In spinor Bose-Einstein condensates, spin-changing collisions are a remarkable proxy to coherently realize macroscopic many-body quantum states. These processes have been, e.g., exploited to generate entanglement, to study dynamical quantum phase tra
nsitions, and proposed for realizing nematic phases in atomic condensates. In the same systems dressed by Raman beams, the coupling between spin and momentum induces a spin dependence in the scattering processes taking place in the gas. Here we show that, at weak couplings, such modulation of the collisions leads to an effective Hamiltonian which is equivalent to the one of an artificial spinor gas with spin-changing collisions that are tunable with the Raman intensity. By exploiting this dressed-basis description, we propose a robust protocol to coherently drive the spin-orbit coupled condensate into the ferromagnetic stripe phase via crossing a quantum phase transition of the effective low-energy model in an excited-state.
We apply a construction developed in a previous paper by the authors in order to obtain a formula which enables us to compute $ell^2$-Betti numbers coming from a family of group algebras representable as crossed product algebras. As an application, w
e obtain a whole family of irrational $ell^2$-Betti numbers arising from the lamplighter group algebra $K[mathbb{Z}_2 wr mathbb{Z}]$, being $K$ a subfield of the complex numbers closed under complex conjugation. This procedure is constructive, in the sense that one has an explicit description of the elements realizing such irrational numbers. This extends the work made by Grabowski, who first computed irrational $ell^2$-Betti numbers from the algebras $mathbb{Q}[mathbb{Z}_n wr mathbb{Z}]$, where $n geq 2$ is a natural number. We also apply the techniques developed to the (generalized) odometer algebra $mathcal{O}(overline{n})$, where $overline{n}$ is a supernatural number. We compute its $*$-regular closure, and this allows us to fully characterize the set of $ell^2$-Betti numbers arising from $mathcal{O}(overline{n})$.
In this paper, we introduce a new technique in the study of the $*$-regular closure of some specific group algebras $KG$ inside $mathcal{U}(G)$, the $*$-algebra of unbounded operators affiliated to the group von Neumann algebra $mathcal{N}(G)$. The m
ain tool we use for this study is a general approximation result for a class of crossed product algebras of the form $C_K(X) rtimes_T mathbb{Z}$, where $X$ is a totally disconnected compact metrizable space, $T$ is a homeomorphism of $X$, and $C_K(X)$ stands for the algebra of locally constant functions on $X$ with values on an arbitrary field $K$. The connection between this class of algebras and a suitable class of group algebras is provided by Fourier transform. Utilizing this machinery, we study an explicit approximation for the lamplighter group algebra. This is used in another paper by the authors to obtain a whole family of $ell^2$-Betti numbers arising from the lamplighter group, most of them transcendental.
Light-induced spin-orbit coupling is a flexible tool to study quantum magnetism with ultracold atoms. In this work we show that spin-orbit coupled Bose gases in a one-dimensional optical lattice can be mapped into a two-leg triangular ladder with sta
ggered flux following a lowest-band truncation of the Hamiltonian. The effective flux and the ratio of the tunneling strengths can be independently adjusted to a wide range of values. We identify a certain regime of parameters where a hard-core boson approximation holds and the system realizes a frustrated triangular spin ladder with tunable flux. We study the properties of the effective spin Hamiltonian using the density-matrix renormalization-group method and determine the phase diagram at half-filling. It displays two phases: a uniform superfluid and a bond-ordered insulator. The latter can be stabilized only for low Raman detuning. Finally, we provide experimentally feasible trajectories across the parameter space of the SOC system that cross the predicted phase transition.
In this paper we consider the algebraic crossed product $mathcal A := C_K(X) rtimes_T mathbb{Z}$ induced by a homeomorphism $T$ on the Cantor set $X$, where $K$ is an arbitrary field and $C_K(X)$ denotes the $K$-algebra of locally constant $K$-valued
functions on $X$. We investigate the possible Sylvester matrix rank functions that one can construct on $mathcal A$ by means of full ergodic $T$-invariant probability measures $mu$ on $X$. To do so, we present a general construction of an approximating sequence of $*$-subalgebras $mathcal A_n$ which are embeddable into a (possibly infinite) product of matrix algebras over $K$. This enables us to obtain a specific embedding of the whole $*$-algebra $mathcal A$ into $mathcal M_K$, the well-known von Neumann continuous factor over $K$, thus obtaining a Sylvester matrix rank function on $mathcal A$ by restricting the unique one defined on $mathcal M_K$. This process gives a way to obtain a Sylvester matrix rank function on $mathcal A$, unique with respect to a certain compatibility property concerning the measure $mu$, namely that the rank of a characteristic function of a clopen subset $U subseteq X$ must equal the measure of $U$.
For a division ring $D$, denote by $mathcal M_D$ the $D$-ring obtained as the completion of the direct limit $varinjlim_n M_{2^n}(D)$ with respect to the metric induced by its unique rank function. We prove that, for any ultramatricial $D$-ring $math
cal B$ and any non-discrete extremal pseudo-rank function $N$ on $mathcal B$, there is an isomorphism of $D$-rings $overline{mathcal B} cong mathcal M_D$, where $overline{mathcal B}$ stands for the completion of $mathcal B$ with respect to the pseudo-metric induced by $N$. This generalizes a result of von Neumann. We also show a corresponding uniqueness result for $*$-algebras over fields $F$ with positive definite involution, where the algebra $mathcal M_F$ is endowed with its natural involution coming from the $*$-transpose involution on each of the factors $M_{2^n}(F)$.
