Durham E-Theses Absolute Absorption and Dispersion in a Thermal Rb Vapour at High Densities and High Magnetic Fields. WELLER, LEE How to cite: WELLER, LEE (2013) Absolute Absorption and Dispersion in a Thermal Rb Vapour at High Densities and High Magnetic Fields., Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/7747/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk 2 Absolute Absorption and Dispersion in a Thermal Rb Vapour at High Densities and High Magnetic Fields. Lee Weller Abstract This thesis presents a comparative study of the measured and calculated ab- solute absorption and dispersion properties of the Rb D lines through a dense thermal vapour in the absence and presence of an applied magnetic field. A detailed theoretical model valid in the weak-probe regime is calculated. The model uses a matrix representation of the atomic Hamiltonian including the magnetic field interaction for Rb in the completely uncoupled basis. Numer- ical diagonalisation allows the frequency detunings and transition strengths to be calculated. The lineshape of each transition is modelled as a Voigt profile, a convolution of the inhomogeneous and homogeneous profiles. The medium’s susceptibility is found by summing over all the electric-dipole- allowed transitions. For dense thermal vapours a modification to the homo- geneous linewidth of each transition, which grows linearly with the number density of atoms, arises due to resonant dipole-dipole interactions between identical atoms in superpositions of the ground and excited terms. In the presence of an applied magnetic field we investigate the Stokes parameters of light propagating through a dense thermal vapour. For fields larger than 0.33 T we enter the hyperfine Paschen-Back regime on the Rb D lines. We present a compact optical isolator based on an atomic vapour, exploiting the spectral region of high transmission and large dispersion where we would normally expect absorption on the Rb D lines. Frequency up-conversion is shown in the fluorescence measurements over the visible and near infra-red regions for strong excitation. Low density transfer arises due to the energy- pooling effect between two identical atoms in their first excited terms. At high densities resonant dipole-dipole interactions give rise to a threshold for the energy transfer. We characterise the threshold behaviour with increasing number density. Absolute Absorption and Dispersion in a Thermal Rb Vapour at High Densities and High Magnetic Fields. Lee Weller A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy Department of Physics Durham University August 9, 2013 Contents Page Abstract i Title ii Contents iii List of Figures viii List of Tables xi Declaration xii Acknowledgements xiv 1 Introduction 1 1.1 Motivation............................. 1 1.2 Atom-lightinteractions . 1 1.3 Magnetic-fieldinteraction . 3 1.4 Opticalisolators. 4 1.5 Cooperativeeffects . 5 1.6 Thesissummary.......................... 6 1.7 Publications............................ 8 1.8 Authorcontributions . 9 I Absolute Absorption and Dispersion 10 2 Model for the absolute susceptibility 11 2.1 Introduction............................ 12 2.2 Susceptibilityofatwo-levelatom . 12 2.2.1 Two-levelatom . 12 2.2.2 Thesusceptibility. 14 2.3 Lineshapefactor ......................... 15 2.3.1 Lorentzian and dispersion lineshapes . 16 2.3.2 Gaussianlineshape . 17 iii Contents iv 2.3.3 Voigtlineshape . 19 2.4 Hamiltonianforamany-levelatom . 20 2.4.1 Atomicstructure . 20 2.4.2 Matrixrepresentation. 26 2.5 Usingthematrixrepresentation . 32 2.5.1 Relativeline-strengthfactors. 32 2.5.2 Transitionfrequencies . 34 2.5.3 Absorption and dispersion coefficients . 35 2.6 Experimentalmethodsandresults. 39 2.6.1 Experimentalapparatus . 39 2.6.2 Comparingtheoryandexperiment . 39 2.7 Discussion ............................. 42 3 Including resonant dipole-dipole interactions 43 3.1 Introduction............................ 44 3.2 Resonantdipole-dipoleinteractions . 44 3.2.1 Formoftheinteraction. 44 3.2.2 Contributiontothesusceptibility . 47 3.2.3 Impactandquasi-staticregimes . 48 3.3 Theself-broadeningparameter. 49 3.3.1 Calculating the self-broadening parameter . 49 3.3.2 TotalLorentzianlinewidth . 51 3.3.3 Homogeneous and inhomogeneous FWHM . 52 3.4 Experimentalmethodsandresults. 53 3.4.1 Experimentalapparatus . 53 3.4.2 2mmcellheater ..................... 54 3.4.3 Comparingtheoryandexperiment . 55 3.5 Modifiedformofthesusceptibility. 56 3.5.1 Modified total homogeneous linewidth . 57 3.5.2 Measuring the self-broadening parameter . 58 3.6 Shiftsandsaturation . 60 3.6.1 Lorentz-Lorenz and collisional shifts. 60 3.6.2 Saturation of the resonance susceptibility . 61 3.7 Discussion ............................. 62 4 Including the magnetic field interaction 63 4.1 Introduction............................ 64 4.2 AtomicHamiltonian . 64 4.3 Matrix representation of HˆZ ................... 66 2 4.3.1 5 S1/2 matrix ....................... 67 2 4.3.2 5 P1/2 matrix ....................... 72 4.3.3 Relative line-strengths and transition frequencies ... 74 4.4 Variousmagneticregimes. 77 4.4.1 HLZandHPBregimes . 77 4.4.2 FPBregime ........................ 79 Contents v 4.4.3 Summary ......................... 79 4.5 Experimentalmethods . 80 4.5.1 Permanentneodymiummagnets . 81 4.5.2 Aluminummagnetholder . 83 4.5.3 Vapourcells........................ 86 4.5.4 Details for moderate magnetic fields. 88 4.5.5 Detailsforlargemagneticfields . 90 4.5.6 Detailsforopticalisolation. 92 4.6 Discussion ............................. 93 5 Measuring the Stokes parameters of light 94 5.1 Introduction............................ 95 5.2 Comparing theory and experiment for moderate fields . 95 5.2.1 Asafunctionofnumberdensity . 95 5.2.2 Asafunctionofmagnetseparation . 98 5.3 Stokesparameters. 101 5.3.1 Circularlyanisotropicmedia . 101 5.3.2 StokesTheory. 103 5.3.3 Results........................... 104 5.4 ThePoincar´erepresentation . 107 5.4.1 Poincar´esphere . 107 5.4.2 Measured and calculated Poincar´esphere . 108 5.5 Discussion ............................. 109 6 The hyperfine Paschen-Back (HPB) regime 110 6.1 Introduction............................ 111 6.2 Absorption and dispersion in the HPB regime . 112 6.2.1 AbsorptionintheHPBregime. 112 6.2.2 DispersionintheHPBregime . 114 6.3 Originoftheweaktransitions . 116 6.3.1 Relativeline-strengthfactors. 116 6.3.2 Transitionfrequencies . 120 6.4 OpticalisolatorintheHPBregime . 120 6.4.1 Propertiesofthemedium . 120 6.4.2 Theoretical and experimental rotation . 122 6.4.3 Characteristics for our optical isolator . 126 6.4.4 Idealparameters . 128 6.5 Discussion ............................. 129 II Cooperative effects 130 7 Cooperative enhancement of energy transfer 131 7.1 Introduction............................ 132 7.2 Experimentalmethods . 132 Contents vi 7.2.1 Method .......................... 132 7.2.2 Confocalmicroscopy . 134 7.2.3 Side-imaging. 134 7.3 Energylevelsandfluorescencespectra. 135 7.4 Theoreticalconsiderations . 138 7.4.1 Potential-energycurves. 138 7.4.2 Kinetictheory. 139 7.4.3 Enhancementintheelectricfield . 144 7.5 Dependencieswithnumberdensity . 145 7.5.1 Confocalmicroscopyresults . 145 7.5.2 Side-imagingresults . 148 7.6 Discussion ............................. 150 8 Conclusions & Outlook 151 8.1 Conclusion............................. 151 8.2 Outlook .............................. 152 Appendices 152 A Rubidium atomic & spectroscopic data 153 A.1 Vapourpressureandnumberdensity . 153 A.2 Atomicdata............................ 154 A.3 Spectroscopicdata . 155 B Matrix calculations 156 B.1 Lˆ Sˆ matrix............................ 156 B.2 Iˆ ·Jˆ matrix ............................ 160 B.3 π ·transitions............................ 163 C Calibration for a single transmission plot 167 C.1 Scalingthefrequencyaxis . 167 C.1.1 Fabry-Perotetalon . 167 C.1.2 Hyperfine/saturated absorption spectroscopy . 169 C.2 Renormalisingthedata. 171 D Extracting optimised parameters and their uncertainties 173 D.1 Errorsfromtheoscilloscope . 173 D.2 Pythonfitting........................... 174 D.3 Covariancematrix . 174 E Breit-Rabi diagrams 176 E.1 Breit-Rabi diagrams for 85Rb .................. 176 E.2 Breit-Rabi diagrams for 87Rb .................. 178 Contents vii F Permanent neodymium magnets 180 F.1 Magneticfieldprofile . 180 F.2 Analyticrotationangle . 183 F.3 Idealparametersforanopticalisolator . 184 Bibliography 185 List of Figures Figure Page 2.1 Two-levelquantumsystem. 13 2.2 Normalised steady-state susceptibility. 17 2.3 Hyperfinestructureandintervals. 24 2.4 Matrixforthe5Pterm. 30 87 2.5 Relative line-strengths for F =1 F ′ = 1 transition in Rb. 34 2.6 Calculated transmission plots for→ a 75 mm cell. 38 2.7 Apparatus for weak beam measurements for a 75 mm cell. 40 2.8 Experimentandtheoryfora75mmcell. 41 3.1