different substrate depends upon the number of layers with the shortest time constant of 150 fs for single layer graphene. So it is appropriate to investigate such nonlinear properties in case of two dimensional single layer graphene (SLG), bilayer graphene (BLG), trilayer graphene (TLG) and multi layer graphene (MLG) where the bands are linear in the case of SLG, parabolic in BLG and cubic in trilayer but all of them possess a property unique to these systems viz. - pseudospin. Therefore, while typically, pump probe spectroscopy is used to investigate incoherent phenomena such as dephasing, it has also been used to study coherent phenomena such as EOSE. The next section ex- plains how to study an equally interesting coherent optical phenomenon viz. anomalous Rabi oscillations using pump probe spectroscopy.
3.3 Experimental results on the relaxation dynamics in graphene and probe pulses. One sees an exponential decay as function of the delay and the decay constant is nothing but the dephasing rate. It is interesting to note that by studying the details of this plot in graphene more carefully, we are able to also infer the presence of anomalous Rabi oscillations which is a coherent phenomenon. This is explained in detail in a later section.
Recently, a large number of experiments on the relaxation of the carrier dynamics in graphene on different substrates has been performed[57,58,62–66,114,119]. Some of the exper- imental efforts are summarized below. Ultrafast relaxation of photogenerated carriers in epitaxially grown graphene layers on SiC substrate[62] depends upon the intraband carrier-carrier and intraband carrier-phonon scattering. The plot between the differen- tial transmission coefficient
T−T0
T0
where, T(T0) is the probe transmission coefficient with (without) pump field, vs time delay between pump and probe pulse shows two type of relaxation (τ1) and (τ2) with different pump pulse energy and different temperature.
The fast relaxation time (τ1) with range nearly (70-120) fs and the slow relaxation time (τ2) with the range (0.4-1.2) ps. Fast relaxation (τ1) is due to the consequence of the intraband carrier-carrier scattering leads to the quaisequilibrium states with a Fermi-Dirac distribution which is consistent with the theoretically predicted intraband carrier-carrier relaxation in graphene[120]. The slower time delay (τ2) is attributed to the further thermalization such as intraband carrier-phonon scattering.
The experimental plot between the differential transmission coefficient
T−T0
T0
vs delay time between pump and probe pulses of carrier relaxation in graphene on mica substrate using femtosecond laser[58] is almost the same as earlier experiments[62] bar- ring some intermittent discrepancies in sign. The plot shows the initial increase (almost linear) in differential transmission coefficient is due to Pauli blocking or due to the repulsion by the initially generated electron-hole density on further generated photo- electrons. The subsequent decrease in transmission coefficient is due to intra-band carrier-carrier and intra-band carrier-phonon scattering. One conclusion of this exper- iment is that carrier relaxation in graphene is almost same as that of graphite. This means coupling between different graphite layers play a minor role in ultrafast carrier relaxation. Wang[66] et. al. studied the relaxation dynamics of hot optical phonons in few-layer and multi-layer graphene grown on a silicon carbide substrate and found that the optical phonon cooling on short time scale are independent of factors such as growth technique, number of graphene layers and type of substrate. Time resolved terahertz spectroscopy has been a powerful tool in the investigation of carrier dynam-
ics in semiconductors[121]. The work on multi-layer graphene nanostructures[122,123] are particularly relevant for applications to optoelectronics.
Another experiment performed by Kumar et.al.[57] on the relaxation dynamics of carriers in graphene with different solvents using a femtosecond laser also shows the two types of relaxation, fast relaxation (τ1) in the range of (130-330) fs and slow relaxation (τ2) in (3.5-4.9) ps. Fast relaxation is related to the carrier-carrier scattering while slower one related to the carrier-phonon scattering.
The experiments performed by the above authors specially Dawalty et.al.[62], Kumar et.al.[57]and Breusing et.al.[58] are the one we shall be studying closely in order to infer the presence of anomalous Rabi oscillations in graphene.
In the following section, we perform a theoretical analysis of the pump-probe ex- periment where we calculate probe susceptibility of the system of interest as a function of the area of the pump pulse assuming the probe is a delta function (broad band) in time. Anomalous Rabi oscillations manifest themselves as periodic oscillations of the probe susceptibility as a function of pump duration at each probe frequency where the pump probe delay is assumed to be zero. The period associated with these oscillations corresponds to the anomalous Rabi frequency. It has been estimated by the authors cited that a quasi-equilibrium state due to Coulomb scattering is reached in graphene in a time scale of ∼ 100−250 fs and phonon assisted energy relaxation takes place in a time scale of 1−2 ps. This means that the phenomena of the present work are seen clearly when a sufficient number of oscillations fit into a time scale of the order of 250 fs. This in turn translates into a constraint on the peak electric field of the pulses used. More detailed numerical estimates will be provided in the results and discussion section.
Subsequent to this, we theoretically analyse the more commonly performed version of the pump-probe experiment where the pump and probe pulses are identical except that the probe pulse is much weaker than the pump pulse and arrives after a certain time delay. The differential transmission coefficient is studied as a function of the pump-probe delay yielding information about relaxation phenomena. We shall see that a closer examination of this dependence also reveals anomalous Rabi oscillations which appears to have been seen also in the actual pump-probe experiment performed by Breusing et.al.[58].