Quantum Heat Engines Described by a Master Equation with Pumping Terms

Eitan Geva, Ronnie Kosloff

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 electro-magnetic 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.

Mohammad H. Ansari, Yuli V. Nazarov

T. E. Humphrey, R. Newbury, Richard P. Taylor +1 more

Yuri V. Rostovtsev, Zoe‐Elizabeth Sariyanni, Marlan O. Scully

A master equation for a single mode radiation field interacting with a coherently prepared beam of atoms is derived. The atomic quantum coherence is built up via interaction with an external microwave field. We discuss various applications of the derived master equation, including lasing without population inversion and the possibility of exceeding the Carnot efficiency of a quantum heat engine fueled by a beam of phase coherent atoms. The main motivation for this paper is to settle subtle technical issues involving the derivation of the master equation.

Gershon Kurizki, Abraham G. Kofman

Eitan Geva, Ronnie Kosloff

The finite-rate performance of a quantum heat engine, constructed from a three-level amplifier, is analyzed. Consistent definitions of thermodynamical quantities in terms of quantum observables are postulated. The performance is analyzed in steady state, where the operation of the amplifier only influences the surroundings. Quantum master equations describe the irreversible dynamics induced by the coupling of the working medium to the reservoirs. It is shown that the standard assumption of field-independent dissipation is inconcistent with thermodynamics. Field-dependent relaxation equations, based upon the semigroup approach, and consistent with thermodynamics, are formulated. These equations are valid if the time scale of the external field is slow compared to that associated with the bath fluctuations. The steady-state values of the thermodynamical quantities are evaluated. The power is found to have maxima as a function of important controls, such as the field amplitude, frequency, and the coupling with the baths. The existence and locations of these maxima differ from those obtained in the standard treatment, where the dissipation is field independent. The irreversible nature of engine operation is due to the finite rate of heat transfer and a genuine ``quantum-friction'' loss term due to dephasing.