Quantum interference shifts in laser spectroscopy with elliptical polarization
We investigate the quantum interference shifts between energetically close states, where the state structure is observed by laser spectroscopy. We report a compact and analytical expression that models the quantum interference induced shift for any admixture of circular polarization of the incident laser and angle of observation. An experimental scenario free of quantum interference can thus be predicted with this formula. Although, this study is exemplified here for muonic deuterium, it can be applied to any other laser spectroscopy measurement of frequencies of a nonrelativistic atomic system, via a scheme.
pacs:32.70.Jz, 36.10.Ee, 32.10.Fn, 32.80.Wr
As it is pointed by Low Low (1952), a spectral line profile can only be described by a conventional Lorentzian profile up to certain limit of accuracy. Beyond this limit, known as the resonant approximation, the full quantum interference between the main resonant channels and other non-resonant channels makes the spectral lines asymmetric. Consequently, if Lorentzian functions are employed to fit the distorted profiles, a mismatch between the obtained centroid frequency and the actual line frequency occurs Labzowsky et al. (1994, 2001, 2009); Brown et al. (2013). These quantum-interference (QI) induced shifts are specific for a particular measurement and its quantification is mandatory for all high-precision spectroscopy experiments aimed for a resolution beyond the resonant approximation. A known example is the case of the transition in hydrogen, which prompt many QI theoretical studies applied to various experimental methods, namely continuous-wave photon scattering Labzowsky et al. (2001); Jentschura and Mohr (2002), two-photon excitation Labzowsky et al. (2007, 2009) and direct two-photon frequency-comb spectroscopy Yost et al. (2014). Other atomic systems that have been considered include the helium fine structure and lithium hyperfine structure, where it is shown that the negligence of the QI effects are the cause of many discrepant measurements Horbatsch and Hessels (2010); Marsman et al. (2012, 2012, 2014, 2015, 2015); Sansonetti et al. (2011); Brown et al. (2013). Recently, the geometric and polarization properties of the QI shifts were investigated in laser spectroscopy, both experimentally and theoretically Sansonetti et al. (2011); Brown et al. (2013), and was found that the QI shifts vanish for a particular angle of linear polarization, so-called “magic angle”. This result has been recently applied to minimize QI shifts in laser spectroscopy of hydrogen Beyer et al. (2015).
The aim of this article is twofold: First, we here continue and conclude the investigation of the QI shifts in laser spectroscopy of muonic atoms Amaro et al. (2015). We confirm that the conclusions of Amaro et al. (2015), namely that the line centers are not affected on a relevant level by QI effects in muonic atoms, holds true even for an hypothetical admixture of circular polarized light. Second, we extend the theoretical description developed for linear polarized photons in Ref. Brown et al. (2013), to elliptical polarized photons. A compact and analytical formula that models the QI shifts for any angle of observation, angle and degree of circular polarization is presented here that forthcoming laser experiments might benefit from.
Laser spectroscopy is often modeled by the physical process of resonant photon scattering Brown et al. (2013); Beyer et al. (2015); Amaro et al. (2015). Here, we quantify the QI effects involved in the precise determination of the transition frequencies, by exciting the transition, and detecting the florescence decay. The overall process to be considered is thus photon scattering. Following the second-order theory of Kramers-Heisenberg Loudon (2000), the differential scattering cross section of photon scattering from an initial state to a final state is given in atomic units by
where and are the initial and final total angular momenta, and and the respective projection along the quantization axis. The second-order amplitude involves a summation over the entire atomic spectrum Safari et al. (2012), which in the near-resonant region comprises only the intermediate states. In the dipole and rotating-wave approximation, it is given by
with being the transition frequencies between and , is the linewidth that is assumed to be independent of the hyperfine state , is the linear momentum, and are the polarization vectors of the incoming and scattered photon, respectively (see below), and is the fine structure constant. Energy conservation sets between the incident () and scattered () frequencies and initial () and final () atomic state frequencies.
The angular distribution of the scattered photon is given by the polar angle , included in the scattering plane defined by both photon momenta ( and ), as illustrated in Fig. 1. We consider the experimental scenario of the second photon’s polarization not being detected, which is often the case in laser spectroscopy experiments Brown et al. (2013); Pohl et al. (2010); Antognini et al. (2013). Following the procedure of Istomin et al. Istomin et al. (2006), we parametrize the incident elliptical polarized photons as . As shown in Fig. 1, is defined with an angle relative to the scattering plane, and . Circular admixture is often quantified by using the degree of circular polarization that is defined by the difference between left and right spherical amplitudes ( and ) of the incident polarization, normalized to the total amplitude (). By using the previous parametrization of , it is related with the admixture parameter () by
By using standard angular algebra Rose (1957), Eq. (1) can be further rearranged in a suitable form for studying the QI shifts in terms of Lorentzian terms and cross-terms Amaro et al. (2015); Brown et al. (2013). The result is given by,
with the quantities defined by
having all the geometrical and polarization dependencies in terms of
Here, is the nuclear spin, contains all radial integrals and
The cross terms (angular momentum quantities are omitted for shortness) contain all interference between neighboring resonances, and if they are zero, then Eq. (4) is reduced to a sum of independent Lorentzian components. As demonstrated by Brown et al. Brown et al. (2013) for the case of linear incident polarized photons (), both theoretically and experimentally, these cross-terms can be parametrized as (angles defined in our geometry Amaro et al. (2015)), where is the second order Legendre polynomial. The coefficient depends on the angular momenta of the states participating in the transition. Therefore, there are particular combinations of and , where QI effects vanish that can be obtained by solving . For the case of , the angle is referred as “magic angle” in the literature Brown et al. (2013).
In order to investigate the role of elliptical polarization on the shifts, and as a continuation of previous investigation Amaro et al. (2015), we choose the resonance with the largest induced shift in muonic deuterium, which is the resonance . Following the procedure in Ref. Amaro et al. (2015), we evaluate the by fitting a simulated spectrum of Eq. (4), that would be observed by a pointlike detector with a sum of Lorentzian profiles. Figure 2 displays the computed in units of the linewidth for all values of , and for three cases of (a), (b) and (c).
As can be evinced in Fig 2(b), is proportional to for linear polarized photons (), as mentioned in Ref. Brown et al. (2013). Consequently, for the angle of polarization . Additionally, the points at \degree() and () with represents the QI shifts listed in Ref. Amaro et al. (2015) (% and %).
As can be observed in Fig. 2(a), for and , there is an additional “magic angle” of observation where vanishes. On the other hand, for the case of and represented in Fig. 2(c), is independent of . This is expected since the dipole pattern of the differential cross section depends only on the angle between polarization and scattered direction, which for \degree is independent of .
Moreover, Fig. 2(b) shows that the contribution of circular admixture to is bounded by the values of and at . Thus, any possible circular admixture reduces the QI contribution relative to the linear case and a point-like detector.
For circular polarized photons (), is independent of (see Fig. 2(b)) since the differential cross section depends only on through the -projection of in the scattering plane, given by , that for is constant. The value of in this setting is the same as at \degree and any or , following the same reasoning.
The symmetry between helicities , displayed in Fig. 2, is a consequence of not considering the scattered polarization and the final magnetic sub-level structure in the measurement scheme.
The laser system employed by the Charge Radius Experiment with Muonic Atoms (CREMA) collaboration was designed for linear polarization Antognini et al. (2005, 2009), but some small admixture of 10% of circular polarization cannot be excluded. Thus, it is worthwhile to evaluate the QI shift with this circular admixture for the CREMA geometry setup , following similar steps as performed in Ref. Amaro et al. (2015). The obtained value of % for can be compared with the value of % Amaro et al. (2015) for (linear polarization). This shift of 0.3% of the linewidth sets a maximum threshold of for all resonances of the muonic atoms considered. Thus, even in the remote case of the laser having a small amount of circular polarization, QI shifts can be neglected for the present experimental resolution of to date measured muonic transitions Pohl et al. (2010); Antognini et al. (2013).
The scattering process considered here is of dipolar type, which is characterized by an angular dependency of the form Loudon (2000). Following the formula of , this dipole angular distribution can always be rewritten as . Thus, the cross-terms can also be expressed as . The analytical forms of and , obtained after evaluation of Eqs. (LABEL:eq:Xi)-(7), can be further rearranged in order to include the and dependencies in . This is accomplish with the help of and with Eq. (3). We found, after this procedure, that the cross-terms have a compact and analytical expression for the angular and polarization properties, which is given by
The coefficient contains the information of the angular quantum numbers involved in a particular transition. The respective values for many resonances in muonic atoms are listed in Ref. Amaro et al. (2015). Equation (8) models the angular and polarization dependency of for any transition in an atomic system, under the premise of nonrelativistic and dipole approximation frameworks. We can thus use Eq. (8) to predict regions of the “magic values” , where , by solving equal to zero. This can be used to design accordingly a spectroscopy experiment insensitive to line pulling effects. For example, the blue contour with in Fig. 2(b), that was computed numerically, is approximately equal to , which for gives . For circular polarization , a quick inspection of Eq. (8) shows that occurs for the angle of observation \degree, as also observed in Figs. 2(a) and 2(c).
Apart from immediate application in laser spectroscopy of atomic systems, Eq. (8) might also be applied to molecular physics and chemistry, where line mixing occurs due to interference of neighborhood molecular states. Without further observation of the internal structure of the target, the dipole pattern of photon scattering is quite general and independent of the target being an atom or a molecule Thirunamachandran (1984). Essentially, the angular and polarization dependency of interference shifts included Eq. (8) might be extended to molecular techniques based on photon scattering, such as resonant x-ray emission spectroscopy (XES) Luo et al. (1996); Horikawa et al. (2010), Raman spectroscopy Lu et al. (2011); Duque et al. (2012), and laser spectroscopy Berman et al. (1997); Mudrich et al. (2008); Goto et al. (2011), where interference effects or line mixing might play a significant role.
In summary, we investigated the contribution of an admixture of circular polarization to the QI shift, by considering incident elliptical-polarized photons. Calculations performed for the CREMA detector setup revealed a negligible impact of QI effects for the maximum expected admixture of circular polarization.
We presented a compact and analytical expression that models the dependency of the angular and polarization properties to the QI shift. As a generalization of a similar expression for linear polarization Brown et al. (2013), this one contains the degree of circular polarization. Although we considered here a particular resonance of muonic deuterium, as an illustrative example, this expression can be applied to any transition in any nonrelativistic atomic system. Thus, this equation can be used to design a spectroscopy apparatus to measure frequencies in a scheme free of quantum interference shifts by optimizing the detector geometry, the laser polarization and the laser direction.
Acknowledgements.This research was supported in part by Fundação para a Ciência e a Tecnologia (FCT), Portugal, through the projects No. PEstOE/FIS/UI0303/2011 and PTDC/FIS/117606/2010, financed by the European Community Fund FEDER through the COMPETE. P. A. acknowledges the support of the FCT, under Contract No. SFRH/BPD/92329/2013. R. P. acknowledges the support from the European Research Council (ERC) through StG. #279765. F. F. acknowledges support by the Austrian Science Fund (FWF) through the START grant Y 591-N16. L. S. acknowledges financial support from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement No. . A. A acknowledges the support of the Swiss National Science Foundation Projects No. 200021L_138175 and No. 200020_159755.
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