Experimental measurement of the Berry curvature from anomalous transport

Abstract

The geometric properties of energy bands underlie fascinating phenomena in many systems, including solid-state, ultracold gases and photonics. The local geometric characteristics such as the Berry curvature 1 can be related to global topological invariants such as those classifying the quantum Hall states or topological insulators. Regardless of the band topology, however, any non-zero Berry curvature can have important consequences, such as in the semi-classical evolution of a coherent wavepacket. Here, we experimentally demonstrate that the wavepacket dynamics can be used to directly map out the Berry curvature. To this end, we use optical pulses in two coupled fibre loops to study the discrete time evolution of a wavepacket in a one-dimensional geometric 'charge' pump, where the Berry curvature leads to an anomalous displacement of the wavepacket. This is both the first direct observation of Berry curvature e ects in an optical system, and a proof-ofprinciple demonstration that wavepacket dynamics can serve as a high-resolution tool for mapping out geometric properties.The Berry curvature is a geometrical property of an energy band, which plays a key role in many physical phenomena as it encodes how eigenstates evolve as a local function of parameters 1 . In a twodimensional (2D) quantum Hall system, for example, the integral of the Berry curvature over the 2D Brillouin zone determines the Chern number: a global topological invariant that underlies the quantization of Hall transport for a 2D filled energy band 2 . As first explained by Thouless 3 , there can be an analogous topological quantization of particle transport in a 1D band insulator, when the lattice potential is 'pumped' , that is, is slowly and periodically modulated in time. Also in this case, the geometrical and topological properties are defined for an effective 2D parameter space, but now one spanned by the 1D Bloch momentum and the external periodic pumping parameter.A local non-zero Berry curvature can have striking physical effects in both 2D systems and 1D pumps, regardless of whether the global topological Chern number is non-trivial. In the simplest case, a semi-classical wavepacket moves as a coherent object governed by classical equations of motion with an additional 'anomalous' Hall velocity due to the geometrical Berry curvature at its centre-ofmass, as an external force is applied or as the control parameter is pumped 1,4 . As highlighted further below, this anomalous transport can be understood physically as the Berry curvature acting like a magnetic field in parameter space [5][6][7] .In recent years, there have been many landmark experiments to engineer and study geometrical and topological energy bands in ultracold gases and photonics . In photonics, for example, topological edge states have been studied in a wide variety of (effectively) 1D set-ups, such as quantum walks 14 and pumping in optical quasicrystals 13,29,30 , as well as in 2D quantum Hall-like systems of photonic crystals 9-11 , propagating waveguides 12 and silico...

Six-dimensional quantum Hall effect and three-dimensional topological pumps

“…This anomalous velocity can be understood as a momentum-space analog of the magnetic Lorentz force, in which the Berry curvature acts like a magnetic field in momentum space [53][54][55] . This term has important physical consequences for semiclassical motion, and can be used to map out the distribution of the Berry curvature over an energy band 56,57,77 . In order to consider higher orders in the perturbing fields, the wave packet must be instead constructed out of perturbed eigenstates.…”

Robust temporal pumping in a magneto-mechanical topological insulator

“…Topological pumps have been implemented in a variety of systems including cold atomic gases [14][15][16][17] and classical metamaterials [18][19][20]. Significant explorations in photonic metamaterials include using topological pumps to map the Berry curvature [21,22], to demonstrate transport of a localized mode in a quasiperiodic waveguide array [23,24], and to probe a four dimensional quantum Hall effect [25]. However, to date, a temporally-controlled topological pump that produces on-demand, disorder-resilient transport has not been demonstrated in any metamaterial system.…”

“…For photonic implementations, the latter requirement necessitates extremely rapid modulation, which is technically very challenging. To date, a workaround has been to use space instead of time as the pumping parameter [9,[22][23][24][25]29]. A time-controlled classical topological pump has remained elusive to date, and as a result, on-demand robust pumping of energy in a classical metamaterial has not yet been achieved.…”

Extracting the quantum metric tensor through periodic driving

“…A similar relation holds for non-interacting fermions (partially) filling the lowest band, in which case the weight |c(k)| 2 in Eq. (19) should be replaced by the density of fermions ρ(k) in this band. Example: Harper-Hofstadter model.…”