Quantum walks: The first detected passage time problem

Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time $\tau$, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time $\tau$ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like ${\left(\text{time}\right)}^{(-3)}$ with superimposed oscillations, with exceptional behavior when the sampling period $\tau$ times the tunneling rate $\gamma$ is a multiple of $\pi /2$. The amplitude of the power-law decay is suppressed as $\tau \to 0$ due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like ${\left(\text{time}\right)}^{-3/2}$, as elucidated by Schrödinger in 1915.

Figure 1

Schematic model of a benzene ring. In the text, measurement is performed on site 0 and we discuss several initial conditions.

Figure 2

The average number of detection attempts $\langle n\rangle$ for a quantum walk on a benzene ring, with initial condition $\left|\psi \right(0)\rangle =|0\rangle$ and projective measurements on the origin, by Eq. (49), $\langle n\rangle =4$, except for the cases when $\gamma \tau$ is a multiple of $\pi /2$ or $2\pi /3$. Here, we plot ${\langle n\rangle }_N={\sum }_{n=1}^{N}n{F}_n$ for $N=500$ and $10000$, the results converging to the analytic result as we increase $N$ further (not shown).

Figure 3

The variance of $n$ versus $\gamma \tau$ [Eq. (50)] for the $0\to 0$ first detection problem on the benzene ring. Divergences of the fluctuations are found close to exceptional sampling times, while for these times themselves the fluctuations are finite (blue circles). For $\gamma \tau =0,2\pi ,\text{Var}\left(n\right)=0$.

Figure 4

The average $\langle n\rangle$ versus $\gamma \tau$ [Eq. (56)]. When $\tau \to 0$ we find $\langle n\rangle \to \infty$ due to the Zeno effect, another expected divergence of $\langle n\rangle$ is found when the sampling time is the full revival time $\gamma \tau =2\pi$. In addition to these two points, we find singularities also for $\pi /2,\pi ,3\pi /2$. Notice the discontinuities of $\langle n\rangle$ for the exceptional sampling times $\gamma \tau =2\pi /3$ and $\gamma \tau =4\pi /3$ which are discussed in the text. Here, $\left|\psi \right(0)\rangle =|3\rangle$ and the detection is on the origin. The plot of $\langle n\rangle$ for this choice of initial condition is vastly different from that presented in Fig. 2. The blue line is the analytic result (56). The blue circles represent the values at the exceptional sampling times. The black dashed line is the result of a numerical evaluation of ${\sum }_{n}n{F}_n$ up to a large finite $N$.

Figure 5

Counterclockwise integration path $C$ in Eq. (26) for evaluation of ${\varphi }_n$ for an unbounded lattice with the sampling rate $\gamma \tau =\pi /2$. Equation (80) avoids the branch cut along the negative real axis when $\left|z\right|>1$. The outer radius approaches infinity.

Figure 6

First detection probability $F_n$ versus $n$ on a log-log plot, for an open system, detection is at the starting point and the sampling rate is $\gamma \tau =\pi /2$. For large $n$, the exact result (dots) converges to the asymptotic power-law behavior (91), $F_n={\left|{\varphi }_n\right|}^2\sim 0.25n^{-3}$ (straight line).

Figure 7

Integration path for the calculation of ${\varphi }_n$ in the complex plane bypasses two branch cuts (dashed lines). Integration along four lines just above and below the two dashed lines is explained in the text, while the integration around the outer circle does not contribute in the limit of an infinite radius. When $2\gamma \tau =k\pi$ the two branch cuts merge $(k=0,1,\cdots )$.

Figure 8

$F_n$ versus $n$ for the rational sampling time $\gamma \tau /\pi =\frac{1}{3}$. Now, the detection probability $F_n$ decays monotonically like $n^{-3}$ with a superimposed periodic oscillation. The lines are the asymptotic theory (95) which for large $n$ nicely match the exact expression (dots).

Figure 9

$F_n$ versus $n$ for the choice of an irrational sampling rate $\gamma \tau /\pi =0.8/\pi$, for a quantum walk on a one-dimensional lattice (note the log linear scale). The numerically exact solution (red circles) nicely matches the theory (the curve) already for moderate values of $n$.

Figure 10

For the one-dimensional tight-binding quantum walk, the probability that the particle is eventually detected $1-S_{\infty }={\sum }_{n=1}^{\infty }{F}_n$ versus the sampling rate $\gamma \tau$ exhibits a nonmonotonic behavior. Unlike its classical random walk counterpart, the quantum walk is not recurrent, unless $\gamma \tau \to 0$, which is the trivial case.

Figure 11

Quantum walk on a ring with $L=100$ sites for a sampling rate $\gamma \tau =0.6$ with detection at the origin of the walk. For small $n$, the system exhibits a behavior similar to that of an infinite system (see Fig. 12). Roughly at $n=70$, there is a sudden increase in $F_n$, probably due to a partial revival of the packet at the origin.

Figure 12

Same as Fig. 11 for an infinite system.

Figure 13

$1-S_N$ versus $\gamma \tau$ for a quantum walk on a line, the particle is launched from the origin.

Figure 14

The ratio $J_0{\left(2n\gamma \tau \right)}^2/F_n$ for $\gamma \tau ={10}^5$.

Figure 15

$1-S_{\infty }$ versus $\gamma \tau$. The approximation (96) is compared with the exact result, and it works reasonably well for large $\gamma \tau$, as expected. However, it does not predict the spiky cusps or the nonmonotonic behavior of $1-S_{\infty }$.

Figure 16

$1-S_{\infty }$ versus $\gamma \tau$. The approximation (D2) works well.