Dynamical Casimir effect and minimal temperature in quantum thermodynamics

We study the fundamental limitations of cooling to absolute zero for a qubit, interacting with a single mode of the electromagnetic field. Our results show that the dynamical Casimir effect, which is unavoidable in any finite-time thermodynamic cycle, forbids the attainability of the absolute zero of temperature, even in the limit of an infinite number of cycles.

Figure 1

Bloch coordinate $z$ of the qubit after the $\mathrm{A}\to \mathrm{B}$ isochore, as a function of the qubit-oscillator interaction time $\tau$ (in units of the swap time ${\tau }_S)$ and the interaction strength $g$.

Figure 2

Evolution of the Bloch ball coordinates as a function of the number of Otto cycles. Top: $z$ coordinate for the initial state of the qubit $|g\rangle \langle g|$ (black full curve), $\frac{1}{2}I$ (red dashed curve), $p|g\rangle \langle g|+(1-p)|e\rangle \langle e|$, with $p=0.6$ (blue dotted curve). In the bottom panel we show, for the state $\frac{1}{\sqrt{2}}\left(\right|g\rangle +|e\rangle )$ the $x$ (black full curve) and the $y\mathrm{coordinate}$ (red dashed curve). Note that for this state $z$ evolves exactly as for the maximally mixed input state. In the inset, for the same initial state, we show the exponential decay of $\left|x\right|$ and $\left|y\right|$.

Figure 3

Asymptotic value $z_{\infty }$ of the $z$ coordinate for the qubit, as a function of the time $\tau$ and the interaction strength $g$.

Figure 4

Asymptotic value $z_{\infty }$ of the $z$ coordinate for the qubit, as a function of the interaction strength $g$, for the rectangular (black full curve) and the Hamming window for $\alpha =1$ (red dashed curve) or $\alpha =2$ (blue dotted curve). The interaction time $\tau ={\tau }_S=\pi /2g$.