Steady-state Hall response and quantum geometry of driven-dissipative lattices

Abstract

We study the effects of the quantum geometric tensor, i.e., the Berry curvature and the FubiniStudy metric, on the steady state of driven-dissipative bosonic lattices. We show that the quantumHall-type response of the steady-state wave function in the presence of an external potential gradient depends on all the components of the quantum geometric tensor. Looking at this steady-state Hall response, one can map out the full quantum geometric tensor of a sufficiently flat band in momentum space using a driving field localized in momentum space. We use the two-dimensional Lieb lattice as an example and numerically demonstrate how to measure the quantum geometric tensor.The developments in the study of topological insulators and superconductors have led to growing interest in understanding the geometry and topology of energy bands in condensed-matter physics [1][2][3]. The prototype of the two-dimensional topological insulator is the quantum Hall system, where the quantized Hall conductance is related to the topological Chern number of bands [4,5]. The Chern number is the integral of the geometrical Berry curvature over the Brillouin zone. This Berry curvature is associated with the adiabatic response of the system under a slow change in parameters [6] and has been directly measured in experiments [7,8].Along with these developments, another geometrical quantity of energy bands, the Fubini-Study metric, has recently attracted great interest . The FubiniStudy metric describes the "distance" between two quantum states in a parameter space and is related to phases acquired under non adiabatic processes [10,11]. Therefore, it affects physical observables under the fast change of external parameters, or when the system is dissipative [9,12]. It is also known to play important roles in the orbital magnetic susceptibility [13][14][15][16][17], the entanglement and many-body properties of quantum systems [18][19][20][21][22], superfluid density [23,24], and quantum information [25][26][27][28]. Both the Berry curvature and the Fubini-Study metric can be understood from a general framework in terms of the quantum geometric tensor, whose real part gives the Fubini-Study metric while the imaginary part gives the Berry curvature [9].In this paper, we find a simple relation between the quantum-Hall-type response in the steady state of drivendissipative bosonic lattices and the full quantum geometric tensor. Our results are relevant to photonic or mechanical lattices with non trivial geometrical or topological band structures, which have lately been an intensive area of study [34][35][36][37]. Previous studies have shown that analogs of the quantum Hall effects take place in certain lattices with loss, through the center-of-mass response of the steady-state when a lattice site is continuously driven [1][2][3]. Here we clarify the relation between the full quantum geometric tensor and the center of mass within linear response for general driven-dissipative lattices, which is valid in any spatial dimensions.Our finding can, in turn...

Kähler geometry and Chern insulators: Relations between topology and the quantum metric

“…As it turns out, the integral of this metric over the Brillouin zone [4] is associated to the spread functional and the localization tensor of the material [3,[5][6][7][8]. There are numerous proposals in the literature for extracting the quantum metric [9][10][11][12][13][14][15]. A proposal [4], making use of the fluctuation-dissipation theorem, shows that the localization tensor can also be measured through spectroscopy in synthetic quantum matter, such as ultracold atomic gases or trapped ions.…”

Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric

“…Importantly, besides the Berry curvature, there exists another gauge-invariant quantity associated with the geometry of quantum states, namely, the quantum metric [31,32]. It is known to play a role in various contexts and phenomena, including the magnetic susceptibility [33], superconducting weight [34], non-Abelian geometric phases [35,36], quantum chains [37], topological insulators [38], steady-state Hall response in drivendissipative lattices [39], semiclassical equations for transport [40][41][42] and bulk incompressibility in fractional quantum Hall states [43]. It has been recently shown that the quantum metric also carries information about the topological charge (Z-monopole) of Weyl and higherdimensional topological semimetals [44].…”

Extracting the quantum metric tensor through periodic driving

“…Closely related to the Berry curvature is the quantum metric tensor (or Fubini-Study metric), which is a distinct geometric property of energy eigenstates that reflects the "distance" between different quantum states [20][21][22]. The significance of the quantum metric was recently identified in a wide range of physical phenomena, including the conductivity in dissipative systems [23][24][25][26][27][28], orbital magnetism [29][30][31][32][33], the superfluid fraction [34][35][36], quantum information [37][38][39][40], entanglement and many-body properties [41][42][43][44][45], interference in Bloch states [46], Lamb-shift-like energy shift in excitons [47] and the mathematical construction of maximally-localized Wannier functions in crystals [22,[48][49][50][51]. Despite the importance of the quantum metric in these various contexts, one still lacks a direct experimental measurement of this geometric object.…”