Enhancement of geometric phase by frustration of decoherence: A Parrondo-like effect

Geometric phase plays an important role in evolution of pure or mixed quantum states. However, when a system undergoes decoherence the development of geometric phase may be inhibited. Here we show that when a quantum system interacts with two competing environments there can be enhancement of geometric phase. This effect is akin to a Parrondo-like effect on the geometric phase which results from quantum frustration of decoherence. Our result suggests that the mechanism of two competing decoherence can be useful in fault-tolerant holonomic quantum computation.

Figure 1

GP [${\gamma }_g\left(\tau \right)$] with respect to $\theta$ and $\varphi$ (a) when ${\alpha }_1=1$ and ${\alpha }_2=0$, that is, for the case of a single bath, (b) when ${\alpha }_1={\alpha }_2=\frac{1}{\sqrt{2}}$, (c) when ${\alpha }_1=\frac{\sqrt{3}}{2}$ and ${\alpha }_2=\frac{1}{2}$, and (d) when ${\alpha }_1={\alpha }_2=\frac{1}{4}$. Here $\omega =2$, and time $t=50$. A comparison between (a), (b), (c), and (d) reiterates the point that the decay of GP gets frustrated when both the baths are acting, and one of the best strategies is seen to be the case where ${\alpha }_1={\alpha }_2=\frac{1}{4}$.

Figure 2

GP for ${\alpha }_1={\alpha }_2=1/4$ with respect to $\theta$ and $\varphi$ for (a) time $t=50$ and (b) time $t=200$, respectively. Here $\omega =2$. Among other sets of ${\alpha }_1$, ${\alpha }_2$, the figure corresponding to the case (a) is found to be optimum in resisting the depletion of GP. Therefore, for different time different ${\alpha }_1$ and ${\alpha }_2$ will help to enhance the GP.

Figure 3

GP for different ${\alpha }_1,{\alpha }_2$ with respect to $\theta$ and $\varphi$ for time $t=50$ and $\omega =2$. The red curve corresponds to ${\alpha }_1=1$, ${\alpha }_2=0$; the green curve corresponds to ${\alpha }_1=0$, ${\alpha }_2=1$; the light blue curve to ${\alpha }_1={\alpha }_2=\frac{1}{4}$; while the dark blue curve corresponds to ${\alpha }_1={\alpha }_2=\frac{1}{2}$. It is clearly evident from the plots that the decay of GP gets frustrated due to the presence of both couplings ${\alpha }_1$ and ${\alpha }_2$.