Finding zeros of the Riemann zeta function by periodic driving of cold atoms

The Riemann hypothesis, which states that the nontrivial zeros of the Riemann zeta function all lie on a certain line in the complex plane, is one of the most important unresolved problems in mathematics. We propose here an approach to finding a physical system to study the Riemann zeros, which is based on applying a time-periodic driving field. This driving allows us to tune the quasienergies of the system (the analog of the eigenenergies for static systems), so that they are directly governed by the zeta function. We further show by numerical simulations that this allows the Riemann zeros to be measured in currently accessible cold-atom experiments.

Figure 1

Surface plot of the time-dependent driving function $f\left(t\right)$ for the smoothed function ${\mathrm{\Xi }}^{*}\left(E\right)$ [Eq. (10)]. Energy is measured in units of the tunneling $J$, and time in units of $J^{-1}$. For small values of $E$ the function is quite featureless, but progressively increases in amplitude and develops more oscillations as $E$ increases. The corresponding driving function for $\mathrm{\Xi }\left(E\right)$ behaves similarly (see Fig. 4).

Figure 2

(a) The driving potential $f\left(t\right)$ is extended to be a periodic function by periodically repeating its behavior over the interval $0\le t<T$. This produces, however, a function with rather poor performance. (b) Construction of a more efficient driving potential. Four copies of $f\left(t\right)$ are joined together, following some reflection transformations, to create a single continuous function, resembling two localized pulses of opposite sign. The vertical dashed lines indicate where the segments are joined. This function is continuous, has a time-average of zero, and $f\left(t\right)=-f(t+T_0/2)$ where $T_0=4T$ is the total period of the signal, satisfying the parity requirement given in the text.

Figure 3

Behavior of $\mathrm{\Phi }\left(t\right)$ and its derivative. Solid black line: $\mathrm{\Phi }\left(t\right)$ has a global maximum at $t=0$, for which $\mathrm{\Phi }\left(0\right)\simeq 0.89339$ and decays rapidly as $t$ increases. Dashed red line: the derivative, ${\mathrm{\Phi }}^{\prime }\left(t\right)={\partial }_t\mathrm{\Phi }\left(t\right)$, evaluated from Eq. (15). The series expansions for both $\mathrm{\Phi }\left(t\right)$ and ${\mathrm{\Phi }}^{\prime }\left(t\right)$ were truncated at 30 terms. Time is measured in units of $J^{-1}$.

Figure 4

Surface of the driving function $f\left(t\right)$ [Eq. (14)], which produces a quasienergy spectrum proportional to the Riemann $\mathrm{\Xi }$ function. The behavior resembles strongly the driving function for the smoothed Riemann function, shown in Fig. 1. As before, energy is measured in units of $J$, and time in units of $J^{-1}$.

Figure 5

Although the driving functions $f\left(t\right)$ that produce ${\mathrm{\Xi }}^{*}\left(E\right)$ and $\mathrm{\Xi }\left(E\right)$ look superficially very similar, they differ in details. We show here the driving functions for $E=4$ for the two Riemann functions we consider.

Figure 6

(a) Change of the driving function as $\mathrm{\Omega }$ is increased. The pulses become progressively narrower and taller, while their spacing remains the same. (b) As $\mathrm{\Omega }$ increases, the quasienergy spectrum asymptotically approaches that of the infinite-frequency limit, and the quasienergy crossings move towards the zeros of the Riemann function. Here we show the quasienergies for a system driven to reproduce the smoothed Riemann function, ${\mathrm{\Xi }}^{*}\left(E\right)$. Note how the first crossing asymptotically approaches the first zero of ${\mathrm{\Xi }}^{*}\left(E\right)$ at $E=8.993$ as $\mathrm{\Omega }$ increases.

Figure 7

(a) Black symbols show the quasienergies of the two-level system driven by the potential $f\left(t\right)$ given by Eq. (10), the dashed (red) line is the smoothed Riemann function ${\mathrm{\Xi }}^{*}\left(E\right)$. The quasienergies are normalized with respect to the quasienergies at $E=0$. The agreement between the two is excellent. (b) As above, but for the true Riemann $\mathrm{\Xi }$ function, with the driving potential given by Eq. (14). Again, the results agree perfectly. (c) Comparison on a logarithmic scale. The first three zeros of ${\mathrm{\Xi }}^{*}\left(E\right)$, corresponding to degeneracies of the quasienergies, are clearly visible as the downward-pointing cusps [black circles, red (thick) dashed line], as are the first two zeros of $\mathrm{\Xi }\left(E\right)$ [blue triangles, blue (thin) dashed line] at $E\simeq 14.1347$ and $21.0220$. The quasienergies reproduce the behavior of the $\mathrm{\Xi }$ functions to a high degree of accuracy.

Figure 8

(a) Expansion of a Gaussian wave packet in an optical lattice with $N=200$ sites under the periodic driving potential $f\left(t\right)$ (10), such that $J_{\mathrm{eff}}\propto {\mathrm{\Xi }}^{*}\left(E\right)$. $T_0$ is the total driving period; $T_0=4T$. For $E=0$ the wave packet spreads quickly, but for $E=4$ the wave packet spreads less rapidly, indicating that $J_{\mathrm{eff}}$ is smaller. For $E=9$ the wave packet hardly expands at all, indicating that CDT is occurring. (b) The solid black curve shows $|{\mathrm{\Xi }}^{*}\left(E\right)|$, and the black circles show the values of $|J_{\mathrm{eff}}|$ measured from the expansion curves. The agreement between the two is excellent. Similarly the dotted red line shows $\left|\mathrm{\Xi }\right(E\left)\right|$, and the red squares show the measured values of $|J_{\mathrm{eff}}|$ for a system driven by the potential (14). Again the agreement is perfect.