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Communications in Theoretical Physics

Sensibility to Changes of Vibrational Modes of Excited Electron: Sum Frequency Signals Versus Difference Frequency Signals

An-Na Gu, Xian-Ting Liang

Published Mar 1, 2011• 0 citations
Publisher

Abstract

In this paper, we investigate a two electronic level system with vibrational modes coupled to a Brownian oscillator bath. The difference frequency generation (DFG) signals and sum frequency generation (SFG) signals are calculated. It is shown that, for the same model, the SFG signals are more sensitive than the DFG signals to the changes of the vibrational modes of the electronic two-level system. Because the SFG conversion efficiency can be improved by using the time-delay method, the findings in this paper predict that the SFG spectrum may probe the changes of the microstructure more effectively.

DOI
10.1088/0253-6102/55/3/25

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In this paper, we investigate a two electronic level system with vibrational modes coupled to a Brownian oscillator bath. The difference frequency generation (DFG) signals and sum frequency generation (SFG) signals are calculated. It is shown that, for the same model, the SFG signals are more sensitive than the DFG signals to the changes of the vibrational modes of the electronic two-level system. Because the SFG conversion efficiency can be improved by using the time-delay method, the findings in this paper predict that the SFG spectrum may probe the changes of the microstructure more effectively.
The theory of time- and frequency-gated (TFG) spontaneous emission (SE) spectra is elaborated. The present formulation generalizes previous derivations, clarifies the interrelations between different existing expressions, and establishes the validity of certain commonly assumed approximations. We obtain various explicit expressions for TFG SE spectra, which are suitable for performing actual calculations for nontrivial systems and which allow us to establish generic (that is, model-independent) properties of TFG spectra. The doorway-window picture of temporally and spectrally resolved spectra is further developed. It is shown that, to the leading order in the pump and probe pulses, the TFG SE signal is equivalent to the stimulated-emission contribution to the integral pump-probe spectrum in the case of nonoverlapping pulses. The theory is illustrated for the example of an electronic two-level system with a single Condon-active harmonic vibrational mode that is coupled to a thermal bath. The effect of imperfect time and frequency resolution is studied. It is pointed out that the TFG SE spectrum carries information not only on the strength of the system-bath coupling, but also on the relative magnitude of the bath correlation time.
Based on an analogy between the macroscopic phenomena and quantum physics, we study the problem of energy conversion for the sum frequency generation (SFG) process in a linear chirped quasi-phase-matched (QPM) crystal. We show that the process of energy conversion from the signal to the sum frequency field is very similar to the process of Landau-Zener tunneling (LZT) in a two-level system. Therefore, the energy conversion mechanism of SFG process can be explained by the theory of LZT. In addition, we obtain the asymptotic and analytical solution of the sum frequency field by solving the coupled equations with the model of Landau-Zener.
In this presentation the underlying theoretical analysis of the four wave mixing (FWM) efficiency is given for the resembling experiment of three-level quantum system (TLQS), where the transition from |1> to |3> correspond to a Raman active vibration mode and is forbidden in the dipole approximation. In this case quantum state |3> (vibration mode) is excited by combined resonance and, subsequently, quantum state |2> is driven by a (delayed) weak field. Notable, that during off-resonance FWM experiments significant excited state probability amplitude and, consequently, FWM exists only as long as the perturbing fields overlap. Actually, in this presentation we discuss efficiency of the FWM process, which exist for long (as compared to the pump and probe pulses) times, and appears dependent on the both electronic and nuclear excitations, stored in the TLQS. The induced material variables (coherences and populations) in the TLS were considered by means of the density matrix, and in order to account for Raman transition |1> → |3> we used an approach closely related to that exploited previously. Considerable impact of the created in TLQS electronic and nuclear coherences on the frequency conversion for specific (negative and positive) time delays will be demonstrated.
Consider sum frequency generation (SFG) ω3 = 2 ω1 + ω2 in a phase matched metal-noble gas mixture, using two lasers ω1 and ω2, where 2 ω1 is tuned to a two photon transition Ωg2 and ω2 is shifted across Rydberg transitions Ω2n (g groundstate, 2 excited state, n Rydberg state of the metal atom). Then the intensity of the SFG spectrum I3 (ω3)~ |χ(3) (ω3)|2 displays resonances and related antiresonances, separated by (1) which are caused respectively by constructive and destructive interference of resonant and nonresonant parts of the 3. order nonlinear susceptibility χ(3) (ω3 ≃ Ωng) = χR(ω3)+ χNR (3) The products μ2nμng:= xn of the dipole transition matrix elements may then be obtained from a simple measurement of the frequency spacings Δωn at a large number of transitions Ωgn and solving iteratively the coupled equation system (1),(3); neither absolute radiation intensity nor vapour density deter mination is needed.
The theory of three wave mixing was formulated in a classic paper by Armstrong, Bloembergen, Ducuing, and Pershan over 25 years ago.1 A theoretical description of three wave mixing for multimode time-dependent input fields (as opposed to cw single frequency time independent input fields) has recently been presented.2 Analytic solutions were developed for die time-dependent intensities of the output fields. These solutions differ dramatically from those for single-mode fields. New frequencies in the sum-frequency output spectrum and in the input-field output spectrum are created and grow in magnitude as the input intensities are increased. The output spectrum broadens under conditions of strong nonlinear coupling, ultimately reaching the phasematched bandwidth limit. Here, we review the results presented in Ref. 2 for sum frequency generation (SFG) and explicidy compare these results with those for second harmonic generation (SHG). We show that SHG cannot be described as a limit of the SFG formalism for multimode fields. The SFG formulation for the interaction of photons close to frequency ω1 with photons close to frequency ω2 producing photons close to frequency ω 3= ω 1+ ω 2 does not allow ω1(ω2) photons to interact with each other to produce photons close to frequency 2ω1(2ω2). This self interaction is crucial for SHG and must be incorporated. Therefore, a separate formulation is used to describe SHG with multimode input fields.
We present expressions for rates of radiationless transitions in two-level systems with arbitrary coupling to a set of damped vibrational modes; calculations are limited to transitions slower than vibrational relaxation. Explicit results are discussed within the harmonic approximation and Condon approximation; systematic methods are introduced for treating non-Condon and anharmonic effects. Results relevant to experiment are (1) significant deviations from the energy-gap law when a small number of vibrational modes is coupled to the transition, (2) dependence of transition rate on homogeneous and inhomogeneous vibrational linewidths, (3) nonexponential decay for transitions with vibrational frequency shifts, and (4) persistence of electronic coherence for long times either at low temperature or when frequency shifts are small. Our approach differs from previous work in the partitioning of the Hamiltonian and in the methods of evaluating operator averages; these differences are discussed in relation to the adiabatic approximation and the irreversibility of the transition.