1 Optically polarized atoms Marcis Auzinsh, University of Latvia Dmitry Budker, UC Berkeley and LBNL Simon M. Rochester, UC Berkeley A 12-T superconducting NMR magnet at the EMSL(PNNL) laboratory, Richland, WA Dr. A. O. Sushkov, May 2007 Censorship

2 1845, Michael Faraday: magneto-optical rotation Chapter 4: Atoms in external fields  Medium Linear Polarization Circular Components Magnetic Field

3 Faraday looked for effect of magnetic field on spectra, but failed to find it 1896, Pieter Zeeman: sodium lines broaden under B 1897, Zeeman observed splitting of Cd lines into three components (“Normal” Zeeman effect) 1897, Hendrik Lorentz: classical explanation of ZE 1898, discovery of Resonant Faraday Effect by Macaluso and Corbino Zeeman effect: a brief history

4 Resonant Faraday Rotation rotation of the plane of linear light polarization by a medium in a magnetic field applied in the direction of light propagation in the vicinity of resonance absorption lines D.Macaluso e O.M.Corbino, Nuovo Cimento 8, 257 (1898) Polarizer Electromagnet Analyzer Flames of Na and Li Diffraction Grating Monochromator Photographic Plate

5 Energy in external field: Consider an atom with S=0  J=L In this case, For magnetic field along z: This is true for other states in the atom If we have an E1 transition,, A transition generally splits into 3 lines This agrees with Lorentz’ classical prediction (normal modes), not the case for S  0 “Normal” Zeeman effect

6 “Normal” Zeeman effect E1 selection rule: DM=0,  1 M= Three lines !

7 “Normal” Zeeman effect Classical Model: electron on a spring Three eigenfrequencies ! B Eigenmodes:

8 The magnetic moment of a state with given J is composed of The magnetic moment of a state with given J is composed of Zeeman effect when S  0

9 Neglect interaction of nuclear magnetic moment with external magnetic field (it is ~2000 x smaller) Neglect interaction of nuclear magnetic moment with external magnetic field (it is ~2000 x smaller) However, average μ now points along F, not J However, average μ now points along F, not J A vector-model calculations a la the one we just did yields: A vector-model calculations a la the one we just did yields: Zeeman effect for hyperfine levels

10 Definition of g F : Definition of g F : The magnetic moment is dominated by the electron, for which we have: The magnetic moment is dominated by the electron, for which we have: To find μ, we need to find the average projection of J on F, so that To find μ, we need to find the average projection of J on F, so that Now, find Now, find Finally, Finally, The actual calculation…

11 Consider atomic states (H, the alkalis, group 1B--Cu, Ag, and Au ground states) Consider 2 S 1/2 atomic states (H, the alkalis, group 1B--Cu, Ag, and Au ground states) L=0; J=S=1/2  F=I  1/2 This can be This can be understood from the fact that the fact that μ comes from comes from J Zeeman effect for hyperfine levels (cont’d)

12 Zeeman effect for hyperfine levels in stronger fields: magnetic decoupling Hyperfine energies are diagonal in the coupled basis: Hyperfine energies are diagonal in the coupled basis: However, Zeeman shifts are diagonal in the uncoupled basis: because However, Zeeman shifts are diagonal in the uncoupled basis: because The bases are related, e.g., for S=I=1/2 (H) The bases are related, e.g., for S=I=1/2 (H) F,M F M S, M I

13 Zeeman effect for hyperfine levels in stronger fields: magnetic decoupling

14 Zeeman effect for hyperfine levels in stronger fields: magnetic decoupling

15 Zeeman effect for hyperfine levels in stronger fields: magnetic decoupling Breit-Rabi diagrams Nonlinear Zeeman Effect (NLZ) But No NLZ for F=I+1/2, |M|=F states Looking more closely at the upper two levels for H : These levels eventually cross! 16.7 T)

16 Atoms in electric field: the Stark effect or LoSurdo phenomenon Johannes Stark ( ) Nazi Fascist

17 Atoms in electric field: the Stark effect or LoSurdo phenomenon Magnetic:Electric: However, things are as different as they can be… Permanent dipole: OKNOT OK (P and T violation) First-order effect Second-order effect

18 Atoms in electric field: the Stark effect Polarizability of a conducting sphere Outside the sphere, the electric field is a sum of the applied uniform field and a dipole field Outside the sphere, the electric field is a sum of the applied uniform field and a dipole field Field lines at the surface are normal, for example, at equator: Field lines at the surface are normal, for example, at equator:

19 Atoms in electric field: the Stark effect Classical insights Natural scale for atomic polarizability is the cube of Bohr radius Natural scale for atomic polarizability is the cube of Bohr radius (a 0 ) 3 is also the atomic unit of polarizability (a 0 ) 3 is also the atomic unit of polarizability In practical units: In practical units:

20 Atoms in electric field: the Stark effect Hydrogen ground state n l m Neglect spin! n l m Neglect spin! Polarizability can be found from Polarizability can be found from

21 Atoms in electric field: the Stark effect Hydrogen ground state (cont’d) The calculation simplifies by approximating The calculation simplifies by approximating =1

22 Atoms in electric field: the Stark effect Hydrogen ground state (cont’d) Alas, this is Hydrogen, so use explicit wavefunction: Alas, this is Hydrogen, so use explicit wavefunction: Finally, our estimate is Finally, our estimate is Exact calculation: Exact calculation:

23 Atoms in electric field: the Stark effect Polarizabilities of Rydberg states The sum is dominated by terms with n i  n k The sum is dominated by terms with n i  n k Better overlap of wavefunctions Better overlap of wavefunctions Smaller energy denominators Smaller energy denominators d ik  n 2. Indeed, d ik  n 2. Indeed, (E k -E i ) -1 scale as n 3 (E k -E i ) -1 scale as n 3

24 Atoms in electric field: the linear Stark effect Stark shifts increase, while energy intervals decrease for large n Stark shifts increase, while energy intervals decrease for large n When shifts are comparable to energy intervals – the nondegenerate perturbation theory no longer works even for lab fields <100 kV/cm  use degenerate perturbation theory When shifts are comparable to energy intervals – the nondegenerate perturbation theory no longer works even for lab fields <100 kV/cm  use degenerate perturbation theory Also in molecules, where opposite-parity levels are separated by rotational energy ~10 -3 Ry Also in molecules, where opposite-parity levels are separated by rotational energy ~10 -3 Ry Also in some special cases in non-Rydberg atoms: H, Dy, Ba… Also in some special cases in non-Rydberg atoms: H, Dy, Ba… In some Ba states, polarizability is >10 6 a.u. In some Ba states, polarizability is >10 6 a.u. C.H. Li, S.M. Rochester, M.G. Kozlov, and D. Budker, Unusually large polarizabilities and "new" atomic states in Ba, Phys. Rev. A 69, (2004)Phys. Rev. A

25 The bizarre Stark effect in Ba Chih-Hao Li Misha Kozlov

26 The bizarre Stark effect in Ba (cont’d)

27 The bizarre Stark effect in Ba (cont’d) C.H. Li, S.M. Rochester, M.G. Kozlov, and D. Budker, Unusually large polarizabilities and "new" atomic states in Ba, Phys. Rev. A 69, (2004)Phys. Rev. A

28 Atoms in electric field: the linear Stark effect Hydrogen 2s-2p states Opposite-parity levels are separated only by the Lamb shift Opposite-parity levels are separated only by the Lamb shift   Secular equation with a 2x2 Hamiltonian: Secular equation with a 2x2 Hamiltonian: Eigenenergies: Eigenenergies: QuadraticLinear Not EDM !

29 Atoms in electric field: the linear Stark effect Hydrogen 2s-2p states (cont’d) Linear shift occurs for Linear shift occurs for Lamb Shift: ω sp /2  1058 GHz Lamb Shift: ω sp /2  1058 GHz  Neglect spin!

30 Atoms in electric field: polarizability formalism Back to quadratic Stark, neglect hfs Back to quadratic Stark, neglect hfs Quantization axis along E  M J is a good quantum # Quantization axis along E  M J is a good quantum # Shift is quadratic in E  same for M J and - M J Shift is quadratic in E  same for M J and - M J A slightly involved symmetry argument based on tensors leads to the most general form of shift A slightly involved symmetry argument based on tensors leads to the most general form of shift Scalar polarizabilityTensor polarizability