The coupled nonlinear dynamics of ultracold quantum matter and electromagnetic field modes in an optical resonator exhibits a wealth of intriguing collective phenomena. Here we study a Λ-type, threecomponent Bose-Einstein condensate coupled to four dynamical running-wave modes of a ring cavity, where only two of the modes are externally pumped. However, the unpumped modes play a crucial role in the dynamics of the system due to coherent backscattering of photons. On a mean-field level we identify three fundamentally different steady-state phases with distinct characteristics in the density and spatial spin textures: a combined density and spin-wave, a continuous spin spiral with a homogeneous density, and a spin spiral with a modulated density. The spin-spiral states, which are topological, are intimately related to cavity-induced spin-orbit coupling emerging beyond a critical pump power. The topologically trivial density-wave-spin-wave state has the characteristics of a supersolid with two broken continuous symmetries. The transitions between different phases are either simultaneously topological and first-order, or second-order. The proposed setup allows the simulation of intriguing many-body quantum phenomena by solely tuning the pump amplitudes and frequencies, with the cavity output fields serving as a built-in nondestructive observation tool. New J. Phys. 21 (2019) 013029 S Ostermann et al *

We examine the accuracy and the validity of the lowest order supersymmetric WKB (SWKB) formula for cyclic shape invariant potentials (CSIPs). For period-2 CSIPs, we show analytically that the SWKB formula can yield exact eigenenergies for either all the even states or all the odd states. Such alternate exactness of the SWKB formula is due to the fact that period-2 CSIPs can also be considered as a kind of translational shape invariant potential. For CSIPs with periods greater than 2, we note that the accuracy of the SWKB formula also demonstrates similar alternating patterns. However, as a consequence of the rapid oscillations in the potential at large distances, the SWKB quantization formula fails to produce highly accurate results even in the high-energy limit.

We study here the self-similar shape invariant potential (SSSIP) proposed by Barclay et al (1993 Phys. Rev. A 48 2786) in the context of supersymmetric quantum mechanics. The superpotential of SSSIP, W(x), obeys an ordinary differential equation involving W and its derivative at two different spatial points, and hence cannot be solved with standard numerical methods. In addition, Taylor series expansion of W(x) about x = 0 also diverges at large x. To provide an effective numerical scheme to construct the superpotential, we use the Padé approximation to express W(x) as a fraction of polynomials in x. We find that the homogeneous two-point Padé approximant can indeed yield accurate values of the superpotential for all x.

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