The column with heading C 1 , C 2 gives the group numbers of the atoms in the same order as they appear in the names of the molecules; the numbers go from 1 to 7. In Table 8. The latera recta go from 0. It is difficult to classify the members within these classes chemically.
For instance, Class 8 includes six dihalides—but there are dihalides in Classes 3 and We will do this with a sample set of graphs showing the calculated loci superposed in all but Fig. No FCF data are known, so the figure can not show how well the two loci agree with the data. There are 40 entries in Class 2 over half of the entries in the table ; for these molecules, both the branches of the Morse locus are above those of the SHM locus.
Two examples of this class are shown in Figures 4 and 5. Class 3 molecules have the opposite appearance: The arms of the Morse locus are below those of the SHM locus. Class 5 molecules have Morse loci curving upward, thus illustrating point 3, about very large anharmonicity constants, in a previous section Fig. Class 8 molecules are represented by Br 2 , whose graphic is quite different than the preceding ones Fig. We infer that the electronic states of those molecules are far from being SHM or Morse potentials.
From this limited study, it is concluded that all classes of molecules except those in Class 9 can be included in the analysis to come, and that the SHM loci appear to represent the locally maximal FCFs. Our study of dihalide molecules would show a nice monotonic behavior for the latera recta were it not broken by F 2 and Br 2 Fig. Conversely, no monotonic behavior in the group numbers is seen when the molecules are sorted against the latera recta in Table 2. Thus, it seems that the analysis of presently collected data for homologous molecules has no promise.
The sequences are shown in Table 2. The results are given in the order of decreasing electron populations. The molecular notations are not consistent with usual practice, so it is necessary to pay attention to which atom is first and which is second in the molecular moniker. They are organized such that as Z of the left atom increases, Z of the right atom decreases, and Z 1 — Z 2 increases.
The data points are fitted with second degree polynomials for the only reason that the polynomials are a simple means of emphasizing the similar behavior from figure to figure; they are in terms of x , which equals Z 1 — Z 2.
The parabolas play no role whatsoever in any of the calculations. In addition to the names and terms of the molecules, the captions state the qualitative appearances of the SHM loci as used in Table 1 , the relative positions of the excited states with respect to the lower states, and the standard deviation of the data from a parabolic fit to the data if there are more than three data.
The figures in this section are similar to those preceding them except that there are only three data and Figs. These two vibration frequencies are plotted in Figures 25 and These steps are shown in Table 3. The second result of this investigation is to place molecular periodicity on a more quantum footing than it was with evidence based on previous graphical studies. Palma's group in Mexico City 13 - 15 has investigated FCFs for arbitrary electronic potentials, but does not appear to have computed Condon loci for the factors.
This investigation began as a yet another search for periodicity in properties of molecules 1 , 3.
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Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Introduction An understanding of molecular periodicity that has a basis in quantum states is highly desired, so this article reports the behaviors of the force constants of the upper and lower states of many band systems for which spectroscopic constants and Franck—Condon factor FCF tables are known. Figure 1 Open in figure viewer PowerPoint.
The simple harmonic approximation Condon locus is shown. Adapted from Herzberg, 4 pg. Figure 2 Open in figure viewer PowerPoint. The uppermost feature represents the probability of finding a classical oscillator at points in its domain. Morse and other anharmonic potential curves The Condon loci for Morse potentials can be calculated in much the same way as the simple harmonic case, except that additional anharmonic spectroscopic constants are necessary. Not all of these molecules are known to have published Franck—Condon factors.
Some of those that do have FCFs have been studied exhaustively and are featured in the next section. This system is the only known entry in Class 1, where the two loci coincide very well in this limited domain. Figure 4 Open in figure viewer PowerPoint. The latus rectum is extremely small, indicating that the electronic potentials have very similar internuclear separations. Figure 5 Open in figure viewer PowerPoint. The upper arm of the SHM locus follows the local maxima for a while and then drops below; the lower arm is always below.
Both arms of the Morse locus are above the two arms of the SHM locus; the lower one by a small amount, which brings it closer to the local maxima, and the upper arm by a very large amount. The SHM locus is about the same shape as the track suggested by the contours but lies at a smaller angle.
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Figure 6 Open in figure viewer PowerPoint. Figure 7 Open in figure viewer PowerPoint. At the same time, the inner curves appear to converge point 4. The SHM latus rectum is rather small, indicating that the electronic potentials have somewhat similar internuclear separations. Figure 8 Open in figure viewer PowerPoint. The Morse locus seems to follow the local maxima rather well but is displaced at about 1.
Secondary Condon loci, of which there have been hints in the previous figures, are clearly visible in the raw data. Figure 9 Open in figure viewer PowerPoint. The SHM locus has an inclination of The latus rectum is very large, indicating very different internuclear separations on the two electronic states. Secondary Condon loci are clearly visible in the raw data. Figure 10 Open in figure viewer PowerPoint. The appearance of this graphic was described in Table 1 as resembling part of a wheel.
Figure 11 Open in figure viewer PowerPoint. Figure 12 Open in figure viewer PowerPoint.
The left to right numbers represent atom 2 as iodine, bromine, chlorine, and fluorine. The band systems are from Classes 3, 8, and 10 of Table 1. Hefferlin, J. Sackett, and J. Tatum, Ch. Figure 13 Open in figure viewer PowerPoint. The molecules AlF, SiO, PN, SC, and ClB have excited A states that are the second, seventh it is the fifth, if one counts the triplet b state as one state , fifth, and second states above the ground state.
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Figure 14 Open in figure viewer PowerPoint. A plot of the latera recta for the band systems are shown in Figure A survey of the last two columns of Table 1 shows that the same is not true within classes. Figure 15 Open in figure viewer PowerPoint. From left to right, the Condon loci are narrow, very wide, wide, wide, wide, and narrow in appearance.
All these B states of the molecules are the second above the ground state. The vertex is at abscissa 8.
Isoelectronic series with several different molecules Figs. Isoelectronic series with three different molecules The figures in this section are similar to those preceding them except that there are only three data and Figs. Figure 16 Open in figure viewer PowerPoint.
Duration Of Molecules In Upper Quantum States
Their Condon loci are narrow, wide, narrow, wide, wide, and narrow in appearance. Direct cooling into the microkelvin regime has also been demonstrated by optoelectrical Sisyphus cooling [ 84 ]. With such remarkable and rapid progress, we can expect the control of ultracold molecules soon to develop to the level required for proof-of-principle quantum simulation.
In this paper we explore the potential of ultracold molecules for quantum simulation. In section 1 we describe the properties of molecules that make them suited to applications in quantum simulation, focussing on the role of the molecular rotation.
Duration of Molecules in Upper Quantum States.
In section 2 we describe the rotational and hyperfine structure of CaF and RbCs, which are important for internal state control using microwave fields. In section 3 we report Ramsey interferometry measurements and investigate the coherence time of a superposition of ground and excited rotational states when the molecules are confined. For both molecules, we show that a judicious choice of molecular hyperfine states minimises the impact of spatially varying shifts in transition frequencies across the trap, providing a route to long interrogation and control times in future experiments.
Finally, we outline our plans for loading, addressing and detecting individual molecules in ordered arrays; this is the next major challenge in the development of molecules for quantum simulation. In this section we introduce the distinctive properties of polar molecules that make them an attractive platform for quantum simulation. We motivate our study of rotational coherence by illustrating how the rotational degrees of freedom can be used to encode models of quantum magnetism.
We limit our discussion to the simple case of diatomic polar molecules in their electronic ground state, pinned on the sites of an optical lattice or tweezer array such that motion and tunnelling between sites can be neglected.
Extensions beyond this simple scenario to include, for example, different trapping geometries, tunnelling between lattice sites, disordered filling, and greater molecular complexity, lead to even richer physics [ 85 — 87 ] but are beyond the scope of this paper. The energies of the rotational states are approximately , where B v is the rotational constant. Typically, in the vibrational ground state and the rotational states can be conveniently manipulated with microwave fields. Molecules also possess rich hyperfine structure, as discussed in section 2 , but this complication does not preclude the use of rotational states for quantum simulation.
It is possible to encode pseudo-spins in molecular rotational states in order to realise various models of quantum magnetism.
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