Weak measurement of quantum superposition states in graphene

We employ a weak measurement approach to demonstrate the very existence of the photoexcited interband superposition states in intrinsic graphene. We propose an optical two-beam setup where such measurements are possible and derive an explicit formula for the differential optical absorption that contains a signature of such states. We provide an interpretation of our results in terms of a non-Markovian weak measurement formalism applied to the pseudospin degree of freedom coupled with an electromagnetic wave.

Figure 1

Direct interband excitations result not only in the photocarrier occupations $f_{11},f_{22}=-f_{11}$ at energies $\pm \hslash \omega /2$ but also in the coherences $f_{12},f_{21}=f_{12}^{*}$ representing the superpositions between conduction- and valence-band eigenstates. The matrix $f_{ij}$ can be seen as a density matrix written in the eigenstate basis of ${\widehat{H}}_0$. In the right diagram the density-matrix components are shown schematically assuming monochromatic radiation ($\lambda =0$, i.e., $f_{ii}$ shrinks to a $\delta$ function). Note that $\mathrm{Re}{f}_{12}$ vanishes faster than $\mathrm{Im}{f}_{12}$ when the energy increases, i.e., the latter dominates in the high-energy tail of the density matrix representing the signature of the quantum superposition states.

Figure 2

Graphene is irradiated by two cw monochromatic light beams having the frequencies $\omega$ and ${\omega }^{\prime }$ and linear polarizations perpendicular to each other. One of the beams creates an initial photocarrier distribution, whereas another is used as a weak measurement probe. The curves represent relative optical absorption difference vs detector's excitation frequency ${\omega }^{\prime }$ in units of $1/\tau$ (dephasing rate) at zero temperature and zero doping, as given by Eq. (13). The differential relative absorption is given in units of $\mathrm{\Gamma }$ [see Eq. (14)]. The verticals show the initial photocarrier occupation positions that have shrunk to zero due to the perfectly monochromatic radiation.