Frequency Shifts and Master Equations for a Quantum Oscillator Coupled to a Reservoir

✍ Scribed by G.W. Ford; R.F. O'Connell

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 128 KB
• Volume: 269
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1006/aphy.1998.5835

✦ Synopsis

For a quantum oscillator coupled to a reservoir, master equations obtained under the assumptions of weak coupling and use of a rotating-wave Hamiltonian (RWA) are known to give incorrect frequency shifts. Here, we show that a calculation which does not invoke the RWA gives results for the frequency shifts, which agree with exact results for Lamb-type (temperature T=0) shifts. However, for non-zero T, we point out that, in general, corresponding energy and free energy shifts for the system require exact treatments since off-resonant contributions (which are automatically excluded in weak coupling calculations) are important in the case of super-Ohmic reservoirs.

1998 Academic Press

Master equations are pervasive in many areas of physics, particularly quantum optics. They give information, primarily, on the time dependence of the reduced density matrix but they also give information on energy shifts (i.e., Lamb-type shifts). Most derivations in the literature assume (a) weak coupling and (b) the rotating-wave approximation (RWA) but it is well-known that the energy shifts resulting from these calculations are not correct. In particular, Ackerhalt et al.

[1], among others, have correctly pointed out that counter-rotating terms (which are dropped in the RWA) make a major contribution to the Lamb shift.

Recently, we have derived a master equation for an oscillator coupled to a very general dissipative environment (in particular, the electromagnetic field) without using the RWA . As expected, the frequency shifts are different from the usual RWA results but the question still remains as to whether or not they are correct. This question can be answered by virtue of the fact that we have previously calculated the exact energy shift for the problem [3] and it is our purpose here to compare the various results.