Dispersion Properties of a Spin-Orbit-Coupled Bose-Einstein Condensate

A Bose–Einstein condensate (BEC) is a state of matter in which a large fraction of the particles of a bosonic gas occupy the lowest-energy quantum state. For such a system, when the particle's spin couples to its orbital angular momentum, a so-called spin-orbit-coupled Bose–Einstein condensate (SOC-BEC) is formed. This state can be achieved experimentally by using Raman lasers to couple two internal states of an atomic BEC while transferring momentum [1]. In the limit of a homogeneous and noninteracting gas, the SOC-BEC can be described by the following momentum-space Hamiltonian:
where is the wavevector, is the mass of the particles, is the Raman coupling strength, is the Raman detuning, and is the Raman wavevector. The equation gives rise to two new energy bands:
In the figures, the lower band is the solid blue line, while the upper band is the thin red line.
As the lower branch is strongly nonparabolic, two -dependent effective mass parameters emerge to describe the particle dynamics when propagating in these bands [2]. We focus on the band with lower energy as the system relaxes toward . The first mass parameter (green dashed line) is related to the particle group velocity (yellow dashed line) and the second mass parameter (purple dashed line) determines the acceleration of the packet when an external force is applied, as well as rate of diffusion of the particle wave packet. The different points at which the mass parameters diverge are labeled on the top axis as . At such points, the particle properties exhibit a peculiar dynamic and separate regions of positive and negative masses, leading to interesting regimes of propagation. You can use this Demonstration to tune the different coupling parameters to control the SOC-BEC dispersion and thus the particle properties.


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The Snapshots show various configurations of the SOC-BEC dispersion.
Snapshot 1: the lower branch only exhibits a large negative mass for positive wavevectors, affecting the value of the velocity
Snapshot 2: the lower branch exhibits both negative mass and for positive wavevectors, affecting both the value and the sign of the velocity
Snapshot 3: the lower branch exhibits a broader range of wavevectors for which there is a negative mass ; this is obtained from Snapshot 2 by using the sliders to increase the Raman detuning and wavevector values
[1] Y.-J. Lin, K. Jiménez-Garcia and I. B. Spielman, "Spin–Orbit-Coupled Bose–Einstein Condensates," Nature, 471(7336), 2001 pp. 83–86. doi:10.1038/nature09887.
[2] D. Colas and F. P. Laussy, "Self-Interfering Wave Packets," Physical Review Letters, 116, 2016 026401. doi:10.1103/PhysRevLett.116.026401.
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