Spectroscopy quantum-beat

Quantum-beat spectroscopy represents not only a beautiful demonstration of the fundamental principles of quantum mechanics, but this Doppler-free technique has also gained increasing importance in atomic and molecular spectroscopy. Whereas commonly used spectroscopy in the frequency domain yields information on the stationary states jk) of atoms and molecules, which are eigenstates of the total Hamiltonian 

If two closely spaced levels 1) and 2) are simultaneously excited from a common lower level / at time t = 0 by a short laser pulse with the pulse width At h/ E2 — E ) (Fig. 12.8a), the wave function of the coherent superposition state 1) -f 2 at = 0 

If the detector measures the total fluorescence emitted from both levels k), the time-dependent signal S(t) is 

C is a constant factor depending on the experimental arrangement, /x = is the dipole operator, and e gives the polarization direction of the emitted light. Inserting (12.25a) into (12.26) yields for equal decay constants y =Y2 = Y of both levels 

This represents an exponential decay exp(—yr) superimposed by a modulation with the frequency co2] = (E2 — Ei)/h, which depends on the energy separation AE21 of the two coherently excited levels (Fig. 12.8b). This modulation is called quantum beats, because it is caused by the interference of the time-dependent wave functions of the two coherently excited levels. 

We have already discussed quantum-beat spectroscopy (QBS) in connection with beam-foil excitation (Fig.6.6). There the case of abrupt excitation upon passage through a foil was discussed. Here we will consider the much more well-defined case of a pulsed optical excitation. If two close-lying levels are populated simultaneously by a short laser pulse, the time-resolved fluorescence intensity will decay exponentially with a superimposed modulation, as illustrated in Fig. 6.6. The modulation, or the quantum beat phenomenon, is due to interference between the transition amplitudes from these coherently excited states. Consider the simultaneous excitation, by a laser pulse, of two eigenstates, 1 and 2, from a common initial state i. In order to achieve coherent excitation of both states by a pulse of duration At, the Fourier-limited spectral bandwidth Au 1/At must be larger than the frequency separation ( - 2)/ = the pulsed excitation occurs at 

The time-dependent fluorescence light intensity from the excited states is determined by the transition matrix element, see (4.28) and (7.26). 

We note, that a modulation is obtained (B 0) only if the matrix elements for the transitions 1 — f and 2 — f are non-zero at the same time. A quantum-mechanical interpretation of the beats is based on the observation that it is impossible to determine whether the atom decayed via the transition 1 — f or 2 f. Then the total probability amplitude is the sum of the two 

Although the general features of the quantum-beat phenomenon can be understood in a simple semi-classical model, a full quantum-mechanical description is required for calculating the correct relation between B and A in (9.23) to determine the beat contrast in the decay curve. Quantum-beat spectroscopy has been discussed in detail in [9.127,128]. 

In Fig. 9.29 an example of Zeeman quantum beats is shown. The geometries for excitation and detection are the same as for a recording of the Hanle effect (Sect. 7.1.5). The signal is also a AM = 2 phenomenon. Zeeman quantum beats can also be explained semiclassically using the same model as 

Although the general features of the quantum-beat phenomenon can be. understood in a simple semiclassical model, a full quantum-mechanical de- 

Hyperfine quantum beats for a P j2 state in recorded for two different polarizer settings [9.157] 

If two or more closely spaced molecular levels are simultaneously excited by a short laser pulse, the time-resolved total fluorescence intensity emitted from these coherently prepared levels shows a modulated exponential decay. The modulation pattern, known as quantwn beats is due to interference between the fluorescence amplitudes emitted from these coherently excited levels. Although a more thorough discussion of quantum beats demands the theoretical framework of quantum electrodynamics [11.33], it is possible to understand the basic principle by using more simple argumentation. 

Assume that two closely spaced levels 1 and 2 of an atom or molecule are simultaneously populated by optical pumping with a short laser pulse from a common initial lower level i (Fig.11.23a). In order to achieve coherent excitation of both levels by a laser pulse with duration aT, the Fourier limited spectral bandwidth Av = a/AT [a is a constant of the order of unity which depends on the pulse profile Ip(t)] must be larger than the frequency separation (E - Ej)/h. 

If the pulse excitation occurs at t = 0, the wave function of the excited states at this time can be written as a linear superposition of the sublevel functions q (k = 1,2,) 

This shows that a modulation of the exponential decay is observed if both matrix elements for the transitions 1 - f and 2 - f are nonzero (Fig.11.23b). The measurement of the modulation frequency allows determination of the energy separation of the two levels, even if their splitting is less than the Doppler width. Quantum beat spectroscopy therefore allows Doppler-free resolution. 

The experimental realization uses either short-pulse lasers, such as the N2 laser pumped dye laser (see Sect.11.1) or mode-locked lasers. The time response of the detection system has to be fast enough to resolve time inter- 

There is one important point to note. The width A5i/2 of the level-crossing signal reflects the average width y = y2) of the two crossing levels. If these lev- 

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When the fine structure frequencies fall below 100 MHz they can also be measured by quantum beat spectroscopy. The basic principle of quantum beat spectroscopy is straightforward. Using a polarized pulsed laser, a coherent superposition of the two fine structure states is excited in a time short compared to the inverse of the fine structure interval. After excitation, the wavefunctions of the two fine structure levels evolve at different rates due to their different energies. For example if the nd3/2 and nd5/2 mf = 3/2 states are coherently excited from the 3p3/2 state at time t = 0, the nd wavefunction at a later time t can be written as40... 

Level crossing spectroscopy has been used by Fredriksson and Svanberg44 to measure the fine structure intervals of several alkali atoms. Level crossing spectroscopy, the Hanle effect, and quantum beat spectroscopy are intimately related. In the above description of quantum beat spectroscopy we implicitly assumed the beat frequency to be high compared to the radiative decay rate T. We show schematically in Fig. 16.11(a) the fluorescent beat signals obtained by... 

Hack, E. and Huber, J.R. (1991). Quantum beat spectroscopy of molecules, International Reviews in Physical Chemistry, 10, 287-317. 

Walther, Th., Bitto, H. and Huber, J.R. (1993). High-resolution quantum beat spectroscopy in the electronic ground state of a polyatomic molecule by IR-UV pump-probe method, Chem. Phys. 

Quantum-beat spectroscopy (see Sections 9.2.1, 9.2.2 and 9.3.2 and reviews by Lombardi, 1988, and by Hack and Huber, 1991) requires preparation of a coherent superposition state, > composed of two eigenstates, +, M)... 

Zeeman quantum beat spectroscopy was used by Gouedard and Lehmann (1979, 1981) to measure the effect of various lu perturbing states on the gj-values [Eq. (6.5.21)] of more than 150 rotational levels of the Se2 B 0+ state (see Section 6.5.2 and Fig. 6.16). In that experiment, the excitation polarization was perpendicular to the applied magnetic field so that quantum beats were observed between nominal B-state components differing in M by 2. The frequencies of these beats increase linearly from 0 MHz at 0 G until the AM — 2 splitting falls... 

The earliest pulsed laser quantum beat experiments were performed with nanosecond pulses (Haroche, et al., 1973 Wallenstein, et al., 1974 see review by Hack and Huber, 1991). Since the coherence width of a temporally smooth Gaussian 5 ns pulse is only 0.003 cm-1, (121/s <-> 121 cm"1 for a Gaussian pulse) nanosecond quantum beat experiments could only be used to measure very small level splittings [e.g. Stark (Vaccaro, et al., 1989) and Zeeman effects (Dupre, et al., 1991), hyperfine, and extremely weak perturbations between accidentally near degenerate levels (Abramson, et al., 1982 Wallenstein, et al., 1974)]. The advent of sub-picosecond lasers has expanded profoundly the scope of quantum beat spectroscopy. In fact, most pump/probe wavepacket dynamics experiments are actually quantum beat experiments cloaked in a different, more pictorial, interpretive framework,... 

Either two or more molecular levels of a molecule are excited coherently by a spectrally broad, short laser pulse (level-crossing and quantum-beat spectroscopy) or a whole ensemble of many atoms or molecules is coherently excited simultaneously into identical levels (photon-echo spectroscopy). This coherent excitation alters the spatial distribution or the time dependence of the total, emitted, or absorbed radiation amplitude, when compared with incoherent excitation. Whereas methods of incoherent spectroscopy measure only the total intensity, which is proportional to the population density and therefore to the square ir of the wave function iff, the coherent techniques, on the other hand, yield additional information on the amplitudes and phases of ir. 

The Fourier analysis of the time-dependent signal (7.27a-7.27b) yields a Doppler-free spectrum /( ), from which the energy spacing A as well as the width y of the two levels ) can be determined, even if A is smaller than the Doppler width of the detected fluorescence (Fig. 7.9c). Quantum-beat spectroscopy therefore allows Doppler-free resolution [868]. 

Because of sub-Doppler resolution, quantum-beat spectroscopy has been used to measure fine or hyperflne structure and Lamb shifts of excited states of neutral atoms and ions [870]. 

An interesting technique for measuring hyperfine splittings of excited atomic levels by quantum-beat spectroscopy has been reported by Leuchs et al. [876]. The pump laser creates a coherent superposition of HFS sublevels in the excited state that are photoionized by a second laser pulse with variable delay. The angular distribution of photoelectrons, measured as a function of the delay time, exhibits a periodic variation because of quantum beats, reflecting the hfs splitting in the intermediate state. 

Because quantum-beat spectroscopy offers Doppler-free spectral resolution, it has gained increasing importance in molecular physics for measurements of Zee-man and Stark splittings or of hyperfine structures and perturbations in excited molecules. The time-resolved measured signals yield not only information on the dynamics and the phase development in excited states but allow the determination of magnetic and electric dipole moments and of Lande g-factors. 

Many other molecules, such as SO2 [880], NO2 [881], or CS2 [882], have been investigated. A fine example of the capabilities of molecular quantum-beat spectroscopy is the determination of the magnitude and orientation of excited-state electric dipole moments in the vibrationless Si state of planar propynal [883]. 

More examples and experimental details as well as theoretical aspects of quantum-beat spectroscopy can be found in several reviews [868, 878, 884], papers [871-885, 887], and a book [886]. 

G. Herzberg, Molecular Spectra and Molecular Structure (van Nostrand, New York, 1950) N. Ochi, H. Watanabe, S. Tsuchiya, RotationaUy resolved laser-induced fluorescence and Zeeman quantum beat spectroscopy of the state of jet-cooled CS2. Chem. Phys. 113,... 

H. Bitto, J.R. Huber, Molecular quantum beat spectroscopy. Opt. Common. 80,184 (1990) R.T. Carter, R. Huber, Quantum beat spectroscopy in chemistry. Chem. Soc. Rev. 29, 305 (2000)... 

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