We use precision microwave spectroscopy of magnetically trapped, ultra-cold
87Rb to characterize intra- and inter-state density correlations. The cold
collision shifts for both normal and condensed clouds are measured. The results
verify the presence of the sometimes controversial "factors of two", in
normal-cloud mean-field energies, both within a particular state and between
two distinct spin species. One might expect that as two spin species decohere,
the inter-state factor of two would revert to unity, but the associated
frequency chirp one naively expects from such a trend is not observed in our
data.Comment: Proceedings of the 18th International Conference on Atomic Physics
  (ICAP 2002

Cornell, E. A.

Harber, D. M.

Lewandowski, H. J.

McGuirk, J. M.

English

arXiv

Crossref

The Mystery of the Ramsey Fringe that Didn’t Chirp

We use precision microwave spectroscopy of magnetically trapped, ultra-cold ⁸⁷Rb to characterize intra- and inter-state density correlations. The cold collision shifts for both normal and condensed clouds are measured. The results verify the presence of the sometimes controversial "factors of two", in normal-cloud mean-field energies, both within a particular state and between two distinct spin species. One might expect that as two spin species decohere, the inter-state factor of two would revert to unity, but the associated frequency chirp one naively expects from such a trend is not observed in our data

Cornell, Eric A.

CU Scholar Institutional Repository

arXiv:cond-mat/0209249v1  [cond-mat.soft]  10 Sep 2002The Mystery of the Ramsey Fringethat Didn’t ChirpD. M. Harber, H. J. Lewandowski, J. M. McGuirk[*], and E. A. Cornell[*]JILA, National Institute of Standards and Technology and Department of Physics,University of Colorado, Boulder, Colorado 80309-0440AbstractWe use precision microwave spectroscopy of magnetically trapped, ultra-cold 87Rbto characterize intra- and inter-state density correlations. The cold collision shiftsfor both normal and condensed clouds are measured. The results verify the presenceof the sometimes controversial “factors of two”, in normal-cloud mean-field energies,both within a particular state and between two distinct spin species. One mightexpect that as two spin species decohere, the inter-state factor of two would revert tounity, but the associated frequency chirp one naively expects from such a trend is notobserved in our data.1 IntroductionWhen one studies the statistics of arrival times of photons emitted from an incoherentsource, one finds that immediately after detecting a photon, one is twice as likelyto detect a second photon than one would expect from the time-averaged detectionrate. This “photon bunching” effect can be understood as arising naturally from thequantum statistics of a noisy bosonic field. If one detects a photon, one can infer thatthe randomly fluctuating boson field is near a peak in its amplitude. Small wonderthen that one is likely to detect another photon soon. Similar effects are seen inthe correlations in the arrival times of bosonic atoms falling from a cold cloud [1].Within an atomic cloud itself, these statistical effects are best thought of as densityfluctuations. Just as the short-time peak in photon arrival statistics is suppressedin a coherent (laser) beam of photons, the density fluctuations in a cloud of bosonicatoms is suppressed if the atoms are Bose-condensed. This effect has been seen in theanalysis of expansion energy of condensates [2, 3] and in the comparison of three-bodyrecombination rates in condensates versus thermal clouds [4].In a recent series of experiments we examined the effects of these density fluctuationson the hyperfine transition frequency in ultra-cold normal and in Bose-condensedrubidium [5]. The MIT hydrogen group performed early work in this area [6]. In thisshorter conference proceedings, we review our spectroscopic study [5] with particularemphasis on the effects of decoherence on the density correlations between two distincthyperfine states. A cloud of thermal atoms begins initially in a single hyperfine state.A microwave pulse coherently splits the cloud into two distinct hyperfine states witha well-defined relative phase. Ramsey spectroscopy, sensitive to the density effects,shows that the two states initially have the same factor of two in their inter-statedensity correlations as each does in its intra-state density correlations. What we findcounter-intuitive is that as time evolves and the two spin species begin to decohere, wesee no corresponding shift in the frequency of the Ramsey fringes. Thus, with apologiesto Arthur Conan Doyle, we came up with the title of the present manuscript.2 Hyperfine SpectroscopySpatial inhomogeneity of the transition frequency was minimized through the use ofa pair of energy levels which experience the same trapping potential. At a magneticfield of ∼ 3.23 G the |1〉 ≡ |F = 1, mf = −1〉 and |2〉 ≡ |F = 2, mf = 1〉 hyperfinelevels of the 5S1/2 ground state of87Rb experience the same first-order Zeeman shift.For a normal cloud at 500 nK, each energy level is Zeeman shifted by ∼10 kHz acrossthe extent of the cloud, however at 3.23 G the differential shift of the two levels acrossthe cloud is ∼1 Hz. Compared to the differential Zeeman shift, the energy shift due tocold collisions is then a relatively large effect at high densities, making measurementsof collisional shifts in this system possible. The small inhomogeneity allows for longcoherence times, ∼2 seconds and longer for low-density clouds, making this systemattractive for precision measurements as well as for the study of coherence in finitetemperature systems.The experimental setup has been previously described [7] and will be briefly summa-rized here. Approximately 109 87Rb atoms are loaded into a vapor cell magneto-opticaltrap (MOT). The atoms are then optically pumped into the |F = 1〉 state by turningoff the repump beam while the MOT beams remain on. Then the trapping beamsare turned off and the MOT coils are ramped to a high current, forming a 250 G/cmgradient to trap |1,−1〉 atoms in the quadrupole field of the coils. The quadrupolecoils are mounted on a linear servo-motor controlled track which then moves the coils44 cm, from the MOT region to a Ioffe-Pritchard trap in the ultra-high vacuum re-gion of the system. The Ioffe-Pritchard trap consists of two permanent magnets whichprovide a 450 G/cm radial gradient. Two pairs of electromagnetic coils, a pinch and abias, provide confinement in the axial direction, which is aligned perpendicular withrespect to gravity. At a typical bias field of 3.23 G atoms in the |1,−1〉 state experi-ence {230, 230, 7} Hz trap frequencies. The sample is further cooled by rf evaporation,and condensates of up to 106 atoms can be formed. Imaging is performed by the useof adiabatic rapid passage to transfer atoms from the |1,−1〉 state to the |2,−2〉 state.Anti-trapped |2,−2〉 atoms rapidly expand for 2-5 ms and then are imaged throughabsorption by a 20 µs pulse of 5S1/2 |2,−2〉 → 5P3/2 |3,−3〉 light.A two-photon microwave-rf transition is used to transfer atoms between the |1〉 and|2〉 states. A detuning of 0.7 MHz from the |2, 0〉 intermediate state provides a two-photon Rabi frequency of ∼ 2.5 kHz. Ramsey spectroscopy of the |1〉 → |2〉 transitionis performed by measuring the total number of atoms remaining in state |1〉 after apair of π2pulses separated by a variable time delay is applied [8]. The frequency of theresulting Ramsey fringes is the difference between the transition frequency ν12 andthe two-photon drive frequency. In previous work we measured local variations of ν12by detecting the number of atoms remaining in state |1〉 at specific spatial locationsalong the axis of the normal cloud [7]. By analyzing the spatio-temporal variationsof ν12, combined with the measured evolution of the |1〉 state after a singleπ2pulse,we were able to spatially resolve the evolution of spin waves [9]. In this work, inorder to perform measurements of ν12 insensitive to spin waves, one of the followingtwo techniques was used. With one technique the entire cloud, rather than specificspatial locations, was monitored to average out the effects of spin waves. AlternativelyRamsey spectroscopy was restricted to interrogation times short compared to the spinwave frequency [10].One effect which shifts the transition frequency ν12 is the differential Zeeman shift.The Breit-Rabi formula predicts a minimum in ν12 at B0 = 3.228917(3) Gauss, thusthe |1〉 and |2〉 energy levels experience an identical Zeeman shift at B = B0. Thedifferential Zeeman shift about B0 can be approximated as ν12 = νmin + β(B − B0)2[11]. Measuring ν12 for different magnetic fields allows us to calibrate our magneticfield from the expected dependence. By working at the vicinity of B = B0 we greatlyreduce spatial inhomogeneity of ν12 and also become first-order insensitive to temporalmagnetic field fluctuations.3 Density ShiftsA second effect which shifts ν12 arises from atom-atom interactions. In the s-waveregime, where the thermal de Broglie wavelength of the atoms is greater than theirscattering length, atoms experience an energy shift equal to α 4πh̄2man, where α isthe two-particle correlation at zero separation, n is atom number density, a is thescattering length, and m is the atom mass. Therefore for a two-component samplethe expected energy shift of each state isδµ1 =4πh̄2m(α11a11n1 + α12a12n2) (1)δµ2 =4πh̄2m(α12a12n1 + α22a22n2), (2)where n1 and n2 are the |1〉 and |2〉 state densities andαij =〈Ψ†iΨ†jΨiΨj〉〈Ψ†iΨi〉〈Ψ†jΨj〉. (3)The shift of the transition frequency in Hz can then be written as∆ν12 = (δµ2 − δµ1)/h=2h̄m(α12a12n1 + α22a22n2 − α11a11n1 − α12a12n2)=h̄mn(α22a22 − α11a11 +(2α12a12 − α11a11 − α22a22)f) (4)wheref =n1 − n2n(5)and n = n1 + n2.For non-condensed, indistinguishable bosons, α = 2 due to exchange symmetry,therefore αnc11 = αnc22 = 2 in a cold normal cloud (where the superscript c or ncrefers to condensed of non-condensed atoms respectively). Distinguishable particlesdo not maintain exchange symmetry, making αnc12 = 1 for a incoherent two-componentmixture. However if a two-component sample is prepared by coherently transferringatoms from a single component, such as in Ramsey spectroscopy, then the excitationprocess maintains exchange symmetry, and we might expect αnc12 = 2 [12]. In thisscenario the collisional shift should be calculated using αnc11 = αnc22 = αnc12 = 2, leadingto a predicted frequency shift of0 2 4 6 8 10 12-25-20-15-10-50  Frequency (Hz)Density (1013 cm-3)Figure 1: Measurement of the cold collision shift. Solid and open circles repre-sent measurements of the normal cloud and condensate respectively. The solidline is a fit to the normal cloud data ∆ν12 = 0.1(0.4) − 3.9(0.3)10−13n; thedashed line is a fit to the condensate data ∆ν12 = −0.1(1.4) − 1.9(0.2)10−13nwhere ∆ν12 is in Hz and n is in cm−3.∆ν12 =2h̄mn(a22 − a11 + (2a12 − a11 − a22)f). (6)This result can also be obtained by solving the transport equation [13, 14]. Fromspectroscopic studies [15] the three 87Rb scattering lengths of interest have been de-termined to be a22 = 95.47a0, a12 = 98.09a0, and a11 = 100.44a0, where a0 is theBohr radius. The frequency shift can then be written as∆ν12 =2h̄ma0n(−4.97 + 0.27f). (7)If on the other hand the |1〉 and |2〉 states do not maintain exchange symmetry, suchthat αnc12 = 1, then the frequency shift would instead be∆ν12 =2h̄ma0n(−4.97 − 97.82f). (8)These two models are clearly distinguished by the dependence of ν12 on f .When we perform Ramsey spectroscopy with a pair of π2pulses, the populations ofthe |1〉 and |2〉 states are equal, and thus f = 0 during the interrogation time. FromEq. (4) it is apparent that with f = 0 the collisional shift is sensitive only to αncii andaii terms. For these measurements the bias field was set to B0, and the transitionfrequency was measured for a range of densities. To adjust density of the sample,the number of atoms in the initial MOT load was varied. All normal cloud data was-1.0 -0.5 0.0 0.5 1.0-10-505  Frequency (Hz)Figure 2: Testing the exchange symmetry between the |1〉 and |2〉 states. Thetransition frequency is measured as f is varied for a normal cloud at fixed peakdensity of 7 × 1012 cm−3 and temperature of 510 nK. The solid line is thefit, which yields αnc12/αnc11,22= 1.01(2), which is to say, inter- and intra-statedensity correlations are quite accurately the same. The dotted line indicatesthe expected slope for αnc12/αnc11,22= 1/2.taken at the same temperature of 480 nK, and all condensate data was taken with highcondensate fractions in order to minimize effects due to the normal cloud. The densityfor the normal cloud was found by fitting Gaussian profiles to absorption images of theclouds and extracting the number, temperature, and density. To measure condensatedensity Thomas-Fermi profiles were fit to absorption images of the condensates andthe total number, N0, in the condensates and the Thomas-Fermi radius along the longaxis, Z, were extracted.The results of this measurement are shown in Fig. 1. Comparing the collisional shiftmeasured for the normal cloud to that measured for a condensate gives αncii /αcii =2.1(2). If instead we assume αncii = 2 and αcii = 1, then the data for both thecondensate and normal cloud can be used to obtain a value for the difference inscattering lengths of a22 − a11 = −4.92(28)a0, in agreement with values determinedfrom molecular spectroscopy [15].4 Inter-state density correlationsExchange symmetry between the |1〉 and |2〉 states can be tested by working at a fixeddensity and varying the relative |1〉 to |2〉 population by varying the length of the firstRamsey pulse [16]. In this case the first term in Eq. (4) will be constant and themeasurement will test αnc12 and a12 as well as the αncii and aii terms. To minimize sys-0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0|1ñ  Number Remaining in       (arb)Time (s)Figure 3: A data set of Ramsey fringes probing for frequency shifts as a functionof coherence. For this measurement normal clouds at a temperature of 480 nKand a peak density of 3.2 × 1013 cm−3 were used.tematics the interrogation times were kept short, making precise frequency determina-tion difficult. Nevertheless, our measurement (Fig. 2) indicates αnc12/αnc11,22 = 1.01(2),where we have used the spectroscopically determined scattering lengths. This clearlyindicates that exchange symmetry is maintained between the |1〉 and |2〉 states.5 Where’s the chirp?As a thought experiment, imagine distinct thermal populations of |1〉 and |2〉 atoms,separately prepared, then mixed together, with the energy of interaction (proportionalto αnc12 ) measured for instance calorimetrically. Surely in this case the density fluctua-tions in state |1〉 and in state |2〉 would be uncorrelated, and αnc12 would be determinedto be 1, not 2. We lack the experimental sensitivity to make such a calorimetric mea-surement, and our Ramsey-fringe method of measuring energy differences obviouslywould not work for incoherent mixtures. We speculated, however, that if αnc12 = 2for coherent superpositions, and if αnc12 = 1 for incoherent mixtures, then for partiallydecohered samples, αnc12 would take on some intermediate value. So by performing ameasurement similar to that in Fig. 2 we might expect to see a more negative slope fora partially decohered sample; alternatively a frequency chirp in the Ramsey fringesmay be seen as the sample decoheres.We probed the time evolution of αnc12 in a way similar to Fig. 2; however rather thanvarying f we set f ≃ 0.8 then measured ν12 with long interrogation times, looking fora frequency chirp as the fringe contrast decreased. This method has the advantagethat there is a relatively small |2〉 state population, so effects arising from |2〉 loss areminimized. Seven data sets were taken for this measurement; an example is shownin Fig. 3. By allowing a linear frequency chirp in the fit of the Ramsey fringes, thefrequency shift can be constrained to −0.2(3) Hz by the time the fringe contrast hasreduced to 1/e [17]. However if we hypothesize that αnc12 goes from 2 to 1 linearly asfringe contrast goes from 100% to 0% we would expect a frequency shift of −20(2) Hzas the fringe decayed, while the experimental limit is a factor of 40 smaller. Clearlythis appealing but unrigorous model is far too naive.6 ConclusionWhere’s the chirp? In truth we don’t know. Our theorist friends tell us that ourconfusion arises from our assuming that the frequency shift ∆ν12 arise from the dif-ference in chemical potentials, µ2 −µ1. Instead, they say, we should directly evaluatethe Boltzmann equations for the spin in an inhomogeneous system [13, 14]. This wehave not as yet done, but even if this solves the mystery in a formal sense, we arereluctant to give up the cherished notion that the frequency of the transverse spinprecession is a direct measurement of the energy difference between spin up and spindown. For those of us raised in the traditions of atomic physics, it is a pleasure to notethat a two-level system can still yield surprises, 75 years after the advent of quantummechanics.We acknowledge useful conversations with the other members of the JILA BECcollaboration. This work was supported by grants from the NSF and NIST.References[*] Quantum Physics Division, National Institute of Standards and Technology.[1] M. Yasuda and F. Shimizu, Phys. Rev. Lett. 77, 3090 (1996).[2] M. J. Holland, D. S. Jin, M. L. Chiofalo, and J. Cooper, Phys. Rev. Lett. 78, 3801(1997).[3] W. Ketterle and H.-J. Miesner, Phys. Rev. A 56, 3291 (1997).[4] E. A. Burt, R. W. Ghrist, C. J. Myatt, M. J. Holland, E. A. Cornell, and C. E.Wieman, Phys. Rev. Lett. 79, 337 (1997).[5] D. M. Harber, H. J. Lewandowski, J. M. McGuirk and E. A. Cornell, cond-mat/0208294 (2002).[6] T. C. Killian, D. G. Fried, L. Willmann, D. Landhuis, S. C. Moss, T. J. Greytak,and D. Kleppner, Phys. Rev. Lett. 81, 3807 (1998); D. G. Fried, T. C. Killian, L.Willmann, D. Landhuis, S. C. Moss, D. Kleppner, and T. J. Greytak, Phys. Rev.Lett. 81, 3811 (1998).[7] H. J. Lewandowski, D. M. Harber, D. L. Whitaker, and E. A. Cornell, Phys. Rev.Lett. 88, 070403 (2002).[8] N. F. Ramsey, Molecular Beams (Clarendon Press, Oxford, 1956).[9] J. M. McGuirk, H. J. Lewandowski, D. M. Harber, and E. A. Cornell, Phys. Rev.Lett. 89, 090402 (2002).[10] At n = 1013cm −3 νspin wave = 5 Hz, so interrogation times of 20 ms areacceptable.[11] Where β = 431.35957(9) Hz/G2 and νmin = 6834678113.59(2) Hz [7].[12] M. Ö. Oktel, T. C. Killian, D. Kleppner and L. S. Levitov, Phys. Rev. A 65,033617 (2002).[13] M. Ö. Oktel and L. S. Levitov, Phys. Rev. A 65, 063604 (2002).[14] J. E. Williams, T. Nikuni and C. W. Clark, Phys. Rev. Lett. 88, 230405 (2002).[15] E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J. Heinzen, and B. J.Verhaar, Phys. Rev. Lett. 88, 093201 (2002).[16] When we changed the length of the first Ramsey pulse, the second pulse lengthwas also changed such that the two pulses combined to form a π pulse.[17] The quoted chirp limit of -0.2(3) Hz includes a correction to account for a smallfrequency change associated with decay of total atom number during the mea-surement time. For each data set the number decay was found by allowing anexponential decay in the fit of the Ramsey fringes (not to be confused with thedecay in Ramsey fringe contrast associated with decoherence), resulting in a cor-rection amounting to -0.74 Hz on average.[18] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. Heinzen, F. L.Moore, M. G. Raizen, and D. J. Wineland, Phys. Rev. A 47, 3554 (1993).

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The Mystery of the Ramsey Fringe that Didn't Chirp

The Mystery of the Ramsey Fringe that Didn't Chirp

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