Zitterbewegung Effect and Quantum Geometry in Non-Hermitian Exciton-Polariton Systems

Abstract

In this work, we analytically derive a semi-classical equation of motion describing the zitterbewegung effects arising in the dynamics of wavepackets in non-Hermitian systems. In Hermitian non-relativistic quantum systems, the zitterbewegung effects can arise due to the spin precession and spin-orbit coupling. Interestingly, the spin dynamics in non-Hermitian systems are qualitatively different because of the effective nonlinear terms induced by the non-Hermitian part of the Hamiltonian. In this work, we show the effects from the non-Hermitian spin dynamics by generalising the description of zitterbewegung effects to non-Hermitian systems. We also uncover novel non-Hermitian correction to the group velocity, which can be expressed in terms of the non-Hermitian quantum metric tensor in the absence of out-of-plane effective field.

Figures (5)

• Figure 1: Real and imaginary parts of band structures of (a, b) the non-Hermitian Dirac model and (c, d) the exciton-polariton model used in this work. The green dots denote the exceptional points, and the pink and purple lines highlight the bulk and imaginary Fermi arcs, respectively.
• Figure 2: Pseudospin dynamics for a (a,b) Hermitian and (c-h) non-Hermitian in an exemplar Hermitian system plotted in different components in (a) in on the Bloch sphere in (b). Here, the blue arrow denotes the trajectory of the pseudospin and the black arrows denote the pseudospins of the two eigenstates. Also showing showing the pseudospin dynamics. (c,d) are away from the Fermi arcs at $\mathbf{k}=(1.5,1.5)\mu$m$^{-1}$, (e,f) are on the bulk Fermi arc at $\mathbf{k}=(0.5,2.685)\mu$m$^{-1}$ and (g,h) are on the imaginary Fermi arc at $\mathbf{k}=(0.9,1.22)\mu$m$^{-1}$. The initial pseudospin configuration is chosen to be $\mathbf{S}=(1/\sqrt{2},1/\sqrt{2},0)$ and $\mathbf{S}=(0,0,1)$ for the Hermitian and non-Hermitian cases, respectively.
• Figure 3: The group velocities (a,b,f,g,k,l) and center-of-mass positions (c,d,h,i,m,n) of a wavepacket initially centred at (a-d) $\mathbf{k}=(0.5,2.685)~\mu$m$^{-1}$ (on the bulk Fermi arc), (f-i) $\mathbf{k}=(1.5,1.5)~\mu$m$^{-1}$ (away from the Fermi arcs) and (k-n) $\mathbf{k}=(0.9,1.22)~\mu$m$^{-1}$ (on the imaginary Fermi arc) as a function of time. Also showing the trajectories in real space in (e,j,o). The lines are analytic results derived from Eq. (\ref{['eq: Hz=0 ZBW']}) and the black dots are the numerical results for a wavepacket with initial with of $0.005~\mu$m$^{-1}$ in momentum space.
• Figure 4: The group velocities (a,b,f,g,k,l) and centre-of-mass positions (c,d,h,i,m,n) of a wavepacket initially centred at (a-d) $\mathbf{k}=(0.5,0)$ (on the bulk Fermi arc), (f-i) $\mathbf{k}=(4,0)$ (on the imaginary Fermi arc) and (k-n) $\mathbf{k}=(2,2)$ (away from the Fermi arcs) as a function of time. Also showing the trajectories in real space in (e,j,o). The lines represent analytic results derived from Eq. (\ref{['eq: Hz=0 ZBW']}) and the black dots represent the numerical results for a wavepacket with initial with of $0.005$ in momentum space.
• Figure 5: The Fourier transform of the group velocities $v_x$ and $v_y$ on the imaginary Fermi arc at $\mathbf{k}=(4,0)$ in the non-Hermitian Dirac model. Here, the primary frequency $\omega$ is shown on both plots along with doubled and tripled frequencies.