Dark Solitons: Unveiling Localized Dips in Bose-Einstein Condensates and their Quantum Information Potential
Bose-Einstein condensates (BECs), exotic states of matter where a significant fraction of atoms occupy the same quantum state, have become a fascinating playground for exploring nonlinear phenomena. Among these, dark solitons stand out as robust, localized dips in the condensate density. This article delves into the physics of dark solitons in BECs, their potential applications in quantum information processing (QIP) and explores their creation through various methods. We will then propose an experiment specifically tailored for the Oqtant system developed by Infleqtion.
The Governing Equation: Gross-Pitaevskii Equation (GPE)
The dynamics of a BEC are elegantly described by the Gross-Pitaevskii equation (GPE), a nonlinear Schrödinger equation [1]. In the presence of a self-interaction between atoms, the GPE takes the form:
Here, ψ is the macroscopic wavefunction of the condensate, m is the atomic mass, V(r) is an external potential, and U0 characterizes the strength of the interatomic interaction. Remarkably, under certain conditions, the GPE admits solutions where the density (proportional to ∣ψ∣2) exhibits a localized dip — a dark soliton.
Speed and the Sound Analogy
Unlike classical solitons that propagate at speeds exceeding the sound speed in their medium, dark solitons in BECs travel slower.
The propagation velocity of a dark soliton, vs, is related to its depth and the sound speed in the condensate, cs, through:
The specific function, f(depth), depends on the details of the system. This subsonic nature arises from the interplay between the nonlinear self-interaction and the quantum mechanical nature of the condensate.
Generating Darkness: Phase Imprinting and Beyond
Creating dark solitons in BECs can be achieved through various techniques. A prominent method is phase imprinting, where a spatially modulated laser pulse imparts a specific phase pattern onto the condensate, effectively creating a density dip. Other methods involve using spatially modulated magnetic fields or exploiting the nonlinearities of Feshbach resonances.
Experimenting with Oqtant: A Tailored Approach
The Oqtant system provides a powerful platform for simulating and manipulating BECs. Here, we propose an experiment to create and observe dark solitons:
- Preparation: Begin with a homogeneous BEC in the Oqtant simulator.
- Phase Imprinting: Apply a spatially modulated laser pulse with a carefully chosen phase profile to create a local density dip.
- Propagation and Observation: Simulate the time evolution of the condensate using Oqtant’s capabilities. Observe the propagation and stability of the generated dark soliton through the density profile.
Conclusion: Unveiling the Potential of Darkness
Dark solitons in BECs offer a unique avenue for exploring nonlinear phenomena and hold promise for QIP applications. Their stability, well-defined propagation characteristics, and potential use as qubits make them intriguing candidates for information storage and processing. We have introduced the GPE framework, explored their generation methods, and proposed an experiment specifically designed for the Oqtant system.
Ready to Dive into Darkness?
Head over to BEConOqtant/Solitons/DarkSolitons.ipynb at main · JoshDumo/BEConOqtant (github.com) to explore a Jupyter notebook implementing the proposed Oqtant experiment. Delve into the world of dark solitons and witness the fascinating interplay of quantum mechanics and nonlinearity firsthand!
[1] Denschlag, J., Simsarian, J. E., Feder, D. L., Clark, C. W., Collins, L. A., Cubizolles, J., Deng, L., Hagley, E. W., Helmerson, K., Reinhardt, W. P., Rolston, S. L., Schneider, B. I., & Phillips, W. D. (2000). Generating solitons by phase engineering of a bose-einstein condensate. Science (New York, N.Y.), 287(5450), 97–101. https://doi.org/10.1126/science.287.5450.97
