Section II: Experimental Analysis

Laser cooling and trapping proves to be a challenging endeavor in the undergraduate laboratory that requires precise experimental setup. Although recent publications suggest an experimental setup costing approximately $3000 for the undergraduate laboratory¹, we chose a slightly more expensive setup hoping to gain better experimental results. The entire apparatus is placed on a 4 by 6 optical table to reduce external vibrations. Figure 2.1 shows the laser cooling and trapping apparatus.

The experimental setup demands two lasers which can reliably produce coherent beams. The obvious choice for the undergraduate laboratory, both for cost and efficiency, is the diode laser. One laser is used for trapping the atoms (trapping laser) while the other is used for optical repumping (repumping laser). Figure 2.2 shows the hyperfine structure of the 6s_1/2 and 6p_3/2 energy levels in cesium that correspond to the frequencies of the trapping and repumping lasers. (A full energy level diagram can be seen in figure 1.1.) In this experiment, we are interested in using a frequency slightly less than the F = 4 to F' = 5 and relying on the Doppler shift to put the atom into the F' = 5 state. Although much less likely, an atom can make an off resonant transition into the F' = 4 state. When this happens, the atom is able to decay into the F = 4 or the F = 3 state. Notice that if the atom goes into the F = 3 level, the trapping laser can no longer excite the atom. The repumping laser, therefore, is necessary to excite the F = 3 to F' = 4 transition in order keep a population of atoms in the F = 4 state.

Figure 2.2. This diagram shows hyperfine structure of the 6s_1/2 and 6p_3/2 energy levels. In the experiment the trapping laser is tuned just below the F = 4 to F' = 5 transition, yet, sometimes the laser excites the F = 4 to F' = 4 transition. This is the need for the repumping laser.

For proper alignment care must be taken when mounting the diode lasers. Furthermore, since the wavelength produced by a diode laser is temperature and current dependent, both factors must be controlled. A detailed description of mounting, temperature control, and current control of diode lasers can be found in a publication by MacAdam.² In order to control the laser temperature, we used an ILX Lightwave Temperature Controller. For current control, an ILX Lightwave Ultra Low Noise Current Source was used for both lasers. The temperature of both lasers was set to a constant value and the current was used to adjust the wavelength. An external voltage divider attached to a function generator was used for finer current control and to ramp the current.

2.1.2 Locking the Diode Lasers

As with most diode lasers, the frequency drifts over time due to temperature drift. One mechanism for controlling laser frequency is through the use of electronic feedback.³ In our experiment, the electronic signal was generated by the use of a non-linear optics technique called saturated absorption spectroscopy. This technique was developed by Schawlow and Hänsch in the 1970's as a way to circumvent Doppler broadening in atomic spectra.⁴ The saturated absorption spectrometer is seen in figure 2.3.

Figure 2.3. A schematic of the typical saturated absorption spectrometer.

In saturated absorption spectroscopy a beam from a diode laser is split into three beams: a pump beam, a probe, and a reference beam. The probe and reference beams are of the same intensity and are passed through a low density cesium vapor cell into a differential photodetector. In the photodetector the signals are subtracted and amplified. The pump beam, which is much more intense than the other two beams, enters the cesium vapor cell from the opposite side and overlaps the probe beam. Given this configuration, when the laser is resonant with a transition the probe and pump beams will scatter off atoms with approximately zero velocity in the direction of the incident beams. Since the pump beam has a higher intensity than the probe beam, most of these atoms will be in the excited state allowing the probe beam to pass through virtually unabsorbed. The reference beam, however, has no overlap and is almost completely absorbed. When this happens, there is a difference in the signals of the probe and reference beams which is amplified by the photodetector. If the laser is scanned over a frequency range, these differences will be seen as spikes in the output signal thus denoting an atomic transition. Figure 2.4 shows a frequency sweep for the trapping and repumping lasers from their respective spectrometers.

Figure 2.4. A screen capture of an oscilloscope displaying the output signals from the trapping and repumping saturated absorption spectrometers. The top and bottom sweeps are not to scale.

The top sweep on the oscilloscope screen is the spectrum from the trapping laser which has a frequency range corresponding to the F = 4 to F' = 3, 4, and 5 transitions in cesium. The bottom sweep is the spectrum from the repumping laser. Notice that it has a frequency range corresponding to the F = 3 to F' = 2, 3, and 4 transitions. Furthermore, notice the structures labeled "Crossover." These structures are by-products of the experimental setup. Crossover occurs when the laser frequency is tuned exactly between any two transitions. It is caused by absorption of the pump beam by two groups of atoms in the vapor cell. The first group has velocities that "red" shift the laser frequency into resonance with the upper transition and the second group has velocities that "blue" shift the laser frequency into resonance with the lower transition.

Since precision is not paramount in the frequency of the repumping beam, the signal from its saturated absorption spectrometer is primarily used as a reference. The frequency of the trapping beam, however, must remain relatively constant and, therefore, the purpose of the signal is twofold. The first is to serve as a reference and the second is to provide a feedback signal for a first derivative lock-in amplifier. Using this type of lock-in amplifier, it is possible to lock the laser frequency to one of the transition peaks in the spectrum. At this point, one may ask: "What good will that do since the trapping laser must be to the 'red' of the transition in order to have cooling?" The answer is to shift the laser frequency.

2.1.3 The Acousto-Optic Modulator

Shifting of the laser frequency can be achieved with the use of an acousto-optic modulator (AOM). The Isomet AOM used in this experiment creates sound waves in a lead molybdate crystal which can diffract an electromagnetic wave. The details of the diffraction of light by sound waves can be found in a text by Yariv.⁵ The AOM used by the repumping laser can shift the laser frequency between 60-100 MHz. The AOM used by the trapping laser has a range between 150-250 MHz. Not only do the AOM's provide an easy way to shift the laser frequencies by small amounts, they use a TTL signal for on/off operation. Thus, the shifted beam can be easily turned on and off. Furthermore, since the AOM allows several orders of diffraction to pass through the aperture, we were able to obtain several different frequencies from one laser! Figure 2.5 shows a typical saturated absorption spectrum from the trapping laser and how locking and shifting can be employed to produce a specific detuning.

Figure 2.5. This is the a portion of the signal from the saturated absorption spectrometer for the trapping laser. Notice that the laser is locked on a crossover peak and then shifted to the desired frequency by the acousto-optic modulator (AOM). The shifted laser frequency is denoted by w_L and the resonant transition frequency is w_o. The detuning, , is the difference of these two.

2.1.4 The Ultra-High Vacuum Chamber and Components

The cesium atoms are contained in an ultra-high vacuum chamber with six main non-reflective coated windows and two probe windows. The chamber is cylindrical in shape with four perpendicular arms extending from its side. Figure 2.6 shows the ultra-high vacuum chamber. The ion pump attached to the chamber maintains a pressure of 1x10^-8 Torr. The cesium atoms are introduced into the chamber via sublimation from a solid sample mounted in a glass tube which is attached to the chamber. The atoms can be viewed by an infrared camera mounted on the side of the chamber.

The trapping beam is routed and split such that it enters three of the windows and is retro-reflected. The three beams are said to be orthogonal and counterpropagating. The repumping beam is sent through two of the windows and is similarly retro-reflected. Before entering (and reflecting back through) a window the beam is circularly polarized using quarter wave plates. The incoming beam is s⁺ and the reflected beam is s-.

A magnetic spherical quadrupole field is created inside the chamber using two coils of wire on the top and bottom of the chamber. Furthermore, Helmholtz coils are placed on each arm of the chamber to help correct beam misalignment and allow movement of the magnetic origin. An FET is employed to allow quick switching on and off of the magnetic field gradient. A similar setup was used by Raab et al.⁶

Figure 2.6. A photograph of the UHV chamber. The laser beams enter the chamber through its windows after passing through quarter wave plates.

2.2 Optical Molasses

Optical molasses was first demonstrated by Chu et al.⁷ In that experiment, atoms, due to the induced dipole moment, were held in the focus of a beam. In this experiment, the creation of optical molasses was due to Doppler cooling by the use of three counterpropagating beams from a trapping laser and the presence of a repumping beam. The trapping beam was detuned and locked at approximately -2G. The repumping laser was set to a frequency corresponding to the F = 3 to F' = 4 transition. A picture of the molasses can be seen in Figure 2.7.

Figure 2.7. This picture shows the optical molasses. The molasses forms at the intersection of the counterpropagating beams. The bright spot at the bottom of the picture is the bottom mirror and quarter wave plate.


The optical molasses in this study, however, was used to show that the trapping and repumping system was operating properly. A more optically dense sample of cold atoms can be created by the use of a magnetic field gradient. The use of Doppler cooling and a magnetic field gradient allows one to create a magneto-optic trap (MOT).

2.3 The Magneto-Optic Trap (MOT)

As mentioned previously, the magneto-optic trap is created by the use of the anti-Helmholtz coils which produce a spherical magnetic quadrupole in the ultra-high vacuum chamber. This field shifts the energy levels of the cesium atoms in such a way that the net force from the circularly polarized laser beams is inward. The MOT has several experimental advantages. The most obvious advantage is that the cold atoms can be compressed to a specific location in the trapping beams and can be moved with the use of other magnetic fields. Another advantage is that the dense cloud is spherically shaped making fluorescence and volume determinations simpler. Figure 2.8 shows the typical setup for experiments using the MOT.

The magneto-optic trap in our experiment was created using a magnetic field gradient of approximately 10 Gauss/cm. The trapping laser had an intensity of about 27 mW/cm² meaning that each of the six trapping beams had I = 9 mW/cm². The frequency was locked using the electronic feedback technique described earlier to frequencies between -G and -4G. The repumping beam was tuned to the F = 3 to F' = 4 transition. A picture of the MOT is the subject of Figure 2.9.

Figure 2.9. This is a picture of the MOT. Notice the trapping beams from all directions. The up and down beam is seen reflected off the top window of the chamber.



2.4 Filling Curve Measurement

When the trapping and repumping beams and the magnetic field gradient are turned on, the MOT does not instantly form. Rather, it grows or fills, in a short period of time. Monroe et al.⁸ described this growth quantitatively as a balance between the capture and loss rates of the trap. The capture rate, R, represents the number of atoms that enter the trap in a second and is determined by the following equation:

where n represents the density of atoms in the chamber, v_c is the capture velocity, V is the trapping volume, m is the atom's mass, k_B is Boltzmann's constant and T is the room temperature. The loss rate of the trap is given as 1/t (don't confuse with the transition lifetime) and is represented by:

where n, again, is the density, and is the cross section for a room temperature atom to knock out a trapped atom. The capture and loss rates can be combined to give the following differential equation:

where N is the number of atoms in the MOT. Assuming that at t = 0, N = 0 the preceding equation can be expressed in the integral form:

where N_s = Rt the steady-state number of atoms.

In order to determine this filling curve the fluorescence signal from the growing cloud must be obtained. This was achieved by placing a photodetector a known distance away from the cloud and recording the signal as the cloud grew. The MOT could easily be turned on and off using a TTL signal applied to the repumping AOM. The MOT could be seen filling and dissipating at regular time intervals. For this particular experiment the trapping beam intensity was 9 mW/cm² in each beam and the detuning was approximately -2.5G. The magnetic field gradient was kept at 10 Gauss/cm

Figure 2.10 shows the data from the filling curve experiment. The solid curve through the data represents a least squares fit. Notice variation in data points as the time increases. This is due to oscillations in the fluorescence signal caused by collective behavior of atoms in the trap.⁹ These fluctuations became worse as the detuning was decreased.

Figure 2.10. Data from the filling curve experiment. The solid line represents a least squares fit analysis. The time constant equals about 1.4 seconds. The curve suggests a maximum capture velocity of about 30 m/s.

Notice that the time constant, t, equals 1.4 seconds. This number represents the mean time that an atom stays in the trap. The steady-state number, N_s, determined from the fit is on the order of 10⁸. Since N_s = Rt, we can use a little algebra to solve for the capture velocity of the trap to obtain the following equation:

where A represents the surface area of the MOT and s is the collisional cross section of cesium in a MOT. The collisional cross section for cesium has an accepted value of s = 2 x 10^-13 cm².¹⁰ Moreover, v_avg is the average velocity of background atoms in the chamber. At room temperature, background atoms in the chamber have v_avg 235 m/s. Thus, given a surface area of the trap equal to 0.69 cm², the capture velocity for the MOT equaled about 30 m/s. So atoms moving at velocities of 30 m/s or less were caught in the trap.

2.5 Number and Density Measurements

The number and the density of atoms in the MOT were obtained using two different methods. The first method measured the number of atoms by analyzing the fluorescence from the MOT. Conversely, the second employs a method of determining the density of the sample by analysis of the absorption spectrum of the cold, dense sample. In both experiments, the volume of the cloud was used to calculate the density and number, respectively.

An infrared camera was used to image the MOT. By sending the camera signal to an oscilloscope, it was possible to obtain an electronic signal corresponding to individual slices of the MOT. Placing a filter in front of the lens of the camera, it was possible to prevent saturation of the camera's signal. Note that in Figure 2.9 the MOT appears to be approximately the size of the beams. The size, however, is distorted due to saturation in the camera. Figure 2.11 shows the actual size of the MOT along with the corresponding camera signal. This signal allowed for a measurement of the cloud's diameter. This was achieved by comparing the width of the MOT's signal with the width of the laser beam's signal. The actual width of the laser beam was known to be 2.54 cm.

Figure 2.11. A filtered picture of the MOT showing its actual size. The figure on the right is the corresponding camera signal. Notice the diameter is about 0.47 cm.

Now that the volume of the MOT can be obtained, the number and density measurements can be discussed. The first method used a similar setup to the filling curve experiment. A photodetector was placed a known distance away from the MOT and the fluorescence signal was analyzed. The voltage signal generated by the photodetector can be converted to an optical power by converting the signal to a current and using the power to current constant for the particular photodetector. The total optical power at the photodetector is given by the relation:

where (/4) is the solid angle subtended by the photodetector, N is the number of atoms, _sc is the scatter rate of photons, and represents the wavelength of the emitted photons. For a detuning of -2G and beam intensity of 9 mW/cm², _sc approximately equals 0.12G. Knowing the optical power from the MOT, the scatter rate, the solid angle, and the energy of the emitted photons, we are able to solve for the number of atoms in the trap. For this experiment we found that the number of atoms equaled approximately 1 x 10⁹ atoms. Given a diameter of 0.47 cm the density is then about 2.3 x 10⁹ atoms/cm³.

The equation for the optical power assumes, however, that each atom in the cloud contributed to the fluorescence seen by the photodetector. This poses a problem. Consider a photodetector positioned on the right side of a MOT. Now consider an atom on the left edge of the MOT. If the photodetector is to see a photon from the atom on the left edge of the cloud, the photon must pass through many atoms to get to the detector. If the sample is sufficiently thick, the likelihood of that photon reaching the detector is low (i.e. the photon will be reabsorbed by a neighboring atom). Thus, the optical power equation may result in a calculation which is less than the true value.

This brings us to the second method used to measure the number of atoms in the MOT. This method does not rely on the fluorescence of the atoms, but on the ability of atoms to absorb photons. From this absorption, a determination of the number of atoms in the sample is possible.

The probe beam is a third diode laser which is linearly polarized and has a frequency sweep over the F = 4 to F' = 3, 4, and 5 transitions. The size of the probe beam was about 2 mm in diameter so that all of the power passed through the MOT. The intensity of the beam was about 60 W/cm² and did not noticeably disturb the MOT. A photodetector was mounted on the other side of the MOT and detected the signal transmitted through the cloud. Figure 2.12 shows the signal from the photodetector during one sweep of the probe beam through the MOT. The result is an absorption spectrum of atomic cesium showing the hyperfine structure of the 6p_3/2 level. These absorptions correspond to the F = 4 to F' = 3, 4, and 5 transitions. The plot shows distinct absorption at -450 MHz, -250 MHz, and 0 MHz detunings which are similar to the spacing between hyperfine energy levels seen in figure 1.1.

The shape and size of spectral lines can reveal a lot about a particular sample. For a determination of the number of atoms in the sample the following equation relating the transmission of the probe beam through the MOT is applicable:

where n is the density of the sample, is the absorption cross-section of cesium, and d is the diameter of the cloud. Magnifying the signal from the F' = 5 absorption spike in figure 2.12, one can obtain the ratio of P to P_o at resonance (zero detuning). From this spectrum, the transmission ratio is 0.04. Rearranging the equation for transmission we can solve for the density of the sample as follows:

This procedure reveals a density equivalent to 2.96 x 10⁹ atoms/cm³. In order to obtain the number of atoms in the MOT, we need to know the volume the MOT occupied. The volume calculation can be determined from the equation for the volume of a sphere, 4/3pr³. Using the radius measured from the camera signal the volume was found to be 0.43 cm³. The number of atoms in the MOT is then equivalent to

1.27 x 10⁹. The number is slightly higher than the number obtained using the fluorescence method. Figure 2.13 shows a summary the number and density measurements.

Trapping detuning (MHz)	-2G	-2G
Volume (cm3)	0.47	0.43
Density (atoms/ cm3 )	2.3 x 109	2.96 x 109
Number	1 x 109	1.27 x 109

Figure 2.13. A summary of the data obtained using the fluorescence and absorption methods.

2.6 Temperature Measurement

The temperature measurement is perhaps the most important and exciting measurement in laser cooling and trapping. There are several different ways to measure the temperature of a MOT. The method used in this study is called time of flight (TOF). The TOF method has proven to be an effective and rather straight forward technique for determining the temperature.¹¹ A diagram of the technique is the subject of figure 2.14.

Essentially, the time of flight method measures the amount of expansion the MOT undergoes a known time after the repumping laser is turned off. If the repumping laser is turned off, the atoms in the MOT are quickly pumped into the F = 3 level and are not scattered by the trapping laser. The atoms due to gravity and their motion fall and expand. Using a 1mm probe beam tuned to the F = 4 to F' = 5 transition positioned beneath the MOT, the atoms scatter photons as they fall. A photodetector was used to detect this scattering. Figure 2.15 shows the signal from the photodetector as the MOT falls and expands.

A simple kinematics analysis of the photodetectors signal yields an approximate temperature of the MOT. First, the MOT's initial diameter must be determined. As explained earlier, this is achieved by using the signal from the infrared camera. The diameter of the MOT in this experiment was found to be D_o = 0.42 cm. Then a determination of the time the cloud fell must be obtained. This time begins when the repumping laser is turned and off and ends when the center of the cloud passes through the probe beam. This time, t_f, is equal to 30 ms. Therefore, the velocity as the center of the cloud passes through the probe beam is v_f = g t_f = 0.29 m/s. The next time to determine is the time the cloud spent in the probe, t_p. This can be determined from the FWHM of the fluorescence signal as the cloud passed through the probe beam and was found to be 20 ms. Using this time and v_f, it is possible to determine the diameter of the cloud in the probe beam. This diameter, D_eff, is equal to 0.58 cm. The difference in D_eff and D_o is the expansion of the cloud due to its thermal motion. Taking this difference and dividing it by the time the cloud fell, we can determine the velocity of this expansion, v_e. Using the thermodynamic relation:

the temperature of the MOT is approximately 0.22 mK = 220 mK. Although much colder temperatures have been obtained, this value is comparable to 265 mK obtained by Sesko et al.¹² using a similar technique. Figure 2.16 shows a summary of the temperature calculation.

Initial diameter of cloud	Do = 4.2 mm (from camera signal)
Time in Probe	tp = 20 ms (from photodetector signal)
Fall time	tf = 30 ms (from photodetector signal)
Velocity due to fall	vf = g tf = 0.29 m/s
Effective diameter of cloud	Deff = vf tp = 5.8 mm
Diameter expansion	Deff - Do = 1.6 mm
Expansion velocity	ve = (Deff - Do) / tf = 0.053 m/s
Temperature of Atoms	T = (M ve2) / (2 kB) = 220 mK

Figure 2.16. This chart shows the calculation involved in the determination of the MOT's temperature.