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Chapter 2. Basic Atomic Physics
Academic and Research Staff
Professor Daniel Kleppner, Professor David E. Pritchard, Professor Wolfgang Ketterle
Visiting Scientists and Research Affiliates
Dr. Theodore W. Ducas,^1 Dr.^ Fred^ L.^ Palmer,^ Dr.^ H.^ Joerg^ Schmiedmayer,^ Stefan^ Wehinger
Graduate Students
Michael R. Andrews, Michael P. Bradley, Michael S. Chapman, Michael W. Courtney, Kendall B. Davis, Frank DiFilippo, Joel C. DeVries, Christopher R. Ekstrom, Troy D. Hammond, Jeffrey R. Holley, Hong Jiao, Michael A. Joffe, Robert I. Lutwak, Marc O. Mewes, Vasant Natarajan, Richard A. Rubenstein, Neal^ W. Spellmeyer
Undergraduate Students
Ilya Entin, Philip M. Hinz, Wan Morshidi, Amrit R. Pant, J. David^ Pelly,^ Abraham^ D.^ Stroock,^ Bridget^ E. Tannian, Stanley H. Thompson, John J. Wuu, Peter S. Yesley
Technical and Support Staff
Carol A. Costa
2.1 Testing Quantum Chaos with
Rydberg Atoms in Strong Fields
Sponsors National Science Foundation Grant PHY 89- Grant PHY 92- U.S. Navy - Office of Naval Research Grant N00014-90-J-
Project Staff Michael W. Courtney, Hong (^) Jiao, Spellmeyer, Professor Daniel Kleppner
Neal W.
Interest in the behavior of Rydberg atoms in strong fields has resulted in the generation of high resolu- tion experimental spectra and accurate calculations for atoms in electric and magnetic fields and also experimental data in the low field regime for electric and magnetic fields. The principal motivation for this work has been to understand atomic systems under external perturbations similar in magnitude (^) to the unperturbed energy. Rydberg atoms are central to this research, because field strengths available in the laboratory are comparable with atomic (^) fields of highly-excited states. We have chosen to work with hydrogen because it is the simplest system to
study. Alkali metal atoms are also of interest, because they break the zero field degeneracy (^) of hydrogen and are much more tractable (^) exper- imentally.
The quantum mechanics of alkali Rydberg atoms in strong electric fields is now believed to be well understood. Much has been learned about diamag- netic hydrogen, but some important regimes remain to be explored, and our knowledge of hydrogen needs to be extended to other alkalis. For example, we know little about the alkali Rydberg atoms in parallel electric and magnetic fields and even less about them when the fields have arbitrary or perpendicular orientation.
With the exception of hydrogen in an electric (^) field, the classical analogues of all these systems undergo a transition from order to chaos. In seeking to understand the connection between quantum mechanics and classical chaos, the problem of hydrogen (^) in a magnetic field has been extensively studied. However, the more general problem of Rydberg atoms in strong fields provides a richer (^) testing ground for theories describing "quantum chaos."
We have developed techniques (^) for carrying out high resolution laser spectroscopy on the lithium
1 Physics Department, Wellesley College, (^) Wellesley, Massachusetts.
209
atom in strong static fields. In addition, we have developed techniques for computing the spectra of alkali atoms in both strong electric and magnetic fields. The differences between lithium and hydrogen are minor for many cases of interest. In other cases, the core potential itself is the cause of classical chaos and becomes essential to consider.
In the experiment, a lithium atomic beam, excited by a laser, transfers atoms from the 2S state to the 3S state by a two-photon transition, and a second laser excites the atoms to Rydberg states.
The Rydberg atoms are detected by electric field ionization. Two methods for relating quantum spectra to the classical dynamics system are (1) periodic orbit theory and (2) use of statistical corre- lations. In the past year, we have^ made^ progress in testing both these methods which characterize quantum chaos.
Classically chaotic systems possess a proliferation of periodic orbits. Periodic orbit spectroscopy is a technique used to relate the quantum mechanical spectrum to the periodic orbits of the corresponding classical system.^2
The hamiltonian of hydrogen in a magnetic field is given by
2 S 1 1 LB + 1 B2p2 (1) 2 r 2 8
This hamiltonian can (^) be rescaled using the relations: r= B-^213 r, p = B 1/^3 p, and L= B-1/3L. gives the scaled hamiltonian,
2 P (^1 1) z + 1 2 2 2 8
This
where -= B- 2 /3H. As a result, the classical dynamics depends only on the parameter (^) EB = B-2/^3 E and not on E and B separately.
The semi-classical Bohr-Sommerfeld quantization
condition for the two non-separable scaled coordi-
nates p, z is
S = (2r) 1 (ppdp + pzdz) = nB- 1/3,
where S is the scaled action of a closed classical orbit. As a result, S = n B- 1/3^ gives rise to equidistant lines on a scale B-^1 /^3 in a spectrum taken at constant scaled energy EB. In other words, each peak in the Fourier transform of a constant scaled energy spectrum corresponds to the scaled action of a periodic orbit.
This has been shown experimentally.^3 In addition, Delos and his colleagues have developed the tech- niques of periodic orbit theory which use classical dynamics to compute the actions of the periodic orbits and semi-classical techniques to compute the strength of the periodicities. The strength of the periodicities is called the recurrence strength, because an outgoing spherical wave will recur at times corresponding to the periods of the periodic orbits of the classical system. Figure 1 shows the Fourier transform of a computed constant scaled energy spectrum compared with the results of peri- odic orbit theory (courtesy of Dr. J.A. Shaw of the College of William and Mary). The recurrences from periodic orbit theory are in good agreement, except for peaks near scaled actions of 4, 8, 12, 16, and 20, which are systematically much stronger in the periodic orbit theory results. These peaks are due to the 5th, 10th, 15th, 20th, and 25th return of the orbit in (^) the z=0 plane. These orbits have a bifurcation very close (^) to the scaled energy shown here which makes their recurrence strength difficult to accurately compute.
We are planning periodic orbit spectroscopy both experimentally and computationally. Periodic orbit spectroscopy requires (^) varying the energy and the magnetic field simultaneously (^) so that scaled energy Es remains constant. Since our computational (^) tech- niques have been verified by our experiments, and since it is easier to accurately vary E and B while maintaining a constant EB numerically than exper- imentally, we have proceeded computationally in the regions where the computations have been veri- fied. In addition, our experiment (^) is limited to exciting only odd-parity states of lithium. Computa- tionally, we have been able to explore both odd and even parity hydrogen and lithium. In this way, we
2 A. Holle, (^) J. Main, G. Wiebusch, H. Rottke, and K.H. (^) Welge, "Quasi-Landau Spectrum of the Chaotic Diamagnetic Hydrogen Atom," Phys. Rev. Lett. 61: (^161) (1988); J.-M. Mao and J.B. Delos, "Hamiltonian Bifurcation (^) Theory of Closed Orbits in the Diamagnetic Kepler Problem," Phys. Rev. A 45: 1746 (1992).
3 A. Holle, J. Main, G. (^) Wiebusch, H. Rottke, and K.H. Welge, "Quasi-Landau (^) Spectrum of the Chaotic Diamagnetic (^) Hydrogen Atom," Phys. Rev. (^) Lett. 61: 161 (1988); T. van der Veldt, W. Vassen, and W. Hogervorst, "Quasi-Landau Structure of Diamagnetic Helium Rydberg Atoms," (^) Europhys. Lett. 21: 9 (1993).
210 RLE Progress Report Number (^136)
Figure 2. Recurrence strength for EB = - 0.7, m = 0, odd parity. The information^ in^ figure^^1 is^ shown^ enlarged^100 times. The arrows point to the orbits in the z = 0 plane.
because classical dynamics does not "know" about parity. As a result, periodic orbit spectroscopy should yield the same periodic orbits for both odd and even parity hydrogen. Similarly, periodic orbit spectroscopy in lithium should give the same peri- odic orbits, regardless of parity. In both hydrogen and lithium, the recurrence strengths may vary between the odd and even parity cases, because these are computed by including parity in the semi- classical methods. However, since odd parity hydrogen and lithium have nearly identical spectra, we expect the periodic orbits of lithium and hydrogen to be nearly identical. This seems to be the case from looking at the recurrences in figure 3.
However, we do not expect a completely^ chaotic system to have the same periodic orbits as a regular system. In addition, from a quantum mechanical point of view, we have no reason to expect that the spectrum of even parity lithium will exhibit the same periodicities as even parity hydrogen. However, figure 4 shows that many of the recurrences are the same. This is a victory for periodic orbit theory. The case of even parity lithium also has many extra recurrences, indicating extra periodic orbits. This is a challenge to periodic orbit theory, because to resolve this paradox, peri- odic orbit theory must find these extra periodic orbits and compute their recurrence strengths in even-parity lithium. Furthermore, it also needs to
be shown that all these extra periodic orbits have a very small recurrence strength in the odd parity case.
In the case of Rydberg atoms in a strong electric field, parity is not a good quantum number. Hydrogen in an electric field is regular and chaos (^) is caused completely by the core. The hamiltonian of hydrogen in an electric field is given by
p
j= 2 12+r Fz,
and this hamiltonian can be rescaled by substituting r = F- 2 r, p = F^1 4 p. As a result, the classical dynamics of hydrogen in^ an^ electric^ field^ depends only on the parameter EF^ =^ F-1/2E^ and not^ on^ E^ and F separately. In analogy with the magnetic field case, each peak in the Fourier transform of a spec- trum taken at constant EF corresponds to the scaled action of a periodic orbit. These classical scaling rules are^ exact^ for^ hydrogen^ and^ good^ approxi- mations for lithium.
In the case of lithium in an electric field, we have done experiments (^) for both the cases m = 0 and m = 1. The m = 1 case is regular and nearly identical to hydrogen, because the angular momentum barrier causes the wavefunction to have a node at the core.
212 RLE Progress Report Number 136
2
1.
1
0
-0.
-1.
- 0 0.5 1 1.5^^2 2.5^3 3.5^4 Scaled Action
The m=0^ case^ is^ chaotic^ for^ all^ regions^ of^ interest. The experimental result for EF - 0.3 is compared with quantum computations for lithium and hydrogen in figures 5 and 6. Notice that lithium and hydrogen have many recurrences in common, but just as in the case of even-parity^ diamagnetic lithium, there are extra recurrences which^ need^ to be computed by periodic orbit theory. Several^ of these extra peaks occur near a scaled action^ of^ 14.
Figure 3. Recurrence strength^ for^ Es^ -^ 0.6,^ m^ -^ 0,^ odd^ parity.^ Notice hydrogen and^ lithium.
Spectral correlations have also been useful in char- acterizing spectra in regions of classical chaos. Computing the spectra in regions of total chaos has enabled the testing of a new class of universal cor- relations in spectra of chaotic systems as a param- eter is changed. Our computations enabled^ MIT Professor Boris Altshuler's group^ to^ verify their^ new class of universal correlations^ on^ a^ real^ physical system. 8 This new class of^ correlations^ has^ been confirmed in the chaotic regions^ of^ both^ diamag- netic hydrogen and lithium.
that it shows the same periodic orbits for
8 B.D. Simons, A. Hashimoto, M. Courtney, D. Kleppner, and B.L. Altshuler, "New Hydrogen in a Magnetic Field," Phys. Rev. Lett. 71: 2899 (1993).
Class of Universal Correlations in the Spectra of
213
1.5e-
le-006 Hydrogen
5e-
0 A
-5-
-le-006 Uthlum
-1.5e- 0 5 10 15 20 Scaled Action
Figure 6. Recurrence strength for EF = - 3.0, m = 0. The experiment for lithium is compared with quantum computations for hydrogen.
2.2 Millimeter-Wave Frequency
Measurement of the Rydberg Constant
Sponsors
Joint Services Electronics Program
Contract DAAL03-92-C-
National Science Foundation
Grant PHY 89-
Grant PHY 92-
Project Staff
Joel C. DeVries, Dr. Theodore W. Ducas, Jeffrey R.
Holley, Robert I. Lutwak, Professor Daniel Kleppner
The Rydberg constant R. relates the wavelengths
of the spectrum of atomic hydrogen to practical lab-
oratory units. It is a so-called fundamental con-
stant, for it relates numerous quantum properties to
laboratory units. Recent advances in optical wave-
length metrology have enabled experiments using
laser spectroscopy to measure R. with accuracy
approaching two parts in 1011.
Although R,. is the most accurately measured fun- damental constant, forthcoming high-precision experiments which will depend on R,. as an auxil- iary constant, demand even more accurate meas- urement. Because the practical limit of measuring wavelengths, about 1 part in 1010, has been reached, progress in the (^) measurement of R~. requires measuring the frequency of spectral lines. Because the speed of light is now defined, by measuring cR., the "Rydberg frequency," a new standard is created.
Our approach involves preparing highly excited "Rydberg" states of atomic hydrogen, n = 29, and measuring millimeter-wave transitions (^) to nearby states. Because the millimeter wave signal is gen- erated coherently from a frequency standard based on an atomic clock, our measurement can make use of the high precision of frequency metrology.
The goals of our experiment are three-fold: First is the reevaluation of R. itself, providing an inde- pendent check, in a different (^) regime, of other evalu- ations based on optical wavelength metrology. Second is the measurement of the ground state
9 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: 1923 (1992); F. Nez, et al., "Precise Frequency Measurement of the 2S-8S/8D Transitions in Atomic Hydrogen: New Determination of the Rydberg Constant,"^ Phys.^ Rev. Lett.^ 69:^2326 (1992).
215
Lamb shift. Because our measurements involve high angular momentum states for (^) which the Lamb shift is extremely small, our results may (^) be com- pared with optical (^) measurements of transitions between low-lying states (^) to yield an improved measurement of the Lamb (^) shift. Third is to provide a frequency calibration (^) of the spectrum of hydrogen, enabling (^) the creation of a comprehensive frequency standard extending from the radio- frequency regime to the ultra-violet.
Our experiment employs an atomic beam configura- tion to reduce Doppler and collisional perturbations. Atomic hydrogen is excited to the low angular momentum n=29, m=0 state by two-photon step- wise absorption. The excited atoms are then trans- ferred to the longer lived n=29, m=28 "circular" state by absorption of circularly polarized radio- frequency radiation. 10 The atoms enter a region of uniform electric field in which the frequency of the resonant transition n=29, m=28 --- n=30, m=29 is measured by the method of separated oscillatory fields. The final state distribution is measured by a state-sensitive electric field ionization detector. The resonance is manifested (^) by a transfer of atoms from the n=29 state to the n=30 state as the millimeter-wave frequency is tuned across the tran- sition.
Figure 7 illustrates the apparatus. Atomic hydrogen or deuterium is produced by dissociating H 2 or D 2 in a radio-frequency discharge. The beam is cooled by collisions in a cryogenic thermalizing channel in
order to slow the beam and thereby (^) increase the interaction time. The atoms enter the circular (^) state production (^) region, where they are excited (^) from the is ground state, through the 2p state, (^) to the n=29,m=0 state by two-photon stepwise (^) excitation. This is performed in an electric (^) field to provide selective population of the (^) particular n=29, m= level required for subsequent microwave excitation of the circular states. The electric field is then rapidly reduced to an intermediate value as the atoms pass through the center of a square config- uration of four electrodes. The electrodes are excited by a 2 GHz RF source with a 90 degree phase delay between the adjacent pairs (figure 8). The circularly polarized field drives the atoms into the m=28 circular state through the stepwise absorption of 28 photons. A detector in the circular state production region monitors the efficiency of the optical excitation and angular momentum transfer processes.
After the atoms are prepared (^) in the n=29 circular state, (^) the beam enters the interaction region. Because Rydberg (^) atoms interact strongly with external fields, accurate measurement of the energy level structure requires careful control of the inter- action environment. Thermal radiation is reduced by cooling the interaction region to ~ 10K by a liquid helium flow system. The ambient magnetic field is shielded out by a double-wall high- permeablility shield. A small electric field, which defines the quantization axis of the atoms, is applied with high uniformity by field plates above
Figure 7. Schematic diagram of the atomic beam apparatus.
10 R. Hulet and D. Kleppner, "Rydberg Atoms in 'Circular' (^) States," Phys. Rev. Lett. 51: 1430 (1983).
216 RLE Progress Report Number 136
Figure 9. Production of "circular" state by absorption of circularly polarized 2 GHz radio-frequency radiation. The initial state, which ionizes at relatively low electric field (^) (900 V/cm) appears on the left. The characteristic ionization signal of the circular state appears at roughly (^1200) V/cm.
2.3 Precision Mass Spectroscopy of
Ions
Sponsors
Joint Services (^) Electronics Program Contract DAAL03-92-C- National Science Foundation Grant PHY 89-
Project Staff
Michael P. Bradley, (^) Frank DiFilippo, Natarajan, Fred L. Palmer, Abraham D. Professor David E. Pritchard
Vasant
Stroock,
In 1993, we developed and applied a (^) new tech- nique for precisely comparing the masses of (^) two ions of widely differing atomic weight. Together with previously developed techniques, we (^) com- pleted a program of mass (^) comparisons designed to determine ten atomic masses important for determi- nation of fundamental constants or metrology. (^) This program resulted (^) in a new high accuracy atomic mass table with typical accuracy about 10-10 (an improvement of one to three orders (^) of magnitude over previously (^) accepted values). It also provided numerous (^) quantitative tests of our internal consist- ency that virtually eliminate the possibility of unknown systematic errors. (^) This capability has allowed us to contribute (^) to several important exper- iments (^) in both fundamental and applied physics, including:
* Contributing to a recalibration of the current
x-ray wavelength standard by weighing the
energy differences that go into y-rays with
wavelengths that will be measured at the
National Institute of Standards and Technology;
* Determination of the molar Planck constant
NAh, by weighing y-rays; this will also provide
an alternate determination of the fine structure
constant;
- Determination (^) of the (^) atomic weight of 28Si,
which is part of a program to replace the
"artifact" kilogram mass standard by a crystal
of isotopically pure 28Si, effectively creating an
atomic standard of mass.
In addition, we are laying the groundwork for two
future advances: (1) increasing the versatility of our
apparatus by loading ions externally, and (2)
improving our level of precision another order of
magnitude. The latter goal will be achieved by
comparing two simultaneously trapped ions (to elim-
inate the problems due to field drift) and using
squeezing techniques to reduce the influence of
thermal noise on the measurements. These
advances will allow us to:
* Measure the^3 H+ - 3 He+ mass difference,
important in ongoing experiments to determine
the electron neutrino rest mass;
218 RLE Progress Report Number (^136)
0.
> 0. -J
S 0.
0.
(^0 500 1000 1500) 2000 Electric Field [V/cm]
* Improve some^ traditional^ applications^ of^ mass
spectrometry due^ to^ our^ orders^ of^ magnitude improvement in both accuracy^ and^ sensitivity;
Determine the molar^ Planck constant^ NAh,^ by measuring the^ atomic^ mass^ and^ recoil^ velocity of an atom that has^ absorbed^ photons^ of known wavelength; and
Determine excitation and^ binding^ energies^ of atomic and^ molecular^ ions^ by^ weighing^ the small decrease^ in^ mass,^ Am^ -^ Ebind/C^
2 (we must
reach our ultimate goal^ of^ a^ precision^ of^ a^ few parts in 10-^12 to make^ this^ a^ generally^ useful technique).
In our experimental approach, we^ measure^ ion cyclotron resonance on a single molecular^ or atomic ion in a Penning trap,^ a^ highly^ uniform^ mag- netic field with confinement along^ the^ magnetic^ field lines provided^ by^ much^ weaker^ electric^ fields.^ We monitor the^ axial^ oscillation^ of^ the^ ion^ by^ detecting the currents induced^ in^ the^ trap^ electrodes. Working with only^ a^ single^ ion^ is^ essential because space charge from other^ ions^ leads^ to^ undesired frequency shifts. This^ work^ in^ trapping^ and^ pre- cision resonance draws^ on^ techniques^ developed by Hans Dehmelt at^ the^ University^ of^ Washington and Norman Ramsey at^ Harvard,^ for^ which^ they shared the 1989 Nobel^ Prize.
We have developed techniques^ for^ driving,^ cooling, and measuring the frequencies^ of^ all^ three^ normal modes of Penning trap^ motion.^ Thus,^ we^ can manipulate the ion position reproducibly^ to^ within^30 microns of the center of^ the^ trap,^ correcting^ for electrostatic shifts in the^ cyclotron^ frequency^ to achieve greater accuracy. We^ use^ a^ Tr-pulse method to coherently swap the phase^ and^ action^ of the cyclotron^ and^ axial^ modes.^
(^12) Therefore,
although we detect^ only^ the^ axial^ motion^ directly, we can^ determine^ cyclotron^ frequency^ by^ meas- uring the phase accumulated in^ the^ cyclotron motion in a known time interval.
In the past two years, we^ have^ rebuilt^ our^ appa- ratus with a new Penning trap^ and^ quieter^ rf^ SQUID detector. We have implemented^ a^ new^ signal^ pro- cessing algorithm to^ improve^ our^ phase^ estimation by a factor of two^ to^ three.^ We^ can^ measure^ the phase of the cyclotron motion to^ within^10 degrees, leading to a^ precision^ of^^1 x^ 10-'1^ for^ a^ one minute measurement. Our^ entire^ ion-making^ process^ has been automated, and the^ computer^ can^ cycle^ from an empty trap to having a^ cooled^ single^ ion^ in^ about
three minutes under optimal conditions. With^ the spatial imperfections in the electric^ and^ magnetic fields made as small^ as^ possible^ our^ systematic errors are well^ below^ 5x10-11.^ However,^ the typical statistical fluctuation^ in^ our magnetic^ field^ is 2.4 x 10-10 between measurements (over^ the smooth drift^ in^ the^ field).^ Thus,^ with^ the^ ability^ to achieve -^20 alternate^ loadings^ of^ two^ different^ ion species in^ one night,^ our overall^ uncertainty^ can^ be as small as 8 x 10-11 (see figure 10).^ We^ have found that^ the^ distribution^ of^ field^ variation^ is^ not Gaussian, but^ instead^ has^ additional outlying^ points. This has led us to use^ robust^ statistical^ analysis^ of field fluctuations; in^ particular,^ a^ generalization^ of least squares fitting^ called "M-estimates"^ in^ which outlying points are^ deweighted^ in^ a^ manner^ deter- mined by the observed excess^ number^ of^ outliers. Using this analysis has^ eliminated^ arbitrary^ deci- sions about dropping^ "bad^ points"^ from^ our^ data sets and has increased^ the^ stability^ of^ the fit.
Figure 10. Cyclotron frequency as a function of time for alternate N 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 in the field common^ to^ both^ ions.
We have also^ developed^ a^ new^ measurement tech- nique to extend our precision comparison^ to^ non- doublets (pairs of ions with substantially^ different mass-to-charge ratios),^ allowing^ us to^ perform^ strin- gent checks on systematics^ using^ such^ known ratios as N /N and Ar /Ar++. This^ technique^ repre- sents a significant advance in^ precision^ mass spectrometry since^ it^ allows^ us^ to^ obtain absolute
12 E.A. Cornell, R.M. Weisskoff, K.R.^ Boyce,^ and^ D.E.^ Pritchard,^ "Mode^ Coupling^ in a^ Penning^ Trap:^ Tr^ Pulses^ and^ a^ Classical^ Avoided Crossing," Phys. Rev.^ A^ 41:^^312 (1990).
219
we described last (^) year.^13 If this works well, the primary source of measurement noise (^) will be the special relativistic mass shift due to thermal fluctu- ations (^) in cyclotron amplitude.
We have proposed a scheme of classical squeezing with parametric drives to reduce amplitude fluctu- ations.^14 In 1993, we were successful in demon- strating a simplified version of the quadrature squeezing of the thermal motion of the ion. Figure 11 shows how the rms fluctuations of the cyclotron amplitude vary when we apply a cyclotron pulse of adjustable phase to a previously squeezed distribu- tion. Note that at 8 = 90 degrees, the fluctuations are reduced to approximately one^ half^ of^ their unsqueezed value. 5
140- 120- (^) * MeasuredNoise 100- ....^ Thermal^ Noise^ Level 80- 0- S60- 40 20
Figure 11. Demonstration of classical amplitude squeezing. The observed amplitude noise is plotted as a function of the relative phase. At a drive phase of 90 degrees, the noise is minimized. A best fit to the data shows that a factor of two reduction has been achieved.
13 E.A. (^) Cornell, K.R. Boyce, D.L.K. Fygenson, (^) and troscopy," Phys. Rev. (^) A 45: 3049 (1992).
14 F. DiFilippo, V. (^) Natarajan, K.R. Boyce, and D.E. Lett. 68: (^2859) (1992).
Thesis
Natarajan, V. Penning Trap Mass Spectroscopy at
0.1 ppb. Ph.D. diss., Dept. of Physics, MIT,
2.4 Atom Interferometry
Sponsors
Joint Services Electronics Program
Contract DAAL03-92-C-
U.S. Army - Office of Scientific Research
Grant DAAL03-92-G-
U.S. Navy - Office of Naval Research
Grant N00014-89-J-
Project Staff
Michael S. Chapman, Christopher R. Ekstrom, Troy
D. Hammond, Amrit R. Pant, Richard A.
Rubenstein, Dr. H. Joerg Schmiedmayer, Bridget E.
Tannian, Stefan Wehinger, Professor David E.
Pritchard
During 1993, we refined our atom interferometerl
and started to perform experiments with^ spatially
separated beams. The interferometer is now oper-
ating with smaller period gratings,1 7^ providing
greater beam separation. The experiments were
performed with the aid of an interaction region that
inserts a thin metal foil between the beams (figure
12). This allowed us to manipulate the atomic
wave function in one arm of the interferometer without affecting the other portion.
D.E. Pritchard, "Two Ions in a Penning Trap: Implications for Precision Mass Spec-
Pritchard, "Classical Amplitude Squeezing for Precision Measurements," Phys. Rev.
15 F. DiFilippo, Precise Atomic Masses for Determining Fundamental Constants, Ph.D. (^) diss., Dept. of Physics, MIT, 1994.
16 D.W. Keith, C.R. Ekstrom, Q.A. (^) Turchette, and D.E. Pritchard, "An Interferometer for Atoms," (^) Phys. Rev. Lett. 66: 2693 (1991).
17 D.W. Keith, M. Rooks, C.R. Ekstrom, D.W. Keith, and D.E. (^) Pritchard, "Atom Optics Using Microfabricated Structures," Appl. Phys. (^) B 54: 369 (1992).
221
0 30 60 90 120 Cyclotron Drive Phase [Degrees]
o150 180
Figure 12. A schematic of our atom interferometer (^) showing the active vibration isolation system and region (not to scale). The 10 pm copper foil (^) is shown between the two arms of the interferometer (thick beams). The optical interferometer (thin lines are He-Ne (^) beams) is used for active vibration isolation. period atom gratings are (^) indicated by a vertical dashed (^) line, line.
Approximately six atom interferometers (^) have been demonstrated since (^) our pioneering demonstration of an interferometer (^16) made with three nanofabricated atom gratings.^17 These instruments will make (^) pos- sible new types of experiments involving (^) inertial effects, (^) novel studies of atomic and molecular (^) prop- erties, and tests (^) of fundamental physics. They could also make possible (^) the development of ultra- small structures using (^) atom lithography. Specif- ically:
The relatively large (^) mass and low velocity of atoms makes atom interferometers (^) especially sensitive to inertial (^) effects such as rotation, acceleration, and gravity. Sagnac rotation (^) has been observed in accord with (^) theoretical pre- dictionss and sensitivity to gravitational (^) accel- eration at the 3 x 10-8 level (^) has been demonstrated.' (^) Atom interferometers may
the interaction lines are atom The 200 nm and the 3.3 pm period optical gratings by a vertical solid
become the best absolute accelerometers and
gravimeters within the next few years.
* Atom interferometers can be applied to a
number of experiments in fundamental physics:
tests of quantum (^) mechanics such as the
Aharonov-Casher effect;^20 geometric phases
and the measurement process; measurement
of the equality of proton and electron charges;
and a precise measurement of the momentum
of a photon. This latter measurement should
produce a new high precision value for the
molar Planck constant NAh.
* Interferometers for atoms and molecules will
provide more accurate ways to measure
intrinsic properties of these particles, like their
polarizability. They will also open up new areas
of study, such as measurements of the "index
(^18) F. Riehle, T. Kisters, A. Witte, J. Helmcke, (^) and C.J. Borde, "Optical Ramsey Spectroscopy (^) in a Rotating Frame: Sagnac (^) Effect Matter-Wave (^) Interferometer," Phys. Rev. Lett. 67: 177 (1991).
19 M. Kasevich (^) and S. Chu, "Atomic Interferometry Using (^) Stimulated Raman Transitions," Phys. Rev. Leftt. (^) 67: 181 (1991).
20 Y. Aharonov and (^) A. Casher, "Topological Quantum Effects for Neutral Particles," (^) Phys. Rev. Lett. 53: 319 (1984).
222 RLE Progress Report Number 136
I
Na/Ar source
12.7cm (^1)
2mm gap^ on^ each^ side Detail of interaction region
L(
10 micron copper foil
random, and the (^) ratio of the real to imaginary parts of the forward scattering (^) amplitude is predicted to be about 0.01 for He, (^) not too far from the measured value (0.1). A (^) van der Waals potential should produce (^) a ratio of 0.73, in rough accord with the rare gas atoms (^) with the largest polarizability, Xe(0.7) (^) and Kr(0.6). Understanding the results for Ne(1.0) and Ar(0.5) will require fuller treatment, based on more realistic interatomic potentials.
Recently, we have changed (^) the velocity of the sodium in our interferometer by changing the mass of the carrier gas in our seeded supersonic oven. This has enabled us to measure the dispersion (^) of the index of refraction (i.e., (^) its dependence on deBroglie wavelength) over (^) a factor of two in wave- length.
We have also applied our atom optics techniques to molecules. Molecules were produced by optimizing the operating parameters of our seeded supersonic source so that sodium dimers constituted (^30) precent of the detected beam (^) intensity. The atoms were (^) then removed from the beam by deflecting them with resonant laser light which was frequency stabilized to our atomic (^) beam. Figure 14 shows diffraction (^) of molecules and atoms and diffraction of only molecules (^) from a 100 nm period diffraction grating.
We have used this molecular beam in our sepa- rated beams interferometer, measuring (^) the ratio of real and imaginary parts of the forward (^) scattering amplitude for collisions of these dimers with He and Ne. The results are the same within experimental error as for atomic sodium.
Lastly, we have demonstrated the (^) Talbot effect, the self-imaging of a periodic structure, with atom waves. (^) We have measured the successive recur- rence of these self-images (^) as a function of the dis- tance from the (^) imaged grating. This is a near field interference effect that has several possible applica- tions (^) in the rapidly growing field of atom lithog- raphy.
2.4.1 Publications
Ekstrom, C.R., (^) J. Schmiedmayer, M.S. Chapman, T.D. (^) Hammond, and D.E. Pritchard. "Measure- ment of the Electric Polarability of Sodium (^) with an Atom Interferometer." (^) Submitted to Phys. Rev. Lett.
Ekstrom, (^) C.R., "Atom Optics App. Phys. B
D.W. Keith, and D.E. Pritchard. Using Microfabricated Structures." 54: 369 (1992).
-1000 -500 0 500 detector position (^) in microns
1000
I I
Figure 14. The (^) top graph shows diffraction of sodium atoms and sodium dimers (^) by a 100 nm period diffraction grating. The fit indicates (^) 16.5 precent of the intensity is molecules, which have half the diffraction (^) angle of the sodium atoms. The bottom graph shows the (^) diffraction pattern (^) after the atoms have been removed from the beam (^) with a deflecting laser beam. The fit is the same as (^) in the top graph, indicating that the small orders in the top graph were entirely molecules.
Pritchard, D.E. (^) "Atom Interferometers." Proceedings of the 13th International Conference on Atomic Physics, Munich, Germany, (^) August 3-7, 1992. Eds. T.W. Hansch (^) and H. Walther. Forthcoming.
Pritchard, D.E., C.R. Ekstrom, J. Schmiedmayer, M.S. Chapman, and T.D. (^) Hammond. "Atom Interferometry." (^) Proceedings of the Eleventh International Conference on Laser Spectros- copy, June 13-18, 1993. Eds. (^) L.A. Bloomfield, T.F. Gallagher, and D.J. (^) Larson. New York:
224 RLE Progress Report Number 136
-1000 -500 0 500 1000 detector position in microns
American Institute of Physics, 1993. Forth- coming.
Pritchard, D.E., T.D. Hammond, J.^ Schmiedmayer, C.R. Ekstrom, and M.S. Chapman. "Atom Inter- ferometry." Proceedings of the Conference Quantum Interferometry,^ Trieste,^ Italy,^ March 2-5, 1993. Eds. F. DeMartini, A. Zeilinger, and G. Denardo. Singapore: World Scientific, 1993. Forthcoming.
Schmiedmayer, J., C.R. Ekstrom, M.S. Chapman, T.D. Hammond, and D.E. Pritchard. "Atom Interferometry." Proceedings^ of^ the^ Seminar^ on Fundamentals of^ Quantum^ Optics^ Ill,^ Kuhtai, Austria, 1993. Forthcoming.
Turchette, Q.A., D.E. Pritchard, and^ D.W.^ Keith. "Numerical Model of a Multiple Grating Interfe- rometer." J. Opt. Soc. Am. B^ 9:^1601 (1992).
Thesis
Ekstrom, C.R. Experiments with a Atom Interferometer. Ph.D. Physics, MIT, 1993.
Separated Beam diss., Dept. of
2.5 Cooling and Trapping^ Neutral
Atoms
Sponsors
Joint Services Electronics Program Contract DAAL03-92-C- U.S. Navy - Office of Naval Research Grant N00014-90-J-
Project Staff Michael R. Andrews, (^) Kendall B. Davis, Ilya Entin, Philip M. Hinz, Michael A. Joffe, Marc O. Mewes, Wan Morshidi, J. David Pelly, Stanley H. Thompson, John J. Wuu, Peter S. Yesley, Pro- fessor Wolfgang Ketterle, Professor David E. Pritchard
Our current objective is to obtain samples of atoms at very high density and ultra-low temperatures. This goal is pursued by using a high flux slower for atoms, a light (^) trap to stop and compress the atoms,
and a magnetic trap for the final confinement and cooling.
Experiments with dense (^) samples of cold neutral atoms promise exciting new discoveries (^) in basic and applied physics. Due to the considerably reduced thermal motion of atoms, they are ideal for high resolution spectroscopy and for more accurate atomic frequency standards. In addition,
Collisions of ultra-cold atoms in such samples are characterized by a long deBroglie wave- length and are dominated by weak long-range interactions. Since the collision duration for slow atoms greatly exceeds the radiative decay time, stimulated and spontaneous radiative transitions can take place during the collision. Slow collisions are^ therefore^ fundamentally^ dif- ferent from fast collisions studied so far and are becoming^ an^ exciting^ new^ field^ of^ atomic physics.2 4
High density samples of atoms open possibil- ities for observing quantum collective effects such as Bose-Einstein condensation and col- lectively enhanced or suppressed radiative decay.
In 1993, we completed the work on a dark light trap which enabled us to confine more than 1010 atoms at densities one to two orders of magnitude higher than achieved before. The key feature of the dark trap is that the atoms spend most of their time in a dark hyperfine state which does not scatter the trapping light.^25
The main^ focus^ of^ our^ work^ has been^ to^ obtain^ high densities and^ long^ trapping times^ in^ a^ magnetic^ trap which was loaded from the dark light trap. The product of density and trapping time is the crucial parameter for achieving evaporative cooling.
2.5.1 Magnetic Trapping of Sodium Atoms
We have captured^ up^ to^^1011 atoms^ in^ a^ dark^ light trap and transferred them into a magnetic trap. By counting the number of atoms before and after the transfer, we determined the transfer efficiency to be 25 percent-in coarse agreement with the 33 percent expected for trapping one out of three equally populated F=1 hyperfine states. The transfer was accomplished in^ a^ "mode-matched"
24 Symposium (^) on Cold Atom Collisions, Institute for Theoretical Atomic and Molecular Physics at the Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, April 26-28, 1992, Extended Abstracts.
25 W. Ketterle, K.B. Davis, M.A. Joffe, A. Martin, and (^) D.E. Pritchard. "High Densities of Cold Atoms in a Dark Spontaneous-force Optical Trap." Phys. Rev. Lett. 70: 2253-2256 (1993).
225
them from the trap. 30 The advantage of^ this^ strategy over simply lowering the magnetic trap depth is^ that confinement is^ not^ weakened.^ So^ far,^ we^ have tested the rf^ setup^ and^ shown^ that^ we can^ change the energy distribution^ function^ of^ trapped^ atoms with rf radiation.
2.5.4 Work^ in^ Progress
We currently load^ our^ trap^ using^ an^ inverted "Zeeman slower" that^ has^ produced the^ largest flux of slow atoms achieved^ to^ date.^31 We^ have a new design that may^ improve^ the^ effective flux by one or two orders of^ magnitude^ by^ over- coming effects due to transverse^ heating.^ The key new idea is to pump the^ atoms^ from^ the weak field^ seeking^ hyperfine^ level^ into^ the strong field seeking one during the slowing process.
- Magnetic Trap with^ High^ Field^ Gradients
We have designed^ a^ new^ magnetic^ trap^ which will produce magnetic^ field^ gradients^ up^ to 1000 G/cm, a factor of ten^ improvement^ over our current setup. Such^ high^ field^ gradients will be used for adiabatic compression^ of the trapped atoms. This will result^ in^ an^ increased speed of evaporative cooling.
2.5.5 (^) Publications
Joffe, M.A., W. Ketterle, A. Martin, and D.E. Pritchard. "Transverse Cooling and^ Deflection of an Atomic Beam inside a Zeeman Slower."^ J. Opt. Soc. Am.^ B^ 10:^ 2257-2262^ (1993).
Joffe, M.A., K.B. Davis, W. Ketterle, M.O. Mewes, D.E. Pritchard. "Experiments with Atoms Cap- tured in a Dark Light Trap." Bull. Am. Phys. Soc. 38(3): 1141 (1993).
Ketterle, W., and D.E. Pritchard. Towards Higher Densities of Cold Atoms: Intense Slow Atom Beams and Dark Light Traps. Fundamentals of Quantum Optics^ Ill///, Proceedings^ of^ the^ Fifth
Figure 15.^ Schematic^ setup^ for^ our^ dark^ light^ trap.^ The standard light trap^ consists^ of^ two^ magnetic^ field^ coils and three orthogonal pairs^ of^ counterpropagating^ laser beams. In the dark light^ trap,^ two^ separate^ "repumping" beams are added which^ have^ a^ dark^ central region.^ This creates a small region^ in^ the^ center^ of the trap where^ the atoms are optically pumped into^ a^ dark^ hyperfine state.
Meeting on Laser Phenomena. Lecture Notes^ in
Physics. Ed.^ F.^ Ehlotzky.^ Berlin:^ Springer,^ 1993,
vol. 420,^ pp.^ 77-89.
Ketterle, W., K.B. Davis,^ M.A.^ Joffe,^ A.^ Martin,^ and
D.E. Pritchard. "High^ Densities^ of^ Cold^ Atoms
in a^ Dark^ Spontaneous-force^ Optical^ Trap."
Phys. Rev. Lett.^ 70:^ 2253-2256^ (1993).
Ketterle, W., K.B. Davis, M.A.^ Joffe,^ A.^ Martin,^ and
D.E. Pritchard. "Dark Spontaneous-force
Optical Trap." OSA^ Ann. Meet.^ Tech.^ Dig.^ 16:
Ketterle, W., K.B.^ Davis,^ M.A.^ Joffe,^ A.^ Martin,^ and
D.E. Pritchard. "Kalte^ Natrium-Atome^ hoher
Dichte in einer^ 'dunklen'^ Lichtdruckfalle."^ Verh.
DPG (VI) 28: 417 (1993).
30 D.E. Pritchard, K. Helmerson, and A.G. Martin, "Atom Traps," Proceedings of the 1 th International Conference on Atomic Physics in Atomic Physics 11, eds. S. Haroche, J.C. Gay, and G. Grynberg (Singapore:^ World^ Scientific,^ 1989),^ pp.^ 179-197. 31 M.A. Joffe, W. Ketterle, A. Martin, and D.E. Pritchard, "Transverse Cooling and Deflection of an Atomic Beam inside a Zeeman Slower," J. Opt. Soc. Am. B 10: 2257-2262 (1993).
227
and
Theses
Joffe, M.A. Trapping (^) and Cooling Atoms at High Densities. Ph.D. (^) diss., Dept. of Physics, (^) MIT,
Pelly, J.D. Magnetostatic Traps for Sodium Atoms. B.S. thesis, (^) Dept. of Physics, MIT, 1993.
2.5.6 Meeting Paper
Ketterle, W., K.B. Davis, M.A. Joffe, A. Martin, and
D.E. Pritchard. "Dark Spontaneous-force
Optical Trap for Sodium Atoms." Quantum
Electronics and Laser Science Conference
(QELS), Baltimore, Maryland, 1993, Technical
Digest Series, Volume 12, p. 222.
228 RLE Progress Report Number 136
