Polarization Quantum Beats

Under even more intense photoexcitation ( 10mJ/cm2), the coherent A g and Eg phonons of Bi and Sb exhibit a collapse-revival in their amplitudes (Fig. 2.10) [42,43], This phenomenon has a clear threshold in the pump density, which is common for the two phonon modes but depends on temperature and the crystal (Bi or Sb). At first glance, the amplitude collapse-revival appears to be analogous to the fractional revival in nuclear wavepackets in molecules [44,45]. However, the pump power dependence may be an indication of a polarization, not quantum, beating between different spatial components of the coherent response within the laser spot [46],... 

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... 

The first measurements of Na nd fine structure intervals using quantum beats were the measurements of Haroche et al41 in which they detected the polarized time resolved nd-3p fluorescence subsequent to polarized laser excitation for n=9 and 10. Specifically, they excited Na atoms in a glass cell with two counterpropa-gating dye laser beams tuned to the 3s1/2—> 3p3/2 and 3p3/2— ndj transitions. The two laser beams had orthogonal linear polarization vectors et and e2 as shown in Fig. 16.9. 

Lange, W. and Mlynek, J. (1978). Quantum beats in transmission by time-resolved polarization spectroscopy, Phys. Rev. Lett., 40, 1373-1375. 

Now, the argument just presented relies on the unproven assumption that rotational quantum beats arising from a thermal sample of isolated molecules will wash each other out. Recently, we examined this assumption by directly simulating the decays associated with thermally averaged rotational beats.47 (Our initial motivation for this work was to try to explain the picosecond pump-probe results of Refs. 51 and 52, which results showed the existence of polarization-dependent early time transients in the decays of t-stilbene.) These theoretical simulations and subsequent picosecond-beam experiments47-50 have revealed that the manifestations of rotational coherence in thermally averaged decays can, in fact, be observed. In this section, we briefly review these results and examine some of their implications with regard to time-resolved studies of IVR. 

The quantum number constraint leads to the two quite distinct polarization classes of QB spectra AM = 0 beats (polarization of excitation and detection parallel to static field, M = M = M" = M ") and AM = 2 beats (perpendicular polarization, M — M + 2 = M" + 1 = M " + 1). The former type is particularly useful for examining an anticrossing between two perturbing states (provided H12 is small enough). The latter displays both intrastate splittings (useful for measurement of -values and dipole moments) and interstate splittings. 

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... 

Ring, et al., (1998 and 1999) have used a time-dependent magnetic field and the combination of a static magnetic field in a direction perpendicular to that of a time-dependent field to create and manipulate novel coherences and to monitor the quantum beats associated with specifiable details of the time evolution of these coherences. The frequencies and decay rates of different classes of coherence (AMj = 2 and 1 polarization beats, AMj = 0 singlet triplet population beats) may be sampled and modified selectively. 

Figure 6.19 Stark quantum beats in BaO A1 +(u = 2, J = 1). The J = 1 level is excited via the R(0) line by radiation from an N2-laser-pumped dye laser. The pump radiation is linearly polarized at 45° to the 5-field direction in order to produce a coherent superposition of At = 0 with M = 1 components. The top trace shows the signal resulting when the polarization of the detected fluorescence is selected to be at 45° to and at 90° to the excitation polarization. The middle trace is for parallel excitation and detection polarizations. The bottom trace is the difference between the two detection geometries. [From Schweda, et ai.(1985).J...

Stark and Zeeman polarization quantum beats are discussed in Section 6.5.3. An external electric or magnetic field destroys the isotropy of space. As a result, the amplitudes for two transition sequences J", M" — J, M = M" 1 —> J ", M" interfere, and the intensity of X or Y (but not Z) polarized fluorescence is modulated at (Fj M =M"+i — Ejim =M"-i)/h. However, it is not necessary to destroy the isotropy of space in order to observe polarization quantum beats. 

In particular, the laboratory frame orientation of the transition moment for spontaneous fluorescence evolves in time. The intensities of z— and (x,y) — polarized fluorescence are modulated 7t/2 out of phase, but the intensity of the total x + y + z polarized fluorescence is not modulated. This is the physical basis for polarization quantum beats (Aleksandrov, 1964 Dodd, et al., 1964) and Rotational Coherence Spectroscopy (Felker and Zewail, 1995). 

Spectroscopic perturbations follow A J = 0, A Mj = 0 selection rules. Therefore perturbations are not a necessary feature of polarization quantum beats. However, such perturbations are essential to a major class of population quantum beats. 

The reason for the name population quantum beats is that the signal intensity (fluorescence, REMPI), integrated over all solid angles and polarization states, oscillates in time after the preparation pulse. It appears as if the population prepared in the excited state at t = 0 vanishes and returns periodically (see Fig. 9.5) (Felker and Zewail, 1984). In fact, the population does not oscillate, but the radiative capability of the time-evolving state prepared at t = 0, k(t), does oscillate. 

Quantum beats can be observed not only in emission but also in the transmitted intensity of a laser beam passing through a coherently prepared absorbing sample. This has first been demonstrated by Lange et al. [872, 873]. The method is based on time-resolved polarization spectroscopy (Sect. 2.4) and uses the pump-and-probe technique discussed in Sect. 6.4. A polarized pump pulse orientates atoms in a cell placed between two crossed polarizers (Fig. 7.12) and generates a coherent superposition of levels involved in the pump transition. This results in an oscillatory time dependence of the transition dipole moment with an oscillation period AF = 1/Av... 

The time development of these coherences corresponds to a time-dependent susceptibility X (0 of the sample, which affects the polarization characteristics of the probe pulse and appears as quantum beats of the transmitted probe pulse intensity. 

One example is the measurement of hyperfine quantum beats in the polyatomic molecule propynal HC=CCHO by Huber and coworkers [877]. In order to simplify the absorption spectrum and to reduce the overlap of absorbing transitions from different lower levels, the molecules were cooled by a supersonic expansion (Sect. 4.2). The Fourier analysis of the complex beat pattern (Fig. 7.14) showed that several upper levels had been excited coherently. Excitation with linear and circular polarization with and without an external magnetic field, allowed the analysis of this complex pattern, which is due to singlet-triplet mixing of the excited levels [877, 878]. 

For measurement of atomic coherence see also "Polarization Selective Detection of Hyperfine Quantum-Beats in Cs", by H. Lehmitz and H. Harde, this symposi+im... 

Fig. 12.1 la,b. Quantum-beat spectroscopy of atomic or molecular ground states measured by time-resolved polarization spectroscopy (a) experimental arrangement and (b) Zee-man quantum beat signal of the Na 3 Si/2 ground state recorded by a transient digitizer with a time resolution of 100 ns. (Single pump pulse, time scale 1 rs/div, magnetic field 1.63 X 10-4 T) [12.40]... 

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