Transition in Splitting Probabilities of Quantum Walks

TL;DR

The paper studies the splitting probability for a monitored continuous-time quantum walk with two absorbing targets. By exploiting parity symmetry, it derives an exact mapping to two independent single-target first-detection problems via auxiliary states $|d_{\pm}\rangle$, allowing a precise analysis of interference-driven splitting outcomes. It shows a universal regime where $P_L=P_R=1/2$ for $0<\tau\le\tau_c$ (with $\tau_c=2\pi/\Delta E$), and a nonuniversal, fluctuating regime for $\tau>\tau_c$ driven by the spectral arrangement of the survival-operator eigenvalues $\{\lambda_-\}$ inside the unit disk. The transition originates from a qualitative reorganization of these eigenvalues as $\tau$ crosses $\tau_c$, including dark-state resonances at special sampling times and a breakdown of the classical proximity effect, with clear experimental implications for quantum walks in qubit, photonic, and trapped-ion platforms.

Abstract

We investigate the splitting probability of a monitored continuous-time quantum walk with two targets and show that, in stark contrast to a classical random walk, it exhibits a nonanalytic, phase-transition-like behavior controlled by the sampling time at the targets. For large systems and sampling times smaller than a critical value $τ_c = 2π/ΔE$, where $ΔE$ is the energy bandwidth, the splitting probability is universal and equal to $1/2$, independent of the initial condition and the sampling time. Above the critical sampling, a nonuniversal regime emerges in which the splitting probability deviates from $1/2$ and develops a fluctuating pattern of pronounced peaks and dips dependent on both the sampling time and the initial condition. These results follow from a nontrivial mapping of the splitting problem onto a pair of single-target detection problems enabled by the superposition principle.

Figures (5)

• Figure 1: A continuous-time quantum walk initially localized at $\ket{\psi_0}=\ket{4}$ moves on a tight-binding chain of size $N=5$, with Hamiltonain in Eq. \ref{['letter-main-eqq-1']}. Projective measurements are performed at $x_L=1$ and $x_R=N$ at regular time intervals $\tau$ adopting the splitting protocol explained in the Letter. The goal is to determine the probabilities of detecting at the left and the right.
• Figure 2: Splitting probability at the left boundary as a function of $\tau$ for $x_0=4$ and $N=5$. Red and blue symbols represent the numerical simulations with blue highlighting the resonant points. Black solid line corresponds to our analytical results in Eqs. \ref{['letter-res-cond']} and \ref{['letter-neq-2']}. When $P_L(4)>1/2$ (shown by the dashed line), the classical proximity effect is broken, as explained in the Letter.
• Figure 3: Flat-to-fluctuating transition of the splitting probability at $\tau _c = \pi/2$ for two initial conditions, $x_0=11$ and $x_0=30$ with $N=400$ using Eq. \ref{['letter-neq-2']}. Resonances at special values of $\tau$, given in Eq. \ref{['letter-res-cond']}, are omitted for clarity.
• Figure 4: Eigenvalues $\{ \lambda_{-} \}$ (in blue) of the survival operator $\mathcal{S}_- = \left( \mathbbm{1}-\ket{d_{-}}\bra{d_{-}} \right)U(\tau)$ in Eq. \ref{['letter-neq-1']} for (a) $\tau =1~(\tau < \tau_c)$, (b) $\tau = \pi/2~(\tau = \tau_c)$ and (c) $\tau = 3.75~(\tau > \tau_c)$ with $N=61$. In (a-b), the two red points represent $a_{1}(\tau) =\exp\!\left(-iE_{\min}\tau\right)$ and $a_2(\tau)=\exp\!\left(-iE_{\max}\tau\right)$, as explained in the Letter. For $x_0=11$, we find $\xi(x_0, N, \tau=1) \approx 0.002$, $\xi(x_0, N, \tau=\tau_c) \approx -0.002$ and $\xi(x_0, N, \tau=3.75) \approx -0.135$.
• Figure 5: Eigenvalues $\{ \lambda_- \}$ (where $|\lambda_- <1|$) of the survival operator $\mathcal{S}_- = \left( 1-\ket{d_-} \bra{d_-}\right)U(\tau)$ for the nearest-neighbor tight-binding Hamiltonian. They are obtained by solving Eq. \ref{['SM-eq-ev-9']} numerically (also see Table \ref{['tab:Spm-eigs']}). Panels (a-e) show these eigenvalues for five different sampling times with $N=61$. For $\tau \leq \tau_c$, they are distributed along a single arc between the points $a_1(\tau) = e^{-iE_{\rm max} \tau}$ and $a_2(\tau) = e^{-iE_{\rm min} \tau}$ [shown in red in (a-b)] where $E_{\rm max} \approx 2$ and $E_{\rm min} \approx -2$ are the maximal and minimal energy eigenvalues for $N \gg 1$. For $\tau>\tau_c$, there is a qualitative change in the eigenvalue arrangement. The single arc breaks into distinct branches in the unit disk, and the arrangement becomes somewhat irregular. This, in turn, leads to qualitative change in $\xi(x_0,N, \tau)$ in Eq. \ref{['pole-eq-4']}. For $x_0=11,~N=61$, we find (a) $\xi(x_0,N,\tau) = 0.002$ for $\tau=1$, (b) $\xi(x_0,N,\tau) = -0.002$ for $\tau=\tau_c$, (c) $\xi(x_0,N,\tau) = -0.03$ for $\tau=2$, (d) $\xi(x_0,N,\tau) = -0.135$ for $\tau=3.75$ and (a) $\xi(x_0,N,\tau) = 0.132$ for $\tau=6$. Plugging these results in Eq. \ref{['supp-letter-neq-2']}, we obtain $P_L(x_0) \approx P_R(x_0) \approx 1/2$ for $\tau \leq \tau _c$ and $P_L(x_0) \neq P_R(x_0) \neq 1/2$ for $\tau > \tau _c$.