Feshbach association

In the field of ultracold molecules, the information on the avoided crossings is fundamentally important to fully describe the molecular spectrum and to understand the coupling mechanisms between different states. At the same time, controlled state transfer near avoided crossings opens up the intriguing possibility of populating many different molecular states that are not directly accessible by Feshbach association. 

Feshbach or compound resonances. These latter systems are bound rotovibra-tional supramolecular states that are coupled to the dissociation continuum in some way so that they have a finite lifetime these states will dissociate on their own, even in the absence of third-body collisions, unless they undergo a radiative transition first into some other pair state. The free-to-free state transitions are associated with broad profiles, which may often be approximated quite closely by certain model line profiles, Section 5.2, p. 270 If bound states are involved, the resulting spectra show more or less striking structures pressure broadened rotovibrational bands of bound-to-bound transitions, e.g., the sharp lines shown in Fig. 3.41 on p. 120, and more or less diffuse structures arising from bound-to-free and free-to-bound transitions which are also noticeable in that figure and in Figs. 6.5 and 6.19. At low spectroscopic resolution or at high pressures, these structures flatten, often to the point of disappearance. Spectral contributions of bound dimer states show absorption dips at the various monomer Raman lines, as in Fig. 6.5. 

The shape resonances have been described by Feshbach in elastic scattering cross-section for the processes of neutron capture and nuclear fission [7] in the cloudy crystal ball model of nuclear reactions. These scattering theory is dealing with configuration interaction in multi-channel processes involving states with different spatial locations. Therefore these resonances can be called also Feshbach shape resonances. These resonances are a clear well established manifestation of the non locality of quantum mechanics and appear in many fields of physics and chemistry [8,192] such as the molecular association and dissociation processes. 

Figure 1 2 10. The reduced Lifshitz parameter"z" - (ET - EF)/(EA- ET), where (EA- Er) is the full energy band dispersion in the c-axis direction, as a function of the number of holes in the G subband in A1 doped MgB2. The quantum uncertainty in the z value is indicated by the error bars that are given by D (

Resonances unassociated with eigenstates of Feshbach s QHQ are often associated with the shape of some effective potential in an open channel, normally a combination of short-range attractive and long-range repulsive potentials, forming a barrier, within which a large part of the wavefunction is kept. These resonances are called "shape resonances" or "potential resonances." They occur at energies above and usually close to the threshold of that open channel. 

Expression (13.80) is called WKBJ (ray theoretic) Green s function, because it is associated with the names of several physicists, G. Wentzel, H. A. Kramers, L. Brillouin, and H. Jeffreys, who independently introduced this approximation in connection with the solution of different physical problems (Morse and Feshbach, 1953). I would also recommend an excellent book by Bleistein et al. (2001), where the interested reader can find a more thorough mathematical analysis of the WKBJ approximation. 

The state selective detection scheme based on RIS has been applied to isolate specific autodetaching channels for doubly excited states in both He" and Li. Here we present the results of recent measurements of the positions and widths of doubly excited states which appear as resonances in partial cross sections below certain excited state thresholds. The 1 s3s4s state in He, for example, lies just below the He(3 S) threshold. A Feshbach resonance associated with the autodetaching decay of this state was observed in both the He(ls2s S)+e(ks) and He(ls2p P)+e(kp) partial cross sectionsfhereafler labeled 2 Sks and 2 Pkp) [20]. Similarly, in the case of Li, resonances due to doubly excited states below the Li(42p) and Li(52p) thresholds were observed in the 32Skp partial cross section [21]. The state selective detection scheme has also been used to study the threshold behavior of the 2 Pks partial cross section. This technique enabled Haeffler et al. [22] to accurately measure the electron affinity of Li. 

The properties of unstable negative-ion states of the kind which play the intermediate complex role in associative-detachment have been the subject of a great deal of attention in recent years, primarily because of their role in electron-molecule collisions. An excellent discussion of such resonances is given by Bardsley and Mandl. " The long-lived N02 intermediate in (29), stabilized on a curve such as 3 in Fig. 8 due to rotational excitation of the NO, is again a nuclear-excited Feshbach resonance. 

The optimum association strategy depends on several factors connected to the specific system under investigation. The main factors to consider are the quantum-statistical nature of the atoms that form the molecule and the particular Feshbach resonance employed. Ultralow temperatures and high phase-space densities are essential requirements for an efficient molecule creation. Therefore all association techniques start with an ultracold trapped atomic sample. 

A very efficient way to detect Feshbach molecules is to convert them back into atom pairs and to take an image of the reconverted atoms with standard absorption imaging techniques [27,28], In principle, each of the above-described association methods can be reverted and turned into a corresponding dissociation scheme. 

A robust and commonly used dissociation scheme is to simply reverse the Feshbach ramp, as illustrated in Figure 9.4c. The reverted ramp is usually chosen to be much faster than the one employed for association. The molecules are then brought into a quasi-bound state above the atomic threshold. Here they quickly dissociate into atom pairs, converting the dissociation energy into kinetic energy. The back-ramp is usually done in free space and can be performed either immediately after release from the trap or after some expansion time ( 10-30msec). The choice of the delay time between the dissociation and the detection sets the type of information that can be... 

Here p, is the reduced mass, and are the Hamiltonians defined in Equation 11.1 for each atom, and Vint is the effective interaction potential depending on the relative position of the atoms, r. For many applications, such as the description of broad scattering resonances and their associated Feshbach molecules, it is sufficient to include in Vint only the rotationally symmetric singlet and triplet Born-Oppenheimer potentials, Vs=o and V5=i, respectively. Their labels 5 = 0 and 5=1 refer to the possible values of the angular-momentum quantum number associated with the total spin of the two atomic valence electrons, S = si -E S2. In this approximation, the interaction part of Equation 11.4 can be represented by [8,29]... 

Resonant enhancements of scattering cross-sections in multichannel collision physics are often described in terms of the Feshbach theory of closed-channel resonance states [57], Feshbach s general formalism involves projecting the stationary Schrddinger equation onto complementary subspaces associated with the open and closed scattering channels. This theory has been applied in the context of the nearthreshold collision physics of ultracold gases consisting of alkali-metal atoms in a variety of different approaches (e.g.. Refs. [9,30,58]). 

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