Measurement-induced quantum walks

We investigate a tight-binding quantum walk on a graph. Repeated stroboscopic measurements of the position of the particle yield a measured “trajectory,” and a combination of classical and quantum mechanical properties for the walk are observed. We explore the effects of the measurements on the spreading of the packet on a one-dimensional line, showing that except for the Zeno limit, the system converges to Gaussian statistics similarly to a classical random walk. A large deviation analysis and an Edgeworth expansion yield quantum corrections to this normal behavior. We then explore the first passage time to a target state using a generating function method, yielding properties like the quantization of the mean first return time. In particular, we study the effects of certain sampling rates that cause remarkable changes in the behavior in the system, such as divergence of the mean detection time in finite systems and decomposition of the phase space into mutually exclusive regions, an effect that mimics ergodicity breaking, whose origin here is the destructive interference in quantum mechanics. For a quantum walk on a line, we show that in our system the first detection probability decays classically like ${\left(\text{time}\right)}^{-3/2}$. This is dramatically different compared to local measurements, which yield a decay rate of ${\left(\text{time}\right)}^{-3}$, indicating that the exponents of the first passage time depend on the type of measurements used.

Figure 1

A segment of the infinite line. For a particle starting at the origin, we allow it to unitarily evolve for $\tau$ time, and then we measure its position. The measurement localizes its wave function to some single site on the lattice, which we record as being the location of the particle at time $\tau$. After this, we allow the particle to freely evolve for $\tau$ time once more before we measure its position again. By repeating this process many times, we generate a “path” that the particle took. For the lattice shown here, an example of a path would be $\{0,2,1,0,-3,\cdots \}$.

Figure 2

Schematic model of a benzene ring. The edges represent jumps between nearest neighbors, all with the same amplitude. In Sec. 7 we examine the properties of a measurement-induced quantum walk on this graph, as well as the first detection problem.

Figure 3

A measurement-induced quantum walk that begins at $|\psi (t=0)\rangle =|{\psi }_{\text{in}}\rangle$. At every step of the walk, the particle is detected at some point on the graph, $|x_n\rangle$, and between each detection the wave function evolves according to the Schrödinger equation with the standard unitary time evolution operator $U\left(\tau \right)=e^{-i\tau H}$. This process can, in principle, continue indefinitely.

Figure 4

A measurement-induced quantum walk which begins at $|\psi (t=0)\rangle =|{\psi }_{\text{in}}\rangle$ and ends after time $t=N\tau$ when the particle is detected at $|{\psi }_{\text{tar}}\rangle$. The underlying physical process is the same as the one that was described in Fig. 3, but in this process we stop the random walk once it is detected at the target state $|{\psi }_{\text{tar}}\rangle$.

Figure 5

A simple illustration of the eigenvalue interlacing theorem. The eigenvalues of $G$ and $G^{\prime }$ for the Hamiltonian (35) plotted on a segment of the real number line. $G$'s eigenvalues are depicted as blue circles, whereas the eigenvalues of $G^{\prime }$ are depicted as blue disks. We find that the eigenvalues of $G^{\prime }$ always interlace between $G$'s eigenvalues just as the theorem predicts.

Figure 6

The eigenvalues of $G$ (denoted $\lambda$) and $G_{\text{tar}}$ (denoted $\mu$) plotted in blue and orange, respectively. For all values of $\gamma \tau$ there is always a $\mu$ between every pair of $\lambda$ aside from zero which does not interlace. The eigenenergies of the system are $0\text{,}\pm \sqrt{2}\gamma$. The exceptional sampling rates are $\gamma \tau =0\text{,}\pi /\sqrt{2}\text{,}\text{and}\sqrt{2}\pi$ plus integer multiples of $\sqrt{2}\pi$. In the figure these are marked by dashed lines, and all of them satisfy $\mathrm{\Delta }E\tau =2\pi k$. Near each of the exceptional sampling rates, we can see that at least one of the $\mu$'s is squeezed between a pair of $\lambda$'s until it equals 1.

Figure 7

The survival probability in the transition from $|0\rangle$ to $|2\rangle$ on the graph whose Hamiltonian is described in Eq. (35). The orange and blue lines correspond to the almost exceptional sampling rates $\gamma \tau =\epsilon$ and $\pi /\sqrt{2}+\epsilon$, where $\epsilon ={10}^{-1}$. The green line corresponds to the nonexceptional sampling rate $\pi /\sqrt{8}$. As explained in the text, the interlacing theorem predicts the slow decay of the survival probability close to the exceptional sampling rates through the analysis of the eigenvalues of $G$. Whenever two eigenvalues of $G$ merge, we get a slow relaxation of the survival probability.

Figure 8

The eigenvalues of $G$ vs $\gamma \tau$ for the benzenelike ring. As with every other system, one of the eigenvalues is a constant one while the others are sinusoidal functions of $\gamma \tau$. Interesting effects that are shown in the following figures are observed when $\lambda =1$ becomes degenerate. The exceptional sampling rates of this Hamiltonian are as follows: $\gamma \tau =0\text{,}2\pi /3\text{,}\pi \text{,}4\pi /3\text{,}2\pi$ plus every integer multiple of $2\pi$. These sampling rates all satisfy $\mathrm{\Delta }E\tau =2\pi k$, where $\mathrm{\Delta }E$ are the non-negative differences between eigenenergies of the Hamiltonian given in Eq. (36). The eigenvalues plotted in orange and green have twofold degeneracy each.

Figure 9

$\langle n\rangle$ for the return problem for a model ring with six sites whose Hamiltonian is given by Eq. (36). As predicted by Eq. (25), $\langle n\rangle =6$ for nonexceptional sampling rates. We find that at every exceptional sampling rate, $\langle n\rangle$ jumps discontinuously to a different integer smaller than 6. In the text we show that this happens because at those values of $\gamma \tau$ the connectivity of the graph is broken, which separates it into several fragmented graphs.

Figure 10

$\mathrm{\Delta }{n}^2=\langle n^2\rangle -{\langle n\rangle }^2$ for the return problem on a model ring with six sites whose Hamiltonian is given by Eq. (36). As predicted in Eq. (26), we find that the variance diverges at every exceptional sampling rate, namely whenever a pair of $\lambda$'s in Fig, 8 coalesce on unity.

Figure 11

$\langle n\rangle$ for the transition problem from $|{\psi }_{\text{in}}\rangle =|0\rangle$ to $|{\psi }_{\text{tar}}\rangle =|3\rangle$ on a model ring with six sites whose Hamiltonian is given by Eq. (36). In the text we explain why it diverges at $\gamma \tau =\pi n$ but only jumps discontinuously at the other sampling rates.

Figure 12

$\mathrm{\Delta }{n}^2=\langle n^2\rangle -{\langle n\rangle }^2$ for the transition problem from $|{\psi }_{\text{in}}\rangle =|0\rangle$ to $|{\psi }_{\text{tar}}\rangle =|3\rangle$ on a model ring with six sites whose Hamiltonian is given by Eq. (36). We find that the variance diverges at every exceptional sampling rate, including $2\pi n/3$, in spite of the fact that the average does not diverge there and only jumps discontinuously.

Figure 13

The possible transitions for various sampling rates on a model ring with six sites whose Hamiltonian is given by Eq. (36). In the text we explain how this result relates to the behavior of the system, and in particular $P_{\text{det}}$ and $\langle n\rangle$. Note that in (c), $n$ is odd.

Figure 14

The probability vector on the infinite line Hamiltonian given in Eq. (38) after the first four measurements plotted as a function of the lattice site $x$ with $\gamma \tau =10$. The shape after the first measurement matches the results obtained in experimental implementations of this type of random walk [38], indicating that the Hamiltonian we use [Eq. (38)] can accurately model such a system. We tested many values of the sampling rate and found that excluding $\gamma \tau \ll 1$, the general shape of the distribution after the first few measurements is the one presented here. The scale of the shape depends on the sampling rate, and we find that the “width” equals $2n\gamma \tau$, as can be seen in this figure. More generally, we can say that the maximum group velocity of the wave packet when evolving without measurement is $2\gamma$, and this is reflected here in the distance at which we can detect the particle. From the fifth measurement onward, the probability vector typically converges to a simple Gaussian shape near the center, as expected. We study the behavior of the tails of the distribution separately later in this section.

Figure 15

A plot of the expression given in Eq. (48) for $\gamma \tau =1$ and $n=10$. The probabilities of detection that were obtained via numerical integration are plotted as blue dots, and the fourth Hermite polynomial is plotted as an orange line. Numerical simulations indicate that typically the two overlap almost completely for $10\le n$.

Figure 16

The rate function $I(x/n)=n^{-1}ln\left(P_{x,n}\right)$ on the infinite 1D line plotted in blue as a function of the lattice site $x$. The small $l$ approximation (Gaussian) is plotted in orange, and the large $l$ approximation [Eq. (49)] is plotted in green. We tested many values of the sampling rate, and we found that, excluding $\gamma \tau \ll 1$, the distribution fits the Gaussian approximation up to around $\left|x\right|=2n\gamma \tau$ and then almost immediately fits the large $l$ approximation.

Figure 17

$F_n$ of the return problem on the infinite 1D line for various values of $\gamma \tau$ plotted as dots alongside their respective asymptotic limits, which were plotted as lines. The choice of sampling rate has a strong effect on the first few first detection probabilities, but after those, they all converge to ${\pi }^{-0.5}\gamma \tau n^{-3/2}$. This convergence is generally faster for smaller values of $\gamma \tau$, but it slows down close to the Zeno limit.

Figure 18

The average number of measurements until first detection in the transition from $|{\psi }_{\text{in}}\rangle =|0\rangle$ to $|{\psi }_{\text{tar}}\rangle =|1\rangle$ for the two-level system model whose Hamiltonian is given in Eq. (E1) as a function of the on-site energy $U$ plotted in blue, where we set $\tau =\pi$ and $\gamma =1$. The average diverges at every exceptional value of $U$ while also tending to increase overall like $U^2/4$ (plotted in orange). The exceptional energy differences are denoted by the dashed vertical lines. The variance (not plotted) behaves very similarly.