PHYSICAL REVIEW A VOLUME 40, NUMBER 6 SEPTEMBER 15, 1989 Limits of weak damping of a quantum harmonic oscillator A. O. Caldeira and Hilda A. Cerdeira Instituto de FIsica "Gleb 8'ataghin, " Uniuersidade Estadual de Carnpinas, 13081-Carnpinas, Sdo Paulo, Brazil R. Ramaswamy School of Physical Sciences, Jaioaharlal Nehru University, Neur Delhi I IO 067, India (Received 27 July 1988; revised manuscript received 28 February 1989) In this Brief Report we analyze the limit of very weak damping of a quantum-mechanical Browni- an oscillator. It is shown that the propagator for the reduced density operator of the oscillator can be written as a double path integral of the same form as that obtained in the high-temperature limit. As a direct consequence, we can write a Fokker-Planck equation for the reduced density operator of the weakly damped oscillator (at any temperature) involving only the damping and a generalized diffusion constant in momentum space. As has already been shown in the literature, the quantum dynamics of a Brownian harmonic oscillator can be ob- tained from the knowledge of its reduced density operator, which can be written as p(x, y, t)= f f dx'dy'J(x, y, t;x', y', 0)p(x', y', 0) (1) where J(x, y, t;x', y', 0) = ff 2)x(t')2)y(t')exp — 'S, fr[x (t'), y(t')]exp — — P[x (t'}, y (t')] x' y' with l' S,s[x(t', y(t')]= ( , 'Mx — , '—My — , '—Mcoox + —, '—Mcooy — myxx+Myyy — Myxy+Myyx)dt' (2) (3) P[x(t'), y(x')]= f vcoth f dr f dcr[x(r) — y(r)]cosv(r — o. )[x(cr) — y(cr)] . 2M' & Av o 2kT o o (4) Here, y, coo, and 0, are the relaxation, the natural, and a cutoff frequency, respectively. T is the temperature of the environment of the oscillator. For alternative ways to express the function J(x, y, t;x', y', 0) the reader is re- ferred to. ' The functional P[x (t'), y(t')] can be simplified when we want to consider temperatures much higher than Acro/k. In this case, we can first perform the frequency integration in (4) using the approximation coth(irtv/2kT) =2kT/Rv (since the typical frequencies of motion are of the order of coo and y) and reduce (4) to y[x(~), y(r)]=, f [x(r) — y(r)]'«. 2M@kT In this limit it has been shown' that the double convo- lution (1) is equivalent to a Fokker-Planck (FP) equation for the Wigner transform of p (see below). Moreover, the functional P[x, y] as written in (5} is responsible for the presence of the diffusive term in that equation. Our main goal in this Report is to show that the kernel in (4) can also be reduced to an instantaneous contribu- tion when we have y «coo regardless of the temperature. In other words, P[x, y] can be written as a single integral either when kT))iricoo (for any y) or when y «coo (for any T). Consequently we can also write a FP equation for the signer transform of p when y «coo with a gen- eralized diffusion constant due to this new instantaneous form of (4). For regimes not described by the two limits above, a generalization of the FP equation is necessary (see Ref. 4). Evaluating the path integral (2) within an infinitesimal time interval ft, t +e] and using this result in (1) we can show that the equation of motion obeyed by p(x, y, t) is (see Ref. 1) Bp fi 8 p irt B p &p Bt 2Mi gx 2Mi Qy ax — y x — y +y(x — y) + Mcoox p — . Mcooy p ap 2MykT x — y p. 3438 1989 The American Physical Society