Probing localization and quantum geometry by spectroscopy

The spatial localization of quantum states plays a central role in condensed-matter phenomena, ranging from many-body localization to topological matter. Building on the dissipation-fluctuation theorem, we propose that the localization properties of a quantum-engineered system can be probed by spectroscopy, namely, by measuring its excitation rate upon a periodic drive. We apply this method to various examples that are of direct experimental relevance in ultracold atomic gases, including Anderson localization, topological edge modes, and interacting particles in a harmonic trap. Moreover, inspired by a relation between quantum fluctuations and the quantum metric, we describe how our scheme can be generalized in view of extracting the full quantum-geometric tensor of many-body systems. Our approach opens an avenue for probing localization, as well as quantum fluctuations, geometry and entanglement, in synthetic quantum matter.Comment: 7 + 2 pages, 4 figures. Published versio

Tensor Berry connections and their topological invariants

The Berry connection plays a central role in our description of the geometric phase and topological phenomena. In condensed matter, it describes the parallel transport of Bloch states and acts as an effective "electromagnetic" vector potential defined in momentum space. Inspired by developments in mathematical physics, where higher-form (Kalb-Ramond) gauge fields were introduced, we hereby explore the existence of "tensor Berry connections" in quantum matter. Our approach consists in a general construction of effective gauge fields, which we ultimately relate to the components of Bloch states. We apply this formalism to various models of topological matter, and we investigate the topological invariants that result from generalized Berry connections. For instance, we introduce the 2D Zak phase of a tensor Berry connection, which we then relate to the more conventional first Chern number; we also reinterpret the winding number characterizing 3D topological insulators to a Dixmier-Douady invariant, which is associated with the curvature of a tensor connection. Besides, our approach identifies the Berry connection of tensor monopoles, which are found in 4D Weyl-type systems [Palumbo and Goldman, Phys. Rev. Lett. 121, 170401 (2018)]. Our work sheds light on the emergence of gauge fields in condensed-matter physics, with direct consequences on the search for novel topological states in solid-state and quantum-engineered systems.Comment: 10 pages, 1 table. Published versio

Space law and space resources

Space industrialization is confronting space law with problems that are changing old and shaping new legal principles. The return to the Moon, the next logical step beyond the space station, will establish a permanent human presence there. Science and engineering, manufacturing and mining will involve the astronauts in the settlement of the solar system. These pioneers, from many nations, will need a legal, political, and social framework to structure their lives and interactions. International and even domestic space law are only the beginning of this framework. Dispute resolution and simple experience will be needed in order to develop, over time, a new social system for the new regime of space

Detecting fractional Chern insulators through circular dichroism

Great efforts are currently devoted to the engineering of topological Bloch bands in ultracold atomic gases. Recent achievements in this direction, together with the possibility of tuning inter-particle interactions, suggest that strongly-correlated states reminiscent of fractional quantum Hall (FQH) liquids could soon be generated in these systems. In this experimental framework, where transport measurements are limited, identifying unambiguous signatures of FQH-type states constitutes a challenge on its own. Here, we demonstrate that the fractional nature of the quantized Hall conductance, a fundamental characteristic of FQH states, could be detected in ultracold gases through a circular-dichroic measurement, namely, by monitoring the energy absorbed by the atomic cloud upon a circular drive. We validate this approach by comparing the circular-dichroic signal to the many-body Chern number, and discuss how such measurements could be performed to distinguish FQH-type states from competing states. Our scheme offers a practical tool for the detection of topologically-ordered states in quantum-engineered systems, with potential applications in solid state.Comment: Revised versio

Detecting Chiral Edge States in the Hofstadter Optical Lattice

We propose a realistic scheme to detect topological edge states in an optical lattice subjected to a synthetic magnetic field, based on a generalization of Bragg spectroscopy sensitive to angular momentum. We demonstrate that using a well-designed laser probe, the Bragg spectra provide an unambiguous signature of the topological edge states that establishes their chiral nature. This signature is present for a variety of boundaries, from a hard wall to a smooth harmonic potential added on top of the optical lattice. Experimentally, the Bragg signal should be very weak. To make it detectable, we introduce a "shelving method", based on Raman transitions, which transfers angular momentum and changes the internal atomic state simultaneously. This scheme allows to detect the weak signal from the selected edge states on a dark background, and drastically improves the detectivity. It also leads to the possibility to directly visualize the topological edge states, using in situ imaging, offering a unique and instructive view on topological insulating phases.Comment: 4 pages, 4 figures, Supplementary material (Appendices A-D). Revised version, accepted in the Physical Review Letter

Quantized Rabi Oscillations and Circular Dichroism in Quantum Hall Systems

The dissipative response of a quantum system upon a time-dependent drive can be exploited as a probe of its geometric and topological properties. In this work, we explore the implications of such phenomena in the context of two-dimensional gases subjected to a uniform magnetic field. It is shown that a filled Landau level exhibits a quantized circular dichroism, which can be traced back to its underlying non-trivial topology. Based on selection rules, we find that this quantized circular dichroism can be suitably described in terms of Rabi oscillations, whose frequencies satisfy simple quantization laws. Moreover, we discuss how these quantized dissipative responses can be probed locally, both in the bulk and at the boundaries of the quantum Hall system. This work suggests alternative forms of topological probes in quantum systems based on circular dichroism.Comment: 7 pages, including 3 figures and Appendi

Tunable axial gauge fields in engineered Weyl semimetals: Semiclassical analysis and optical lattice implementations

In this work, we describe a toolbox to realize and probe synthetic axial gauge fields in engineered Weyl semimetals. These synthetic electromagnetic fields, which are sensitive to the chirality associated with Weyl nodes, emerge due to spatially and temporally dependent shifts of the corresponding Weyl momenta. First, we introduce two realistic models, inspired by recent cold-atom developments, which are particularly suitable for the exploration of these synthetic axial gauge fields. Second, we describe how to realize and measure the effects of such axial fields through center-of-mass observables, based on semiclassical equations of motion and exact numerical simulations. In particular, we suggest realistic protocols to reveal an axial Hall response due to the axial electric field $\mathbf{E}_5$, and also, the axial cyclotron orbits and chiral pseudo-magnetic effect due to the axial magnetic field $\mathbf{B}_5$.Comment: 16 pages, 6 figures, published versio

Lasing, quantum geometry and coherence in non-Hermitian flat bands

We show that lasing in flat band lattices can be stabilized by means of the geometrical properties of the Bloch states, in settings where the single-particle dispersion is flat in both its real and imaginary parts. We illustrate a general projection method and compute the collective excitations, which are shown to display a diffusive behavior ruled by quantum geometry through a peculiar coefficient involving gain, losses and interactions. Then, we analytically show that the phase dynamics display a surprising cancellation of the Kardar-Parisi-Zhang nonlinearity at the leading order. Because of the relevance of Kardar-Parisi-Zhang universality in one-dimensional geometries, we focus our study on the diamond chain and provide confirmation of these results through full numerical simulations.Comment: Added Ref. 17,18,2

Extracting the quantum metric tensor through periodic driving

We propose a generic protocol to experimentally measure the quantum metric tensor, a fundamental geometric property of quantum states. Our method is based on the observation that the excitation rate of a quantum state directly relates to components of the quantum metric upon applying a proper time-periodic modulation. We discuss the applicability of this scheme to generic two-level systems, where the Hamiltonian's parameters can be externally tuned, and also to the context of Bloch bands associated with lattice systems. As an illustration, we extract the quantum metric of the multi-band Hofstadter model. Moreover, we demonstrate how this method can be used to directly probe the spread functional, a quantity which sets the lower bound on the spread of Wannier functions and signals phase transitions. Our proposal offers a universal probe for quantum geometry, which could be readily applied in a wide range of physical settings, ranging from circuit-QED systems to ultracold atomic gases.Comment: 6 + 1 pages, 3 figure