In this Chapter we have studied the implementation of a quantum sensing device based on a weakly interacting BEC trapped in a 2D ring potential. We have started by deriving, in the context of the GPE description of the BEC, the general set of cou-pled non-linear equations that govern the time evolution of the amplitudes of the OAM modes. Then, we have focused on an initial state consisting of an imbalanced
superposi-tion of the two counter-propagating OAMl= 1 modes in the weakly interacting regime.
The density profile of this state has a minimal line that rotates due to a time-dependent relative phase between the counter-propagating modes induced by the non-linear cou-plings. By comparison with a direct integration of the full 2D GPE, we have shown that it is enough to restrict the general equations of motion to the OAM l = 1 and l = 3 modes to obtain a very good description of the dynamics of the BEC. Within this truncated model, we have obtained a simple analytical expression that relates the atom-atom interaction strength to the rotation frequency of the minimal density line
Harnessing the latter analytical relation, we have proposed a protocol to use the system as a sensor of two-body interactions in dilute BECs. The experimental deter-mination of all the quantities involved in the measurement could be done by analysing fluorescence images of the BEC density profile. We have tested the protocol by compar-ing the numerical results of the analysis of the BEC density profiles with the ab initio known values of the interaction strength used in the simulations. In the weakly inter-acting regime, the protocol provides accurate estimations of the interaction strength.
In the presence of a Feshbach resonance, the system could also be used as a magnetic field sensor by measuring the change in the rotation frequency of the nodal line induced by variations of the s-wave scattering length caused by magnetic field fluctuations. For atomic species with low three-body losses close to the resonant value of the magnetic field, the sensitivity would be enhanced around this region. We have also discussed the use of the device as a sensor of external rotations, which could be measured as the dif-ference between the experimentally observed rotation frequency of the minimal density line and the one predicted by the model.
Finally, we have discussed two possible ways to prepare the initial state consisting of an imbalanced superposition of counter-propagating OAM l = 1 modes. The most straightforward -although experimentally challenging- approach would be to directly imprint the phase and density profiles corresponding to this state onto a BEC loaded in the ground state of the 2D ring potential. Alternatively, one could initially load only one of the OAM modes and induce a coupling with the mode with opposite circulation by momentarily breaking the cylindrical symmetry of the potential. We have performed full GPE numerical simulations which confirm the feasibility of this latter approach.
Topological edge states and Aharonov-Bohm caging with ultracold atoms carrying orbital angular momentum in a diamond chain
In this Chapter, we study the single-particle properties of a system formed by ul-tracold atoms loaded into the manifold of l = 1 Orbital Angular Momentum (OAM) states of an optical lattice with a diamond chain geometry. We find that this system has a topologically non-trivial band structure and exhibits robust edge states that persist across the gap closing points, indicating the absence of a topological transition. We dis-cuss how to perform the topological characterization of the model with a generalization of the Zak’s phase and we show that this system constitutes a realization of a square-root topological insulator. In addition, we demonstrate that quantum interference between the different tunneling processes involved in the dynamics may lead to Aharonov-Bohm caging in the system.
The Chapter is organized as follows. In Sec. 4.1 we give a brief overview on recent progress on the study of topological systems with ultracold atoms and photonic plat-forms. In Sec. 4.2 we describe in detail the physical system that we consider and we derive the tight-binding model that we use to describe its single-particle properties. We also compute the band structure and discuss the differences with the model of a dia-mond chain without the OAM degree of freedom. Next, we introduce three successive analytical mappings that allow to unravel the main features of the model. In particular, the basis rotation introduced in Sec. 4.3 decouples the original chain with two states per site into two independent chains with one orbital per site and a net π flux through the plaquettes. Then, in Sec. 4.4 we map each of these independent chains into a
modi-67
68 Chapter 4 – Topological edge states and AB caging in a diamond chain fied Su-Schrieffer-Heeger (SSH) model. This mapping allows to understand the different types of eigenstates of the system and the occurrence of Aharonov-Bohm caging in the limit when all the bands are flat. To complete the analytical analysis of the model, in Sec. 4.5 we perform a third mapping that allows to characterize the topology of the system. In Sec. 4.6 we support the analytical findings discussed in the previous sections with numerical results. Finally, in Sec. 4.7 we summarize our conclusions and outline some further perspectives for this work.
4.1 Introduction
Since the observation of the quantum Hall effect in two-dimensional electron gases [68, 106] and the discovery of its relation with topology [70], the study of systems with non-trivial topological properties has become a central topic in condensed matter physics. A very interesting example of such exotic phases of matter are topological insulators [67], which are materials that exhibit insulating properties on their bulk but possess a bulk-boundary correspondence that correlates non-trivial topological indices of the bulk energy bands, such as, e.g., the Berry phase [152], with the existence of conducting edge states under open boundary conditions. There are many different types of topological insulators, which can be systematically classified in terms of their symmetries and dimensionality [79].
In recent years, many efforts have been devoted to implementing topologically non-trivial models in clean and highly controllable systems. Topological states have been observed and characterized in light-based platforms [204] such as photonic crystals [205–
209] and photonic quantum walks [210,211]. Ultracold atoms in optical lattices are also a well-suited environment to implement topological phases of matter [83]. Remarkable achievements in this platform include the realisation of the Haldane [94] and Hofstadter [46, 47] models, the demonstration of a link between topology and out-of-equilibrium dynamics [104], the observation of a many-body topological phase with Rydberg atoms [107], the experimental measurement [103] of the Zak’s phase [157], the detection of topological states [100, 212] or the observation of a topological Anderson insulator [81].
There are a wide range of further theoretical proposals for the observation of topological phenomena in cold atoms [213–219], most of which are based around the realisation of artificial gauge fields by laser dressing [46, 47, 220], or periodically driving the lattice system [85, 89]. In both photonic and ultracold atom systems, the possibility to use synthetic dimensions provides a powerful way to explore topological matter [96, 97].
In this Chapter, we explore topologically non-trivial multi-level models that arise naturally for ultracold atoms in excited OAM states of a one-dimensional (1D) chain. We study a concrete example of a diamond chain to demonstrate how this model is rendered
topologically non-trivial due to relative phases in the tunneling amplitudes between OAM l = 1 states with opposite circulations. Such a system could be experimentally realized, for instance, by exciting the atoms to thep-band of a conventional optical lattice [58, 135,221,222] or by optically transferring OAM [109, 122] to atoms confined to an arrangement of ring-shaped potentials, which can be created by a variety of techniques [109–114,116]. Remarkably, we find that topological states exist regardless of the values of the parameters of the model, with no topological transition across the gap closing points. This system constitutes an unusual example of a topological insulator with non-quantized values of the Zak’s phase due to the inversion axes not crossing the center of any choice of unit cell [157]. In order to circumvent this difficulty, we make use of a recently developed technique [223] to perform the topological characterization.
Furthermore, the model belongs to a new class of square-root topological insulators [224, 225], in which the quantized values of the Zak’s phases are recovered after taking the square of the bulk Hamiltonian. Fundamentally, this behaviour arises because the OAMl= 1 states are equivalent to thepxandpy orbitals in optical lattices [58,135,221], which have been shown to naturally display non-trivial topological properties in one-[222]
and two-[226,227] dimensional systems due to the parity of their wave functions. In the OAM l = 1 basis, the mechanism that yields topological properties is the appearance of relative phases in the tunnelling amplitudes, which are controllable by tuning the geometry of the lattice [137].
Additionally, a proper tuning of the inter-site separation and the central angle can lead to Aharonov-Bohm caging, which consists in the confinement of wave packets due to quantum interference [225,228–231]. A distinctive advantage regarding the realization of Aharonov-Bohm caging in this model is that, at variance with other proposals [225,230–
232], one does not need to rely on creating synthetic gauge fields [84–86,233] to produce the magnetic flux required for Aharonov-Bohm caging. Instead, in our OAMl = 1 model complex phases with values controlled by the central angle appear naturally in some of the tunneling parameters [137, 234], giving rise to an effective magnetic flux.