT ENGINE MODEL OF TWO NONINTERACTING FERMIONS IN A 1 D BOX TRAP

IV. OPTIMIZATION ON THE PERFORMANCE OF THE HEAT ENGINE

V. RELATIONSHIP BETWEEN EFFICIENCY OF THE QUANTUM ENGINE CYCLE AND THAT OF THE CORRESPONDING QUANTUM CARNOT CYCLE

VI. CONCLUSIONS

### Key Topics

- Entropy
- 6.0
- Excited states
- 4.0
- Recurrence relations
- 4.0
- Boltzmann equations
- 3.0
- Adiabatic theorem
- 2.0

## Figures

Graphic sketch of a three-level quantum heat engine model working with two noninteracting fermions in a 1 D box trap. At instants 1 and 4, the two particles stays in the lowest two energy levels, while at instants 2 and 3, the two particles stay in the first and second excited energy levels. In the process 1 → 2 (3 → 4) the system absorbs (releases) energy from (to) energy bath I (II) and the energy of the system is kept unchanged at a constant energy *E _{H} * (

*E*). In the adiabatic processes 2 → 3 and 4 → 1 the two-particle system is decoupled from the energy bath and stays in fixed states.

_{C}Graphic sketch of a three-level quantum heat engine model working with two noninteracting fermions in a 1 D box trap. At instants 1 and 4, the two particles stays in the lowest two energy levels, while at instants 2 and 3, the two particles stay in the first and second excited energy levels. In the process 1 → 2 (3 → 4) the system absorbs (releases) energy from (to) energy bath I (II) and the energy of the system is kept unchanged at a constant energy *E _{H} * (

*E*). In the adiabatic processes 2 → 3 and 4 → 1 the two-particle system is decoupled from the energy bath and stays in fixed states.

_{C}Schematic diagram of a quantum heat engine cycle in the plane of the width *L* and force *F*(*L*). The quantum states of the two particles and the values of the potential width at the four special instants are as follows: |*u* _{1}⟩, |*u* _{2}⟩ and *L* _{1} at the starting instant 1 of the process 1 → 2, |*u* _{2}⟩, |*u* _{3}⟩ and at the starting instant 2 of the process 2 → 3, |*u* _{2}⟩, |*u* _{3}⟩ and *L* _{3} at the starting instant 3 of the process 3 → 4, and |*u* _{1}⟩, |*u* _{2}⟩ and at the starting instant 4 of the process 4 → 1.

Schematic diagram of a quantum heat engine cycle in the plane of the width *L* and force *F*(*L*). The quantum states of the two particles and the values of the potential width at the four special instants are as follows: |*u* _{1}⟩, |*u* _{2}⟩ and *L* _{1} at the starting instant 1 of the process 1 → 2, |*u* _{2}⟩, |*u* _{3}⟩ and at the starting instant 2 of the process 2 → 3, |*u* _{2}⟩, |*u* _{3}⟩ and *L* _{3} at the starting instant 3 of the process 3 → 4, and |*u* _{1}⟩, |*u* _{2}⟩ and at the starting instant 4 of the process 4 → 1.

The dimensionless power output *P** (left) and efficiency *η* (right) vs the parameter *r* with *r* ≤ 50. In the left side, the inset figure *P** vs *r* with *r* ≤ 7; in the right side, curve a (*α* = 0), curve b (*α* = 0.03), curve c (*α* = 0.08), and curve d (*α* = 0.15) are presented.

The dimensionless power output *P** (left) and efficiency *η* (right) vs the parameter *r* with *r* ≤ 50. In the left side, the inset figure *P** vs *r* with *r* ≤ 7; in the right side, curve a (*α* = 0), curve b (*α* = 0.03), curve c (*α* = 0.08), and curve d (*α* = 0.15) are presented.

The dimensionless power output *P** vs the efficiency *η*. Curve a (*α* = 0), curve b (*α* = 0.03), curve c (*α* = 0.08), and curve d (*α* = 0.15) are presented.

The dimensionless power output *P** vs the efficiency *η*. Curve a (*α* = 0), curve b (*α* = 0.03), curve c (*α* = 0.08), and curve d (*α* = 0.15) are presented.

## Tables

The values of *r _{m} _{η} *,

*η*and

_{mp},*η*for given parameter

_{max}*α.*

The values of *r _{m} _{η} *,

*η*and

_{mp},*η*for given parameter

_{max}*α.*

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