Magnetic sublevel independent magic wavelengths: Application in the Rb and Cs atoms
A generic scheme to trap atoms at the magic wavelengths (s) that are independent of vector and tensor components of the interactions of the atoms with the external electric field is presented. The s for the laser cooling D2 lines in the Rb and Cs atoms are demonstrated and their corresponding polarizability values without vector and tensor contributions are given. Consequently, these s are independent of magnetic sublevels and hyperfine levels of the atomic states involved in the transition, thus, can offer unique approaches to carry out many high precision measurements with minimal systematics. Inevitably, the proposed technique can also be used for electronic or hyperfine transitions in other atomic systems.
pacs:32.60.+i, 37.10.Jk, 32.10.Dk, 32.70.Cs
Introduction: Techniques to cool and trap atoms using laser light have revolutionized modern experimental procedures. They are applied not only to carry out very high precision spectroscopy measurements, but also to probe many subtle signatures like parity violation Wood et al. (1997), Lorentz symmetry invariance Pruttivarasin et al. (2015), quantum phase transitions Tiecke et al. (2014) etc. Vogl and Weitz had demonstrated cooling of Rb atoms by resonating the trap laser light with their D-lines Vogl and Weitz (2009), while Monroe et. al. had observed the clock transition in Cs by cooing the atom using the D2 line Monroe et al. (1991). As demonstrated in Ref. Katori et al. (1999), trapping atoms at s is the foremost process today in a number of applications such as constructing optical lattice clocks. Following this, a number of experimental and theoretical studies have been reported s in the neutral atoms Lundblad et al. (2007); Safronova et al. (2012); Sahoo and Arora (2013); Arora et al. (2007); Takamoto et al. (2005); Yi et al. (2011); Barber et al. (2006); Ovsiannikov et al. (2007), and recently in the singly charged alkaline earth ions Tang et al. (2013); Kaur et al. (2015). In a remarkable work, Katori et. al. Takamoto and Katori (2003) had demonstrated use of magic wavelengths (s) for Sr atoms, to reduce the systematics in the measurements. Use of s for trapping and controlling atoms inside high-Q cavities in the strong coupling regime with minimum decoherence for the D2 line of Cs atom have been demonstrated by McKeever et. al. McKeever et al. (2003). Liu and co-workers experimentally demonstrated the existence of s for the Ca clock transitions Liu et al. (2015).
A linearly polarized light is predominantly used to trap the atoms which is free from the contribution of vector component of the interaction between atomic states and electric fields. A substantial drawback of these s is that they are magnetic sublevel dependent for the transitions involving states with angular momentum greater than . It has also been argued that considering circularly polarized light for trapping could be advantageous due to dominant role played by the vector polarizability in the ac Stark shifts Sahoo and Arora (2013); Arora and Sahoo (2012). This may help augmenting the number of s in some cases but at the same time requires magnetic sublevel selective trapping. The dependence of magic wavelengths on magnetic sublevels demands for state selective traps. To circumvent this problem, it is imperative to find out s independent of magnetic sublevel.
In this paper, we propose a scheme to trap atoms and ions at the s that are independent of the atomic magnetic and hyperfine levels. They can be used in a number of the applications discussed above. Just for the demonstration purpose, we present here s of the widely used D2 transitions of the Rb and Cs atoms. They are useful for optical communications where lasers are tuned to their D-lines to trap and repump the atoms in order to prevent them from accumulating in the ground state Fox et al. (1993). Moreover, D2 lines of Rb and Cs are used for studying their microwave spectroscopy Vogl and Weitz (2009); Monroe et al. (1991); Yang et al. (2012); Entin and Ryabtsev (2004), quantum logic gates Friebel et al. (1998) and to assert the accuracy of the fine structure constant Gerginov et al. (2006). In this proposal, we only presume that the atomic systems are trapped in sufficiently strong magnetic fields.
Theory: The ac Stark shift for any state with angular momentum of an atom placed in an oscillating electric field with polarization vector is given as Manakov et al. (1986)
where is the total dynamic polarizability for the state with its magnetic projection as
where with are the scalar, vector and tensor components of the frequency dependent polarizability respectively.In the above expression and can be replaced suitably by either the atomic angular momentum or hyperfine angular momentum with their corresponding magnetic projection or , depending upon the consideration of atomic or hyperfine states, respectively. The and are defined as Beloy (2009)
with the quantization axis unit vector . The differential Stark shift of a transition between states to is, hence, given by
It is obvious from the above expression that for obtaining null differential Stark shift, it is necessary that either independent components cancel out each other or the net contribution nullifies which prominently depends on the choices of , and values. By adequately selecting the experimental configuration such that the and values are zero, it is possible to remove the vector and tensor components. As implied from above considerations and Eq. (5), the differential ac Stark shift depends only on the scalar polarizabilities of the associated states. Thus, it is independent of the magnetic sublevels. Moreover, the scalar polarizabilities are same for the atomic and hyperfine levels; i.e. (for all the allowed values). Hence, these s are also independent of the hyperfine splittings of the participating atomic states. Therefore, s obtained by applying the above conditions will be independent of the choice of , and quantum numbers. On account of above, we elucidate a lab frame in which null values for and can be accomplished.
Discussions: We start our analysis by defining a coordinate system with the components , and , where and are the real components of the complex unit polarization vector and describe the system such that
Here parameter is analogous to the degree of polarization () and is a real quantity representing an arbitrary phase. Conveniently this can be represented by Fig. 1, where the electric field vector sweeps out an ellipse in a unit period about the axis of wave vector with semi-major and semi-minor axes of the ellipse aligned along and , respectively. The ratio of the semi-minor width to semi-major width of the ellipse needs to be . Furthermore using Eqs. (3) and (6), one can express , where . The biased magnetic field is along the quantization axis and can technically be aligned in any direction. Without loss of generality, it can be assumed to lie in the plane for the present requirement. Parameters , and are the angles between the respective unit vectors and , respectively, as shown in Fig. 1. In terms of these geometrical parameters, one can conveniently express and satisfying the relations
Substituting explicit form of and in Eq.(2), the expression for the polarizability is given by
In this description, it reduces to and for the linearly polarized light with , where is the angle between the quantization axis and direction of polarization vector. Similarly, one can simplify the above expression for the circularly polarized light by using either or .
To eliminate the dependence of on values in Eq. (5), one can choose a suitable combination of the above parameters so that null values for both and can be achieved. This obviously corresponds to and , which can be brought about by suitably setting up the , and parameters. One can achieve by fixing the quantization axis at right angle to the wave vector; i.e. one can assume to repose in the plane. On the otherhand, many possible freedom exist to achieve . For example, we plot values (note that and are related) versus in Fig. 2, where each point on the graph represents a set of and that can yield . As mentioned previously, is a measure of the polarization and can be adjusted by setting the eccentricity ‘’ of the ellipse . It is evident from Fig. 2 that for the values and 135, none of the pair of angles and can offer . This means that the above criterion cannot be attained by applying the circularly polarized light. However as has been reported in Kotochigova and DeMille (2010), this condition can be attained with = 0 and = 54.74 for a linearly polarized light. This critical condition seems to be too demanding and could be hard to achieve in an experimental set-up. On the otherhand, one could get a more relaxed experimental conditions by using an elliptical polarized light as it offers more freedom to choose from a number of and combinations as shown in Fig. 2. In fact, we observe that these estimated values of s will change maximum up to 1% for the variation of by one degree when is constant and vice versa. Therefore, it seems to be feasible to prepare a trap geometry using our proposed criteria for the elliptically polarized light aptly.
|Resonant transition||Resonant transition|
Results: It looks straightforward to achieve s for any atomic or hyperfine transitions in a given atomic system by maintaining the above geometry for trapping atoms provided that the differential polarizabilities of the considered transition nullifies within the resonance lines. We subsequently demonstrate below these s, specifically for the D2 lines of the Rb and Cs atoms.
Rb atom: In Fig. 3, we have plotted scalar dipole polarizabilities of the and states of Rb with respect to wavelength of the external electric field. These values were obtained in our previous work where we presented s for the D lines of Rb using the linearly and circularly polarized light Sahoo and Arora (2013). As can be seen from the figure, a number of s represented by the crossings of and polarizabilities have been predicted for this transition and are presented in Table 1 along with the resonance lines to highlight their locations. Two s are found at 615.2 nm and 627.2 nm, which belong to the visible region, while the other five s are located at 740.4 nm, 775.8 nm, 791.3 nm and 1397.1 nm. One more probable in between the and resonance lines seems to exist, but we have not listed it in the above table due to inability to identify it distinctly. All the s mentioned in Table. 1, except the one at 1397.1 nm, support blue-detuned trapping scheme. We, however, recommend the use of at 791.3 nm for a blue-detuned and 1397.1 nm for a red-detuned trap for the experimental purposes, since these wavelengths are far from the resonant transitions.
Cs atom: Adopting a similar procedure as in Ref. Arora et al. (2007), we have evaluated dynamic polarizabilities of the ground and states of Cs. We have also plotted the frequency dependent scalar polarizabilities of the ground and states of this atom in Fig. 4 to find out s that are independent of the magnetic sublevels and hyperfine levels of the atomic states of the D2 line. Table 1 enlists the s for this transition which lie within the wavelength range of 600-1500 nm. As demonstrated in Ref. Arora et al. (2007), s exist between every two resonances for the linearly polarized light. We correspondingly locate six wavelengths at 602.9 nm, 614.8 nm, 621.9 nm, 657.7 nm, 685.9 nm and 698.5 nm in the visible region using the proposed geometry for an elliptically polarized light. We also found two s at 793.6 nm and 886.4 nm, which belong to infra-red region. They all support dark or blue-detuned traps. Two more s in the infrared region are located at 920.6 nm and 936.2 nm that can support red-detuned traps. In this case, we intend to recommend the use of s at 920.6 nm and 936.2 nm for the red-detuned trapping and 685.9 nm for the blue-detuned trapping due to the availability of lasers at these wavelengths.
Conclusion: A novel trap geometry has been proposed using an elliptically polarized light in a sufficiently large magnetic field that can produce null differential Stark shifts among the transitions and can be exclusively applicable among any magnetic sublevels and hyperfine levels. Their applications in the D2 lines of Rb and Cs atoms have been highlighted and the corresponding magic wavelengths are reported. Furthermore, we have also recommended the magic wavelengths that are suitable for both the blue and red detuned traps of Rb and Cs atoms. These magic wavelengths will be immensely useful in a number of high precision measurements.
Acknowledgement: S.S. acknowledges financial support from UGC-BSR scheme. The work of B.A. is supported by CSIR Grant No. 03(1268)/13/EMR-II, India. We gratefully acknowledge helpful discussions with Dr. K. Beloy, Jasmeet Kaur and Kiranpreet Kaur.
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