A NEW QUANTUM FORMALISM FOR DAMPING AND GAIN IN HARMONIC-OSCILLATORS

A new formulation of loss or gain in the quantum theory of harmonic oscillators is put forward using a non-passive reactive circuit which can be readily quantized. The analysis is based on the electrical circuit theory and demonstrates how a circuit, with negative inductance –Ln and negative capacitance –Cn, coupled to a conventional harmonic oscillator circuit, of positive inductance L and positive capacitance C, can act as a source or sink of energy and allow for both gain and loss. Classically this series circuit is indistinguishable in its transients from either a + G or –G conductance shunted across a main LC oscillator circuit. However, unlike the resistive circuit, this coupled circuit can be quantized, maintaining the uncertainty principle. A two-valued solution is found, dependent on whether the circuits are in a state to receive energy or a state to absorb energy. A full correspondence, including secondorder frequency shifts, is found between the quantum and the classical solutions with states which are appropriate to thermodynamic equilibrium of a conductance at a temperature T as well as to the classical-like coherent states. While the accessible mode in the + L + C circuit does not exhibit any squeezing directly, the system as a whole is an example of two-mode squeezing discussed by other authors. © 1990 Taylor & Francis Ltd.