1. FFK-14, Dubna, December 3, 2014 11 Higher-order constraints on precision of the time-frequency metrology of atoms in optical lattices V. D. Ovsiannikov Physics Department, Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia V. G. Pal'chikov Institute of Metrology for Time and Space at National Research Institute for Physical--Technical and Radiotechnical Measurements, Mendeleevo, Moscow Region 141579, Russia

2. FFK-14, Dubna, December 3, 2014 22 Contents 1. Principal goal: to determine irremovable clock-frequency shifts induced by multipole, nonlinear and anharmonic interaction of neutral Sr, Yb and Hg atoms with an optical lattice of a magic wavelength (MWL). 2. Attractive lattice of a Red-detuned MWL: a) Spatial distribution of atom-lattice interaction. b) Lattice potential wells. c) Lattice-induced clock-frequency shift. d) Numerical estimates of electromagnetic susceptibilities and clock-frequency shifts of neutral Sr, Yb and Hg atoms in a lattice of a red-detuned MWL. e) MWL for an atom in a traveling wave (TW). f) MWL for an atom in a standing wave (SW). g) MWL for equal dipole polarizabilities (EDP) in ground and excited clock states h) MWL precision. 3. Elimination of nonlinear effects in a lattice of Sr blue-detuned MWL of λ m =389.889 nm. a) Spatial distribution of interaction between atom and a repulsive lattice. b) Motion-insensitive standing-wave MWL (SW MWL). c) Numerical estimates of the blue-detuned-lattice-induced shifts

3. FFK-14, Dubna, December 3, 2014 3 «Clock» transition Typical structure of energy levels in alkaline-earth and alkaline-earth-like atoms (Mg, Ca, Sr, Zn, Cd, Yb, Hg) Radiation transitions between metastable and ground states, stimulated in odd isotopes by the hyperfine interaction, is strictly forbidden in even isotopes. This prohibition makes extremely narrow the line of the clock transition, which may be stimulated by an external magnetic field or by the circularly polarized lattice wave. This transition may be used as an oscillator with extremely high quality The width of the oscillator depends on (and may be regulated by) the intensity of the lattice wave or a static magnetic field. M2 E1 2ω(M1+E1) (ΔS=1)

4. FFK-14, Dubna, December 3, 2014 4 Natural isotope composition Even isotopes Odd isotopes (J=0) abundance abundance (J≠0) 24,26 Mg: 90% 25 Mg: 10% (J=5/2) 40→48 Ca: 98.7% 43 Ca: 1.3% (J=7/2) 84,86,88 Sr: 93% 87 Sr: 7% (J=9/2) 168→176 Yb: 73% 171,173 Yb: 27% (J=1/2, 5/2) 196→204 Hg: 69.8% 199,201 Hg: 30.2% (J=1/2,3/2) 106→116 Cd: 75% 111,113 Cd: 25% (J=1/2) 64→70 Zn:95.9% 67 Zn: 4.1% (J=5/2)

5. FFK-14, Dubna, December 3, 2014 55 2. Red-detuned MWL 2.a) Spatial distribution of atom-lattice interaction

6. FFK-14, Dubna, December 3, 2014 6

7. 7 2b) Lattice potential wells. Clock-level shift is the Lattice-trap potential energy

8. FFK-14, Dubna, December 3, 2014 8 n=0 1 2 3 4 Stark-trap potential and vibration-state energies of an atom in a standing wave of a lattice field 5

9. FFK-14, Dubna, December 3, 2014 99 depth harmonic oscillations anharmonic energy is the recoil energy of a lattice photon

10. FFK-14, Dubna, December 3, 2014 10 The strict magic-wavelength condition should imply the equality To hold this condition, the equality should hold for the susceptibilities: The most important of which is the E1 polarizability, so the primitive MWL condition implied

11. FFK-14, Dubna, December 3, 2014 11 Wavelength dependences of the linear in the lattice-laser intensity Stark shifts for Yb atoms in their upper 6s6p 3 P 0 (e) and lower 6s 2 1 S 0 (g) clock states at 10kW/cm 2. λ mag =762.3 nm (theory) λ mag =759.3537 nm (experiment) nm kHz

12. FFK-14, Dubna, December 3, 2014 12 Wavelength dependence of the linear in the lattice-laser intensity I =25 kW/cm 2 Stark shifts ΔE/kHz of Hg atoms in their upper 6s6p 3 P 0 (e) and lower 6s 2 1 S 0 (g) clock states. λ mag =364 nm (theory) λ mag =362.53 nm (experiment) nm ΔE/kHz

13. FFK-14, Dubna, December 3, 2014 13 The wavelength dependence of Stark shifts ΔE/kHz of Mg clock levels. The shifts of the ground state 3s 2 1 S 0 (red solid line) and the excited state 3s3p 3 P 0 (black dashed curve) in a lattice field of a laser intensity I =40 kW/cm^2 (chosen provisionally to provide the Stark trapping potential depth of about 40-50 photon recoil energies). The magic wavelength λ mag ≈453 nm is determined by the point of intersection of the lines. nm kHz

14. FFK-14, Dubna, December 3, 2014 14 Stark shifts of magnesium clock levels in case of a right-handed circular polarization of lattice. Red solid line is for the ground state 3s 2 1 S 0, all the rest for different magnetic sublevels of the excited 3s3p 3 P 1 state in a lattice field of a laser intensity I =40 kW/cm^2 (about 40-50 photon recoil energies). The magic wavelengths (MWL) are 419.5 nm for M=-1 and 448.1 nm for M=0 magnetic substates of the upper clock level 3s3p( 3 P 1 ), correspondingly. There is no MWL for the state M=1 in a circularly polarized lattice. kHz nm

15. FFK-14, Dubna, December 3, 2014 15 Stark shifts of magnesium clock levels in case of a linearly polarized lattice wave of the laser intensity I =40 kW/cm^2. The shifts of states 3s3p 3 P 1 (M=±1) are identical and completely equivalent to that of the state M=0 in a circularly polarized lattice beam with the MWL 448.1 nm, which is nearly equal to the MWL 453.5 nm for an averaged over M, independent of polarization (scalar) shift; the MWL for the M=0 state is 527 nm. The shifts of upper clock states experience the resonance enhancement on the 3s4s( 3 S 1 )-state at 517 nm, except for the state M=0 in the case of linearly polarized lattice and M=1 (M=-1) state in the right-handed (left-handed) case of circular polarization nm kHz

16. FFK-14, Dubna, December 3, 2014 16 2c) Lattice-induced clock-frequency shift.

17. FFK-14, Dubna, December 3, 2014 17 If then

18. FFK-14, Dubna, December 3, 2014 18 2.d) Numerical estimates of electromagnetic susceptibilities and clock-frequency uncertainties AtomSrYbHg /nm813.42727389.889759.35374362.53 45.2– 92.740.55.70 1.38– 13.6-8.068.25 –200.01150– 366.3– 2.50 02.4804.34 – 311.01550240.22.53 02.3706.37 25.0574.818.0313.1 0.25410.30.7200.134 3.4715.12.007.57 Table 1

19. FFK-14, Dubna, December 3, 2014 19 The wavelength (in nanometers) dependence of the hyperpolarizability (in μHz/(kW/cm 2 ) 2 ) of clock transition in Yb atoms for the linear (red dashed curve) and circular (black solid curve) polarization of the lattice-laser wave. The vertical lines indicate positions of two-photon resonances on 6s8p( 3 P 2 ) state at 754.226 nm, 6s8p( 3 P 0 ) state at 759.71 nm (this resonance appears only for linear polarization) and 6s5f( 3 F 2 ) state at 764.953 nm nm μHz/(kW/cm 2 ) 2 3P23P2 3P03P0 3F23F2

20. FFK-14, Dubna, December 3, 2014 20 The wavelength (in nanometers) dependence of the hyperpolarizability (in μHz/(kW/cm 2 ) 2 ) of clock transition in Sr atoms for the linear (red dashed curve) and circular (black solid curve) polarization of the lattice-laser wave. The vertical lines indicate positions of two-photon resonances on 5s7p( 3 P 2 ) state at 795.5 nm, 5s7p( 3 P 0 ) state at 797 nm (this resonance does not appear for circular polarization) and 5s4f( 3 F 2 ) state at 818.6 nm nm μHz/(kW/cm 2 ) 2 3P03P0 3P23P2 3F23F2

21. FFK-14, Dubna, December 3, 2014 21 2.e) MWL for an atom in a traveling wave Due to homogeneous spatial distribution of intensity in a traveling wave, the second-order shift of clock levels is determined by the sum of E1, E2 and M1 polarizabilities So, the MWL is determined from the equality

22. FFK-14, Dubna, December 3, 2014 22 At this condition, and coefficients for the intensity dependence of the shift are

23. FFK-14, Dubna, December 3, 2014 23 (a) Sr TW MWL (n=0)(b) Yb TW MWL (n=0) Intensity I /(kW/cm 2 ) dependence of the lattice-induced clock-frequency shift (Δν/mHz) for (a) Sr and (b) Yb atoms in a linearly (red solid), elliptically (green dotted) and circularly (black dashed) polarized lattice of a traveling-wave MWL kW/cm 2 mHz kW/cm 2 mHz

24. FFK-14, Dubna, December 3, 2014 24 Intensity _ I _/(kW/cm 2 )) dependence of the lattice-induced clock-frequency shift (Δν/mHz) in Hg atoms in a linearly (red solid), elliptically (green dotted) and circularly (black dashed) polarized lattice of a traveling-wave MWL. The imaginary part – clock- frequency broadening for linear (black solid) and circular (red dashed) polarizations are negative values (thin curves at the plot bottom). Hg TW MWL (n=0) mHz kW/cm 2

25. FFK-14, Dubna, December 3, 2014 25 2.f) MWL for an atom in a standing wave of an optical lattice (motion-insensitive MWL) At this condition,

26. FFK-14, Dubna, December 3, 2014 26 (a) Sr SW MWL (n=0) (b) Yb SW MWL (n=0) Intensity _ I _/(kW/cm 2 )) dependence of the lattice-induced clock-frequency shift (Δν/mHz) for (a) Sr and (b) Yb atoms in a linearly (red solid) elliptically (green dotted) and circularly (black dashed) polarized lattice of a standing-wave MWL kW/cm 2 mHz

27. FFK-14, Dubna, December 3, 2014 27 Intensity _ I _/(kW/cm 2 )) dependence of the lattice-induced clock-frequency shift (Δν/mHz) in Hg atoms in a linearly (red solid), elliptically (green dotted) and circularly (blue dashed) polarized lattice of a standing-wave MWL. The imaginary part – clock- frequency broadening for linear (red solid) and circular (black dashed) polarizations are negative values (thin curves at the plot top). Hg SW MWL (n=0) kW/cm 2 mHz

28. FFK-14, Dubna, December 3, 2014 28 2.g) MWL for equal dipole polarizabilities in ground and excited clock states At this condition,

29. FFK-14, Dubna, December 3, 2014 29 Intensity I /(kW/cm 2 )) dependence of the lattice-induced clock-frequency shift (Δν/mHz) for: (a) Sr and (b) Yb atoms in a linearly (red solid), elliptically (green dotted) and circularly (black dashed) polarized lattice of an “equal dipole polarizabilities” MWL. (a) Sr EDP MWL (n=0)(b) Yb EDP MWL (n=0) mHz kW/cm 2

30. FFK-14, Dubna, December 3, 2014 30 Intensity _ I _/(kW/cm 2 )) dependence of the lattice-induced clock-frequency shift, Re(Δν/mHz) in Hg atoms in a linearly (red solid), elliptically (green dotted) and circularly (blue dashed) polarized lattice of an “equal dipole polarizabilities” MWL. The imaginary part – clock-frequency broadening for linear (red solid) and circular (black dashed) polarizations are negative values. Hg EDP MWL (n=0) kW/cm 2 mHz

31. FFK-14, Dubna, December 3, 2014 31 Dependence of the lattice-induced clock-frequency shift on the lattice intensity, circular polarization degree ξ and on the vibration quantum number n in Yb 1. For the TW MWL ( ): 2. For the SW MWL ( ): 3. For the ED MWL ( ):

32. FFK-14, Dubna, December 3, 2014 32 2.h) MWL precision Uncertainties of the clock frequency are directly proportional to the uncertainties of the MWL: The principal contribution to the derivative comes from the E1 polarizability in the lattice well depth and in the frequency of harmonic vibrations:

33. FFK-14, Dubna, December 3, 2014 33 A 15% precision estimate of frequency derivatives for polarizabilities in Sr atoms gives: For I=10 kW/cm 2 the departure from the magic frequency Δω m < 100 kHz provides the fractional uncertainty of the clock frequency at the level

34. FFK-14, Dubna, December 3, 2014 34 Conclusions 1 (Red-detuned MWL) 1. At least 3 different methods may be used for determining MWL for the red- detuned optical lattice, providing MWL, and their mean value (in Sr, ). These MWLs provide different lattice-induced shifts and uncertainties on the clock frequency, with different dependencies on the lattice laser intensity. 2. The polarizabilities contribute only to the lattice potential depth and harmonic oscillation frequencies and never contribute to the anharmonic terms, where the contributions come from hyperpolarizabilities only. 3.The hyperpolarizability provides quadratic, power 3/2 and linear contributions to the lattice-potential depth, frequency of vibrations and anharmonic interaction, correspondingly. At I>10 kW/cm 2 the hyperpolarizability contribution to the lattice-induced shift in Sr and Yb atoms becomes comparable or exceeding that of polarizability. In Hg atoms the hyperpolarizability terms do not exceed 10% of polarizability terms at I<100 kW/ cm 2.

35. FFK-14, Dubna, December 3, 2014 35 3. Elimination of nonlinear effects in a lattice of Sr blue-detuned MWL of λ m =389.889 nm 3.1. Spatial distribution of interaction between atom and a repulsive lattice. Trapped atoms locate near nodes of the lattice field: Atom-lattice interaction:

36. FFK-14, Dubna, December 3, 2014 36 The second-order term is linear in the laser intensity I and is determined by the E1 and multipole polarizabilities ( E2, M1…) : The fourth-order term is quadratic in the laser intensity I and is determined by the dipole hyperpolarizability:

37. FFK-14, Dubna, December 3, 2014 37 The Stark-effect energy determines the trap potential energy for excited and ground-state atom: is the depth of the lattice well, quite similar to the red-detuned lattice, but the position- independent energy shift involves only the E2-M1 polarizability, in contrast to the red-detuned MWL, where both E1 polarizability and hyperpolarizability were involved. The difference between top (X= λ/4) and bottom (X=0) of the trap potential

38. FFK-14, Dubna, December 3, 2014 38 bottomharmonic oscillationsanharmonic energy with the energy

39. FFK-14, Dubna, December 3, 2014 39 Lattice-induced clock-frequency shift is where

40. FFK-14, Dubna, December 3, 2014 40 3.2. Motion-insensitive standing-wave MWL (SW MWL) the lattice-induced clock-frequency shift is The hyperpolarizability effects in the shift and broadening (caused by two-photon ionization) are strongly reduced by the factor (as follows from the data of table 1 for the blue MWL, where intensity is in kW/cm 2 ). is determined by the equality

41. FFK-14, Dubna, December 3, 2014 41 From the data of table 1 for the Sr blue-detuned MWL we have 3.3. Numerical estimates of the blue-detuned-lattice- induced shifts The hyperpolarizability effects in the shift and broadening (caused by two-photon ionization) are strongly reduced by the factor (as follows from the data of table 1 for the blue MWL, where intensity is in kW/cm 2 ). In the blue-detuned lattice of Sr atoms the shift of the clock frequency is directly proportional to the lattice-laser intensity and is mainly determined by the difference of E2-M1 polarizabilities of the clock levels. The influence of hyperpolarizability appears only in the third digit number. The broadening (imaginary part of the shift) is more than 4 orders smaller than the shift. For I=10 kW/cm 2 the lattice-induced shift is about 137 mHz, the lattice- induced width is about 6 μHz.

42. FFK-14, Dubna, December 3, 2014 42 Conclusions 2 1.The motion-insensitive blue-detuned MWL depends on only the polarizabilities and is not influenced by hyperpolarizability effects. 2.The hyperpolarizability effects on the clock levels appear only in anharmonic interaction of atom with lattice. 3.The intensity of the lattice laser is sufficient to trap atoms cooled to 1 μK at the lowest vibrational state. 4.To achieve the clock frequency precision at the 18 th decimal place, the irremovable multipole-interaction-induced shift by the field of optical lattice should be taken into account with uncertainty below 1.0%.

43. FFK-14, Dubna, December 3, 2014 43 Thank you for attention!