Perfect wave-packet splitting and reconstruction in a one-dimensional lattice

Particle delocalization is a common feature of quantum random walks in arbitrary lattices. However, in the typical scenario a particle spreads over multiple sites and its evolution is not directly useful for controlled quantum interferometry, as may be required for technological applications. In this paper we devise a strategy to perfectly split the wave packet of an incoming particle into two components, each propagating in opposite directions, which reconstruct the shape of the initial wavefunction after a particular time $t^{*}$. Therefore, a particle in a $\delta$-like initial state becomes exactly delocalized between two distant sites after $t^{*}$. We find the mathematical conditions to achieve the perfect splitting, which are satisfied by viable example Hamiltonians with static site-dependent interaction strengths. Our results pave the way for the generation of peculiar many-body interference patterns in a many-site atomic chain (such as the Hanbury Brown and Twiss and quantum Talbot effects) as well as for the distribution of entanglement between remote sites. Thus, as for the case of perfect state transfer, the perfect wave-packet splitting can be a new tool for varied applications.

Figure 1

Comparison between the perfect wave-packet splitting couplings $J_n^{\mathrm{split}}$ and the perfect state transfer couplings $J_n^{\text{PST}}$ in Eq. (4) for an even chain (a) and for an odd chain (b). Only in the latter case the perfect splitting requires also the engineering of a field profile $B_n^{\text{split}}$.

Figure 2

Relative variation of the bunching probability $P_{11}(t=t^{*})$ in the noninteracting regime $U_n=0$, in the presence of (a) random hopping noise and (b) random diagonal noise, respectively, with coupling strength $\epsilon$ and $\eta$. Several chain lengths $L$ are considered.

Figure 3

Quantum carpet: space-time evolution of $\langle n_j^2\left(t\right)\rangle$ for a $L=40$ one-dimensional chain initialized in $|{\psi }_0\rangle =a_{10}^{†}{a}_{31}^{†}|0\rangle$. We consider the free-boson regime $U_n=0$.

Figure 4

Quantum carpet due to four-particle interference. The initial state $|{\psi }_0\rangle =a_2^{†}{a}_6^{†}{a}_9^{†}{a}_{13}^{†}|0\rangle$ contains four bosonic particles. We consider the space-time evolution of $\langle n_j\left(t\right)\rangle$ in the top row and the space-time evolution of $\langle n_j^2\left(t\right)\rangle$ in the bottom row. The chain length is $L=14$ and several values of the onsite interaction $U_n=U$ are considered: $U=0$ (left column), $U=1$ (center column), and $U=30$ (right column). Here $t^{*}$ (expressed in J units) is the fractional revival time, while $2t^{*}$ is the full revival time. The difference between the first and second row is due to bunching and antibunching effects. Note the transition from bosonic $\left(U=0\right)$ to fermionic and hard-core boson $\left(U=\infty \right)$ behavior as a function of $U$.

Figure 5

Plot of the (a) mean occupation number and of the (b) mean-square occupation number of site 2 as a function of time $t$ for several values of the onsite interaction. Here $L=14$ and the initial state is $|\psi \left(0\right)\rangle =a_2^{†}{a}_6^{†}{a}_9^{†}{a}_{13}^{†}|0\rangle$. Note the transition from bosonic $\left(U=0\right)$ to fermionic and hard-core boson $\left(U=\infty \right)$ behavior as a function of $U$.

Figure 6

Plot of the mean occupation number and of the mean-square occupation number for each site at $t=t^{*}/4$ for $U/J=0$. Here $L=14$ and the initial state is $|\psi \left(0\right)\rangle =a_2^{†}{a}_6^{†}{a}_9^{†}{a}_{13}^{†}|0\rangle$.