The traditional Raman process alluded to in Chap. 13 is a three-state sequence of transitions in which radiative excitation (induced by a pump field) is followed by spontaneous emission that produces a final state differing from the initial state [Her50b] [Sho90, § 17.5]. When the final state of the sequence is more energetic than the initial state the resulting emission line (to the red of the pump wavelength) is known as a Stokes spectral line. The difference between the pump frequency and the Stokes frequency, the Raman frequency , defines the excitation energy of the final state relative to the initial state. When, instead, the final state has lower energy than the initial state, as can occur when the initial quantum state is already excited, then the emission is an anti-Stokes line, at a bluer wavelength than the pump field. The overall Raman scattering is a two-photon process. Typically Raman spectroscopy deals with molecules; the two-photon transitions are then between vibrational-rotational states, through electronically excited intermediate states, that are characterized in part by vibrational quantum number v and rotational angular momentum quantum numbers J, M . From any given excited electronic state there are many fluorescing transitions, corresponding to various vibrational and rotational quantum numbers of the final state. The wavelengths of the various Stokes and anti-Stokes lines (i.e. the Raman frequencies) characterize the particular molecular species, and so they have provided a valuable diagnostic tool for spectroscopists.
The traditional Raman process alluded to in Chap. 13 is a three-state sequence of transitions in which radiative excitation (induced by a pump field) is followed by spontaneous emission that produces a final state differing from the initial state [Her50b] [Sho90, § 17.5]. When the final state of the sequence is more energetic than the initial state the resulting emission line (to the red of the pump wavelength) is known as a Stokes spectral line. The difference between the pump frequency and the Stokes frequency, the Raman frequency , defines the excitation energy of the final state relative to the initial state. When, instead, the final state has lower energy than the initial state, as can occur when the initial quantum state is already excited, then the emission is an anti-Stokes line, at a bluer wavelength than the pump field. The overall Raman scattering is a two-photon process. Typically Raman spectroscopy deals with molecules; the two-photon transitions are then between vibrational-rotational states, through electronically excited intermediate states, that are characterized in part by vibrational quantum number v and rotational angular momentum quantum numbers J, M . From any given excited electronic state there are many fluorescing transitions, corresponding to various vibrational and rotational quantum numbers of the final state. The wavelengths of the various Stokes and anti-Stokes lines (i.e. the Raman frequencies) characterize the particular molecular species, and so they have provided a valuable diagnostic tool for spectroscopists.
Raman scattering is a two-photon process [1]. As illustrated in Fig. la, an incident photon is annihilated and a new photon of a different frequency is created. This implies that the scattering is inelastic, and the medium is left in a different energy state. When the scattered photon is lower in energy, or frequency, it is said to be Stokes shifted. The shift in frequency of the Stokes photon is related to a characteristic frequency of the medium. This is illustrated in Fig. 2a. Calling this characteristic material excitation frequency ω)v, the Stokes frequency is then ωs = ΩL — ωv, where ωL is the incident laser frequency. Thus, Raman (Stokes) scattering results in an excitation from the ground state of the medium to an excited state mediated by a two-photon inelastic scattering. The internal material excitation can involve the creation of an excited electronic state, an excited vibrational-rotational state, a lattice vibration, a spin flip in semiconductors, or electron waves in plasmas [2]. The phenomenon occurs in gases, liquids, solids, and plasmas.
A simple description of the stimulated Raman emission of Stokes radiation, and of the induced absorption found by Stoicheff at the anti-Stokes frequency, is based on the semiclassical theory of radiation. This theory gives the well-known formulas for absorption and stimulated emission when carried to the first order in the time-dependent perturbation due to the interaction of the light beam with the molecules. If there are two light beams of frequencies ωL and ω, the second-order perturbation may be associated with stimulated emission at the Stokes frequency ω=ω−1=ωL—ωR and with induced absorption at the anti-Stokes frequency ω=ω1=ωL+ωR, where ωL is the laser frequency and ωR a Raman-active vibrational frequency; in both cases, there is a molecular transition to the excited vibrational state. The highly directional and very sharp anti-Stokes emission at ω1 arises from the conversion by molecules in intense beams at ωL and ω−1 of two ωL photons into an ω−1, ω1 pair. Like spontaneous emission from excited states, the production of this anti-Stokes radiation requires the full quantum theory of radiation for an adequate description. The emission of ω1 and the observed directional absorption of ω−1 due to the reverse conversion ω−1+ω1→2ωL are explained. The anti-Stokes emission is proportional to the Stokes intensity and to the square of that of the laser, and to the square of the molecular number density; its sharpness is due to the absence of a molecular transition. The generation of photon pairs ω−n+ωn, where ωn=ωL+nωR, can occur for all values of n, but it is most efficient for n=1, due to resonance in the appropriate molecular hyperpolarizability and to the intensity of the Stokes beam at ω−1.
Quantum changes of three-state systems have some similarities with those of two-state systems. When subject to steady illumination the populations may undergo oscillations similar to the Rabi cycling of two-state systems, and various forms of adiabatic following are possible. Analytic solutions to the relevant TDSE exist [Sho90, Chap. 23]. The additional degree of freedom, typically allowing controllable parameters of a second laser pulse, allows a wider variety of controlled excitation. The resulting differences and similarities to two-state systems have been discussed at length [Whi76; Sho77; Rad82b; Yoo85; Car87]. Three-state linkages Two-field linkages . The simplest extension of two-state excitation allows two laser fields, here identified by letters P (for pump) and S (for Stokes), as befits the stimulated Raman process discussed in Chap. 14. The carrier frequencies of the two fields, ω P and ω S , are each assumed to be close to resonance with one, and only one, Bohr frequency, so that each field can be uniquely identified with a particular transition (failure of this restriction, and the resulting linkage ambiguity, is discussed in [Una00]). I will assume that the P field is (near) resonance only with the 1–2 transition, while the S field is (near) resonant only with the 2–3 transition; these interactions thereby form a two-step linkage chain. This system has three possible linkage patterns, shown in Fig. 13.1. The linkage patterns (sometimes called configurations) differ by the ordering of the energies of the linked states.With the assumption that population initially occupies state 1, as in Fig. 13.1, the definitions are: Ladder : The ladder system has the energy ordering E 1 < E 2 < E 3 .
After the sequence of introductory chapters that collected the necessary tools, we are now ready to discuss complete experiments. The first type of experiment we consider is the creation and detection of order in multilevel atoms through coherent Raman processes. The atoms that we use to study these processes are rare earth ions in a crystal matrix. Enclosing them in a crystal allows long interaction times, but also makes it necessary to consider their interaction with neighbouring atoms. Overview Raman processes Introduction In the preceding chapters, we mentioned several types of Raman processes. Their common feature is a resonant change of the energy of the photons that interact with the material system. The energy of the scattered photons may be lower (Stokes process) or higher (anti-Stokes) than that of the incident photons. The energy difference is transferred to the material system, where it must match an energy level separation. The photon energy itself, however, does not have to match exactly a transition frequency of the medium. This is commonly expressed by the statement that the Raman scattering proceeds through a virtual state, represented by the dashed line in Figure 7.1. The presence of a real state of the atom, indicated by the full line, nevertheless increases the coupling efficiency, as discussed in Chapter 3. The earlier sections on three-level effects and optical anisotropy dealt with the mathematical formalism of Raman processes, using generic level systems to describe them.
The general method developed in a previous paper (see ibid., vol.10, p.345 (1977)) is applied to the problem of resonance Raman scattering in very intense laser fields. At very low intensities the spectrum consists of three monochromatic lines (Rayleigh, Raman Stokes and anti-Stokes) which split into a triplet and two doublets when the laser light, tuned on one of the two transitions connecting the upper state to the two lower sublevels, has a sufficiently high intensity to saturate this transition but not the second one. The Raman and Rayleigh lines are completely mixed when the intensity becomes so high that both transitions are saturated. The corresponding evolution of the spectrum is determined. Optical-pumping effects leading to asymmetries in the fluorescence spectrum are taken into account.
The non-classical effects of squeezing of light in coherent anti-Stokes Raman scattering (CARS) and coherent anti-Stokes hyper-Raman scattering (CAHRS) are investigated under the short-time approximation based on a fully quantum mechanical approach. In an idealized model CARS, the interaction is looked upon as a process involving absorption of a pump photon and emission of a Stokes photon, which is followed by absorption of a pump photon and subsequently emission of an anti-Stokes photon at a different frequency, whereas CAHRS is looked upon as a process involving absorption of two pump photons and emission of a Stokes photon, which is followed by absorption of two more pump photons and subsequent emission of an anti-Stokes photon at a different frequency. The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established. It is shown that squeezing in the Stokes and anti-Stokes mode in CARS is found to be dependent on amplitude-squared squeezing of the initial pump field while, in CAHRS, it is dependent on the fourth-order amplitude squeezing of the initial pump field. Higher-order squeezing can be converted to linear squeezing through these processes. It is also shown that squeezing is greater in CAHRS than the corresponding squeezing in CARS.