Subharmonic Generation and Squeezing in a Damped Oscillator

Abstract

Various approximations and assumptions are recalled on which the conventional quantum optical approach to quantum fluctuations in parametric processes with dissipation is based (weak dissipation, i.e. weak coupling of the basic oscillator to the environment so that lowest order perturbation theory applies, Markov approximation, ‘high’ temperatures k_BT» ħκ where κ is the dissipation rate, assumption of negligible influence of the parametric driving on the oscillator-reservoir interaction). Some of these approximations may not always be satisfied and all of them, taken together, do not satisfy exactly a basic fundamental principle, the fluctuation-dissipation relation in thermodynamic equilibrium. To remedy this unsatisfactory situation we develop here a general theory of parametrically driven, linear, dissipative quantum oscillators which is exact within a specified framework, and which respects all fundamental physical principles. The conventional approach is contained as a limiting case. The theory is applied to the decay of a squeezed state of a quantum oscillator where we reproduce in a very easy and direct manner results obtained earlier in the literature by a more complicated method involving a triple functional integral. The theory is also applied to subharmonic generation. The exact result for the subharmonic fluctuation intensities still involves time-integrals over response functions, which have to be calculated by solving a Mathieu equation. An approximate evaluation, valid near the subharmonic instability of the Mathieu equation, is given.