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DAMOP 2008 focus session: Atomic polarization and dispersion. Polarizabilities, Atomic Clocks, and Magic Wavelengths. May 29, 2008. Marianna Safronova Bindiya arora. Charles W. clark NIST, Gaithersburg. Outline. Motivation Method Applications

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## Polarizabilities, Atomic Clocks, and Magic Wavelengths

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**DAMOP 2008 focus session:**Atomic polarization and dispersion Polarizabilities, Atomic Clocks, and Magic Wavelengths May 29, 2008 Marianna Safronova Bindiya arora Charles W. clark NIST, Gaithersburg**Outline**• Motivation • Method • Applications • Frequency-dependent polarizabilities of alkali atoms • and magic frequencies • Atomic clocks: blackbody radiation shifts • Future studies**Motivation: 1Optically trapped atoms**Atom in state B sees potential UB Atom in state A sees potential UA State-insensitive cooling and trapping for quantum information processing**Motivation: 2**Atomic clocks: Next Generation Microwave Transitions Optical Transitions http://tf.nist.gov/cesium/fountain.htm, NIST Yb atomic clock**Motivation: 3**Parity violation studies with heavy atoms & search for Electron electric-dipole moment http://CPEPweb.org, http://public.web.cern.ch/, Cs experiment, University of Colorado**Motivation**• Development of the high-precision methodologies • Benchmark tests of theory and experiment • Cross-checks of various experiments • Data for astrophysics • Long-range interactions • Determination of nuclear magnetic and anapole moments • Variation of fundamental constants with time**Polarizability of an alkali atom in a statev**Valence term (dominant) Core term Compensation term Electric-dipole reduced matrix element Example: Scalar dipole polarizability**How to accurately calculate various matrix elements ?**Very precise calculation of atomic properties We also need to evaluate uncertainties of theoretical values!**How to accurately calculate various matrix elements ?**Very precise calculation of atomic properties WANTED! We also need to evaluate uncertainties of theoretical values!**All-order atomic wave function (SD)**core valence electron any excited orbital Core Lowest order Single-particle excitations Double-particle excitations**All-order atomic wave function (SD)**core valence electron any excited orbital Core Lowest order Single-particle excitations Double-particle excitations**Actual implementation: codes that write formulas**The derivation gets really complicated if you add triples! Solution: develop analytical codes that do all the work for you! Input: ASCII input of terms of the type Output: final simplified formula in LATEX to be used in the all-order equation**Problem with all-order extensions: TOO MANY TERMS**The complexity of the equations increases. Same issue with third-order MBPT for two-particle systems (hundreds of terms) . What to do with large number of terms? Solution: automated code generation !**Automated code generation**Codes that write formulas Codes that write codes Input: list of formulas to be programmed Output: final code (need to be put into a main shell) Features: simple input, essentially just type in a formula!**Results for alkali-metal atoms**Experiment Na,K,Rb: U. Volz and H. Schmoranzer, Phys. Scr. T65, 48 (1996), Cs: R.J. Rafac et al., Phys. Rev. A 60, 3648 (1999), Fr: J.E. Simsarian et al., Phys. Rev. A 57, 2448 (1998) Theory M.S. Safronova, W.R. Johnson, and A. Derevianko, Phys. Rev. A 60, 4476 (1999)**Theory: evaluation of the uncertainty**HOW TO ESTIMATE WHAT YOU DO NOT KNOW? • I. Ab initio calculations in different approximations: • Evaluation of the size of the correlation corrections • Importance of the high-order contributions • Distribution of the correlation correction • II. Semi-empirical scaling: estimate missing terms**Polarizabilities: Applications**• Optimizing the Rydberg gate • Identification of wavelengths at which two different alkali atoms have the same oscillation frequency for simultaneous optical trapping of two different alkali species. • Detection of inconsistencies in Cs lifetime and Stark shift experiments • Benchmark determination of some K and Rb properties • Calculation of “magic frequencies” for state-insensitive cooling and trapping • Atomic clocks: problem of the BBR shift • …**Polarizabilities: Applications**• Optimizing the Rydberg gate • Identification of wavelengths at which two different alkali atoms have the same oscillation frequency for simultaneous optical trapping of two different alkali species. • Detection of inconsistencies in Cs lifetime and Stark shift experiments • Benchmark determination of some K and Rb properties • Calculation of “magic frequencies” for state-insensitive cooling and trapping • Atomic clocks: problem of the BBR shift • …**ApplicationsFrequency-dependent polarizabilities of alkali**atoms from ultraviolet through infrared spectral regions Goal: First-principles calculations of the frequency-dependent polarizabilities of ground and excited states of alkali-metal atoms Determination of magic wavelengths**Magic wavelengths**Excited states: determination of magic frequencies in alkali-metal atoms for state-insensitive cooling and trapping, i.e. When does the ground state and excited np state has the same ac Stark shift? Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 052509 (2007) Na, K, Rb, and Cs**What is magic wavelength?**Atom in state B sees potential UB Atom in state A sees potential UA Magic wavelength lmagic is the wavelength for which the optical potential U experienced by an atom is independent on its state**α(l)**S State P State wavelength Locating magic wavelength**What do we need?**Lots and lots of matrix elements!**56 matrix elements in**What do we need? Lots and lots of matrix elements! Cs**What do we need?**Lots and lots of matrix elements! All-order “database”: over 700 matrix elements for alkali-metal atoms and other monovalent systems**Theory (This work)**Experiment* w=0 (3P1/2) Na 359.9(4) 359.2(6) (3P3/2) 361.6(4) 360.4(7) (3P3/2) -88.4(10) -88.3 (4) (4P1/2) K 606.7(6) 606(6) (4P3/2) 616(6) 614 (10) (4P3/2) -109(2) -107 (2) (5P1/2) Rb 807(14) 810.6(6) 869(14) (5P3/2) 857 (10) -166(3) (5P3/2) -163(3) *Zhu et al. PRA 70 03733(2004) Excellent agreement with experiments !**MJ = ±3/2**MJ = ±1/2 Frequency-dependent polarizabilities of Naatom in the ground and 3p3/2 states. The arrows show the magic wavelengths**Magic wavelengths for the 3p1/2 - 3s and 3p3/2 - 3s**transition of Na.**ac Stark shifts for the transition from 5p3/2F′=3**M′sublevels to 5s FM sublevels in Rb.The electric field intensity is taken to be 1 MW/cm2.**MJ = ±3/2**MJ = ±1/2 Magic wavelength for Cs lmagic Other* a0+ a2 lmagic around 935nm a0- a2 * Kimble et al. PRL 90(13), 133602(2003)**ac Stark shifts for the transition from 6p3/2F′=5**M′sublevels to 6s FM sublevels in Cs.The electric field intensity is taken to be 1 MW/cm2.**atomic clocksblack-body radiation ( BBR ) shift**Motivation: BBR shift gives the larges uncertainties for some of the optical atomic clock schemes, such as with Ca+**Blackbody radiation shift in optical frequency standard with**43Ca+ ion Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, 064501 (2007)**Motivation**For Ca+, the contribution from Blackbody radiation gives the largest uncertainty to the frequency standard at T = 300K • DBBR = 0.39(0.27) Hz [1] [1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004)**Frequency standard**Level B Clock transition Level A T = 0 K Transition frequency should be corrected to account for the effect of the black body radiation at T=300K.**Frequency standard**Level B Clock transition DBBR Level A T = 300 K Transition frequency should be corrected to account for the effect of the black body radiation at T=300K.**Why Ca+ ion?**The clock transition involved is 4s1/2F=4 MF=0→ 3d5/2 F=6 MF=0 4p3/2 Easily produced by non-bulky solid state or diode lasers 854 nm 4p1/2 3d5/2 866 nm Lifetime~1.2 s 393 nm 3d3/2 397 nm E2 729 nm 732 nm 4s1/2**BBR shift of a level**• The temperature-dependent electric field created by the blackbody radiation is described by (in a.u.) : • Frequency shift caused by this electric field is: Dynamic polarizability**BBR shift and polarizability**BBR shift can be expressed in terms of a scalar static polarizability: Dynamic correction Dynamic correction ~10-3 Hz. At the present level of accuracy the dynamic correction can be neglected. Vector & tensor polarizability average out due to the isotropic nature of field.**BBR shift for a transition**Effect on the frequency of clock transition is calculated as the difference between the BBR shifts of individual states. 3d5/2 729 nm 4s1/2**Need BBR shifts**Need ground and excited state scalar static polarizability NOTE: Tensor polarizability calculated in this work is also of experimental interest.**Contributions to the 4s1/2 scalar polarizability ( )**43Ca+(w = 0) Stail 6p1/2 6p3/2 0.01 0.01 0.06 5p3/2 5p1/2 0.01 0.01 4p1/2 4p3/2 24.4 48.4 Total: 76.1 ± 1.1 4s 3.3 Core**Contributions to the 3d5/2 scalar polarizability ( )**43Ca+ nf7/2 7-12f7/2 nf5/2 6f7/2 1.7 np3/2 tail 0.2 0.5 0.3 0.01 5f7/2 5p3/2 0.01 0.8 4f7/2 4p3/2 2.4 22.8 3d5/2 Total: 32.0 ± 1.1 3.3 Core**Comparison of our results for scalar static polarizabilities**for the 4s1/2 and 3d5/2 states of 43Ca+ ion with other available results [1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004) [2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005) [3] C.E. Theodosiou et. al. Phys. Rev. A 52, 3677 (1995)**Black body radiation shift**Comparison of black body radiation shift (Hz) for the 4s1/2- 3d5/2 transition of 43Ca+ ion at T=300K (E=831.9 V/m). An order of magnitude improvement is achieved with comparison to previous calculations [1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004) [2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005)**Black body radiation shift**Comparison of black body radiation shift (Hz) for the 4s1/2- 3d5/2 transition of 43Ca+ ion at T=300K (E=831.9 V/m). Sufficient accuracy to establish The uncertainty limits for the Ca+ scheme [1] C. Champenois et. al. Phys. Lett. A 331, 298 (2004) [2] Masatoshi Kajit et. al. Phys. Rev. A 72, 043404, (2005)

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