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1 Introduction↩︎

Quantum walks (QWs) are powerful frameworks for modeling quantum transport and developing quantum algorithms [1]–[5]. Their framework encompasses both discrete-time (DTQWs [6]) and continuous-time (CTQWs [7]) variants. Conventional versions of the DTQW and CTQW exhibit remarkable properties such as ballistic spread and bimodal distributions, but a richer phenomenology emerges with nonstandard versions of these models [8]–[13]. QWs can be simulated from local states or extended (also called nonlocal or delocalized) states  [14]–[32].

The first studies with an explicit focus on QWs from nonlocal initial conditions emerged in 2006, with key contributions from Refs. [14]–[17]. In [14] the authors showed that delocalized initial conditions tend to enhance the entanglement between the internal (spin) and external (position) degrees of freedom in DTQWs. The authors of [15] analyzed the scaling of the survival probability of DTQWs with initial delocalized states. In turn, [16], [17] studied CTQW and DTQWs from extended initial states. The ability to tailor the propagation of QWs through extended initial conditions was systematically shown in [18]. The use of QWs from a Gaussian initial state to generate cat states was reported in [21]. In [29], DTQWs with a fully delocalized initial state were studied in the context of rogue waves. It was shown in [30] that both CTQWs and DTQWs from a delocalized state can serve as platforms for state transfer. The first explicit experimental demonstration of QWs from initial superposition states was presented in [31].

Recently, Ref. [32] investigated CTQWs initiated from both fully localized (\(|0\rangle\)) and fully delocalized (\(|\pm k\rangle\)) states. While these extreme cases have been thoroughly examined in the literature, the intermediate regime remains largely unexplored. Our work addresses this gap.

2 Model↩︎

We analyze a continuous-time quantum walk (CTQW) on a one-dimensional infinite lattice. The system dynamics are governed by the Hamiltonian \[\begin{align} H = -\gamma \sum_{x=-\infty}^{\infty} \left( e^{\mathrm{i}\alpha}\ket{x+1}\bra{x} + e^{-\mathrm{i}\alpha}\ket{x}\bra{x+1} \right) \end{align}\] where \(\gamma > 0\) is the hopping rate and \(\alpha\) is a phase that breaks time-reversal symmetry, leading to complex hopping amplitudes. We set \(\hbar=1\) throughout. For \(\gamma=1\), this model reduces to the one analyzed in Ref. [32]. Our work thus extends the investigation of CTQWs with complex hoppings [32]–[37].

Unlike all the works mentioned in the previous section, we define a new class of tunable delocalized initial states: \[\begin{align} \label{eq:initial95condition} |\Psi(0)\rangle = \sqrt{1-D}|0\rangle + \sqrt{\frac{D}{2}}(|1\rangle + |-1\rangle). \end{align}\tag{1}\] The parameter \(D \in [0, 1]\) controls the degree of delocalization: \(D=0\) corresponds to a state localized at the origin, \(|0\rangle\), while \(D=1\) describes a state equally delocalized over the sites \(x = \pm1\). The state defined by Eq. (1 ) is normalized for all \(D\), as \(\langle\Psi(0)|\Psi(0)\rangle = 1\).

3 Results↩︎

3.1 Wavefunction↩︎

Due to translational symmetry, the Hamiltonian is diagonal in the momentum basis. The momentum eigenstates \(|k\rangle\) are Fourier transforms of the position basis states \(|x\rangle\): \[\begin{align} |k\rangle = \frac{1}{\sqrt{2\pi}} \sum_{x=-\infty}^{\infty} e^{\mathrm{i}kx} |x\rangle, \end{align}\] where the quasimomentum \(k\) is in the first Brillouin zone, \(k \in [-\pi, \pi]\). Applying the Hamiltonian to \(|k\rangle\) yields the energy dispersion relation: \(E(k) = -2\gamma \cos(\alpha-k).\)

In the momentum basis, the initial state is \[\begin{align} \label{eq:initial95condition95kspace} \psi(k,0) = \frac{1}{\sqrt{2\pi}} \left( \sqrt{1-D} + \sqrt{2D} \cos k \right). \end{align}\tag{2}\] The time-evolved wavefunction is therefore \[\begin{align} \psi(k, t) = \frac{e^{\mathrm{i} 2\gamma t \cos(\alpha-k)}}{\sqrt{2\pi}} \left( \sqrt{1-D} + \sqrt{2D} \cos k \right). \end{align}\]

The wavefunction in position space is obtained via the inverse Fourier transform of \(\psi(k,t)\): \[\begin{align} \psi(x,t) = \frac{1}{\sqrt{2\pi}} \int_{-\pi}^{\pi} e^{\mathrm{i} kx} \psi(k,t) dk. \end{align}\] Using the Jacobi-Anger expansion to evaluate this integral, the final form of the wavefunction is found to be: \[\begin{gather} \label{eq:wave95function95main} \psi(x,t) = \sqrt{1-D} \tilde{J}_x(2\gamma t) \\ + \sqrt{\frac{D}{2}} \left(e^{\mathrm{i}\alpha}\tilde{J}_{x-1}(2\gamma t) + e^{-\mathrm{i}\alpha}\tilde{J}_{x+1}(2\gamma t)\right), \end{gather}\tag{3}\] where \(\tilde{J}_n(z) \equiv \mathrm{i}^n J_n(z)\) and \(J_n(z)\) is the Bessel function of the first kind. The probability distribution is then given by \(P(x,t) = \left| \psi(x,t) \right|^2\).

The probability distribution depends on the parameters of the initial conditions (\(D\)) and the Hamiltonian (\(\alpha\)). The nonlocal initial conditions (\(D>0\)) introduce two additional terms in the probability amplitude, which are modulated by a phase factor.

Figure 1 shows the analytically obtained probability distributions, which are in excellent agreement with the numerical results. For the phase value considered (\(\alpha=0.5\)), the localized (\(D=0\)) and fully delocalized (\(D=1\)) initial setups, which are commonly considered in the literature, both exhibit symmetric distributions during the system’s evolution. The fully delocalized case leads to a lower probability in the central region. In contrast, the intermediate delocalization (\(D=0.5\)) leads to a biased spreading, despite the same phase factor value and an initially symmetric state. To further analyze this asymmetrical spreading, we calculate the average position as a function of these parameters.

3.2 Average Position↩︎

The average position \(\langle X \rangle(t)\) is given by Ehrenfest’s theorem, \(d\langle X \rangle/dt = \langle v_g \rangle(t)\), where \(v_g(k) = \partial E(k)/\partial k = 2\gamma\sin(\alpha-k)\) is the group velocity. The expectation value of the group velocity is \[\begin{align} \langle v_g \rangle(t) = \int_{-\pi}^{\pi} |\psi(k,t)|^2 v_g(k) dk. \end{align}\] Since the momentum distribution is time independent, \(|\psi(k,t)|^2 = |\psi(k,0)|^2\), the average group velocity \(\langle v_g \rangle\) is constant.

The initial state distribution \(|\psi(k,0)|^2\) is an even function of \(k\). The group velocity \(v_g(k) = 2\gamma(\sin k \cos\alpha - \cos k \sin\alpha)\) has odd and even parts. Only its even part, \(-2\gamma \sin\alpha \cos k\), contributes to the integral, leading to \[\begin{align} \label{eq:average95main} \langle v_g \rangle = -2\gamma \sin\alpha \sqrt{2D(1-D)}. \end{align}\tag{4}\] A net drift, characterized by \(\langle v_g \rangle \neq 0\), emerges from the interplay between the initial state’s delocalization (\(D\)) and the Hamiltonian phase (\(\alpha\)). The drift vanishes for the extreme cases \(D=0\) and \(D=1\) but is maximized for intermediate delocalization, as shown in Fig. 2. Given that the initial average position is zero, \(\langle X \rangle(0)=0\), The average position is then simply \(\langle X \rangle(t) = \langle v_g \rangle t.\)

Table 1 summarizes the findings related to transport properties where we see clearly that a combination between the initial state delocalization and the Hamiltonian phase allows a control of directional spreading.

3.3 Mean Square Displacement↩︎

The mean square displacement (MSD), defined as \(\mathrm{MSD}(t) = \langle (X - X_0)^2 \rangle(t)\), measures the wavepacket’s spreading around its initial average position \(X_0 \equiv \langle X \rangle(0)\). For our initial state, symmetry dictates \(\langle X \rangle(0) = 0\), which simplifies the expression: \[\begin{align} \mathrm{MSD}(t) = \langle X^2 \rangle(t) - \langle X \rangle(0)^2 = \langle X^2 \rangle(t). \end{align}\]

Using \(X=\mathrm{i}\,\partial_k\) and \(\psi(k,t)=e^{-\mathrm{i} E(k)t}\,\psi(k,0)\), we find that \(\left|\mathrm{i}\,\partial_k\psi(k,t)\right|^2 = (t\,\partial_k E)^2\,|\psi(k,0)|^2 + \left|\partial_k\psi(k,0)\right|^2\), leading directly to \(\langle X^2 \rangle(t) = \langle X^2 \rangle(0) + t^2\langle v_g^2 \rangle\).

The initial MSD, \(\langle X^2 \rangle(0) = D\), quantifies the initial delocalization. This result is intuitive: it equals 0 for a state perfectly localized at the origin (\(D=0\)) and 1 for a state delocalized over sites \(x = \pm1\) (\(D=1\)).

The average squared group velocity is given by \(\langle v_g^2 \rangle = \int v_g(k)^2 |\psi(k,0)|^2 dk\). Substituting \(v_g(k) = -2\gamma \sin(\alpha-k)\) and the expression for \(|\psi(k,0)|^2\) from Eq. (2 ), we evaluate the integral to obtain: \[\begin{align} \langle v_g^2 \rangle = 2\gamma^2 \left( 1 - \frac{D}{2} + D\sin^2\alpha \right). \end{align}\]

Combining these results yields the exact analytical expression for the MSD: \[\begin{align} \label{eq:MSD95main} \mathrm{MSD}(t) = D + 2\gamma^2 t^2 \left( 1 - \frac{D}{2} + D\sin^2\alpha \right). \end{align}\tag{5}\] Figure 3 shows that Eq. (5 ) captures the full time-evolution of the wavepacket’s spatial extent, from its initial value to its long-term ballistic spreading (\(\sim t^2\)), and highlights the non-trivial coupling between the initial condition parameter \(D\) and the Hamiltonian’s phase \(\alpha\).

We define \(t_{\mathrm{cross}}\) as the instant when the \(\mathrm{MSD}(t)\) curves for different \(D\) intersect, i.e., when the MSD becomes independent of the initial condition. This is obtained by setting \(\partial \mathrm{MSD}/\partial D = 0\) in Eq. (5 ): \[\label{eq:t95cross} \gamma t_{\mathrm{cross}}(\alpha) = \frac{1}{\sqrt{1 - 2\sin^{2}\alpha}}.\tag{6}\] Thus, \(t_{\mathrm{cross}}\) exists only for \(1 - 2\sin^{2}\alpha > 0\) (i.e., outside the interval \(\alpha \in [\pi/4,\,3\pi/4]\) modulo \(\pi\)) and diverges at \(\alpha=\pi/4\) and \(3\pi/4\).

The existence of a finite crossing time reveals a regime of counterintuitive dynamics. For a fixed time \(t > t_{\mathrm{cross}}(\alpha)\) within the interval \(0 < \alpha < \pi/4\), Eq. (5 ) shows that the MSD becomes a decreasing function of the delocalization parameter \(D\). This means that for these times, a more delocalized initial state (higher \(D\)) results in a smaller spatial spread of the wavepacket, the opposite of the intuitively expected outcome. This counterintuitive phenomenon, where the strategy of increasing initial delocalization to enhance spreading instead weakens it after a characteristic time, is formally analogous to the “Backfire Effect” known in cognitive science [38]. There, corrective evidence can paradoxically strengthen a person’s misconceptions. Here, a more delocalized initial state improves the short-time spreading, but leads to worse asymptotic spreading performance for a proper combination of parameters. We therefore designate this as a Quantum Backfire Effect.

This effect is clearly visible in the post-crossing regime (blue shading in Fig. 4 for \(\sin^2\alpha < 1/2\)), where the MSD for \(D = 1\) becomes lower than for \(D = 0.5\), which in turn becomes lower than for \(D = 0\), as shown in the top panel of Fig. 3. For times \(t < t_{\mathrm{cross}}(\alpha)\), the intuitive ordering, where greater initial delocalization leads to greater MSD, still holds (see Fig. 3).

3.4 Survival Probability↩︎

The survival probability, \(P_{\text{surv}}\), is an important physical quantity for classical and quantum systems [39]–[44]. In the context of CTQW this quantity is defined as the sum of the probabilities of finding the particle in each site of a given lattice region. In this work, we are interested in the central region of the lattice, and therefore we have: \[\begin{align} P_{\text{surv}}(t) = \sum_{x \in \{-1, 0, 1\}} P(x,t). \end{align}\] Starting with an initial condition where \(P_{\text{surv}}(0)= 1\), the wavepacket spreading leads to a decrease in this quantity. The decay analysis of this value informs how fast the particle leaves the region. Using the probability distribution we get: \[\begin{align} \label{eq:survival95main} \begin{aligned} P_{\text{surv}}(t) = J_0^2(2\gamma t) &+ 2(1 - D\sin^2\alpha)J_1^2(2\gamma t)\\ + D J_2^2(2\gamma t) & - 2D\cos(2\alpha)J_0(2\gamma t)J_2(2\gamma t). \end{aligned} \end{align}\tag{7}\] This result allows for a precise analysis of the asymptotic decay. For the generic case (\(D \neq 1\)), the long-time behavior is governed by the squared Bessel function terms, \(J_n(z)^2 \approx 1/(\pi z)\), while the oscillatory cross-term \(J_0J_2\) averages to zero. This leads to the decay \(P_{\text{surv}}(t) \approx (3 - 2D\sin^2\alpha + D)/(2\pi\gamma t)\).

However, a exception occurs for the fine-tuned parameters \(D=1\) and \(\alpha = \pi/2 + m\pi\) (where \(m \in \mathbb{Z}\)). In this case, Eq. (7 ) simplifies perfectly to \(P_{\text{surv}}(t) = (J_0(2\gamma t) + J_2(2\gamma t))^2\). Using the Bessel function recurrence relation \(J_{n-1}(z) + J_{n+1}(z) = (2n/z)J_n(z)\), this becomes: \(P_{\text{surv}}(t) = J_1(2\gamma t)^2/(\gamma^2 t^2)\). For long times, this expression decays as: \(P_{\text{surv}}(t) \approx 1/(\pi\gamma^3 t^3)\).

These analytical results show that the enhanced temporal decay \(t^{-3}\) occurs only in the case \(D=1\) for specific values of \(\alpha\). Other combinations of \(D\) and \(\alpha\) lead to the usual scaling with \(t^{-1}\). These findings are numerically confirmed in Fig. 5. Thus, we clarify that in a previous work [32] the breakdown of the standard scaling for \(P_{\mathrm{surv}}(t)\) was a result of the special choice of the initial condition.

4 Final Remarks↩︎

We presented an analytical treatment for CTQWs with a tunable delocalization parameter \(D\). Our main results are given by Eqs. (4 ,5 ,7 ). All these equations are associated with properties that correspond to experimentally accessible observables, making our proposal potentially testable in quantum platforms [45]. The key insights from our work are threefold.

First, we demonstrated an emerging directional spreading from unbiased initial states with intermediate delocalization, \(0 < D < 1\). This result comes from the synergy between the initial state’s delocalization and a phase parameter intrinsic to the Hamiltonian.

Second, our analysis revealed a Quantum Backfire Effect: a phenomenon where an increase in the initial state’s delocalization can increase the short-time spreading, but can decrease the long-term spatial spread. This effect occurs after a crossing time, \(t_{\mathrm{cross}}\).

Third, by computing the exact survival probability, we elucidated that the transition from a standard scaling \(\sim t^{-1}\) to an enhanced decay \(\sim t^{-3}\) is a fine-tuned effect that occurs for the specific case of a fully extended initial state (\(D=1\)) and for specific Hamiltonian phases.

The control of transport dynamics in QWs is an important topic in quantum information science [3], [46]–[57]. Taking a broad perspective, our findings offer novel insights into how the interplay between complex hopping and delocalized initial states can lead to surprising phenomena related to transport properties.

While significant progress has been made in understanding CTQWs on networks, many questions remain [58]. A promising avenue for future work would be to investigate how the phenomena we have observed in 1D systems translate to 2D lattices and more complex network structures [3], [59].

4.0.0.1 Acknowledgements

MAP acknowledges financial support from CNPq (310093/2025-2).

References↩︎