4.2 The field-free system
4.2.4 Inclusion of additional basis states
Every analysis we have performed so far was restricted to the degenerate hydrogenic manifold n= 35, l ≥3. In this paragraph expand the considered basis set by systematically including basis states in the energetic vicinity of the considered hydrogenic manifold. For every single basis set we perform exact diagonalization in order to achieve convergence. As we are interested in studying the potential curve in the Σ symmetry sector we diagonalize VΣ = Vens +Venp1 using the Rydberg wave functions ψnl0(r). We obtain for the matrix elements
[VΣ(r,R)]n′l′,nl =hψn′l′0(r)|VΣ(r,R)|ψnl0(r)ir, ψnl0(r) =Rnl(r)Yl0(ϑ, ϕ). (4.45) In the convergence study performed in this chapter we concentrate on the results for the X3Σ potential curve as the results of the A3Σ curve just differ quantitatively but not qualitatively.
We start our analysis by defining the basis sets (1)-(1c) in the energetic vicinity of the n = 35, l ≥ 3 hydrogenic manifold as they are presented in Table 4.1. Hereby basis set (1) denotes the pure hydrogenic manifold while the other basis sets (1b) and (1c) contain the quantum defect states which lie below, respectively above the considered manifold. As discussed in Section 1.3.2 the atomic Rydberg states withl= 0,1,2 possesses a finite hyperfine splitting. However, this splitting is of the order of 200 MHz which is much less than the energetic separation of the Σ potential curves to these energy levels. For this reason, we neglect the spin-orbit coupling for these states and use the single electronic Rydberg nlstates instead. In Fig. (4.7)(a) we present the potential energy curves which have been calculated using the different basis sets (1)-(1c). We see that the inclusion of the quantum defect states below and above the n = 35, l ≥ 3 hydrogenic manifold hardly changes the X3Σ potential curve. In particular, with the inclusion of additional states, we obtain a relative deviation of 10−3.
Next we start to include the neighbored degenerate hydrogenic manifolds as well as all the
basis set hydrogenic manifold quantum defect states
(1) n= 35, l≥3
-(1b) n= 35, l≥3 37p,36d,38s (1c) n= 35, l≥3 37p,36d,38s,38p,37d,39s
Table 4.1:Basis set (1)-(1b) in the energetic vicinity of then= 35, l≥3 hydrogenic manifold. Basis set (1a) denote the pure hydrogenic manifold. Basis set (1b)((1c)) contain the quantum defect states which lie energetically below (above) the considered manifold.
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−7.5
−7
−6.5
−6
−5.5 (a)
R(a0)
Energy/h(GHz)
(1) (1b)
(1c) −150
−100
−50 0 50 100 150
Energy/h(GHz)
n= 35, l≥3 n= 36, l≥3 (b)
n= 34, l≥3 38s
36d 37p 39s 37d 38p
Figure 4.7: (a) Potential energy curves for then= 35, l≥3 hydrogenic manifold calculated with the basis sets (1)-(1c) (see Table 4.1). We see that including the neighboring quantum defect states hardly changes the potential curve. Figure (b) shows the energetic level scheme for the basis set (2).
nearby quantum defect states. For this we define the basis sets (2)-(6) as they are presented in Table 4.2. These basis states are characterized by an integer number ∆nwhich denotes the number of hydrogenic manifolds in the vicinity of the n= 35 manifold which are included into the basis set. For instance, ∆n = 1 includes the n = 34, l ≥ 3 till n = 36, l ≥ 3 hydrogenic manifolds including all quantum defect states (see Section 1.3). In Fig. (4.7)(b) the energetic level scheme for the basis set (2) is shown. In Fig. 4.8 we present the potential energy curves which have been calculated using the basis sets (1)-(6). In contrast to the previous analysis now the curves show a strong energy shift with an increasing number of basis states. With increasing radial distanceRthe effect of including additional basis states becomes more dominant. Although we have included the states of five degenerate manifolds below and above the consideredn= 35, l≥3 manifold we have not yet achieved convergence. To get a possible explanation for the observed level shift we remark
basis set hydrogenic manifold quantum defect states
(1) ∆n= 0
-(2) ∆n= 1 37p−39s
(3) ∆n= 2 36p−40s
(4) ∆n= 3 35p−41s
(5) ∆n= 4 34p−42s
(6) ∆n= 5 33p−43s
Table 4.2: Basis set (2)-(6) in the energetic vicinity of the n= 35, l ≥3 hydrogenic manifold. The number ∆ndenote the range of hydrogenic manifolds which are taken into the basis set. All quantum defect states which lie between the hydrogenic manifolds are included (see 1.3 and Fig. 4.7(b)).
4.2 The field-free system 45
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−7.5
−7
−6.5
−6
−5.5
R(a0)
Energy/h(GHz)
(1) (2) (3) (4) (5) (6)
Figure 4.8: Potential energy curves calculated with the basis sets (1)-(6). With an increasing number of basis states the potential curve continues moving energetically downwards.
(a) basis set quantum defect states
(1)
-(2)qds 37p−39s
(3)qds 36p−40s
(4)qds 35p−41s
(5)qds 34p−42s
(6)qds 33p−43s
(b) basis set hydrogenic manifolds
(1) ∆n= 0
(2)nqd ∆n= 1
(3)nqd ∆n= 2
(4)nqd ∆n= 3
(5)nqd ∆n= 4
(6)nqd ∆n= 5
Table 4.3: In table (a) we present the basis sets which only include quantum defect states nmin p-nmaxs. In contrast table (b) only includes basis sets which contain degenerate hydrogenic manifolds withnmin/max= 35±∆n.
that we can distinguish two different subsets of basis states. First, we have quantum defect states which are non degenerate. In contrast, the hydrogenic manifolds are highly degenerate possessing levels of degeneracy of n−3. For this reason we conclude that the constant level shift observed in Fig. 4.8 is related to the inclusion of highly degenerate hydrogenic manifolds. To analyze this issue more systematically we perform additional studies where we define two novel basis sets which are presented in Table 4.3(a,b). The first basis set just contains the quantum defect states between the degenerate hydrogenic manifolds. The basis states are denoted as (n)qds (see Table 4.3(a)). The second analysis is performed with basis sets which only contain degenerate hydrogenic manifolds.
These basis states are denoted as (n)nqdand labels the number of neighboring degenerate manifolds with minimal and maximal principal quantum number nmin/max= 35±∆n(see Table 4.3(b)). In Fig. 4.9(a) we present the potential curves which are obtained by exact diagonalization using the basis sets (1)-(6)qds. Similar to the previous study using basis set (1)-(1c) (see Fig. 4.7(a)) the potential curves are hardly affected by increasing the number of basis states. Again we obtain relative deviations of 10−3. In contrast in Fig. 4.9(b) we present the potential curves obtained from the diagonalization in the basis sets (1)-(6)nqd. Here we obtain similar results to the study using the basis states (1)-(6). With increasing the number of degenerate hydrogenic manifolds, the potential curve continue to shift with a constant relative deviation up to 0.9. This strengths the idea that the constant energy shift is caused by the mixing of the degenerate hydrogenic states by the contact interaction Ven. However, a satisfactory explanation of this behavior could not be found. For this reason we remark that this issue requires additional research. Throughout this thesis we have used a basis set consisting of then= 35, l≥3 hydrogenic manifold and the 38s,36dand 37pstates. As
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−7
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−6
−5.5 (a)
R(a0)
Energy/h(GHz)
(1) (2)qds
(3)qds
(4)qds
(5)qds
(6)qds
1600 1800 2000 2200
−7.5
−7
−6.5
−6
−5.5
R(a0)
Energy/h(GHz)
(b)
(1) (2)nqd
(3)nqd
(4)nqd
(5)nqd
(6)nqd
Figure 4.9: Figure (a) shows the potential curves obtained via exact diagonalization using the basis sets (1)-(6)qds. Including additional basis states hardly affects the potential curves. Figure (b) shows the convergence studies using the basis states (1)-(6)nqd only containing degenerate hydrogenic manifolds.
Increasing the number of included hydrogenic manifolds causes the potential curves to shift energetically downwards.
we expect the Σ potential curves to lower in energy we have included the quantum defect states energetically below the hydrogenic n = 35, l ≥ 3 manifold. In this case we denote the field free eigenstates as Ξi(r;R)i, this means
Hel(r;R)|Ξi(r;R)i=ǫ(i)0 (R)|Ξi(r;R)i. (4.46) For sufficiently large electric and magnetic field strengths we expect the energy level to approach cross each other. As we expect this regime to provided a plethora of interesting physics the inclusion of these states is essential.
Chapter 5
Electrically dressed ultralong-range Rydberg molecules
5.1 Introduction
In this chapter we perform a study of the impact of an electric field on the structure and dynamics of high-l ultralong-range diatomic Rubidium molecules. We hereby proceed as follows. Section 5.2 provides a formulation of the problem presenting the working Hamiltonian and a discussion of the underlying interactions. Our analysis goes beyond the s-wave approximation and takes into account the next order p-wave term of the Fermi-pseudopotential. Section 5.3 contains our methodology and a qualitative discussion of the effects of an external electric field strength. In Section 5.4 we analyze the evolution of the topology of the potential energy surfaces (PES) with varying electric field strength. Besides a numerical exact diagonalization scheme, we also study the PES in two approximative approaches. The resulting PES show a strongly oscillatory behavior with bound states in the MHz and GHz regime. With increasing field strength the diatomic molecular equilibrium distance shifts substantially in a range of the order of thousand Bohr radii.
In Section 5.5 we analyze the behavior of the corresponding electric dipole moment. Thereby we realize molecular states with a dipole moment up to several kDebye. Based on these properties and thes-wave admixture via the external electric field a preparation scheme for high-lpolar molecular electronic states via a two photon excitation process is presented. Finally, in Section 5.6 we provide an analysis of the vibrational spectra which exhibit spacings of the order of several MHz.