Theory of "magic" optical traps for Zeeman-insensitive clock transitions in alkalis
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r Theory of “magic” optical traps for Zeeman-insensitive clock transitions in alkalis
Andrei Derevianko ∗ Department of Physics, University of Nevada, Reno NV 89557 (Dated: November 9, 2018)Precision measurements and quantum information processing with cold atoms may benefit fromtrapping atoms with specially engineered, “magic” optical fields. At the magic trapping conditions,the relevant atomic properties remain immune to strong perturbations by the trapping fields. Herewe develop a theoretical analysis of magic trapping for especially valuable Zeeman-insensitive clocktransitions in alkali-metal atoms. The involved mechanism relies on applying “magic” bias B-fieldalong a circularly polarized trapping laser field. We map out these B-fields as a function of trappinglaser wavelength for all commonly-used alkalis. We also highlight a common error in evaluatingStark shifts of hyperfine manifolds.
PACS numbers: 37.10.Jk, 06.30.Ft
A recurring theme in modern precision measurementsand quantum information processing with cold atoms andmolecules are the so-called “magic” traps [1]. At themagic trapping conditions, the relevant atomic proper-ties remain immune to strong perturbations by opticaltrapping fields. For example, in optical lattice clocks,the atoms are held using laser fields operating at magicwavelengths [2]. The clock levels are shifted due to thedynamic Stark effect that depends on the trapping laserwavelength. At the specially-chosen, “magic”, wave-length, both clock levels are perturbed identically; there-fore the differential effect of trapping fields simply van-ishes for the clock transition. This turned out to bea powerful idea: lattice clocks based on the alkaline-earth atom Sr have recently outperformed the primaryfrequency standards [3].Finding similar magic conditions for ubiquitous alkali-metal atoms employed in a majority of cold-atom ex-periments remains an open challenge. Especially valu-able are the microwave transitions in the ground-statehyperfine manifold. Finding magic conditions here, forexample, would enable developing microMagic clocks [4]:microwave clocks with the active clockwork area of a fewmicrometers across. In addition, the hyperfine manifoldsare used to store quantum information in a large fractionof quantum computing proposals with ultracold alkalis.Here the strong perturbation due to trapping fields isdetrimental. Namely the dynamic differential Stark shiftsis the limiting experimental factor for realizing long-livedquantum memory [5]. Qualitatively, as an atom moves inthe trap, it randomly samples various intensities of thelaser field; this leads to an accumulation of uncontrolledphase difference between the two qubit states. For verycold samples, the accumulation of uncontrolled phasesmay arise because the interrogation by the microwavesis an ensemble average over the spatial distribution ofatoms across the trap. Magic conditions rectify theseproblems, as both qubit states see the very same optical ∗ Electronic address: [email protected] potential and do not accumulate differential phase at all.In other words, we engineer a decoherence-free trap.Initial steps in identifying magic conditions for hyper-fine transitions in alkali-metal atoms have been made inRefs. [6–8]. The proposals [7, 8] identified magic condi-tions for M F = 0 states. Due to non-vanishing magneticmoments, these states, however, are sensitive to straymagnetic fields which would lead to clock inaccuraciesand decoherences (except for special cases of relativelylarge bias fields, see below). Recently, it has been re-alized by Lundblad et al.[9] that magic conditions maybe attained for the Zeeman-insensitive M F = 0 states aswell. Here the bias magnetic field is tuned to make theconditions “magic” for a given trapping laser wavelength.These authors experimentally demonstrated these con-ditions for lattice-confined Rb atoms at a single wave-length. As demonstrated below, mapping out values ofmagic bias B-fields for a wide range of wavelengths re-quires full-scale structure calculations. Below I carry outsuch calculations and point out common pitfalls in eval-uating differential polarizabilities of hyperfine manifolds.In this work, we are interested in the clock tran-sition of frequency ν between two hyperfine states | F ′ = I + 1 / , M ′ F = 0 i and | F = I − / , M F = 0 i at-tached to the ground electronic nS / state of an alkali-metal atom ( I is the nuclear spin). Here and below wedenote the upper clock state as | F ′ i and the lower stateas | F i . The magic conditions are defined as the clockfrequency being independent on the perturbing trappingoptical field.We start by reviewing the Zeeman effect for the clockstates. The Zeeman Hamiltonian reads H Z = − µ z B , µ being the magnetic moment operator. The permanentmagnetic moments of the M F = 0 states vanish, so theeffect arises in the second order. We need to diagonalizethe following Hamiltonian H Z eff = (cid:18) hν H ZF ′ F H ZF F ′ (cid:19) . (1)The leading effect is due to off-diagonal coupling H ZF F ′ = h F ′ , M ′ F = 0 | H Z | F, M F = 0 i . In case of alkalis,( µ z ) F F ′ ≈ µ B , where µ B is the Bohr magneton. Theresulting Zeeman substates repeal each other and in suf-ficiently weak B-fields, µ B B ≪ hν , the shift of the tran-sition frequency is quadratic in magnetic field, δν Z ( B ) ν ≈ (cid:18) µ B hν B (cid:19) . (2)Since atoms are trapped by a laser field, the atomiclevels are shifted due to the dynamic Stark effect (see,e.g., a review [10]). The relevant energy-shift operatorreads ˆ U ( ω L ) = − ˆ α ( ω L ) (cid:18) E L (cid:19) , where E L is the amplitude of the laser field and ˆ α ( ω L )is the operator of dynamic atomic polarizability; it de-pends on the laser frequency. Notice that ˆ U may haveboth diagonal and off-diagonal matrix elements betweenatomic states of the same parity.Now we add the Stark shift couplings to the Hamilto-nian (1). The Stark shift operator has both the diagonaland off-diagonal matrix elements in the clock basis. Tofind the perturbed energy levels, we diagonalize the ef-fective Hamiltonian H eff = (cid:18) hν + U F ′ F ′ U F ′ F + H ZF ′ F U F F ′ + H ZF F ′ U F F (cid:19) . (3)For sufficiently weak fields, the resulting shift of the clockfrequency reads δν clock ( ω L , B, E L ) = δν Z ( B ) + δν S ( ω L , B, E L ) (4)with the Stark shift δν S ( ω L , B, E L ) = 1 h { α F ′ F ′ ( ω L ) − α F F ( ω L ) − (cid:18) µ F F ′ Bhν (cid:19) α F ′ F ( ω L ) (cid:27) (cid:18) E L (cid:19) . (5)The “magic” conditions are attained when δν S ( ω L , B, E L ) = 0 for any value of the laser am-plitude, i.e., simply when the combination inside thecurly brackets vanishes.At this point one may evaluate the dynamic polariz-abilities and deduce the magic B-field. Before proceedingwith the analysis, I would like to address common pitfallsin evaluating polarizabilities of hyperfine-manifold states,so the reader appreciates the necessity of full-scale cal-culations. A generic expression for the polarizability of | nF M F i state reads α (0) F F ( ω ) = X i = | n i F i M i i h nF M F | D z | i ih i | D z | nF M F i E nF M F − E i + ω + ... (6)where the omitted term differs by ω → − ω , and D is thedipole operator. All the involved states are the hyperfinestates. While this requires that the energies include hy-perfine splittings, it also means that the wave-functions incorporate hyperfine interaction (HFI) to all-ordersof perturbation theory. Including the experimentally-known hyperfine splittings in the summations is straight-forward and unsophisticated practitioners stop at that,completely neglecting the HFI corrections to the wave-functions. This is hardly justified as both contributionsare of the same order.I would like to remind the reader of a recent contro-versy: neglecting the HFI correction to wave-functionshas already lead to a (even qualitatively) wrong iden-tification of magic conditions. The authors of Ref.[11]employed the simplified approach and (for B = 0) founda multitude of magic wavelengths for clock transitions inCs. The prediction was in a contradiction with a subse-quent fountain clock measurement; the full-scale calcula-tions have found that in fact there are no magic wave-lengths at B = 0, Ref. [6]. To reinforce this point inthe context of this paper, in Fig. 1, I compare results oftwo calculations of magic B-fields for Rb as a functionof laser frequency. In the first calculation, I neglectedthe HFI correction to the wave-functions (while includ-ing the hyperfine corrections to the energies), and thesecond result comes the full-scale calculation describedbelow. We clearly see that the simplified approach is offby as much as a factor of two. Only near the resonancethe two approaches produce similar results.
FIG. 1: (Color online) Importance of full-scale calculations.Dependence of magic B-field (in Gauss) on laser frequency(in atomic units) for Rb. Full-scale calculations (solid blueline) are compared with approximate “experimentalist” com-putations which neglect the HFI contribution to atomic wave-functions (dashed red line).
These two examples should convince the reader thatthe full-scale calculations are indeed required for reliablypredicting magic fields. A consistent approach to evalu-ating dynamic polarizabilities of hyperfine states was de-veloped in Ref.[6]. The HFI correction to wave-functionsand energies was included to the leading order; this leadsto a third-order analysis quadratic in dipole couplingsand linear in the HFI. Below I simplify the magic fieldconditions using the formalism of Ref.[6].We may decompose the polarizability into a sum over0-, 1-, and 2-rank tensorsˆ α ( ω L ) = ˆ α (0) ( ω L ) + A ˆ α (1) ( ω L ) + ˆ α (2) ( ω L ) . (7)These terms are conventionally referred to as the scalar,vector (axial), and tensor contributions. We also explic-itly factored out the degree of circular polarization A ofthe wave ( A = ± σ ± light). The direction of thebias B-field defines the quantization axis. We also fixedthe direction of the wave propagation ˆk to be parallel tothe B-field. Notice that the circular polarization of theoptical field is defined with respect to the quantizationaxis (not ˆk ).Below we show that the “magic” value of the magneticfield may be represented as B m ( ω L ) ≈ − µ B I + 12 I α (0) , HFI
F F ( ω L ) Aα anS / ( ω L ) hν . (8)It depends on the laser frequency and the degree ofcircular polarization A , | A | ≤ α (0) , HFI
F F ( ω L ) is thescalar HFI-mediated third-order polarizability of thelower clock state, F = I − / α ( ω L ) = α F ′ F ′ ( ω L ) − α F F ( ω L ),entering Eq. (5), comes only through the hyperfine-mediated interactions: ∆ α ( ω L ) = α HFI F ′ F ′ ( ω L ) − α HFI
F F ( ω L ). This reflects the fact that both hyperfinelevels belong to the same electronic configuration - thesymmetry in responding to fields is only broken whenthe HFI is included. Moreover, for alkalis α F F and α F ′ F ′ are dominated by the scalar part of polarizabil-ity: ∆ α ( ω L ) ≈ α (0) , HFI F ′ F ′ ( ω L ) − α (0) , HFI
F F ( ω L ). Thesetwo polarizabilities never intersect – they are strictlyproportional to each other: α (0) , HFI F ′ F ′ ( ω L ) = − ( I +1) /I α (0) , HFI
F F ( ω L ).Now we turn to simplifying the off-diagonal ma-trix element α F ′ F ( ω L ) entering Eq. (5). It is dom-inated by the vector part of polarizability. Indeed, h F ′ , M ′ F | ˆ α (0) | F, M F i = 0 due to the angular selec-tion rules ( F ′ = F ). While the tensor contribution h F ′ , M ′ F | ˆ α (2) | F, M F i does not vanish, the electronic mo-mentum of the ground state nS / is J = 1 /
2; there-fore (since h J = 1 / | ˆ α (2) | J = 1 / i ≡
0) this ma-trix element requires the HFI admixture and becomesstrongly suppressed. By contrast, the vector contribu-tion h F ′ , M ′ F | ˆ α (1) | F, M F i does not vanish even if thehyperfine couplings are neglected. It is worth mention-ing that it arises only due to relativistic effects, since theorbital angular momentum L = 0 for the ground state;e.g., vector polarizability is much smaller in Li than inCs. The off-diagonal matrix element of the rank-1 polar-izability may be expressed as α (1) F ′ F ( ω L ) = α anS / ( ω L ),where α aJ ( ω L ) is the conventionally-defined second-ordervector polarizability of the ground nS / state.To evaluate the polarizabilities, we used a blend of rel-ativistic many-body techniques of atomic structure, as described in [12]. To improve upon the accuracy, high-precision experimental data were used where available.To ensure the quality of the calculations, a compari-son with the experimental literature data on static Starkshifts of the clock transitions was made. Overall, we ex-pect the theoretical errors not to exceed 1% for Cs and tobe at the level of a few 0.1% for lighter alkalis. If required,better accuracies may be reached with many-body meth-ods developed for analyzing atomic parity violation [13]. FIG. 2: (Color online) Dependence of magic B-field (in Gauss)on laser frequency (in atomic units) for Na (dashed greenline), Rb (solid blue line), and
Cs (dot-dashed red line).Magic B-fields for other isotopes of the same element may beobtained using the scaling law, Eq. (9).
Our computed dependence of magic B-field on laser fre-quency for representative alkalis ( Na, Rb, and
Cs)is shown in Fig. 2. We also carried out similar calcu-lations for K and Li. Results for several laser wave-lengths are presented in Table I.
TABLE I: Values of magic B-fields for representative laserwavelengths. The optical field is assumed to be purely cir-cularly polarized. Values of the clock transition frequencies ν and the second-order Zeeman frequency shift coefficients δν Z /B are listed in the second and the third columns, respec-tively. Magic B-fields for other isotopes of the same elementmay be obtained with the scaling law, Eq. (9). ν δν Z /B “magic” B (Gauss)(GHz) (kHz/G ) 10.6 µ m 1.065 µ m 811.5 nm Li 0.80 4.9 - 144 64.9 Na 1.77 2.2 47.4 5.07 4.05 K 0.46 8.5 0.782 0.0848 0.0672 Rb 6.83 0.57 41.0 4.39 3.62
Cs 9.19 0.43 27.3 3.00 3.81
From Fig. 2 we observe that below the resonances,magic B-fields grow smaller with increasing laser fre-quency. This is a reflection of the fact that at small ω L , the HFI-mediated polarizability approaches a con-stant value, while the vector polarizability ∝ ω L . Thus, B m ∝ /ω L in accord with Fig. 2. As the frequencyis increased, the B m ( ω L ) increases near the atomic reso-nance (fine-structure doublet). This leads to a prominentelbow-like minimum in the B m ( ω L ) curves.Magic B-field has been recently measured for opticallattice-confined Rb at 811.5 nm, Ref. [9]. At this wave-length and degree of circular polarization A = 0 . B m = 4 . B m =3 . .
62 Gauss,is 1 . σ larger than the measured value.A quick glance through the Table I reveals that therequired B-fields for K are much weaker than for otheralkalis; this is related to the fact that the nuclear momentof this isotope is almost an order of magnitude smallerthan that of other species. An additional suppression isdue to the magic B-fields being quadratic in hyperfinesplitting (clock frequency).Notice that if the B-field and the direction of laserpropagation are set at an angle θ , then A → A cos θ inEq.(8) (see Ref. [7]). This angle provides an additionalexperimental handle on reaching the magic conditions.Increasing the angle and reducing the degree of circularpolarization raise values of the magic B-field.Generically, the ratio α (0) , HFI
F F ( ω L ) /α anS / ( ω L ) is inthe order of a ratio of the hyperfine splitting to the fine-structure splitting in the nearest P -state manifold, i.e.,it is much smaller than unity. This reinforces the validityof the weak-field approximation used to derive Eqs.(5,8).Notice, however, that lim ω L → α anS / ( ω L ) →
0; this maylead to unreasonably large magic B-fields for very low-frequency fields. Such a breakdown occurs for Li at10.6 µ m in Table I.It is worth pointing out that the results of Fig. 2 andTable I may be extended to other, e.g., unstable, isotopes.An analysis of the third-order expressions for the HFI-mediated polarizabilities shows that the magic B-fieldsscales with the nuclear spin and g-factor as B m ∝ g I I (2 I + 1) (2 I + 2) / . (9) Finally, I would like to comment on the magic con-ditions for the M F = 0 states discussed in our earlierwork [7]. The idea there was to rotate the bias B-fieldwith respect to the laser propagation. At a certain laser-frequency-dependent magic “angle”, θ ≈ ◦ , contribu-tions of the HFI-meditated scalar polarizability and therotationally-suppressed vector polarizability were com-pensating each other. Notice that we may attain theZeeman-insensitivity even in this case. Indeed, in mag-netic field, two hyperfine levels | F = I + 1 / , M F i and | F = I − / , M F i repel each other through off-diagonalZeeman coupling. In addition, the g-factors of the twolevels have opposite signs. This leads to a minimum inthe clock-frequency dependence on B-fields. These min-ima occur at relatively large magnetic fields, e.g., about2 kGauss for Rb. This dν ( B ) /dB = 0 condition fixes“magic” B-field values for the proposal [7].It is anticipated that a variety of applications couldtake advantage of the magic conditions computed in thispaper. For example, the dynamic Stark shift is the pri-mary factor limiting lifetime of quantum memory [5];here an advance may be made by switching to the magicB-fields. It remains to be seen if the microMagic latticeclock can be developed; here one needs to investigate thefeasibility of stabilizing bias magnetic fields at the magicvalues. In this regard, notice that we still have a choiceof fixing laser wavelength/polarization/rotation angle tooptimize clock accuracy with respect to drifts in the B-field. Acknowledgements —
I would like to thank Trey Porto,Nathan Lundblad, and Alex Kuzmich for discussions.This work was supported in part by the US NSF andby the US NASA under Grant/Cooperative AgreementNo. NNX07AT65A issued by the Nevada NASA EP-SCoR program. [1] J. Ye, H. J. Kimble, and H. Katori, Science , 1734(2008).[2] H. Katori, M. Takamoto, V. G. Pal’chikov, and V. D.Ovsiannikov, Phys. Rev. Lett. , 173005 (2003).[3] A. D. Ludlow, T. Zelevinsky, G. K. Campbell, S. Blatt,M. M. Boyd, M. H. G. de Miranda, M. J. Martin, J. W.Thomsen, S. M. Foreman, J. Ye, et al., Science , 1805(2008).[4] K. Beloy, A. Derevianko, V. A. Dzuba,and V. V. Flambaum, Phys. Rev. Lett. , 120801 (pages 4) (2009), URL http://link.aps.org/abstract/PRL/v102/e120801 .[5] R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Camp-bell, D. N. Matsukevich, T. A. B. Kennedy, andA. Kuzmich, Nat. Phys , 100 (2009), ISSN 1745-2473, URL http://dx.doi.org/10.1038/nphys1152 .[6] P. Rosenbusch, S. Ghezali, V. A. Dzuba, V. V.Flambaum, K. Beloy, and A. Derevianko, Phys.Rev. A , 013404 (pages 8) (2009), URL http://link.aps.org/abstract/PRA/v79/e013404 .[7] V. V. Flambaum, V. A. Dzuba, and A. Derevianko, Phys.Rev. Lett. , 220801 (2008).[8] J. M. Choi and D. Cho, Journal of Physics:Conference Series , 012037 (6pp) (2007), URL http://stacks.iop.org/1742-6596/80/012037 .[9] N. Lundblad, M. Schlosser, and J. V. Porto, Experimen-tal observation of magic-wavelength behavior in opticallattice-trapped Rb (2009), arXiv.org:0912.1528, URL .[10] N. L. Manakov, V. D. Ovsiannikov, and L. P. Rapoport, Phys. Rep. , 319 (1986).[11] X. Zhou, X. Chen, and J. Chen (2005), arXiv:0512244.[12] K. Beloy, U. I. Safronova, and A. Derevianko, Phys. Rev.Lett. , 040801 (2006). [13] S. G. Porsev, K. Beloy, and A. Derevianko, Phys.Rev. Lett. , 181601 (pages 4) (2009), URL http://link.aps.org/abstract/PRL/v102/e181601http://link.aps.org/abstract/PRL/v102/e181601