Exactly solvable coupled-channel potential models of atom-atom magnetic Feshbach resonances from supersymmetric quantum mechanics

Abstract

Starting from a system of N radial Schrödinger equations with a vanishing potential and finite threshold differences between the channels, a coupled N × N exactly-solvable potential model is obtained with the help of a single non-conservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as well as its Jost matrix. It depends on N (N + 1)/2 unconstrained parameters and on one upperbounded parameter, the factorization energy. A detailed study of the model is done for the 2×2 case: a geometrical analysis of the zeros of the Jost-matrix determinant shows that the model has 0, 1 or 2 bound states, and 0 or 1 resonance; the potential parameters are explicitly expressed in terms of its bound-state energies, of its resonance energy and width, or of the open-channel scattering length, which solves schematic inverse problems. As a first physical application, exactly-solvable 2 × 2 atom-atom interaction potentials are constructed, for cases where a magnetic Feshbach resonance interplays with a bound or virtual state close to threshold, which results in a large background scattering length.

Dynamical regimes of a quantum SWAP gate beyond the Fermi golden rule

“…It also may appear in many theoretical models: e.g., describing the decay of superdeformed nuclei [33], phase transitions and avoided level crossings [34,35,36], coupling of bound states to open decay channels [37], geomagnetic polarity reversal [38], tunneling between quantum dots [39,40], optical microcavity [41], vibrational surface modes [42], and in the context of the crossing of two Coulomb blockade resonances [43]. On the other hand, the virtual-localized transition has been addressed in the context of the famous n − p singlet system [20], models of stabilization of quantum mechanical binding by potential barriers [44], Feshbach resonances [45,46] and stability of atomic and molecular states [47,48]. This paper is organized as follows.…”

Shadow poles in coupled-channel problems calculated with the Berggren basis

“…(31) of Ref. [48] we can get k r p r < 0, while the sign of p i is determined by the condition p r p i < 0 for E i > 0 (which also implies k i p i < 0). Considering these restrictions we have…”

“…Taking for example E r = 0.4 and E i = 0.01 as in [48] we get k r = 0.632550, k i = −0.007905 and p r = −0.006455 p i = 0.774624. The exact solutions for the upper sign in Eqs.…”

“…(52) of Ref. [48] is wrong). Using the threshold condition k 2 − p 2 = ∆ we can determine p r and p i with p 1 = p r + ip i and p 2 = −p * 1 = −p r + ip i .…”

“…A very convenient approach was introduced in Ref. [48]. In this approach one first fixes two solutions and the other two solutions are obtained in closed form.…”