Designing Atomtronic Circuits via Superfluid Dynamics

TL;DR

The paper addresses the design of atomtronic circuits using Bose-Einstein condensates in trap-like geometries controlled by mobile barriers, aiming to realize a universal set of classical logic gates. It adopts a classical-field simulation approach to model 2D condensates with static and mobile barriers, mapping the system to a lattice description and extracting gate inputs/outputs from barrier-driven currents and density imbalances. Key contributions include implementing a 2-input AND gate, a 4-input AND circuit, a NOT gate via Josephson oscillations, and a NAND gate, thereby establishing a universal gate set and detailing barrier protocols for circuit operation. The work demonstrates a feasible route to robust, low-dissipation classical computation using atomtronic circuits in ultracold atomic systems and provides a blueprint for scalable integration of logical elements in BEC-based devices.

Abstract

We propose to design atomtronic circuits with Bose-Einstein condensates (BECs) in circuit-like traps that are controlled via mobile barriers. Using classical-field simulations, we demonstrate a universal set of logical gates and show how to assemble them into circuits. We first demonstrate an AND gate based on a T-shaped BEC, utilizing a combination of mobile and static barriers. The mobile barriers provide the logical input of the gate, while the static barrier functions as a Josephson junction that generates the AND output of the gate via a density imbalance across the barrier. Next we show how to combine three AND gates into a circuit, with a design composed of two T-shapes and an H-shape. Furthermore, we demonstrate how to use Josephson oscillations to create a NOT gate and combine it with an AND gate, thereby showcasing a universal set of gates and their assembly into circuits.

Figures (8)

• Figure 1: Dynamical regimes of an atomtronic 2-input AND gate. (a) Density distribution $n(x, y)$, averaged over the initial ensemble, is shown at $t_{\text{end}}=650 \text{ ms}$ after the constant movement of two mobile barriers, labeled A and B (marked in red), with a velocity of $v_0= 0.1 \text{ mm/s}$. These barriers move from the edge of the condensate to their final positions. Each mobile barrier has a height of $\tilde{V} _{0, \mathrm{m}}=3$ and a width of $\tilde{\sigma} _\mathrm{m} = 8$. The static barrier, marked in orange, has a height of $\tilde{V} _{0, \mathrm{s}}=2.5$ and width of $\tilde{\sigma} _\mathrm{s} = 3$. (b) The same parameter values are used to display the density distribution when only the vertical barrier is moved. (c) $n(x, y)$ is shown for the case of horizontal barrier movement. (d) The barrier protocol. All Gaussian barriers are linearly ramped up over $100$ ms, followed by a $50\,\text{ms}$ waiting period. After this, the barriers move at a constant speed for $500$ ms. Finally, we calculate the density imbalance using densities across the static barrier within the region $80\,\mu\text{m} < x < 180\,\mu\text{m}$ and $0\,\mu\text{m} < y < 40\,\mu\text{m}$.
• Figure 2: Implementation of an atomtronic AND gate. (a) Sketch of a standard AND gate is shown alongside the corresponding truth table for the atomtronic AND gate. $v_A$ and $v_B$ represent the velocities of the individual barrier movements, while $z(t)$ denotes the density imbalance across the static barrier. (b) Time evolution of the imbalance $z(t)$ is shown after the completion of the barrier movements; see text. When only the barrier A is moved or only the barrier B is moved, the imbalance remains relatively small. However, a significant imbalance is observed when both barriers A and B are moved together.
• Figure 3: Implementation of an atomtronic circuit to create a 4-input AND gate. (a) We show the density distribution $n(x,y)$ at time $t=650\text{ ms}$, for the case of moving all four mobile barriers (A, B, C and D) simultaneously at a constant velocity of $v_0= 0.1 \text{ mm/s}$ from the edges of the condensate to their final positions. Each two-input AND gate consists of two mobile barriers with heights $\tilde{V} _{0, \mathrm{m}}=3$ and widths $\tilde{\sigma} _\mathrm{m}= 8$, and one static barrier with a height $\tilde{V} _{0, \mathrm{s}}=2.5$ and width $\tilde{\sigma} _\mathrm{s}= 3$ (orange dashed lines). Four Gaussian barriers, each with a height of $\tilde{V} _{0, \mathrm{s}}=5$ and a width of $\tilde{\sigma} _\mathrm{s} = 3$, separate the two-input AND gates from the central part, functioning as left and right output gates. A static barrier with a height of $\tilde{V} _{0, \mathrm{s}}=2$ and width $\tilde{\sigma} _\mathrm{s}= 2$ (white dashed lines) is placed in the central part, acting as the output channel. (b) At $t=900\text{ ms}$, $n(x,y)$ is shown after the right output gates are opened. (c) At $t=1095\text{ ms}$, $n(x,y)$ is displayed after the left output gates are opened. (d) Timeline of the barrier protocol. Following the AND-gate procedure shown in Fig. \ref{['fig1']}(d), the left and right output gates are linearly ramp down to analyze the resulting density buildup within the area of $80\,\mu\text{m} < x < 180\,\mu\text{m}$ and $130\,\mu\text{m} < y < 170\,\mu\text{m}$.
• Figure 4: Results of the logical atomtronic 4-input AND gate. (a) Sketch of a 4-input AND gate. (b) Time evolution of the density imbalance $z(t)$ is shown for all relevant cases among the 16 possible combinations of the 4 mobile input barriers (A, B, C and D). In brackets are the cases that behave similarly due to the symmetry of the setup.
• Figure 5: (a) NOT gate and the corresponding truth table for the atomtronic NOT gate. $z(0)$ represents the initial imbalance, and $z(t_\text{end})$ is the final imbalance. (b) Time evolution of the imbalance $z(t)$ is shown following a phase imprint of $\phi_0 = 0.25\, \pi$ on the left subsystem. $z(t)$ undergoes half a Josephson oscillation (JO), with regions I and II (each representing a quarter of the JO) illustrating the imbalance dynamics for the two NOT-gate operations.(c) Barrier protocol. A static barrier is linearly ramped up at the center of the cloud over $100$ ms, followed by a $50\,\text{ms}$ waiting period. At $t=150$, we imprint a phase $\phi_0 = 0.25\, \pi$ on the left subsystem.