The coherent excitation of a three level atom by one- and two-photon resonances coupled by a common level is treated by the method of averages. The equations for the density matrix of the three resonant levels are solved analytically for square pulses of duration shorter than relaxation times. Double free induction decay occuring when both exciting fields are switched out of resonance simultaneously is exhibited. The emitted radiation is also calculated for an experiment in which the field driving the one-photon resonance remains on after the other field ends, and it is shown that experiments at several detunings and amplitudes would allow determination of various combinations of relaxation rates. Numerical studies for nonsquare pulses are briefly described. The results show that the main qualitative features of the square-pulse case are retained. 15 references.
The coherent excitation of a three level atom by one- and two-photon resonances coupled by a common level is treated by the method of averages. The equations for the density matrix of the three resonant levels are solved analytically for square pulses of duration shorter than relaxation times. Double free induction decay occuring when both exciting fields are switched out of resonance simultaneously is exhibited. The emitted radiation is also calculated for an experiment in which the field driving the one-photon resonance remains on after the other field ends, and it is shown that experiments at several detunings and amplitudes would allow determination of various combinations of relaxation rates. Numerical studies for nonsquare pulses are briefly described. The results show that the main qualitative features of the square-pulse case are retained. 15 references.
A theory of photon echoes using double-resonance optical pulses (PEDROP) is presented. Two double-resonance pulses, separated by a time interval T, are used to cause transitions among three states of an atom. Each double-resonance pulse consists of two simultaneously applied laser pulses that drive coupled transitions in the atom. It is shown that for time t > T, six photon echoes are produced (three on each of the coupled transitions) at five distinct times. Moreover, a nonradiating macroscopic coherence is also produced between states of the same parity. With the use of an interrogation pulse to monitor this coherence, three additional photon echoes can be produced on either of the coupled transitions. A Doppler phase diagram is constructed that enables one easily to predict the times and conditions of echo formation. PEDROP combines features of both two-photon and trilevel echoes but offers advantages over these methods for studying relaxation phenomena.
Resonant two-photon absorption is discussed for the case in which the exciting field may be intense enough and near enough to resonance to cause appreciable saturation. A single pair of levels in an otherwise quite general atom is assumed to satisfy the appropriate resonance condition, and equations are derived governing the time evolution of the associated 2 \ifmmode\times\else\texttimes\fi{} 2 submatrix of the full atomic density matrix. Atomic relaxation is treated explicitly. The analysis, which treats both inversion-symmetric and inversion-nonsymmetric cases within the same general formalism, is developed systematically, the necessary approximations receiving careful discussion. The results are analogous to those for the corresponding one-photon process, but with a shift in the resonance frequency proportional to the intensity of the exciting field. This frequency shift is not identical to the one calculated by time-independent (second-order) perturbation theory, and becomes large when the energy of an intermediate state of the atom approaches the mean of the energies of the active states.
The coherent interaction of two smoothly varying, near-resonant, two-photon pulses with a three-level system can be described by "two-photon damped Bloch equations" which are analogous to those for a one-photon transition in a two-level system except for the presence of a two-photon coupling and a frequency shift. These equations are solved for the cases ${\ensuremath{\gamma}}_{1}$, ${\ensuremath{\gamma}}_{2}\ensuremath{\ll}\ensuremath{\Omega}$, ${\ensuremath{\gamma}}_{1}={\ensuremath{\gamma}}_{2}$, and $\frac{{\ensuremath{\gamma}}_{2}{k}^{2}{\ensuremath{\epsilon}}^{4}}{{\ensuremath{\Omega}}^{2}}$, ${\ensuremath{\gamma}}_{1}\ensuremath{\ll}\ensuremath{\Omega}$, where ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{2}$ are the atomic energy and phase relaxation widths, respectively, and $\ensuremath{\Omega}$ is the Rabi frequency. The leading contribution to the refractive index is intensity dependent, caused by the level shifts inherent in multiphoton processes; it includes a relaxation dependent part which is important at times shorter than ${\ensuremath{\gamma}}_{1}^{\ensuremath{-}1}$. The second-order contributions depend on the square of the intensity and the time-integrated square of the intensity. The latter contribution, which is relaxation dependent, causes line asymmetry at the long-wavelength wing; it consists of a term proportional to ${\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$ and only important at early times and a term proportional to $2{\ensuremath{\gamma}}_{2}\ensuremath{-}{\ensuremath{\gamma}}_{1}$.
We consider the excitation spectrum of a three-level atom interacting with an electromagnetic field whose two initially populated modes ω a and ω b are in resonance with two atomic transition frequencies while the remaining unpopulated modes are simultaneously coupled to the atomic states. It is found that at high photon densities, the Green's functions describing the lower and upper systems corresponding to the transition frequencies ω a and ω b respectively are-independent of each other. The spectral functions for the lower and upper systems consist of seven lorentzian lines describing the central peak and three pairs of sidebands where each pair is symmetrically located with respect to the transition frequency. For the lower system, the lines are peaked at frequencies ω = ω a , ω = ω a ± Ω a , ω = ω a ± Ω b and ω = ω a ± Ω ab with radiative widths γa 0/2, 3γa 0/4, (γa 0 + γb 0)/4 and (γa 0 + 3γb 0)/4 respectively, where Ω a and Ω b are the energy shifts of the populated modes ω a and ω b , Ω ab 2 = Ω b 2 + Ω a 2/4 and γa 0/2 is equal to the natural width of a photon spontaneously emitted from an isolated atom. For the upper system, the lines occur at ω = ω b , ω = ω b ± Ω b , ω = ω b ± Ω a and ω = ω b ± Ω ba with radiative widths (γa 0 + γb 0)/2, (γa 0 + 3γb 0)/4, (γa 0 + 2γb 0)/4, and 3γa 0/4 respectively. Relations between Ω a and Ω b are derived for which the heights of the lorentzian lines of the corresponding bands become positive or negative indicating attenuation or amplification of the signal field respectively. It is also shown that when certain relations exist between Ω a and Ω b , the signal field is attenuated in the lower system while it is simultaneously amplified in the upper system at different frequencies and vice versa. Spectral functions of the lower and upper systems are derived and discussed for equally spaced levels and equally populated modes Ω a = Ω b , as well as for the limiting case when Ω b 2 = 3Ω a 2/4 or Ω a 2 = 3Ω b 2/4.
A new method of picosecond and femtosecond transient spectroscopy yielding information about ultrafast relaxation processes without requiring ultrashort light pulses is presented. It is based on the present theoretical analysis of the resonant degenerate four-wave-mixing process excited by two temporally incoherent light beams with wave vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ and ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}$ which are originated from a single beam at frequency $\ensuremath{\omega}$ but have mutual time delay $\ensuremath{\tau}$. Under the assumption that the incoherent light field has Gaussian random complex amplitude and that the resonant material consists of the usual two-level atoms, the statistically averaged intensity of the output light field with ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{3}=2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ at $\ensuremath{\omega}$ is calculated as a function of $\ensuremath{\tau}$. Even with the light having a much longer duration than both T1 (the longitudinal relaxation time) and T2 (the transverse relaxation time), the correlation trace, i.e., output intensity versus $\ensuremath{\tau}$, represents a decay profile determined mainly by T2 for both homogeneously and inhomogeneously broadened transitions, as long as the light correlation time ${\ensuremath{\tau}}_{c}$ is much shorter than the relaxation times. The correlation trace does not always represent a single-exponential decay but is sometimes slightly deformed by the T1 effect. However, it does not cause a significant error in the determination of T2. Moreover, as $\frac{{T}_{2}}{{T}_{1}}\ensuremath{\rightarrow}0$, the trace becomes a single-exponential decay curve determined only by T2. The feature of the results obtained by the present method is similar to that obtained by the conventional coherent transient spectroscopy with short pulses, such as the photon echo. The time resolution in the present method, however, is limited only by ${\ensuremath{\tau}}_{c}$ much shorter than the light duration. By regarding the incoherent light as a series of random ultrashort pulses, the present four-wave-mixing process is also interpreted as the ensemble of numerous transient four-wave-mixing processes caused by various combinations of these pulses.
The dynamics of a three-level atom in a cascade (or ladder) configuration with both transitions coupled to a single structured reservoir of quantized electromagnetic field modes is treated using Laplace transform methods applied to the coupled amplitude equations. In this system two-photon excitation of the reservoir occurs, and both sequences for emitting the two photons are allowed and included in the theory. An integral equation is found to govern the complex amplitudes of interest. It is shown that the dynamics of the atomic system is completely determined in terms of reservoir structure functions, which are products of the mode density with the coupling constant squared. This dependence on reservoir structure functions rather than on the mode density or coupling constants alone, shows that it may be possible to extend pseudomode theory to treat multiphoton excitation of a structured reservoir---pseudomodes being introduced in one-one correspondence with the poles of reservoir structure functions in the complex frequency plane. A general numerical method for solving the integral equations based on discretizing frequency space, and applicable to different structured reservoirs such as high-Q cavities and photonic band-gap systems, is presented. An application to a high-Q-cavity case with identical Lorentzian reservoir structure functions is made, and the non-Markovian decay of the excited state shown. A formal solution to the integral equations in terms of right and left eigenfunctions of a non-Hermitian kernel is also given. The dynamics of the cascade atom, with the two transitions coupled to two separate structured reservoirs of quantized electromagnetic field modes, is treated similarly to the single structured reservoir situation. Again the dynamics only depends on reservoir structure functions. As only one sequence of emitting the two photons now occurs, the integral equation for the amplitudes can be solved analytically. The non-Markovian decay of the excited state is shown for the same high-Q-cavity case of identical Lorentzian reservoir structure functions, and differs from that for the single reservoir situation.