Magic wavelengths for the transition in helium
We have calculated ac polarizabilities of the and states of both He and He in the range 318 nm to 2.5 m and determined the magic wavelengths at which these polarizabilities are equal for either isotope. The calculations, only based on available ab initio tables of level energies and Einstein A coefficients, do not require advanced theoretical techniques. The polarizability contribution of the continuum is calculated using a simple extrapolation beyond the ionization limit, yet the results agree to better than with such advanced techniques. Several promising magic wavelengths are identified around 320 nm with sufficient accuracy to design an appropriate laser system. The extension of the calculations to He is complicated due to the additional hyperfine structure, but we show that the magic wavelength candidates around 320 nm are predominantly shifted by the isotope shift.
pacs:31.15.ap, 32.30.-r, 37.10.Jk, 42.62.Fi
In recent years a growing number of experimental tests of QED in atomic physics have surpassed the accuracy of theory, allowing new determinations of fundamental constants. High-precision spectroscopy in atomic hydrogen has been achieved with sufficient accuracy to allow a determination of the proton size from QED calculations P.J. Mohr et al. (2012), and spectroscopy in muonic hydrogen has allowed an even more accurate determination R. Pohl et al. (2010); A. Antognini et al. (2013). Interestingly, the muonic hydrogen result currently differs by from the proton size determined by hydrogen spectroscopy and electron-proton collision experiments. So far there has not been a satisfying explanation for this discrepancy, which is aptly named the proton radius puzzle R. Pohl et al. (2013). Research in this field has expanded to measurements in muonic helium ions, a hydrogenic system which has a different nuclear charge radius T. Nebel et al. (2012). As this work is done for both naturally occurring isotopes of helium (He and He), the absolute charge radii of the -particle and the helion may be determined at an aimed relative precision of (0.5 attometer), providing a very interesting testing ground for both QED and few-body nuclear physics.
Parallel to these developments, high-precision spectroscopy in neutral helium has become an additional contribution to this field in recent years. Although QED calculations for three-body systems are not as accurate as for hydrogen(ic) systems, mass-independent uncertainties cancel when considering the isotope shift G.W.F. Drake et al. (2005); D.C. Morton et al. (2006a). Therefore isotope-shift measurements in neutral helium can provide a crucial comparison of the nuclear charge radius difference determined in the muonic helium ion and planned electronic helium ion measurements.
High-precision spectroscopy in helium is a well-established field, and transitions ranging from wavelengths of 51 nm to 2058 nm P. Cancio Pastor et al. (2004); T. Zelevinsky et al. (2005); E.E. Eyler et al. (2008); J.S. Borbely et al. (2009); D.Z. Kandula et al. (2010); M. Smiciklas and D. Shiner (2010); R. van Rooij et al. (2011); P. Cancio Pastor et al. (2012); P.-L. Luo et al. (2013a); *Luo1_erratum; P.-L. Luo et al. (2013c); R.P.M.J.W. Notermans and W. Vassen (2014) have been measured in recent years both from the ground state and from several (metastable) excited states. Only two transitions have been measured in both helium isotopes with sufficient precision for accurate nuclear charge radius difference determinations. The transition at 1083 nm P. Cancio Pastor et al. (2012) and the doubly-forbidden transition at 1557 nm K.A.H. van Leeuwen and W. Vassen (2006); R. van Rooij et al. (2011) are measured at accuracies exceeding , providing an extracted nuclear charge radius difference with 0.3% and 1.1% precision, respectively. Interestingly, the determined nuclear charge radius differences from both experiments currently disagree by P. Cancio Pastor et al. (2012).
In order to determine the nuclear charge radius difference with a precision comparable to the muonic helium ion goal, we aim to measure the transition with sub-kHz precision. One major improvement to be implemented is the elimination of the ac Stark shift induced by the optical dipole trap (ODT) in which the transition is measured. Many high-precision measurements involving optical (lattice) traps solve this problem by implementation of a so-called magic wavelength trap N. Poli et al. (2013); A.D. Ludlow et al. (2014). In a magic wavelength trap the wavelength is chosen such that the ac polarizabilities of both the initial and final states of the measured transition are equal, thereby cancelling the differential ac Stark shift.
In this paper we calculate the wavelength-dependent (ac) polarizabilities of both metastable (lifetime s) and (lifetime ms) states and identify wavelengths at which both are equal for either He or He. Generally one will find multiple magic wavelengths over a broad wavelength range, but our goal is to identify the most useful magic wavelength for our experiment. Currently R. van Rooij et al. (2011); R.P.M.J.W. Notermans and W. Vassen (2014) we employ a 1557 nm ODT at a power of a few 100 mW, providing a trap depth of a few K and a scattering lifetime of 100 s (the actual lifetime in the trap is limited to 10’s of seconds due to background collisions). A good overview on calculating trap depths and scattering rates in ODTs is given in R. Grimm et al. (2000), and the specific calculations for our ODT are discussed in the Appendix. For our future magic wavelength trap we need to produce a similar trap depth with sufficient laser power at that wavelength. Furthermore, the scattering rate should be low enough to have a lifetime of at least a few seconds, providing enough time to excite the atoms with a 1557-nm laser.
The purpose of this paper is to show that it is possible to calculate magic wavelengths with sufficient accuracy to design an appropriate laser system solely based on ab initio level energies and Einstein A coefficients without having to resort to advanced theoretical techniques Z.-C. Yan and J.F. Babb (1998); J. Mitroy et al. (2010). Based on the calculations reported here, we are currently building a laser system at 319.82 nm with a tuning range of 300 GHz based on similar designs A.C. Wilson et al. (2011); H.-Y. Lo et al. (2014).
The polarizabilities for the and states of He are presented over a wavelength range from 318 nm to 2.5 m. In this range all magic wavelengths including estimated required ODT powers and corresponding trap lifetimes are calculated. From these results we identify our best candidate for a magic wavelength trap. A lot of work, both theoretical and experimental, has been done for the dc polarizability of the and states (see Table 1 for an overview). Therefore these are used as a benchmark for our calculations by also calculating the polarizabilities in the dc limit , as discussed in Sec. IV. Calculations of the ac polarizability of the and states J. Mitroy and L.-Y. Tang (2013); M.-K. Chen (1995) states allows for comparison of the polarizability calculations at finite wavelengths.
Finally we present a simple extension to He which has a hyperfine structure that needs to be taken into account. Although different theoretical challenges arise due to the hyperfine interaction, we can get an estimation of the He magic wavelength candidates and show that they are equal to the He results approximately shifted by the hyperfine and isotope shift.
Ii Theory for He
For an atomic state with angular momentum J and magnetic projection , the polarizability induced by an electromagnetic wave with polarization state () and angular frequency due to a single opposite parity state is Sobel’man (1972)
Here is the transition frequency and the Einstein A coefficient of the transition. The term between two brackets represents the symbol of the transition. The total polarizability is given by a sum over all opposite-parity states as
In a general way the polarizability can be written as the sum of a scalar polarizability, independent of , and a tensorial part describing the splitting of the levels J. Mitroy et al. (2010); J. Mitroy and M.W.J. Bromley (2004). Within the coupling scheme the tensor polarizability of the and states in He is zero and the polarizability is defined by averaging over all states and therefore independent of . As our experimental work specifically concerns the spin-stretched state R. van Rooij et al. (2011); R.P.M.J.W. Notermans and W. Vassen (2014), Eqns. 1 and 2 are used to calculate the polarizability for the state assuming linearly polarized light (). For He the calculations specifically concern the spin-stretched and states.
The higher-order contribution to the Stark shift, the hyperpolarizability, is estimated using calculations of a similar system M. Takamoto et al. (2009). The contribution is many orders of magnitude smaller than the accuracy of our calculations and therefore neglected.
The summation in Eqn. 2 can be explicitly calculated for transitions up to , as accurate ab initio energy level data and Einstein A coefficients are available G.W.F. Drake and D.C. Morton (2007). Extrapolation of both the energy levels and the Einstein A coefficients is required to calculate contributions of dipole transition matrix elements with states beyond . A straightforward quantum defect extrapolation can be used to determine the energies using the effective quantum number G.W.F. Drake and R.A. Swainson (1991):
where are fit parameters and the quantity is commonly referred to as the quantum defect. For both the singlet and triplet series, Eqn. 3 is used to fit the literature data up to and to extrapolate to arbitrary . This method is tested using a dataset provided by Drake G.W.F. Drake and R.A. Swainson (1991).
Extrapolation of the Einstein A coefficients is more complicated as there is no relation such as Eqn. 3 for Einstein A coefficients. Furthermore, the sum-over-states method does not provide straightforward extrapolation beyond the ionization limit, as the energy levels converge to the ionization limit for . Both problems can be solved by calculating the polarizability contribution of a single transition (or ) as given in Eqn. 2 and defining the polarizability density per upper state energy interval as
which is evaluated at . and are the energies of the neighbouring upper states with the same value of . The energies are given by the Rydberg formula , where is the ionization potential of the ground state. For ease of notation we have omitted all the dependent variables of as defined in Eqn. 1. The polarizability density is a function of energy and can not only be used to calculate the polarizability contribution from dipole transition matrix elements to highly excited (Rydberg) states, but additionally allows extrapolation beyond the ionization potential. Using the Rydberg formula, the polarizability density becomes
where we have made the approximation that is constant for increasing . This approximation already works better than for . In the limit , the polarizability contribution per energy interval can be written as
where we define
As there is no exact analytical model for as function of energy, the method of extrapolation is based on a simple low-order polynomial fit of the as function of for the levels. The result is a function that is used to extrapolate to arbitrary upper states and calculate the corresponding polarizability contributions. This method can be used to calculate the finite polarizability contributions of all Rydberg states for . As the general behaviour of the Einstein A coefficients is proportional to for the Rydberg states, will have a finite value at the ionization potential indicating that contributions from the continuum have to be taken into account as well. As the extrapolation is a function of energy, it is extended beyond the ionization potential to calculate additional continuum contributions to the polarizability. This omits all higher order effects such as resonances to doubly-excited states or two-photon excitations into the continuum, and it should be considered as an approximation of the continuum.
For a large enough quantum number , the discrete sum-over-states method smoothly continues as an integration-over-states method following Eqn. 6. The ionization potential serves as a natural choice as the energy at which the calculation would switch from the discrete sum to the integration method. But even for large enough there is a negligible numerical error in varying the exact cutoff energy at which we switch between these methods. The calculation of the total polarizability is therefore performed using the sum-over-states method to an arbitrary cutoff at and continued with an integration over the remaining states as
where is the energy of the corresponding state. A low-order polynomial fit of Eqn. 7 is used to calculate such that the integral of Eqn. 8 provides an analytical solution. The total polarizability is therefore easily calculated as a sum-over-states part and an analytical expression
Iii Numerical uncertainties
In this section we discuss the sources of any numerical errors in our calculations, which are purely based on the technical execution of our method. The accuracy of our calculations due to our estimation of the continuum contribution will be discussed in Sec. IV where our results are compared to other calculations.
The numerical convergence of Eqn. 9 is tested by varying . The polarizability converges as and even for the polarizability is within a fraction of the polarizability calculated using . The computation of Eqn. 9 therefore poses no numerical problems.
A more crucial matter is the fact that our calculations are based on two extrapolations: that of the level energies and the Einstein A coefficients. For the levels in helium the ab initio calculations of the level energies and Einstein A coefficients are used G.W.F. Drake and D.C. Morton (2007). The higher level energies are extrapolated using Eqn. 3 and include up to fifth order () contributions. Variation of the total number of orders () or using a different dataset (such as the NIST database A. Kramida et al. (2014) as used in other recent work J. Mitroy and L.-Y. Tang (2013)) affects the polarizabilities at the level and is negligible.
The limiting factor in the accuracy of the calculations is the choice of extrapolation of the Einstein A coefficients through extrapolation of . As mentioned before, no advanced methods are used to calculate transtion matrix elements to higher states or doubly excited states in the continuum. The heuristic approach we use instead, is to choose an extrapolation function that is smooth, continuous and provides a convergent integral in Eqn. 8. A number of different functions have been tried which provide a similar quality of the fit, and their effect on the calculation of the continuum contribution can lead to a polarizability shift which is a significant fraction of the continuum contribution itself. In our calculations this is the limiting factor in the accuracy of the calculated magic wavelengths. A second order polynomial function is chosen to extrapolate as it has the additional advantage of providing an analytical solution of the continuum contributions.
In order to discuss the absolute accuracy of the calculations, we first present our polarizabilities calculated in the dc limit as a lot of literature is available for these calculations. After comparison with the dc polarizabilities in Sec. IV.1, the ac polarizabilities are given in Sec. IV.2 including the magic wavelengths at which they are equal for the and states. Experimental characteristics, such as the required trapping power and scattering lifetime at the magic wavelengths, are estimated in order to discuss which magic wavelength candidate is most suitable for our experiment. In Sec. IV.3 the tune-out wavelength (where the polarizability is zero) of the state near 414 nm is compared to the result calculated by Mitroy and Tang J. Mitroy and L.-Y. Tang (2013).
iv.1 dc polarizabilities
An overview of previously calculated and measured dc polarizabilities for the and states of He is given in Table 1 together with our results. For convenience the polarizabilities are given in atomic units ( is the Bohr radius), but they can be converted to SI units through multiplication by JVm. Furthermore, the dc polarizabilities are calculated using the common convention of averaging over all states and all possible polarizations J. Mitroy et al. (2010).
There is general agreement between our results and previously calculated dc polarizabilities, but comparison with the work of Yan and Babb Z.-C. Yan and J.F. Babb (1998), which provides the most accurate calculated dc polarizabilities to date, shows that both our and dc polarizabilities are slightly larger ( and , respectively). The difference is comparable to the uncertainty in the calculated continuum contributions as discussed in Sec. III, and we conclude that our absolute accuracy is indeed limited by the exact calculation of the continuum contributions. It should be noted that the continuum contributions in the dc limit are and , respectively. This only contributes to the total polarizability in contrast to e.g. ground-state hydrogen for which the continuum contribution is 20% of the total polarizability L. Castillejo et al. (1960).
|Crosby and Zorn (1977) [Experiment]||D.A. Crosby and J.C. Zorn (1977)|
|Ekstrom et al. (1995) [Experiment]||C.R. Ekstrom et al. (1995); R.W. Molof et al. (1974)|
|Chung and Hurst (1966)||K.T. Chung and R.P. Hurst (1966)|
|Drake (1972)||G.W.F. Drake (1972)|
|Chung (1977)||K.T. Chung (1977)|
|Glover and Weinhold (1977)||R.M. Glover and F. Weinhold (1977)|
|Lamm and Szabo (1980)||G. Lamm and A. Szabo (1980)|
|Bishop and Pipin (1993)||D.M. Bishop and J. Pipin (1993)|
|Rérat et al. (1993)||M. Rérat et al. (1993)|
|Chen (1995)||M.-K. Chen (1995)|
|Chen and Chung (1996), B Spline||M.-K. Chen and K.T. Chung (1996)|
|Chen and Chung (1996), Slater||M.-K. Chen and K.T. Chung (1996)|
|Yan and Babb (1998)||Z.-C. Yan and J.F. Babb (1998)|
|Mitroy and Tang (2013), hybrid||J. Mitroy and L.-Y. Tang (2013)|
|Mitroy and Tang (2013), CPM||J. Mitroy and L.-Y. Tang (2013)|
|Laser power [W]||Lifetime [s]||Nearest transition|
iv.2 Magic wavelengths
We have calculated the ac polarizabilities of the and states in the range of 318 nm to 2.5 m and an overview of the identified magic wavelengths is shown in Table 2. The slope of the differential polarizability is also given in order to estimate the sensitivity of the determined magic wavelength due to the accuracy of the calculated polarizabilities. Table 2 furthermore provides the trapping beam power required to produce a trap depth of K and the corresponding scattering lifetime (see the Appendix) to indicate the experimental feasibility of each magic wavelength.
The magic wavelengths in the range 318-327 nm, as shown in Fig. 1, are mainly due to the many resonances in the singlet series. The most promising magic wavelength for application in the experiment is at 319.815 nm, as the polarizability is large enough to provide sufficient trap depth at reasonable laser powers while the estimated scattering lifetime is still acceptable (see Table 2).
The magic wavelengths at 318.611 nm and 326.672 nm are not useful for our experiment as the absolute polarizability is negative and therefore a focused laser beam does not provide a trapping potential. There are more magic wavelengths for nm, but the polarizability of the state will stay negative until the ionization wavelength of the state around 312 nm. In the range 327-420 nm, shown in Fig. 2, there are four more magic wavelengths. The magic wavelength at 411.863 nm, previously predicted with nm accuracy E.E. Eyler et al. (2008), is the only one in this region with a small yet positive polarizability (see inset in Fig. 2). There are no more magic wavelengths in the range 420 nm-2.5 m, which is shown in Fig. 3, and the polarizabilities converge to the dc polarizabilities for m.
The ac polarizability of the state can be compared to previous polarizability calculations from dc to 506 nm M.-K. Chen (1995). Combined with the dc polarizability comparison and the tune-out wavelength result for the state, as discussed in the Sect. IV.3, we find that the accuracy of our calculations is limited by the exact calculation of the continuum contributions. We note that around 320 nm the absolute continuum contributions ( and for the and states, respectively) and the corresponding uncertainty have increased, as the shorter wavelengths are closer to the ionization limit at 312 nm. The uncertainty in the absolute value of the polarizabilities translates to an uncertainty in the absolute value of the magic wavelength through the slope of the differential polarizability at the zero crossing. For the magic wavelength at 319.815 nm this gives a frequency uncertainty of 10 GHz (0.003 nm), yet for the magic wavelength near 412 nm the uncertainty is approximately 1 nm due to the very small slope at the zero crossing. However, the latter magic wavelength is not suitable for our experiment as the absolute polarizability is very small.
iv.3 Tune-out wavelength of the state
The zero crossings of the absolute polarizability of a single state occur at so-called tune-out wavelengths. Mitroy and Tang calculated several tune-out wavelengths for the state J. Mitroy and L.-Y. Tang (2013), of which the candidate at 413.02 nm is the most sensitive to the absolute value of the polarizability due to a very small slope at the zero crossing. We find this tune-out wavelength at 414.197 nm (see inset in Fig. 2), which is considerably larger. However, the slope of the polarizability at the zero crossing can be used to calculate that the difference in tune-out wavelength is equivalent to a difference in the calculated absolute polarizabilities. Comparison of the calculated dc polarizabilities (see Table 1) shows a similar difference, so within a constant offset of the absolute polarizability our tune-out wavelength is in agreement with Mitroy and Tang’s result.
V Extension to He
The transition is also measured in He in order to determine the isotope shift of the transition frequency R. van Rooij et al. (2011). Hence a magic wavelength trap for He will be required as well. As He has a nuclear spin (), the measured hyperfine transition is and the magic wavelengths need to be calculated for these two spin-stretched states.
The mass-dependent (isotope) shift of the energy levels is taken into account by using He energy level data D.C. Morton et al. (2006b) and recalculating the quantum defects using Eqn. 3. The Einstein A coefficients of the transitions also change due to the different reduced mass of the system G.W.F. Drake and D.C. Morton (2007), but this effect is negligible compared to the accuracy of the calculations. In total, the mass-dependent shift of the magic wavelengths is dominated by the shift of the nearest transitions and is approximately -45 GHz.
The fine-structure splitting decreases as whereas the hyperfine splitting converges to a constant value for increasing W. Vassen and W. Hogervorst (1989). In this regime the coupling scheme is not the best coupling scheme because is no longer a good quantum number. Instead an alternative coupling scheme is used which first couples the nuclear spin quantum number and total electron spin to a new quantum number I.A. Sulai et al. (2008). This new quantum number then couples to to form the total angular momentum . In this coupling scheme the transition strengths can be calculated with better precision compared to the coupling scheme, and can be applied for states with . Although this coupling scheme does not work perfectly for (which in any case is far-detuned from the magic wavelengths), it provides an estimate of the transition strengths that is sufficiently accurate for our purposes.
For increasing , the strong nuclear spin interaction with the electron becomes comparable with the exchange interaction between the and electrons W. Vassen and W. Hogervorst (1989). This leads to mixing of the singlet and triplet states as the total electron spin is no longer a good quantum number. The solution requires exact diagonalization of the Rydberg states, which provides the singlet-triplet mixing and the energy shifts of the states. The mixing parameter is then used to correct the Einstein A coefficients and the energies of the states. Although this is implemented in the calculations, these corrections lead to shifts in the magic wavelengths that are below the absolute accuracy of the calculations.
Due to the two hyperfine states of He in the ground state, there are two Rydberg series in the He atom. For even higher than discussed before, this leads to mixing of Rydberg states with different W. Vassen and W. Hogervorst (1989). The resulting shifts in the polarizabilities are well below the accuracy of the calculations and are therefore neglected.
Using the aforementioned adaptations, the polarizability of the and states can be calculated using Eqn. 1, but with substituted quantum numbers , Einstein A coefficients and transition frequencies. The numerical calculation of the polarizabilities and discussion of the numerical accuracies is similar to the He case. An additional uncertainty of is added in the calculation of the polarizabilities of the He states based on a conservative estimate of the shifts caused by the hyperfine interaction. It should be noted that the states of interest, and , both have angular momentum and both are in the fully spin-stretched state. Therefore neither He nor He has a tensor polarizability for the states discussed in this paper.
A comparison between the He and He magic wavelengths is presented in Table 3. Magic wavelengths up to 330 nm are all shifted by the isotope shift with small corrections due the abovementioned effects. The frequency difference between the two isotopes (third column of Table 3) grows with increasing wavelengths because decreases and the results become more sensitive to the absolute accuracy () of the calculations, as can be seen from the growing uncertainties associated with the shifts. The isotope shifts for magic wavelengths with nm have been omitted in Table 3 as they are not useful due to the large relative uncertainty.
The difference of the magic wavelengths between the two isotopes is well within the tuning range of our designed laser system near 320 nm. Furthermore there is no significant change in the absolute polarizability or the slope at the magic wavelengths. This means that an ODT at these wavelengths has a comparable performance for either isotope.
We have calculated the dc and ac polarizabilities of the and states for both He and He in the wavelength range of 318 nm to m and determined the magic wavelengths at which these polarizabilities are equal for either isotope. The accuracy of our simple method is limited by the extrapolation of the polarizability contributions of the continuum states. This is less than achievable through more sophisticated methods which calculate the transition matrix elements explicitly. However, the purpose of this paper is to show that using a simple extrapolation method it is possible to achieve an accuracy on the order of 10 GHz for the magic wavelengths that are of experimental interest, which is required to design an appropriate laser system for the required wavelengths.
Most experimentally feasible magic wavelength candidates are in the range of 319-324 nm, as the absolute polarizability of the state in this range is positive and large enough to create reasonable (K) trap depths in a crossed-beam ODT with a few Watts of laser power. The estimated scattering rates at these wavelengths and intensities are low enough to perform spectroscopy on the doubly-forbidden transition.
The calculations are extended to also calculate magic wavelengths in He. Although the hyperfine structure, which is absent in He, leads to complications in the calculation of the polarizabilities, these effects are very limited for the and states. The magic wavelengths of interest, around 320 nm, are shifted relative to the He magic wavelengths by predominantly the isotope shift.
Acknowledgements.This work is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is financially supported by the Netherlands Organisation for Scientific Research (NWO).
Appendix A a crossed-beam optical dipole trap
An overview of optical dipole traps (ODTs) and the equations used in this Appendix can be found in R. Grimm et al. (2000). The depth of a crossed-beam ODT, as currently used in our experiment R. van Rooij et al. (2011); R.P.M.J.W. Notermans and W. Vassen (2014), is
where is the polarizability of the state, the power of the incident trapping laser beam and the beam waist. In our experiment, the first ODT beam is reused by refocusing it through the original focus (m) at an angle of with respect to the original beam. At the currently used ODT wavelength of 1557 nm the polarizability is (see Table 2) which gives a trap depth of approximately K at an ODT beam power of mW. In Table 2 we used Eqn. 10 to calculate the trapping power at the different magic wavelengths corresponding to a trap depth of K to indicate the required beam power that should be produced at that magic wavelength.
As a good approximation of the lifetime of the atoms in the ODT due to scattering, one can take the nearest transition into account to calculate the corresponding scattering rate. The scattering rate is
where is the total intensity of the light, the angular frequency of the trapping light and and the transition frequency and linewidth (all in ). The nearest transitions are given in Table 2, and the lifetime is given for each magic wavelength trap using the required trapping beam power calculated to provide a K deep trap.
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