Ultracold atom-electron interaction: from two to many-body physics
Anita Gaj, Alexander T. Krupp, Jonathan B. Balewski, Robert Löw, Sebastian Hofferberth, Tilman Pfau
aa r X i v : . [ phy s i c s . a t o m - ph ] A p r Ultracold atom-electron interaction: from two to many-body physics
A. Gaj, ∗ A. T. Krupp, J. B. Balewski, R. L¨ow, S. Hofferberth, and T. Pfau †
5. Physikalisches Institut, Universit¨at Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany (Dated: 15 March 2014)
The transition from a few-body system to a many-bodysystem can result in new length scales, novel collectivephenomena or even in a phase transition. Such a thresh-old behavior was shown for example in He droplets,where He turns into a superfluid for a specific number ofparticles [1]. A particularly interesting question in thiscontext is at which point a few-body theory can be sub-stituted by a mean field model, i. e. where the discretenumber of particles can be treated as a continuous quan-tity. Such a transition from two non-interacting fermionicparticles to a Fermi sea was demonstrated recently [2]. Inthis letter, we study a similar crossover to a many-bodyregime based on ultralong-range Rydberg molecules [3]forming a model system with binary interactions. Thisclass of exotic molecules shows very weak binding ener-gies, similar to magneto-associated Feshbach molecules[4, 5], and thus requires ultracold temperatures. A widerange of fascinating phenomena, starting from the co-herent creation and breaking of chemical bonds [6] toa permanent electric dipole moment in a homonuclearmolecule [7], has been studied. Dimers, consisting of asingle atom in the Rydberg state and one atom in theground state, have been observed in an ultracold gas ofRb in the Rydberg S -state [8], D -state [9, 10] and P -state[11], and of Cs [12] in the Rydberg S -state. The many-body regime, where the Rydberg electron experiences amean shift by thousands of atoms within its orbit, hasbeen studied at very high densities in a BEC, leading toelectron-phonon coupling [13], and in a hot vapor [14].Here, we trace the transition between the two regimes inan ultracold cloud with a constant density by tuning onlyone parameter: the principal quantum number n of theexcited Rydberg state. We probe the border of the meanfield limit, where the energy and length scales of thesemolecules become extreme. While the binding energiesare the smallest ever observed for this type of molecules,the size reaches the dimensions of macroscopic biologicalobjects like viruses or bacteria.The bond in ultralong-range Rydberg molecules resultsfrom the scattering of a slow Rydberg electron from aneutral atom with a negative scattering length a [3]. Thetheoretical approach is based on Fermi’s original conceptof the pseudopotential [15] V pseudo ( r , R ) = 2 π ~ am e δ ( r − R ) (1)describing such a scattering event between an electron ofmass m e at position r and an atom at R . If the scatter-ing length a is negative, the interaction is attractive and ground state atoms can be bound in a potential V ( R ) = 2 π ~ am e | Ψ( R ) | , (2)where | Ψ( R ) | is the local electron density. In case of Rb the theoretical value of the scattering length a fortriplet scattering is − a [16], where a is the Bohrradius. The singlet scattering length is positive andtherefore does not lead to a bound state. Momentum-dependent corrections to these values can be estimatedusing a semiclassical approximation [17]. Additionally, a p -wave shape resonance can cause a substantial modifica-tion of the molecular potential, leading to butterfly-typemolecules [18] and states bound by internal quantum re-flection [19]. These corrections are largely negligible forhigh principal quantum numbers and large distances fromthe Rydberg core, where the relative motion of the Ryd-berg electron and the perturber is slow. The ground statewavefunction of the Rydberg molecules with the highestbinding energy per atom, is localized mainly at the posi-tion of the outermost lobe of the electron wavefunction,i. e. near the classical turning point of the electron (see n =51 n =62 n =71 n =80 R E / h B Rb + a FIG. 1:
Scattering potentials calculated from equa-tion (2) for n =51, 62, 71 and 80. Dimers are predomi-nantly bound in the vibrational ground state in the outermostwell (light blue). For n =80 a single molecular state localizedin the outermost lobe cannot be resolved in the experiment,because the mean binding energy per atom becomes too low(light blue dashed line) and thus a change from a few-bodyto a many-body description is required. The spheres belowillustrate the highest order poly-mer observed for each princi-pal quantum number. Orange dashed line indicates the meanshift for the experimental density. Fig. 1).Molecules containing more than one ground state atomcan be described with the same formalism. Two groundstate atoms inside the electron orbit essentially do notinteract with each other, since the Rb atom-atom scat-tering length [20] is much smaller than the mean particledistance in a dilute thermal cloud. Hence, the bindingenergy of an i -atomic molecule, i ∈ N , is ( i −
1) timeslarger than the binding energy of a dimer. Only whenthe number of ground state atoms inside the Rydbergatom becomes very large, a change in the description ofthe system from discrete bound states to a mean fieldapproximation is required (Fig. 1). In this case no indi-vidual bound states are resolved, but the Rydberg line isshifted by∆ E = Z Z d r d R V pseudo ( r , R ) | Ψ( r ) | ρ ( R )= Z d R V ( R ) ρ ( R ) = 2 π ~ am e ¯ ρ. (3)If higher order corrections to the zero-energy scatteringlength can be neglected, the mean shift depends only onthe value of a and the density ¯ ρ averaged over the volumeof the Rydberg atom.We perform the experiment in a magnetically trappedultracold cloud of Rb atoms in the 5 S / , F=2, m F =2ground state with typical temperatures of 2 µ K and den-sities on the order of 10 cm − . Detailed informationabout the setup can be found in [21]. After the prepara-tion of the ultracold cloud we excite the atoms in a twophoton process via the 6 P / state to the nS / Ryd-berg state. For our high precision spectroscopy measure-ments we use narrow bandwidth lasers ( ≤
30 kHz), whichare locked to a high finesse ULE reference cavity. The420 nm light, driving the lower transition, is blue detunedfrom the intermediate state by 80 MHz to avoid absorp-tion and heating of the cloud. It is sent to the exper-iment in 50 µ s pulses with a repetition rate of 167 Hz.During the sequence the 1016 nm laser light driving theupper transition is on constantly. After the excitation wefield-ionize the Rydberg atoms and collect the ions on amicrochannel plate detector. In a single atomic cloudwe perform typically 400 cycles of Rydberg excitationand detection while scanning the frequency of the bluelaser light. In order to realize high spectral resolutionwe choose long excitation pulses of 50 µ s. Taking fur-ther into account the laser bandwidth, Doppler broaden-ing and natural linewidth this results in an experimentalresolution of around 60 kHz. In order to obtain the bestvisibility while changing the principal quantum numberof the excited Rydberg state, we adjust the power of theblue laser to account for power broadening. Only thespectrum of n =51 was taken with higher laser power andthus in this case the atomic line is slightly broadened. -2 -1.5 -1 -0.5 05162718090100111 Relative frequency (MHz) -1 -0.5 0 0123Relative frequency (MHz)Fit71S data-1.5 R y de r g s i gna l ( a r b . u . )
Overview of the S / to nS / excitationspectra showing the dimer - poly-mer transition forincreasing principal quantum number. The origin ofthe relative frequency axis corresponds to the center of theatomic line (dashed line) for n ≤
71, where molecular lines aredistinguishable. The gray shaded area between −
30 kHz and30 kHz indicates the laser bandwidth. Spectra at n >
71 arehorizontally shifted such that their centers of gravity overlapwith the mean center of gravity h cg i (orange dashed line) ofthe first three spectra ( n =51, 62, 71). All data was taken atsimilar cloud parameters, therefore the density induced shiftfor all spectra is constant to first approximation. Moleculeswith up to three bound ground state atoms for 62 S and up tofour for 71 S are resolvable in the spectra. Colored diamondsindicate the positions of the dimers (red), trimers (violet) etc.following the power law scaling of the binding energies fittedto the first three spectra. In the inset the molecular spectrumfor the n =71 Rydberg state is shown. A multilorentzian fit(green line), assuming a constant spacing between the molec-ular peaks is plotted to indicate the positions of the higherorder molecular lines. The spectrum for n =40 is not shown,because the binding energy of the dimer is larger than theplotting range. Each spectrum is an average over 20 indepen-dent measurements with standard deviation error bars. In Fig. 2 excitation spectra from the 5 S / to the nS / state, where n is ranging from 51 to 111, are presented.The shape of the obtained Rydberg spectra varies signif-icantly for different n . For low principal quantum num-bers clearly distinguishable molecular lines are present onthe red side of the atomic peak, which is situated at theorigin. In the spectrum of n =51 the peak at − − n ∗ with the effective principal quantum num-ber n ∗ = n − δ , where δ is the quantum defect. Hence,higher order molecules are formed more likely at higher n .At the same time the binding energy per atom decreasesand thus polyatomic molecular lines become visible in thespectra of n =62 and 71. The binding energy E B can bedirectly measured as the difference between the atomicand the molecular line in the spectrum. At n =62 lines upto the tetramer together with corresponding vibrationalexcited states are visible. The broadening of the tetramerline may be caused by the presence of these excited states,possibly with a reduced lifetime [22]. At n =71 only vibra-tional ground states are resolved. Polyatomic moleculesup to a pentamer can be identified. The size of such amolecule becomes enormous due to the Rydberg electronorbit radius reaching almost 10 000 a . For large n thebinding energy E B decreases until it is below the exper-imental resolution. This manifests in a non-resolvableshoulder and finally in an inhomogeneously broadenedspectral line.The experimental binding energies can be calculatedbased on the molecular potential (2) and the mean shift(3). We use corrections to the s -wave scattering lengthincluding terms linear in the relative momentum of twoscattering partners based on a semiclassical approxima-tion [17]. This approximation is valid for large distancesfrom the ionic core. Therefore we restrict the analysisonly to the lowest bound state. The discussion of highervibrational states can be found in [19]. We solve theSchr¨odinger equation for the ground state atom in themolecular potential using Numerov’s method and fit thezero-energy scattering length a to the binding energiesof the dimers. In this paper the best agreement withthe experimental data is obtained for − a , which isvery close to the theoretically predicted value of − a [16].For n >
71, where no distinct molecular lines can beidentified, a mean field description is required. Further-more, the spectral position of pure Rydberg atoms, andthus the zero position, cannot be identified directly fromthe signal. However it can be determined from the cen-ter of gravity cg of the spectra, taken in a non-blockadedsample, which is constant for a given density. Intu-itively, this result in the first approximation can be ex-
40 50 60 70 80 90 100 110-100-10-1-0.1-0.01 Principal quantum number n Power law fitExperiment B i nd i ng ene r g y M H z () m a n b o d y y d i m e r t r i m e r t e t r a m e r Diameter ( m) m mean shift FIG. 3:
Measured binding energies (red points) ver-sus principal quantum number (bottom axis) and di-ameters (top axis) of the molecules.
The data for n ≤ n =40, 43, 51 the frequency range chosenin the experiment was too small to photoassociate moleculeswith larger binding energies than a dimer. The power law n ∗ b (blue lines) fitted to the measured data and multiplied byfactors i − i ∈ N , ( i =2 for a dimer) shows that for n >
75 thebinding energy of the dimer becomes smaller than the exper-imental resolution. The increasing number of ground stateatoms inside the electron orbit leads eventually to a meanshift of the Rydberg line. Calculated values of E B are notshown in the plot, because they are hardly distinguishablefrom the experimental data on this scale. The error bars aredetermined as the standard deviation of the fit. plained by the fact, that while the mean potential depth¯ V = R d R V ( R ) R d R averaged over the volume of the Rydbergatom decreases with the effective principal number as n ∗− , the probability to find an atom inside the Rydbergelectron orbit increases with n ∗ . In the experiment alldata was taken at a fixed density. Thus, in our analysiswe overlap the center of gravity of the spectra at n ≥ h cg i = −
300 kHz determined at low princi-pal quantum numbers. Doing so we can identify the zeroposition in the top panels of Fig. 2. Assuming the scatter-ing length a to be constant, we determine the effectivedensity to be 3 · cm − , which is close to the peakdensity obtained from a Gaussian fit to absorption im-ages of the thermal cloud. This indicates that moleculesare most likely created in regions of high density. Onlyfor n ≤
80 the highest signal originates from pure Ryd-berg atoms. Already for n =80 there is on average oneground state atom inside the electron orbit, leading to ahigh probability to excite dimers instead of pure Rydbergatoms. On average there are four atoms inside the 100Selectron orbit and eight for the 111S state. Therefore,the atomic line is suppressed, while the molecular linesare not resolvable in the experiment any more, causedby their very low binding energies. According to thecentral limit theorem, for even higher principal quantumnumbers the shape of the spectrum is expected to be-come Gaussian, with the maximum at the position of themean shift ( h cg i in Fig. 2). The density dependent de-phasing resulting from the existence of many molecularlines within the Rydberg line envelope sets a fundamentallimit for the number of atoms inside the blockade radiusi. e. the optical thickness. This fact is of importance forevery experiment taken at high principal quantum num-bers and high densities, in particular for quantum opticsexperiments in ultracold clouds [23–25].The binding energies of all observed molecules are plot-ted in Fig. 3. From the extrapolated molecular bindingenergies the transition from the few particle descriptionof discrete bound states to a mean field shift becomes vis-ible. A power law fitted to the n ≥
40 data shows a scalingwith the effective principal quantum number n ∗ to thepower of − . ± .
12, close to the value of −
6, expectedfrom the size scaling argument. The deviation can beexplained by the dependence of the scattering length onthe relative momentum and the fact, that with increas-ing principal quantum number the shape of the outer-most well of the molecular potential changes. Taking thecorresponding zero-point energy and momentum depen-dent corrections to the scattering length into account,we obtain an exponent of − .
37, which is in very goodagreement with our experimental data. Contributions ofhigher order partial waves and p -wave shape resonancesto the molecular potential can be neglected since the ki-netic energy associated with the relative motion of twoscattering partners is small in the region of interest.Rydberg molecules in an ultracold cloud constitute atunable model system to study the transition from a few-body to a many-body regime. They offer a unique toolto address few-body subsystems with control on a sin-gle particle level by changing the detuning of the excita-tion laser light. Complementary to this work the numberof constituents and the interaction strength can be alsovaried independently by changing the density and theprincipal quantum number of the excited Rydberg atom.Furthermore, the analysis of the relative strength of themolecular lines opens up the possibility to measure corre-lations in a bosonic gas. In high density gases and for lowprincipal quantum numbers, where the size of the Ryd-berg atom is comparable to the de Broglie wavelength,extracting the g (2) correlation function [26] of thermaland Bose-condensed gases is feasible. In addition to pre-vious measurements [27, 28] also higher order correlationfunctions can be studied using polyatomic molecules.We acknowledge support from Deutsche Forschungs-gemeinschaft (DFG) within the SFB/TRR21 and theproject PF 381/4-2. Parts of this work was also foundedby ERC under contract number 267100. A.G. acknowl- edges support from E.U. Marie Curie program ITN-Coherence 265031 and S.H. from DFG through theproject HO 4787/1-1.The experiment was conceived by A.G., A.T.K., R.L.,S.H. and T.P. and carried out by A.G. and A.T.K. ; dataanalysis was accomplished A.G. and A.T.K.; numericalcalculation is by A.G. and J.B.B. and A.G. wrote themanuscript with contributions from all authors.The authors declare that they have no competing finan-cial interests. ∗ Electronic address: [email protected] † Electronic address: [email protected][1] Hartman, M., Miller, R. E., Toennies, J. P. & Vilesov,A. F. High-resolution molecular spectroscopy of van derWaals clusters in liquid helium droplets.
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