Comment on “The Lax-Onsager Regression “Theorem” revisited”

Abstract

Ž We have just read the paper by Lax to appear in w x. this volume 1 and we are pleased to note that Lax and ourselves are in basic agreement; the Lax formula is correct and extremely useful for systems where the coupling is weak and only frequencies near resonance are of interest. Indeed, it is the only approximation we have for the strongly driven systems of quantum optics. We should stress that we never had objections to necessary and appropriate approximations such as the Lax formula; our objection is when these approximations are elevated to Ž general principles not by Lax himself who, in his w x Ž . initial 2 referred to as Q II by Lax and subsequent w x Ž . w x publications 3 referred to as Q III by Lax 4 Ž . w x Ž referred to as Q XI by Lax 5 referred to as Q IV . by Lax was careful to note the domain of applicabil. ity of his formula . However, we are puzzled when Lax says that the Ž . w x key statement viz. the title of Ref. 6 can be w x misleading. All we did in Ref. 6 was present an exactly solvable example demonstrating that the Onsager regression hypothesis fails in the quantum case. We then went on to point out that there is a correct and broadly applicable quantum generalization of the Onsager hypothesis: the fluctuation-disw x sipation theorem of Callen and Welton 7 . We do not agree that this is simply a mathematical statement. It is true that much of our work has involved the message that quantum noise is universal: its spectrum is that of Planck plus, where appropriate, zeropoint noise. It might be helpful to note that for classical stochastic processes, as described by a Langevin equation, there are two equivalent characŽ . terizations of a Markov process: a no memory in Ž . the dissipative term and b white noise power spectrum. In fact, in 1965, using a model of coupled oscillators to describe a heat bath, Ford, Kac and w x Mazur 8 showed that, in the context of the quantum Ž . Ž . Langevin equation, one can have a but never b . The Lax model for a maser or laser is based on the idea that accurate results are possible even if the noise is not white provided that it does not vary too much over the width of the line. Rather than trying to characterize the nature of this approximation, we feel it is best to refer to the detailed exposition given w x by Lax 4,5 and summarized by him in his present w x paper 1 . For a detailed and exact treatment of a quantum particle moving in an arbitrary potential w x and coupled to a heat bath, we refer to Ref. 9 , which also discusses various models not found in the quantum optics literature. In addition, as well as presenting a general result for the symmetric autocorrelation of the random force, we also wrote down its nonequal-time commutator so that the correspondŽ ing result for the normally ordered operators used to