Quantum coherence in the dynamical Casimir effect

We propose to use quantum coherence as the ultimate proof of the quantum nature of the radiation that appears by means of the dynamical Casimir effect in experiments with superconducting microwave waveguides. We show that, unlike previously considered measurements such as entanglement and discord, quantum coherence does not require a threshold value of the external pump amplitude and is highly robust to thermal noise.

Figure 1

Negativity $E$, quantum discord $\sqrt{D}$, and quantum coherence $\sqrt{C}$ as a function of the driving frequency for given average input thermal photons for (a) $T=30$ mK ($n^{th}=6.71\times {10}^{-4}$), (b) $T=50$ mK ($n^{th}=1.67\times {10}^{-2}$), and (c) $T=70$ mK ($n^{th}=6.71\times {10}^{-2}$). The plots are realized with experimental parameters ${\omega }_d=20\pi$ GHz, $L_{\mathrm{eff}}^0=0.5$ mm, and $v=1.2\times {10}^8$ m/s, and the range of the epsilon guarantees that the small parameter $f<0.08$ is well within the perturbative regime. While entanglement and discord have a threshold value of $\epsilon$ below which they are zero, quantum coherence is finite for any value of $\epsilon$. Note that $\epsilon =\frac{2v}{L_{\mathrm{eff}}^0{\omega }_d}f$.

Figure 2

Negativity $E$, quantum discord $\sqrt{D}$, and quantum coherence $\sqrt{C}$ as a function of the average input thermal photons for a given driving amplitude for (a) $\epsilon =0.1$ and (b) $\epsilon =0.5$). The plots are realized with experimental parameters ${\omega }_d=20\pi$ GHz, $L_{\mathrm{eff}}^0=0.5$ mm, and $v=1.2\times {10}^8$ m/s. While quantum entanglement and quantum discord vanish for a given thermal noise threshold, coherence does not disappear in that regime of temperatures and driving amplitudes. Also, quantum entanglement is less robust than discord: In (a), entanglement vanishes at $T\approx 48$ mK, and discord survives up to $T\approx 67$ mK, while in (b), the threshold is displaced to $T\approx 70$ mK for the entanglement and $T\approx 100$ mK for the discord. Note that coherence is always larger than the other two.