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Chapter 2. Basic Atomic Physics


Academic and Research Staff

Professor Daniel Kleppner, Professor David E. Pritchard, Professor Wolfgang Ketterle, Dr. Alan L. Lenef, Dr.Fred L. Palmer, Dr. Nicolaas J. van Druten

Visiting Scientists and Research Affiliates

Dr. John R. Brandenberger,1 Dr. Theodore W. Ducas,2 Dr. Marek L. Czachor,3 Dr. Bernd S. Rohwedder,4 Dr.H. Jbrg Schmiedmayer

Graduate Students

Michael R. Andrews, Michael P. Bradley, Michael S. Chapman, Michael W. Courtney, Kendall B. Davis, JoelC. DeVries, Frank DiFilippo, Dallin S. Durfee, Troy D. Hammond, Jeffrey R. Holley, Ljubomir M. Ilic, HongJiao, David A. Kokorowski, Dan M. Kurn, Robert I. Lutwak, Marc-O. Mewes, Daniel Ripin, Richard A.Rubenstein, Edward T. Smith, Neal W. Spellmeyer

Undergraduate Students

Ilya Entin, David Garrison, Philip M. Hinz, Everest W. Huang, Amrit R. Pant, Till P. Rosenband, Szymon M.Rusinkiewicz, Charles K. Sestok, Stanley H. Thompson, Peter S. Yesley

Technical and Support Staff

Carol A. Costa

2.1 Studies in Quantum Chaos:Rydberg Atoms in Strong Fields

Sponsors

Joint Services Electronics ProgramGrant DAAH04-95-1-0038

National Science FoundationGrant PHY 92-21489

U.S. Navy - Office of Naval ResearchGrant N00014-90-J-1322

Project Staff

Michael W. Courtney, Hong Jiao, Neal W.Spellmeyer, Dr. John R. Brandenberger, ProfessorDaniel Kleppner

The goal of this research is to understand the struc-ture and dynamics of atomic systems in externalfields that are comparable to the interior fields and

in which the classical motion may be irregular.Rydberg atoms are central to this research becauselaboratory fields can be comparable to their atomicfields. In addition, the classical counterparts ofthese systems often undergo a transition from orderto chaos as the field or energy is varied. Conse-quently, Rydberg atoms in applied fields provide anatural testing ground for studying the (1) con-nections between quantum mechanics and classicaldynamics and (2) quantum behavior of chaoticsystems. Our understanding of this system hasbeen advanced by high resolution spectroscopy,development of efficient quantum calculations foratoms in electric and magnetic fields, and theore-tical advances in classical dynamics.

Periodic orbit theory provides a unifying principle forrelating a quantum spectrum to the periodic orbitsof the corresponding classical system in the semi-classical limit. According to periodic orbit theory,

1 Professor, Lawrence University, Appleton, Wisconsin.

2 Professor, Physics Department, Wellesley College, Wellesley, Massachusetts.

3 Fulbright fellow, Warsaw, Poland.

4 Universidad Catolica de Chile.

5 Professor, Physics Department, Innsbruck University, Innsbruck, Austria.

217


each periodic orbit produces a sinusoidal modu-lation in the density of states. A spherical wave sentout from the origin will recur (overlap with the ori-ginal outgoing wave) at times corresponding to theperiods of the periodic orbits of the classicalsystem. The magnitude of this overlap gives thestrength of the spectrum modulation.

In our laboratory, we have recently studied lithiumin parallel electric and magnetic fields.6 In this work,we were concerned with only the low action orbitsand ignored differences between hydrogen andlithium caused by the lithium core.7

The Hamiltonian of hydrogen in parallel electric andmagnetic fields is

1 2 1 1 2H 2 1 p2 - Fz.2 r 8

This can be rescaledr - B2/3 r, p = B-11/3p, tscaled Hamiltonian

using the substitutionsBt, and f - FB - 4/3. The

1 -2 1 1 ~2H=-p + p -fz2 r 8

depends only on the scaled energy H = E = EB - 2/3

and scaled field f = FB- 4/3. Consequently, the clas-sical dynamics also depends only on these twoparameters, not on E, F, and B separately. In thelimit of large f, this is just the hydrogen Starksystem for which the classical motion is regular. Asf approaches zero, the system evolves intodiamagnetic hydrogen, which is strongly chaotic forE > - 0.1. An examination of the system at interme-diate values of f should reveal the system becomingchaotic. We experimentally measured the recur-rence spectrum at a scaled energy of E O0. At thisenergy, the system is in the continuum and themotion is unbound.

The scaled action of a particular closed orbit,

Sk - B1/3 Sk, depends only on E and f. If the spectrum

is recorded while the field and the energy are

varied simultaneously to keep E and f constant, theclassical dynamics remains the same. Thisapproach has been used previously,8 but this is itsfirst experimental application to the parallel fieldssystem. An orbit's scaled action and recurrencestrength can be obtained directly from the Fouriertransform of the spectrum. This Fourier transform iscalled the recurrence spectrum because each peakis located at the scaled action of a periodic orbit,and the height of each peak is proportional to thatorbit's recurrence strength.

Our experiment employs a lithium atomic beam thattravels along the axis of a split-coil superconductingmagnet. A pair of field plates provides an electricfield parallel to the atomic beam. Laser beamsintersect the atomic beam at a right angle. A 735nm laser excites the 2S -, 3S two-photon transi-tion, and a second laser polarized parallel to theapplied fields excites m=0 continuum states thatrapidly ionize. The ions are detected by a micro-channel plate. To generate a scaled-energy spec-trum, the magnetic field is ramped between 0.5 Tand 3.0 T while the electric field is varied to main-tain f constant. In general, the laser frequencymust also be varied to maintain constant E, but bychoosing E - 0. we could operate at fixed laser fre-quency. The fractional uncertainty in scaled energyand scaled field are 0.15 percent and 0.2 percent,respectively.

Understanding the experimental spectrum requiresa good understanding of the classical dynamics.The traditional tool for studying chaos is thePoincar6 surface of section. However, this is notwell suited to the continuum where phase space isunbounded and the trajectories rapidly ionize. Asan alternative, we have found that the concept ofclassical ionization time provides a useful signatureof the transition from regular to chaotic behavior.Following Main and Wunner,9 we define the clas-sical ionization time Tion to be the time that an elec-tron excited near the nucleus at an initial angle 80relative to the z axis requires to reach the electricfield saddle point. Figure 1 shows how Tion varieswith 80 for f = 1.0, 0.4, and 0.05.

6 H. Jiao, Experimental and Theoretical Aspects of Quantum Chaos in Rydberg Atoms in Strong Fields, Ph.D. diss., Dept. of Physics,MIT, 1996.

7 U. Eichmann, K. Richter, D. Wintgen, and W. Sander, Phys. Rev. Lett. 61: 2438 (1988); M. Courtney, H. Jiao, N. Spellmeyer, and D.Kleppner, Phys. Rev. Lett. 73: 1340 (1994).

8 U. Eichmann, K. Richter, D. Wintgen, and W. Sander, Phys. Rev. Lett. 61: 2438 (1988); T. van der Veldt, W. Vassen, and W.Hogervorst, Europhys. Lett. 21: 9 (1993); A. Holle, J. Main, G. Wiebusch, H. Rottke, and K.H. Welge, Phys. Rev. Lett. 61: 161(1988).

9 J. Main and G. Wunner, Phys. Rev. Lett. 69: 586 (1992).

218 RLE Progress Report Number 138


In the pure Stark problem, the only periodic orbit is40 the "uphill" orbit, along the axis with 80 = 180

(a) degrees. At other angles, the electron leavesessentially immediately. This behavior is seen in

Ton figure la, except that Tion remains large down to170 degrees due to the stabilizing effect of the

0 _ - magnetic field. At f = 0.4 (figure lb), the stable0 30 60 90 120 150 180 region extends down to about 140 degrees, but

40 below this angle Tion varies erratically for about 15(b) degrees. At f = 0.05 (figure ic), the erratic

behavior extends to below 60 degrees, with theTon system displaying wild and random fluctuations.

Successive magnifications (figures id and le) show0 , , , ., that the randomly varying pattern extends to ever0 30 60 90 120 150 180 finer scales: fractal behavior is evident. Similar

40 behavior has been observed in nuclear scattering(c) from model potentials, where the phenomenon has

been dubbed chaotic scattering. 10 Following MainTon- and Wunner, we adopt the term chaotic ionization

to describe the ionization process.

0 30 60 9 120 150 180 The chaotic ionization process is intimately linked to40 the existence of closed orbits in the system. Figure

(d) 2 shows the locations of closed orbits with scaled

action up to S = 10, and Tion as functions of 80.Ton -The fractal structures appear to be localized near

the closed orbits. The closed orbits seem to exert0o - a trapping effect on nearby trajectories: the sensi-94 96 100 102 104 106 tivity of these trajectories to 80 is the source of the

40 fractal behavior in Tion 11 Because chaotic ionization(e) develops as f approaches zero, we expect to see

the development of new closed orbits in the recur-Ton rence spectrum of the system. This is what we

observe experimentally.

97.5 97.6 97.7 97.8 97.9 98

o0 (degrees)

Figure 1. The classical ionization time Ton (in units ofscaled time) as a function of the initial angle 00 at E = 0for various values of f. (a) f = 1.0 , (b) f = 0.4, (c) f =0.05. (d) and (e) successive blowups of regions indicated.

10 B.P. Koch and B. Bruhn, J. Phys. A 25: 3945 (1992).

11 B. Eckhardt and C. Jung, J. Phys. A 19: L829 (1986); C. Jung and H.J. Scholz, J. Phys. A 20: 3607 (1987).

219


Figure 2. Above: the classical ionization time (in units of scaled time) at E = 0 and f =0.05. Below: location of closedorbits with S !: 10.

0

0.2

4 3 0.4

0.6 f2

3 0.8- 4

S 5 6

Figure 3. Experimental recurrence spectra at E = 0 for scaled fields ranging from f = 0.05 to f = 1. The curved linesare the calculated scaled actions of the parallel orbit and its repetitions. Locations of bifurcations are marked with smallovals.


80 100,, (degrees)

180


Figure 3 shows experimental recurrence spectra forvalues of f between 1.0 and 0.05. At large f, therecurrences are those of the parallel "uphill" orbitand its repetitions. As f is reduced, new recur-rences become visible. These are created bybifurcations of the parallel orbit and its repetitions.Near the bifurcation points, the orbits often gen-erate large recurrences. This enhancement,caused by the focusing effects of nearby orbits, hasbeen described quantitatively.12

In summary, we have shown that one can charac-terize classical chaos in the continuum through theconcept of chaotic ionization, and we have demon-strated that closed-orbit theory provides a naturalway to interpret the continuum spectra of a chaoticsystem. These results help to establish a usefulconnection between the classical and quantum des-criptions of an unbound chaotic system.

In the coming year, we plan to approach thisproblem in a fundamentally new way: by applying aresonant RF field, we anticipate being able tomanipulate individual classical closed orbits in thesystem. Work is now under way.

2.1.1 Publications

Courtney, M., N. Spellmeyer, H. Jiao, and D.Kleppner. "Classical, Semiclassical, andQuantum Dynamics in the Lithium StarkSystem." Phys. Rev. A 51: 3604 (1995).

Courtney, M., H. Jiao, N. Spellmeyer, D. Kleppner,and J.B. Delos. "Closed Orbit Bifurcations inContinuum Stark Spectra," Phys. Rev. Lett. 74:1538 (1995).

Courtney, M., H. Jiao, N. Spellmeyer, and D.Kleppner. "Long-period Orbits in the StarkSpectrum of Lithium. Phys. Rev. Lett. 73: 1340(1994).

Courtney, M., H. Jiao, N. Spellmeyer, and D.Kleppner. "Quantum Chaos and RydbergAtoms in Strong Fields." Proceedings of theDrexel Conference. Forthcoming.

Shaw, J.A., J.B. Delos, M. Courtney, and D.Kleppner, "Recurrences Associated with Clas-

sical Orbits in the Nodes of Quantum Wavefunc-tions." Phys. Rev. A 52: 3695 (1995).

Theses

Courtney, M. Rydberg Atoms in Strong Fields: ATesting Ground for Quantum Chaos. Ph.D. diss.,Dept. of Physics, MIT, 1995.

Jiao, H. Experimental and Theoretical Aspects ofQuantum Chaos in Rydberg Atoms in StrongFields. Ph.D. diss., Dept. of Physics, MIT, 1996.

2.2 Determination of the RydbergFrequency

Sponsors



Project Staff

Joel C. DeVries, Dr. Theodore W. Ducas, Jeffrey R.Holley, Robert I. Lutwak, Professor Daniel Kleppner

The Rydberg constant, R,, relates the wavelengthsof the spectrum of atomic hydrogen to practical lab-oratory units. As such, R. is the natural unit formeasurements of atomic energies, and appears asan auxiliary constant in many spectroscopic mea-surements. Recent advances in optical wavelengthmetrology have made possible measurements ofR. with accuracy approaching 2 parts in 1011.13 TheRydberg frequency, cRo, similarly relates the atomicunit of frequency to laboratory units. Although thespeed of light c is an exactly defined quantity, therelation between the Rydberg constant and theRydberg frequency is not merely formal. The pre-cision with which a frequency can be measured islimited in principle to the precision of atomic clocks,which currently exceeds 1 part in 1014 and isexpected to grow even larger. In contrast, wave-length metrology appears to have reached its limitof precision, somewhat less than 1 part in 1011.

To make full use of the precision of lasers andmodern laser spectroscopy and for applications incommunications, control, and metrology, we need

12 M. Courtney, H. Jiao, N. Spellmeyer, D. Kleppner, J. Gao, and J.B. Delos, Phys. Rev. Lett. 74: 1538 (1995).

13 T. Andreae et al., "Absolute Frequency Measurement of the Hydrogen 1S-2S Transition and a New Value of the Rydberg Constant,"Phys. Rev. Lett. 69(13): 1923-1926 (1992); F. Nez et al., "Precise Frequency Measurement of the 2S-8S/8D Transitions in AtomicHydrogen: New Determination of the Rydberg Constant," Phys. Rev. Lett. 69(16): 2326-2329 (1992).

221


to develop general techniques for measuring thefrequency of light. As part of this effort, we proposeto help establish an atomic optical frequency scaleby measuring the Rydberg frequency directly.

The goals of our experiment are three-fold. First isthe re-evaluation of Ro itself, providing an indepen-dent check, in a different regime and using a totallydifferent technique from previous measurements.Second is the determination of the ground stateLamb shift. Because our method employs highangular momentum states for which the Lamb shiftis extremely small and effectively exactly known,our results may be compared with optical measure-ments of transitions between low-lying states toyield an improved measurement of the Lamb shift.Third is to provide a frequency calibration of thespectrum of hydrogen, enabling the creation of acomprehensive frequency standard extending fromthe radio frequency regime to the ultraviolet.

Our approach involves measuring the frequency oftransitions in atomic hydrogen in a regime wherethe frequency can be synthesized directly from anatomic clock. The experiment explores transitionsbetween highly excited "Rydberg" states of atomichydrogen in the neighborhood of n - 29. The tran-sition n=29 - 30 occurs at approximately 256 GHz.

The experiment employs an atomic beam configura-tion to reduce Doppler and collisional perturbations.Atomic hydrogen is excited to the low angularmomentum n=29,m=0 state by a two-step process.The excited atoms are then transferred to thelonger lived n=29, m=28 circular state by absorp-tion of circularly polarized radio frequencyradiation. 14 The atoms enter a region of uniformelectric field in which the frequency of the transition(n=29,m=28)--(n=30,m=29) is measured by themethod of separated oscillatory fields. The finalstate distribution is analyzed by a state-sensitiveelectric field ionization detector. The resonancesignal appears as a transfer of atoms from then=29 state to the n=30 state as the millimeter-wavefrequency is tuned across the transition.

Figure 4 illustrates the main features of the appa-ratus. Atomic hydrogen or deuterium is dissociatedfrom H2 or D2 in a radio frequency discharge. Thebeam is cooled by collisions with a cryogenicthermalizing channel in order to slow the atoms andthereby increase the interaction time. After thebeam is collimated, the atoms pass through twolayers of magnetic shielding and an 80 K cryogenicshield before entering the 4 K interaction region.The interaction region is logically divided into three

sections:separatedThese are

the circular state production region, thefields region, and the detection region.described briefly below.

In the circular state production region, the hydrogenatoms are excited from the is ground state to the2p3/2 state, and then to one of the n=29,m=0 states.This takes place in an electric field to provide selec-tive population of one particular n=29, m=0 Starksub-level. The electric field is then rapidly reducedto an intermediate value as the atoms pass throughthe center of a circle of four electrodes. These formtwo pairs of antennas which are fed by a 2 GHz RFsource with a 90 degree phase offset. This createsa circularly polarized field that drives the atoms intothe n=29, m=28 circular state by the absorption of28 photons. A pulsed electric field ionization (EFI)detector in the circular state production region mon-itors the efficiency of the optical excitation andangular momentum transfer processes.

After the atoms are prepared in the n=29 circularstate, the beam enters the millimeter-wave sepa-rated fields region. Because Rydberg atomsinteract strongly with external fields, accurate meas-urement of the energy level structure requirescareful control of the interaction environment.Thermal radiation is reduced by cooling the inter-action region to - 4 K with a liquid helium flowsystem. The ambient magnetic field is reduced bythe double-wall high-permeability shields. To definethe quantization axis, a small electric field is appliedwith high uniformity by field plates above and belowthe atomic beam. The millimeter-waves intersectthe atomic beam at two locations separated by 50cm. The millimeter-wave optical system wasdescribed in the 1990 RLE Progress Report.

The state distribution of the atoms emerging fromthe interaction region is analyzed by a state-selective electric field ionization detector. Withinthe detector, the atoms enter a region of increasingelectric field produced by a pair of symmetricramped plates held at constant potential. Atoms indifferent states are selectively ionized at differentfields and the charged nuclei are detected at dif-ferent positions. The detection electronics recordthe state and arrival time of each atom to reach thedetector. Because the laser system is pulsed, thetime resolution of the ionization signal allows con-tributions to the resonance pattern from each vel-ocity class to be analyzed individually, providing avaluable check on possible systematic errors.

A preliminary resonance measurement was pre-sented in the 1992 RLE Progress Report. The pre-

14 R. Hulet and D. Kleppner, "Rydberg Atoms in 'Circular' States," Phys. Rev. Lett. 51(16): 1430-1433 (1983).



Liquid N2 Millimeter-wave interaction points(50 cm separation)

Hbeam---, I -

Two-channel detector

regionN2 -cooled shield Magnetic shields

Figure 4. Main chamber of the atomic beam apparatus.

cision of that measurement was limited by lowsignal levels. Since that time, we have built asecond-generation version of the apparatusdesigned to improve the signal-to-noise ratio byseveral orders of magnitude. We have completedthe new apparatus and have recently begun takingdata.

Two accomplishments this year have brought us tothe point where we are now beginning to explorethe systematics of the new apparatus. The firstwas the completion and installation of thecryogenically-cooled interaction region. The designof the interaction region was discussed in lastyear's RLE Progress Report, and construction andassembly brought few surprises. The secondachievement was the development of the lasersystem, which is depicted schematically in figure 5.

The laser system consists of two high-powernarrow-linewidth dye lasers. The heart of each dyelaser is the tunable oscillator, which is shown infigure 6. The gain medium is a quartz cell throughwhich a flourescent dye (DMQ dissolved inP-Dioxane) flows at roughly 3 I/min. The dye is

excited by the excimer laser beam which is focusedto a line with cylindrical optics. The oscillator cavityis basically of the grazing incidence design, 15 withthe addition of an intra-cavity etalon and additionalline-narrowing components. The grating-tuningmirror combination acts as a narrow-band mirror,which along with the 60 percent output coupler,defines a plane-parallel Fabry-Perot cavity. Thefour-prism anamorphic achromatic beam expanderprovides expansion of 25 times in the horizontalplane which ensures that the 3600 line/mm gratingis uniformly illuminated, providing a linewidth ofabout 0.15 cm-1 . The oscillator linewidth is furtherreduced to about 0.04 cm - 1 by the intra-cavityetalon. The solid quartz plane-parallel etalon has afree spectral range of 1.3 cm- 1. Because the etalonlinewidth increases with the divergence of the inci-dent beam, several steps were taken to reducedivergence within the cavity. The prism beamexpander reduces the horizontal divergence by afactor of 25 and the pinhole aperture eliminateshigh order cavity modes. The oscillator frequencyis tuned by rotating the grating and the etalonsimultaneously along a trajectory which maintainsthe centering of the etalon mode on the grating line.

15 M. Littman and H. Metcalf, "Spectrally Narrow Pulsed Dye Laser Without Beam Expander," Appl. Opt. 17(14): 2224-2227 (1978).

223


Delay Line

366.4 nmI mJ

364.5 nm 121.5nmI( () -0 (nJ

Stepping MotorController

IEEE-4m------4 > LPC-AT

Figure 5. Schematic of the high power ultraviolet laser system.

Figure 6. Narrow band tunable dye laser oscillator cavity with intra-cavity etalon.

In order to maintain stability, all fixed oscillator com-ponents are rigidly attached to a one-inch thick alu-minum base. The tunable components aremounted on double-row high precision industrial

bearing sets which are fixed into the base.Because the grating and etalon are quite sensitiveto temperature changes, the cavity is enclosed in aplexiglass box and the sensitive components are



actively stabilized to ± 5 mK. The rotating compo-nents are driven by 100-pitch lead screws, whichpush against extended lever arms. An additional100:1 anti-backlash worm and gear set is incorpo-rated in the grating drive mechanism. The leadscrews are turned by stepping motors with 400steps per revolution, which are driven by thestepping-motor controller. The angular resolution ofthe drive system is 10 prad for the etalon and 100nano-rad for the grating. An AT-compatible per-sonal computer which operates the stepping-motorcontroller through its parallel printer port maintainsthe synchronous tuning of the grating and etalon.The controlling software allows for keyboard manip-ulation of the tuning curves as well as an IEEE-488interface for remote control of the laser tuning.

The oscillator is pumped by 25 mJ of excimer laserenergy and produces roughly 250 pJ of tunableradiation around 364 nm. The tunable radiation isthen amplified to 1 mJ in a preamplifier, pumped byan additional 25 mJ of excimer power which isdelayed by 5 ns to allow for the oscillator start-up

1600

1400

E+ 1200

a 1000 -

. 800 -2pl/2 2 P3/2

600

400 -

200

0

0 20 40 60 80

Laser Tuning [GHz]

Figure 7. EFI spectrumstate.

delay. This 1 mJ is sufficient for the 2p -- n = 29laser at 365 nm. The Lyman-oa radiation for theIs - 2p transition is generated by four-wave mixingof a signal from a laser that is tuned to one-third ofthe Lyman-o( frequency. This signal passesthrough a second amplifier stage, which is pumpedwith roughly 150 mJ of excimer power, to produce a10 mJ pulse at 364.5 nm. The radiation is thenfocused into a high-purity gas cell containing aphase-matched mixture of krypton and argon gas,where the 364.5 nm radiation is tripled by degen-erate four-wave mixing to produce the 121.5 nmLyman-oa radiation.

A typical scan of the Lyman-oc laser is presented infigure 7. For this scan the second laser was tunedto the n=70 state of hydrogen. A 10 V/cm electricfield was applied to wash out the resonant structureof the upper state. The Lyman-oa radiation wasthen tuned through the 2pl/2 and 2p3/2 states whilethe excited state population was monitored by thepulsed electric field ionization detector located inthe circular state production region.

100 120 140

of the 2p states of atomic hydrogen obtained by two-photon excitation of the n-70 Rydberg

225


Currently, we are fine-tuning the process of trans-ferring the atoms from the n=29, m=0 laser-excitedstate to the n=29, m=28 circular state. We expectto begin circular-state spectroscopy and to makemeasurements below the 10-10 level this year.

Conference Proceedings

Lutwak, R., J. Holley, J. DeVries, and D. Kleppner,"Millimeter-Wave Measurement of the RydbergFrequency." Proceedings of the Symposium onFrequency Standards and Metrology, WoodsHole, Massachusetts, October 1995.

2.3 Precision Mass Spectrometry ofIons

Sponsors



Project Staff

Michael P. Bradley, David Garrison, Ljubomir M.Ilic, Dr. Fred L. Palmer, Daniel Ripin, Szymon M.Rusinkiewicz, Professor David E. Pritchard

In 1995, we published papers which confirmed thatour atomic mass measurements are the most accu-rate in the world. 16 Our mass table consists of tenatomic masses of particular importance to physicsor metrology.' 7 The accuracy of these masses, typi-cally 10-10, represents one to three orders of mag-nitude improvement over previously acceptedvalues. Furthermore, a wide variety of self consist-ency checks has allowed us to virtually eliminatethe possibility of unknown systematic errors. Theseresults contribute to several important experimentsin both fundamental and applied physics, including:

* Recalibration of the current x-ray wavelengthstandard by weighing the energy differencesassociated with the neutron capture gammarays of 14N, which are widely used as cali-bration lines; and

* Determination of the atomic weight of 28Si, partof a program to replace the artifact kilogram

mass standard by a crystal of pure silicon, ineffect creating an atomic standard of mass.

We have also developed several new techniquesthat will improve the accuracy and versatility of ourapparatus. These include methods for loading awider variety of ions into the trap, hardware andsoftware for directly controling the amplitude andphase of the the ions' magnetron orbits, the demon-stration of a technique for dramatically improvingthe accuracy of our mass comparisons by com-paring two simultaneously trapped ions (to eliminatethe problems due to field drift), and the demon-stration of a squeezing technique to reduce theinfluence of thermal noise on the measurements.These advances open the door to another dramaticimprovement in mass resolution that would allow:

* Determination of excitation and binding ener-gies of atomic and molecular ions by weighingthe associated small decrease in mass,Am - Ebin/c 2 (we must reach our ultimate goalof a few times 10- 1

2 to make this a generallyuseful technique);

* Determination of the molar Planck constantNAh by weighing neutron capture y-rays ofsilicon or sulphur, whose wavelengths will bemeasured by a NIST group; this will alsoprovide an independent determination of thefine structure constant;

* Measurement of the 3H - 3He mass difference,important in ongoing experiments to determinethe electron neutrino rest mass;

* Improvement of some traditional applications ofmass spectrometry resulting from our orders ofmagnitude improvement in both accuracy andsensitivity; and

* Determination of the molar Planck constantNAb, by measuring the atomic mass and recoilvelocity of an atom that has absorbed photonsof known wavelength.

Our experimental approach is to measure ioncyclotron resonance on a single molecular oratomic ion in a Penning trap, a highly uniform mag-netic field in which confinement along the magneticfield lines is provided by much weaker electricfields. We monitor the ion's axial oscillation bydetecting the currents induced in the trap electrodesusing ultrasensitive superconducting electronics

16 F. DiFilippo, V. Natarajan, M. Bradley, F. Palmer, and D.E. Pritchard, Physica Scripta T59: 144-54 (1995); F. DiFilippo, V. Natarajan,M. Bradley, F. Palmer, and D.E. Pritchard, in Atomic Physics 14 (New York: American Institute of Physics, 1995).

17 F. DiFilippo, V. Natarajan, K. Boyce, and D. E. Pritchard, Phys. Rev. Lett. 73: 1481 (1994).



developed for this application.'8 This work in trap-ping and precision resonance draws on Nobel Prize(1989) winning techniques developed by Dr. HansDehmelt at the University of Washington and Dr.Norman Ramsey at Harvard University.

We have developed techniques for driving, cooling,and measuring the frequencies of all three normalmodes of ion motion in a Penning trap. Thus wecan manipulate the ion position reproducibly towithin 30 microns of the center of the trap, cor-recting for electrostatic shifts in the cyclotron fre-quency to great accuracy. We use a Tr-pulsemethod to coherently swap the phase and action ofthe cyclotron and axial modes. 19 Therefore,although we detect only the axial motion directly,we can determine cyclotron frequency by meas-uring the phase accumulated in the cyclotronmotion in a known time interval. We can measurethe phase of the cyclotron motion to within 10degrees, leading to a precision of 10-10 for a oneminute measurement. Our entire ion-making

process is fully automated, and the computer cancycle from an empty trap to having a cooled singleion in about three minutes under optimal conditions.

Careful shimming of the electric and magnetic fieldsallows us to keep our systematic errors well below5 x 10-11, but unfortunately, the typical statisticalfluctuation in our magnetic field between measure-ments is 2.4 x 10-10. Thus even with the ability toachieve - 20 alternate loadings of two different ionsin one night, our overall accuracy is at best8x 10- 11 (see figure 8). We have found that thedistribution of field variation is not Gaussian, butrather has too many outlying points. This has ledus to use a generalization of least squares fittingcalled robust statistics in which a Hampel estimateis used to deweight outlying points in a mannerdetermined by the observed excess number ofoutliers. This has eliminated arbitrary decisionsabout dropping "bad points" from our data sets andhas increased the stability of our fits.

Figure 8. Cyclotron frequency as a function of time for alternate N2 + and CO+ ions in our Penning trap. The fre-quencies are obtained after a 50s integration of cyclotron phase (see text). The solid line is a polynomial fit to the drift inthe field common to both ions.

18 R. Weisskoff, G. Lafyatis, K. Boyce, E. Cornell, R. Flanagan, Jr., and D.E. Pritchard, J. Appl. Phys. 63: 4599 (1988).

19 E.A. Cornell, R.M. Weisskoff, K. Boyce, and D.E. Pritchard, Phys. Rev. A 41: 312 (1992).

227

0.94 - - 0.60

00.93- , -0.59 +

0

0.92 - 0.58

0.91- 0.57

S- N2 -fit

CU 0.90- 0 CO+ 0.56

Z--- CO fit

0.89 - 0.55

2:30 AM 3:30 AM 4:30 AM9/26/93

Measurement time


We have performed a wide variety of stringentchecks for systematic errors. In fact, every mass inthe table has been obtained from at least two inde-pendent sets of mass ratios. Several of the checksemploy our new technique for comparison of non-doublets,20 pairs of ions whose nominal mass tocharge ratios are unequal. This technique repres-ents a significant advance in precision massspectrometry since it allows us to obtain absolutemasses by direct comparison to carbon, which isdefined to have an atomic mass of exactly 12. Italso provides an absolute check of our accuracy byallowing measurements that should verify knownratios such as N2 +/N+ and Ar /Ar++. We have com-pared CD 4 + and CD 3 + to C to obtain two indepen-dent determinations of the atomic weight ofdeuterium. The results are consistent with thevalues for the hydrogen masses we obtained fromconventional doublet measurements. An extensiveseries of checks using only doublet ratios was alsoperformed. These include repeated checks of iden-tical ratios (some interspaced by a significantreduction of our field inhomogeneities), checks ofcircular ratios (eg., A/B, B/C, and C/A), and checksof related ratios (e.g., CO/N 2 and C0 2/N 20). Theconsistency of all these checks practically guaran-tees that our errors are within the quoted limits.

We have made major advances toward our longterm goal of improving our accuracy by more thanan order of magnitude, which will be achieved bymeasuring the cyclotron frequencies of two ionstrapped in the same field at the same time to elimi-nate the problem of field fluctuations, and by usingsqueezing techniques to reduce thermal noise. Wehave modified our apparatus to allow excitation anddetection of two ions at the same time, and demon-strated the capability of two ion measurements toreduce the effects of field fluctuations (figure 9).We have expanded our previous theoretical under-standing 21 of two ion dynamics to include the effectsof minute imperfections in the trapping fields, identi-fied particular magnetron orbits where these effectsare minimized, and developed hardware and soft-ware to place ions in those orbits.

Figure 9. Each dot represents one measured cyclotronphase of simultaneously trapped single N2, and CO, ions16 seconds after excitation. Daytime field fluctuationshave virtually randomized the phases of the individualions but the phase difference between the two remainswell defined.

We have also studied the possibility of placing twoions in adjacent traps in case unexpected problemsoccur when placing them in the same trap. Wehave evaluated the leading sources of error anddesigned an arrangement of superconducting coilsthat will prevent the magnetic fields in the two trapsfrom fluctuating independently.

With either two-ion scheme, the primary source ofmeasurement noise will be the special relativisticmass shift due to thermal fluctuations in cyclotronamplitude. We have proposed several methods ofclassical squeezing with parametric drives to reduceamplitude fluctuations22 and demonstrated the sim-plest of these,23 reducing thermal noise by about afactor of two.

We have also improved the efficiency and versatilityof our basic apparatus. We have built a specializedloader for alkali atoms, as well as an external ionloader with associated ion optics to guide ions intothe trap. We have implemented a new signal pro-

20 V. Natarajan, K. Boyce, F. DiFilippo, and D.E. Pritchard, Phys. Rev. Lett. 71:. 1998 (1993).

21 E.A. Cornell, K. Boyce, D.L.K. Fygenson, and D.E. Pritchard, Phys. Rev. A 45: 3049 (1992).

22 F. DiFilippo, V. Natarajan, K. Boyce, and D.E. Pritchard, Phys. Rev. Lett. 68: 2859 (1992).

23 V. Natarajan, F. DiFilippo, and D.E. Pritchard, Phys. Rev. Lett. 74: 2855 (1995).



cessing algorithm that quickly detects unwantedions in the trap, and a new data analysis routinethat more accurately extracts mass ratios from the

data. Finally, we have begun construction of a newdetector based on a DC SQUID that will increaseour sensitivity by a factor of three or more.

Table 1. Atomic mass table. The center column lists the atomic masses determined from our exper-iment, and the right column lists the accepted atomic masses from the 1983 evaluation.

Atom Mass (amu) Accepted values (amu)

1H 1.007 825 031 6 (5) 1.007 825 035 0 (120)

n 1.008 664 923 5 (23) 1.008 664 919 0 (140)

2H 2.014 101 777 9 (5) 2.014 101 779 0 (240)

13C 13.003 354 838 1 (10) 13.003 354 826 0 (170)

14N 14.003 074 004 0 (12) 14.003 074 002 0 (260)

15N 15.000 108 897 7 (11) 15.000 108 970 0 (400)

160 15.994 914 619 5 (21) 15.994 914 630 0 (500)

20Ne 19.992 440 175 4 (23) 19.992 435 600 0 (22000)

28Si 27.976 926 532 4 (20) 27.976 927 100 0 (7000)40Ar 39.962 383 122 0 (33) 39.962 383 700 0 (14000)

2.4 Atom Interferometry

DiFilippo, F., V. Natarajan, M. Bradley, F. Palmer,and D.E. Pritchard. "Accurate Atomic MassMeasurements from Penning Trap Mass Com-parisons of Individual Ions." Physica ScriptaT59: 144-54 (1995).

DiFilippo, F., V. Natarajan, M. Bradley, F. Palmer,and D.E. Pritchard. "Accurate Atomic MassMeasurements from Penning Trap Mass Com-parisons of Individual Ions." In Atomic Physics14. New York: American Institute of Physics,1995.

Natarajan, V., F. DiFilippo, and D.E. Pritchard."Squeezed States of Classical Motion in aPenning Trap." Phys. Rev. Lett. 74: 2855(1995).

SponsorsCharles S. Draper Laboratory

Contract DL-H-4847759Joint Services Electronics Program

Grant DAAH04-95-1-0038U.S. Army - Office of Scientific Research

Grant DAAL03-92-G-0229Grant DAAL01-92-6-0197

U.S. Navy - Office of Naval ResearchGrant N00014-89-J-1207

Project Staff

Michael S. Chapman, Troy D. Hammond, Dr. AlanL. Lenef, Edward T. Smith,24 Richard A.Rubenstein, Dr. H. Jorg Schmiedmayer,25 David A.Kokorowski, Dr. Marek L. Czachor, Bernd S.Rohwedder, Allen Jordan, Professor David E.Pritchard

24 Partially supported by a National Science Foundation Fellowship.

25 Partial support from the University of Innsbruck and an APART Fellowship of the Austrian Academy of Sciences.

229

2.3.1 Publications


Atom interferometers, in which atom or molecularde Broglie waves are coherently split and thenrecombined to produce interference fringes, haveopened up exciting new possibilities for precisionand fundamental measurements with complex parti-cles. The ability to accurately measure interactionsthat displace the de Broglie wave phase has led toqualitatively new measurements in atomic andmolecular physics, fundamental tests of quantummechanics, and new ways to measure accelerationand rotation:

* Atom interferometers permit completely newinvestigations of ground state properties ofatoms and molecules including precision mea-surements of atomic polarizabilities to testatomic structure models, determination of longrange forces important in cold collisions andBose-Einstein condensation, and measure-ments of molecular polarizability tensor compo-nents.

* Atom interferometers can make fundamentalmeasurements in quantum mechanics. Theseinclude measurements of topological andgeometric phases, loss of coherence from areservoir, quantum measurement, and investi-gations of multiparticle interferometry andentanglement.

* The large mass and low velocities of atomsmakes atom interferometers especially useful ininertial sensing applications, both as precisionaccelerometers and as gyroscopes. They have

a potential sensitivity to rotations ~ 1010greater than in optical interferometers of thesame area.

* Atom interferometers may have significantapplications to condensed matter physics,including measurements of atom-surface inter-actions and lithography using coherentlymanipulated fringe patterns that are directlydeposited onto substrates.

Our group has pioneered many of these areas,including the first (and only) atom interferometryexperiments that employ physically separated pathsto make precision measurements. Several majorpapers have been published this year on a varietyof topics underscoring the versatility and usefulnessof our interferometer. These topics includequantum coherences,2 6 atomic polarizability mea-surements.2 7 index of refraction of matter waves,28

near field imaging (atomic Talbot effect),29 moleculeoptics and interferometry, 30 and a new velocityselection scheme for precision measurements. 1

The investigations have proved to be of wide-spread general interest to the scientific communityand have received many write-ups in the popularscientific press.32

During 1995, we made significant improvements toour interferometer's performance, completed exper-iments in quantum measurement theory, and under-took an experiment demonstrating the high inertialsensitivity of atom interferometers.

26 M.S. Chapman, T.D. Hammond, A. Lenef, J. Schmiedmayer, R.A. Rubenstein, E.T. Smith, and D.E. Pritchard, "Photon Scatteringfrom Atoms in an Atom Interferometer: Coherence Lost and Regained," Phys. Rev. Lett. 75: 3783 (1985).

27 C.R. Ekstrom, J. Schmiedmayer, M.S. Chapman, T.D. Hammond, and D.E. Pritchard, "Measurement of the Electric Polarizability ofSodium with an Atom Interferometer," Phys. Rev. A 51: 3883 (1995).

28 J. Schmiedmayer, M.S. Chapman, C.R. Ekstrom, T.D. Hammond, S. Wehinger, and D.E. Pritchard, "Index of Refraction of VariousGases for Sodium Matter Waves," Phys. Rev. Lett. 74: 1043 (1995).

29 M.S. Chapman, C.R. Ekstrom, T.D. Hammond, J. Schmiedmayer, B.E. Tannian, S. Wehinger. and D.E. Pritchard, "Near FieldImaging of Atom Diffraction Gratings: the Atomic Talbot Effect," Phys. Rev. A 51: R14 (1995).

30 M.S. Chapman, C.R. Ekstrom, T.D. Hammond, R. Rubenstein, J. Schmiedmayer, S. Wehinger, and D.E. Pritchard, "Optics andInterferometry with Molecules," Phys. Rev. Lett. 74: 4783 (1996).

31 T.D. Hammond, D.E. Pritchard, M.S. Chapman, A. Lenef, and J. Schmiedmayer, "Multiplex Velocity Selection for Precision MatterWave Interferometry," Appl. Phys. B 60: 193 (1995).

32 Articles on recent work performed by our interferometer group have appeared in several journals: J. Maddox, Nature February(1995); I. Peterson, Sci. News, 147: 116 (1995); P. Yam, Sci. Am. 272(4): 30-31 (1995); R. Pool, Sci. 268: 1129 May (1995); S.Werner, Phys. World, 8(6): 25 (1995); B. Levi, Phys. Today, 48(7): 17-18 (1995); P.F. Schewe and B. Stein, A/P Phys. Bull. on Phys.News, January 4, 1996; M. Browne, "It's a Molecule. No, It's More Like a Wave," New York Times Science Section, August 15, 1995.



2.4.1 Improvements to the Apparatus

This year, we have increased the sensitivity of ourapparatus through an improved fabrication processfor the 200 nm period de Broglie wave transmissiongratings that are its most important component. Incollaboration with the Cornell NanofabricationFacility, where we make our gratings, we havedeveloped a novel registration technique that iscapable of producing gratings with phase uniformityover five times the usable area of our old gratings. 33

With the new gratings, we achieve an atom countrate of 200 kcount/sec and a contrast of 17 percent,giving a ot noise limited sensitivity of5 millirad/ min . We have also fabricated 140 nmand 160 nm period gratings which will furtherimprove sensitivity and allow for separated beamexperiments with more massive atoms andmolecules. Currently, efforts are underway in theNanoStructures Laboratory here at MIT to develop100 nm large scale silicon nitride (1 cm 2) gratingswith unprecedented phase uniformity usingachromatic interferometric (optical) lithography.34

2.4.2 Quantum Interference and Coherence

We have investigated the loss of coherencebetween the two separated de Broglie wave compo-nents of our interferometer by scattering resonantphotons, A, in an optically pumped sodium beam.This experiment realized a which-path gedankenexperiment contemplated by Feynman 35 and quanti-fies the transition between coherence and loss ofcoherence in an interferometer. We measured thefringe contrast due to scattering as a function ofseparation between the two paths and found that itdecreased significantly when the separationexceeded A/4, in accordance with the expectationsof Bohr's complimentarity principle. Contrast atlarger separations, however, exhibited strongrevivals (figure 10a), as predicted by our theory.The phase shift of the fringes was also measuredand suffered predicted discontinuities (figure 10b).

In an extension of this decoherence experiment, weexplicitly showed that scattered photons do not nec-essarily destroy coherence between the two deBroglie wave components of the interferometer. Byusing a highly collimated beam and reducing theacceptance angle of our detector, we were able toobserve atoms which experienced a photon recoilconfined to a small scattering range. For theseatoms, we found that the fringe contrast persistsover significantly larger path separations than in theprevious experiment where all scattering eventswere detected.

2.4.3 Rotation Sensing

Experiments have been performed to measure boththe interferometer's response to rotations as well asits rotational noise, demonstrating the inherent sen-sitivity of matter-wave interferometers to inertialeffects. 36 Small rotations of a few earthrates(earthrate, e -= 7.3 x 10-5rad/sec) or less wereapplied to the apparatus, and the resultinginterferometer phase measured (figure 11). Sincethe interferometer has a linear phase response torotations, a rotation rate was easily inferred andcompared to the rotation rate determined from twoaccelerometers attached to the vacuum housing ofour apparatus. A comparison was then made forrotation amplitudes ranging from - 4 to + 4 e, andplotted in figure 12a.

We have shown the response to have better than 1percent agreement with theory, and directlyobserved rotations below 50 mQe with 20 secondintegration times. Rotational noise was shot-noiselimited for short times (less than two seconds) andwas measured to be 13 mQe in 80 seconds (figure12b). These results are three orders of magnitudemore sensitive than previous measurements of rota-tion using atom interferometers and demonstratethe promise of using atom interferometers forinertial navigation systems.

33 M.J. Rooks, R.C. Tiberio, M.S. Chapman, T.D. Hammond, E.T. Smith, A. Lenef, R.A. Rubenstein, D.E. Pritchard, and S. Adams,"Coherence of Large Gratings and Electron-beam Fabrication Techniques for Atom-Wave Interferometry," J. Vac. Sci. Technol. B 13:2745 (1995); M.J. Rooks, R.C. Tiberio, M.S. Chapman, T.D. Hammond, E.T. Smith, A. Lenef, R.A Rubenstein, D.E. Pritchard, andS. Adams, "Coherence and Structural Design of Free-standing Gratings for Atom-Wave Optics," submitted to the Jap. J. Appl. Phys.

34 T.A. Savas, S.N. Shah, M.L. Schattenburg, J.M. Carter, and H.I. Smith, "Achromatic Interferometric Lithography for 100 nm-PeriodGratings and Grids," J. Vac. Sci. Technol. B 13: 2732 (1995).

35 R. Feynman, R. Leighton, and M. Sands, "The Feynman Lecture Notes," vol. 3 (Boston: Addison-Wesley, 1966).

36 A. Lenef, et al, "Rotation Sensing with an Atom Interferometer," to be submitted.

231


Figure 10. Results of light scattering experiment. (a) Contrast versus separation between de Broglie wave componentsin units of optical wavelengths. Decoherence data (circles) shows fringe contrast decreasing rapidly, but with revivals.Recoherence data (triangles) show a more gradual decrease in fringe contrast from sub-photon recoil momentumresolved detection of atoms. (b) Phase versus separation between de Broglie wave components in units of opticalwavelengths for decoherence data (circles) and recoherence data (triangles).



a - 2D

t = LL

Figure 11. The interferometer in motion under the influ-ence of a transverse acceleration. The atomic beamtravels from left to right in the laboratory frame but inter-acts with the progressively displaced gratings of themoving apparatus. Because a center-line (short dash)between the atom beam paths passes through the middleof the first grating at t = 0, and is offset by a transversevelocity v, = 1/2T, it also passes through the middle ofthe displaced second grating at t = T. The dashed curve(long dash) represents the displacement of theinterferometer due to acceleration. The center-line of theaccelerating interferometer is shown (short/long dash) at t= 0, and t = 2T where fringes have a relative displace-ment of -D.

2.4.4 Ongoing Investigations

Longitudinal Momentum Correlations

Recent advances in atom optics and interfer-ometers exploit the transverse coherence of atomsources. Longitudinal coherence is possible, butsince atoms with different (but correlated) velocitieshave different kinetic energies, it is an inherentlytime dependent phenomenon. Currently, we arepursuing theory with time varying potentials tounderstand ways to measure, create, and manipu-late the longitudinal coherences in our beam.

Understanding and being able to manipulate thesecoherences will allow us to employ our apparatus toperform studies of a wide range of intriguing time-dependent interactions, including forces whichdifferentially alter the momentum of the atoms inthe two legs of the interferometer. One of these,the Anandan force,37 that arises when a quantumdipole passes through a region of uniform electricand magnetic field, is the subject of considerabletheoretical debate.

Velocity Dependent Index of Refraction of a Gas

The refractive index of a gas to passage .of asodium de Broglie wave arises from the collision-induced phase shift between the ground statesodium atoms of the beam and the molecules in thegas cell through which they pass. We plan toextend our recent experiment by varying the inci-dent beam energy, providing a powerful means forfurther exploration of the intermolecular potentialsinvolved in the collisions.

Dedicated Rotation Sensing Device

The impressive results of our rotation response andnoise measurements have motivated the study of adedicated inertial sensing device based on a highflux atom interferometer with fabricated gratings.Such a device using an effusive Cs beam wouldproduce detected fluxes of about 1010 atoms/secand have -a shot noise limited sensitivity of5x 10- 8./hr .

Velocity Selection through Velocity-multiplexing

The strength of applied potentials in atominterferometry is typically limited because theresulting dispersive phase shifts wash out the inter-ference pattern. In velocity-multiplexing, the vel-ocity distribution of the atomic beam is chopped intoa series of narrow peaks such that the velocitydependence of the phase shifts result in rephasingof the interference signal for certain strengths ofapplied potential. The technique overcomes limita-tions due to wide and/or poorly known velocity dis-tributions and thus allows a more precisedetermination of the applied interaction with com-plete independence from the initial velocity distribu-tion of the beam. We plan to use this scheme toperform experiments requiring precision beyond the0.1 percent level.

Geometric Phases

Precision measurements of the Aharonov-Casher(AC) phase are planned. This will allow a study ofthe dependence on the interfering particle's dipoleorientation for the first time. Modifications made toour interaction region (a device which allows dif-ferent potentials to be applied to either arm of theinterferometer) to introduce spatially varying mag-netic fields will permit investigation of Berry's phaseas well.

37 J. Anandan, Phys. Rev. Lett. 48: 1660 (1982).

233


Rotation Rate Inferred From Accelerometers (Q,.)

4.

2

0

-2

-4

0.2-

0.0

-02 -

0.1-9--

7.

6.

5-

4-

3-

2-

0.01-

*......- i.........

4

4 i

: i

.. .....i .. ,.., ......................-

.... ... . .. -. . . . . . . . . .

% I

.............. .... ..... ... ..

S..

.. .. .. . . . .. ... ..... , ,a . .

Si 'l I I I I ' ''II I , I I I i 1!

Integration Time (sec)

Figure 12. (a) A plot of the measured interferometer phase, 0nas, verses the inferred phase from the accelerometerreadings, qro,, from a combination of 20 second runs totaling approximately 400 seconds of data (-- 10 sec of data perpoint). There is a 0.8 percent difference between these measurements with a total error of 1 percent. (b) Reproducibilityof the interferometer measurement when driven by a 4.6 Hz sinusoidal motion. The standard deviation of the fluctu-ations in the measured rotation of the 4.6 Hz spectral peak is plotted verses integration time (circles). This is comparedto the shot noise limited error verses integration time (dashed line).


I I 1 I I-4 -2 0 2 4

Phase Inferred From Accelerometers, , ,, ,,(rad)

................................. i.......... .. ..... ..... ... - -- .. ................................ ..................... .......... .. ...

.. .. ... .. ............................

. .. .. . . . . . .. . . . .. ... . . . . .. . . . ....... ................. . .................. ...-- -- - ................ .... ... .. .I .

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2.4.5 Publications

Chapman, M.S., C.R. Ekstrom, T.D. Hammond, R.Rubenstein, J. Schmiedmayer, S. Wehinger, andD.E. Pritchard. "Optics and Interferometry withMolecules." Phys. Rev. Lett. 74: 4783 (1995).

Chapman, M.S., T.D. Hammond, A. Lenef, J.Schmiedmayer, R.A. Rubinstein, E.T. Smith, andD.E. Pritchard. "Photon Scattering from Atomsin an Atom Interferometer: Coherence Lost andRegained." Phys. Rev. Lett. 75: 3783 (1995).

Schmiedmayer, J., et al, "Atom and Molecule Inter-ferometry with Separated Beams." In Atom Inter-ferometry. Ed. Paul R. Berman, New York:Academic Press. Forthcoming.

Thesis

Chapman, M.S. Photon Induced Coherence Loss inAtom Interferometry. Ph.D. diss., Dept. ofPhysics, MIT, 1995.

2.5 Cooling and Trapping NeutralAtoms

Sponsors

Alfred P. Sloan FoundationNational Science Foundation

Grant PHY 95-01984U.S. Army Research Office

Contract DAAL01-92-C-0001U.S. Navy - Office of Naval Research

Grant N00014-90-J-1642Grant N00014-94-1-0807

Project Staff

Professor Wolfgang Ketterle, Dr. Nicolaas J. vanDruten, Michael R. Andrews, Kendall B. Davis,Dallin S. Durfee, Ilya A. Entin, Philip M. Hinz,Everest W. Huang, Dan M. Kurn, Marc O. Mewes,Till P. Rosenband, Charles K. Sestok, Stanley H.Thompson, Peter S. Yesley

2.5.1 Introduction and Summary

Cooling and trapping of neutral atoms offersexciting new possibilities. Many are related to thefact that the deBroglie wavelengthincreases withdecreasing temperature T as 1/T T. When thedeBroglie wavelength is comparable to atomicdimensions (range of the interaction potential), colli-sions can no longer be treated classically. Theyare dominated by weak long-range interactions.Dramatic effects happen at even colder temper-atures and higher densities, when the deBrogliewavelength becomes comparable to the interatomicspacing. In this case, the atomic waves overlap,and a novel type of highly correlated quantummatter is observed. Fermions would form a corre-lated Fermi sea where collisions and light scatteringare suppressed if most of the final states for theseprocesses are already occupied.

Einstein predicted in 1925 that a gas of bosonicparticles will undergo a phase transition into aBose-Einstein condensate (BEC) if they are cooledto a very low temperature.3 8 This new phase ischaracterized by a macroscopic population of thelowest energy levels. Bose-Einstein condensationis therefore a macroscopic quantum phenomenonlike superfluidity and superconductivity. It is distin-guished by the coherence of atoms; all atoms in thecondensate are indistinguishable and behave iden-tically.

The observation of Bose-Einstein condensation hasbeen one of the major goals in atomic physics inthe last few years. This goal has been achievedrecently by three groups using different atoms anddifferent techniques.3 9 The successful production ofBose condensates is the starting point for a thor-ough study of this long-sought novel form of matter.The study of Bose-Einstein condensation is impor-tant because:

* Bose-Einstein condensation is a paradigm ofquantum statistical physics that was predictedmore than 70 years ago.38

* It becomes possible to perform crucial tests oftheories on the weakly interacting Bosecondensate. In contrast to other macroscopicquantum phenomena like superconductivity andsuperfluidity, Bose condensation in atomicgases can be studied at such low densities that

38 A. Einstein, Sitzungsber. Preuss. Akad. Wiss. 18 (1925).

39 M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Sci. 269: 198 (1995); C.C. Bradley, C.A. Sackett, J.J.Toilet, and R.G. Hulet, Phys. Rev. Lett. 75: 1687 (1995); K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee,D.M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75: 3969 (1995).

235


perturbative approaches and mean-field theo-ries are accurate.40 A detailed theoreticalunderstanding of Bose condensation based onfirst principles might advance our under-standing of more complex systems like liquidhelium or high-Tc superconductors.

A Bose condensate is the ultimate source ofultracold atoms. The kinetic energy of a Bosecondensate is on the order of tens ofnanokelvin. Such ultracold atoms are ideal forprecision experiments (determination of funda-mental constants, tests of fundamental symme-tries) because the slow motion eliminates mostsystematic effects. Furthermore, such samplesof atoms have potential applications in the fieldof atom optics, such as the creation of micro-scopic structures by direct-write lithography oratom microscopy. Structures as small as 65nm were obtained recently by laser-focusedatomic deposition, 4 1 mainly limited by the trans-verse collimation and the thermal velocityspread of the atomic beam. Improvements byan order of magnitude should be possible.With ultracold atoms, one could realize the ulti-mate resolution in focusing atoms. This isanalogous to the diffraction limit in optics. ABose condensate would also find application inmetrology, improving frequency standards andatom interferometry. Cesium clocks usingmicrokelvin atoms might improve the accuracyof the current frequency standard by two ordersof magnitude.42 Nanokelvin atoms would offeradditional improvements.

* A Bose condensate consists of coherentatoms. Such a state of matter has many anal-ogies with coherent photons or laser light andcan be regarded as the first realization of anatom laser. The accumulation of atoms in theBose condensate can be interpreted as stimu-lated emission of matter waves. The study ofcoherent matter waves might become anexciting new subfield of atomic physics.

* A Bose condensate is a novel form of matterwith many unknown properties. The ideal(non-interacting) case is well understood, butdoes not include even the qualitative features

of real, weakly interacting systems.40 In thewake of the progress of the last few years,many theories have been developed on the for-mation of a condensate, and its collisional andoptical properties. Most theories are approxi-mate and have to be tested experimentally.Some theories even lead to contradictory con-clusions.

* When a new regime of ultracold temperaturesis explored, there is always the hope of findingsomething completely unexpected!

Despite the rapid progress in laser cooling over thelast few years, the closest approach towards BECwas still missing four to five orders of magnitude inphase space density. Laser cooling is limited in thelowest temperature by heating due to spontaneousemission and in density by absorption of scatteredlight and excited state collisions. Current researchactivities try to push the limits of laser cooling evenfurther.43

Our approach is to use laser cooling as a first stepand then continue with evaporative cooling whichdoes not involve light and therefore does not sufferfrom the above mentioned limitations. The coolingis accomplished by selectively removing atoms withthe highest energy from a magnetic trap and thenallowing the rest of the sample to rethermalizethrough elastic collisions. In 1994, we were able torealize for the first time this combination of lasercooling and evaporative cooling.44 The cooling waslimited by trap loss (Majorana flops) in the magnetictrap. In 1995, we could eliminate the trap loss in anovel trap, the optically plugged magneticquadrupole trap, and cool sodium atoms to Bose-Einstein condensation.

2.5.2 Optically Plugged MagneticQuadrupole Trap

In 1995, we demonstrated a novel atom trap whichconsists of a combination of magnetic fields andfar-off-resonant light. This trap offers a superiorcombination of large trapping volume and tight con-finement. It allowed us to obtain samples of ultra-cold atoms at unprecedented densities (1014 cm- 3)

40 K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1987), pp. 286-302.

41 J.J. McClelland, R.E. Scholten, E.C. Palm, and R.C. Celotta, Sci. 262: 877 (1993).

42 K. Gibble and S. Chu, Metrologia 29: 201 (1992).

43 J. Lawall, S. Kulin, B. Saubamea, N. Bigelow, M. Leduc, and C. Cohen-Tannoudji, Phys. Rev. Lett. (1995), forthcoming.

44 K.B. Davis, M.-O. Mewes, M.A. Joffe, M.R. Andrews, and W. Ketterle, Phys. Rev. Lett. 74: 5202 (1995).



and to evaporatively cool atoms to Bose-Einsteincondensation.45

Evaporative cooling requires an atom trap which istightly confining and stable. So far, magnetic andoptical dipole traps have been used. Optical dipoletraps provide tight confinement, but have only avery small trapping volume (10-8 cm3 ). The tightestconfinement in a magnetic trap is achieved with aspherical quadrupole potential (linear confinement);however, atoms are lost from this trap due to non-adiabatic spinflips as the atoms pass near thecenter where the field rapidly changes direction.This region constitutes a hole in the trap of micro-meter dimension. The recently demonstrated TOPtrap suppresses this trap loss, but at the cost oflower confinement. 46

We suppressed the trap loss by adding a repulsivepotential around the zero of the magnetic field, liter-ally plugging the hole. This was accomplished bytightly focusing an intense blue-detuned laser thatgenerated a repulsive optical dipole force. Theoptical plug was created by an argon ion laserbeam (514 nm) of 3.5 W focused to a diameter of30 pm. Heating due to photon scattering was sup-pressed by using far-off-resonant light, and by thefact that the atoms are repelled from the regionwhere the laser intensity is highest.

The total potential experienced by the atoms is acombination of the dipole potential, the quadrupoletrapping potential and the rf dressed atom potential.The latter can be understood qualitatively with asimple argument. At the point where atoms are inresonance with the rf, the trapped state undergoesan avoided crossing with the untrapped states, andas a result the trapping potential bends over. Thetotal potential orthogonal to the propagation direc-tion of the blue detuned laser is depicted in figure13.

At temperatures above 15 pK, the observedtrapped clouds were elliptical with an aspect ratio of2:1 due to the symmetry of the quadrupole field(figure 14). At the position of the optical plug, therewas a hole, which was used for fine alignment. Amisalignment of the optical plug by - 20 pmresulted in increased trap loss and prevented usfrom cooling below 50 pK. This is evidence that theMajorana spin flips are localized in a very smallregion around the center of the trap. At temper-atures below 15 pK, the cloud separated into two

45 K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S.(1995).

Figure 13. Adiabatic potential due to the magneticquadrupole field, the optical plug, and the rf. This cut ofthe three-dimensional potential is orthogonal to the prop-agation direction (y) of the blue detuned laser. The sym-metry axis of the quadrupole field is the z-axis.

pockets (see figure 14b) at the two minima in thepotential of figure 13.

2.5.3 Bose-Einstein Condensation ofSodium Atoms

We have recently observed Bose-Einstein conden-sation in a dilute gas of sodium atoms. 45 The atomswere trapped in the optically plugged magneticquadrupole trap and cooled by evaporation. Theessential condition for evaporative cooling is thatthe collision rate be sufficiently high for many colli-sions to occur within the lifetime of the atoms in thetrap. Beside high initial density and long trappingtimes, evaporative cooling requires a method forselectively removing hot atoms from the trap. In rfinduced evaporation, atoms are spin-flipped to anuntrapped state when they are in resonance with anapplied rf field (figure 15). Since this resonancefrequency is a function of magnetic field B, atomsare selectively removed at a specific value of B. Inthe case of transitions between magnetic sublevelsmF, the resonance frequency is wrf= g BB/h,where g is the g-factor and PB the Bohr magneton.Since the trapping potential is given by mFIgpB B(r),only atoms which have a total energy E > hw oImFwill evaporate. Evaporative cooling is forced bylowering the rf frequency. Within seven seconds,this reduced the temperature by a factor of 100,and increased the phase space density by sixorders of magnitude, leading to Bose-Einstein con-densation of the remaining atoms.

Durfee, D.M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75: 3969

46 W. Petrich, M.H. Anderson, J.R. Ensher, and E.A. Cornell, Phys. Rev. Lett. 74: 3352 (1995).

237


Figure 14. Absorption images of atom clouds trapped inthe optically plugged trap. Cloud (a) is already colderthan was attainable without the plug (Ar ion laser beam).Cloud (b) shows the break-up of the cloud into twopockets in the two minima of the potential in figure 13.The size of cloud (c) reaches the optical resolution of theimaging system (< 10 mm) still absorbing 90 percent ofthe probe light. This sets an upper bound on temper-ature (<10pK) and a lower bound on density(5 x 1012 cm-3 ).

Condensates contained up to 5 x 105 atoms at den-sities exceeding 1014 cm - 3 . The striking signature ofBose condensation was the sudden appearance ofa bimodal velocity distribution when the sample wascooled below the critical temperature of - 2pK.The distribution consisted of an isotropic thermaldistribution and an elliptical core attributed to theexpansion of a dense condensate (figure 16).

Figure 15. Experimental setup for cooling atoms toBose-Einstein condensation. Sodium atoms are trappedby a strong magnetic field, generated by two coils. In thecenter, the magnetic field vanishes, allowing the atoms tospin-flip and escape. Therefore, the atoms are keptaway from the center of the trap by a strong (3.5 W) +laser beam (optical plug) which exerts a repulsive forceon the atoms. Evaporative cooling is controlled by radio-frequency radiation from an antenna. The rf selectivelyflips the spins of the most energetic atoms. Theremaining atoms rethermalize (at a lower temperature) bycollisions among themselves. Evaporative cooling isforced by lowering the rf frequency.

This experiment was one of three independentapproaches which succeeded in creating BEC inthe last few months.47 Our results on sodium aredistinguished by a production rate of Bose con-densed atoms which is three orders of magnitudelarger than in the two previous experiments. Densi-ties higher than 1014 atoms/cm3 exceed previousresults by more than a factor of ten and are prom-ising for the study of collisions and interactions ofdense ultracold atoms.

We are currently preparing experiments to study theshape, stability, and optical properties of thecondensate. Of special interest are experimentswhich might reveal the coherent nature of theatoms. Such experiments could be coherentexcitations of the condensate by modulating thetrapping potential,48 and interference between twocondensates. 49

47 M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell, Sci. 269: 198 (1995); C.C. Bradley, C.A. Sackett, J.J.Toilet, and R.G. Hulet, Phys. Rev. Lett. 75: 1687 (1995); K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee,D.M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75: 3969 (1995).

48 P.A. Ruprecht, M.J. Holland, K. Burnett, and M. Edwards, Phys. Rev. A 51: 4704 (1995).

49 J. Javanainen and S.M. Yoo, Phys. Rev. Lett. 76: 161 (1996).



Figure 16. Two-dimensional probe absorption images, after 6 ms time of flight, showing evidence for BEC. (a) is thevelocity distribution of a cloud cooled to just above the transition point, (b) just after the condensate appeared, (c) afterfurther evaporative cooling has left an almost pure condensate. The figure shows the difference between the isotropicthermal distribution and an elliptical core attributed to the expansion of a dense condensate. The width of the images is0.87 mm.

2.5.4 Publications

Davis, K.B., M.-O. Mewes,Druten, D.S. Durfee,Ketterle. "Bose-Einsteinof Sodium Atoms."3969-3973 (1995).

M.R. Andrews,D.M. Kurn,CondensationPhys. Rev.

Davis, K.B., M.-O. Mewes, M.A.Andrews, and W. Ketterle.Cooling of Sodium Atoms." Phys.5202 (1995); Erratum: Phys. Rev.(1995").

N.J. vanand W.in a Gas

Lett. 75:

Joffe, M.R."Evaporative

Rev. Lett. 74:Lett. 75: 2909

Davis, K.B., M.-O. Mewes, and W. Ketterle: "AnAnalytical Model for Evaporative Cooling ofAtoms." Appl. Phys. B 60: 155-159 (1995).

Conference Contributions with PublishedAbstracts

Davis, K.B., M.-O. Mewes, M.A. Joffe, M.R.Andrews, and W. Ketterle. "Evaporative Coolingof Sodium Atoms." In Research Conference onBose-Einstein Condensation, Mont Ste. Odile,June 1995, Book of Abstracts, p. 53.

Ketterle, W. "Evaporative Cooling of -MagneticallyTrapped Sodium." Bull. Am. Phys. Soc. 40:1269 (1995).

239


Theses

Davis, K.B. Evaporative Cooling of Sodium Atoms.Ph.D. diss., Dept. of Physics, MIT, 1995.

Entin, I.A. Magnetic Trapping of Neutral SodiumAtoms. S.B. thesis, Dept. of Physics, MIT,1995.

Thompson, S.H., Jr. Radio Frequency InducedEvaporative Cooling of Magnetically TrappedNeutral Sodium Atoms. S.B. thesis, Dept. ofPhysics, MIT, 1995.

Yesley, P.S. The Design and Testing of Novel,Spin-Flip, Zeeman Slowing Technique. S.B.thesis, Dept. of Physics, MIT, 1995.


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