Hyperfine Energy Levels of 85Rb & 87Rb

Hyperﬁne Energy Levels of 85Rb & 87Rb

Chance J. Haycock, Fernando H. Jabaloyes, and Joe R. Gibbs University of Warwick, UK (Dated: February 13, 2019) The technique of optical pumping was used to study the hyperfine energy level splitting of Rubidium-85 and Rubidium-87. The low field and quadratic Zeeman effects were demonstrated in order to verify the theoretical values given by the Breit-Rabi equation. The atomic Lande g- 85 87 factor gF was obtained for the isotopes Rb and Rb and were found to be 0.3323 ± 0.0006 and 0.5008±0.0014 respectively. These values agree with the predictions for both isotopes, giving a good agreement of the expected low field Zeeman effect. The instantaneous dynamics of a system under- going a transition from a pumped to depumped state were also investigated, however, the analysis was limited by the sources of error.

I. INTRODUCTION defined as F = I + J (1) French physicist Alfred Kastler is credited with the development of optical pumping techniques in the 1950’s where J = L+S is the electron’s total angular momentum [1][2]. The process of optical pumping uses a focused and I is the nucleus’ spin angular momentum. For both beam of coherent, high frequency light with particular isotopes of Rubidium, quantum mechanical properties. These imposed proper- 1 1 ties allow us to manipulate the distribution of electrons L = 0, S = , J = , 2 2 amongst energy levels and is particularly useful for the construction of lasers; creating a population inversion of but their nuclear spin values, I, are distinct. In the ab- the system [3]. In this case, the system exists in a state sence of an external magnetic field, the hyperfine levels in which more electrons occupy higher energy states than can not be well resolved. In the presence of an exter- lower ones. The de-excitation of these electrons, and the nal magnetic field, each F level splits into 2F + 1 (where subsequent emission of photons, then produce a coherent F = | F |) Zeeman levels. Then, the Hamiltonian is light source. The technique for measuring the hyper- µ µ H = A(I · J) − J (J · B) − I (I · B) (2) fine energy level transitions of atoms will be explored in J I depth throughout the paper. In particular, the splitting of the 5s energy level of Rubidium-85 and Rubidium-87 where B is the external magnetic field, A is the hyperfine subject to an external magnetic field will be investigated. splitting frequency for B = 0, J =| J |, I =| I | and Two main regimes are focused on, namely the Low Field µJ , µI are the electron and nuclear magnetic moments Zeeman Effect and the Quadratic Zeeman Effect [4]. respectively. In the weak field regime, the Hamiltonian of equation (2) can be resolved for the energy eigenvalues:

E = mF gF µBB (3) −24 −1 II. THEORY where µB = 9.27 × 10 JT [5] is the Bohr Magneton and mF = −F, −F + 1,...,F − 1,F . We also obtain In an atom, the magnetic dipole moment of the J(J + 1) + S(S + 1) − L(L + 1) g = g nucleus interacts with the magnetic field generated by F J 2J(J + 1) the orbiting electrons. The alignment of spins in the atom cause the electron energy levels to split. This is F (F + 1) + J(J + 1) − L(L + 1) known as hyperfine splitting. In order to quantitatively gJ = 1 + investigate this splitting, the role and interaction of the 2F (F + 1) electron’s angular momentum and intrinsic spin angular Equation (3) can be recast as momentum must be discussed. g µ B ν = F B [6] (4) Rubidium has an electron configuration given by h [Kr]5s1. This configuration consists of filled inner elec- and is known as the weak-field Zeeman equation. For tron orbitals, which are stable and spatially symmetric, 1 stronger fields and the special case of J = 2 , the eigen- and one valence electron. Due to screening, we can ig- values of (2) are given by the Breit-Rabi Equation nore the effects of these inner electrons and the atom can −∆E µ ∆E r 4m x be modelled as hydrogen-like. Taking into account all mF hf I hf F 2 EF =I± 1 = − BmF ± 1 + + x contributions to angular momentum from the outer elec- 2 2(2I + 1) I 2 2I + 1 tron and the nucleus, the total angular momentum, F, is (5) 2 where,

(gJ − gI )µBB −µI x := gI := ∆Ehf µBI and ∆Ehf is the hyperfine splitting energy at zero magnetic field. This theoretical result shows an approxi- mately linear relationship for weak fields, and a non- linear relation for larger fields. Lastly, in a transitioning system from a pumped to depumped state or vice-versa in a very short period of time, we expect the populations to overshoot giving rise to ringing. This is referred to as having transient effects. It is predicted that in these cases, the frequency of the ringing will be proportional to the RF Amplitude applied [7].

III. EXPERIMENTS

1. Experimental Setup FIG. 2. Energy level schemes of 87Rb (left) and 85Rb (right) showing fine structure, hyperfine structure and Zeeman split- The presented setup, as shown in Figure 1, uses a Ru- ting. For a given energy level, it is shown how the application bidium discharge lamp in series with an interference fil- of a magnetic field B increases the number of allowed tran- ter, a linear polariser and a quarter-wave plate to achieve sitions via the division of states of a single F into 2F + 1 the conditions required for this experiment. From the states. theory and Figure 2 it can be seen that each F level splits into 2F + 1 Zeeman levels. In order to pump the system, the selection rules that are required are as follows: One of the major difficulties with measuring transitions between hyperfine level transitions in a magnetic field is ∆F = 0 ∆mF = +1 (6) that there is nearly no distinction between the popula- This will allow electrons to be excited into higher energy tion size of electrons in two adjacent energy levels. The states except those which are trapped by a lack of higher ratio of the populations of two adjacent levels with an energy levels available to transition to for which selection energy difference ∆E, is determined by a Boltzmann dis- −∆E rules (6) hold. tribution, exp . Taking for example, a splitting kBT of 1MHz at room temperature (about 300K), this gives a ratio of ≈ 1 − 2 × 10−7 ≈ 1.

2. Calibrations of Fields

The predicted value for the Earth’s magnetic field at the location of the laboratory was ≈ 49µT [8]. How- FIG. 1. Apparatus used. From left to right: 1 is the Ru- ever, the setting in which the experiment took place had, bidium discharge lamp; 2 is an interference filter; 3 is a linear itself, a resultant magnetic field measured to vary in ori- polariser; 4 is a quarter wave plate; 5 is the Rubidium absorp- entation throughout the laboratory. Thus, the resultant tion cell kept at a constant 323K; 6 is the optical detector. background magnetic field was different from simply that of the Earth’s magnetic field. The coils used satisfied For this pumping to occur, the combined use of the the Helmholtz condition, and hence the residual field at IN linear polariser, quarter wave plate and the interference the coils can be calculated from B = k Gauss [7] filter is vital. The first pair will achieve right hand cir- R cularly polarised light, and the interference filter is used where k = 8.991 × 10−3 Gauss m, N is the number of to ensure that the transmitted light is of approximate turns on each side and R is the mean radius of the coils. wavelength λ = 795nm. These are the conditions that The magnitude of the Earth’s magnetic field, (though it are necessary in order to stimulate the D1 transition, is more precisely the resultant background field; it will visualised in Figure 2. still be referred to as BEarth) at the coils was measured to be 3

BEarth = 0.2137 ± 0.00005 Gauss (7)

As the known gF values are either integer or half- integer, the approximated value for Bcoil is suﬃcient for the estimation of gF values. For later experiments, a greater accuracy than that obtained from the geometry of the coils was required. For the calibration of the sweep coils, the knowledge of the gF values for both isotopes is assumed and the mea- surements of the value of current at which the resonances occurred are used to arrive at the relationship

Bsweep = 0.621Isweep − 0.222

Similarly, for the main coils

B = 7.1628I (8) main main FIG. 3. Low Field Zeeman Effect at 70kHz (top graphic) and 120kHz (bottom graphic). Largest dip is the zero resultant B The condition Bmain(0) = 0 was imposed. These calibra- field and the following 2 on either side correspond to the 85Rb tions were used throughout the experiment. and 87Rb. Note: As the frequency increases the symmetry is maintained but the isotope peaks are further away from the central peak. 3. Low Field Zeeman Effect

The current at which resonances occurred was mea- 5. Transient Effects sured for each isotope at frequencies in the range 70- 150kHz. Following on from the calculation of the Earth’s Lastly, the behaviour of electrons for a system in tran- field strength (see result (7)), the geometry of the coils sition from a pumped to depumped state was investigated. was used to find the value of the magnetic field, Bcoil, These conditions were achieved by filtering the RF field NIcoil within it using Bcoil = k [7]. Then, the total field with a square wave of pulse frequency νsquare = 5 Hz, R generating a B field that is continuously gated. at the coils was related with applied frequency using the When the RF is off, no transitions should occur between weak field equation (4), giving Zeeman levels and hence no absorption of photons; the h transmitted light intensity here is a maximum. Upon B = BEarth + Bcoil = ν (9) switching on the RF, the Zeeman levels split and ab- g µ F B sorption can take place; the transmitted light here is a minimum. This relation can be used to plot hν/µB against B, which should give a linear relationship. Applying a To investigate the effect of the RF amplitude in partic- linear regression, the graph can be used to find the ular, the switching-on of the RF field was aligned with gradient and obtain an estimate for both g values. the resonance peak as shown in Figure (6). Due to the F short rise time of the square wave pulse, an exponen- From the theory, the expected ratio, R, of the gF values will be roughly 3/2. tially damped ringing period of the light intensity before it reaches its equilibrium de-pumped state was observed using an oscilloscope. The frequency of this ringing is expected to be proportional to the RF amplitude[7]. The 4. Quadratic Zeeman Effect frequency was plotted against the RF amplitude in order to yield a gradient proportional to gF . The constant of proportionality is unknown so g itself cannot be deter- The current through the sweep coils is measured at F mined however the ratio between the values for 85Rb and each of the 2F + 1 resonances for a fixed RF field fre- 87Rb can still be obtained. quency of 4 Mhz. These recorded currents will be com- pared to the expected values produced by the equation 87 3 1 (5). For Rb, this corresponds to F = 2 ± 2 and hence 85 5 IV. RESULTS 2F = 6 resonances while for Rb, I = 2 and hence F = 2, 3 giving 2F = 10 resonances. From here, the main field calibration in result (8) will lead to a measurement From the linear regression shown in Figure 4, the value 85 of the total magnetic field. gF = 0.3323 ± 0.0006 was obtained for Rb, which is 4

1 in reasonable agreement with the theoretical value of 3 87 TABLE II. Quadratic Zeeman Effect data vs. Breit-Rabi Cal- and gF = 0.5008 ± 0.0014 for Rb, which is in excellent 85 1 culation for Rb with B = 9.186 Gauss and ν = 4 MHz. agreement with the theoretical value of 2 . The ratio of The third column shows the measured frequencies of the 10 the two measured gF values, R, is given by resonances and the 2 double quantum transitions, whilst the 0.5008 ± 0.0014 fourth column shows the corresponded calculated value from R = = 1.507 ± 0.005 (10) the Breit-Rabi equation. Note how the BR equation does not 0.3323 ± 0.0006 calculate these double quantum transitions. which is in reasonable agreement with the expected value Isweep (A) Btotal (G) νmeasured (MHz) νcalculated(MHz) 3 of 2 . 0.293 9.145 4.263 4.311 From tables 1 & 2, the following results for quadratic 0.305 9.153 4.266 4.299 Zeeman effect were extracted. For 87Rb the correspond- 0.310* 9.156 4.267 - ing magnetic field strength is of B = 6.258 Gauss. Simi- 0.328 9.167 4.273 4.299 larly, for 85Rb, the main field calibration leads to a total 0.341 9.175 4.276 4.287 magnetic field of B = 9.186 Gauss. 0.365 9.190 4.283 4.287 0.382 9.201 4.288 4.275 0.400 9.212 4.293 4.275 TABLE I. Quadratic Zeeman Effect data vs. Breit-Rabi Cal- 0.418 9.223 4.299 4.263 culation for 87Rb with B = 6.258 Gauss and ν = 4 MHz. 0.436 9.234 4.304 4.263 The third column shows the measured frequencies of the 6 0.451* 9.244 4.308 - resonances, whilst the fourth column shows the corresponded 0.471 9.256 4.315 4.251 calculated value from the Breit-Rabi equation.

Isweep (A) Btotal (G) νmeasured (MHz) νcalculated(MHz) 0.170 6.141 4.298 4.378 The time period of each oscillation was measured for 0.183 6.149 4.303 4.372 a range of RF amplitudes for each isotope. From theory, 0.197 6.158 4.310 4.383 there should be an inverse relationship between the two. 0.208 6.165 4.314 4.378 The ratio of the two gradients, R, was found to be. 0.220 6.172 4.320 4.372 44 ± 2 0.224 6.175 4.321 4.366 R = = 1.45 ± 0.05 (11) 31.0 ± 0.3 Figure 7 clearly shows noise from an external source This ratio is in excellent agreement with the theoreti- whilst investigating the transient effects. This heavily cal ratio of 1.50, however there is an unpredicted lower complicated the measurement of the period of the oscil- frequency oscillation. One potential source of this error lation. Ideally, this period should have been measured is due to the strange behaviour of the ambient magnetic more accurately by fitting an exponentially damped, si- field in the laboratory. The noise from the AC source con- nusoidal curve. However, due to the noise, this was found siderably contributed to the error in the measurement. to not be possible. The overlaid frequency of the noise For future experiments, signal processing techniques such was measured to be νnoise = 50.51 ± 0.125 Hz and hence was most likely due to the AC power source as it so closely matches mains current frequency.

85 87 FIG. 4. Low Field Zeeman data for Rb () and Rb (N). The expected linear relationship between the terms of equation (9) is clearly demonstrated. The linear ﬁt for 85Rb has FIG. 5. Two plots at diﬀerent RF gains for Rubidium-85. an intercept (2.4 ± 1.4) × 10−4 which suggests there is a small As gain is increased, the main resonances broaden and the systematic error involved. double quantum transitions sharpen. 5

enough time.

V. CONCLUSION

In conclusion, from the experiment it was found that the values for the atomic Landég-factor, gF , for both 85Rb and 87Rb match very closely with their respective theoretical values and obtained clear, qualitative proof for the theoretical predictions for Quadratic Zeeman Ef- fect. Lastly, the estimated value for the Transient Effects is within the error and so matches the theoretical value very well. However, it should be noted that, possibly 1 FIG. 6. Demonstrates the characteristic e form of light in- due to certain shortcomings of the apparatus, the pro- tensity as the system is being pumped. The rapid decay of cess of calculating the ratio of gF values was complicated the ringing can be seen in the immediate time following the by amount of noise which was present. turning off of the square wave. The persisting sinusoidal wave far after this square wave shut-off point is a result of noise in the apparatus.

FIG. 8. The frequency of the damped transient oscillations 85 85 for Rb () and Rb (N) are shown to be proportional to the RF frequency as expected. The gradients of the linear fits for 85Rb and 87Rb were 31.0 ± 0.3 and 45 ± 2 respectively. FIG. 7. On this scale, the effects of a gated RF field can Three outliers, shown in red, were excluded from the linear clearly be seen. The period of oscillation can be approxi- fit. mately measured via the markers on the oscilloscope. The oscillation on the right is from the noise. ACKNOWLEDGMENTS as using FFT to filter out the identified noise may have We wish to acknowledge the support of James Gott for helped resolve this error and provided more accurate re- offering suggestions and encouragement throughout the sults. These techniques were not pursued for a lack of experiment.

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