Using Hilbert transform and classical chains to simulate quantum walks

We propose a simulation strategy which uses a classical device of linearly coupled chain of springs to simulate quantum dynamics, in particular quantum walks. Through this strategy, we obtain the quantum wave function from the classical evolution. Specially, this goal is achieved with the classical momenta of the particles on the chain and their Hilbert transform, from which we construct the many-body momentum and Hilbert transformed momentum pair correlation functions yielding the real and imaginary parts of the wave function, respectively. With such a wave function, we show that the classical chain's energy and heat spreading densities can be related to the wave function's modulus square. This relation provides a new perspective to understand ballistic heat transport. The results here may give a definite answer to Feynman's idea of using a classical device to simulate quantum physics.

Figure 1

The wave function's squared modulus ${\left|{\psi }_m\left(t\right)\right|}^2$ from (a) the prediction of Eqs. (2, 3, 4) with the dispersion relation (15) and (b) simulations using the Hilbert transform as explained in the text, for the classical harmonic chain. Several short time results from $t=10$ to $t=50$ are plotted.

Figure 2

The rescaled densities for the harmonic chain ($t=600$), obtained from plugging Eq. (15) into Eqs. (2, 3, 4), are compared with the prediction from the Arcsine distribution and simulations.

Figure 3

Dispersion relations [(a), (f), and (k)], rescaled densities from the predictions of Eqs. (2, 3, 4) with ${\omega }_q$ in Table 1 [(b), (g), and (l)] and simulations $[{\rho }_E(m,t)$: (c), (h), and (m); ${\rho }_Q(m,t)$: (d), (i), and (n)], and the predicted densities from QPVA [(e), (j), and (o)] for Model II (a)–(e); Model III (f)–(j), and Model IV (k)–(o), respectively. For all the densities, three long times' results [solid ($t=200$), dashed ($t=400$), and dotted ($t=600$)] are rescaled for comparison. In (k) the harmonic chain's dispersion relation (dashed) is plotted for comparison. Note that ${\rho }_Q(m,t)$ and ${\rho }_E(m,t)$ are not mathematically identical, though both are well approximated by $\rho (m,t)$.

Figure 4

Dispersion relations (a), rescaled densities [solid ($t=200$), dashed ($t=400$), and dotted ($t=600$)] for the Toda chain, from the predictions of Eqs. (2, 3, 4) [(b)–(d)], and simulations [(f)–(h)]. (e) The side peaks in (f)–(h) are fitted by Gaussian distributions $N(\nu ,{\sigma }^2)$ with means $\nu$ and variances ${\sigma }^2$: $T=0.05(1.04,0.00025)$, $T=0.5(1.19,0.004)$, and $T=1.5(1.39,0.015)$. In the inset of (d) the $y$ axis is logarithmic.

Figure 5

The rescaled density of the NN tight-binding quantum walk. Dotted, dashed, and solid lines correspond to $t=200,400$, and 600, respectively. The inset ($t=600$) shows the oscillations (interference) in detail.

Figure 6

${\tilde{p}}_m\left(t\right)$ and $p_m\left(t\right)$ versus $t$ for a harmonic chain, here we use the particle with the label of $m=1000$, for example.

Figure 7

Comparison of the wave function's real and imaginary part with ${\rho }_p(m,t)$ and ${\rho }_{\tilde{p}}(m,t)$ from simulations for a harmonic chain ($t=300$, for example).

Figure 8

The kinetic energy correlation function for the harmonic chain: (a) Prediction ${\left\{\mathrm{Re}\left[{\psi }_m\left(t\right)\right]\right\}}^2$. (b) Simulation ${\rho }_{E_k}^S(m,t)$; here the results of $t=300$ are plotted.

Figure 9

The stretch correlation function [(a) and (c)], the stretch-momentum cross-correlation function [(b) and (d)] for a harmonic chain: (a)–(b) from predictions; (c)–(d) from simulations ($t=300$).

Figure 10

The potential correlation function [(a) and (c)], the non-normalized total energy correlation function [(b) and (d)] for a harmonic chain: (a) and (b) from predictions; (c) and (d) from simulations ($t=300$).

Figure 11

The phonon dispersion relation for Model II.

Figure 12

The rescaled ${\rho }^{-}(m,t)$ (a) and ${\rho }^{+}(m,t)$ (b) for Model II ($t=600$).

Figure 13

The rescaled ${\rho }^T(m,t)$ (a), $\rho (m,t)$ (b), ${\rho }_E^S(m,t)$ (c), and ${\rho }_Q^S(m,t)$ (d), for Model II ($t=600$).