Rabi frequency oscillation

The quantum levels of a real atom or molecule are usually degenerate. In that case, the description of the interaction between a field and degenerate two-level system becomes more complicated. In particular, there is no longer any simple, graphic picture of the particle s Rabi-frequency oscillations between the two levels. There are, instead, the particle s oscillations between individual sublevels with frequencies of their own, which combine to smooth out the oscillations of the net level populations. 

If the variation of the population as a function of delay time features damped oscillations with Rabi frequency, it is expected to see the Rabi splitting. Can this be observed ... 

Figures 6a-c show the population dynamics encountered in a three-level system (see Fig. 4) interacting resonantly with two Fourier-transform-limited laser pulses with three different delay times between the two pulses. The calculation was done assuming that the chosen Rabi frequencies fulfill the relation > 1/pulse duration) in all three cases. This relation ensures that the typical time for a Rabi oscillation of the population in an isolated two-level system is shorter than the pulse duration. Ionization from level 2 was introduced as a fast laser intensity-dependent decay of level 2 [6, 60], and resonant laser frequencies were assumed.

We see that / (i, t) displays damped Rabi-type oscillations between the initial state and the final continuum states, where the damping is given by resonance decay rate TJ2h. Although the frequency of the Rabi oscillations is a function of the fieldib strength 0, the branching ratio between channels is independent of the laser para- 4... 

Though the integrands are rapidly oscillating, a certain smoothness can be observed in those integrals, especially for F t). In such a case, we judge that coarse-graining (CG) is appropriate. Note that F(t) is a linear function of t when the CG Rabi frequency is constant. 

The population oscillates with the Rabi frequency of the g) —. v) transition and at certain times Ps(t) = 1, indicating that all the population is in the symmetric state. This happens at times... 

In fact, if we think of oscillations as involving a factor exp(iClt), then if the Rabi frequency is taken as complex, its real part will be the bound state contribution, and the imaginary part can be used to represent loss of population into the continuum. 

The microwave detected MODR scheme closely resembles pulsed nuclear magnetic resonance (Hahn, 1950), optical coherent transients by Stark switching (Brewer and Shoemaker, 1971) and laser frequency switching (Brewer and Genack, 1976). The on-resonance microwave radiation field, ojq = ( 2 — Ei)/H, creates an oscillating bulk electric dipole polarization (off-diagonal element of the density matrix, pi2(t)). The oscillation is at u>o u>r, where ojr is the (Mj-dependent) Rabi frequency,... 

Figure 8.2 Time dependence of the probability Pe(t) of observing the spontaneously decaying two-level system in its excited state at the center of a closed spherical cavity The number of resonantly interacting field modes is of the order of rR/ 7rc and depends on the size of the cavity R. For FR/c = 10 (upper figure) a spatially localized photon wave packet is generated by spontaneous emission and can be reabsorbed again by the two-level system at the center of the cavity at later times. For FR/c = 1 (lower figure) only a small number of cavity modes interact resonantly and the two-level system performs approximate Rabi oscillations governed by the vacuum Rabi frequency.

The Stokes laser generates a coherent superposition of the wavefunctions of levels 2) and 3). The states 2) and 3) are, however, not occupied before the pump pulse arrives. The wavefunction oscillates between levels 2) and 3) with the Rabi frequency which depends on the intensity of the Stokes pulse and its detuning from resonance. Now the pump pulse comes with a time delay At with respect to the Stokes pulse, where At is smaller than the width of the Stokes pulse, which means that the two pulses still overlap (Fig. 7.15b). This places the molecule at a coherent superposition of levels 1) and 2) and 2) and 3). If the delay At, the detuning A v and the intensities of the two lasers are correctly chosen, the population in level 11) can be completely transferred into level 13) without creating a population in level 2) (Fig. 7.15c). The coherently excited levels 1) and 2) are described by the wavefunction... 

In this case, the populations of states 1 and 2 oscillate at the Rabi frequency, as shown in Fig. 2.6. The only difference is that the oscillations decay exponentially during the phase relaxation time T2- If condition (2.62) is satisfied, the interaction of the two-level system with the laser-light field is said to be coherent. 

Another experimental proof of the localization of cold atoms at the minima of a periodic optical potential was obtained by recording the resonance fluorescence spectra of cesium atoms trapped in three-dimensional optical molasses (Westbrook et al. 1990) and rubidium atoms in a one-dimensional optical potential (Jessen et al. 1992) The resonance fluorescence spectrum of a motionless two-level atom consists of the well-known Mollow triplet, which includes a central peak at the laser frequency u> and two side components displaced to the red and blue sides by an amount equal to the Rabi frequency (Mollow 1969). For a two-level atom oscillating in a potential well at a frequency lower than the Rabi frequency, each component of the Mollow triplet is split into side components corresponding to changes in the vibrational state of the atom. If the ratio between the oscillation amplitude of the atom in the potential well and the radiation wavelength (the Lamb-Dicke factor) is small, each component of the... 

If the laser pulse applied to the sample molecules is sufficiently long and intense, a molecule (represented by a two-level system) will be driven back and forth between the two levels at the Rabi flopping frequency (2.134). The time-dependent probability amplitudes a (t) and a (t) are now periodic functions of time and we have the situation depicted in Fig.2.30. Since the laser beam is alternatively absorbed (induced absorption E ) and amplified (induced emission E2-> E ), the intensity of the transmitted beam will display an oscillation. Because of relaxation effects this oscillation is damped and the transmitted intensity reaches a steady state determined by the ratio of induced to relaxation transitions. According to (2.133) the Rabi frequency depends on the laser intensity and on the detuning molecular eigenfrequency o) 2 laser frequency co. This detuning can... 

Rabi is the Rabi frequency which is determined by intrinsic atomic properties and by the intensity of the laser. By proper timing of the laser pulse any rotation between the two states ) and e) can be performed. For example, by choosing Rabi = 7t/2 transformation (6.8) is realized. As a sideremark we note that for microwave transitions Rabi oscillations can be induced by exposing the atom to a microwave field. 

The electron hopping frequency may be estimated from time-dependent perturbation theory. If Hab is treated as a constant perturbation, the system will start to oscillate between the two diabatic states once the perturbation is turned on. In a bimolecular reaction, for example, the perturbation is turned on upon formation of the precursor complex, while in a covalently attached (bridged) binuclear system it can be turned on upon reduction (oxidation) of one end of the fully oxidized (reduced) system by an external reagent or by photoexcitation. If the system is in the diabatic reactant state at / = 0, then the probability of it being in the product state at some later time t is given by the Rabi formula [27]. 

We can explain these features by considering the equations of motion (96) for the density matrix elements. When A = 0, and the laser is tuned to the middle of the upper levels splitting the states 1) and 3) are equally driven by the laser and the coherences p12 and p32 oscillate in phase with frequency A/. The coherences are directly coupled by the cross-damping term I ) 2. However, for a strong driving field (fl 3> F) the Rabi oscillations dominate over the spontaneous exchange of photons, resulting in independent oscillations of the atomic dipole moments. 

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