Magnetic Dipole Transition in Rubidium87

Introduction

Rubidium atoms undergo two types of optical transitions. A transition from a ground state to a excited state is called a D1 transition whereas excitation to the state is called a D2 transition. In this experiment we study the D2 transition by shining a c.w polarized laser beam though a cell containing Rubidium vapors. The ground state of Rubidium has a Hyperfine splitting of F=1 and F=2. If we start from the F=1 ground state and excite the atoms, it is possible to pump all the Rubidium atoms to the level of the ground state. Due to such kind of pumping of the population there is absence of transitions starting from the levels which decreases the overall absorption of Rb. In the presence of a magnetic field the sublevels for F=1 ground state undergoes splitting due to Zeeman effect. The main aim of our experiment is to use a transverse RF magnetic field that would cause magnetic dipolar transitions between these Zeeman levels thereby populating the levels that were left vacant due to optical pumping. We expect to observe an increase in the absorption by Rb due these transitions. Assuming that when the RF frequency matches the Zeeman splitting (resonance) the transitions have maximum probability we expect to observe a peak absorption that would correspond to the resonant RF frequency.

D2 Transition in Rubidium and Optical pumping

A D2 transition in Rubidium occurs between the (ground state) and the (excited state). The difference between the levels is 780.2nm (384.23THz). The ground state has hyperfine levels F=1 and F=2 separated by a gap of 6.83 GHz whereas the excited state has the hyperfine levels F=0,1,2,3 spread over a relatively less energy spread of 495.8 MHz. For our convenience we choose to start with the F=1 level in the ground state. Since the hyperfine gap between F=1 and F=2 in the ground state is relatively higher it is possible to fix F=1 as the ground state by selecting the laser frequency at around 780nm (laser line width ~1MHz) such that absorption takes place. However, it is rather difficult to choose any one of the F levels in the excited state since at room temperature the Doppler broadening is of the same order as the energy spread of the hyperfine states in the excited states. But as long as the ground state is fixed to F=1 state the excitation to any of the allowed hyperfine levels in the level followed by spontaneous emission should be able to give rise to optical pumping to the level of the ground state. For our convenience we choose the F=2 hyperfine level in the excited state to illustrate optical pumping of our system of Rb atoms (see fig. 1).

When we apply a magnetic field in the z direction we define the quantization axis of the Rubidium atoms. Since the laser is also directed along the direction of the field it is important to note that the c.w polarization of the laser (s+) allows E1 (electric dipole) transitions involving due to addition of angular momentum of the light quanta absorbed. It is because of this that transitions lead to a higher state (see fig. 1) which eventually causes the pumping. The Bz field also causes Zeeman splitting among the sublevels of the F=1 ground state with different projections of the total angular momentum along the z direction. The typical splitting energy is -0.70MHz/G. Therefore in the presence of field of the order of a few Gauss the Zeeman splitting is much smaller than the thermal energy at room temperature. Also because the Zeeman levels represent the ground state hence there is no decay from this state to any other. This ensures that magnetic dipole transitions (with energy range in MHz) are solely responsible for distribution of population among the Zeeman levels after we treat the optically pumped sample with the RF.

Absorption				Spontaneous emission
Initial state		Final state		Initial state		Final state

Magnetic Dipole transition and Resonance

The Zeeman splitting in the F=1 ground state in the presence of the magnetic field in the is given by . Where is the overall g-factor and is Bohr-magneton. Therefore if we choose two consecutive Zeeman levels (and or, and ) the resulting Hamiltonian could be assumed to be of the form,

Since the eigen function of this Hamiltonian are states with well defined angular momentum projection along the quantization axis (z) therefore if we apply a magnetic field in the perpendicular direction( say along x) the resulting Hamiltonian has an additional energy term due to the interaction of the magnetic dipole with the external field given by where is the magnetic dipole moment. In our eigen basis (,) H’ would have off-diagonal terms. Since in this experiment we choose to apply a transverse RF magnetic field which interacts with the magnetic dipole the resultant Hamiltonian in the original basis could be written as,

where is the amplitude of the interaction energy between the dipole and the field. The time-dependent Schrodinger equation gives a solution to the equation where the probability of finding the atom in the upper state as a function of time is given by,

We see that it suggests that the amplitude of the probability of finding the atom in the upper state is a Lorentzian function of the RF frequency and is maximum when (resonance).

Thus the magnetic dipole transition is strong at resonance since the amplitude of the probability of finding the atom in the other two Zeeman levels is maximum. With respect to our experiment it indicates how strongly the atoms are driven from the optically pumped state to the state and then from state to the state. As a direct consequence of this we expect to observe a peak in the laser absorption (E1 transition) during resonance of the RF with the Zeeman splitting because now it is possible to drive E1 transitions from all the Zeeman levels. This might also explain why the laser absorption is expected to have a lorentzian distribution as a function of the RF frequency.

Experimental setup

A cylindrical rubidium cell (~2cm dia. and ~5cm length; approx 10^11 atoms/cc) is mounted at the center of a three pairs of Helmholtz coils having their axes along the three orthogonal directions. Currents through each pair of coils (for Bx,By and Bz) are driven by separate current sources. Figure 2 shows only the pair of coils that produce the field along the z direction. The laser is passed through a polarizing beam-splitter and a quarter wave plate to produce c.w. polarized light. We use several mirrors to guide the laser along the axis of the cell. A 50-50 beam splitter is used to send half of the beam through the cell and allowing the unabsorbed beam to behave as a reference to the absoption of the other beam. Both the beams are collected by two separate photodetectors and fed into a differential trans-impedance amplifier the output of which goes into a lock in amplifier for noise reduction.

Fig. 2. a)The experimental setup. b) The Rb cell is mounted at the center of the three orthogonal Helmholtz coils.

The Rf transverse field is produced by a copper coil (10 turns, ~3cm dia.) placed close to the Rb cell . The Rf is generated using a synthesized function generator (DS345 30MHz) which sweeps the RF over a frequency window of 1100Khz in 10 secs. For noise reduction we need to modulate the signal periodically. To do this we use a programmable waveform synthesizer (Wavetek-178) to modulate the Rf amplitude which in turn modulates the signal (absorption). Since the absorption is measured within the RF frequency window of 1100 KHz a reasonable modulation frequency is set at 5KHz. The time constant set in the lock-in amplifier is 100ms which allows around 500cyles of signal modulation to eliminate the noise. The time constant cannot be freely increased since the RF keeps changing its frequency over time and we are interested in obtaining the nose filtered signal for a particular value of RF rather than over a significantly wide range.

D2 Transition in the absence of Rf

As a first step we need to fix the laser at the same frequency as the D2 transition line between the F=1 in the ground state to the F=2 (784.2 nm approx) level in the excited state. The typical line width of the laser is of the order of 1MHz. However at room temperature since the average speed of atoms in the Rb vapor is of the order of 300 m/s therefore there is a corresponding Doppler broadening of 400-500 Mhz. Since the overall hyperfine splitting of the excited state (F=0 to F=3) is of the same order (494Mhz) hence there is ample probability that the atom gets excited to any of the hyperfine levels in the excited state. For this part we do not use the RF. The output from the differential amplifier is fed directly to the oscilloscope with a trigger from the laser source. To bring about the D2 transition we do the following:

a) Shine the c.w. polarized laser through the Rb cell and observe the absorption (within a frequency window) by changing the frequency from the laser source.

b) At resonance when absorption peaks are observed the frequency is nearly close to the transition frequency of the D2 excitation. The laser source is fixed at this frequency.

Since the Laser frequency has a small enough line width (~1Mhz) compared to the Doppler broadening all the E1 transitions from the F=1 ground state to all the F=0,1,2 states in the excited state is purely due to Doppler broadening. However since we are only interested to optically pump the population of atoms to the sublevel of the F=1 ground state hence it does not depend on which F state the laser excites the atom as long as spontaneous emission from any of those levels brings back the atoms to the F=1 ground state.

Optimization of the output

After the laser is fixed for D2 transition we proceed further to study Magnetic dipole transition between the Zeeman levels of the ground states. The main aim is to observe the absorption peak (E1 transition from the ground to the excited state) as a function of the RF (which de-pumps the ground state from +1 state to all other Zeeman levels) and the corresponding width associated with it. It is expected that as we increase the Zeeman splitting (by increasing the current Iz in the Helmholtz coil) there should be a shift in the RF resonant frequency at which absorption is maximum (resonance). However, before we can study this behavior it is important to optimize the signal so that the output has minimum width and noise riding the signal.

To compute the width we start fitting the observed plot with the expected Lorentzian distribution as explained in the preceding section. A typical output that shows the behavior of the absorption over an RF range is shown in figure 5.

Due to the presence of background fields the magnetic field is not uniform inside the cell this might lead to inhomogeneous broadening. To produce a uniform field aligned along the z direction we pass current inside the other two Helmholtz coils to produce bias fields in the x and the y direction. It is observed that specific values of current in these coils help minimize the width (see figure). The other parameters kept fixed during this part of the experiment are shown in the following table.

Iz (Bz)	Rf amp	Laser power
1.6A (3.2G)	7V p-p	120E-6W

Fig 3. Optimization of the bias fields for minimization of signal width and noise

Here the noise is calculated as the root mean square error for each data point upon fitting with the appropriate Lorentzian distribution. We have to minimize the power broadening due to laser and also due to the RF . Fig 4. shows the width as a function of the laser power and the amplitude of the RF where the other parameters that are fixed are shown in the following table.

Iz (Bz)	Ix(Bx)	Iy(By)	Laser Power	Rf amplitude
1.6A (3.2G)	0.7A (1.5G)	0.4A (1.6G)	120E-6W ---opt--	---opt--- 7.0V p-p

Fig 4. The power broadening due to the laser intensity and the RF amplitude

The parameters decided after optimization of the bias field and the power are as follows,

Ix	Iy	Laser power	Rf amp.
0.7 A	0.4 A	120E-6W	8.0 V p-p

Resonance frequency Vs. Zeeman Splitting

At resonance the absorption is higher since the atomic population is maximally distributed among all the levels in the F=1 ground state. Fixing the parameters after optimization of the signal from the previous step of the experiment we start observing the peaks in absorption for different Bz used for the splitting as a function of the resonant RF. The typical window for a decent plot that can be fitted is chosen to be 1100KHz. We vary the z field current from Iz = 1.6 amp-2.4amp and extrapolate to get a linear fit that pass through the origin. Smaller values of the current are not investigated because the Zeeman field then has a magnitude comparable to the bias fields and that might not give a linear response with the resonance frequency. The RF resonance frequency is observed to behave linearly for higher values of the current as indicated by the following plots.

Figure 5. Absorption Vs. Rf for Bz=3.2 G (1.6 amp) Figure 6. Resonance frequency of RF Vs. Bz(Iz)

Results

· The resonance frequency behaves linearly with the Zeeman splitting as shown in Fig 6. When extrapolated it passes through the origin. We however we do not measure resonance at lower values of the Bz field since we also have Bx and By bias field applied to reduce inhomogeneity of the resultant field inside the cell. Also at lower values of Bz the quantization axis for the Rb atoms might get tilted from the z direction due to the bias fields and we cannot expect any behavior predicted by our theory. However it is possible to make a check of the accuracy of our measurements in determining the resonance frequency with the Zeeman splitting. For the given plot (Fig 5) we have Bx=1.5G, By=1.6G and Bz=3.2 G which gives the total B=3.88G. This corresponds to a Zeeman splitting of 2.715MHz where the observed value is 2.735 MHz (well within the half width of the distribution).

· We also make a cross-check on the width at RF amplitude = 0V. In the absence of the RF we expect a broadening due to finite time of interaction of atoms with light. Given the average velocity of Rb atoms at room temperature ~300m/s and the laser width ~3mm the typical interaction time is 1E-5 sec (assuming transverse motion of atoms so that no Doppler effects happen) which gives a line width of ~50-100Khz. On making a linear extrapolation of the RF power-broadening plot we obtain a broadening of similar order for no RF.

· In this experiment we use the Laser as a probe and the RF determines the absorption with a very narrow width compared to the frequency of the laser. Therefore it might be predicted that this technique can be used in making high precision measurements.

Acknowledgements

Thanks to Prof. Luis Orozco for providing the project and workspace in his lab.

Thanks to Mr. Dong Sheng for his supervision and help with the experiment.

Thanks again to Prof. Luis Orozco for organizing the course.