Experiment S-10: Optical Pumping. Physics 510

Transcription

1 Experiment S-10: Opticl Pumping Physics 510 Drren Puigh Dted: Februry 27, 2006) This experiment used the process of opticl pumping to investigte the hyperfine splitting of two rubidium isotopes in the presence of n externl mgnetic field. The mgnetic field cused splitting in the energy of the m F mgnetic sublevels. Shining source of opticl photons through chmber contining 85 Rb nd 87 Rb vpor resulted in the toms being pumped into the stte of highest m F. The level to which the smple hd been pumped ws determined by mesuring the intensity of opticl photons tht pssed through the vpor. When photons of the right energy corresponding to rdio frequencies) intercted with the toms, they would trnsition out of the stte of highest m F. The bsorption of the pumping photons would then increse, nd the photodiode current would decrese. Utilizing this technique, the Lndé g F fctors for 85 Rb nd 87 Rb were found to be ± nd ± 0.006, respectively, in good greement with theoreticl vlues of nd Simultneously, the mgnetic field of the Erth ws mesured to be ± Guss. Investigtions into the chrcteristic time required to estblish stedy stte polriztion of the spins yielded tht this time ws independent of the light intensity. The chrcteristic times for r.f. fields of frequency MHz, MHz, nd MHz were determined to be 11.5 ±0.4 ms, 12.6 ±0.6 ms, nd 12.8 ± 0.6 ms, respectively. In ddition, it ws lso observed tht there ws n pprent resonnce when the totl field seen by the smple ws swept through zero. By vrying the mplitude of the r.f. field, the difference between sudden nd dibtic pssge through resonnce ws studied quntittively. INTRODUCTION The process of opticl pumping hs been round for more thn hlf century. Its nme refers to the preprtion of the smple. It indictes tht n equilibrium popultion of sttes will be pumped upon until only the higher energy sttes re occupied. This process is ccomplished through the bsorption/emission of opticl photons. Once this nonequilibrium popultion is estblished, it cn be used in vriety of wys. Opticl pumping hs led to mny importnt developments, such s lsers, sensitive mgnetometers, nd tomic clocks. For this experiment, it ws used to exmine the hyperfine structure of tomic sttes. It is well known tht there re mny corrections to the Bohr tom. There is fine structure to the hydrogen tom energies due to two different mechnisms: electron spin-orbit coupling nd reltivistic correction. Due to this coupling of the spin, S, nd orbitl ngulr momentum, L, we find it necessry to use eigensttes of totl ngulr momentum J, where J = L + S [1]. Once these corrections hve been tken into ccount, we see tht the fine structure breks the degenercy in l; the energies re now determined by the quntum numbers n nd j. In ddition to the fine structure, there is lso correction due to the spin of the nucleus nd its mgnetic dipole moment. Just s before, this interction cuses neither the nucler spin, I, nor the electron s totl ngulr momentum, J to be constnts of motion they re not eigensttes of the perturbtion). Only the totl ngulr momentum of the tom, F, is conserved, where F = I+ J. There is now splitting in the energy levels of different F. This is the hyperfine splitting. The ppliction of wek externl mgnetic field results in the Zeemn spliting of these F sttes, lifting the degenercy in the m F mgnetic sublevels. Now, the only orienttions of F llowed re the ones tht hve projections long the direction of the mgnetic field, m F, tht tke on n integer vlue between F nd - F. In principle, higher order multipole terms other thn the mgnetic dipole would lso contribute to the hyperfine structure. However, since the nucler dimensions re so much smller thn tomic dimensions, the contribution to the energy drops rpidly with multipole order [2]. Figure 1 displys the level digrm for 87 Rb when considering only the dipole hyperfine structure. One gol of our experiment ws to determine the reltion between the energy splittings in the m F mgnetic sublevels nd the pplied mgnetic field using rubidium. In order to mesure the trnsition between one sublevel nd nother, we first hd to prepre the toms such tht only one sublevel ws occupied. This is complicted by the fct tht the relevnt energy differences between djcent sublevels correspond to photons in the rdio frequency bnd of the electromgnetic spectrum. The seprtion is so smll tht therml excittions nd collisions will cuse ech level to be nerly eqully populted. This is where opticl pumping becomes importnt. Opticl pumping is bsed on energy nd momentum conservtion, s well s the selection rules for mgnetic dipole trnsitions. Consider the cse of 87 Rb I = 3/2). Imgine tht the smple toms re ll in the 2 S 1/2 stte with F = 0 or 1. Initilly, ll sublevels re nerly eqully occupied in the presence of wek mgnetic field. The

2 However, such stte does exist when trnsitioning from the ground stte to 2 P 3/2 is llowed through bsorption of D 2 rdition). Therefore, D 2 light must be filtered to prevent this escpe route from the m F = 2 stte. The previous discussion ws bsed on 87 Rb, but nturl rubidium contins both 85 Rb nd 87 Rb, with nucler spins 5/2 nd 3/2, in the rtio 72 to 28%, respectively. Thus, with one chmber of rubidium vpor, it ws possible to investigte the field dependence of the Zeemn splittings of both isotopes simultneously. 2 THEORETICAL BACKGROUND Zeemn Splitting of Hyperfine Structure FIG. 1: Level Digrm for 87 Rb I = 3/2) s different corrections re considered. For simplicity, the hyperfine structure nd Zeemn splitting of the 2 P 3/2 stte re not shown. Included in the digrm re the possible trnsitions of n tom initilly in the F = 2, m F = 0 sublevel of the ground stte fter bsorption of photon with one unit of ngulr momentum nd spontneously emitting photon of rndom polriztion. objective is to get ll toms into the F = 2, m F = 2 stte of the groundstte. One cn ccomplish this by first shining circulrly polrized light through the vpor corresponding to trnsition between the 2 S 1/2 stte nd the 2 P 1/2 stte D 1 rdition). The selection rules involving the bsorption or emission of mgnetic dipole rdition re F = 0 or ±1 nd m F = 0, or ±1 [3]. If the light is right circulrly polrized nd trveling prllel to the mgnetic field, it is denoted σ +. From momentum conservtion, bsorption of σ + light by n tom rises m F by 1 unit. Once the tom is in the excited stte, it will spontneously emit rdition of rndom polriztion resulting in n equl probbility of m F incresing by 1, decresing by 1, or stying the sme. The net result of the bsorption nd then emission of one D 1 photon is m F > 0 with 2/3 probbility or m F = 0 with 1/3 probbility. Figure 1 illustrtes one such cycle. This process is repeted with ech incident photon. If relxtion processes re slow compred to the rte of pumping, the toms will ll be pumped into the F = 2, m F = 2 stte of the groundstte, s desired. The toms in this stte cn no longer bsorb the D 1 rdition, becuse there is no m F = 3 sublevel in the 2 P 1/2 stte. When we determine the reltion between Zeemn energy splittings nd wek mgnetic fields, specificlly we re mesuring the Lndé g-fctor, lso known s the gyromgnetic rtio. The g-fctor reltes the mgnetic dipole moment to the ngulr momentum of quntum stte. With ngulr momentum in units of h, we hve the following equtions relting mgnetic dipole moment to ngulr momentum [4]: µ S = g S µ B S, 1) µ L = g L µ B L, 2) µ J = g J µ B J, 3) µ I = g I µ N I, 4) µ F = g F µ B F, 5) where µ B nd µ N re the Bohr nd nucler mgneton, respectively. Given the reltions 1) - 3) nd J = L+ S, nd using the properties of the dot product, one obtins g J J = g L L cos θ JL + g S S cos θ JS 6) with θ AB representing the ngle between the vectors A nd B. Similrly, one cn use the eqution for the totl ngulr momentum of the electron J to get L 2 = S 2 + J 2 2SJ cos θ JS, 7) S 2 = L 2 + J 2 2LJ cos θ JL. 8) Quntum mechniclly, the expecttion vlue for the ngulr momentum of A 2 is AA+1). In the end, solving 7) nd 8) for the cosine of the ngles nd substituting the result into 6) yields JJ + 1) + SS + 1) LL + 1) g J = g S + 2JJ + 1) JJ + 1) + LL + 1) SS + 1) g L 2JJ + 1) 9) For this experiment, we re studying the Zeemn splitting of the groundstte. Here L = 0 nd J = S = 1/2.

3 3 Therefore, g J = g S. Similrly, one cn use the exct sme procedure for the totl ngulr momentum of the tom to find tht it hs g-fctor g F = g J FF + 1) + JJ + 1) II + 1) 2FF + 1) g I FF + 1) + II + 1) JJ + 1) 2FF + 1) 10) We re interested in the cse where F = 2 nd J = 1/2. Substituting these vlues nd g J = g S into 10) for 85 Rb I = 5/2), we find nd for 87 Rb I = 3/2) we find g F = 1 6 g S + 7g I ), 11) g F = 1 4 g S 3g I ). 12) The negtive sign on 11) indictes tht decresing vlues of m F yield higher energy levels. Using the method of opticl pumping, we will be determining the bsolute vlue of g F. Thus, we will only be quoting positive vlues of g F. Experimentlly, the gyromgnetic rtio of the electron hs been mesured, yielding vlue g S = [5]. In ddition, the nucler to electronic g-fctor rtio, g I /g J, for 85 Rb hs been determined to be ) 10 4, independent of mgnetic field [6]. Assuming tht 87 Rb hs similrly smll g-fctor rtio, then 11) nd 12) provide theoreticl predictions to 3 significnt figures) of g F = nd g F = for 85 Rb nd 87 Rb, respectively. The number of significnt figures ws chosen to correspond with the precision to which this experiment ws performed nd to emphsize the fct tht clcultions re vlid only to first order. Finlly, the Zeemn energy splitting of m F sublevels in wek mgnetic field is to first order) given by [1] E = g F e h 2m e B m F = g F µ B B m F. 13) Estblishing Stedy Stte Polriztion After the rubidium toms hve been subjected to rdio frequency r.f.) field of the right energy to induce trnsition, the intensity of the D 1 rdition pssing through the rubidium vpor will drop shrply. Once this r.f. field is removed, the toms will be pumped on gin until stedy stte polriztion is estblished. There is chrcteristic time ssocited with this phse. Following Benumof [4], we cn determine the chnge in popultion of the m F = 3 groundstte of 87 Rb reltive to the other sublevels. This cn be written s dn dt = nw d + NW u 14) where n is the popultion of the m F = 3 sublevel nd N is the popultion of every other sublevel. The rte of trnsitioning down out of the pumped stte is given by W d, nd W u is the rte of trnsitioning into m F = 3. The chrcteristic time to estblish stedy stte polriztion is found from 14). Do we expect this time to depend on the intensity of the light? Clerly, the relxtion processes cusing the trnsition out of the pumped stte, W d, should not depend on the light intensity. W d is result of the toms colliding with the glss wlls. Next, we cn brek up W u into two different steps. This rte depends both on the rte of trnsition into the 2 P 1/2 excited stte, Γ, nd the rte of trnsition bck down into the m F = 3 groundstte, Γ. The longer rte, Γ or Γ, will determine W u. The intensity of the light is mesure of rte tht opticl photons re incident on the smple. Since these photons re responsible for excittion of the groundstte, Γ will depend directly on the intensity. However, the trnsition bck to the groundstte is cused by the spontneous emission of the opticl photon. From the Fermi s Golden Rule [2], the spontneous emission should depend on the frequency of the emitted photon nd not the intensity of the light. We expect timescles of spontneous emission to be much longer thn those of photon bsorption from the light. Therefore, the chrcteristic time should not depend on intensity. Dynmic Response of Spins to Time Vrying Mgnetic Fields We wish to first discuss pssge through zero field pplied field cnceling the Erth s Field) in the bsence of ny r.f. field. In sttic coordinte system, the nucler spins obey the clssicl eqution of motion [7] di ) dt = γ I B. 15) Due to the vrition of B with time, this is difficult problem to solve. For instructive purposes, we exmine limiting cses. The mgnetic field vector cn be seprted into components perpendiculr nd prllel to the xis of the primry Helmholtz coils. The two cses [8] re dibtic pssge throug zero field given by nd sudden pssge given by 1 db γb, 16) B dt 1 db γb. 17) B dt If the mgnetic field is vrying slowly, then 16) is stisfied, nd I will sty nerly prllel with B. As B is swept through zero, so is I, nd the z-component of I

4 4 will reverse sign. This will be s if the pumping light ws reversed [4]. Light tht ws right-circulrly polrized nd resulted in rising m F by one unit will now remove one unit of ngulr momentum. The pumping process will begin gin, pumping toms in the opposite wy s before, nd bsorption of D 1 light will increse drmticlly. When this hppens, there will be lrge opticl signl. On the other hnd, it might be the cse tht the sweep rte of the mgnetic field is too fst for the mgnetiztion to follow. In this sitution, 17) is stisfied. Now, mgnetiztion remins lwys pointed in the sme direction. So, when the field chnges sign, it did so quickly enough to not ffect the spins. Therefore, right-circulrly polrized light will still increse m F by one unit. The net result is tht pumping remins unchnged, so the opticl signl is reltively stedy. When considering resonnce in the presence of r.f. mgnetic field, it is useful to trnform into rotting coordinte system. Here, we wish to trnsform into coordinte frme tht rottes t the frequency of the r.f. field. The rte of chnge for ny vector in fixed reference frme is given by [9] di ) dt fixed = ω I + di ) dt rotting. 18) Using this trnsltion long with 15), it cn be shown [8] tht B becomes replced by B r.f.. If we define the quntity X µ B γ 1 B 2 r.f. db dt, 19) then the dibtic nd sudden conditions cn be written s X 1 nd X 1, respectively. Although the rgument bove ws bsed on clssicl mechnics, the result holds quntum mechniclly [7]. EXPERIMENTAL SETUP The lyout for this experiment is shown in Figure 2. Two sets of Helmholtz coils were used to provide the wek externl mgnetic field. The primry coils shown in the figure) were ligned so tht they would produce mgnetic field either prllel or nti-prllel to the Erth s mgnetic field. Secondry coils were plced t right ngles to these fields in n effort to minimize the trnsverse field nd its inhomogeneities. Current through the primry coils ws controlled with trpezoidl sweep by using function genertor nd control circuit. Production of circulrly polrized wves ws chieved by using liner polrizer nd qurter-wve plte. The two were ligned to obtin mximl circulr polriztion FIG. 2: Digrm of opticl pumping setup. Only the primry Helmholtz coils re shown in this digrm. The secondry coils used to trim the trnsverse fields to zero re not shown. of opticl light from lmp. The D 2 rdition ws removed from the light for the previously mentioned resons. A converging lens focused the light going through resonnce bulb full of rubidium vpor. Due to the low melting point of rubidium 38.5 C), the resonnce bulb ws heted by blowing hot ir onto it to chieve optimum vpor pressures. A signl genertor supplied the rdio frequency field necessry to induce trnsitions out of the pumped stte. The mgnitude of the r.f. field ws determined by mesuring the emf induced in smll pickup coil. For ech frequency, the mgnetic field from the primry Helmholtz coils ws swept through rnge of vlues. When the mgnetic field pssed through resonnce for given isotope stisfying 13), bsorption of the D 1 rdition incresed shrply. Since the sweep of the primry coils ws trpezoidl, there were two peks for ech isotope during one period of the sweep. Figure 3 shows both the sweep nd the resonnt peks for ech isotope t fixed frequency. For ll mesurements, the oscilloscope ws run in Averge 256 mode to reduce the mount of noise in the signl. In order to determine the effect of light intensity on the chrcteristic time to estblish stedy stte polriztion, it ws necessry to vry the mount of incident light. This ws chieved by plcing second liner polrizer between the lmp nd the first liner polrizer. By vrying the the ngle between the two polrizers, it ws possible to control the intensity of light shining on the smple.

5 d l e Fi e t n g x 1 e 4 R b 8 5 R b T ) i c M E n e r g y J ) x 1 e 2 7 FIG. 3: Oscilloscope trce of resonnt peks t frequency of MHz. Ch. 1 - Trpezoidl sweep for the primry Helmholtz coils. Ch. 2 - Signl from the photodiode. Peks 1) nd 1 ) correspond to 85 Rb, nd peks 2) nd 2 ) correspond to 87 Rb. Note the ppernce of n extr pir of peks 3) nd 3 ) corresponding to zero pplied field. RESULTS AND DISCUSSION The mgnetic field due to the primry Helmholtz coils ws determined from the geometry of the coils nd the mount of current flowing through them. These coils were designed to provide s uniform of field s possible. To tht end, the seprtion between the coils ws mesured to be the sme up to uncertinty) s the rdius of ech coil. Therefore, the mgnetic field in Tesl) in between nd on the xis the coils is given by [10] B = ) 3/2 4 µ 0 NI, 20) 5 r where N is the number of turns in the coil nd r is the rdius of the coils. The energy level splitting ws determined by mesuring the frequency of the r.f. field nd using E = hν. The reltionship between the Zeemn energy splitting nd pplied mgnetic field for both isotopes of rubidium is shown in Figure 4. The two lines were found using lest-squres weighted fit [11]. The mgnetic field B ws grphed vs. the energy E, insted of the reverse, due to the lrger error in mesurements of B. The slope of ech line determined the vlue of g F µ B ) 1, from which the g-fctor could be obtined. In ddition, the y-intercept of ech line is mesure of the Erth s mgnetic field. According to 13), one would expect the zero of energy FIG. 4: Finding the g F vlues for 85 Rb nd 87 Rb. The slope of these lines represent the g F µ B) 1. The y-intercept of ech line not shown) ws used to infer the mgnitude of the Erth s mgnetic field. Resonnce mesurements for twenty different frequencies were used. TABLE I: G-fctor mesurements for two rubidium isotopes. Also included is the mgnitude of the Erth s mgnetic field determined using ech isotope. Rb Isotope g F B Erth G) ± ± ± ± Weighted verge yields 0.496±0.029 G. splitting to occur in the presence of zero pplied mgnetic field. The fct tht the y-intercept is not zero indictes tht there is n dditionl mgnetic field other thn the one produced by the Helmholtz coils. This extr field is ttributed solely to field of the Erth. From the y- intercepts of both lines, we obtin weighted verge for the Erth s mgnetic field of ± G. Tble I summrizes the results. The chrcteristic time, τ, to estblish stedy stte polriztion ws mesured for different frequencies nd intensities. Assuming n exponentil decy of photodiode signl, the logrithm of the voltge ws grphed versus time so tht the points would fit stright line with slope τ 1. Agin, ech line ws creted using lest-squres weighted fit. Figure 5 displys the results t fixed frequency of MHz. Notice tht ech line hs pproximtely the sme slope. Tble II provides the chrcteristic time for different polriztions. These chrcteristic times re ll in good greement, indicting tht τ does not depend upon the light intensity, s hypothesized. The mesured vlues of τ nd their uncertinties for ech intensity nd for three different frequencies re detiled in Tble III. Note tht ech of the vritions of τ

6 d e g r e e s 2 5 d e g r e e s 4 3 d e g r e e s 6 0 d e g r e e s TABLE IV: Mesurements of the pek voltge for different r.f. field strengths. Included is the defined vrible X which provides mesure of sudden pssge X 1) nd dibtic pssge X 1). V ) m ge l t V o k P e g L o T i m e m s e c ) FIG. 5: Dependence of τ on light intensity t MHz. The different lines indicte different intensities. It ws ssumed tht the decy of the voltge reding from the photodiode ws exponentil. Therefore, the slope of ech line is τ 1. The fct tht these lines re roughly prllel suggests tht τ is insensitive to chnges in intensity. TABLE II: Chrcteristic time to estblish stedy stte polriztion t MHz. θ deg) I b I 0 τms) ± ± ± ± 2.8 Approximte. Ech vlue hs n estimted uncertinty of 3 b Intensity relted to ngle between polrizers by I = I 0 cos 2 θ). over the rnge of intensities gree with zero, suggesting tht there is no dependence of τ on light intensity. In ddition to the peks corresponding to the two rubidium isotopes, it ws observed tht there ws nother pprent resonnce when the field ws swept through zero pplied field cnceling the erth s field) in the bsence of n r.f. field. See the Theory section for explntion. Figure 6 shows the dependence of the pek voltge on TABLE III: Chrcteristic time nd its vrition per degree seprtion between liner polrizers for three frequencies. frequency MHz) τ ms) τ θ ± ± ± ± ± ± Weighted verge from ech polriztion. Pek mv) B r.f T) X b Uncertinty on ech pek mesurement estimted to be 4 mv. b db See eqution 19) for definition. = T/s. dt the mgnitude of the r.f. field. For lrge vlues of the r.f. field, the dibtic condition X 1) is stisfied nd the pek voltges rech constnt vlue s the the nuclei preserve their orienttion reltive to the mgnetic field. As the r.f. field gets smller, the sudden condition X 1) is stisfied nd the pek voltge goes to zero s the mgnetiztion hs no opportunity to follow the fields. Tble IV lists the mesured pek voltge for 16 different vlues of the r.f. field. The vlue of X 19) is listed for ech pek entry. Error Anlysis There were mny sources of error in this experiment. The bove figures were displyed such tht the quntity with the lrgest uncertinty ws used s the y-xis. In the mesurement of g F for ech isotope, the biggest source of error ws in the mgnetic field corresponding to resonnce. Due to the nonuniformities of the mgnetic field, it ws difficult to scertin where exctly resonnce occurred. It ws necessry to verge the loction of the peks on both sides of the trpezoidl sweep to chieve the most ccurte vlue of the mgnetic field. Once vlue ws decided upon, it still hd two sources of uncertinty: voltge mesurements on the oscilloscope nd the rdius of the coils. Voltge redings from the oscilloscope hd to be trnslted into current in the primry coils. Then, the current ws used to determine the

7 7 V ) m ge l t V o k P e B r. f. T ) x 1 e 7 Rubidium is n excellent specimen for studying both the hyperfine structure of tomic sttes nd the process of opticl pumping. The Lndé g F fctors for 85 Rb nd 87 Rb were found to be ± nd ± 0.006, respectively, in good greement with theoreticl predictions of nd At the sme time, we were ble to mesure the Erth s mgnetic field in the lbortory to be ± G. Investigtions into chrcteristic times of estblishing stedy stte polriztions yielded tht these times were independent of the light intensity. Finlly, we exmined the behvior of the spins in the limiting cses of dibtic nd sudden pssge through both resonnce nd zero field. There re couple of wys in which this experiment could be expnded upon. First, dt nlysis would be fster nd more ccurte if the oscilloscope could output dt file of the curves insted of requiring reding vlues from the screen. It would lso be interesting to use stronger mgnetic field. This would provide study of the hyperfine spcing between different F levels. FIG. 6: Sudden vs. dibtic pssge through resonnce. Above certin r.f. field strength, the pek voltges were roughly equl dibtic pssge). However, decresing the r.f. field eventully led to ttenution of the pek voltges sudden pssge). mgnetic field. Using stndrd procedures of error propgtion [11], it ws possible to determine resonble mesure of uncertinty in ech mesurement. However, it is still likely tht the error ws under-estimted. For instnce, it ws ssumed tht the clibrtion between the voltge reding on the oscilloscope nd the current in the primry coils ws perfect. Clerly, ny tiny error in this clibrtion will ply role in the error of the mgnetic field. For mesurements of τ, the lrgest source of uncertinty cme from the voltge mesurements. Even with the oscilloscope in Averge mode, there ws fir mount of fluctution round n idel exponentilly decresing curve. This error ws most pronounced when mesuring smll peks for low intensities. CONCLUSION Acknowledgments I would like to thnk Professor Hrtill nd Johnnes Heinonen for their explntions of the bsic experimentl setup. In ddition, I would like to thnk Bryn Dniels for his helpful suggestions concerning this report. Specil thnks to Tien Le for her continul support. [1] D. J. Griffiths, Introduction to Quntum Mechnics Prentice Hll, 1995). [2] K. Gottfried nd T. M. Yn, Quntum Mechnics: Fundmentls Springer, 2004), 2nd ed. [3] R. L. de Zfr, Am. Jour. Phys. 28, ). [4] R. Benumof, Am. Jour. Phys. 33, ). [5] R. S. V. D. Jr., P. B. Schwinberg, nd H. G. Dehmelt, Phys. Rev. Lett. 59, ). [6] N. P. Economou, S. J. Lipson, nd D. J. Lrson, Phys. Rev. Lett. 38, ). [7] I. I. Rbi, N. F. Rmsey, nd J. Schwinger, Rev. Mod. Phys. 26, ). [8] R. H. Silsbee 1972). [9] S. T. Thornton nd J. B. Mrion, Clssicl Dynmics of Prticles nd Systems Thomson Brooks/Cole, 2004), 5th ed. [10] D. J. Griffiths, Introduction to Electrodynmics Prentice Hll, 1999), 3rd ed. [11] J. R. Tylor, An Introduction to Error Anlysis University Science Books, 1997), 2nd ed.