## The critical coupling constant *G*_{c} (in units of ω_{y}) as a function of the chemical potential μ for the spherical γ = 1 case (upper panel) and deformed cases with the frequency ratios γ = 2 (middle panel) and γ = 1.57 (lower panel)

_{c}

#### figure

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**Figure 6.** The critical coupling constant *G _{c}* (in units of ω

_{y}) as a function of the chemical potential μ for the spherical γ = 1 case (upper panel) and deformed cases with the frequency ratios γ = 2 (middle panel) and γ = 1.57 (lower panel). The smearing parameter δ and the dimensionless Rashba coupling parameter β are given in the panels. The notation in the legend indicates that G_\Sigma is obtained by doing summation over levels, while

*G*

_{0.5}and

*G*

_{1}indicate that we have used smeared distribution with δ = 0.5 and δ = 1 respectively. Here we set ω = ω

_{y}.

**Abstract**

We consider a spin–orbit coupled system of particles in an external trap that is represented by a deformed harmonic oscillator potential. The spin–orbit interaction is a Rashba interaction that does not commute with the trapping potential and requires a full numerical treatment in order to obtain the spectrum. The effect of a Zeeman term is also considered. Our results demonstrate that variable spectral gaps occur as a function of strength of the Rashba interaction and deformation of the harmonic trapping potential. The single-particle density of states and the critical strength for superfluidity vary tremendously with the interaction parameter. The strong variations with Rashba coupling and deformation imply that the few- and many-body physics of spin–orbit coupled systems can be manipulated by variation of these parameters.