Recent experiments with dilute trapped Fermi gases observed that weak interactions can drastically modify spin transport dynamics and give rise to robust collective effects including global demagnetization, macroscopic spin waves, spin segregation, and spin self-rephasing. In this work we develop a framework for studying the dynamics of weakly interacting fermionic gases following a spin-dependent change of the trapping potential which illuminates the interplay between spin, motion, Fermi statistics, and interactions. The key idea is the projection of the state of the system onto a set of lattice spin models defined on the single-particle mode space. Collective phenomena, including the global spreading of quantum correlations in real space, arise as a consequence of the long-ranged character of the spin model couplings. This approach achieves good agreement with prior measurements and suggests a number of directions for future experiments.The interplay between spin and motional degrees of freedom in interacting electron systems has been a longstanding research topic in condensed matter physics. Interactions can modify the behavior of individual electrons and give rise to emergent collective phenomena such as superconductivity and colossal magnetoresistance [1]. Theoretical understanding of non-equilibrium dynamics in interacting fermionic matter is limited, however, and many open questions remain. Ultracold atomic Fermi gases, with precisely controllable parameters, offer an outstanding opportunity to investigate the emergence of collective behavior in out-of-equilibrium settings.Progress in this direction has been made in recent experiments with ultracold spin-1/2 fermionic vapors, where initially spin-polarized gases were subjected to a spin-dependent trapping potential ( Fig. 1) implemented by a magnetic field gradient [2-4], or a spin-dependent harmonic trapping frequency [5-8] -equivalent to a spatially-varying gradient. Even in the weakly interacting regime, drastic modifications of the single-particle dynamics were reported. Moreover, despite the local character of the interactions, collective phenomena were observed, including demagnetization and transverse spinwaves in the former, and a time-dependent separation (segregation) of the spin densities and spin self-rephasing in the latter. Although mean-field and kinetic theory formulations have explained some of these phenomena [8][9][10][11][12][13][14][15][16][17][18], a theory capable of describing all the time scales and the interplay between spin, motion, and interactions has not been developed.In this work, we develop a framework that accounts for the coupling of spin and motion in weakly interacting Fermi gases. We qualitatively reproduce and explain all phenomena of the aforementioned experiments and obtain quantitative agreement with the results of Ref. [7]. In this formulation the state of the system is represented as a superposition of spin configurations which live on lattices whose sites correspond to modes of the underlying single-particle sy...

Motivated by several experimental efforts to understand spin diffusion and transport in ultracold fermionic gases, we study the spin dynamics of initially spin-polarized ensembles of harmonically trapped non-interacting spin-1/2 fermionic atoms, subjected to a magnetic field gradient. We obtain simple analytic expressions for spin observables in the presence of both constant and linear magnetic field gradients, with and without a spin-echo pulse, and at zero and finite temperatures. The analysis shows the relevance of spin-motional coupling in the non-interacting regime where the demagnetization decay rate at short times can be faster than the experimentally measured rates in the strongly interacting regime under similar trapping conditions. Our calculations also show that particle motion limits the ability of a spin-echo pulse to remove the effect of magnetic field inhomogeneity, and that a spin-echo pulse can instead lead to an increased decay of magnetization at times comparable to the trapping period.

We prove that the corona product of two graphs has no Laplacian perfect state transfer whenever the first graph has at least two vertices. This complements a result of Coutinho and Liu who showed that no tree of size greater than two has Laplacian perfect state transfer. In contrast, we prove that the corona product of two graphs exhibits Laplacian pretty good state transfer, under some mild conditions. This provides the first known examples of families of graphs with Laplacian pretty good state transfer. Our result extends of the work of Fan and Godsil on double stars to the Laplacian setting. Moreover, we also show that the corona product of any cocktail party graph with a single vertex graph has Laplacian pretty good state transfer, even though odd cocktail party graphs have no perfect state transfer.

We study state transfer in quantum walk on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer.1 maximum degree k which have perfect state transfer. Therefore, the following relaxation of this notion is often more useful to consider. The state transfer between u and v is called "pretty good" (see Godsil [18]) or "almost perfect" (see Vinet and Zhedanov [27]) if the (u, v)-entry of the unitary matrix U(t) can be made arbitrarily close to one.Christandl et al. [9,8] observed that the path P n on n vertices has antipodal perfect state transfer if and only if n = 2, 3. In a striking result, Godsil et al. [21] proved that P n has antipodal pretty good state transfer if and only if n + 1 is a prime, twice a prime, or a power of two. This provides the first family of graphs with pretty good state transfer which correspond to the quantum spin chains originally studied by Bose.Shortly after, Fan and Godsil [12] studied a family of graphs obtained by taking two cones K 1 +K m and then connecting the two conical vertices. They showed that these graphs, which are called double stars, have no perfect state transfer, but have pretty good state transfer between the two conical vertices if and only if 4m + 1 is a perfect square. These graphs provide the second family of graphs known to have pretty good state transfer.In this work, we provide new families of graphs with pretty good state transfer. Our constructions are based on a natural generalization of Fan and Godsil's double stars. The corona product of an n-vertex graph G with another graph H, typically denoted G • H, is obtained by taking n copies of the cone K 1 + H and by connecting the conical vertices according to G. In a corona product G • H, we sometimes call G the base graph and H the pendant graph. This graph product was introduced by Frucht and Harary [15] in their study of automorphism groups of graphs which are obtained by wreath products.We first observe that perfect state transfer on corona products is extremely rare. This is mainly due to the specific forms of the corona eigenvalues (which unsurprisingly resemble the eigenvalues of cones) coupled with the fact that periodicity is a necessary condition for perfect state transfer. Our negative results apply to corona families G • H when H is either the empty or the complete graph, under suitable conditions on G. In a companion work [1], we observed an optimal negative result which holds for all H but in a Laplacian setting.Given that perfect state transfer is rare, our subsequent results mainly focus on pretty good state transfer. We prove that the family of graphs K 2 • K m , which are called barbell graphs (see Ghosh et al. [16]), admit pretty good state transfer for all m. Here, state transfer occurs between the two vertices of K 2 . This is in contrast to the double stars K 2 • K m where pretty good state transfer requires number-theoretic conditions on m.We observe something curious for corona products when the base graph is c...

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