Damped oscillator with quantum noise

by R. F. Streater, Jour. of Physics, A15, 1477-1485, 1982.

ABSTRACT

We find the general form of positive-energy Gaussian noise so that a linear damped (Bose or Fermi) oscillator obeying a quantum Langevin equation should remain canonical for all time. That is, the quantum operators p(t), q(t) obey Heisenberg's commutation relations for all times t. We show that in a heat bath the system converges to a KMS state as time goes to infinity.

KMS stands for Kubo, Martin and Schwinger.

NOTE

The noise is first taken to be a boson field in a general quasifree state, defined by its two-point function (the quantum covariance). The condition on the weight-function of the covariance of the noise, needed so that the time-evolution is unitary, is solved.

Then the same method is used to add noise to the Fermi oscillator, in such a way that the solution to the Fermi Langevin equation obeys the CAR for all time.

Using this method, and taking the asymmetric case (Section 5 of the present paper), Klauder et al are able to reproduce the process of Lewis and Thomas (the quantum Ford-Kac-Mazur process) in the limit as gamma_1 goes to zero. From our point of view this is a singular limit, as the canonical variables cease to be defined self-adjoint operators at each time; their commutator becomes a distribution in time. It leads to a Langevin equation with no noise in the equation dQ/dt=omega P, which describes a particle at position Q (and not a field quantum with noise acting on both Q and P).

This work was done while I was visiting the Mittag-Leffler Institute, Stockholm, for some months in 1981. This was called the "sleeping beauty" by Garding, referring to its under-use as a haven in which to do research. It was so obscure that the taxi-driver who took me there could not find the institute from the address given on the letter head, as it gave no street name, only Djursholm, the name of the suburb. I shared a flat with Garding, which was separated from that of the Burbea family by an internal door. Mrs. Burbea told me that we were as quiet as mice. I had agreed to talk to the visiting mathematicians on the problem of adding quantum noise to the dynamics of the oscillator, and the audience would have Garding, Malliavin, Burbea and Hejhal, among others. But as the time of the seminar approached, I realised that the condition I had obtained, for the Heisenberg commutation relations to be independent of time, led to an integral equation whose solution I had to find pdq. Fortunately, I found all the solutions just in time. Garding assured me that the work could be made rigorous. I gave a second seminar on the Ito-Clifford integral, which went down well with Malliavin.