Entropy production and fluctuation theorems in a continuously monitored optical cavity at zero temperature
Michael J. Kewming, Sally Shrapnel
Abstract
Fluctuation theorems allow one to make generalised statements about the behaviour of thermodynamic quantities in systems that are driven far from thermal equilibrium. In this article we use Crooks' fluctuation theorem to understand the entropy production of a continuously measured, zero temperature quantum system; namely an optical cavity measured via homodyne detection. At zero temperature, if one uses the classical definition of inverse temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B2;</mml:mi></mml:math>, then the entropy production becomes divergent. Our analysis shows that the entropy production can be well defined at zero temperature by considering the entropy produced in the measurement record leading to an effective inverse temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x03B2;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">e</mml:mi><mml:mi mathvariant="normal">f</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub></mml:math> which does not diverge. We link this result to the Cramér-Rao inequality and show that the product of the Fisher information of the work distribution with the entropy production is bounded below by half of the square of the effective inverse temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>&#x03B2;</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">e</mml:mi><mml:mi mathvariant="normal">f</mml:mi><mml:mi mathvariant="normal">f</mml:mi></mml:mrow></mml:msub></mml:math>. This inequality indicates that there is a minimal amount of entropy production that is paid to acquire information about the work done to a quantum system driven far from equilibrium.