The quantum heat engine and heat pump: An irreversible thermodynamic analysis of the three-level amplifier

Abstract

The manifestations of the three laws of thermodynamics are explored in a model of an irreversible quantum heat engine. The engine is composed of a three-level system simultaneously coupled to hot and cold heat baths, and driven by an oscillating external field. General quantum heat baths are considered, which are weakly coupled to the three-level system. The work reservoir is modeled by a classical electromagnetic field of arbitrary intensity, which is driving the three-level system. The first law of thermodynamics is related to the rate of change of energy obtained from the quantum master equation in the Heisenberg picture. The fluxes of the thermodynamic heat and work are then directly related to the expectation values of quantum observables. An analysis of the standard quantum master equation for the amplifier, first introduced by Lamb, is shown to be thermodynamically inconsistent when strong driving fields are used. A generalized master equation is rigorously derived, starting from the underlying quantum dynamics, which includes relaxation terms that explicitly depend upon the field. For weak fields the generalized master equation reduces to the standard equation. In very intense fields the amplifier splits into two heat engines. One engine accelerates as the field intensifies, while the other slows down and eventually switches direction to become a heat pump. The relative weight of the slower engine increases with the field intensity, leading to a maximum in power as a function of the field intensity. The amplifier is shown to go through four ''phases'' as the driving field is intensified, throughout all of which the second law of thermodynamics is generally satisfied. One phase corresponds to a ''refrigeration window'' which allows for the extraction of heat out of a cold bath of temperatures down to the absolute zero. This window disappears at absolute zero, which is conjectured to be a dynamical manifestation of the third law of thermodynamics.

Performance of discrete heat engines and heat pumps in finite time

“…This is a reduced description in which the dynamical response of the bath is cast in kinetic terms [18]. Since the dynamics has been described previously [20] only a brief summary of the main points is presented here, emphasizing the differences in the energy exchanges on the 'adiabats'.…”

“…The Carnot efficiency is a well established limit for the efficiency of lasers as well as other quantum engines [10][11][12][13][14]. Moreover, even the irreversible operation of quantum engines with finite power output has many similarities to macroscopic endo-reversible engines [15][16][17][18][19].…”

Thermodynamic analysis of quantum light amplification

“…Our model offers a statistical description for the reservoirs, and it allows us to derive thermodynamic fluxes, which in turn yield Scovil and Schulz-DuBois's efficiency formula. We note that Geva and Kosloff [8] also considered a semiclassical model for a three-level amplifier. The main difference between their model and the semiclassical ED-JCM is that in Geva and Kosloff's model the time dependence of the classical field affects the dissipative superoperator.…”

“…In principle, cavity losses can be introduced to the ED-JCM. However, we do not consider field damping in this paper, which allows us to compare the thermodynamical fluxes in the quantum ED-JCM with their analog in a semiclassical ED-JCM (section VII) and a similar model by Geva and Kosloff in which field damping is not included [7] [8]. The differences between our model and that of the SL model will be seen below to play a crucial role in our ability to give a thermodynamic foundation of amplification.…”

“…Based on Alicki's definitions for heat and work, Kosloff analyzed two coupled oscillators interacting with hot and cold thermal reservoirs in the framework of a heat engine, and showed that the engine's efficiency complies with the second law of thermodynamics [6]. In later work, Geva and Kosloff studied a three-level amplifier coupled to two heat reservoirs [7] [8]. In their model the external field influences the dissipative terms, and the second law of thermodynamics is generally satisfied.…”