Quantum walks assisted by particle-number fluctuations

We study the spreading of a quantum particle placed in a single site of a lattice or binary tree with the Hamiltonian permitting particle-number changes. We show that the particle-number-changing interactions accelerate the spreading beyond the ballistic expansion limit by inducing off-resonant Rabi oscillations between states of different numbers of particles. We consider the effect of perturbative number-changing couplings on Anderson localization in one-dimensional disordered lattices and show that they lead to decrease of localization. The effect of these couplings is shown to be larger at larger disorder strength, which is a consequence of the disorder-induced broadening of the particle dispersion bands.

Figure 1

Time dependence of the standard deviation (in units of lattice constant) of the wave packet for a particle initially placed in a single site of a one-dimensional ideal lattice. The solid black curves represent the ballistic expansion governed by the Hamiltonian (1) with $v=0$ and ${\widehat{V}}_{\mathrm{nc}}=0$. The oscillating curves show the size of the wave packets governed by the Hamiltonian (1) with $v=0,\gamma =0,\mathrm{\Delta }/t=1,t=1,\mathrm{\Delta }\varepsilon =20/t$ and a Hilbert space with different number of particles, 1 particle (black), 1 and 3 particles (red), and 1, 3, and 5 (blue). The upper inset shows the logarithm of the particle probability distributions in a disordered 1D lattice with 19 sites. The results are averaged over 50 realizations of disorder and are time-independent with $w=10/t$. And the lower inset depicts the average number of particles $\langle n\rangle$ as a function of time for the three different wave packets.

Figure 2

Time dependence of the standard deviation (in units of lattice constant) of the wave packet for a particle initially placed in a single site of a one-dimensional ideal lattice. The solid black curves represent the ballistic expansion governed by the Hamiltonian (1) with $v=0$ and ${\widehat{V}}_{\mathrm{nc}}=0$. (Upper panel) The oscillating curves show the size of the wave packets governed by the Hamiltonian (1) with $v=0,\gamma =0,\mathrm{\Delta }/t=1,t=1$ and two values of $\mathrm{\Delta }\varepsilon$: $\mathrm{\Delta }\varepsilon =10/t$ (blue) and $\mathrm{\Delta }\varepsilon =20/t$ (red). (Lower panel) The oscillating curves show the size of the wave packets governed by the Hamiltonian (1) with $v=\pm 1,\mathrm{\Delta }/t=1,t=1$, and $\mathrm{\Delta }\varepsilon =20/t$. The insets show the average number of particles $\langle n\rangle$ as a function of time for the corresponding Hamiltonian parameters. Notice that for $\mathrm{\Delta }\varepsilon =20/t,\langle n\rangle$ stays below 1.2 at all times.

Figure 3

Time dependence of the standard deviation (in units of lattice constant) of the wave packet for a particle initially placed in a single site of a one-dimensional ideal lattice. The solid black line represents the ballistic expansion governed by the Hamiltonian (1) with $v=0,\mathrm{\Delta }=0$, and $\gamma =0$. For the dotted red curve $\mathrm{\Delta }=0$ and $\gamma =t$, while for the blue solid curve $\mathrm{\Delta }=t$ and $\gamma =0$. For all of these calculations, $\mathrm{\Delta }\varepsilon =20/t$. The inset shows the average number of particles $\langle n\rangle$ as a function of time for $\mathrm{\Delta }=0$ and $\gamma =t$ (dotted red curve), and $\mathrm{\Delta }=t$ and $\gamma =0$ (blue solid curve).

Figure 4

(Upper panel) The logarithm of the particle probability distributions in a disordered 1D lattice with 41 sites: diamonds – $\mathrm{\Delta }/t=1$, squares – $\mathrm{\Delta }/t=1/2$, circles – $\mathrm{\Delta }/t=1/10$. The results are averaged over 100 realizations of disorder and are time independent. The dashed line is an exponential fit to the $\mathrm{\Delta }/t=1/10$ results. (Lower panel) The long-time limit of the IPR defined in Eq. (3) averaged over 100 instances of disorder as a function of the disorder strength $w$: solid line – $\mathrm{\Delta }=0$, dashed line – $\mathrm{\Delta }/t=1$. The inset shows the IPR averaged over 100 realizations of disorder for two disorder strengths $w=\{5/t,10/t\}$ as functions of time: the solid black curves – $\mathrm{\Delta }=0$; the dotted and dot-dashed curves – $\mathrm{\Delta }=t$.

Figure 5

Schematic diagram of an ideal binary tree with depth 5 ($\mathcal{G}5$).

Figure 6

(Upper panel) The growth of the wave packet for a single particle placed at the root of the tree ($\mathcal{G}5$): black line – $\mathrm{\Delta }=0$; the oscillating curves – $\mathrm{\Delta }/t=1$ with $\mathrm{\Delta }\varepsilon =10/t$ (blue) and $\mathrm{\Delta }\varepsilon =20/t$ (red). The inset shows the probability of reaching the last node (in layer 5) of the binary tree as a function of time. The curves are color-coded in the same ways as in the main plot. (Lower panel) The $L1$ norm between the probability distribution of the wave packet and the uniform distribution as a function of time. The uniform distribution is defined by the value $1/\dim \left(\mathcal{G}5\right)$ for each node. The solid black curve is for $\mathrm{\Delta }=0$. The oscillating curves are for $\mathrm{\Delta }=t$ with $\mathrm{\Delta }\varepsilon =10/t$ (blue) or $\mathrm{\Delta }\varepsilon =20/t$ (red). For all curves in both figures $\gamma =0$.

Figure 7

The squares represent the stationary distribution ${\pi }_S\left(\nu \right)$ for a quantum walk on the $\mathcal{G}5$ tree with $\mathrm{\Delta }=0$. The circles and diamonds are the stationary distributions ${\pi }_S\left(\nu \right)$ for quantum walks with $\mathrm{\Delta }/t=1$ and $\mathrm{\Delta }\varepsilon =10/t$ and $\mathrm{\Delta }\varepsilon =20/t$, respectively.

Figure 8

Schematic diagram of ideal glued binary trees with depth 4 ($\mathcal{GBT}4$).

Figure 9

Average particle probability distribution in a disordered $\mathcal{GBT}4$ graph with strength $w=5/t$. The wave packet for a single particle is initially placed in the head node of a $\mathcal{GBT}4$ graph shown in Figure 8. The upper panel displays the probability (14) summed over all nodes of layer $j=3$; the lower panel shows $p_j\left(t\right)$ for the bottom node $j=6$. Insets: The particle probability distributions for an ideal $\mathcal{GBT}4$ graph with $\mathrm{\Delta }\epsilon =10/t$ (blue dot-dashed) and $\mathrm{\Delta }\epsilon =20/t$ (red dashed). The broken curves show the results obtained with $\mathrm{\Delta }/t=1$ and the full black lines – $\mathrm{\Delta }/t=0$.

Figure 10

Average particle probability distribution in a disordered $\mathcal{GBT}4$ graph: upper panel $-\mathrm{\Delta }/t=0$, middle panel $-\mathrm{\Delta }\epsilon =10/t$, and lower panel $-\mathrm{\Delta }\epsilon =20/t$. For all panels we consider 100 realizations of disorder with a strength of $w=5/t$ and $\gamma =0$. The wave packet for a particle is initially placed in the head node of a $\mathcal{GBT}4$ graph.