Dispersion Properties of a SpinOrbitCoupled BoseEinstein Condensate
Dispersion Properties of a SpinOrbitCoupled BoseEinstein 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 lowestenergy quantum state. For such a system, when the particle's spin couples to its orbital angular momentum, a socalled spinorbitcoupled Bose–Einstein condensate (SOCBEC) 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 SOCBEC can be described by the following momentumspace Hamiltonian:
H=
ℏk(k+2 k R 2m δ 2  Ω 2 
Ω 2  ℏk(k+2 k R 2m δ 2 
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:
k
m
Ω
δ
k
R
E
±
ℏ
2
k
2m
2
ℏ
k
R
m
δ
2
2
Ω
2
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 (k)=k (green dashed line) is related to the particle group velocity (yellow dashed line) and the second mass parameter (k)=(k) (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 SOCBEC dispersion and thus the particle properties.
k
E

m
1
2
ℏ
1
((k))
∂
k
E

v=(k)
∂
k
E

m
2
1
2
ℏ
∂
k,k
E

i
n