Quantum-defect theory for type of interactions
We present a quantum-defect theory (QDT) for the type of long-range potential, as a foundation for a systematic understanding of charge-neutral quantum systems such as ion-atom, ion-molecule, electron-atom, and positron-atom interactions. The theory incorporates both conceptual and mathematical advances since earlier formulations of the theory. It also includes more detailed discussions of the concept of resonance spectrum and its representations, universal properties in charge-neutral quantum systems, and the QDT description of scattering resonances that is applicable to any potential with .
The quantum-defect theory (QDT) for type of interactions, if broadly defined as a quantum theory that explicitly takes advantage of the universality due to the long-range potential, has existed in various forms for decades O’Malley et al. (1961); Watanabe and Greene (1980); Fabrikant (1986). Notably, the theory of O’Malley et al. O’Malley et al. (1961) gives an analytic description of ultracold electron-atom and ion-atom collision that has stood for many years. The theory of Fabrikant Fabrikant (1986) gives a theory of scattering that takes further advantage of the modified Mathieu functions Holzwarth (1973); Khrebtukov (1993); Olver et al. (2010). The theory of Watanabe and Greene Watanabe and Greene (1980) gives a more complete QDT formulation for potential in that it treats both positive and negative energies in a consistent QDT manner which is important for, e.g., its application in a multichannel formulation to describe Fano-Feshbach resonances. Together, these theories have provided a solid theoretical backbone for our understanding of charge-neutral quantum systems in low-energy regimes or around a dissociation (detachment) threshold, and have served us well for many years, including in more recent applications such as Rydberg molecules Greene et al. (2000); Bendkowsky et al. (2009, 2010); Löw et al. (2012).
Renewed interest in QDT for polarization potential has arisen with the emergence of cold ion-atom Côté and Dalgarno (2000); Grier et al. (2009); Idziaszek et al. (2009); Gao (2010a); Zipkes et al. (2010a, b); Schmid et al. (2010); Rellergert et al. (2011); Hall et al. (2011); Idziaszek et al. (2011) and ion-molecule interactions and reactions Willitsch et al. (2008); Staanum et al. (2008); Roth et al. (2008); Hudson (2009); Gao (2011a); Willitsch (2012). For such heavier systems, the QDT takes on a different magnitude of importance for three primary reasons. First, a same long-range polarization potential, which in the case of electron-atom interaction can only bind a few states or lead to a few resonances Buckman and Clark (1994), can bind many more states and lead to many more resonances. Their existence implies a rapid energy variation induced by the long-range interaction, making the QDT description far more important and necessary. Mathematically, this greater importance of long-range potential for heavier systems is reflected in the length scale, , and the corresponding energy scale, , associated with a interaction. The length scale scales with the reduced mass as , and is hundreds times greater for ion-atom Gao (2010a) and ion-molecule Gao (2011a) systems than for electron-atom systems. The energy scale scales with the reduced mass as , and is typically times smaller Gao (2010a, 2011a). Between a fixed energy and the dissociation threshold , the number of bound states or resonances due to the long-range potential is determined by the scaled energy (see Ref. Gao (2010a) and Sec. III.2.3), which is vastly greater for heavier systems. Second, at any fixed positive energy above the threshold, vastly many more partial waves, of the order of , contribute to ion-atom and ion-molecule interactions than to electron-atom interactions. At room temperature, e.g., hundreds of partial waves contribute to typical ion-atom scattering, while only one or a few to typical electron-atom scattering. An efficient and unified description of a large number of partial waves Gao (2001, 2008) is thus critically important to any systematic quantum theory of heavier charge-neutral systems that intends to cover a wide range of energies such as from absolute zero temperature to the room temperature. We emphasize that large number of partial waves and seemingly small de Broglie wave length do not guarantee classical behavior, there are subtle quantum effects such as shape resonances that can persist even when such conditions seem well satisfied. These “high temperature” resonances can be expected to play an important role in chemistry such as molecule formation in a dilute environment Chandler (2010); Barinovs and van Hemert (2006), and in thermodynamics. Third, ion-atom and ion-molecule interactions are extremely sensitive to the short-range potential Hechtfischer et al. (2002); Li and Gao (2012), due again to their large reduced mass Gao (1996). With the exception of H+H and its isotopic variations Igarashi and Lin (1999); Esry et al. (2000); Krstić et al. (2004); Bodo et al. (2008), this sensitive dependence makes most theoretical predictions based on ab initio potentials unreliable. QDT and related multichannel quantum-defect theory (MQDT), especially though their partial-wave insensitive formulations Gao (2001); Gao et al. (2005), offer a prospect to overcome this difficulty by reducing their description to very few parameters that can be determined with a few experimental data points Carrington et al. (1988, 1993, 1995a, 1995b); Hechtfischer et al. (2002), without relying on the precise knowledge of the short-range potential Li and Gao (2012).
We present here a version of the QDT for the potential that incorporates both recent conceptual advances in QDT Gao (2008, 2010a) and mathematical advances in the understanding of the modified Mathieu functions Gao (2010a); Idziaszek et al. (2011). A brief account of the theory, and its initial applications to ion-atom interactions and charge-neutral reactions, have been presented in Refs. Gao (2010a, 2011a); Li and Gao (2012). This work presents the details of the underlying QDT formulation, in preparation for its further applications. It includes derivations of the quantum reflection and transmission amplitudes for the potential (important for understanding charge-neutral reactive processes Gao (2011a)), a more detailed discussion of the concept of resonance spectrum Gao (2010a) and its representations, universal properties in charge-neutral quantum systems especially ion-atom interactions, and the QDT description of scattering resonances that is applicable to any potential with . This presentation also serves as a concrete example showing how the general QDT structure of Ref. Gao (2008) is actually realized for a particular .
We begin by recalling the keys steps in constructing a QDT for a () potential Gao (2008). We want to first find solutions of the Schrödinger equation
where is the reduced mass, and is the energy. After scaling the length by the length scale
and the energy by a corresponding energy scale . Eq. (1) takes the dimensionless form of
where is a scaled radius, and is a scaled energy. We would like to find a pair of linearly independent solutions of Eq. (3), the so-called QDT base pair Gao (2008), defined by the small- asymptotic behavior of
for all energies. Here .
The large- asymptotic behaviors of such an pair, in the limit of , defines the matrix for positive energies, and the matrix for negative energies, the combination of which gives one formulation of QDT for potential Gao (2008). From the and matrices, one can derive the quantum reflection and transmission amplitudes associated with the long-range potential, from which a different QDT formulation can be constructed Gao (2008). This latter formulation, namely QDT in terms of reflection and transmission amplitudes, especially its multichannel generalization Gao (2010b), is playing an important role in applications of QDT in reactions and inelastic processes Gao (2010b, 2011a).
For , the solutions of Eq. (3) are given in terms of the modified Mathieu functions Holzwarth (1973); Khrebtukov (1993); Olver et al. (2010). While they are well-known mathematical special functions, their understanding and application in physics have been somewhat limited by their relative complexity. Our QDT for potential includes an alternative method of solving and understanding Mathieu class of functions that may help to stimulate their further applications. The method further emphasizes the structural similarities of the solutions to solutions for Gao (1998a) and Gao (1999a) potentials, which should be helpful in understanding all such solutions.
The paper is organized as follows. In Sec. II, we present the QDT functions for potential, including the reference wave functions, the and matrices, the quantum reflection and transmission amplitudes, and the corresponding QDT functions for negative energies, such as the quantum order parameter introduced in Ref. Gao (2008). The key results of the corresponding single-channel QDT for interaction are presented in Sec. III. It includes a unified understanding of both the bound spectrum and the resonance spectrum, their different representations, and a QDT description of scattering resonance that is applicable to any potential with . Section IV discusses the single-channel universal behaviors for charge-neutral quantum systems, especially ion-atom systems, as implied in the QDT formulation. In Sec. V, we briefly discuss the differences in applying the theory to ion-atom and to electron-atom interactions. Section VI concludes the article.
Ii QDT functions for potential
ii.1 The math reference pair
Specializing to the potential with , the length scale becomes , and Eq. (3) becomes
One pair of its solutions, which we call the math pair, is given in terms of the modified Mathieu functions Holzwarth (1973)
Here , and and are the modified Mathieu functions with Laurent expansions Holzwarth (1973)
Here , and is the characteristic exponent for the potential, discussed further in the Appendix A. We have solved this set of recurrence relations using the method developed earlier for Gao (1998a) and Gao (1999a) potentials to give
in which is given by a continued fraction
With analytic expressions for as given by Eqs. (13) and (14), the asymptotic behaviors of the math pair, for both small and large , can be derived directly from their Laurent expansions, using a method that is similar to what led to the large- behaviors of the solutions Gao (1998a). For small , we obtain
where for , are the Bessel functions Olver et al. (2010), and
For large , the asymptotic behaviors of the math pair are given for by
where , and for by
where , and are the modified Bessel functions Olver et al. (2010). An equivalent pair of solutions has been found independently by Idziaszek et al. Idziaszek et al. (2011), using a similar method Gao (1998a, 1999a).
ii.2 The QDT base pair and the and matrices
From the definitions of and , and the small- asymptotic behaviors of the math pair, it is straightforward to show that the QDT base pair is given in terms of the math pair by
From the solutions for , , and the definition of , one can verify that , , and hence and , are entire functions of . Physically, this is what ensures that the short-range matrix, defined in reference to the QDT base pair, being meromorphic in energy Gao (2008). Mathematically, it allows analytic continuation of the base pair to negative energies (and complex energies if necessary) without explicitly solutions of the math pair for such energies.
The large- asymptotic behaviors of the QDT base pair, which give the and the matrices, follow from Eqs. (28) and (29), and the large- behaviors of the math pair, as given by Eqs. (21)-(24). For , we obtain
For , we obtain
In the expressions for the and matrices, we have used the definition
for both the positive and the negative energies. We note the subtle difference between the , defined by Eq. (42), and the , defined by Eq. (19), which is the result of careful analytic continuation through entire functions 111The negative energy solutions have also been independently verified through explicit solutions of Eq. (6) for negative energies..
The function, which is function of the scaled energy , is one of the most important mathematical entities in QDT for the potential. All QDT functions of physical interest for the potential can be represented in terms of and the characteristic exponent , which is itself a function of (see Appendix A). Furthermore, all singular behaviors at are isolated to the factor within .
The and matrices, which are constrained by and , give one formulation of the QDT for potential Gao (2008, 2009), to be discussed further in Sec. III. We note that the determinant constraints on and are automatically ensured by their representations in terms of and .
The elements of and matrices are illustrated for in Figs. 1 and 2, respectively, on the natural energy scales of for positive energies and for negative energies. More generally for a potential, there exist natural energy scales of for positive energies and for negative energies. They are energy scales associated with the semiclassical behaviors away from the threshold Le Roy and Bernstein (1970); Gao (1999b); Flambaum et al. (1999); Friedrich and Trost (2004). In later figures covering both positive and negative energies, the natural energy scale for the potential will be represented as , with defined by
ii.3 Quantum reflection and transmission amplitudes
Instead of the matrices, QDT for positive energies can also be constructed in terms of the quantum reflection and transmission amplitudes associated with the long-range potential Gao (2008). Such a formulation, especially its multichannel generalization Gao (2010b), has clearer physical interpretation and has proven to be especially effective in treating and understanding reactive and inelastic processes Gao (2010b, 2011a).
There are four such amplitudes for each partial wave . The two for reflection by the long-range potential can be written as Gao (2008)
where and represent the reflection amplitudes by the long-range potential for particles going outside-in (approaching each other) and inside-out (moving away from each other), respectively. The two amplitudes for transmission can be written as Gao (2008)
where and represent the transmission amplitudes through the long-range potential for particles going outside-in and inside-out, respectively. Equations (45)-(47) imply that the two transmission amplitudes are always equal, while the two reflection amplitudes generally differ by a phase. All amplitudes can be determined from three independent functions: (a) the quantum reflection probability or the quantum transmission probability , which are related by , (b) the long-range (transmission) phase shift , and (c) the reflection phase shift , all of which can be determined from the matrix Gao (2008).
From the matrix of the previous section, we obtain for potential the quantum reflection probability
The related transmission probability is given by
It is illustrated in Fig. 3 for the first few partial waves.
The quantum reflection probability, which also serves as a quantum order parameter for , is illustrated in Fig. 4 together with the quantum order parameter for Gao (2008), to be discussed further in Sec. II.4. Together, they specify a range of energies, where they differ substantially from zero, as the region in which the quantum effects are important Gao (2008). In the semiclassical region defined by , the effect of the long-range potential on scattering is fully characterized by the long-range phase shift . In the quantum region, even a single channel scattering has contributions from multiple paths, which interfere with each other Gao (2008) to give rise to phenomena such as the shape resonance. A complete characterization of the effects of the long-range interaction on scattering in the quantum regime require all three QDT functions Gao (2008).
From again the matrix elements, the long-range (transmission) phase shift and the reflection phase shift , are determined, to within a , by
from which we obtain , independent of energy. This value for implies that, for the potential, the two reflection amplitudes are related by (where ).
ii.4 Quantum order parameter and other QDT functions below the threshold
As discussed in Ref. Gao (2008), the QDT for negative energies can be formulated using either the matrix, or three QDT functions including two phases and , and one amplitude . The latter formulation is convenient for, e.g., understanding the semiclassical limit away from the threshold.