Collective Excitation Interferometry with a Toroidal Bose-Einstein Condensate
CCollective Excitation Interferometry with a Toroidal Bose-Einstein Condensate
G. Edward Marti, ∗ Ryan Olf, † and Dan M. Stamper-Kurn
1, 2 Department of Physics, University of California, Berkeley, California 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Dated: July 10, 2018)The precision of compact inertial sensing schemes using trapped- and guided-atom interferometershas been limited by uncontrolled phase errors caused by trapping potentials and interactions. Here,we propose an acoustic interferometer that uses sound waves in a toroidal Bose-Einstein condensateto measure rotation, and we demonstrate experimentally several key aspects of this type of interfer-ometer. We use spatially patterned light beams to excite counter-propagating sound waves withinthe condensate and use in situ absorption imaging to characterize their evolution. We present ananalysis technique by which we extract separately the oscillation frequencies of the standing-waveacoustic modes, the frequency splitting caused by static imperfections in the trapping potential, andthe characteristic precession of the standing-wave pattern due to rotation. Supported by analyticand numerical calculations, we interpret the noise in our measurements, which is dominated byatom shot noise, in terms of rotation noise. While the noise of our acoustic interferometric sensor,at the level of ∼ rad s − / √ Hz, is high owing to rapid acoustic damping and the small radius of thetrap, the proof-of-concept device does operate at 10 − times higher density and in a volume10 times smaller than free-falling atom interferometers. PACS numbers: 37.25.+k,03.75.-b,03.75.Kk,03.75.DgKeywords: matter wave interferometry, collective excitations, ring trap, toroidal trap, gyroscopy, Bose-Einstein condensates
Conventional atom interferometers measure accelera-tion [1] and rotation [2–4] by interfering dilute atomicwavepackets that traverse distinct paths in free fall [5].The impressive sensitivity of these devices scales withthe area enclosed by the arms of the interferometer, fa-voring larger interferometers that average measurementson centimeter length scales [6]. Extending the capabil-ity of atom interferometers to probe shorter length scalescould address fundamental questions, such as how grav-ity operates at short range; tackle practical problems,such as non-invasive material characterization; and aidin the development of miniaturized atomic sensors [7].Trapped- or guided-atom interferometers may allow sen-sitive, localized inertial measurements by allowing inter-ferometer arms to enclose the same area multiple times,gaining precision while remaining compact. However, foran atom interferometer to reach high signal-to-noise andprobe short length scales, it is critical to develop an in-terferometric scheme compatible with high densities andrealistic trapping potentials.Trapped quantum degenerate gases offer a brightsource for atom interferometry in small volumes, reach-ing number densities of 10 cm − that are at least fourorders of magnitude higher than those utilized in free-falling-atom devices [6]. However, the price of high den-sity is uncontrolled phase shifts and damped atomic mo-tion [8–11]. Also, in spite of high densities, the numberflux of ultracold atoms through a trapped-atom interfer-ometer is typically low, making it highly desirable that ∗ [email protected] † [email protected] the readout noise of such interferometers reach, or evensurpass [12–15], the atom-shot-noise limit.Here, we propose and demonstrate a new type of in-terferometer that circumvents many challenges of high-density atom interferometry: interfering collective exci-tations of a dense, trapped sample to measure force orrotation. In this proof-of-principle work, we interferephonons, our chosen collective excitation, in a toroidalBose-Einstein condensate (BEC) and extract a signalthat is sensitive to rotations. Our scheme is similarto those of hemispherical resonator [16] and superfluid-helium gyroscopes [17]. In analogy to an optical gyro-scope, phonons play the role of light, traveling throughthe vacuum mode of the BEC. The effects of trap in-homogeneity and of atomic interactions are amelioratedin two ways. First, atomic interactions themselves areused to suppress the effect of trap inhomogeneity onsound propagation. We demonstrate this fact experi-mentally by showing that significant trap inhomogeneityleads only to weak coupling between counter propagatingsound modes. Second, we develop an analysis techniquethat allows us to isolate a rotation-sensitive signal fromthe dynamical evolution of sound waves in the toroidalBEC in a manner that is largely independent of the effectsof interactions and of trap inhomogeneity. This analysistechnique is applied to experimental data to quantify thenoise in a rotation-rate measurement, and to data gen-erated by numerical simulations to quantify the rotationsensitivity.We demonstrate several key advantages of collective-excitation interferometry by constructing a high-density(1 × cm − ) but small (16 µ m radius) sample. Ascollective modes of an interacting Bose-Einstein conden-sate, sound waves can, in principle, propagate over long a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un distances. Interactions in a superfluid can enhance thelifetime of the sound mode even in the presence of dis-order [18] and suppress systematic biases that arise fromweak disorder in the potential. Time reversal symme-try guarantees that linear forces, atom number varia-tions, interaction energy shifts, and static trap inhomo-geneities cannot distinguish counter-propagating acous-tic modes; these effects primarily introduce common-mode phase shifts that do not deteriorate the signal.Further, the irrotational nature of the superfluid pro-vides a non-rotating frame for the propagating soundwaves, against which the slow rotation of an observer canbe measured absolutely. Our compact device presentlyachieves a rotation sensitivity of only ∼ rad s − / √ Hz,far inferior to available sensors. Extending our schemeto circular waveguides of millimeter dimensions [19] andreducing damping to gain longer propagation times wouldbe necessary to improve sensitivity.Low-order collective modes of BEC’s have been usedto measure Casimir-Polder forces [20] and quantized cir-culation [21–23]. We extend this work by using higher-order standing-wave acoustic modes to increase sensitiv-ity, overcome technical noise limitations, and reach atom-shot-noise-limited detection. Since we lack the sensitiv-ity to measure rotation directly, the goal of this paperis to validate our proposal by matching the noise of therotation signal to the expected atom shot noise and byidentifying systematic biases. The most critical bias weinvestigate results from azimuthal perturbations in thetrap potential that add frequency shifts to the standing-wave eigenmodes, which could appear as rotation signal ifnot properly accounted for. We characterize and correctthe rotational signal for these effects.We begin in Sec. I by describing our experimentalsystem for producing toroidal-shaped Bose-Einstein con-densed gases of Rb, our optical method for excitingstanding-wave acoustic modes of various angular ordersand initial angular positions, and our measurement ofthe evolution of these modes through in-situ absorptionimaging. We characterise the frequency and spatial pat-tern of several acoustic collective modes. We also observethese modes to be damped more rapidly than expectedbased on Landau damping. In Sec. II, we show how theseparate effects of static trap inhomogeneities and of ro-tation can be isolated in our data analysis. The resultingrotation-rate measurements, based on the precession ofstanding-wave modes of different orders, are presented.Because we lack sensitivity to measure rotation rates thatwe can reasonably apply to our experiment, we quan-tify the variance in these measurements as the noise ina rotation-rate measurement. For rotation sensing usinghigher-order acoustic modes, we find this measurementnoise to be consistent with that expected for atom-shot-noise-limited measurements. Finally, in Section III weexamine the limits of our scheme in the zero-temperature,mean-field limit by numerically simulating a representa-tion of our system in the presence of rotation. In concert,this experimental and numerical work highlights many of (c) m Ω tt R e A m ][I m A m ][ gg (a)(b) FIG. 1. A toroidal optical dipole trap for a Rb Bose-Einstein condensed gas is formed by intersecting three lightfields, as indicated by the side (a) and top (b) views of thetrap, with the direction of gravitational acceleration indi-cated. A standing acoustic wave oscillates initially at itseigenfrequency (c) with fast density oscillation at the antin-ode (Re[ A m ], blue arrow and curve) and no amplitude at thenode (Im[ A m ], red arrow and curve). Rotation induces a sig-nal in Im[ A m ], whose envelope (gray) increases in proportionto the rotation rate and mode number. Density shifts shouldnot alter this envelope. the advantages of collective-excitation interferometry andcan serve to guide future work in this area. I. CREATION AND EVOLUTION OF SOUNDWAVES
In our experiment, we prepare a degenerate, spin-polarized gas of Rb atoms in a far-detuned opticaldipole trap, following a procedure similar to that ofRef. 24. We then transfer the atoms into an overlaintoroidal optical potential formed by the intersection ofthree light beams (Fig. 1(a), (b)). An attractive lightsheet, with an optical power of 12 mW, a wavelengthof λ = 836 nm, and an elliptical focus with 1 /e radiiof 10 . µ m and 400 µ m, confines atoms to a horizontalplane. An annular potential, created by coaxial attrac-tive (400 µ W, λ =830 nm, 1 /e radius of 26 µ m) andrepulsive (3 mW, λ =532 nm, 1 /e radius of 11 µ m) ver-tically propagating light beams provides in-plane confine-ment. To maximize the stability of the optical system,we deliver the coaxial vertical beams via the same large-mode-area fiber. We adjust the alignment and beam radiiat the trap location using a telescope with adjustable ax-ial and lateral chromatic shifts. The optical potentialminimum lies in a circle of radius 16 µ m, about whichthe atoms experience radial and vertical trap frequen-cies of ( ω z , ω r ) = 2 π × (260 ,
86) Hz. Evaporative coolingfrom the optical potential yields samples with 8 × condensed atoms with a 20% thermal fraction, Thomas-Fermi radii of 6 µ m and 1.5 µ m in the radial and ver-tical directions, respectively, and a chemical potential of µ = h ×
700 Hz. Similar optical ring traps have beendemonstrated previously, albeit with different optical se-tups [25–27].The excitations of a cylindrically symmetric mediummay be characterized by the integer azimuthal quantumnumber m , as well as the transverse (radial and axial)quantum numbers. We excite a specific superposition ofthe ± m lowest-transverse-order sound modes by apply-ing an additional optical potential to the trapped BECthat establishes an initial density modulation with highspatial overlap with the selected mode. To create thispotential, we illuminate a chrome optical mask of an m -fold propeller pattern (Fig. 3(e), icons) with a 400 µ W, λ =532 nm light beam. The masked light is then imagedonto the plane of the trapped atoms, using a 1:10 imagingsystem with a resolution of 6 µ m, imposing a repulsive op-tical potential while the atoms are evaporatively cooledto the final atom number and temperature. The light isthen suddenly extinguished, allowing the perturbed con-densate to evolve freely.After a variable hold time t , we measure the in situ condensate density with a high signal-to-noise absorptionimaging protocol. We first reduce the optical density ofthe gas by exciting only 10% −
25% of the atoms to the | F =2 (cid:105) ground hyperfine state with a short (20 µ s), weak(2% of saturation intensity) light pulse detuned by about5 linewidths from the | F =1 (cid:105) → | F (cid:48) =2 (cid:105) D2 transition [28].Detuning the light from resonance makes the sample op-tically thin and ensures that it is uniformly excited. Wethen apply a resonant probe pulse to the cycling tran-sition for 50 µ s at saturation intensity [29] and imagethe unscattered light onto a CCD camera. We foundthis approach to give greater sensitivity than dispersiveimaging.The acoustic excitations reveal themselves as oscillat-ing standing waves of the condensate density. From themeasured column density distribution ˜ n ( x, y ), we extractthe azimuthal spatial Fourier coefficient A m = (cid:82) r 6, shown as the diagonal grayline. A large negative shift of mode m =7 is expected fromcoupling to the first radial mode (dashed black line). Theinset shows the quality factor Q for each mode. dent of mode number for modes 2-7 (inset of Fig. 3(e)).We note that, for our experimental conditions, free-particle (high momentum) excitations would be dampedwithin a distance of around d = ( nσ ) − = 14 µ m, shorterthan that traversed by the phonon excitations studied inthis work; here, n = 10 cm − is the peak condensatedensity and σ = 7 × − cm is the scattering crosssection.The measured damping rates are larger than those ex-pected from Landau damping, which is a form of thermalde-excitation. The invariance of the excitation qualityfactor with mode number is predicted for Landau damp-ing in a homogeneous gas at temperatures greater thanthe chemical potential [33, 34]. However, our system’sanisotropic potential and near-equality of the thermal en-ergy and chemical potential ( k B T ≈ . µ ) place it outsidethe regime in which Landau damping has been studiedand is understood. Indeed, the damping rate we observeis a factor of 3 − π ∆ f m / m (a)(b) O cc u r e n ce s −5 0 5 Rotation rate (rad/s)Mode σ / σ A S N (c) Ω A m FIG. 4. (a) The frequency splitting is determined by fitting,on average, 500 images and 18 orientations of the optical maskper mode to a three-mode dynamical model. Error bars aredetermined from a jackknifing procedure that excludes a sin-gle orientation from the dataset. The units here, in rad/sper mode number, can be compared directly to the measuredrotation rate. (b) Rotation estimates are binned in 1 rad/sintervals. Each value is the result of a fit to the rotation ratefor ∼ 30 points taken at each orientation of the optical mask.(c) The standard error σ of the rotation estimates (closed cir-cles) and noise in each component of A m (open squares) areclose to the atom-shot-noise limit σ ASN for modes m =4 − m =2 and 3 show an excess of noise, likely from tech-nical fluctuations in the toroidal potential. produced inadvertently in the annulus of our ring con-tribute to the observed damping, and that reducing theirnumber would enhance the lifetime of our acoustic oscil-lations. II. PROOF-OF-CONCEPT ROTATION NOISE A rotation of the lab will create a precession in the ori-entation of the standing acoustic mode (see Sec. III). In asmooth ring, the standing wave orientation remains fixedin the inertial frame and precesses in the rotating labframe as the clockwise ( − m ) and counterclockwise (+ m )propagating modes appear to be split in frequency by therotation. The equations of motion of the standing waveare identical to those a Foucault pendulum [36], wherethe two orthogonal standing waves (real and imaginarycomponents of A m ) are identified with the in-plane ( x and y ) components of the pendulum’s position [35]. Thissituation differs from phonons in a rigid object, where astanding wave precesses at a rate dependent on the ma-terial shape and slower than the lab rotation rate [37].In contrast, static trap distortions couple, rather thansplit, the ± m modes. Owing to such distortions, the de-generacy between the sine- and cosine-like superpositionsof the ± m modes is broken, and the ± m modes them-selves are no longer the eigenmodes of the system. Forinstance, in a ring of radius r , a static potential perturba-tion with azimuthal dependence V ( φ ) = Re[ (cid:80) m V m e imφ ]yields (to first order in V m ) a frequency splitting betweenthe two m -node standing acoustic modes of∆ f m ∝ ( | V m | /h )( ξ/λ ) , (2)where ξ ∝ n − / is the healing length [34], n is the atomicdensity, and λ is the acoustic wavelength. The effects ofsuch distortions are thus suppressed by the high den-sity of the atomic medium [35]; for our system we ob-serve a suppression factor of (∆ f m /m ) / ( V m /h ) (cid:39) / V m = 2 µ | A static2 m | .While the acoustic modes are thereby affected both byrotation and by static trap distortions, the separate ef-fects of each of these influences are distinct, akin to theseparate effects of circular and linear birefringence, re-spectively, on the polarization of light. To quantify eacheffect, we excite each mode m =2 through 6 at a largenumber of angles (a small selection is shown in Fig. 2(a)),and fit the data at each m to a three-mode model (un-perturbed superfluid and its ± m acoustic excitations) ofthe temporal evolution described by the Hamiltonian H m = (cid:126) ω ( a † x a x + a † y a y + 1)+ (cid:126) π ∆ f m ( a † x a x − a † y a y )+ i (cid:126) m Ω( a † x a y − a † y a x ) , (3)where a x and a y are annihilation operators for the twonon-rotating, sine- and cosine-like standing wave eigen-modes. From this fit, we obtain the eigenmode principalaxes selected by the static trap distortion (Fig. 4, top)and the eigenmode frequencies ω x/y = ω ± π ∆ f m , as wellas the rotation rate Ω. The simplified three-mode de-scription of our system is consistent with the dynamicspredicted by numerically integrating the Gross-Pitaevskiiequation for our system parameters in the limit of lowamplitude excitations of a single m -mode, as demon-strated in Section III.To extract a rotation rate, we fit the data at eachmask orientation ( ∼ 30 points, corresponding to a rowin Fig. 2(a)) to a prediction of the sound wave evolutionin a rotating frame. The extracted rates are plotted asa histogram in Fig. 4(b). The only free parameters arethe rotation rate and the amplitude of the initial exci-tation. Mechanical properties of the system such as thefrequency, frequency splitting, and phase are fixed fromfits to the rest of the data set.The measured rotation noise is at the level of 1 rad/s.Considering that these measurements are obtained in around 30 repetitions of the experiment, each of whichhas an average measurement time of tens of millisec-onds, this rotation noise can also be expressed as ap-proximately 1 rad s − / √ Hz where we do not account forthe low duty cycle of the experiment. This measure-ment noise matches well with the expected atom-shot-noise limit (Fig. 4, bottom). In imaging the distributionof N uncorrelated atoms, atom-shot-noise yields an un-certainty ∆ A m = (2 N ) − / in extracting either the realor imaginary component of A m in a single run. The atomshot noise limits the rotation signal to an uncertainty of∆Ω ASN = α Γ mA √ N N r , (4)where Γ is the decay rate of the standing wave, A isthe fractional density modulation excited in operatingthe interferometer (measured by the Fourier amplitude A m defined in Eq. 1), and N r is the number of inde-pendent experimental realizations. The prefactor α de-pends on how measurements are distributed in time; here, α = 3 . / Γ [35]. For measurements using the higher-order sound modes, the reduced technical noise in ourimages at higher spatial frequencies allows us to achievethe atomic shot-noise limit (Fig. 4(c)). Unfortunately,because the quality factor of acoustic standing-wave os-cillations is found to be roughly independent of the modenumber, the use of higher order modes does not improvesensitivity in our apparatus beyond overcoming technicalnoise.The fundamental sensitivity due to atom shot noise inour device, based on interfering phonons, takes a verysimilar form to a conventional free-space atom interfer-ometer, based on interfering atoms. In the latter case,the noise can be written as∆Ω free = T − π ( L/λ ) √ N , (5)where L is the distance between interaction regions, λ is the wavelength of the optical or material grating usedas an atomic beam splitter, and T is the atomic traveltime between beam splitters [3]. Written this way, ouracoustic interferometer acts as a device that measureshow far a feature of azimuthal size λ/L ∼ /m rotatesover a time T ∼ / Γ. III. MEAN-FIELD NUMERICS The short lifetime of acoustic modes in our appara-tus makes evaluating the limits and efficacy of the three-mode model of Eq. 3, and our sensing scheme in gen-eral, difficult. Thus, we explore the feasibility of ourscheme in the absence of anomalous damping by simu-lating it numerically. These results validate the essentialpoints of our proposal, namely that interferometers basedon collective excitations within gaseous superfluids sup-press the impacts of common sources of systematic bias Time (ms) R e [ A ] 20 40 60 80 100 120−0.01 0 0.01 0.02 Time (ms) 20 40 60 80 100 120 01020ASN FIG. 5. A is extracted from snapshots of the simulated condensate column density both with (right) and without (left) theaddition of simulated atom shot noise. For each trap configuration, 300 samples over 793 ms at each of 5 angles of the excitationpropeller are analyzed using the three-mode model. Oscillation of the m =3 mode is exaggerated by a factor of 4 in the includedimages, for clarity. and, being irrotational, provide an absolute non-rotatingframe of reference.We modeled our system using the Gross-Pitaevskii(GP) equation, which we discretized via a sym-metric split-step Fourier method [38, 39] on a N x × N y × N z =64 × × ∼ . µ m and local temporal ac-curacy to O (∆ t ), with ∆ t the contemporaneous stepsize. Error thresholds were chosen such that accumu-lated errors throughout a typical simulated experimentwere negligible.Each simulated experiment began by finding theground state of 10 Rb atoms in a non-rotating poten-tial, including an m -fold repulsive propeller and a sim-ulated 3-beam optical dipole trap, possibly with staticinhomogeneity, via evolution in imaginary time. Fromthis starting point, we excited acoustic oscillations bysuddenly removing the m -fold propeller and propagatingthe system in real time, rotating the simulated lab ata rate Ω . We implemented the rotation of the lab byrotating the trapping potentials, including any static in-homogeneity, with respect to the non-rotating frame ofthe simulation.We chose a non-rotating ground state as our start-ing point because of its compatibility with the split-stepFourier method. For small rotation rates, the differ-ence between rotating and non-rotating ground statesis small. This difference manifests itself in our simula-tions as an additional acoustic oscillation that we wouldnot expect to be present in our experimental apparatus.These acoustic oscillations can be observed on their ownby running simulated experiments in which the propellerpattern is not turned off immediately. Owing to its ir-rotational nature, the superfluid is not dragged by therotating propeller pattern.To produce data analogous to the images from our ex-perimental apparatus, we saved snapshots of the sim-ulated condensate column density, integrated along theaxial dimension, at various times. To understand andverify our assumptions about the role of atom shot noise, A ]Re[ A ] Small ∆ f Large ∆ f FIG. 6. Sample simulated data (no atom shot noise) with aninitial excitation along an m =3 principal axis at two differentvalues of frequency splitting ∆ f . The rotation rate in bothdatasets is Ω =0 . V ) and 6-fold ( V ) perturbationsof h × 11 Hz (small ∆ f ) and h × 59 Hz (large ∆ f ). we derived noisy measurements of the condensate columndensity from these snapshots by adding Gaussian noiseat a level consistent with the atom shot noise limit forimages including 2 × atoms. We extracted A m fromboth the noisy and noiseless snapshots in the rotatinglab frame (Fig. 5), mirroring the experimental procedureand analysis presented in Section II.In our simulations, we chose to focus on excitations ofan m =3 propeller in a trapping potential with parame-ters similar to those of our experimental apparatus, butwith additional variable 5-fold and 6-fold angular pertur-bations of the trapping potential. The simulated lightsheet, modeled numerically as if formed by a focusedGaussian light beam, naturally includes a moderate 2-fold inhomogeneity.The sample data shown in Fig. 6 demonstrate im-portant qualitative features of the sensing scheme andthe three-mode model (Eq. 3). For small ∆ f (small 6- −10 −3 −10 −4 −10 −5 −5 −4 −3 −2 −1 Ω (b) −5 0 5 Ω − Ω × −3 π ∆ f / (a) Actual Rotation Ω (Hz) σ / σ A S N −5 −4 −3 −2 −1 Ω A With shot noise FIG. 7. (a) Frequency splitting, (b) rotation rate Ω, and(inset) Ω − Ω residuals of the m =3 mode extracted from sim-ulated data with (open circles) and without (closed circles)atom shot noise. Error bars apply to the open circle datapoints. The filled gray area delineates the uncertainty in pa-rameters extracted from the noiseless data and represents thelimits of the three-mode description of the full mean-field dy-namics in the simulated trap and with the simulated exci-tation, both of which were set to match real experimentalparameters. The three-mode model distinguishes between ro-tation and frequency splitting, and can extract both from thesame data stream. The measured 2 π ∆ f / . 13 is close tothe value 0.10 expected based on the static trap perturbation V (Eq. 2). (c) The standard deviation of Ω and A m extractedfrom noisy snapshots is consistent with the expected fit-freeatom shot noise limit (Eq. 4). fold perturbation in trapping potential, V ), the rotationΩ =0 . . When ∆ f is large (large V ), the exci-tation remains pinned along its initial axis. The offsetin Re[ A ] in the data for large ∆ f is due to the larger5-fold trap perturbation, V , which gives a small static3-fold perturbation when combined with the 2-fold per-turbation of the simulated light sheet (one of the threeoptical trapping beams, as described in Sec. I).Our analysis scheme based on the three-mode model,applied to the results of our full numerical simulation, isable to extract the mechanical parameters of the oscil-lator, including the mode frequency splitting, as well asthe rotation rate. Figure 7 shows the results of fittingthe three-mode model to 300 snapshots over 793 ms ateach of 5 angles of a 3-fold propeller. All model parame- ters were fit simultaneously, with error bars determinedby a delete-one jackknife procedure [40]. Errors obtainedvia this procedure were consistent with errors obtaineddirectly from fits.The extracted rotation rates are well-calibrated for alarge range of rotation values. As expected from the irro-tational flow of the superfluid, small rotation rates can bemeasured absolutely, without bias drift or pinning. Forlarge rotation rates, the extracted values are consistentwith the actual rotation rate beyond 0.1 Hz. At 1 Hz, themeasured rotation rate slips to 0.998 Hz. At this rotationrate, the true ground-state of the rotating system has ahigh probability of entraining a vortex, but our simula-tions begin with a condensate in the non-rotating groundstate; the additional acoustic oscillations caused by thelarge deviation of our simulated system from the trueground-state at this high rate of rotation are, evidently,no longer negligible and they likely account for our fail-ure to accurately measure rotations in this limit. Whilephysical systems based on this implementation would notsuffer the same limitation, they would likely need to ac-count for the probability of entraining a quantized vortexin the model used to extract rotation to be accurate athigh rotation rates.Uncertainty in the values of parameters of the three-mode model persist, even in the absence of noise, as in-dicated by the filled gray areas in Fig. 7. This residualuncertainty indicates the degree to which the three-modemodel accurately describes the temporal evolution of theacoustic excitation in the mean-field limit. Our simula-tions have shown that the three-mode model works bestfor low-amplitude excitations wherein the excitation pro-peller excites only the lowest order of the desired m -foldperturbation. IV. CONCLUSION For a trapped atom sample to reach sensitivities com-petitive to free-space interferometers, short wavelengths(higher m ), large atom numbers, long propagation times,and very selective excitation are required. In our currentsetup, both atom number and m are limited by the smallsize of our ring. However, rings have been demonstratedwith ≈ greater enclosed area [19]. Further, collectivemodes of BECs have been observed with temperature-limited quality factors ten times greater than those re-ported here [41, 42]. Fundamental zero-temperaturedamping of these modes, Beliaev damping, should allowfor subhertz damping rates for wavelengths a few timesgreater than the healing length [43].More importantly, we emphasize that collective exci-tation interferometry has applications beyond rotationsensing with phonons. Other collective excitations andgeometries could be employed in a diverse range of sen-sors. For example, magnons in a ferromagnetic spinorBEC are predicted to show free-particle behavior andcould form the basis of a compact, short-range interfero-metric magnetic sensor.We thank A. ¨Ottl and M. Solarz for help with de-signing and building the cold atom setup and G. Dunnand S. Lourette for technical assistance. This work wassupported by the Defense Advanced Research Project Agency (Grant No. 49467-PHDRP) and the DefenseThreat Reduction Agency (Contract No. HDTRA1-09-1-0020). GEM acknowledges support from the Hertz Foun-dation. [1] J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden,and M. A. Kasevich, Phys. Rev. A , 033608 (2002).[2] T. L. Gustavson, P. Bouyer, and M. A. Kasevich, Phys.Rev. Lett. , 2046 (1997).[3] T. L. Gustavson, A. Landragin, and M. A. Kasevich,Classical and Quantum Gravity , 2385 (2000).[4] D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, Phys.Rev. Lett. , 240801 (2006).[5] A. D. Cronin, J. Schmiedmayer, and D. E. Pritchard,Rev. Mod. Phys. , 1051 (2009).[6] H. M¨uller, S. W. Chiow, S. Herrmann, S. Chu, and K.-Y.Chung, Phys. Rev. Lett. , 031101 (2008).[7] Daniel M. Farkas, Kai M. Hudek, Evan A. Salim, StephenR. Segal, Matthew B. Squires, and Dana Z. Anderson,Appl. Phys. Lett. , 093102 (2010).[8] Y.-J. Wang, D. Z. Anderson, V. M. Bright, E. A. Cornell,Q. Diot, T. Kishimoto, M. Prentiss, R. A. Saravanan,S. R. Segal, and S. Wu, Phys. Rev. Lett. , 090405(2005).[9] O. Garcia, B. Deissler, K. J. Hughes, J. M. Reeves, andC. A. Sackett, Phys. Rev. A , 031601 (2006).[10] Y. Shin, M. Saba, T. A. Pasquini, W. Ketterle, D. E.Pritchard, and A. E. Leanhardt, Phys. Rev. Lett. ,050405 (2004).[11] G.-B. Jo, Y. Shin, S. Will, T. A. Pasquini, M. Saba,W. Ketterle, D. E. Pritchard, M. Vengalattore, andM. Prentiss, Phys. Rev. Lett. , 030407 (2007).[12] J. Esteve, C. Gross, A.Weller, S.Giovanazzi, and M. K.Oberthaler, Nature (London) , 1216 (2008).[13] J. Appel, P. J. Windpassinger, D. Oblak, U. B. Hoff,N. Kjrgaard, and E. S. Polzik, Proc. Natl. Acad. Sci. , 10960 (2009).[14] I. D. Leroux, M. H. Schleier-Smith, and V. Vuleti´c, Phys.Rev. Lett. , 073602 (2010).[15] M. H. Schleier-Smith, I. D. Leroux, and V. Vuleti´c, Phys.Rev. Lett. , 073604 (2010).[16] D. M. Rozelle, in Spaceflight Mechanics , Advances inthe Astronautical Sciences, Vol. 134, edited by A. M.Segerman, P. C. Lai, M. P. Wilkins, and M. E. Pit-telkau, American Astronautical Society (Univelt, Inc.,2009) Proceedings of the 19th AAS/AIAA Space FlightMechanics Meeting.[17] R. E. Packard and S. Vitale, Phys. Rev. B , 3540(1992).[18] D. Dries, S. E. Pollack, J. M. Hitchcock, and R. G. Hulet,Phys. Rev. A , 033603 (2010).[19] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, andD. M. Stamper-Kurn, Phys. Rev. Lett. , 143201 (2005).[20] D. M. Harber, J. M. Obrecht, J. M. McGuirk, and E. A.Cornell, Phys. Rev. A , 033610 (2005).[21] A. E. Leanhardt, A. G¨orlitz, A. P. Chikkatur, D. Kielpin-ski, Y. Shin, D. E. Pritchard, and W. Ketterle, Phys.Rev. Lett. , 190403 (2002).[22] M. Cozzini, S. Stringari, V. Bretin, P. Rosenbusch, and J. Dalibard, Phys. Rev. A , 021602 (2003).[23] V. Bretin, P. Rosenbusch, F. Chevy, G. V. Shlyapnikov,and J. Dalibard, Phys. Rev. Lett. , 100403 (2003).[24] Y.-J. Lin, A. R. Perry, R. L. Compton, I. B. Spielman,and J. V. Porto, Phys. Rev. A , 063631 (2009).[25] K. Henderson, C. Ryu, C. MacCormick, and M. G.Boshier, New J. of Phys. , 043030 (2009).[26] A. Ramanathan, K. C. Wright, S. R. Muniz, M. Ze-lan, W. T. Hill, C. J. Lobb, K. Helmerson, W. D.Phillips, and G. K. Campbell, Phys. Rev. Lett. ,130401 (2011).[27] S. Moulder, S. Beattie, R. P. Smith, N. Tammuz, andZ. Hadzibabic, Phys. Rev. A , 013629 (2012).[28] A. Ramanathan, S. R. Muniz, K. C. Wright, R. P. Ander-son, W. D. Phillips, K. Helmerson, and G. K. Campbell,Rev. Sci. Instrum. , 083119 (2012).[29] G. Reinaudi, T. Lahaye, Z. Wang, and D. Gu´ery Odelin,Opt. Lett. , 3143 (2007).[30] The center of the polar coordinates is determined by fit-ting each image to an ideal toroidal Thomas-Fermi witha dipole angular perturbation.[31] R. Dubessy, T. Liennard, P. Pedri, and H. Perrin, Phys.Rev. A , 011602 (2012).[32] The error given for the speed of sound is the 1 σ statisticalerror and does not include the systematic error on theimaging system calibration.[33] P. O. Fedichev, G. V. Shlyapnikov, and J. T. M. Wal-raven, Phys. Rev. Lett. , 2269 (1998).[34] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. , 463 (1999).[35] See Supplemental Material for a description of the equa-tions of motion in our three-mode model, a derivationof the fractional splitting between sine- and cosine-likestanding wave modes caused by static trap inhomo-geneities, a derivation of the atomic shot-noise limit forour rotation sensor, and a discussion of thermal andzero-temperature damping of phonon modes in a Bose-Einstein condensate.[36] L. Foucault, Comptes rendus hebdomadaires des seancesde l’Acadmie des Sciences , 135 (1851).[37] G. Bryan, Proc. of Cambridge Phil. Soc. 1890 , 101(1890).[38] M. Feit, J. Fleck, and A. Steiger, Journal of Computa-tional Physics , 412 (1982).[39] T. R. Taha and M. I. Ablowitz, Journal of ComputationalPhysics , 203 (1984).[40] R. G. Miller, Biometrika , 1 (1974).[41] D. S. Jin, M. R. Matthews, J. R. Ensher, C. E. Wieman,and E. A. Cornell, Phys. Rev. Lett. , 764 (1997).[42] D. M. Stamper-Kurn, H.-J. Miesner, S. Inouye, M. R.Andrews, and W. Ketterle, Phys. Rev. Lett. , 500(1998).[43] N. Katz, J. Steinhauer, R. Ozeri, and N. Davidson, Phys.Rev. Lett. , 220401 (2002). ollective Excitation Interferometry with a Toroidal Bose-Einstein Condensate:Supplemental Material G. Edward Marti, ∗ Ryan Olf, † and Dan M. Stamper-Kurn 1, 2 Department of Physics, University of California, Berkeley, California 94720, USA Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA S1. EQUATION OF MOTION OF A m The motion of the sound waves with mode number m is identical to that of an anisotropic two-dimensionaloscillator (of mass M ), with natural frequencies of ω ± δ ,that is rotating at the rate m Ω, where Ω is the lab-framerotation rate for the sound-wave system. The classicalHamiltonian for this oscillator system is H = p x M + p y M − m Ω( xp y − yp x )+ M ω + δ ) x + M ω − δ ) y . We can define (unperturbed) annihilation operators a x and a y for the two non-rotating eigenmodes to findthe quantum Hamiltonian of Eq. (3) in the main text.The classical result can be recovered by considering athree state system, | , i , | , i and | , i , written in the | n x , n y i basis. The Hamiltonian can then be representedby a matrix, H ⇔ ~ ω + δ − i Ω 0 i Ω ω − δ 00 0 0 , and the two position operators are, x ∝ y ∝ . It is sufficient to solve for the response of the systemwhen kicked along an eigenaxis. Linearity will let uscombine solutions to account for an arbitrary kick. Wedefine A σν as the response along the ν axis to a systeminitially oscillating along the σ axis. Hence, if the pen-dulum starts along x , then we find: A xx = cos( ωt ) cos(Ω t ) − δ Ω sin( ωt ) sin(Ω t ) A xy = ΩΩ cos( ωt ) sin(Ω t ) , (S1) ∗ [email protected] † [email protected] where Ω = √ Ω + δ . For an oscillator kicked along the y direction, A yx = − ΩΩ cos( ωt ) sin(Ω t ) A yy = cos( ωt ) cos(Ω t ) + δ Ω sin( ωt ) sin(Ω t ) . We emphasize that these results, as well as the equa-tions of motion calculated for Bogoliubov modes, areidentical to the classical ones. S2. DERIVATION OF MEAN-FIELDSPLITTING Consider a one-dimensional circular waveguide. Foruniform radial perturbations, we estimate the poten-tial as a function of radius r and azimuthal angle φ as V ( r, φ ) = V ( r ) + Re[ P m V m e imφ ]. The effective (chan-nel) speed of sound is c eff = s µ − V ( φ ) ζ m Rb ≈ c (cid:18) − V ( φ )2 µ (cid:19) , (S2)where µ is the chemical potential, m Rb is the atomicmass, and ζ is a constant that depends on the form of V ( r ) (1 for a square potential, 2 for a harmonic potential,and ∼ . m modes have the perturbed frequencies ω m = ω m, (cid:18) ± | V m | µ (cid:19) , (S3)with the unperturbed frequency ω m, = mc eff /r . Wereport the frequency f m = ω m / π and the frequencysplitting ∆ f m = δ/π . In Fig. 3(c) of the main text, thetheory line can be rewritten as∆ f m m = V m f m /m µ = V m h ξr √ ζ where ξ = ~ / √ m Rb µ is the healing length [1]. S3. DERIVATION OF ATOM-SHOT-NOISELIMIT We first derive the atom-shot-noise limit in measuring A m , and then in the rotation rate, Ω. The two quadra-tures of A m have equal noise, so without loss of generality,we calculate only the noise in the real quadrature. Foreach pixel i we detect n i atoms with an atom-shot-noisevariance of ¯ n i . For a large number of atoms ( N (cid:29) n i ),Re[ A m ] = P i n i cos mφ i P i n i var Re[ A m ] ≈ N X i var ( n i cos mφ i )= 1 N X i ¯ n i cos mφ i ≈ N . To estimate ∆Ω ASN , we make the short-time, smallrotation rate approximation, Ω t (cid:28) δt (cid:28) 1, forwhich a kick along the real axis leads to a response alongthe imaginary axis. The relevant signal is the imaginarycomponent of A m ,Im[ A m ] = A e − Γ t cos ωt sin Ω mt. In each iteration of the experiment, the atom-shot-noise uncertainty in Ω is∆Ω( t ) = std Im[ A m ] | ∂ Im[ A m ] /∂ Ω | = 1 / √ NAmt e − Γ t | cos ωt | . We choose to uniformly sample our data on an interval t ∈ [0 , b/ Γ], where b is between 2 and 3. For a largenumber of samples, the linear least-squares estimate of the error is∆Ω ASN = "X i ∆Ω( t i ) − − / = " Γ b Z b/ Γ0 dt N A m t e − t cos ωt − / = Γ Am √ N s b R b u e − u d u ≈ (3.2 to 3.6) Γ Am √ N . S4. THERMAL AND ZERO-TEMPERATURESCATTERING Landau damping, collisions with thermally excited col-lective modes, is expected to be the main form of damp-ing of sound in our system. A collective excitation in ahomogeneous Bose-Einstein condensate at chemical po-tential µ and temperature T has a quality factor Q inde-pendent of mode number and inversely proportional totemperature, Q = ω 2Γ = 43 π ~ ck B T a , where ω is the (real) mode frequency, Γ is the damp-ing rate (in radians per second), c is the speed of soundand a is the s-wave scattering length. This approximateassumes ~ ω (cid:28) µ (cid:28) k B T , which does not hold in our sys-tem. Regardless, we find this description qualitativelyagrees with our data but predicts a quality factor 3 − µ ≈ k B T , have been considered theoretically [2], and thepredicted damping rates under such conditions area sim-ilarly smaller than what we observe experimentally.At zero temperature, scattering of collective modesshould be dominated by Beliaev scattering, a four-wavemixing process (see Ref. 3 for calculations relevant toBose-Einstein condensates). This form of damping ischaracterized by a very strong dependence of the damp-ing rate on the wavenumber, Γ ∼ k . For wavelengthsas short as one micron, much shorter than used in thiswork, the damping rate should be below 1 s − and farbelow what is experimentally observed. [1] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari,Rev. Mod. Phys. , 463 (1999).[2] P. O. Fedichev, G. V. Shlyapnikov, and J. T. M. Wal-raven, Phys. Rev. Lett. , 2269 (1998).[3] N. Katz, J. Steinhauer, R. Ozeri, and N. Davidson, Phys.Rev. Lett.89