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Higher-order constraints on precision of the time-frequency metrology of atoms in optical lattices

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

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Higher-order constraints on precision of the time-frequency metrology of atoms in optical lattices

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  1. 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 FFK-14, Dubna, December 3, 2014 1

  2. Contents 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) . Attractive lattice of a Red-detuned MWL: Spatial distribution of atom-lattice interaction. Lattice potential wells. Lattice-induced clock-frequency shift. Numerical estimates of electromagnetic susceptibilities and clock-frequency shifts of neutral Sr, Yb and Hg atoms in a lattice of a red-detuned MWL. MWL for an atom in a traveling wave (TW). MWL for an atom in a standing wave (SW). MWL for equal dipole polarizabilities (EDP) in ground and excited clock states MWL precision. Elimination of nonlinear effects in a lattice of Sr blue-detuned MWL of λm=389.889 nm. Spatial distribution of interaction between atom and a repulsive lattice. Motion-insensitive standing-wave MWL (SW MWL). Numerical estimates of the blue-detuned-lattice-induced shifts FFK-14, Dubna, December 3, 2014 2

  3. 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. «Clock» transition M2 2ω(M1+E1) E1 (ΔS=1) FFK-14, Dubna, December 3, 2014

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

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

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

  7. 2b) Lattice potential wells. Clock-level shift is the Lattice-trap potential energy FFK-14, Dubna, December 3, 2014

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

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

  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 FFK-14, Dubna, December 3, 2014

  11. kHz nm Wavelength dependences of the linear in the lattice-laser intensity Stark shifts forYbatoms in their upper 6s6p3P0 (e) and lower 6s21S0 (g) clock states at 10kW/cm2. λmag =762.3 nm (theory) λmag=759.3537 nm(experiment) FFK-14, Dubna, December 3, 2014

  12. ΔE/kHz nm Wavelength dependence of the linear in the lattice-laser intensity I=25 kW/cm2 Stark shifts ΔE/kHz ofHg atoms in their upper 6s6p3P0 (e) and lower 6s21S0 (g) clock states. λmag=364 nm (theory)λmag=362.53 nm (experiment) FFK-14, Dubna, December 3, 2014

  13. kHz nm The wavelength dependence of Stark shifts ΔE/kHz of Mg clock levels. The shifts of the ground state 3s21S0 (red solid line) and the excited state 3s3p 3P0(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 λmag≈453 nm is determined by the point of intersection of the lines. FFK-14, Dubna, December 3, 2014

  14. kHz nm Stark shifts of magnesium clock levelsin case of a right-handed circular polarization of lattice. Red solid line is for the ground state 3s21S0, all the rest for different magnetic sublevels of the excited 3s3p 3P1 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(3P1), correspondingly. There is no MWL for the state M=1 in a circularly polarized lattice. FFK-14, Dubna, December 3, 2014

  15. kHz nm Stark shifts of magnesium clock levels in case of a linearly polarized lattice wave ofthelaser intensity I=40 kW/cm^2. The shifts of states 3s3p 3P1 (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 MWLfor the M=0 state is 527 nm. The shifts of upper clock states experience the resonance enhancement on the 3s4s(3S1)-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 FFK-14, Dubna, December 3, 2014

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

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

  18. 2.d) Numerical estimates of electromagnetic susceptibilities and clock-frequency uncertainties Table 1 FFK-14, Dubna, December 3, 2014 18

  19. μHz/(kW/cm2)2 nm 3P0 3F2 3P2 The wavelength (in nanometers) dependence of the hyperpolarizability (in μHz/(kW/cm2)2) of clock transition inYb 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(3P2) state at 754.226 nm, 6s8p(3P0) state at 759.71 nm (this resonance appears only for linear polarization) and 6s5f(3F2) state at 764.953 nm FFK-14, Dubna, December 3, 2014

  20. μHz/(kW/cm2)2 nm 3P2 3P0 3F2 The wavelength (in nanometers) dependence of the hyperpolarizability (in μHz/(kW/cm2)2) of clock transition in Sratoms 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(3P2) state at 795.5 nm, 5s7p(3P0) state at 797 nm (this resonance does not appear for circular polarization) and 5s4f(3F2) state at 818.6 nm 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 FFK-14, Dubna, December 3, 2014 21

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

  23. (a) Sr TW MWL (n=0) (b) Yb TW MWL (n=0) mHz mHz mHz kW/cm2 kW/cm2 Intensity I/(kW/cm2) 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 FFK-14, Dubna, December 3, 2014

  24. Hg TW MWL (n=0) mHz kW/cm2 Intensity _I_/(kW/cm2)) dependence of the lattice-induced clock-frequency shift (Δν/mHz) inHg 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). 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, FFK-14, Dubna, December 3, 2014 25

  26. (b) Yb SW MWL (n=0) (a) Sr SW MWL (n=0) mHz mHz kW/cm2 kW/cm2 Intensity _I_/(kW/cm2)) 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 FFK-14, Dubna, December 3, 2014

  27. Hg SW MWL (n=0) kW/cm2 mHz Intensity _I_/(kW/cm2)) 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). FFK-14, Dubna, December 3, 2014

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

  29. (a) Sr EDP MWL (n=0) (b) Yb EDP MWL (n=0) mHz mHz kW/cm2 kW/cm2 Intensity I/(kW/cm2)) 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. FFK-14, Dubna, December 3, 2014

  30. Hg EDP MWL (n=0) kW/cm2 mHz Intensity _I_/(kW/cm2)) 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. 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 ( ): FFK-14, Dubna, December 3, 2014 31

  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: FFK-14, Dubna, December 3, 2014 32

  33. A 15% precision estimate of frequency derivatives for polarizabilities in Sr atoms gives: For I=10 kW/cm2 the departure from the magic frequency Δωm < 100 kHz provides the fractional uncertainty of the clock frequency at the level FFK-14, Dubna, December 3, 2014 33

  34. Conclusions 1 (Red-detuned MWL) • At least 3 different methodsmay 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. • 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. • 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/cm2 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/cm2. FFK-14, Dubna, December 3, 2014 34

  35. 3. Elimination of nonlinear effects in a lattice of Sr blue-detuned MWL of λm=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: FFK-14, Dubna, December 3, 2014 35

  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: FFK-14, Dubna, December 3, 2014 36

  37. The Stark-effect energy determines the trap potential energy for excited and ground-state atom: The difference between top (X=λ/4) and bottom (X=0) of the trap potential 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. FFK-14, Dubna, December 3, 2014

  38. with the energy bottom harmonic oscillations anharmonic energy FFK-14, Dubna, December 3, 2014 38

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

  40. 3.2. Motion-insensitive standing-wave MWL (SW MWL) is determined by the equality 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/cm2). FFK-14, Dubna, December 3, 2014 40

  41. 3.3. Numerical estimates of the blue-detuned-lattice-induced shifts From the data of table 1 for the Sr blue-detuned MWL we have 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/cm2 the lattice-induced shift is about 137 mHz, the lattice-induced width is about 6 μHz. 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/cm2). FFK-14, Dubna, December 3, 2014 41

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

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

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