Bounds on the dynamics of periodic quantum walks and emergence of the gapless and gapped Dirac equation

We study the dynamics of discrete-time quantum walk using quantum coin operations, $\widehat{C}\left({\theta }_1\right)$ and $\widehat{C}\left({\theta }_2\right)$, in time-dependent periodic sequence. For the two-period quantum walk with the parameters ${\theta }_1$ and ${\theta }_2$ in the coin operations we show that the standard deviation $\left[{\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)\right]$ is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations ${\theta }_1$ or ${\theta }_2$, ${\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)=\min \{{\sigma }_{{\theta }_1}\left(t\right),{\sigma }_{{\theta }_2}\left(t\right)\}$. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one- and two-period quantum walks, we present the bounds on the dynamics of three- and higher-period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using ${\theta }_1$ and ${\theta }_2$. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter $\theta =0$, here we show the recovery of the massless Dirac equation with nonzero $\theta$ parameters contributing to the intriguing interference in the dynamics in a totally nonrelativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.

Figure 1

Probability distribution after 200 steps of the quantum walk using a different combination of quantum coin operations and a corresponding standard deviation as a function of time. In (a), (c), and (e) we have plotted the probability distribution in position space for both one- and two-period quantum walks. We can notice that the spread of the probability for the two-period case after $t$ steps is bounded by $\pm \min \left\{t\right|cos\left({\theta }_1\right)|,t|cos\left({\theta }_2\right)\left|\right\}$. The standard deviation plot in (b) and (d) shows that ${\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)={\sigma }_{{\theta }_2}\left(t\right)$ and in (f) ${\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)={\sigma }_{{\theta }_1}\left(t\right)$. However, the interference pattern is clearly distinct with prominent oscillations for the two-period case.

Figure 2

Standard deviation ($\sigma$) as a function of ${\theta }_1$ when ${\theta }_2$ is fixed. With increase in ${\theta }_1$ we notice that ${\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)=\min \left\{t\right|cos\left({\theta }_1\right)|,t|cos\left({\theta }_2\right)\left|\right\}$.

Figure 3

Standard deviation as a function of ${\theta }_1$ and ${\theta }_2$ after 25 steps of the quantum walk. With increase in both, ${\theta }_1$ and ${\theta }_2$ we note that ${\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)=\min \left\{t\right|cos\left({\theta }_1\right)|,t|cos\left({\theta }_2\right)\left|\right\}$.

Figure 4

Group velocity obtained from the dispersion relation as a function of ${\theta }_1$ and ${\theta }_2$ for the two-period quantum walk. The group velocity obtained in the continuum limit of evolution for each step of the walk when multiplied by the number of steps of the walk it matches with the overall pattern of standard deviation obtained in discrete evolution of the walk.

Figure 5

Probability distribution after 200 steps of the quantum walk for a different combination of quantum coin operations and a corresponding standard deviation as a function of time. In (a), (c), and (e) we have plotted the probability distribution in position space for both the one- and three-period quantum walks. We can notice that the spread of the probability for the three-period case after $t$ steps is always lower than $\pm \max \left\{t\right|cos\left({\theta }_1\right)|,t|cos\left({\theta }_2\right)\left|\right\}$ but not bounded by the minimum of the two as it was for the two-period quantum walk. The standard deviation plot in (b), (d), and (f) shows that ${\sigma }_{{\theta }_1,{\theta }_2}\left(t\right)$ will be around $\min \{{\sigma }_{{\theta }_1}\left(t\right),{\sigma }_{{\theta }_2}\left(t\right)\}$.

Figure 6

Standard deviation as function of ${\theta }_1$ and ${\theta }_2$ after 45 steps of the three-period quantum walk. Except for $\theta$'s where $|cos\left({\theta }_1\right)|\approx |cos\left({\theta }_2\right)|$ and close to unity, the standard deviation is very low. This can be attributed to multiple peaks in the distribution where peaks with higher probability are closer to the origin.

Figure 7

Spread of the probability distribution in position space after 100 steps of the three-period quantum walk as a function of ${\theta }_1$ and ${\theta }_2$. This bound on the spread is obtained from the maximum of group velocity $v_3^g$.

Figure 8

Probability distribution and standard deviation after 200 steps of the $n$-period quantum walk. In (a) the probability distribution for three-period, four-period, and fifty-period quantum walks is shown. The inset shows the position probability distribution for the two-period walk and when the coin is homogeneous (one-period) with the coin parameters ${\theta }_1$ and ${\theta }_2$. The standard deviation (b) shows only the two-period evolution is bounded by ${\theta }_1$; for the three-period and four-period evolutions it is bounded between ${\theta }_2$ and ${\theta }_1$. We can verify that the spread in probability distribution is bounded by a maximum of group velocity for all $n$-period quantum walks.

Figure 9

Entanglement between the particle and position for 200 steps for one-, two-, three- and fifty-period quantum walks. For the two-period quantum walk (a), in contrast to the standard deviation, the mean value of entanglement is bounded around the maximum of the two on-period quantum walk. For the three-period quantum walk, entanglement reaches a maximum possible value and from the larger $n$-period quantum walk we can see how the enhancement happens when the quantum coin operation with ${\theta }_2$ is introduced periodically.