Exponential bound in the quest for absolute zero

In most studies for the quantification of the third law of thermodynamics, the minimum temperature which can be achieved with a long but finite-time process scales as a negative power of the process duration. In this article, we use our recent complete solution for the optimal control problem of the quantum parametric oscillator to show that the minimum temperature which can be obtained in this system scales exponentially with the available time. The present work is expected to motivate further research in the active quest for absolute zero.

Figure 1

A typical extremal trajectory. The blue solid line corresponds to segments with constant control $u=u_1$ and the red dashed line to segments with $u=u_2$. The switching points lie on two straight lines passing through the origin, which are symmetric with respect to the $x_1$ axis.

Figure 2

Solutions of transcendental equation (35) (middle red line), with the $+$ sign and for the integer $n=N$ defined in (40), for various values of increasing $\gamma$. The lower (blue line) and upper (black line) bounds from (45) are also displayed. Observe that the solution converges to the limiting value $1/4$.