Regular Article

Positivity violations of the density operator in the Caldeira-Leggett master equation

¹ Department of Physics of Complex Systems, Eötvös Loránd University, ELTE, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary
² Department of Physics, University of Malta, Msida MSD 2080, Malta
³ Institut für Angewandte Physik, Technische Universität Darmstadt, 64289 Darmstadt, Germany
⁴ Axioplex Ltd., Külső Szilágyi út 36, H-1048, Budapest, Hungary

^a e-mail: zsolt.bernad@um.edu.mt

Received: 1 November 2018
Received in final form: 22 January 2019
Published online: 12 March 2019

Abstract

The Caldeira-Leggett master equation as an example of Markovian master equation without Lindblad form is investigated for mathematical consistency. We explore situations both analytically and numerically where the positivity violations of the density operator occur. We reinforce some known knowledge about this problem but also find new surprising cases. Our analytical results are based on the full solution of the Caldeira-Leggett master equation obtained via the method of characteristics. The preservation of positivity is mainly investigated with the help of the density operator’s purity and we give also some numerical results about the violation of the Robertson-Schrödinger uncertainty relation.