Two-Level System with Dissipation. Off-Resonance Amplification at Nonequilibrium Positive Temperatures

Abstract

Previously derived equations of motion---in terms of the expectation values of the Pauli spin matrices---for a two-level system (TLS) coupled to a loss mechanism (LM) are generalized to include higher order interactions which describe the reaction of the LM to the effect of the driving field on the TLS. This generalization results in the addition of a driving term to the equations of motion which is approximately independent of the state of the TLS and involves parameters of the LM that characterize its response at the difference between the TLS frequency $\ensuremath{\omega}$ and the driving-field frequency $\ensuremath{\nu}$. The new driving term is negligible unless the TLS possesses a permanent dipole moment, and, in the case of a magnetic TLS, unless the driving field has a transverse component. A solution of the equations of motion for a magnetic TLS being driven by a transverse rotating field shows that amplification of the driving field may be obtained without population inversion of the TLS provided $\ensuremath{\nu}<\ensuremath{\omega}$, provided the loss constant $\ensuremath{\eta}$ (which is a measure of the response of the LM) has a frequency dependence such that $\ensuremath{\eta}(\ensuremath{\omega}\ensuremath{-}\ensuremath{\nu})$ is significant compared to $\ensuremath{\eta}(\ensuremath{\nu})$, and provided the TLS temperature exceeds the LM temperature by an appropriate amount that is minimized when $\frac{\ensuremath{\eta}(\ensuremath{\nu})}{\ensuremath{\eta}(\ensuremath{\omega}\ensuremath{-}\ensuremath{\nu})}$ is a minimum. If $\frac{\ensuremath{\eta}(\ensuremath{\nu})}{\ensuremath{\eta}(\ensuremath{\omega}\ensuremath{-}\ensuremath{\nu})}$ is small compared with unity, amplification is obtained when the TLS temperature is larger than the LM temperature by a factor $\frac{\ensuremath{\omega}}{(\ensuremath{\omega}\ensuremath{-}\ensuremath{\nu})}$. The induced emission power is discussed.