LIST OF FIGURES 2.7 This figure shows the variation of both the slow and fast varying. 108 4.2 The above figures show the collapse times (left figure) and the generation time. right figure) versus the average number of photons ¯n.
Definition
Graphene: 2 dimensional or NOT !
Peculiar properties of graphene
- Atomic or crystal structure
- Electronic band structure: Tight binding approach
- Alternative approach to graphene band structure
- Electronic and transport properties
- Optical properties
- Theoretical study
- Experimental study
- Other interesting properties
Γ is the center point and 'K', 'K0' are the edges of the first Brillouin zone formed by the reciprocal lattice of graphene. 2dc−c (qx−iqy) The final expression for the low-energy spectrum of the Hamiltonian is given by,.
Method of obtaining graphene
Mechanical method: Scotch tape
Epitaxial growth on various substrate
Graphene layers can also be grown on a metal substrate such as Rubedium, Platinum and Iridium via chemical vapor deposition (CVD)[83] within a high temperature environment. High-quality graphene layers can be obtained by the chemical vapor deposition method in which the deposition is done on a thin film of nickel and methane used as a carbon source, then these layers can be transferred to different substrates leading to applications of many electronic [84,85].
Recent experimental study on graphene based systems
But these methods are very expensive due to the requirement of high temperature and very high vacuum.
Graphene dreams: Applications
Nonlinear optics of graphene is a new field judging by the small number of articles in this field so far [110,111]. The non-resonant/non-linear optical response of graphene in the time domain was investigated by Ishikawa [55].
Problem formulation
Bloch equations of graphene in presence of continuous optical pump field 25
Solution of Bloch equations
At the outset, it is clear that if one includes only the slowly varying terms and drops all the rapidly oscillating terms, the conventional Rabi resonance is lost. The inclusion of higher harmonics leads to newer anomalous resonances in contrast to conventional Rabi oscillations, where the inclusion of higher harmonics merely causes a shift in the conventional resonant frequency. Exactly at the Dirac point the Bloch equations become (where momentum . 2.4 Rabi oscillations in the graph vanishes).
Difference between conventional RWA and ARWA
Comparison between the results obtained in RWA and ARWA
After back substitution, we can see that the slowly varying part of the polarization density for ω ωR oscillates with the anomalous Rabi frequency 2ωωR2. It describes the dependence of the effective or generalized Rabi frequency on the band energy ek = vF|k| for a fixed value of the external frequency (ω = 100) and Rabi frequency for zero detuning (ωR = 10) in the generalized unit (~ = 1). For the sake of comparison, the two currents in the frequency domain and the time domain are given in Fig. (2.2) and Fig. (2.3).
Analysis under length gauge: The question of gauge invariance
Computations of gauge invariant quantities
In order to give this concept a universal meaning, we must therefore focus on measurement-invariant quantities. Therefore, we need to somehow solve for these quantities and then check whether they exhibit oscillations that could be characterized as Rabi oscillations. From this we can derive the equation for the slowly changing part of the basic quantities of this part.
Toy model: Analogous to graphene except “pseudospin”
At the other extreme, we get a result similar to the case of graphene, but with some differences. In the large frequency limit, we can expand the effective Rabi frequency in powers of ω−1 as,. We expect a similar observation to hold in the case of graphene, where the equations cannot be solved exactly due to pseudospin.
Anomalous Rabi oscillations with pulsed non-resonant excitation
The only difference is that instead of ΩRt we must write Φ(t) defined by the following. which is nothing but the area of the pulse. Thus we can see that the cyclic nature is completely determined by the range of the pulse. When the pulse is off, near the bottom of the vF|k| bands ≈ 0 we see that the excess population freezes to a value entirely determined by the total pulse area Φmax =Rtf.
Issue of the vector potential: Real vs complex
This means that the Rabi frequency is what is found in the context of two-level atoms, i.e. Existence Theorem of ARO: Even when the vector potential is chosen real and sinusoidal, anomalous Rabi oscillations are seen even though the dependence on the anomalous Rabi frequency of the components of the electric field depend on whether the fields are treated as (i) complex and exponential or (ii) real and sinusoidal. We now evaluate the conventional Rabi frequency and see whether the same relationship between conventional and nonstandard Rabi frequency holds when the fields are real and sinusoidal.
Interpolation between RWA and ARWA regimes
Also of interest is the correction term that shows deviation from the prediction of RWA. Now it is natural to get curious about what is happening in the region between RWA and ARWA. The answer to the same question is also shown when frequency doubling effects are included but triple frequencies are excluded.
Numerical solution of the Bloch equation
Results and discussion
Fig.(2.6) shows a plot of Rabi effective frequency density versus band energy vF|k| and the Rabi frequency for zero detuningωR (related to the radiation intensity). The numerically obtained anomalous Rabi frequency (frequency of the slow part) agrees with the quoted analytical result. Moreover, the anomalous Rabi frequency has a well-defined dependence on the external frequency, while the excitonic energies are independent of the external frequency.
Conclusions
This chapter is devoted to the study of the above predicted anomalous Rabi oscillations [53] in graphene using pump spectroscopy. Commonly incoherent optical properties, such as optical dephasing and relaxation of band electrons in semiconductors, are studied using a pump-probe experiment [45,46] in which two consecutive laser pulses are used, one to prepare the system in a specific way called a pump pulse, and one for testing after a variable time delay called a probe pulse. While pump-probe spectroscopy is commonly used to investigate incoherent phenomena such as dephasing, it is also used to study coherent phenomena such as EOSE.
Problem formulation
Thus, it is appropriate to investigate such nonlinear properties in the case of two-dimensional single-layer graphene (SLG), double-layer graphene (BLG), three-layer graphene (TLG) and multi-layer graphene (MLG), where the bands are linear in the case of SLG. parabolic in BLG and cubic in three-layers, but all possess a property unique to these systems, viz. The next section explains how to study an equally interesting coherent optical phenomenon, viz.
Experimental results on the relaxation dynamics in graphene
Recently, a large number of attempts have been made to relax the carrier dynamics in graphene on various substrates. Anomalous Rabi oscillations manifest as periodic fluctuations 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 differential transfer coefficient is studied as a function of pump-probe delay, which provides information on relaxation phenomena.
Coherent Bloch equations in presence of an intense optical pump pulse 75
The main reason for this assumption is that we only want the anomalous Rabi frequency to contribute to the area of the pulse and not the free particle scattering (even ifk is small 2vF|k|(tpr−Tf) may not be small if tpr Tf ). The Rabi frequency that we derive (both anomalous and conventional) are known as Floquet exponents. Therefore, our analytical approach, although approximate in contrast to the numerical approach, is necessary to establish the general relationships between the Rabi frequency and models under investigation and regimes considered.
Linear response due to the probe pulse
Linear response of graphene
To derive the above linearized Bloch equations, we ignore terms like zA(t)×δp(~k, t), zA(t)×δp(~k, t), zA(t)×δzA( t) since we assume in the previous section, that there is a finite time duration (very short) between the pump pulse and the probe pulse, i.e. after solving the above equations in the presence of a probe field we get the following expression for δndif f(~ k, t) and δp (~ k, t) as,. We can see that the sensitivity of the probe depends on the area of the pump pulse and that it exhibits an oscillatory behavior as a function of the duration of the pump field.
Experimental verification of ARO’s: Differential transmission
In the graph, we think of the induced field as proportional to the induced current, which in turn is proportional to the polarization. However, some differences are expected here, since both the pump pulse and the probe pulse have the same duration, which means that harmonics of the abnormal Rabi frequency are also seen since an expression like cos2ω2. If we also ensure that 2ωωR2τ ~ 10, so that there are about 10 anomalous Rabi oscillations in the duration between pump and probe, this phenomenon is clearly seen in the periodic variation of the differential transmission coefficient as a function of delay.
Results and discussion
This is two orders of magnitude smaller than the maximum achievable using THz pulses. Alternatively, simulated graphene using cold atoms[126] may also be a possibility where such limitations can be circumvented as experiments can be tailored to the specifications of a theory. The analysis of the present paper shows that even in such systems (with or without a gap), anomalous Rabi oscillations can be expected.
Conclusions
By treating the radiation field quantum mechanically[47–49], some new phenomena are observed in Rabi oscillations known as quantum 'collapse'. Rabi oscillations in graphene were studied by Mischenko[54] and Ishikawa[55] using the well-known rotating wave approximation[45](RWA) close to resonance. Recently, we predicted anomalous Rabi oscillations in graphene that occur far from resonance with an alternative to RWA that we called asymptotic rotating wave approximation [53] (ARWA).
Problem formulation
Revival refers to the revival of previously extinguished oscillations due to destructive interferences that gradually diminish, progressively revealing the original oscillations. Rabi oscillations have long been studied also in semiconductors[46], where energy levels are replaced by bands. In this chapter, we would like to show that the phenomenon of anomalous Rabi oscillations occurs even when the EM field is quantum, clearly proving that AROs are due to pseudospin and not due to the approximations or assumptions we have made.
Quantum Rabi oscillations in graphene: Probability amplitude equation 89
Single photon anomalous Rabi oscillations
The equation for generalized abnormal Rabi frequency Eq. (4.8) suggests that the same effect also survives in the single (or even zero) photon limit. In this case of a single photon limit, the expression for the generalized Rabi frequency is given by the following expression. Now we want to check whether conventional Rabi oscillations are also present in the single photon limit.
Conventional Rabi oscillations
- Conventional Rabi oscillations: Single photon limit
We can thus see that conventional Rabi oscillations are also present in the single photon limit in graphene (details missing from this section are included in the Appendix.
Collapse and revival
- Collapse and revival time
We want to study these phenomena both in the precisely solvable pseudospinless model (JC model) and in graphene. This expression shows that there is a threshold behavior in the current density[53]in the frequency domain where the threshold frequency (with = 0) is given by ΩAR,0 = 2(¯n+1/2)ω |λ|2 . We see that the current density in the time domain exhibits an oscillatory behavior determined by the threshold anomalous Rabi frequency ΩAR,0 and a power law decay envelope (∼ t−1) with exponent −1.
Exactly solvable analogue of graphene: “Pseudospinless” graphene
Equation of motion for the probability amplitude
Unlike the graph equations Eq.(4.2) and Eq.(4.3), which require domain-specific approximate methods for their solution (to be discussed subsequently), the equations for the pseudospinless model, i.e. Eq.(4.18) and Eq.(4.19) are exactly solvable by virtue of being the same as the equations for the Jaynes Cummings model. Since we can now set n −1 ≈ n, instead of being an infinite tower (in the variable n) of coupled equations, the equations Eq.(4.2) and Eq.(4.3) become just two coupled equations.
Analysis of ‘pseudospinless’ probability amplitude equations within
Substituting the above values of probability amplitudes into equation (4.20) and extracting the coefficient e−iωt within the ARWA limit allows us to conclude that there are no abnormal Rabi oscillations in this situation. We can then proceed to study them using the conventional rotary wave approximation (RWA) or the newly introduced asymptotic RWA [53], which is valid far from resonance, where we encountered the phenomenon of anomalous Rabi oscillation.
Interpolation
This leads to an incorrect result for the Rabi frequency when ω ∼ 2 (conventional RWA), but it is convenient enough to obtain the ARWA result when → 0. Thus, the Rabi frequency associated with the probability amplitude comes out as (n+ 1)ω |λ|2 which is ωωR2 for n large compared to unity. The Rabi frequency associated with polarization and current is twice this value - a result that is consistent with that obtained in previous work[53].
Results and discussion
Conclusions
In this thesis, we proved that the existence of anomalous Rabi oscillations is a consequence of the pseudo-spin nature of graphene. We studied the phenomenon of crossing Rabi oscillations as a function of the announcement - the difference between the frequency of the incident wave and the interband energy (2vF|k|). Unlike conventional Rabi oscillations, anomalous Rabi oscillations are unique to graphene (and possibly to the surface states of topological insulators (TIs)), which can be attributed to the pseudospin (conventional spin for TI) degrees of freedom and the Dirac-fermionic character of the graphene system.
Calculation for the value of population excess and polarization at time
Current density calculations
Analysis of Bloch equation in RWA regime
The goal is to find the probability amplitude equation for the state h0,1, n|φ(t)iin and h1,0, n|φ(t)iin. First, we find the general probability amplitude equation for hnA, nB, nν|φ(t)iin using the Hamiltonian (4.1) and the following Schodinger/Dirac equation. Upendra Kumar, Enamullah, Vipin Kumar and Girish S Setlur, published in the conference proceedings "Advances in Nanotechnology and Renewable Energy" with ISBN page no. 46.
