† Corresponding author. E-mail:

Project supported by the National Natural Science Foundation of China (Grant Nos. 11304063 and 11174066) and the Youth Foundation of Liaoning Normal University, China (Grant No. LS2014L002).

An approximation formula is developed to determine the tune-out wavelengths for the ground states of the alkaline-metal atoms lithium, sodium and cesium from the existing relativistic reduced matrix elements and experimental energies. The first longest tune-out wavelengths for Li, Na, and Cs are 670.971 nm, 589.557 nm, and 880.237 nm, respectively. This is in good agreement with the previous high precise results of 670.971626 nm, 589.5565 nm, and 880.25 nm from the relativistic all-order many-body perturbation theory (RMBPT) calculation [*Phys. Rev. A*

The realization of mixture of trapped ultracold atomic gases in optical lattice is one of the most important improvements in the experimental side in recent years.^{[1–5]} This opens new paths toward the formation of ultracold diatomic molecule and some other targets that the physicists have been seeking for a couple of years.^{[6–10]} The atom can be trapped in the optical lattice by the optical dipole force and will experience an energy shift due to the ac stark effect. Both effects are proportional to the dynamic polarizability of the atom and can be cancelled at certain wavelength where the dynamic polarizability of the atom goes to zero. This means that the atom is unaffected by the presence of the electromagnetic field and also can be released from the trap simply while the other atom in the mixture are still strongly trapped. Such wavelength is termed as the “tune-out wavelength”^{[11,12]} and has attracted much attention recently.

A number of experimental measurements on the determination of the tune-out wavelengths have been performed in the past few years. Since the measurement is a null experiment, it is not subject to the strength of an electric field or the intensity of a laser field. The tune-out wavelengths for the Rb and K atoms have been measured by different groups.^{[13–15]} Holmgren *et al.*^{[14]} measured the longest tune-out wavelength of potassium to an uncertainty of 1.5 pm by using an atom interfermeter. This is more accurate than the uncertainty of 3 pm which is performed by using state-of-the-art atomic theory.^{[12]} Some tune-out wavelengths for rubidium have also been measured through the diffraction of a Bose-Einstein condensate off a sequence of standing wave pulses.^{[13]} On the theoretical side, the most reliable prediction of the tune-out wavelength for the alkaline-metal atoms is the relativistic all-order many-body perturbation theory (RMBPT) calculation.^{[12]}

In this paper, we present an estimate of the longest tune-out wavelengths for the alkaline-metal atoms: lithium, sodium, and cesium using a formula which is obtained through the analysis of an exact non-relativistic calculation of the tune-out wavelengths for lithium and sodium. This non-relativistic calculation of the tune-out wavelengths are computed using the configuration interaction plus core polarization (CICP) method in which a semiempirical potential is adopted to describe the polarization effect between the core and valence electrons. This method has been successfully applied to the description of many one- and two-electron atoms.^{[16–22]} The detailed analysis of the exact CICP calculation shows that the tune-out wavelengths are determined by explicit calculation of the dynamic polarizability at a series of *ω* values and they all tend to be close to the transition energies of *np* states for the alkaline-metal atoms. The fine structure split is not considered in CICP calculation and thus it cannot be used to give the accurate values of tune-out wavelengths. However an approximation formula can be built since that the CICP calculation shows that the values of the tune-out wavelengths are mostly determined by the data of only a few transitions. Using this formula, one can use the existing experimental and theoretical transition information to predict the tune-out wavelengths for the alkaline-metal atom.

The present relativistic estimate of the three longest tune-out wavelengths for lithium and four longest wavelengths for sodium are given in this paper along with the related non-relativistic calculation as a validity. Due to the strong relativistic effect, we do not give the CICP calculation about Cs, but only present the relativistic estimates of the five longest tune-out wavelengths in Section 3.

The details of the non-relativistic CICP method have been given elsewhere^{[17,18]} and here we only briefly outline this method. The CICP method was a frozen-core procedure. The core electron was represented by a Hartree–Fock wavefunction that was calculated exactly using Slater-type orbitals (STOs). The effective Hamiltonian for the valence electrons was essentially a fixed core Hamiltonian with the addition of semiempirical potentials to allow for the polarization interaction with the core.^{[23]} The direct and exchange interactions of the valence with the Hartree–Fock core electrons were computed without approximation. The semiempirical polarization potential was tuned so that they reproduced the binding energies of the *ns* ground state, the *np* and *nd* excited states.^{[23]} The effective Hamiltonian was diagonalized in a very large orbital basis of Laguerre-type orbitals for each *l* value due to the superior linear dependence properties. The semiempirical core polarization potential *V*_{p} has the functional form:

*α*

_{core}is the static dipole polarizability of the core,

*ρ*

_{l}is the cutoff parameter.

A good test of the CICP method comes from the tabulation of the energy in Table ^{[24]} The agreement with experiment for the lowest states of each symmetry is excellent since the polarization cutoff parameters are tuned to reproduce the experimental binding energy of these states. The agreement level of the excited states is also good which indicates the validity of the CICP method.

The dynamic polarizability is defined as

*f*

_{0n}is the oscillator strength for the dipole transition from the ground state and

*ε*

_{0n}is the transition energy. When the frequency goes to zero, the polarizability is the static polarizability. As another validation of the CICP method, the static dipole polarizabilities for the ground states of Li, Na and Cs produced in the CICP calculation are listed in Table

^{[29–32]}All the static polarizabilities are computed using experimental energy differences for the lowest energy excited states and the core contributions are also added. The value of 403.6(81) a.u. (The unit a.u. is short for atomic unit) in Cs is weighted average of experimental data from Refs. [31] and [32]. The CICP static polarizabilities are in good agreement with the experimental measurement except for Cs which is as expected since the relativistic effect is important for Cs.

Table *n*p excited states. Taking lithium as an example, the first tune-out wavelength is located between the 2s–2p resonant transition and the 2s–3p excitation energies and mainly dominated by these two terms. The 3p term has the opposite sign and cancels out most of the 2p contribution. The remainder and core terms make a small contribution to the total polarizability. In other situations, the tune-out wavelengths all occur just below the excitation energies of the 4p, 5p, and 6p states. This is due to the fact that the polarizability is proportional to the square of the reduced matrix elements. The magnitude of the 2s–2p matrix element is more than one order larger than the 2s–*n*p (*n* = 4,5,6) matrix element, then these tune-out wavelengths must be very close the corresponding 2s–*n*p excitation energies so that the contribution to the total polarizability from the 2s–*n*p term can cancel the contribution from 2s–2p term. From Table

As can be seen from Table

*α*

_{0}is the background polarizability arising from all transitions except the transition most close to the tune-out wavelength. Then the background polarizability

*α*

_{0}is dominated by contribution from the resonant transition and can be represented as

*α*

_{rest}is the rest polarizability, only accounts for a few percent of the resonance transition, and varies slowly with the tune-out frequency. Therefore, we assume that

*α*

_{rest}has the same value at

*ω*= 0 and

*ω*

_{to}. Then the value of

*α*

_{rest}can be calculated as

*α*is the static polarizability of the ground states.

_{d}*f*

_{resonant},

*f*, Δ

*E*

_{resonant}, and Δ

*E*are the oscillator strengths and transition energies of the resonant transition and the transition near the tune-out wavelength respectively. Using Eq. (

^{[24]}and oscillator strengths come from the CICP calculation. Table

Based on the analysis of the non-relativistic calculation and the fact that the fine structure split of the energy levels is not considered, the present non-relativistic calculational methodology cannot be applied to the determination of the accurate tune-out wavelengths. However, equation (

_{1/2}resonant transition and the transition 2s→ 2p

_{3/2}that is most close to the tune-out wavelength,

*n*≥ 3.

Similar form is adopted for the sodium atom with the 2p_{j} state replaced by the 3p_{j} state, and for cesium with the 2p_{j} state replaced by the 6p_{j} state.

All the fundamental information adopted here to make an estimate of the tune-out wavelength for lithium, sodium and cesium is listed in Table ^{[33]} SD matrix elements,^{[25]} and experimental transition energies of NIST tabulation.^{[24]} The longest tune-out wavelengths for lithium and sodium with the CCSD(T) matrix elements and the experimental energies^{[24]} are listed in Tables ^{[12]} the differences are 6.2 × 10^{−4} nm, 1.9 × 10^{−2} nm, and 2.8 × 10^{−2} nm. The predicted four longest tune-out wavelengths for sodium are 589.557 nm, 331.839 nm, 330.372 nm, and 285.581 nm. One can note that our values for the first longest tune-out wavelengths compared with the Arora *et al*.^{[12]} calculations achieve an accuracy of 0.001-nm level.

The element Cs has been the subject of various theoretical calculations and experiments in atomic clock research.^{[34,35]} The value of tune-out wavelength for Cs could be very useful for experimentalist. The measurements of tune-out wavelengths are also used to test the precision of the excited-state matrix elements which are difficult to measure or calculate. Due to the large correlation corrections, the transition matrix elements are hard to calculate accurately. The measurement of 6s–7p_{j} transition in Cs has been performed in Ref. [36]. The longest tune-out wavelengths and the contributions to the polarizability at the tune-out frequency for Cs are given in Table ^{[12]}

An estimate of the uncertainties for the first longest tune-out wavelength of Li, Na, and Cs can be made through Eq. (*α* = 0, then equation (

*X*

_{shift}=

*f*/(2

*α*

_{0}Δ

*E*

^{2}), one has

*X*

_{shift}makes an estimate of the relative difference between the transition frequency and tune-out frequency. The uncertainty in

*X*

_{shift}only considers the uncertainties in oscillator strengths and background polarizabilities, one can have

*β*= 137.035999 a.u. is the velocity of light in atomic units. The uncertainties of the oscillator strengths and polarizabilities are made by examination of the adopted data and existing experimental and theoretical information.

Another useful parameter for the measurement of tune-out wavelength is the energy window Δ*ω*. We suppose that the condition for determination of the tune-out wavelength is that the polarizability be set to zero with an uncertainty of ±0.1 a.u. The photon energy range at which the dynamic polarizability is 0 ± 0.1 a.u. is termed as the energy window Δ*ω*,

*ω*= 1.95 × 10

^{−14}a.u. and

*X*

_{shift}is 1.51 × 10

^{−5}. The uncertainty of the first longest tune-out wavelength for lithium is

*δλ*

_{to}= 4.74 × 10

^{−6}nm. Similarly, for sodium, the energy window is Δ

*ω*= 4.35 × 10

^{−11}a.u.,

*X*

_{shift}is 6.76 × 10

^{−4}, and the uncertainty of the first longest tune-out wavelength is

*δλ*

_{to}= 0.0937 nm. For Cs, the energy window is Δ

*ω*= 2.80 × 10

^{−8}a.u.,

*X*

_{shift}is 0.0312, and the uncertainty of the first longest tune-out wavelength is

*δλ*

_{to}= 0.388 nm. The energy windows for lithium and sodium are too narrow and difficult to achieve with existing technology, while the energy window for cesium is relatively larger and more feasible.

Using a similar method, we could also obtain the uncertainties for other longest tune-out wavelengths. The uncertainties for the second and third longest tune-out wavelengths of lithium are 0.013 nm and 0.009 nm, respectively. The corresponding energy windows are 1.49 × 10^{−6} a.u. and 2.71 × 10^{−6} a.u. For sodium, the uncertainties for the second, the third, and the fourth tune-out wavelengths are 0.063 nm, 0.0001 nm, and 0.013 nm. The corresponding energy windows are 1.74 × 10^{−6} a.u., 6.26 × 10^{−10} a.u., and 5.22 × 10^{−7} a.u. For the cesium atom, the uncertainties for the second and third longest wavelengths are 0.559 nm and 0.748 nm which are relatively larger since the matrix elements are more difficult to calculate precisely. The corresponding energy windows are 7.49 × 10^{−8} a.u. and 1.16 × 10^{−7} a.u.

The longest tune-out wavelengths for the lithium, sodium and cesium atoms are computed by an approximation formula. The accuracy of the calculation is tested by comparing with the all-order RMBPT calculation.^{[12]} The first longest tune-out wavelengths for Li, Na, and Cs are 670.971 nm, 589.557 nm, and 880.237 nm, respectively. The agreement of the first longest tune-out wavelength is at the 0.001-nm level. Some other longest tune-out wavelengths are also calculated, and there is a reasonable agreement with the RMBPT results.^{[12]} The wavelengths near some resonances which are in the ultraviolet are not calculated, since they are difficultly detected in most laboratories. The uncertainties of the predicted tune-out wavelengths are derived from the difference among the existing transition matrix elements. An parameter termed as the energy window of tune-out wavelength is developed as an indication of the technology required to measure these wavelengths. The longest tune-out wavelengths for lithium and sodium are difficult to measure with existing technology due to their too narrow energy windows while the first three tune-out wavelengths for atomic cesium and the tune-out wavelengths for lithium positioned at 324.161 nm and 274.883 nm and for sodium positioned at 331.839 nm are more feasible choices.

**Reference**

1 | Science 291 2570 |

2 | Phys. Rev. Lett. 87 080403 |

3 | Science 294 1320 |

4 | |

5 | Chin. Phys. B 22 090308 |

6 | Phys. Rev. Lett. 97 180404 |

7 | Phys. Rev. Lett. 97 120402 |

8 | J. Phys. B: At. Mol. Opt. Phys. 41 203001 |

9 | Science 322 231 |

10 | Phys. Rev. Lett. 102 020405 |

11 | Phys. Rev. A 75 053612 |

12 | Phys. Rev. A 84 043401 |

13 | Phys. Rev. Lett. 109 243003 |

14 | Phys. Rev. Lett. 109 243004 |

15 | |

16 | Phys. Rev. A 38 3339 |

17 | Phys. Rev. A 68 052714 |

18 | Phys. Rev. A 79 012513 |

19 | Eur. Phys. J. D 53 15 |

20 | Phys. Rev. A 87 032518 |

21 | Phys. Rev. A 88 022511 |

22 | Chin. Phys. Lett. 32 123102 |

23 | Phys. Rev. A 68 035201 |

24 | |

25 | Phys. Rev. A 60 4476 |

26 | Phys. Rev. B 90 245405 |

27 | Chin. Phys. B 23 063101 |

28 | Phys. Rev. A 49 982 |

29 | Eur. Phys. J. D 38 353 |

30 | Phys. Rev. A 51 3883 |

31 | Phys. Rev. A 10 1141 |

32 | Phys. Rev. A 10 1131 |

33 | Phys. Rev. A 87 023402 |

34 | Phys. Rev. A 71 043807 |

35 | Phys. Rev. A 79 013404 |

36 | Phys. Rev. A 66 020101 |