Universal behavior of diatomic halo states and the mass sensitivity of their properties

The formation of a quantum halo state is a phenomenon associated with loosely bound particles held in short-range potential wells. Halo states extend over an unusually large space and many of their properties hinge on the tail of their wavefunctions exhibiting unusual scattering properties [1]. For instance, they allow for the formation of three body systems exhibiting the Efimov effect [2]; the long-range asymptote of the wavefunction of such a state implies quite distant correlations which are relevant for the formation of Bose–Einstein condensates [3] and sympathetic cooling [4]. The actual scattering is characterized by an s-wave scattering length that is much larger than the range of the particle interactions. It is therefore conceivable that the scattering length a and its related properties, such as the binding energy D₀ or the average value of the internuclear separation $\langle R\rangle$, can exhibit highly anomalous mass dependencies, thus serving as promising probes of a possible variation of the electron-to-proton mass ratio β = m_e/m_p. Interestingly, as noted by Abraham et al [5] and analyzed in detail by Chin and Flambaum [6], this dependence can be dramatically enhanced in collisions of atoms near narrow Feshbach resonances.

The development of laser-cooling techniques enables long-range halo states to be studied with very high spectral resolution, for example, using photoassociation spectroscopy [7]. Moreover, as was recently shown for He₂ in its ground electronic state [8], the square $| {\rm{\Psi }}{| }^{2}$ of the wavefunction of the halo state can be measured by recording a large number of Coulomb explosion events using cold target recoil ion momentum spectroscopy [9]. Since the properties of halo states are expected to be extremely sensitive to tiny variations of the interaction potential, we have performed calculations for the halo states of the ground and excited ${}^{5}{{\rm{\Sigma }}}_{g}^{+}$ electronic states of the helium dimers, for which precise theoretical interaction potentials [10, 11] and appropriate experimental bond lengths and binding energies are available. For the electronic ground state these data were obtained by molecular beam diffraction from a transmission grating [12] and from the above mentioned Coulomb explosion experiment [8]; while for the ${}^{5}{{\rm{\Sigma }}}_{g}^{+}$ electronic state these data were obtained from a two-photon photoassociation experiment [13].

In our approach, the potentials are used to generate relationships between the scattering properties of the probed states and their actual values, from which the mass sensitivities can be determined. This is done through numerical solution of the radial Schrödinger equation for a set of effective potentials ${V}^{{sc}}(R)$, obtained by scaling the interaction potential V(R) with a constant scaling factor f (for further details see [14]),

$$\begin{eqnarray}&&{V}^{{sc}}(R)=f\cdot V(R).\end{eqnarray} \tag{ 1 }$$

The scaled potentials are then utilized to establish the f-dependence of the scattering properties, and by combining these dependencies, a direct relationship between the probed properties can be found. For example, establishing the two relations D₀ = D₀(f) and a = a(f) for a set of values of f, e.g. f_i (i = 1, 2, ...), allows us to obtain the pairs $\{{D}_{0}({f}_{i}),a({f}_{i})\}$, which subsequently defines the one-to-one relations a = F(D₀).

Calculations were performed using the effective vibrational Hamiltonian of Herman and Asgharian [15] for nuclear motion in ${}^{1}{\rm{\Sigma }}$ state molecules,

$$\begin{eqnarray}&&{H}_{\mathrm{eff}}=-\displaystyle \frac{{{\hslash }}^{2}}{2m}\displaystyle \frac{{\rm{d}}}{{\rm{d}}R}\left(1+\beta {g}_{v}(R)\right)\displaystyle \frac{{\rm{d}}}{{\rm{d}}R}+{V}_{\mathrm{ad}}(R)+{V}^{{\prime} }(R),\end{eqnarray} \tag{ 2 }$$

where m is the appropriate nuclear reduced mass, V_ad is the 'adiabatic' part of the molecular potential energy function (assumed to include Born–Oppenheimer, adiabatic, relativistic, quantum electrodynamical (QED) and residual retardation terms), and the terms ${V}^{{\prime} }(R)$ and g_v(R) account for nonadiabatic effects. The so-called vibrational g_v-factor is fixed to its ab initio value, and the effective potential energy function $V={V}_{{ad}}(R)+{V}^{{\prime} }(R)$ is determined either from first principles or from fitting to the available experimental data.

The mass sensitivity of the probed properties, for example, the binding energy D₀ and the scattering length a, are described by the following expressions,

$$\begin{eqnarray}&&{K}_{\beta }=\displaystyle \frac{\beta }{{D}_{0}}\displaystyle \frac{{\rm{d}}{D}_{0}}{{\rm{d}}\beta },\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{T}_{\beta }=\displaystyle \frac{\beta }{a}\displaystyle \frac{{\rm{d}}a}{{\rm{d}}\beta }.\end{eqnarray} \tag{ 4 }$$

The resulting sensitivity coefficients, K_β and T_β, can then be used to determine the induced shift of the respective property,

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{\Delta }}\nu }{{\nu }_{0}}={P}_{\beta }\displaystyle \frac{{\rm{\Delta }}\beta }{{\beta }_{0}}\qquad (P=K,T,..),\end{eqnarray} \tag{ 5 }$$

where ${\rm{\Delta }}\nu ={\nu }_{\mathrm{earlier}}-{\nu }_{0}$ is the change in the property, and ${\rm{\Delta }}\beta ={\beta }_{\mathrm{earlier}}-{\beta }_{0}$ is the change in β, both with respect to their present day values ν₀ and β₀.

Alternative approaches for relating the scattering properties of diatomic molecules do exist, however, these methods have certain drawbacks because of the assumptions they employ: the analytical formulas based on semiclassical and quantum defect theories, and specific $-{C}_{n}/{R}^{n}$ asymptotes of interaction potentials (see, e.g. [16–21]), are quantitative only in the case of very large scattering lengths (see below). Whereas the spectroscopic approach, which constructs potentials from spectroscopic data and then extracts the sought information from the relevant wavefunctions, or from extrapolating the phases of the last bound levels towards the dissociation limit (see, e.g. [22–24]), may be hampered by the dependence of the least bound state on the interaction potential; for example, a tiny variation of V(R) may introduce or remove a bound state causing the value of a to pass between $\pm \infty$ as a result (see, e.g. the studies on ⁶Li [22] and ⁸⁴Sr⁸⁸Sr [23]).

The ground state potential of [11], denoted PCJS, supports a bound state only for the heaviest stable isotopomer ⁴He₂. As seen in the top left panel of figure 1, this state is a true halo state. The much deeper potential of the ${}^{5}{{\rm{\Sigma }}}_{g}^{+}$ electronic state from [10], denoted PJ, supports a number of bound ro-vibrational states for each of the He₂ isotopomers. However, only the least bound vibrational states (namely v = 12, 13 and 14 for ³He₂, ⁴He³He and ⁴He₂, respectively) exhibit the behavior of a quantum halo state (see the top right panel of figure 1, and note that quantum halo states are defined as bound states of particles with a radius extending into classically forbidden regions [1]).

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The most important helium dimer appears to be that of the ground electronic state. This stems from the role of helium in the planned redefinition of the Kelvin unit of thermodynamic temperature in terms of the Boltzmann constant [28]. Since the thermophysical properties of helium computed from first principles are more accurate than their experimental counterparts, the present thermometry relies on the theoretical values (see, e.g. [29]). These thermophysical properties are defined in terms of the same interaction potential as the binding energy and scattering length, quantities which can be very accurately determined in experiment. The thermophysical properties can therefore be checked, and perhaps predicted more accurately, using the experimental values of D₀ and a and a key aspect in this process is the interaction potential.

The best available potential for the helium dimer was recently computed [11] and possesses submillikelvin uncertainties. The potential accounts for all relevant 'higher-order' effects (adiabatic, relativistic, QED and retardation corrections), allowing for a direct and detailed analysis of each contribution. Calculations with these potentials, represented by table 2 of [11] and table 5 of [10], reveal the following important facts (note that we adopt the same notation here as the aforementioned tables): (a) the most rigorous theoretical potential predicts the binding energy within the error bars of the experimental value deduced from Coulomb explosion measurements, (b) the adiabatic and relativistic effects contribute significantly meaning that the effect of retardation is small, (c) accurately accounting for nonadiabatic effects is possible through adiabatic calculations with atomic mass values (instead of nuclear masses).

Interestingly, see the middle and bottom panels of figure 1, tables 1 and 2, upon scaling of the interaction potential defined by equation (1), the different potentials all provide closely coinciding relations between the calculated properties. Furthermore, as expected from our previous calculations [14], the same relations are also established for the empirical potentials obtained from fitting to the thermochemical data [25, 26]. This demonstrates the 'universality' in two body systems [30], where the knowledge of only one of the probed scattering properties, in conjunction with a moderately accurate interaction potential, allows for a quantitative prediction of all the remaining properties.

Table 1.  Scattering lengths a of ⁴He₂ in its ground (${}^{1}{{\rm{\Sigma }}}_{g}^{+}$) and excited (${}^{5}{{\rm{\Sigma }}}_{g}^{+}$) electronic states (in Å).

Potential	Calc-1	Calc-2	Calc-3	Calc-4
${}^{1}{{\rm{\Sigma }}}_{g}^{+}$
${V}_{\mathrm{BO}}$	87.847	88.634	86.718(+3.889, −3.409)	86.730(+3.889, −3.410)
${V}_{\mathrm{BO}}+{V}_{\mathrm{ad}}$	85.550	86.295	86.718(+3.889, −3.409)	86.730(+3.889, −3.410)
${V}_{\mathrm{BO}}+{V}_{\mathrm{rel}}$	93.827	94.730	86.715(+3.889, −3.409)	86.726(+3.889, −3.410)
${V}_{\mathrm{BO}}+{V}_{\mathrm{ad}}+{V}_{\mathrm{rel}}$	91.194	92.045	86.715(+3.889, −3.409)	86.726(+3.889, −3.410)
V	90.376	91.211	86.716(+3.889, −3.409)	86.728(+3.889, −3.409)
V + Vret	90.502	91.339	86.716(+3.888, −3.409)	86.728(+3.889, −3.409)
Hurly	88.323	89.119	86.715(+3.889, −3.409)	86.727(+3.889, −3.410)
HFDHe2	124.304	125.909	86.723(+3.889, −3.409)	86.735(+3.889, −3.410)
${}^{5}{{\rm{\Sigma }}}_{g}^{+}$
VU	76.893	78.287	75.116(18)	75.124(18)
V	75.456	76.787	75.115(18)	75.122(18)
VL	74.082	75.355	75.113(18)	75.121(18)
$V+\delta {V}_{\mathrm{ad}}$	73.252	74.491	75.116(19)	75.124(18)
$V+\delta {V}_{\mathrm{rel}}$	77.302	78.713	75.115(18)	75.122(18)
$V+\delta {V}_{\mathrm{ad}}+\delta {V}_{\mathrm{rel}}$	74.968	76.278	75.117(18)	75.124(18)
GLD	162.86	170.92	75.268(19)	75.276(19)

Note. If not stated otherwise the interaction potentials and notation are from [10, 11]. D₀ = 36.73 MHz [8] and 91.35 MHz [13] for the ground and excited state of ⁴He₂, respectively. Calc-1 and Calc-2: derived from the 'zero collision energy' wavefunctions using atomic (${m}_{\mathrm{atom}}=4.002\,603\,254\,13$ amu) and nuclear (m_nuc = 4.001 506 179 125 amu) masses, respectively; Calc-3 and Calc-4: evaluated using the 'a versus D₀' relations with atomic and nuclear masses, respectively. Hurly: obtained using the potential of [25]. HFDHe2: obtained using the potential of [26]. GLD: obtained using the potential of [27].

Table 2.  Average internuclear separations $\langle R\rangle$ of ⁴He₂ in its ground (${}^{1}{{\rm{\Sigma }}}_{g}^{+}$) and excited (${}^{5}{{\rm{\Sigma }}}_{g}^{+}$) electronic states (in Å).

Potential	Calc-1	Calc-2	Calc-3	Calc-4
${}^{1}{{\rm{\Sigma }}}_{g}^{+}$
VBO	45.802	46.195	45.237(+1.946, −1.706)	45.243(+1.946, −1.706)
${V}_{\mathrm{BO}}+{V}_{\mathrm{ad}}$	44.653	45.025	45.237(+1.946, −1.706)	45.243(+1.946, −1.706)
${V}_{\mathrm{BO}}+{V}_{\mathrm{rel}}$	48.793	49.244	45.234(+1.963, −1.706)	45.240(+1.946, −1.706)
${V}_{\mathrm{BO}}+{V}_{\mathrm{ad}}+{V}_{\mathrm{rel}}$	47.475	47.901	45.234(+1.946, −1.706)	45.239(+1.946, −1.706)
V	47.067	47.485	45.236(+1.946, −1.706)	45.243(+1.946, −1.706)
V + Vret	47.129	47.548	45.235(+1.946, −1.706)	45.241(+1.946, −1.706)
Hurly	46.040	46.438	45.235(+1.946, −1.706)	45.241(+1.946, −1.706)
HFDHe2	64.026	64.827	45.242(+1.946, −1.706)	45.247(+1.946, −1.706)
${}^{5}{{\rm{\Sigma }}}_{g}^{+}$
VU	49.289	50.012	48.366(10)	48.370(9)
V	48.542	49.234	48.365(9)	48.369(9)
VL	47.827	48.489	48.363(9)	48.367(9)
$V+\delta {V}_{\mathrm{ad}}$	47.397	48.041	48.367(9)	48.371(9)
$V+\delta {V}_{\mathrm{rel}}$	49.501	50.233	48.365(9)	48.369(9)
$V+\delta {V}_{\mathrm{ad}}+\delta {V}_{\mathrm{rel}}$	48.290	48.970	48.367(9)	48.371(9)
GLD	93.189	97.265	48.410(10)	48.405(9)

Note. If not stated otherwise the interaction potentials and notation are from [10, 11]. Calc-1 and Calc-2: derived from the 'zero collision energy' wavefunctions using atomic and nuclear masses, respectively; Calc-3 and Calc-4: evaluated using the 'a versus D₀' relations with atomic and nuclear masses, respectively. Hurly: obtained using the potential of [25]. HFDHe2: obtained using the potential of [26]. GLD: obtained using the potential of [27].

The 'universality' of the simple scaling defined by equation (1) is also deeply reflected in the fitting of the experimental Coulomb explosion data for the helium dimer wavefunction Ψ of [8]. In figure 2, the top panels illustrate the dispersion of the theoretical Ψ values evaluated using different interaction potentials; the middle panels show the agreement of the theoretical Ψ values obtained with potentials scaled to produce the experimental binding energy D₀ = 36.73 MHz, deduced from experimental data corrected for electron-recoil effects; the bottom panels reproduce the uncorrected experimental data using potentials scaled to provide the best least squares fittings (the effective binding energies corresponding to the best reproductions are around 46 MHz, giving insight into the role of electron-recoil effects).

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Generally speaking, the close agreement of the scaled Ψ values is also present in other halo states, for example, the spin-polarized helium atoms in the ${2}^{3}{S}_{1}$ metastable state (see figure 3). Importantly, as seen in the previous figures, the fitted wavefunctions closely coincide over the whole interval of internuclear separations, demonstrating the robustness of the scaling approach defined by equation (1) over the whole range of relevant binding energies and scattering lengths.

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In figure 4, we see that for small binding energies the presented scheme is in excellent agreement with the results obtained using quantum defect theory, which is usually employed in the literature [20, 21]. However, for higher binding energies associated with scattering lengths tending to $-\infty$, these approaches do not provide reliable asymptotes and thus fail to provide reliable mass sensitivities in these energy regions, particularly for scattering lengths exhibiting a nonlinear and discontinuous energy dependence. These regions appear to be particularly promising for probing the mass sensitivity of the scattering properties (see figure 5), however, as shown in table 3, the dimensionless scaling of the 'global' interaction potential should be adequate for the entire energy region. It should be stressed that only globally accurate potentials appear to be adequate for this critical region.

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Table 3.  The mass sensitivity coefficients K_β and T_β of ⁴He₂ in its ground (${}^{1}{{\rm{\Sigma }}}_{g}^{+}$) and excited (${}^{5}{{\rm{\Sigma }}}_{g}^{+}$) electronic states.

Potential	K1a	K1n	K2a	K2n	T1a	T1n	T2a	T2n
${}^{1}{{\rm{\Sigma }}}_{g}^{+}$	4He2
VBO	66.611	67.228	65.726	65.735	−32.35	−32.66	−31.91	−31.91
${V}_{\mathrm{BO}}+{V}_{\mathrm{ad}}$	64.809	65.393	65.725	65.734	−31.45	−31.75	−31.92	−31.92
${V}_{\mathrm{BO}}+{V}_{\mathrm{rel}}$	71.374	72.083	65.787	65.796	−34.73	−35.09	−31.94	−31.95
${V}_{\mathrm{BO}}+{V}_{\mathrm{ad}}+{V}_{\mathrm{rel}}$	69.305	69.974	65.787	65.796	−33.70	−34.03	−31.94	−31.95
V	68.637	69.292	65.763	65.773	−33.37	−33.69	−31.93	−31.94
V + Vret	68.744	69.402	65.771	65.780	−33.42	−33.75	−31.93	−31.94
Hurly	67.035	67.660	65.772	65.781	−32.57	−32.88	−31.94	−31.94
HFDHe2	94.936	96.193	65.532	65.541	−46.50	−47.13	−31.82	−31.82
${}^{5}{{\rm{\Sigma }}}_{g}^{+}$	4He2
VU	174.11	177.74	169.49	169.51	−64.69	−66.34	−62.63	−62.63
V	170.39	173.85	169.50	169.52	−63.04	−64.57	−62.63	−62.64
VL	166.83	170.14	169.51	169.53	−61.44	−62.93	−62.64	−62.65
$V+\delta {V}_{\mathrm{ad}}$	164.64	167.86	169.49	169.51	−60.46	−61.90	−62.62	−62.63
$V+\delta {V}_{\mathrm{rel}}$	175.19	178.86	169.50	169.52	−65.18	−66.87	−62.63	−62.64
$V+\delta {V}_{\mathrm{ad}}+\delta {V}_{\mathrm{rel}}$	169.10	172.51	169.49	169.51	−62.45	−63.99	−62.62	−62.63
GLD	398.06	419.13	169.24	169.26	−171.3	−181.7	−62.25	−62.25
HST	78.55	79.22	172.28	175.83	−26.39	−13.51	−70.22	−72.15
${}^{5}{{\rm{\Sigma }}}_{g}^{+}$	3He2
V	94.31	95.64	94.05	95.37	−31.88	−32.42	−31.78	−32.30
$V+\delta {V}_{\mathrm{ad}}$	92.60	93.88	94.04	95.37	−31.21	−31.72	−31.78	−32.30
$V+\delta {V}_{\mathrm{rel}}$	95.69	97.06	94.05	95.37	−32.43	−32.98	−31.78	−32.30
$V+\delta {V}_{\mathrm{ad}}+\delta {V}_{\mathrm{rel}}$	93.93	95.25	94.04	95.37	−31.73	−32.26	−31.78	−32.30
GLD	134.3	137.2	94.10	95.42	−48.79	−50.04	−31.80	−32.32
HST	65.92	66.54	90.82	92.05	−22.10	−22.31	−31.78	−32.31
${}^{5}{{\rm{\Sigma }}}_{g}^{+}$	3He4He
V	668.4	746.1	653.4	727.3	−306.3	−344.6	−298.8	−335.4
$V+\delta {V}_{\mathrm{ad}}$	579.9	636.8	653.2	727.0	−262.4	−290.6	−298.7	−335.2
$V+\delta {V}_{\mathrm{rel}}$	759.6	863.6	653.4	727.3	−351.6	−402.4	−298.8	−335.3
$V+\delta {V}_{\mathrm{ad}}+\delta {V}_{\mathrm{rel}}$	646.8	719.1	653.2	727.0	−295.6	−331.3	−298.7	−335.2
GLD	17.51	17.55	652.6	726.1	248.3	233.4	−305.8	−345.3
HST	127.3	129.5	492.8	532.9	−47.93	−49.02	−298.8	−335.5

Note. If not stated otherwise the interaction potentials and notation are from [10, 11]. ${D}_{0}=36.73\,\mathrm{MHz}$ [8] and 91.35 MHz [13] for the ground and excited state of ⁴He₂, respectively. K₁^a/T₁^a and K₁ⁿ/T₁ⁿ: derived from the 'zero collision energy' wavefunctions using atomic and nuclear masses, respectively; K₂^a/T₂^a and K₂ⁿ/T₂ⁿ: evaluated using 'a versus D₀' with atomic and nuclear masses, respectively. Hurly: obtained using the potential of [25]. HFDHe2: obtained using the potential of [26]. GLD: obtained using the potential of [27]. HST: calculations performed using the V_HST effective potential of [21] with r₀ = 3.8Å and C₆=3 276.68 a.u. [10]. In the case of ³He₂ and ³He⁴He, where D₀ is not available, V_HST was scaled so that it provides the same T₂^a as its ab initio counterpart V.

The scattering properties of diatomic halo states can be extracted from one-parameter only f-relationships. These relations were derived from numerically exact solutions of the radial Schrödinger equation for a set of effective potentials, obtained by a multiplicative scaling of the 'generic' interaction potential. The calculated scattering properties are as accurate as their unscaled f = 1 counterparts, which are derived from the best available interaction potentials. Furthermore, the predicted values appear somewhat independent of the underlying potential, suggesting that any moderately accurate potential can be utilized in the presented approach. As anticipated, the mass sensitivities of the scattering lengths and of the related properties grow significantly when increasing the classically forbidden part of the halo state wavefunction. For example, the sensitivities of the delocalized state of ⁴He³He are about one order of magnitude larger than those relating to a less delocalized state of ³He₂.

Most importantly, a single-parameter scaling based on equation (1) allows for the close fitting of the square of the experimental halo state wavefunction $| {\rm{\Psi }}{| }^{2}$, thus reflecting the universality of quantum diatomic halo states of diatomic systems (see, e.g. [30]). This universality means that all the parameters characterizing the low-energy scattering of atoms can be determined with high precision if only one of these characteristics is accurately known. Moreover, scattering properties and their mass sensitivities, which appear promising for investigating the stability of the electron-to-proton mass ratio β, can be fully determined. All one requires is a potential energy curve of moderate accuracy and an accurate experimental value for a pertinent scattering property. In principle, unlike the semiclassical [19] or quantum defect theories [32, 33] which can be viewed as the standard alternatives for deriving the 'a versus D₀' relations, the scaling described by equation (1) can be used straightforwardly for any type of long-range potential asymptote.

Currently, the best (spectroscopic) measurements of the scattering length can probe a variation of β at the level of 10⁻¹³–10⁻¹⁶ yr⁻¹ [6]. Given that the scattering phase shift δ₀ can be measured with a precision that yields scattering lengths with 1ppm accuracy [34] (see also [35]), one can expect to investigate temporal variations of β at the level of 10⁻¹⁵–10⁻¹⁸ yr⁻¹. The presented approach can cope with this accuracy by providing accurate mass sensitivities even in cases where a lack of spectral data prevents the use of standard approaches. However, it should be noted that the accuracy of the experimental setup will dictate future investigations of β, and large sensitivity coefficients may not always be transferable to experiment.

We thank Stefan Zeller and his coauthors for the Coulomb explosion data. This work was a part of the research project RVO:61388963 (IOCB) and was supported by the Czech Science Foundation (grant P209/18-00918S). AO gratefully acknowledges a fellowship from the Alexander von Humboldt Foundation and support from the Deutsche Forschungsgemeinschaft (DFG) through the excellence cluster 'The Hamburg Center for Ultrafast Imaging—Structure, Dynamics and Control of Matter at the Atomic Scale' (CUI, EXC1074).