temperature, such devices would be very different from what we make today in silicon.”
Scientists at the University of Vienna reported that calcium doped graphene is a superconducting material[87], but they are still trying to estimate the superconduct- ing critical temperature. They performed angle-resolved photoemission spectroscopy (ARPES) to find an electron donor for graphene that is capable of inducing strong electron-phonon coupling and superconductivity. However, phonon mediated supercon- ductivity has already been observed in graphene by lithium deposition[88].
1.6 Graphene dreams: Applications
Graphene is not only of immense interest to physicists due to its unique electronic and other properties but it has important applications as well. It presents an opportunity for enabling new classes of electronics, optoelectronics and electromechanical devices.
However, for industrial applications, efficient and cheap methods of large scale pro- duction is yet to achieved. From an application point of view, graphene can be used in to the field of electronics, sensors, data storage devices. The experimental result shows that the transistors made from graphene nanoribbons make efficient magnetic field sensors[89]. It has also been reported that the carbon nanotube is an excellent material for the solid state gas sensors[90]. A finite band gap in graphene is obtained in a strained sample[91]which may be used to make switchable devices. It can also be used as spin valve and superconducting field effect transistors as the recent report describing magneto-resistance[92] and substantial bipolar supercurrents[93]. Bilayer graphene may also be used to make a sensitive detector of infrared light[86] with applications includ- ing remote detection of chemical and biochemical weapons. Experiments on graphene nanoribbons show unusual ballistic transport properties[44]. This property can be ex- ploited in the construction of optical waveguides and quantum dots.
Chapter 2
Coherent Rabi oscillations in graphene
2.1 Introduction
The study of interaction of light with matter has always led to novel outcomes. One of the most well-studied coherent phenomenon in nonlinear optics is the phenomenon of Rabi oscillations[94]. A coherent periodic exchange of energy between a two level system and the applied optical field is known as Rabi oscillation. This represents the oscilla- tions in the population and polarization of carries (with a given wave-vector in case of a band) with a frequency ωR determined by the intensity of the externally applied optical field. This frequency ωR is typically much smaller than the optical frequency ω itself. This phenomenon is most easily understood in two level systems studied in atomic physics[45,94]. The same phenomenon manifests itself in semiconductors, which have bands instead of energy levels and where there is a mixing of these energies due to long-range Coulomb interactions. This leads to, among other effects, the phenomenon of excitonic quantum beats and excitonic optical Stark effect[46]. It is therefore, appro- priate to investigate the same effects in graphene where the bands are linear instead of parabolic and the system is two dimensional instead of three and there is pseudospin character. A recurring theme in the subject of nonlinear optics of semiconductors has been the comparison between nonlinear optical processes that occur in two-level atoms[45] and conventional semiconductors as described in the extensive literature on the subject[46,95–104]. With the advent of graphene, the comparison is now between conventional semiconductors and graphene. This comparative effort is well underway as may be seen from the emergence of several papers on the subject[54,55,104–111]. It is
our intention therefore, to contribute to this growing literature and the present inves- tigation being a modest start. Nonlinear optics of graphene is a nascent field judging by the small number of papers in the field to date[110,111]. Indeed Romanets et.al.[110] even make a remark to the effect that there are very few works either experimental or theoretical in the field of nonlinear optics of graphene in the introduction to their paper. Mischenko[54] has studied the nonlinear response of graphene and investigated saturation effects in the current due to relaxation (at the single photon level). These works have studied detailed many body effects including relaxation processes when new physics at the level of coherent phenomena are still unexplored. Hendry et.al.[105] and Mikhailov et.al.[106–108] have stressed the importance of including higher harmon- ics in the evaluation of nonlinear optical response in graphene. It is the purpose of this chapter to highlight simple yet important new phenomena at the level of coherent mean-field approximations. We also find new phenomena upon inclusion of higher har- monics. Non-resonant/nonlinear optical response of graphene in the time domain has been investigated by Ishikawa[55]. The resonant nonlinear dynamics of establishment of electron-hole coherent superpositions states in graphene by multi-photon resonant excitation of inter- band transitions in laser fields with corresponding Rabi oscillations of Fermi-Dirac sea has been considered by Avetissian et al.[109]. These works are closest in spirit to the present work although our methods are considerably different.
The main advancement reported here is the use of an alternative to the rotating wave approximation (RWA) (which we have called ‘asymptotic rotating wave approximation’) used in the context of two-level systems. This alternative approach is able to show that for systems with the pseudospin such as graphene, a second anomalous Rabi frequency is present, and the current density exhibits a crossover from one type of singular behavior close to the normal Rabi frequency to another type of singular behavior close to the (typically smaller) ‘anomalous’ Rabi frequency. A comparison with an exactly solvable model identical to graphene in all respects except lacking in pseudospin shows that the anomalous Rabi frequency is peculiar to models with pseudospin. This exactly solvable model also validates the new technique we refer to as asymptotic rotating wave approximation (ARWA) (the ARWA on this model agrees with the appropriate limit of the exact solution). Lastly, we introduce an analytical technique that interpolates between these two regimes (RWA and ARWA) and express the nonlinear current in a closed form. The numerically exact solution of coherent Bloch equations also reconfirm these findings.