Effective Theory of Nonadiabatic Quantum Evolution Based on the Quantum Geometric Tensor

We study the role of the quantum geometric tensor (QGT) in the evolution of two-band quantum systems. We show that all its components play an important role on the extra phase acquired by a spinor and on the trajectory of an accelerated wave packet in any realistic finite-duration experiment. While the adiabatic phase is determined by the Berry curvature (the imaginary part of the tensor), the nonadiabaticity is determined by the quantum metric (the real part of the tensor). We derive, for geodesic trajectories (corresponding to acceleration from zero initial velocity), the semiclassical equations of motion with nonadiabatic corrections. The particular case of a planar microcavity in the strong coupling regime allows us to extract the QGT components by direct light polarization measurements and to check their effects on the quantum evolution.

Figure 1

(a) Bloch sphere with the spin (red arrow) and the magnetic field $\mathrm{\Omega }$ (blue arrow), adiabatic trajectory (blue), and real trajectory (red dashed line). (b) Mechanical analog: adiabatic trajectory of an infinitely small wheel (blue), cycloid trajectory of a point on a wheel (red). $\mathrm{\Omega }$—wheel rotation frequency, $v$—wheel velocity, $\omega$—shaft center rotation. (c) Total extra phase for one full spinor rotation as a function of the rotation time. (d) Deviation from the adiabatic Berry phase: numerical calculation (black) and analytical correction exhibiting $1/T$ decay (red dashed).

Figure 2

(a) LPB split by Zeeman field and TE-TM SOC. (b) Pseudospin texture of the lower eigenstate: In-plane pseudospin projection (arrows) and $S_Z$ (color).

Figure 3

QGT components: $B_Z$ (black), $g_{kk}$ (red), and $g_{\varphi \varphi }$ (green) calculated analytically (solid lines) and extracted from numerical experiment (dashed lines).

Figure 4

(a) WP trajectories in real space: Numerical (black) and analytical (red dashed, uncorrected–blue dash-dotted) for 3 values of TE-TM SOC $\beta$ ($\mathrm{\Delta }=0.06\mathrm{meV}$). (b) Final lateral shift as a function of $\beta$: adiabatic (blue dash-dotted line), corrected (red solid line), and numerical (black dots). Here, $\mathrm{\Delta }=0.03\mathrm{meV}$.