Laboratory Courses on Laser Spectroscopy and Atom Trapping

H. Beica, S. Winter, C. Mok, B. Barrett, R. Berthiaume, A. Vorozcovs, F. Yachoua, N.Afkhami-Jeddi, M. Aggarwal, A.Pouliot, G.Carlse, K.Borsos, R. Marants, and A. Kumarakrishnan

Department of Physics & Astronomy, York University, 4700 Keele St. Toronto, ON M3J 1P3

We present an overview of experiments covered in two, semester-length, laboratory courses ded-icated to laser spectroscopy and atom trapping. These courses constitute a powerful approach forteaching experimental physics in a manner that is both contemporary and capable of providingthe background and skills relevant to a variety of research laboratories. The courses are designedto be accessible for all undergraduate streams in physics and applied physics as well as incominggraduate students. In the introductory course, students carry out several experiments in atomic andlaser physics. In a follow up course, students trap atoms in a magneto-optical trap and carry outpreliminary investigations of the properties of laser cooled atoms based on the expertise acquiredin the first course. We discuss details of experiments, impact, possible course formats, budgetaryrequirements, and challenges related to long-term maintenance.

PACS numbers: 01.40.gb, 07.57.-c, 37.10.De, 37.10.Jk

I. INTRODUCTION

Since the invention of the laser in the 1960s, they havebeen integrated into a myriad of applications, from pre-cision spectroscopy to industrial machining and surgery.As a result, it has become important to offer specializedinstructional training opportunities for students and re-searchers entering these fields. In this paper we discussa powerful range of educational experiments, organizedinto two laboratory courses that provide a detailed in-troduction to laser spectroscopy and atom trapping, andimpart the necessary experitse. These courses, whichhave been operational for ten years, have established ex-cellent performance indicators such as reliability of ex-periments and high rates of completion. The courseswere inspired by the robustness of the most commonlyused laser-based neutral atom trap, namely a magneto-optical trap (MOT) [1, 2]. The simplicity in the designand construction of MOTs has led to their proliferationin research labs. MOTs have become easy to operate,in part due to the improvement in the stability of diodelasers [3, 4]. Diode lasers have found widespread com-mercial use in DVD players, scanners and optical com-munication, and can be readily adapted for experimentsin optical physics [3–9]. Typically, students entering thethird year of study in an undergraduate physics programhave a background in electromagnetism, optics, and mod-ern physics. As a result, students can grasp the ba-sic principles involved in understanding and operatinga MOT, making laser cooling one of the most popularsubjects for research projects. Another reason for thispopularity is that the techniques that students learn inthis field are widely applicable in other areas such asobservational astronomy, experimental particle physics,biological physics, engineering physics, and in industriallabs dedicated to research and development in photon-ics. The typical upper year undergraduate curriculumat most universities has included classical experimentsthat allow students to understand and test fundamental

principles in physics. At York University, key experi-ments have included measurements of the speed of light,the inverse square laws in gravity or electromagnetism,charge quantization, discreteness of atomic energy lev-els, electron spin resonance, Fourier optics, gamma rayspectroscopy and alpha-particle scattering. These ex-periments have utilized equipment that is generally notused in research labs. Although students learned aboutpioneering measurements, it was clear that they wouldbenefit from a background that included exposure andtraining in a research environment using conventionaltechniques. With the proliferation of research labs ded-icated to atom trapping, it has been fairly straightfor-ward to provide training in this research area. This hasensured that students acquire a thorough background inexperimental physics that has included training in ad-vanced data analysis and hands-on skills with contem-porary equipment. The technology that has lead to thedevelopment of exciting trends in atomic physics includ-ing MOTs provides an ideal background.

It is also significant that laser cooling and trappingof neutral atoms has emerged as one of the most excit-ing areas of physics during the last two decades, withseveral Nobel prizes awarded for developments relatedto this field [10–22]. Atom traps containing ultra-coldsamples with characteristic temperatures of ∼ 100 µKhave resulted in high precision in experiments relatingto inertial sensing such as measurements of gravitationalacceleration [23, 24], gravity gradients [25], and rotation[26]. These experiments have inspired the developmentof portable sensors using cold atoms that can be utilizedfor oil and mineral prospecting and for correcting tidalcharts [27–29]. Other fascinating results include preci-sion measurements of the atomic fine structure constant[30–33], tests of the equivalence principle by measuringthe differential acceleration of two laser cooled isotopes[34], measurements of the gravitational redshift [35, 36],and low-energy tests of the standard model [37]. Atomicfountain clocks using cold atoms [38, 39] have replacedatomic beam based national time standards and are used

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to synchronize a network of global positioning satellites.The development of a femtosecond frequency comb thatcan be referenced to an atomic clock has also made itfeasible to develop a new generation of frequency stan-dards [40, 41]. Trapped atoms have also been cooled tonanokelvin temperatures at which Bose-Einstein conden-sation (BEC) can be observed [42, 43]. Obtaining BEChas made it possible to realize coherent sources of atoms[44, 45] and carry out interesting tests of atomic andcondensed matter physics at a level considered impossi-ble only a few years ago [46, 47]. Recent experiments inwhich a BEC is dropped over a 100 meter path-lengthhave made it possible to consider precision tests of theequivalence principle [48].

This paper provides a detailed overview of instruc-tional experiments adopted to laboratory courses and re-lated to MOTs. We begin with a discussion of the courseformat in section II and provide a brief overview of home-built diode lasers and tapered amplifiers that can reducethe cost of course development. In section III, we discussdetails of the experiments related to laser spectroscopyand atom trapping with examples of typical data. Thepaper concludes with a discussion of impact, alternateapproaches for course development, challenges related tothe training of teaching assistants, and long-term main-tenance in section IV.

II. COURSE FORMAT

At York University, two one semester courses, desig-nated as PHYS 4061 (Experimental Techniques in LaserPhysics) and PHYS 4062 (Atom Trapping Laboratory),have been part of the curriculum since 2007. The in-troductory course (PHYS 4061) involves six laboratoryhours and two hours of classroom instruction per weekand nine experiments related to techniques in atomicphysics and laser spectroscopy. The experiments inPHYS 4061 are based on a number of undergraduateresearch projects [49–58]. The objective of the course isfor students to acquire hands-on experience with conven-tional equipment and contemporary experimental tech-niques. Therefore, PHYS 4061 stresses short-form lab re-ports with an emphasis on data analysis. Although thereare a number of platforms for symbolic computation, wehave adopted Mathematica since it seems well suited forbatch processing and curve fitting. As a result, the coursebegins with a comprehensive Mathematica tutorial thatis spread over four three-hour laboratory sessions. Duringthese sessions, students complete a laser safety tutorialand become familiar with the flash card system for down-loading data from oscilloscopes. For the remainder of thecourse, students work in groups of two and cycle throughthe spectroscopy experiments on a weekly basis. Eachof these experiments has been designed to be completedduring two three-hour laboratory sessions.

Topics required to understand these experiments arecovered in a variety of upper level physics courses such

as electromagnetism, modern physics and quantum me-chanics. Additionally, a comprehensive laboratory man-ual [59] has been developed to cover theoretical conceptsassociated with each experiment. The lecture componentof this course includes twelve lectures that cover theoret-ical background related to instrumentation such as gasand diode lasers, LEDs, electro-optic and acousto-opticmodulators, Fabry-Pérot interferometers, and propertiesof Gaussian beams. Each lecture is supported by a tu-torial that covers the background for experiments. Thecourse has been designed to accommodate 20 students inone section.

PHYS 4062 focuses on atom trapping and allows stu-dents to apply skills acquired in PHYS 4061 to trap atomsin a MOT and carry out simple investigations of the prop-erties of laser cooled atoms. The lecture component ofPHYS 4062 describes the formalism related to the radi-ation pressure force, magnetic trapping and the opticaldipole force. The tutorials describe techniques relevantto cold atom experiments. Students are provided withtwo preassembled vacuum chambers for atom trappingthat have been pumped down to the required base pres-sure. This course has a laboratory component amountingto six hours per week for a period of eight weeks. Duringthe last four weeks of the term, students are required tocomplete a long-form lab report as in traditional labora-tory courses.

The student experience in this course closely resemblesa research environment with open-ended experiments.Our approach has been different from atom trapping ex-periments at other universities because it provides stu-dents with a greater degree of hands-on experience.

Four of the nine experiments in PHYS 4061 are basedon the use of external cavity diode lasers (ECDLs). Sofar, we have relied entirely on commercial diode lasersystems to support these courses. However, we have con-structed home-built ECDLs to support course expansionin the future. Graduate teaching assistants have foundthe commercial systems to be straightforward to main-tain since the only requirement is the replacement of laserdiodes and re-alignment of the laser head (typically oncein two years). However, investment in these systems canhave a significant impact on the equipment budget re-quired to set up the courses, as discussed further in thissection and section IV.

ECDLs can be assembled from readily available com-mercial components from a number of robust and inex-pensive designs in Refs. 3, 5, 7–9. These lasers have spec-ifications that are similar to commercial systems, such asan output power of up to 150 mW and a linewidth of∼ 1 MHz. We have built a number of laser heads basedon the design in Ref. 3, which uses a diffraction gratingin the Littrow configuration to provide optical feedback.More recently, we have had even greater success employ-ing interference filter (IF) stabilized ECDLs [Figure 1],based on the design in Refs. [8, 9]. These lasers can bevacuum sealed to improve long-term frequency stability[60] and they have been successfully integrated into in-

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dustrial gravimeters for accurate measurements of grav-itational acceleration [61]. Additionally, we have devel-oped temperature and current control modules to stabi-lize the frquency of ECDLs. Home-built locking modulesfor active frequency stabilization have allowed peak lock-ing and side locking [55] as well as auto-locking withoutthe need for human intervention [60, 61]. We find thatthe cost of a home-built laser system, including electron-ics, is about 1/4th of the cost of a similar commercialsystem ($20,000). The home-built lasers have workedreliably in atom trapping experiments conducted in a re-search laboratory, but require more expertise to maintainthan commericial systems.

Based on the design in Ref. 62, we have also amplifiedthe power outputs of these lasers to 2 W using home-built systems that incorporate tapered amplifier waveg-uides [63]. Optical fibre coupling, with typical couplingefficiencis of 35%, can be used to improve the spatial pro-file at the cost of optical power. Such a Master Oscilla-tor Power Amplifier (MOPA) system can be constructedfor 1/5th of the commercial cost ($30,000). AlthoughECDLs provide sufficient power for aotm trapping, powerlimitations due to optical losses and the need for inten-sities that are large compared to the saturation inten-sity for atoms (∼1 mW/cm2) restrict the diameter of thetrapping laser beams to a few millimeters. In contrast,a single MOPA system can be used to operate severalMOT setups in a student lab with trapping beam diam-eters of several centimeters. The number of atoms thatcan be loaded into a vapor cell MOT with a laser beamdiameter d is known to scale as dx, where x is typicallylarger than 3 [64]. Therefore, alignment of a MOT canbe made easier with large diameter laser beams, apartfrom acheiving a high signal-to-noise ratio in cold atomexperiments. Therefore, the impact of a large beam di-ameter for experiments in an instructional lab must beconsidered.

ECDLs and MOPA systems must be protected fromoptical feedback that can damage both laser diodes andtapered amplifiers. In our experience, home-built Fara-day isolators developed on the basis of an undergraduateresearch project [58] have been cost effective solutions.These devices have provided optical isolation of > 30 dBfor 1/5th of the commercial cost.

III. DESCRIPTION OF EXPERIMENTS

Four experiments in PHYS 4061 use Rb spectroscopyand ECDLs operating at 780 nm. They involve top-ics such as absorption and fluorescence spectroscopy,electro-optic modulation, Zeeman shift, laser linewidth,and laser frequency stabilization. We have divided thearticle into subsections that provide an overview of theseexperiments with each subsection concluding with re-marks about the relevance of these experiments to atomtrapping in PHYS 4062. Four additional experimentsthat involve diode laser modules operating at other wave-

LD

CL VAPBS

IF FL M

Piezostack

λ/2

(a)

(b)

FIG. 1. (a) Layout of IF laser. The optical elements are: LD- laser diode, CL - collimating lens, VA - variable aperture,λ/2 - half wave-plate, PBS - polarizing beam splitter, IF -interference filter, FL - focusing lens, M - mirror. (b) Home-built IF-laser head.

lengths are then described. These additional experimentsare related to optical heterodyne detection, Gaussianbeam propagation, optical detectors, and optical tweez-ers. We then describe a separate experiment that usesa pumping station to introduce vacuum techniques. Allexperiments reviewed in this section provide a suitablebackground for experiments with atoms in a MOT.

Most of the laser spectroscopy experiments in thecourse are carried out in Rb vapor at room temperatureusing ECDLs. The energy level diagrams for the D2-linesin 85Rb and 87Rb are shown in Fig. 2(a). The Dopplerbroadened absorption spectrum obtained by scanning theECDL several GHz over the Rb resonances in a 5 cmlong vapor cell at room temperature is shown in Fig.2(b). The Doppler widths of the individual transitions(for example, the F = 3 → F ′ = 2, 3, 4 resonances in85Rb) can be obtained from background subtracted fits.The calculated FWHM for an isolated two-level atomicsystem at a wavelength of 780 nm and a temperature of300 K is 511.2 MHz. It is an interesting exercise to under-stand this discrepancy using known frequency splittingsbetween individual transitions that contribute to the fit-ted line shape and the corresponding transition proba-bilities. The Doppler-free saturated absorption spectrumcorresponding to the aforementioned resonance is shownin Fig. 2(c). A fit to the F = 3 → F ′ = 4 resonance isshown in Fig. 2(d).

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FIG. 2. (a) Energy level diagrams for 85Rb and 87Rb. TheRb reference cell contains the two naturally occurring iso-topes in the number ratio of 72:28. (b) Oscilloscope traceof Doppler broadened absorption spectra in 85Rb and 87Rbobserved by scanning the ECDL. The ground states corre-sponding to each peak are labeled. The frequency inter-val between the two ground states in 85Rb is marked. Thetime axis can be converted to frequency units based on theknown separation between energy levels. The full width athalf maximum (FWHM) extracted from a Gaussian fit of theF = 3 → F ′ = 2, 3, 4 resonance is 735 ± 5 MHz. Basedon Beer’s law, I = I0e

−α(ν)l where α(ν) is the absorptioncoefficient, l is the length of the Rb cell, I0 is the incidentintensity and I is the transmitted intensity. We find thatthe maximum absorption corresponds to an optical depthofα0l = 0.516 ± 0.003, where α0 = α(ν0) and ν0 is the reso-nance frequency. (c) Oscilloscope trace of Doppler-free spec-trum for the 85Rb F = 3 → F ′ = 2, 3, 4 transitions. Thefrequency interval between the F ′ = 4 peak and the adjacentcrossover resonance is 60.4 MHz. (d) An expanded plot ofthe F = 3 → F ′ = 4 resonance line. A FWHM of 12.7 ± 0.9MHz is obtained from a Lorentzian fit. In general, this valueshould have contributions from the natural broadening (∼ 6MHz), power broadening, and laser linewidth.

C

BA Probe

beam

Glass slideBeamblock

Laser

MirrorMirror

Beamblock

Solenoid

Rb cellλ/4 λ/4

λ/2Polarizing beamsplitter

Pump beamEOM

FIG. 3. Saturated absorption setup used in emission andabsorption spectroscopy, Zeeman shift, and laser frequencystabilization experiments. A quarter-wave plate representedby λ/4 is used to produce circularly polarized laser beams. Ahalf-wave plate, represented by λ/2, is used to control the ra-tio of power reflected and transmitted by the polarizing beamsplitter. “A” and “B” represent photodiodes used for record-ing absorption spectra. The absorption of a weak probe beamthat serves as a reference is recorded by photodiode A. Theabsorption of a weak probe beam in the presence of a strongcounter-propagating pump beam is recorded by photodiode B.Electronic subtraction of the signals from these photodiodesgives a Doppler-free spectrum shown in Fig. 2(c). Photodi-ode C is used to record right-angle fluorescence. Locations ofthe EOM used in the emission and absorption spectroscopyexperiment and the solenoid used in the Zeeman shift exper-iment are shown by dashed lines. In experiments with theEOM, the two weak probe beams are blocked and only thepump beam remains on.

A. Emission and Absorption Spectroscopy and TheElectro-Optic Phase Modulator

Students are introduced to atomic structure and ab-sorption and fluorescence spectroscopy in the first partof the experiment. The two measurement techniques arecompared and contrasted. A schematic of the experi-mental setup is shown in Fig. 3. A weak probe beam isscanned across Rb resonances in a vapor cell and the ab-sorption is detected by photodiode A. In this manner, theDoppler broadened absorption spectra from the groundstates of 85Rb and 87Rb can be measured. The spectraare fitted using Gaussian functions is shown in Fig. 2(b).The absorption of a weak probe beam in the presence ofa strong counter propagating pump beam is detected byphotodiode B showing Doppler-free resonances as shownin Fig. 2(c). A Lorentzian fit to a single resonance lineis shown in Fig. 2(d).

In a separate experiment, an electro-optic phase mod-ulator (EOM) is placed in the pump laser with theprobe beams blocked [52, 57]. This commercially avail-able modulator is tuned to a resonance frequency of2.915 GHz. This value was chosen corresponding to thedifference between the F = 3 → F ′ = 4 and F = 2 →F ′ = 3 transitions in 85Rb, shown in Fig. 2(a).

In the EOM experiment, the pump beam intensity isfixed and the fluorescence spectra from a detector at anangle of 90 (photodiode C in Fig. 3) with respect to thelaser beam is monitored. Spectra are recorded with the

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EOM turned on and off, as shown in Fig. 4. New spectralfeatures that are displaced by the operating frequency ofthe EOM are produced. These features can be used toinfer the efficiency of the EOM [52]. In addition, IT isan interesting challenge to interpret the relative changesin amplitudes of other spectral lines [52].

This experiment can also be carried out using a home-built EOM with a resonance frequency of ∼ 20 MHz.The cost of the home-built EOM is approximately onesixth of the cost of the commercial EOM [57]. For ex-periments with low frequency EOMs, it is necessary tomeasure Doppler-free absorption spectra (in the presenceof pump and probe lasers) to resolve the sidebands.

FIG. 4. Fluorescence spectra from a pump laser beam passingthrough an EOM and a Rb vapor cell. The laser is scannedover several GHz and a fluorescence signal is recorded by pho-todiode C in Fig. 3. The black curve shows the fluorescencewith the EOM off. The grey curve is obtained with the EOMon. The operating frequency of the EOM is 2.915 GHz whichrepresents the energy difference between the F = 3 → F ′ = 4and F = 2 → F ′ = 3 transitions in 85Rb.

A comprehensive introduction to atomic level struc-ture, Doppler-free spectra, and cross-over resonances isprovided in the manual [59] so that students can inter-pret the results. A description of the line shapes due toDoppler broadening and natural broadening is also pro-vided. A brief description of electro-optic phase modula-tion and applications of EOMs is provided in connectionwith the last part of the experiment.

Knowledge of atomic level structure and frequencyshifts of laser beams are crucial for experiments in atomtrapping since these experiments require frequency sta-bilization and frequency tuning of a laser beam with re-spect to an atomic spectral line. Trapping 85Rb atoms ina MOT requires a trapping laser to rapidly cycle atomsbetween the F = 3 and F ′ = 4 states and a second (re-pump) laser to optically pump the F = 2 ground state.The laser frequencies can be derived from the same ECDLusing an EOM with an appropriate operating frequency.

B. Zeeman Shift

In this experiment, the frequency shift of an atomictransition in Rb is studied as a function of magnetic fieldstrength [65]. The saturated absorption setup shown inFig. 3 is used to observe the Doppler-free Rb spectrumshown in Fig. 2(c). A solenoid is placed around the Rbvapor cell (see Fig. 3) to produce a nearly uniform mag-netic field. The laser is scanned over the F = 3 → F ′ = 4atomic resonance as shown in Fig. 2(a) and Fig. 2(d). Tomeasure the Zeeman shift of this spectral line, spectra aresimultaneously recorded with reference to a separate Rbvapor cell under ambient conditions. This procedure iscrucial for eliminating the effect of laser frequency drifts.The magnetic field has been sufficiently large (> 1 Gauss)to overwhelm the effect of the Earth’s field, but not largeenough (< 30 Gauss) to produce level mixing. It is alsoimportant to ensure that the circular polarization of bothpump and probe beams are set so that they drive thesame transition.

An example of the measured Zeeman shift plotted asa function of magnetic field is shown in Fig. 5. Theslope is consistent with the expected Zeeman shift forthe F = 3,mF = 3 → F ′ = 4,mF ′ = 4 transition. Thissuggests that a system consisting of seven independenttransitions between the F = 3 and F ′ = 4 levels can bemodeled by a two level system. We attribute this behav-ior to two effects [65]. Firstly, for circular polarization,the transition probabilities increase for transitions withhigher magnetic quantum numbers. For example, thetransition probability for the mF = 3 → mF ′ = 4 spec-tral line is 28 times larger than the transition probabilityfor the mF = −3 → mF ′ = −2 spectral line. Secondly,if the beam diameter is sufficiently large, such that thetransit time is comparable to the optical pumping time,optical pumping during the scan time of the pump lasercan preferentially populate the F = 3,mF = 3 groundstate. It is instructive to record the Zeeman shifts of bothDoppler-free and cross-over resonances with different po-larization settings for the pump and probe and comparethe measured shifts with theory.

It is notable that the narrow laser linewidth andDoppler-free spectroscopy allow the Zeeman shift to becharacterized using a small wire-wound solenoid and acompact setup. In contrast, the Zeeman shift is typi-cally studied in upper year laboratories using a dischargelamp source and a large water-cooled solenoid to generatemagnetic fields of ∼ 1 T.

In the final part of the experiment, the dependence ofthe width of a spectral line on laser intensity is stud-ied. In this case, the width of the Doppler-free resonanceis recorded as a function of the pump beam intensity,which is varied by using a λ/2-wave plate and polarizingbeam splitter at the entrance of the saturated absorp-tion setup. Alternatively, optical neutral density filterscan be introduced into the pump beam. It is possible toinfer the natural linewidth of the spectral line as well asthe saturation intensity of the transition from this data

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if the vapor cell is magnetically shielded.

0 5 10 15 20 25 30

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FIG. 5. The frequency shift of the F = 3 → F ′ = 4 tran-sition as a function of magnetic field. The frequency shift isobtained by recording the position of the spectral line in a Rbcell placed in a magnetic field relative to the position of thespectral line without a magnetic field. A value of 1.49± 0.08MHz/G for the Zeeman shift is found from the slope of astraight line fit. The expected value for the Zeeman shift ofthe mF = 3 → mF ′ = 4 sublevel transition is 1.4 MHz/G.The horizontal error bars are estimated to be ±5% and thevertical error bars are comparable to the size of the points.

Background concepts related to the Zeeman shift, cal-culation of Landé g-factors and power broadening aredescribed in the manual. In this way, students are intro-duced to the Zeeman shift, which is responsible for therestoring force in a MOT. Additionally, the frequencyshifts of laser beams used in atom trapping can be con-trolled by Zeeman shifting spectral lines to which thelasers are locked [65].

C. Faraday Isolator and Fabry-PérotInterferometry

Two separate experiments have been combined intothis lab. In the first part, the function of polarizingelements such as linear polarizers and wave plates arereviewed. Linearly and circularly polarized light are dis-tinguished by measuring the transmission through a ro-tating polarizer attached to a compact motor. Since op-tical feedback can alter or damage laser diodes, the util-ity of using an optical isolator consisting of a polarizingcube and λ/4-wave plate to reduce optical feedback isdemonstrated. This arrangement reduces optical feed-back only for a particular polarization. Subsequently,students align a home-built Faraday isolator [58], a de-vice that eliminates optical feedback for all polarizations.The rotation angle and the isolation ratio of the isolatorare measured in the experiment.

In the second part of the experiment, light from theECDL (linewidth ∼ 1 MHz) is sent through a Fabry-Pérot interferometer (FPI) with a resolution of ∼ 60MHz. The ECDL grating that provides optical feed-back is bypassed so that the linewidth of the laser be-comes larger than the FPI resolution. The laser fre-quency is fixed and the transmission spectrum of the FPI

Laser system

Removablemirror

Laser diode

Grating

FIG. 6. Experimental setup for measurement of the laserlinewidth using a FPI. The grating in the ECDL cavity isbypassed by inserting a mirror attached to a removable mountso that the linewidth of the free running laser diode can bedetermined. This configuration is shown by the dotted box.When the mirror is removed, the linewidth of the laser is muchsmaller (∼ 1 MHz) due to the presence of grating feedback.Since the FPI resolution is ∼ 60 MHz, only the instrumentlimited linewidth is measured in this case.

is recorded by scanning the spacing between the cavitymirrors. The linewidth of the free-running laser diodecan be estimated from the transmission spectrum of theFPI. The experiment is repeated after the reintroductionof the grating. In this case, the instrument limited res-olution rather than the laser linewidth is measured. Aschematic of the experiment is shown in Fig. 6 and rep-resentative FPI spectra are shown in Fig. 7. The modecharacteristics of the ECDL can also be investigated byvarying the diode current and temperature.

(a)

FPI R

espo

nse

[arb

. uni

ts]

Time [ms]

(b)

Time [ms]

FPI R

espo

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[arb

. uni

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FIG. 7. Transmission peaks obtained by operating the diodelaser at a fixed frequency and scanning the FPI. (a) FPI peakswithout grating feedback. The measured FWHM is 133.3± 5MHz. (b) FPI peaks with grating feedback. The measuredFWHM of 66.7± 3 MHz is comparable to the calculated res-olution of the FPI. The value of the free spectral range of thespherical mirror FPI (c/4L), calculated on the basis of theknown cavity length of L = 7.5 cm is ∼ 1 GHz. Since theseparation between adjacent transmission peaks is equal tothe free spectral range, it is possible to convert the time axisinto frequency units.

The operating principles and design of a Faraday iso-lator and essential properties of the FPI are reviewed inthe lab manual. The students are exposed to the proper-ties and characteristics of narrow linewidth ECDLs thatare used for atom trapping.

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D. Laser Frequency Stabilization and the Lock-InAmplifier

This experiment focuses on the use of a lock-in ampli-fier for frequency stabilization of a diode laser. The satu-rated absorption setup used in this experiment is shownin Fig. 3. The schematic for the feedback loop requiredto lock the laser frequency to the peak of an atomic spec-tral line is shown in Fig. 8.

Modulation

Laser

Errorsignal

Lock-inamplifier

Absorption signal(modulated resonance peak)

Saturatedabsorption

setup

To experiment

FIG. 8. Setup for laser frequency stabilization using a lock-inamplifier. The modulation was provided by a function gen-erator operating at ∼ 7 kHz. A scan controller is used toscan the laser frequency at a rate of ∼ 10 Hz. An exam-ple of the saturated absorption setup is shown in Fig. 3. Atrace of the saturated absorption signal obtained by scanningthe laser over a single Doppler-free transition is shown. Forillustrative purposes, this trace was recorded with a large am-plitude for frequency modulation. The output of the lock-inamplifier generates the corresponding dispersion shaped errorsignal. To lock the laser to the peak of this resonance line,the laser frequency is centered on the resonance peak, and thescan amplitude is gradually reduced (ideally to zero) beforeengaging the feedback loop.

In the first part of this experiment, students are guidedthrough the essential components of a lock-in amplifier,shown in Fig. 9. Exercises include studying the perfor-mance of a phase shifter and radio frequency (RF) mixer.The phase shifter consists of an analog circuit with a vari-able resistor that allows the phase shift imposed on anRF waveform to be varied and calibrated. Two RF in-puts, consisting of the output of the phase shifter andthe modulated absorption signal, are used as inputs tothe mixer. The mixer output is verified to scale as thecosine of the phase difference between the two RF inputs.

In the second part of the experiment, a commerciallock-in module attached to the laser controller is used tostudy the dispersion-shaped error signal shown in Fig.9. For this experiment, the modulated saturated absorp-tion signal and the modulation from a built-in functiongenerator are used as inputs to the lock-in. The depen-dence of the error signal on lock-in phase is studied byscanning the laser across a suitable atomic resonance, asshown in Fig. 10. To lock the laser, the error signal isfirst minimized by varying the phase, as shown in Fig.10(b). The phase is incremented by ±90 to maximizethe error signal, as shown in Fig. 10(a) and 10(c). It can

be determined that the laser will remain locked for oneof these settings and that it will be forced away from res-onance for the other setting. To lock the laser, the scanamplitude is reduced and the feedback loop is engagedby sending the error signal to the laser. After locking thelaser to a particular spectral line, the frequency stabilityof the laser is determined by recording the fluctuationsin the DC level of the error signal, as shown in Fig. 11.This figure also illustrates the frequency excursions of thelaser without feedback.

INPUTS OUTPUT

Error signal

Gainamplifier

Low-passfilterMixer

Phaseshifter

Modulation

Absorptionsignal

Lock-in amplifier

FIG. 9. Schematic diagram showing the essential componentsof a lock-in amplifier.

Atom trapping experiments require frequency stabi-lized lasers to drive specific atomic transitions. The labmanual presents both a physical model of the error signal[49], as well as a more mathematical treatment [55].

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Time [ms]

(c)0.0

-0.2

-0.4-0.6-0.8-1.0

45 50 55 60

FIG. 10. Lock-in output (error signal) obtained by scanningthe laser across a resonance line for various phase shifts be-tween the absorption signal and the modulation: (a) ϕ = 0,(b) ϕ = 90 and (c) ϕ = 180.

E. Radio Frequency Components and OpticalHeterodyne Detection

Students are introduced to the function of RF com-ponents in the first part of this experiment. Simpleexperiments that utilize a voltage-controlled oscillator

8

Time

Sign

al [a

rb. u

nits

]

10 s

0 0

Lock On Lock Off

FIG. 11. Lock-in error signal recorded with and without en-gaging the feedback loop. For both parts of the trace, thescan amplitude has been turned off, whereas the modulationremains on. The amplitude variations are mainly due to laserfrequency fluctuations, which are greatly reduced with thefeedback loop engaged (lock on). Under typical operatingconditions, the diode laser remains locked for about 30 min-utes.

(VCO) are used to understand working principles of am-plifiers, mixers, power splitters, frequency doublers, andtransistor-transistor logic (TTL) switches. For example,students can design a frequency doubler by connectingtwo outputs of a power splitter into a mixer and filteringout the low frequency component. Other experiments in-clude measuring the frequency range and power outputof a VCO and measuring the on/off ratio of RF pulsesusing a mixer or a TTL switch.

In the primary experiment, a continuous wave (CW)RF signal from an amplifier is used to drive anacousto-optic modulator (AOM). Light from one of theECDL-based experiments described previously is alignedthrough the AOM. The first order diffracted beam is ren-dered co-propagating with the undiffracted beam usinga combination of mirrors and a 50/50 beam splitter, asshown in Fig. 12(a). The undiffracted beam from theAOM serves as an optical local oscillator (LO). A photo-diode placed in the path of overlapped beams records aheterodyne beat note at the RF frequency used to drivethe AOM. Under typical operating conditions, the inten-sity of the diffracted beam is comparable to that of theundiffracted beam. If the AOM is driven at low power,the diffracted beam can represent a weak signal. Withthe LO blocked, the photodiode can be used to detect thesignal as a change in intensity. With the LO unblocked,the signal can be detected by the presence of the beatnote. Other extensions of this experiment could includestudying the size of the beat signal versus the power inthe diffracted beam and comparing the smallest signalsthat can be measured using heterodyne and intensity de-tection.

The remainder of the experiment involves pulsing theRF signal driving the AOM and recording the diffractedpulse detected by the photodiode. Pulsing the amplitudeof the RF signal from a VCO is accomplished using afunction generator and a TTL switch, as shown in Fig.12(a). The photodiode records a time-delayed optical

pulse, as shown in Fig. 12(b). The envelope of the op-tical pulse exhibits the heterodyne beat note. The timedelay between the trigger pulse used to drive the AOMand the corresponding optical pulse is measured. Thedistance between the transducer attached to the AOMcrystal and the laser beam is recorded. These measure-ments allow the speed of sound through the AOM crystalto be inferred. The rise time of the diffracted pulse canbe used to measure the size of the laser beam passingthrough the AOM. Additional exercises relating to thisexperiment could include demonstrations of setting upAOMs in a dual pass configuration [51].

The background covered in the lab manual includes thetheory of Bragg diffraction in an AOM and a derivationof the frequency shift of the diffracted beam based onthe first order Doppler shift for sound waves. In theatom trapping course, students use an AOM to derivethe frequency-shifted trapping laser beam from a diodelaser locked to an atomic spectral line. The trappinglaser can be amplitude modulated for a range of coldatom experiments.

F. Optical Detectors

This experiment is designed to provide a workingknowledge of three different optical detectors: photodi-odes, photomultiplier tubes (PMT), and charged coupleddevices (CCD cameras). In a preliminary experiment,CW light intensity is measured using both photodiodesand PMTs so that the dynamic range, saturation charac-teristics, and signal-to-noise ratios associated with thesedetectors can be estimated. Using an AOM, it is alsopossible to demonstrate pulsed detection. The responsetime of an optical detector with a 1 MΩ termination re-sistance is measured. The time constant of the detectionsystem is modified by changing the termination resistorso that the rise time of the optical pulse from the AOMcan be measured.

A triggered CCD camera and frame grabber are used tomeasure the fluorescence generated by an electron beammoving across the phosphor screen of an analog oscil-loscope, as shown in Fig. 13. The characteristic de-cay in fluorescence intensity as the beam sweeps acrossthe screen is shown in Fig. 14. We find that the decaytime of the phosphor coating on the oscilloscope screen isstrongly dependent on both the electron beam intensityand the oscilloscope time-base settings. For this mea-surement, we used an oscilloscope with a P31 phosphor.

Triggering the CCD prepares students for atom trap-ping experiments since photographing the electron beamfluorescence is analogous to photographing the ballisticexpansion of a cloud of laser cooled atoms after turn-ing off the confining forces. CCD imaging is a standardtechnique used for measuring the temperature of trappedatoms [56]. PMTs and photodiodes are also extensivelyused in atom trapping. For example, a PMT can be usedto measure trap fluorescence and estimate the number of

9

VCO

Switch

(b)

FIG. 12. (a) Heterodyne detection setup. The incoming beamis generated by an ECDL operating at 780 nm. The opticallocal oscillator (LO) is the undiffracted beam from the AOM.The first order diffracted beam from the AOM is rendered co-propagating with the LO using a beam splitter and mirrorsso that a heterodyne beat signal can be detected by the pho-todiode. The RF input to the AOM is either CW or pulsed,and the control electronics are shown. The output of the TTLswitch is an RF pulse that drives the AOM. (b) The squarepulse used to amplitude-modulate the AOM and correspond-ing time-delayed optical pulse detected by the photodiode.The square pulse is the signal input to the electronic TTLswitch, which also triggers the oscilloscope. The optical pulseexhibits a beat note at the AOM drive frequency. The timedelay between the onset of the square pulse and the beginningof the optical pulse is 3.45± 0.05µs. This delay is consistentwith the estimated time for an acoustic wave to propagatefrom the transducer at the edge of the AOM crystal to thelocation of the laser beam passing through the AOM.

trapped atoms and a photodiode can be used for measur-ing the absorption spectrum of the cold sample [54].

G. Gaussian Beam Propagation and Optical FiberCoupling

This lab builds on earlier experiments in second yearoptics where students are introduced to techniques for

CCD image after trigger

shutter closedshutter open

Control Pulse

FIG. 13. A CCD camera and frame grabber are used to mea-sure the fluorescence of an electron beam as it sweeps acrossthe screen of an analog oscilloscope. A pulse generator is usedto generate a TTL pulse that controls the shutter of the CCDcamera.

0 1 2 3 40

200

400

600

800

1000

Time [ms]

Fluo

resc

ence

[arb

. uni

ts]

FIG. 14. Intensity of fluorescence due to an electron beamsweeping across the screen of a line triggered oscilloscoperecorded by a shutter-controlled CCD camera. The dips inintensity correspond to the grid lines on the oscilloscope. Thespacing between the grid lines is determined by the time basesetting of the oscilloscope, thereby allowing the pixel axis tobe converted to a time scale. The solid curve is the best fitto the data based on a decaying exponential. The fit gives adecay time of 1.30± 0.03 ms.

obtaining the spatial profiles of laser beams. These tech-niques include scanning a pinhole or knife edge across theprofile while measuring the transmitted intensity or mea-suring the intensity through a series of calibrated aper-tures centered on the beam. In this paper, all measure-ments of the spatial profile are carried out using a com-mercially available scanning knife-edge beam profiler.

In the first part of the experiment, the spatial profileof a focused He-Ne laser is measured along two orthog-onal directions perpendicular to the direction of propa-gation of the laser beam. The measured beam size of afocused laser beam as a function of position is shown inFig. 15(a). The Rayleigh range and beam divergence canbe determined from fits to the beam profile as a functionof the distance from the position of the focus.

The second part of the lab demonstrates how spatialfiltering can be used to improve the beam profile. Bothpinhole filtering and optical fiber coupling are demon-strated. Since the experiment involves very few opticalelements and the laser beam propagates over a limiteddistance on the optical table, a scratched cover slide isused to distort the beam profile. The effect of spatiallyfiltering this beam using an optical fiber is shown in Fig.15(b).

10

(a)

Position z [mm]

Bea

m ra

dius

w [m

m]

-0.15 -0.05 0.150.060.080.100.120.140.160.180.20

0.05Time [ms]

Inte

nsity

[mV

]

(b)

-4 -2 0 2 40

20

40

60

80

FIG. 15. (a) Beam radius, w, orthogonal to the directionof propagation as a function of distance (z) from a 20 cmfocal length plano-convex lens. The beam size was measuredusing a scanning knife edge beam profiler. The position z = 0represents the focal spot. The solid line is a fit to the equationw = w0[1 + (z − z1)

2/z20 ]1/2, where z0 = πw2

0/λ. Here, w0 isthe minimum beam radius, z0 is the Rayleigh range and z1is an offset that specifies the location of the minimum beamradius from the origin. The fit gives w0 = (58.7 ± 1.3)µm,z0 = (5.8 ± 0.2) cm, and z1 = (4.6 ± 0.8) mm. (b) Spatialprofile of a laser beam before (black) and after (grey) passingthrough an optical fiber. The profile at the entrance to thefiber was obtained by distorting the beam using a scratchedcover slide. The profiles represent the analog outputs of ascanning knife edge spatial profiler.

Further appreciation of spatial profiles is emphasizedin the atom trapping experiment since the number ofatoms that can be trapped in a MOT depends criticallyon the beam profile.

H. Vacuum Components and Pumping Techniques

This experiment provides a working knowledge of somevacuum pumps and pumping techniques. The experi-mental setup is shown in Fig. 16. In the first part of theexperiment, students use a roughing pump to pump on avacuum chamber with a volume of ∼ 13 L. The pumpingspeed is investigated with different bellows hose lengths,apertures and leaks. In all these cases, the decrease in thefore line pressure is measured as a function of time usinga Pirani gauge and a stopwatch. We find that the changein pressure can be suitably modeled as a decaying expo-nential immediately after the pump down commences.For the geometry with the highest pumping speed, thetransition from the viscous flow regime to the molecularflow regime is estimated to occur for pressures less than10 mTorr. In the second part of the experiment, the vac-uum chamber is pumped using a roughing pump in serieswith a turbo pump to a pressure of ∼ 10−6 Torr. Theeffect of out gassing is illustrated by valving off the turbopump and by heating parts of the vacuum chamber.

The theoretical background in the laboratory manualincludes a discussion of both continuum and molecularflow regimes, mean free path, and the effect of the di-ameter and the length of a hose on the conductance.The principles of operation of roughing pumps, turbopumps, diffusion pumps, and entrapment devices suchas ion pumps and cryogenic pumps are reviewed. A few

Roughingpump

Bellows

Piranigauge

Chamber

Iongauge

Turbopump

Valve A

Valve B

Valve C

FIG. 16. Vacuum system used in the experiment. In the firststage of the experiment, valves B and C are closed and thechamber is pumped directly using the roughing pump. In thesecond stage, valve A is closed and the chamber is pumpedby the turbo pump in series with the roughing pump.

common types of pressure gauges are discussed.This experiment provides a suitable background to

understand the conditions in the small volume vacuumchamber used for atom trapping experiments.

I. Optical Tweezers

We have also introduced an optical tweezers experi-ment based on Ref. 66. A suspension of polystyrenebeads (∼ 1−2 µm in diameter) in saline solution is placedin a channel on a cover slide attached to a translationstage (shown in Fig. 17). The beads can be viewed ona CCD camera using a backlight for shadow imaging, asshown in Fig. 18(a). The beads are trapped using lightfrom a diode laser that outputs ∼ 50 mW of power at awavelength λ = 532 nm. Most of the laser light incidenton a dichroic mirror (oriented at 45 as shown in Fig. 17)is focused onto the sample using a microscope objective.A lens mounted on a rail is used to adjust the size ofthe beam at the entrance to the objective. A small frac-tion of the back-scattered light from a trapped bead willpass through the dichroic mirror, along the line of sightassociated with the backlight. The position of the beadcan be tracked using the CCD camera. A LabVIEW in-terface displays an image of the trapped bead obtainedby subtracting background light, as shown in Fig. 18(b).The interface allows the data associated with the trappedbead to be displayed either in real time or after the acqui-sition of a number of frames. Figure 18(b) also displaysthe spatial profiles of light reflected off a trapped beadalong two orthogonal directions. These plots representpixel readouts along directions marked by cursors placedon the image of the trapped bead. Due to the relativelyslow processing time of the CCD camera, the data rep-resenting the bead positions are well separated in time(∼ 15 ms) so that correlation effects can be ignored.

Figure 19(a) shows the mean-squared displacement ofthe bead (calculated in software) as a function of theinverse laser power. The spring constant, ks, of theharmonic potential associated with the dipole force trapis proportional to laser intensity. Therefore, the mean-

11

squared displacement is inversely related to laser powerin a manner consistent with expectations. The springconstant of the dipole force trap as a function of thelaser power is shown in Fig. 19(b). The spring constantwas calculated by using the equipartition theorem andthe measured mean squared displacement. As expected,we find that the spring constant varies linearly with laserpower.

Dichroicfilter

FIG. 17. Schematic diagram of the optical tweezers experi-ment. The laser source is a diode laser operating at λ = 532nm. The dichroic mirror ensures that most of the laser lightreaches the microscope objective. The rail lens is used to varythe size of the laser beam. The objective focuses the laserbeam to a spot size of ∼ 3 µm. The sample of polystyreneglass beads in saline solution is placed at the focal spot of thelaser. The sample is uniformly illuminated by a backlight.Light from this lamp source passes through a condenser lensand the sample. After passing through the microscope objec-tive, it is imaged on a CCD camera. With the laser blocked,the CCD camera can image the glass beads suspended in so-lution. When the optical dipole force trap is aligned, thedichroic mirror and dichroic filter ensure that a small frac-tion of the light scattered by the trapped beads reaches theCCD camera. The scattered light follows the same opticalbeam path as the backlight and is sufficiently intense for thetrapped bead to be imaged.

The same setup can also be used to measure the au-tocorrelation function of a trapped bead. In this case, aposition sensitive detector (PSD) with a response time of∼ 100 µs is used to track the bead position. The PSDis placed at the location of the back light. To ensure asufficient signal-to-noise ratio, an iris is placed in frontof the PSD to eliminate background light. Figure 19(c)shows the bead displacement measured by the PSD overa time-scale of 45 ms. An autocorrelation function canbe calculated from this data, as shown in Fig. 19(d).The correlation time τc is inferred by fitting the data toa Lorentzian function [66]. Figure 19(e) shows the cor-

relation time τc as a function of the laser power. Theinverse dependence is consistent with the prediction forτc = γ/ks. Here, γ is the rate of collisions between thesolute molecules and the trapped bead.

An even more popular viariant of this experiment hasbeen added to the course, which allows students to trapmicron-sized particles in free-space using the same diodelaser [Figure 20(a)]. The simplest particles to trap arecarbon based ink particles ablated from the surface of apermanent marker or saturated cotton swab. The parti-cles are trapped near the location of the minimum beamwaist. A plastic enclosure with glass slide “windows”should be used to surround the region around the trapto shield the droplets from air currents. Particles canalso be introduced to the trapping beam by atomizationthrough one of the windows. Students are able to mea-sure the spring constant of the traps for different particlessuch as charcoal, graphite and carbon black ink [Figure20(b)].

The lab manual provides an introduction to the ori-gin of the optical dipole force (ODF) on glass beads anddiscusses the mechanism by which the ODF can trapneutral atoms. The techniques used in this experimentare widely applied in biophysics to manipulate proteins,DNA, and other organic molecules that can be tetheredto glass beads [67, 68]. This experiment has additionalrelevance to atom trapping since neutral atoms that arelaser cooled and trapped in a MOT can also been loadedinto an ODF trap, also known as a far off resonance trap(FORT). FORTs can be designed to have large poten-tial well depths and have the advantage of low scatteringrates in comparison to MOTs.

(a)

(b)

FIG. 18. (Color online) Display of the LabVIEW inter-face used to record and analyze CCD images of the trappedbeads.(a) Image of beads suspended in solution. (b) The up-per frame shows the image of a 2 µm diameter trapped bead.The corresponding intensity profile is displayed in the lowerframe.

12

1/P [m ]W-1

0.00 0.02 0.04 0.06 0.08 0.10

0.000

0.005

0.010

0.015

(a) (b)

P [mW]

k s[

N/m

]

10 20 30 40 50 60 70 80

02000400060008000

10 000

-

+

(c)

0 20 40-0.2-0.1

0.00.1

PSD

Out

put [

V]

Time [ms]

(d)

0 1 20

0.0010

0.0020

0.0030

Aut

ocor

rela

tion

Func

tion

[V ]

t [ms]0.5 1.5

2

Power [mW]

t [m

s]

20 40 60 80

(e)

c

FIG. 19. (a) Mean squared displacement as a function of in-verse laser power. The power represents the measured valueat the entrance to the microscope objective. Only ∼ 50% ofthe light reaches the sample. (b) Spring constant of the dipoleforce trap as a function of laser power. The horizontal axishas not been corrected for the power loss at the microscopeobjective. The points representing the spring constant arecalculated from the data in (a) on the basis of the equiparti-tion theorem. (c) Signal from the PSD which represents theposition of the bead as a function of time. (d) Autocorrelationfunction (black) calculated from the PSD signal by fitting toa Lorentzian (grey) for a small section of the data such as infigure (c). The horizontal axis represents the time lag betweenthe data points in the record. The fit gives a correlation timeτc = 0.422 ± 0.005 ms. (e) Correlation time as a function oflaser power.

J. Atom Trapping in a MOT

The overall goals of the atom trapping course (PHYS4062) is to provide students with hands-on expertise andexperience that is similar to that of a research lab. PHYS4062 is an elective course with an enrollment of approxi-mately ten students. The course operates with two inde-pendent setups consisting of two groups of five students.Each student group is guided by a teaching assistant.We find that students routinely trap atoms in three tofour weeks and can focus on cold atom experiments forabout four weeks. After the completion of laboratorywork, students are allotted three weeks to submit a for-mal long-form lab report that requires interpretation ofdata and a comprehensive discussion.

Figure 21(a) shows a picture of the atom trapping ap-paratus. The stainless steel vacuum chamber has eight

beam

Horizontal plane

Gaussian beam profile

/2

Focusing lens

(a)

(b)

P [mW]

k s[

N/m

]-

+

FIG. 20. (a) Modified experimental set-up for free-space trap-ping experiment. (b) Spring constant of the dipole force trapas a function of laser power for a charcoal particle confinedin free-space. The horizontal axis has been corrected for thepower loss of the microscope objective.

symmetrically aligned viewports in the horizontal planeand two viewports along the vertical axis. Six viewportsare used by the three pairs of orthogonally directed trap-ping beams. Two viewports in the horizontal plane areused for CCD imaging of the atom cloud [shown in Fig.21(b)] and detection of trap fluorescence using a photo-diode. The two remaining viewports can be used for ex-periments such as absorption spectroscopy with a probelaser.

Typically, it takes a few weeks to prepare a vacuumsystem that is suitable for atom trapping. Therefore, itis preferable for students to work with a chamber thathas already been pumped out. The vacuum system isinitially pumped down using a turbo pump and bakedat ∼ 100C for a few weeks so that the base pressure is∼ 10−9 Torr at the end of the bake-out. The turbo pumpis then disconnected and an ion pump with a pumpingspeed of 8 liters per second is used to maintain the vac-uum. The Rb source consists of a sealed ampoule that is

13

fused to a glass-to-metal adapting flange. The Rb sourcecan be fabricated in a glass shop using a commerciallyavailable ampoule and a glass-to-metal adapter. The sealcan be broken in vacuum by using a magnet to manipu-late a glass-enclosed magnetic striker. During the atomtrapping experiments, the typical vapor pressure of Rbatoms in this chamber is ∼ 5× 10−8 Torr.

The number of atoms that can be accumulated in aMOT can be optimized by maximizing the laser coolingforce and the atom trapping force. The trapping beamdiameter, which is set by the available power and laserdetuning, determines the laser cooling force [69]. Typi-cally, the atom trapping force can be maximized underconditions in which the Zeeman shift is comparable tothe Doppler shift. For 85Rb atoms, this is achieved us-ing a magnetic field gradient of ∼ 10 G/cm. The anti-Helmholtz coils of the MOT shown in Fig. 21(a) (diame-ter ∼ 20 cm, ∼ 140 turns) can produce the required mag-netic field gradient with an operating current of ∼ 5 A.The coils can constructed by winding magnet wire on aplastic or metal frame.

(a)

(b)

FIG. 21. (a) A portion of the atom trapping setup. A pair ofanti-Helmholtz coils is used to apply a magnetic field gradient.Two pairs of the trapping beams are aligned in the horizontalplane between the anti-Helmholtz coils. An orthogonal trap-ping beam is aligned along the vertical. (b) Image of trapped85Rb atoms recorded by a CCD camera. An optimized traphas a 1/e2 cloud radius of ∼ 1 mm and holds ∼ 107 atoms.The CCD image of the trap in (b) is saturated causing bloom-ing.

For a multi-level atom such as 85Rb exposed to a high-intensity laser beam, the laser cooling force is maximized[70] for a trap laser detuning of ∼ 15 MHz below theF = 3 → F ′ = 4 cycling transition [see Fig. 2(a)]. A sec-ond (repump) laser is tuned to the F = 2 → F ′ = 3 tran-sition to optically pump atoms out of the F = 2 groundstate. The recommended power in the trapping laser is> 50 mW. For a trapping beam with 50 mW of powerand a diameter of ∼ 1 cm, we find that a repump laserpower of ∼ 1 mW is sufficient. We find that the experi-ments described in the next section of this paper can becarried out without the need to actively lock the repumplaser due to the power available (∼ 50 mW) . However,the repump laser frequency is carefully monitored to en-

sure that it coincides with the resonant frequency. Thenumber of trapped atoms can be maximized if the re-pump laser is combined with the trapping beams alongall three axes using polarizing beam splitters. However,it is possible to operate the MOT even if the repumplaser is aligned along a separate axis through the centerof the chamber.

The trapping laser is aligned along three orthogonaldirections of the trapping chamber using polarizing cubebeam splitters and λ/2-plates to control the relative in-tensities in each direction. At the entrance to the trap-ping chamber, each incident beam is circularly polarizedusing a λ/4-wave plate and aligned through the centers ofthe viewports. After passing through the chamber, eachbeam is directed through another λ/4-wave plate andretro-reflected. This alignment ensures that the polariza-tion of the MOT laser beams is set to a σ+σ− configura-tion along each orthogonal direction. The incident andretro-reflected beams have opposite angular momentumwith respect to the axis defined by the incident beam.Due to quantum mechanical selection rules, these beamsdrive atomic transitions between hyperfine states withmagnetic quantum numbers that differ by ∆mF = ±1,provided that the quantization axis is collinear with theaxis of propagation.

Maxwell’s equations ensure that the magnetic field gra-dients along the three orthogonal directions satisfy thecondition that ∇ · B = 0. Therefore, it is necessary toensure that two of the incident trapping beams in thehorizontal plane have the same angular momentum dueto cylindrical symmetry. The third incident beam (alongthe vertical axis in this case) is set with the opposite an-gular momentum. The orientations of the first λ/4-waveplate in each of the incident beams should be set carefullyto ensure circular polarization. However, the σ+σ− con-figuration for a MOT can be achieved for any orientationof the second λ/4-wave plate.

The MOT is formed at the location where the mag-netic field is zero. Therefore, it is important to align theanti-Helmholtz coils so that the field is zero at the cen-ter of the apparatus. It is also important to ensure thatthe laser beams are centered at this location. Ideally theEarth’s magnetic field, which shifts the position of zerofield, should be canceled at the center of the chamberusing three pairs of Helmholtz coils. However, cancel-lation of ambient fields is generally not required in aninstructional experiment.

We have described the standard configuration used fora vapor cell loaded MOT [1] in which the repump laseris derived from a separate diode laser. The advantageof this approach is that the experimental setup is wellsuited to develop hands-on experience and promotes aninteractive approach. In this arrangement, we find thatstudents are able to make mistakes and correct them withrelative ease.

Other possible configurations include a system of pyra-midal mirrors that ensure the MOT is automaticallyaligned if the incident beams are directed along the ap-

14

propriate axis [71]. It is notable that miniaturized vac-uum chambers using epoxied glass sheets have been used[72]. It is also possible to introduce an EOM in the trap-ping laser to generate a sideband that corresponds to therepump transition frequency.

We now provide a step-wise description of the stagesinvolved in the atom trapping experiment. A small frac-tion of laser light is required for setting up saturated ab-sorption for both trap and repump lasers. The repumplaser is tuned to resonance with the F = 2 → F ′ = 3transition in 85Rb. The trap laser is locked to the peakof a crossover transition ∼ 60 MHz below the F = 3 →F ′ = 4 transition in 85Rb [see Fig. 2(c)]. Most of the traplaser light is aligned through a single pass AOM operat-ing with a frequency upshift of 45 MHz so that the laserfrequency of the diffracted beam is ∼ 15 MHz below theF = 3 → F ′ = 4 resonance. The diffracted beam servesas the trapping laser and it can be conveniently ampli-tude modulated using the AOM. This beam is combinedwith the repump laser using a polarizing beam split-ter. The combined beams are split into three paths andaligned through the vacuum chamber. Ideally, the retro-reflected beams should be aligned collinear within a fewmilliradians with respect to the incident beams. Spatialfiltering of the beams using fibers can result in a signifi-cant increase in the number of trapped atoms. However,we avoid the use of fibers since there is a 50% reduc-tion in laser power and increased experimental complex-ity. The polarization of the three orthogonal trappinglaser beams can be conveniently determined throughoutthe beam paths by measuring the optical transmissionthrough a rotating polarizer.

The next step involves the calibration and alignment ofthe anti-Helmholtz coils. The Rb source is gently heatedso that the fluorescence associated with laser beams pass-ing through the vacuum chamber is visible on a monitoror on a hand-held view scope sensitive to near-infraredlight. If the frequencies of the lasers are adequately con-trolled and the optical alignment is satisfactory, a MOTcan be observed with very few permutations to the exper-imental configuration. These involve switching the polar-ization state of the vertical trapping beams between twoorthogonal circular polarizations and reversing the direc-tion of current through the anti-Helmholtz coils.

K. Experiments with Laser Cooled Atoms

After optimizing the MOT, students carry out a num-ber of straightforward experiments relating to the prop-erties of cold atoms. All these experiments can be carriedout by monitoring the fluorescence of the MOT using aPMT or amplified photodiode and determining the cloudradius using steady-state images captured by the CCDcamera, similar to Fig. 21(b). Examples include tun-ing the lasers to trap 85Rb and 87Rb isotopes, studyingthe atom number as a function of the magnetic field gra-dient and laser frequency, measuring the atom number

and atomic density using the trap fluorescence and cloudradius, determining the temperature of trapped atomsand gravitational acceleration by photographing the bal-listic expansion of the trapped cloud, investigating op-tical pumping by turning off either the trap or repumplasers, measuring the effect of collisions from the loadingrate of the trap, probe laser spectroscopy, and investiga-tion of the spring constant of the trap. The results helpstudents understand important properties of atom trapsthat were first revealed in several classic papers in thefield [1, 39, 44, 64, 69, 73–75]. We now review a selectionof experiments performed by students using 85Rb atoms.

Ato

m N

umbe

r[a

rb. u

nits

]

Trap Power [mW]

(a)

10 20 30 40 50 60 700

50100

150200

Repump Power [mW]Ato

m N

umbe

r[a

rb. u

nits

] (b)

0.2 0.4 0.6 0.8 1.0

50

100150200

FIG. 22. (a) Steady state atom number as a function of traplaser power. The atom number is inferred by recording thetrap fluorescence on a photodiode. The voltage recorded bythe detector can be converted into atom number. The maxi-mum voltage corresponds to an atom number of ∼ 107. Sincethe trap laser is retro-reflected through the vacuum cham-ber, the laser power incident on the atoms is approximatelytwice the trap laser power. (b) Atom number as a functionof repump laser power. For both of these experiments, thetrap laser detuning was fixed at 15 MHz below resonance, thetrapping beam diameter was ∼ 0.90 cm and the magnetic fieldgradient was ∼ 7 G/cm.

Fig. 22(a) shows the steady state atom number, Nss,as a function of trap laser power for a fixed trap laserdetuning and trapping beam diameter. The experimentis carried out with trap laser intensities that are signifi-cantly larger than the saturation intensity Isat for 85Rbatoms (Isat = 1.67 mW/cm2 for the F = 3,mF = 3 →F ′ = 4,mF ′ = 4 transition). Under these conditions,it has been predicted that the optimal detuning shouldbe ∆ = −2.5Γ [69, 70], which is the value used in thisexperiment. The data shows that the atom number satu-rates as the trap laser power is increased. This is a resultof the excited state population saturating for intensitiesI ≫ Isat. The ratio I/Isat can be used to quantify theradiation pressure force on atoms.

To understand the steady state atom number, we notethat the MOT operates by slowing and trapping atomsin the low velocity tail of the speed distribution of thebackground Rb vapor. The trapping beam diameter de-termines the maximum velocity of atoms (the capturevelocity, vc) that can be trapped. The steady stateatom number is related to the capture velocity throughNss = (d2/σ)(vc/u)

4 [39, 64], where d is the diameter ofthe trapping beams, σ is the cross-section for collisionswith “hot” background atoms, and u is the most prob-

15

able speed of the background atoms (u ∼ 242 m/s for85Rb at T = 300 K).

The repump laser is combined with the trapping laseralong all three orthogonal directions. Figure 22(b) showsNss as a function of the repump laser power. Under theseconditions, atoms can be cooled over the entire volumeof Rb vapor that is illuminated by both trap and re-pump lasers. The data shows that the atom number sat-urates with an increase in repump laser power, therebyconfirming that the atom number is not limited by re-pump laser intensity. Under these conditions, the ex-periment achieves complete optical pumping out of theF = 2 ground state.

0 5 10 150

10

20

30

40

50

60

0 2 4 6 8 10 1202468

Position [mm]

Inte

nsity

[arb

. uni

ts] (b)

Atom Number [ x10 ]7

Ave

rage

Ato

m D

ensi

ty [

x10

/cm

]9

3

(a)

FIG. 23. (Color online) Variation in average density as afunction of atom number. The atom number, inferred fromtrap fluorescence, was varied by changing the trap laser power.The trap laser detuning was fixed at 15 MHz below resonance,the trapping beam diameter was ∼ 0.90 cm and the magneticfield gradient was ∼ 7 G/cm. Three different repump laserpowers were used, 0.6± 0.06 mW (red squares), 1.0± 0.1 mW(green circles), and 50 ± 5 mW (blue triangles). The cloudradius was determined from fits along two dimensions thatwere summed in quadrature. (b) Spatial profile of the cloudobtained from a CCD camera image for a trap laser power of75±7 mW, repump laser power of 50±5 mW, and a magneticfield gradient of ∼ 7 G/cm. The vertical axis represents theintensity obtained by integrating pixel values. A Gaussian fitgives a 1/e2 cloud radius of 0.66± 0.01 mm.

For dilute traps (optical depth less than unity), pho-tons that are absorbed and emitted by trapped atoms arenot re-absorbed by other trapped atoms. Under theseconditions, the density of the trap should increase as theatom number is increased [56]. As radiation trapping be-comes important, the density of the cloud is expected tosaturate, indicating that the cloud volume increases withan increase in atom number due to the outward pressurefrom re-scattered photons [73]. The data in Fig. 23 isconsistent with these expectations. Here, the atom num-ber is varied by changing the trap laser power. Figure 23also shows that the data is independent of repump laserpower. The average density is calculated by combining

the atom number measured from trap fluorescence andthe cloud radius measured using a CCD camera. Sincethe trap density is relatively low, the power requirementsfor the repump are very modest.

It is possible to obtain atom numbers of ∼ 109 byusing larger trapping beam diameters. Under these con-ditions, it is possible to observe a “bounce trap” [76] ifthe repump laser intensity lowered. In this situation, therepump light is highly attenuated at the center of thecloud due to absorption. Only atoms in the outer shellthat are cycled on the F = 3 → F ′ = 4 transition by thetrapping laser contribute to the trap fluorescence, leav-ing the atoms at the center of the cloud predominantlyoptically pumped into the F = 2 ground state. As aconsequence, the trap fluorescence can scale linearly asa function of the surface area of the cloud. In contrast,if the repump laser intensity is large, the trap fluores-cence can scale linearly with the cloud volume. However,to study these trends, it is important to consider a suit-able method of changing the steady state atom numberwithout independently changing the cloud size.

Iris diameter [cm]

Ato

m n

umbe

r [ar

b. u

nits

]

0.0 0.2 0.4 0.6 0.80

50

100

150

(a)

3 4 5 6 7 8 9 10

0.08

0.10

0.12

0.14

0.16

Magnetic Field Gradient [G/cm]

Trap

Rad

ius [

cm]

(b)

FIG. 24. (a) Atom number as a function of iris diameter, d.The data shows a sensitive dependence on the beam size. Fora well aligned trap and a well centered iris, the atom numbercan be expected exhibit a power law dependence shown bythe solid line. For this experiment, the trap detuning wasset to 15 MHz below resonance, the trap laser power was∼ 65 mW, the repump laser power was set at ∼ 15 mW,and the magnetic field gradient was ∼ 7 G/cm. (b) Cloudradius (1/e) as a function of magnetic field gradient along thevertical direction. For this data, the trap laser power wasfixed at ∼ 75 mW and the trap laser detuning was ∼ 15 MHzbelow resonance with a beam diameter of ∼ 1.0 cm. Therepump power was ∼ 12 mW.

The capture velocity for a MOT is a non-linear func-tion of the trapping beam diameter. As a consequence,the atom number is predicted te scale as dx, where d is thetrapping beam diameter and x typically ranges from ∼ 3to 4 [64, 69]. The variation in x arises because the atomnumber must be optimized for a given trapping beamdiameter by varying the detuning, trap laser intensity,and magnetic field gradient. The sensitive dependenceof the atom number on the trapping beam diameter canbe studied by varying the size of an aperture that is cen-tered on the trapping beam. Figure 24(a) shows a strongdependence of the atom number on beam diameter. Theatom number falls to zero for a finite beam diameter dueto misalignment.

16

For small atom number, the linear variation of the den-sity with atom number in Fig. 23(a) suggests that radi-ation trapping does not affect the cloud radius for thisdata set. As a result, we assume that the temperatureof the trapped atoms is not affected by the absorption ofre-scattered photons [56]. Careful measurements of thetemperature in Ref. 56 suggest that the effects of ra-diation trapping influence the temperature even in thelinear regime.

If the temperature is assumed constant, the equiparti-tion theorem can be used to argue that the cloud radiusshould scale as

√1/(dB/dz) [56, 77]. This is because

kBT = ksr2, where ks is the spring constant of the trap

(which is directly proportional to the magnetic field gra-dient [70]) and r represents the cloud radius. Although itcan be challenging to confirm the predicted trend, typicaldata shown in Fig. 24(b) qualitatively confirms expecta-tions that the cloud radius decreases monotonically withincreasing magnetic field gradient. This is because thestiffness of the trap is proportional to the magnetic fieldgradient. For an optimally aligned trap, the position ofthe centroid of the cloud should not change when themagnetic field gradient is increased. Therefore, studiesas a function of magnetic field gradient are also useful asa test of alignment.

50 100 150 2000

0.05

0.10

0.15

0.20

0.25

Time [μs]

Trap

Las

er F

luor

esce

nce

[arb

. uni

ts]

FIG. 25. Fluorescence from the F = 4 → F ′ = 3 transitionas a function of time after blocking the repump laser. Anexponential fit gives a time constant for the decay of 42.42±0.05 µs. The trap laser power was 75 ± 7 mW, the repumppower was 50 ± 5 mW, and the magnetic field gradient was∼ 7 G/cm.

Figure 25 shows the decay of the fluorescence from thetrap after turning off the repump light reaching the trapusing an AOM. The AOM has a switching time of < 1µs. The decay represents optical pumping to the F = 2ground state. Atoms in this state do not interact with thetrap laser. The data is fitted to an exponential decay todetermine the optical pumping time. The value obtainedfrom the fit is consistent with the expected time for op-tical pumping based on a three-level model. This datais a clear signature of optical pumping as the timescalefor atoms to leave the trapping volume is much longer.This experiment also illustrates the importance of usinga repump laser. An important precaution for this exper-

iment is the time for shutting off the repump laser mustbe much shorter than the optical pumping time.

(a)

Ato

m N

umbe

r [a

rb. u

nits

]

(b)

Background Rb Fluorescence [mV]

FIG. 26. (a) Loading curve for the MOT. The data wasrecorded for a fixed nRb. A fit to the functional form de-scribed in text gives Γ−1 = 1.17± 0.03 s. (b) Variation of thecollisional rate with background fluorescence, which is pro-portional to nRb.

Figure 26(a) shows a measurement of the loading timeof the trap obtained with a separate setup. To recordthis data, the magnetic field gradient was initially turnedoff, with the trap and repump lasers on. In this man-ner, the trapped atoms were allowed to escape from thetrapping volume. The magnetic field gradient was thenturned on and the atom number was recorded as a func-tion of time. Alternatively, the magnetic field gradientand repump lasers can be left on, and the trap laser canbe amplitude modulated. This approach makes it easierto trigger the experiment. However, it is necessary in-troduce a RF switch and pulse generator for amplitudemodulation.

The loading rate of a vapor cell loaded MOT can bemodeled by a differential equation of the form dN/dt =R − ΓN [1]. Here, N is the atom number, R =nRbd

2v4c/u3 is the capture rate of atoms from the back-

ground, nRb is the background density of Rb atoms andΓ is the collisional rate of cold atoms with background Rbvapor. This equation assumes that there are no collisionsbetween cold Rb atoms that can also result in a changein the atom number. The solution to this differentialequation is N(t) = Nss(1 − e−Γt), where Nss = R/Γ isthe steady-state atom number. The data shown in Fig.26(a) is well modeled by N(t). The collisional rate Γcan be extracted by fitting the data to the prediction forN(t).

Loading curves such as those shown in Fig. 26(a)can be recorded for different background Rb densitiesby varying the temperature of the Rb source. Since Rbhas an appreciable vapor pressure at room temperature(3×10−7 Torr) and since the melting point of Rb is 312.5K, only a small variation in the temperature of the Rbsource is required to affect a significant change in thebackground density. Figure 26(b) shows that the colli-sional rate is a linear function of the background Rb den-sity, nRb. The relative density was monitored by mea-suring the fluorescence at the center of the cell. Themeasured trend is consistent with expectations, since thevariation of Γ as a function of density can be described byΓ = nRbσvrel. Here, σ is the cross-section for collisions

17

between background and laser cooled Rb atoms and vrelis the relative velocity between colliding atoms.

It is also an interesting exercise to monitor Nss as afunction of the change in background fluorescence due toan increase in the temperature of the Rb source. For lowbackground density, Nss increases because the increasein the capture rate R is larger than the increase in thecollisional rate Γ. As the background density increasesfurther, the trap loads faster, and Nss can be observedto decrease.

(a)

Trap

Flu

ores

cenc

e [a

rb. u

nits

]

51015202530

Time [s]0

0

0.5 1.0 1.5 2.0 Frac

tion

of A

tom

s Rem

aini

ng(b)

0Expansion Time [ms]

FIG. 27. (a) Measurement of the optical molasses decay time.A fit to a decaying exponential gives a 1/e time constantof 0.165 ± 0.002 s. (b) Measurement of the temperature oftrapped atoms using the release and re-capture method. Thevertical axis represents the fraction of re-captured atoms. Thehorizontal axis is the free expansion time. A determinationof the most probable speed u from the fit gives a temperatureT = 420± 20 µK.

If the MOT is carefully aligned, it is possible to ob-serve an optical molasses due to the damping force ofthe laser beams by turning off the magnetic field gradi-ent. Fig. 27(a) shows the molasses decay time for typicaloperating conditions. The decay is monitored by record-ing the fluorescence associated with cold atoms after themagnetic field gradient is turned off. The relatively fastdecrease in trap fluorescence is primarily due to imbal-ances in the forces due to trapping beams and the failureto cancel background magnetic fields. Typically, the mo-lasses decay time can be extended to several seconds byimproving the alignment and canceling background mag-netic fields such as the Earth’s field. For these reasons,monitoring the molasses decay time can be a sensitivetest of MOT alignment. This experiment was carriedout by turning off the power supply driving the anti-Helmholtz coils. The time constant for turning off thecurrent is determined primarily by the self-inductance ofthese coils. However, this time scale is generally muchsmaller than the time constant associated with molassesdecay. It should be noted that in research labs, magneticfield gradients can be turned off on sub-millisecond timescales using field effect transistors (FETs) or insulatedgate bipolar transistors (IGBTs).

Figure 27(b) shows a measurement of the temperatureof the trap using the release and re-capture technique[78]. With the repump laser and magnetic field gradi-ent on, the trap laser is pulsed off, allowing atoms toleave the central region of the MOT before the remain-ing atoms are recaptured by the trap light turning back

on. The atom number remaining in the central region ismeasured by varying the “off” time of trap laser light andrecording the fluorescence at the time of trap laser turnon. For this experiment, the central region is carefullyimaged on an optical detector. Ideally, the repump laserand the magnetic field gradient should also be turnedoff. However, it is not critical to do so since these pa-rameters have a small effect on the temperature. Turningoff the magnetic field gradient and repump laser requiresadditional equipment. The temperature of the trappedatoms, T , is determined by fitting to an expression of theform [Rd/R(t)]3, where R(t) =

√R2

0 + (ut)2. Here, Rd isthe radius of the detection volume, R0 is the initial cloudradius, u =

√2kBT/M is the most probable speed, and

M is the atomic mass. A more complete expression forR(t) can be derived on the basis of Refs. 50 and 53.

We note that there are several methods for measur-ing the temperature of the trapped atoms. For example,Refs. 50 and 53 describe time of flight techniques used inRefs. 56 and 77. However, these experiments require ad-ditional lasers, changes in alignment, and synchronizedtiming for CCD imaging. Therefore, we find that therelease and re-capture technique is generally best suitedfor an instructional environment.

MOTs are also well suited for several additional ex-periments including trapping 87Rb and spectroscopy ofcold atoms. Absorption spectroscopy can be carried outby scanning the frequency of a probe laser across Rbresonances and comparing with the saturated absorptionspectrum at room temperature. Alternatively, fluores-cence measurements can be carried out by slowly sweep-ing the trap laser across resonance after the atoms havebeen trapped.

IV. CONCLUSIONS

The development of these courses was made possibleby the participation of six graduate students who vali-dated experimental concepts, completed troubleshooting,and developed a laboratory manual [59]. This model forcourse development was inspired by an earlier experiencein which a number of undergraduate students were ableto revitalize second year laboratory courses on electro-magnetism and optics with the introduction of severalnew experiments.

Although the impact is difficult to measure in quanti-tative terms, it is evident that students who have com-pleted these courses are better prepared in advanced dataanalysis techniques and have adequate exposure in ex-perimental physics for graduate studies or industrial ca-reers. Due to its impact, the introductory course onlaser spectroscopy is currently a compulsory requirementfor all physics streams at York University. Demand forthis course has led to an expansion to two sections, eachof which can accommodate 20 students. The follow upcourse on atom trapping has consistently experienced en-rollments close to its capacity of 10 students. In all, over

18

a ten year period, more than 300 students have receivedthe specialized training offered through these courses.

The popularity of these courses seems to be related, inpart, to the accessibility of the background material andthe reliability of experiments. Students find the introduc-tory course to be demanding, but describe the hands-onexperience and the exposeure to sophisticated data anal-ysis as both new and rewarding.

The follow-up course on atom trapping seems to repli-cate a typical research lab environment. This course en-courages students to develop data taking strategies anddata interpretation skills through intense discussions.Therefore it is clear that students develop the indepen-dence and teamwork required in a research laboratory.

Both courses have consistently received superior stu-dent evaluations in comparison to traditional lab coursesand have been repeatedly described as one of the mostconsequential courses within the physics program. Theseoutcomes and positive student feedback are due in partto the nature of the experiments, emphasis on data ac-quisition techniques, and troubleshooting experience thatare notably different from other upper-level laboratorycourses.

The individual experiments described here have servedas effective platforms to guide upper-level students indesign concepts and research topics related to ECDLs[3, 60], tapered amplifiers [62, 63], Faraday isolators [58],electro-optic modulators [57], and analog circuits for laserfrequency stabilization [55]. Other examples of develop-ing relevant instrumentation such as Fabry-Pérot inter-ferometers are described in Ref. 79.

These courses have had an unmistakable impact on re-cruiting and imparting training relevant to research inultra-cold atoms. The course has contributed to the pro-fessional development of 30 undergraduate and 10 gradu-ate students associated with a single research group overa ten-year period. The quality of these students can beinferred because 80% of these graduate students and 70%of these undergraduates have been recipients of some ofCanada’s most exclusive scholarships. These studentshave completed a variety of research projects at the un-dergraduate level [49–58]. The contributions of graduatestudents are reviewed in Refs 61 and 80.

The average operational budget required to maintainthese courses is approximately $3,000 per year. Such anallocation is sufficient for minor equipment and for cov-ering expenses associated with equipment failure. Typ-ically, items that require replacement are laser diodes(costing $1,000 for commercial lasers) and control elec-tronics such as power supplies. Additionally, ion pumps(costing $1,000) that are used in the atom trapping vac-uum chambers may need to be replaced on a timescaleof 10 years. This budget is comparable to the cost ofmaintaining other upper year laboratory courses.

Ideally, two trained graduate teaching assistants with a

research concentration in atomic physics and one techni-cal support staff member would be required to maintainthese courses. We find that a small fraction of grad-uate students who have enrolled in the courses can beretained as teaching assistants. Over the ten-year pe-riod, the courses have involved nine teaching asistants ofwhom five have made long-term contributions.

We estimate that the cost of equipment required to setup all experiments described in this paper (based on com-mercially available components) is ∼ $150, 000. The bud-get is dominated by the cost of four diode laser systems(∼ $80, 000). The equipment budget can be substan-tially reduced by using home-built ECDLs and MOPAs,as described in section II. As an example, the cost offour home-built ECDLs and a single MOPA system isabout $26,000. Other home-built components could in-clude Faraday isolators, locking circuits, and Rb refer-ence cells. A list of equipment relating to individual ex-periments and the laboratory manual [59] are availableupon request.

In conclusion, we expect that this paper will serve asa useful model for developing contemporary experimentsrelated to laser spectroscopy and atom trapping. A fewleading institutions such as California Institute of Tech-nology, University of Michigan and Stony Brook Univer-sity and several liberal arts colleges have also developedcourses in related areas. Although we have described acomprehensive approach in which a range of experimentsare carried out over fixed laboratory hours, alternativeformats that may be equally effective could involve theintroduction of even a small subset of these experiments.A laboratory course with one or two ECDLs and openhours as in the traditional format, may be well suited toexpose students to the key topics.

V. ACKNOWLEDGMENTS

This work was supported by the Canada Foundationfor Innovation, Ontario Innovation Trust, the NaturalSciences and Engineering Research Council of Canada,Ontario Centres of Excellence, and York University. Weacknowledge the role of graduate teaching assistants M.Weel, S. Beattie, I. Chan, E. Rotberg, A. Carew, andundergraduate students S. Chudasama, D. Gosset, K.Sowka, V. Popovici, H. Morrison, A. Sibilia, J. PerezGarcia, B. Barron, S. Chudasama, P. Dowling, S. Jack-son, and A. Schiff-Kearn, who have helped develop dif-ferent aspects of experiments. We thank C. Guimaraesfor providing data in Fig. 15. The courses were devel-oped as a result of a gift of $125, 000 from Optech Inc.and matching funds from the Faculty of Science and En-gineering at York University.

19

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