Atoms and Molecules as Quantum Attosecond Processors

TL;DR

The paper addresses the fundamental limit on temporal resolution in optical resonators and proposes atoms and molecules as attosecond-scale processors that maintain long operation times. By solving Bloch equations and showing that the atomic polarization acts as the integral of the incident-field envelope, it achieves attosecond-resolution processing with orders-of-magnitude higher precision than conventional resonators. The work validates the integration mechanism through multiple Bloch formulations, analyzes multi-level reductions, and identifies practical atomic platforms (e.g., Rb, Sr, Er, He) along with experimental schemes for emission- and pump-probe configurations. This establishes a new paradigm for ultrafast computation, signal modulation, and optical switching at atomic length scales with potential impact on high-speed data processing and fundamental ultrafast science, enabled by realizable transitions and modern atomic-array platforms.

Abstract

Advancing temporal resolution in computation, signal modulation, and measurement is crucial for pushing the frontiers of modern science and technology. Optical resonators have recently demonstrated computational operations at frequencies beyond the gigahertz range, surpassing conventional electronics, yet remain constrained by an inherent trade-off between temporal resolution and operation time -- limiting performance to the picosecond scale. Here we show that atoms and molecules can overcome this limitation, enabling attosecond-level temporal resolution with over 100,000-fold higher precision than state-of-the-art optical resonators while sustaining long operation times. When resonantly driven, these systems naturally perform temporal integration of the incident field envelope -- a process verified by solving the Bloch equations using four independent formulations in excellent agreement with analytic predictions. We identify feasible atomic transitions and excitation schemes realizable with current technology. Furthermore, we suggest techniques to differentiate and generate waveforms at such resolution. These results establish a new paradigm for attosecond-resolution optical computation, signal modulation, and ultrafast control in atomic and quantum systems.

Figures (6)

• Figure 1: Types of passive resonators, which have loss and no gain: subwavelength particles (a), electric circuits (b), and larger-than-wavelength slabs (c). These resonators are characterized by a complex resonance frequency $\omega+i\Gamma.$ As we show atoms and molecules behave similarly to this class of passive resonators and perform integration of the incoming-field envelope (d). Slabs exhibit picosecond resolution and can integrate Gaussian pulses of 1 ns duration; however, integration of shorter pulses, such as 1 ps, is limited by the long round-trip time. Atoms have dramatically improved performance with attosecond resolution, enabling them to integrate pulses with modulation frequencies on the order of $10^{17}\,(1/\mathrm{s})$ or higher, and pulses of 200 (as)-10 (fs), corresponding to UV and optical frequencies, and at the same time they can process long pulses of 1 ns duration (processing 200 (as) pulse requires 10 (as) resolution).
• Figure 2: The response of two atoms to external excitations of $E\propto \cos(\omega t)\exp(-\Gamma t)$ and $E\propto f(t) \cos(\omega t)$. The response of an atom with $\omega_\mathrm{ge}=3.76\cdot 10^{15}\,\mathrm{(1/s)},\,\,\Gamma=10^8\,\mathrm{(1/s)},\,\,\mathrm{and}\,\,\Omega_R=10^{10}\,\mathrm{(1/s)}$ to external excitations: (a) Im$(\rho_{12}(t))$ (b) $P(t)$ and (c) $\rho_{22}(t)-\rho_{11}(t)$ for the excitation $E\propto \cos(\omega t)\exp(-\Gamma t)$. The response of $\propto t\exp(i\omega t)$ in (b) is a signature of a pole in the transfer function of the atom. (d) Im$(\rho_{12}(t))$ in response to the excitation $E\propto (1/(1+t^2))\cos(\omega t)\exp(-\Gamma t)$ or $E\propto (1/(1+t^2))\cos(\omega t),$ which is in very good agreement with its integral $\arctan(t).$ Due to the high Q factor of the atom, the effect of $\exp(-\Gamma t)$ in the input at short times is negligible. We also calculate these quantities for an atom with $\Omega=10^7\,(\mathrm{1/s})$ and $\Gamma=1\,(\mathrm{1/s})$ in (e)-(h). As can be seen in (f) and (h) the pole regime time lasts more than $10\,\mathrm{ns}$ and the ultrahigh temporal resolution is maintained throughout this time, which is on the order of $4\cdot 10^7$ optical cycles. Here, too, there is excellent agreement between the analytic integration and the atomic integration.
• Figure 3: The response of a lossless slab to $E_\mathrm{inc}=e^{-\Gamma t+i\omega t}$. (a) The initial response of a slab with $l=1800\lambda, r_1=0.99,\,\, n_1=1.4,\lambda=550\mathrm{nm},\,\,\mathrm{and}\,\, Q=3.68\cdot 10^6$ to the complex-resonant-frequency incoming field $E_\mathrm{inc}=\exp(-i\omega t -\Gamma t)\theta(t).$ Unlike the previous case, here the long roundtrip results in discretization in the response, which reduces the temporal resolution by more than $10^6$. (b) The scattered field for the complex-frequency excitation for $t<4/\Gamma,$ of the form $E_\mathrm{scat}\propto t\exp(-i\omega t -\Gamma t)\theta(t).$
• Figure 4: Comparison of $v(t)$ for $f(t)=\frac{1}{1+at^2},$ where $a=10^{15} \,(\mathrm{Hz}),$ by numerically solving differential Bloch equations to the analytic integral $\frac{\Omega}{a} \mathrm{atan}(at)$ with very good agreement.
• Figure 5: The response of a lossless slab to a complex-frequency resonant excitation. (a) The initial response of a slab with $l=10\lambda, r_1=0.99,\,\, n_1=1.4,\lambda=550\mathrm{nm},\,\,\mathrm{and}\,\, Q=2810$ to the complex-resonant-frequency incoming field $E_\mathrm{inc}=\exp(-i\omega t -\Gamma t)\theta(t).$ Unlike the previous case, here the long roundtrip results in discretization in the response, which significantly limits the temporal resolution. (b) The scattered field for the complex-frequency excitation for $t<4/\Gamma,$ of the form $E_\mathrm{scat}\propto t\exp(-i\omega t -\Gamma t)\theta(t).$