Measuring the quantum geometric tensor in two-dimensional photonic and exciton-polariton systems

We propose theoretically a method that allows to measure all the components of the quantum geometric tensor (the metric tensor and the Berry curvature) in a photonic system. The method is based on standard optical measurements. It applies to two-band systems, which can be mapped to a pseudospin, and to four-band systems, which can be described by two entangled pseudospins. We apply this method to several specific cases. We consider a 2D planar cavity with two polarization eigenmodes, where the pseudospin measurement can be performed via polarization-resolved photoluminescence. We also consider the $s$ band of a staggered honeycomb lattice with polarization-degenerate modes (scalar photons), where the sublattice pseudospin can be measured by performing spatially resolved interferometric measurements. We finally consider the $s$ band of a honeycomb lattice with polarized (spinor) photons as an example of a four-band model. We simulate realistic experimental situations in all cases. We find the photon eigenstates by solving the Schrödinger equation including pumping and finite lifetime, and then simulate the measurements to finally extract realistic mappings of the k-dependent tensor components.

Figure 1

(a) Sketch of a planar microcavity showing two distributed Bragg reflectors (DBRs) formed by interleaving $\lambda /4$ layers and an active region with quantum wells (QWs). (b) Staggered honeycomb lattices can be made out of etched planar microcavities. The panel shows a top view of such lattice. Each sublattice ($A/B$) is composed of 0D cavities (micropillars) with different radius.

Figure 2

Dispersion of the planar microcavity with crossed effective magnetic fields (XY, Zeeman, and TE-TM splittings). (a) $k_y$ cross-section and (b) $k_x$ cross-section.

Figure 3

Trace of the metric tensor $g_{xx}+g_{yy}$ of the LPB in a cavity system without Zeeman splitting: without (a) and with XY splitting (b) from the analytical formula (15).

Figure 4

Berry curvature of the LPB in a cavity system: without (a) and with XY splitting (b) from the analytical formula (15).

Figure 5

Berry curvature extracted from numerical simulations using Schrödinger equation and Eq. (13).

Figure 6

Staggered honeycomb lattice tight-binding dispersion (${\mathrm{\Delta }}_{AB}/J=0.2$). Dashed vertical lines mark high symmetry points in the first Brillouin zone.

Figure 7

Quantum geometric tensor components in staggered honeycomb lattice (tight-binding results). (a) $g_{xx}$, (b) $g_{yy}$, (c) $g_{xx}+g_{yy}$, (d) Berry curvature $B_z$ (lower band, ${\mathrm{\Delta }}_{AB}/J=0.2$). Dashed red squares around $K$ point show the zoomed region for the numeric QGT extraction.

Figure 8

Examples of interference pattern in real space used in order to reconstruct phase difference between A and B pillars for two opposite values of $q$. The thick white lines mark the contours of the numerical staggered honeycomb potential used in the simulations. Red (blue) dashed circles highlight the A (B) pillars.

Figure 9

Berry curvature (a) and metric tensor component $g_{xx}$ (c), extracted from numerical simulations based on the Schrödinger equation (16), compared with corresponding tight-binding results [(b) and (d)]. Zoom around $K$ point (dashed red square in Fig. 7).

Figure 10

Tight-binding dispersion of regular honeycomb lattice with TE-TM SOC and Zeeman field (${\mathrm{\Delta }}_z/J=0.1, \delta J/J=0.2$). Dashed vertical lines mark high symmetry points in the first Brillouin zone.

Figure 11

Quantum geometric tensor components in regular honeycomb lattice (first band, tight-binding results). (a) $g_{xx}^{\left(1\right)}$, (b) $g_{yy}^{\left(1\right)}$, (c) $g_{xx}^{\left(1\right)}+g_{yy}^{\left(1\right)}$, (d) Berry curvature $B_z^{\left(1\right)}$. (First band, ${\mathrm{\Delta }}_z/J=0.1, \delta J/J=0.2$.)

Figure 12

Quantum geometric tensor components in regular honeycomb lattice (tight-binding results). (a) $g_{xx}^{\left(2\right)}$, (b) $g_{yy}^{\left(2\right)}$, (c) $g_{xx}^{\left(2\right)}+g_{yy}^{\left(2\right)}$, (d) Berry curvature $B_z^{\left(2\right)}$. (2nd band, ${\mathrm{\Delta }}_z/J=0.1, \delta J/J=0.2.$)

Figure 13

Berry curvature ${\mathrm{\Omega }}_z^{\left(2\right)}$ (a) and metric component $g_{xx}^{\left(2\right)}$ (c) extracted from numerical simulations based on (16) using (21) and (22). (b)–(d) Tight-binding results. Zoom around $K$ point.