Atomic calculations: recent advances and modern applications JQI seminar July 8, 2010 Marianna Safronova

Selected applications of atomic calculations Study of fundamental symmetries Atomic clocks Optical cooling and trapping and quantum information Examples: magic wavelengths Present status of theory How accurate are theory values? Present challenges Development of CI+all-order method for group II atoms Future prospects Outline

Atomic calculations for study of fundamental symmetries

Transformations and symmetries Translation Momentum conservation Translation in timeEnergy conservation RotationConservation of angular momentum [C] Charge conjugationC-invariance [P] Spatial inversionParity conservation (P-invariance) [T] Time reversalT-invariance [CP] [CPT]

Transformations and symmetries Translation Momentum conservation Translation in timeEnergy conservation RotationConservation of angular momentum [C] Charge conjugationC-invariance [P] Spatial inversionParity conservation (P-invariance) [T] Time reversalT-invariance [CP] [CPT]

Parity-transformed world: Turn the mirror image upside down. The parity-transformed world is not identical with the real world. Parity is not conserved. Parity Violation r  ─ r

Parity violation in atoms Nuclear spin-independent PNC: Searches for new physics beyond the Standard Model Weak Charge Q W Nuclear spin-dependent PNC: Study of PNC In the nucleus e e q q Z0Z0 a Nuclear anapole moment Nuclear anapole moment

Standard Model

(1) Search for new processes or particles directly (2) Study (very precisely!) quantities which Standard Model predicts and compare the result with its prediction High energies Cs experiment, University of Colorado Searches for New Physics Beyond the Standard Model Weak charge Q W Low energies

C.S. Wood et al. Science 275, 1759 (1997) 0.3% accuracy The most precise measurement of PNC amplitude (in cesium) 6s 7s F=4 F=3 F=4 F=3 1 2 Stark interference scheme to measure ratio of the PNC amplitude and the Stark-induced amplitude  1 2

Parity violation in atoms Nuclear spin-dependent PNC: Study of PNC In the nucleus a Nuclear anapole moment Nuclear anapole moment Present status: agreement with the Standard Model Nuclear spin-independent PNC: Searches for new physics beyond the Standard Model

Valence nucleon density Parity-violating nuclear moment Anapole moment Spin-dependent parity violation: Nuclear anapole moment Nuclear anapole moment is parity-odd, time-reversal-even E1 moment of the electromagnetic current operator. 6s 7s F=4 F=3 F=4 F=3 1 2 a

Constraints on nuclear weak coupling contants W. C. Haxton and C. E. Wieman, Ann. Rev. Nucl. Part. Sci. 51, 261 (2001)

Nuclear anapole moment: test of hadronic weak interations *M.S. Safronova, Rupsi Pal, Dansha Jiang, M.G. Kozlov, W.R. Johnson, and U.I. Safronova, Nuclear Physics A 827 (2009) 411c NEED NEW EXPERIMENTS!!! The constraints obtained from the Cs experiment inconsistent were found to be inconsistent with constraints from other nuclear PNC measurements, which favor a smaller value of the 133 Cs anapole moment. All-order (LCCSD) calculation of spin-dependent PNC amplitude: k = 0.107(16)* [ 1% theory accuracy ] No significant difference with previous value k = 0.112(16) is found. Fr, Yb, Ra +

Why do we need Atomic calculations to study parity violation? (1)Present experiments can not be analyzed without theoretical value of PNC amplitude in terms of Q w. (2)Theoretical calculation of spin-dependent PNC amplitude is needed to determine anapole moment from experiment. (3) Other parity conserving quantities are needed. Note: need to know theoretical uncertainties!

Transformations and symmetries Translation Momentum conservation Translation in timeEnergy conservation RotationConservation of angular momentum [C] Charge conjugationC-invariance [P] Spatial inversionParity conservation (P-invariance) [T] Time reversalT-invariance [CP] [CPT]

Permanent electric-dipole moment ( EDM ) Time-reversal invariance must be violated for an elementary particle or atom to possess a permanent EDM. S d S d

EDM and New physics Many theories beyond the Standard Model predict EDM within or just beyond the present experimental capabilities. David DeMille, Yale PANIC 2005

Atomic calculations and search for EDM EDM effects are enhanced in some heavy atoms and molecules. Theory is needed to calculate enhancement factors and search for new systems for EDM detection. Recent new limit on the EDM of 199 Hg Phys. Rev. Lett. 102, (2009) | d( 199 Hg) | < 3.1 x e cm

Atomic clocks Microwave Transitions Optical Transitions Blackbody Radiation Shifts and Theoretical Contributions to Atomic Clock Research, M. S. Safronova, Dansha Jiang, Bindiya Arora, Charles W. Clark, M. G. Kozlov, U. I. Safronova, and W. R. Johnson, Special Issue of IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 57, 94 (2010).

(1)Prediction of atomic properties required for new clock proposals New clock proposals require both estimation of the atomic properties for details of the proposals (transition rates, lifetimes, branching rations, magic wavelength, scattering rates, etc.) and evaluation of the systematic shifts (Zeeman shift, electric quadruple shift, blackbody radiation shift, ac Stark shifts due to various lasers fields, etc.). (2)Determination of the quantities contributing to the uncertainty budget of the existing schemes. In the case of the well-developed proposals, one of the main current uncertainty issues is the blackbody radiation shift. Atomic calculations & more precise clocks

Blackbody radiation shift T = 300 K Clock transition Level A Level B  BBR T = 0 K Transition frequency should be corrected to account for the effect of the black body radiation at T=300K.

BBR shift and polarizability BBR shift of atomic level can be expressed in terms of a scalar static polarizability to a good approximation [1]: [1] Sergey Porsev and Andrei Derevianko, Physical Review A 74, R (2006) Dynamic correction is generally small. Multipolar corrections (M1 and E2) are suppressed by  2 [1]. Vector & tensor polarizability average out due to the isotropic nature of field. Dynamic correction

microWave transitions optical transitions 4d 5/2 Sr + Lowest-order polarizability 5s 1/2 Cs 6s F=3 6s F=4 In lowest (second) order the polarizabilities of ground hyperfine 6s 1/2 F=4 and F=3 states are the same. Therefore, the third-order F-dependent polarizability  F (0) has to be calculated. terms term

Blackbody radiation shifts in optical frequency standards: (1) monovalent systems (2) divalent systems (3) other, more complicated systems Mg, Ca, Zn, Cd, Sr, Al +, In +, Yb, Hg ( ns 2 1 S 0 – nsnp 3 P) Hg + (5d 10 6s – 5d 9 6s 2 ) Yb + (4f 14 6s – 4f 13 6s 2 )

Example: BBR shift in s r + PresentRef.[1]Ref. [2]  (5s 1/2 → 4d 5/2 ) 0.250(9)0.33(12)0.33(9) Present  0 (5s 1/2 ) 91.3(9)  0 (4d 5/2 ) 62.0(5) Sr + : Dansha Jiang, Bindiya Arora, M. S. Safronova, and Charles W. Clark, J. Phys. B (2010). Ca + : Bindiya Arora, M.S. Safronova, and Charles W. Clark, Phys. Rev. A 76, (2007) nf tail contribution issue has been resolved Need precise lifetime measurements 1% Dynamic correction, E2 and M1 corrections negligible [1] A. A. Madej et al., PRA 70, (2004); [2] H. S. Margolis et al., Science 306, 19 (2004).

Summary of the fractional uncertainties   due to BBR shift and the fractional error in the absolute transition frequency induced by the BBR shift uncertainty at T = 300 K in various frequency standards. M. S. Safronova et al., IEEE - TUFFC 57, 94 (2010). 5  Present

Atoms in optical lattices Cancellations of ac Start shifts: state-insensitive optical cooling and trapping State-insensitive bichromatic optical trapping schemes Simultaneous optical trapping of two different alkali-metal species Determinations of wavelength where atoms will not be trapped Calculations of relevant atomic properties: dipole matrix elements, atomic polarizabilities, magic wavelengths, scattering rates, lifetimes, etc. Optimizing the fast Rydberg quantum gate, M.S. Safronova, C. J. Williams, and C. W. Clark, Phys. Rev. A 67, (2003). Frequency-dependent polarizabilities of alkali atoms from ultraviolet through infrared spectral regions, M.S. Safronova, Bindiya Arora, and Charles W. Clark, Phys. Rev. A 73, (2006) Magic wavelengths for the ns-np transitions in alkali-metal atoms, Bindiya Arora, M.S. Safronova, and C. W. Clark, Phys. Rev. A 76, (2007). Theory and applications of atomic and ionic polarizabilities (review paper), J. Mitroy, M.S. Safronova, and Charles W. Clark, submitted to J. Phys. B (2010) State-insensitive bichromatic optical trapping, Bindiya Arora, M.S. Safronova, and C. W. Clark, submitted to Phys. Rev. A (2010)

Magic wavelength magic is the wavelength for which the optical potential U experienced by an atom is independent on its state Magic wavelength magic is the wavelength for which the optical potential U experienced by an atom is independent on its state Atom in state A sees potential U A Atom in state B sees potential U B magic wavelength

wavelength α  S State P State Locating magic wavelength

Core term Valence term (dominant) Compensation term Example: Scalar dipole polarizability Electric-dipole reduced matrix element Polarizability of an alkali atom in a state v

many-body perturbation theory Sum over infinite sets of many-body perturbation theory (MBPT) terms. Calculate the atomic wave functions and energies Calculate various matrix elements Calculate “derived” properties useful for particular problems Scheme: Relativistic all-order method

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) Results for alkali-metal atoms: E1 transition matrix elements in

Example: “Best set” Rb matrix elements

Magic wavelengths for the 5p 3/2 - 5s transition of R b.

ac Stark shifts for the transition from 5p 3/2 F ′ =3 M ′ sublevels to 5s FM sublevels in R b. The electric field intensity is taken to be 1 MW/ cm 2.

 0 -  2  0 +  2 magic M J = ±3/2 M J = ±1/2 Other* magic around 935nm * Kimble et al. PRL 90(13), (2003) Magic wavelength for Cs

A combination of trapping and control lasers is used to minimize the variance of the potential experienced by the atom in ground and excited states. Trap laser Control laser atom bichromatic optical trapping

Surface plot for the 5s and 5p 3/2 |m| = 1/2 state polarizabilities as a function of laser wavelengths 1 and 2 for equal intensities of both lasers.

Magic wavelengths for the the 5s and 5p 3/2 | m| = 1/2 states for 1 = nm and 2 =2 1 for various intensities of both lasers. The intensity ratio (  1 /  2 ) 2 ranges from 1 to 2.

Other applications Variation of fundamental constants Long-range interaction coefficients Data for astrophysics Actinide ion studies for chemistry models Benchmark tests of theory and experiment Cross-checks of various experiments Determination of nuclear magnetic moment in Fr Calculation of isotope shifts …

MONovalent systems: Very brief summary of what we calculated with all-order method Properties Energies Transition matrix elements (E1, E2, E3, M1) Static and dynamic polarizabilities & applications Dipole (scalar and tensor) Quadrupole, Octupole Light shifts Black-body radiation shifts Magic wavelengths Hyperfine constants C 3 and C 6 coefficients Parity-nonconserving amplitudes (derived weak charge and anapole moment) Isotope shifts (field shift and one-body part of specific mass shift) Atomic quadrupole moments Nuclear magnetic moment (Fr), from hyperfine data Systems Li, Na, Mg II, Al III, Si IV, P V, S VI, K, Ca II, In, In-like ions, Ga, Ga-like ions, Rb, Cs, Ba II, Tl, Fr, Th IV, U V, other Fr-like ions, Ra II

how to evaluate uncertainty of theoretical calculations?

Theory: evaluation of the uncertainty HOW TO ESTIMATE WHAT YOU DO NOT KNOW? I. Ab initio calculations in different approximations: (a) Evaluation of the size of the correlation corrections (b) Importance of the high-order contributions (c) Distribution of the correlation correction II. Semi-empirical scaling: estimate missing terms

Example: quadrupole moment of 3d 5/2 state in C a + Electric quadrupole moments of metastable states of Ca+, Sr+, and Ba+, Dansha Jiang and Bindiya Arora and M. S. Safronova, Phys. Rev. A 78, (2008)

Lowest order D 5/2 quadrupole moment in Ca +

Third order Lowest order D 5/2 quadrupole moment in Ca +

All order (SD) Third order Lowest order D 5/2 quadrupole moment in Ca +

All order (SDpT) All order (SD) Third order Lowest order D 5/2 quadrupole moment in Ca +

Coupled-cluster SD (CCSD) All order (SDpT) All order (SD) Third order Lowest order D 5/2 quadrupole moment in Ca +

Coupled-cluster SD (CCSD) All order (SDpT) All order (SD) Third order Lowest order D 5/2 quadrupole moment in Ca + Estimate omitted corrections

All order (SD), scaled All-order (CCSD), scaled All order (SDpT) All order (SDpT), scaled Third order Final results: 3d 5/2 quadrupole moment Lowest order (13)

All order (SD), scaled All-order (CCSD), scaled All order (SDpT) All order (SDpT), scaled Third order Final results: 3d 5/2 quadrupole moment Lowest order (13) Experiment 1.83(1) Experiment: C. F. Roos, M. Chwalla, K. Kim, M. Riebe, and R. Blatt, Nature 443, 316 (2006).

Development of high-precision methods Present status of theory and need for further development Development of high-precision methods Present status of theory and need for further development

All-order Correlation potential CI+MBPT

Atomic clocks Study of parity violation (Yb) Search for EDM (Ra) Degenerate quantum gases, alkali-group II mixtures Quantum information Variation of fundamental constants Motivation: study of group II – type systems Divalent ions: Al +, In +, etc. Mg Ca Sr Ba Ra Zn Cd Hg Yb

Summary of theory methods for atomic structure Configuration interaction (CI) Many-body perturbation theory Relativistic all-order method (coupled-cluster) Correlation - potential method Configuration interaction + second-order MBPT Configuration interaction + all-order method* *under development

GOAL of the present project: calculate properties of group II atoms with precision comparable to alkali-metal atoms

Configuration interaction method Single-electron valence basis states Example: two particle system:

Configuration interaction + all-order method CI works for systems with many valence electrons but can not accurately account for core-valence and core-core correlations. All-order (coupled-cluster) method can not accurately describe valence-valence correlation for large systems but accounts well for core-core and core-valence correlations. Therefore, two methods are combined to acquire benefits from both approaches.

Configuration interaction method + all-order H eff is modified using all-order calculation are obtained using all-order method used for alkali-metal atoms with appropriate modification In the all-order method, dominant correlation corrections are summed to all orders of perturbation theory.

CI + ALL-ORDER RESULTS Atom CI CI + MBPT CI + All-order Mg 1.9% 0.11%0.03% Ca 4.1% 0.7%0.3% Zn 8.0% 0.7%0.4 % Sr 5.2% 1.0%0.4% Cd 9.6% 1.4%0.2% Ba 6.4% 1.9%0.6% Hg 11.8% 2.5%0.5% Ra 7.3% 2.3%0.67% Two-electron binding energies, differences with experiment Development of a configuration-interaction plus all-order method for atomic calculations, M.S. Safronova, M. G. Kozlov, W.R. Johnson, Dansha Jiang, Phys. Rev. A 80, (2009).

C d, Z n, and S r Polarizabilities, preliminary results (a.u.) ZnCICI+MBPTCI + All-order 4s 2 1 S s4p 3 P CdCICI+MBPTCI+All-order 5s 2 1 S s5p 3 P *From expt. matrix elements, S. G. Porsev and A. Derevianko, PRA 74, R (2006). SrCI +MBPTCI+all-orderRecomm.* 5s 2 1 S (2) 5s5p 3 P (3.6)

C d, Z n, S r, and Hg magic wavelengths, preliminary results (nm) [1] A. D. Ludlow et al., Science 319, 1805 (2008) [2] H. Hachisu et al., Phys. Rev. Lett. 100, (2008)

Conclusion Atomic Clocks S 1/2 P 1/2 D 5/2 Qubit Parity Violation Quantum information Future: New Systems New Methods, New Problems