Letter

Nature 458, 1005-1008 (23 April 2009) | doi:10.1038/nature07945; Received 17 September 2008; Accepted 27 February 2009

Observation of ultralong-range Rydberg molecules

Vera Bendkowsky1, Björn Butscher1, Johannes Nipper1, James P. Shaffer1,2, Robert Löw1 & Tilman Pfau1

  1. 5. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
  2. University of Oklahoma, Homer L. Dodge Department of Physics and Astronomy, Norman, Oklahoma 73072, USA

Correspondence to: Vera Bendkowsky1Tilman Pfau1 Correspondence and requests for materials should be addressed to V.B. (Email: v.bendkowsky@physik.uni-stuttgart.de) or T.P. (Email: t.pfau@physik.uni-stuttgart.de).

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Rydberg atoms have an electron in a state with a very high principal quantum number, and as a result can exhibit unusually long-range interactions. One example is the bonding of two such atoms by multipole forces to form Rydberg–Rydberg molecules with very large internuclear distances1, 2, 3. Notably, bonding interactions can also arise from the low-energy scattering of a Rydberg electron with negative scattering length from a ground-state atom4, 5. In this case, the scattering-induced attractive interaction binds the ground-state atom to the Rydberg atom at a well-localized position within the Rydberg electron wavefunction and thereby yields giant molecules that can have internuclear separations of several thousand Bohr radii6, 7, 8. Here we report the spectroscopic characterization of such exotic molecular states formed by rubidium Rydberg atoms that are in the spherically symmetric s state and have principal quantum numbers, n, between 34 and 40. We find that the spectra of the vibrational ground state and of the first excited state of the Rydberg molecule, the rubidium dimer Rb(5s)–Rb(ns), agree well with simple model predictions. The data allow us to extract the s-wave scattering length for scattering between the Rydberg electron and the ground-state atom, Rb(5s), in the low-energy regime (kinetic energy, <100 meV), and to determine the lifetimes and the polarizabilities of the Rydberg molecules. Given our successful characterization of s-wave bound Rydberg states, we anticipate that p-wave bound states9, trimer states10 and bound states involving a Rydberg electron with large angular momentum—so-called trilobite molecules5—will also be realized and directly probed in the near future.

In 1934, Fermi introduced the ideas of scattering length and pseudopotential to describe the scattering of a low-energy electron from a neutral atom4. Although the polarization potential for electron–atom interaction is always attractive, he realized that quantum mechanical s-wave scattering can give rise to either a positive or a negative scattering length depending on the relative phase between the ingoing and the scattered electron waves. Taking this idea farther, Greene et al.5 predicted a novel molecular binding mechanism arising from a low-energy Rydberg electron scattering from an atom with negative scattering length.

Fermi's approach to characterizing the binding interaction that arises from scattering of a Rydberg electron from a ground-state atom requires that the binding energy (in frequency units) be smaller than the Kepler frequency of the Rydberg electron, and that the size of the electron wavefunction, proportional ton2, be much larger than the range of interaction, r (which in units of the Bohr radius (a0 approximately 0.529 Å) is given by r = Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com (ref. 11), where alpha is the polarizability of the ground-state atom). Averaged over many scattering events and weighted with the local electron density, |Psin,l,m|2, the approach effectively leads to a mean-field potential, VMF, between the scattering partners. If R is the position of the ground-state atom relative to the ionic core of the Rydberg atom, then the potential is given by

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

and can, depending on the scattering length, a(k(R)), be repulsive (a > 0) or attractive (a < 0)12. Evidence for these molecular potential curves was found in theoretical work on alkali/rare-gas scattering13, 14 as well as in spectroscopic data of rubidium at high temperatures, where inhomogeneous line broadenings were observed for low principal quantum numbers15.

In a semi-classical approximation, the scattering length is a function of the relative momentum, k(R), of the two scattering partners. This k dependence can be expressed as

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

where aatom is the zero-energy scattering length12, 16. The scattering length depends on R because the momentum, k, of the Rydberg electron changes with its position in the Coulomb potential of the nucleus. Owing to the correspondence principle for large principal quantum numbers, n, a reasonable ansatz for k(R) (where R = |R|) is the classical equation given in ref. 5:

Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

Our focus in this study is on rubidium in its simplest Rydberg state, the s state (angular quantum number, l = 0). Figure 1 shows the mean-field potential given by equation (1) and the electron probability density calculated for the 87Rb(35s) state. (The densities were calculated using Numerov's method, including quantum defect corrections17, 18. Energy levels and wavefunctions of the molecular potential were computed using a numerical solver19.) The molecular potential, VMF(R), is proportional to the Rydberg electron probability density, so the expected bond length to be given by the size of the Rydberg wavefunction; for example, the size of the Rydberg wavefunction of the 87Rb(40s) state is 2,556a0.

Figure 1: Electron probability density and molecular potential for the 35s state.
Figure 1 : Electron probability density and molecular potential for the 35s state. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

The surface plot shows the spherically symmetric density distribution of the Rydberg electron in the Rphi plane, (R/2pi)|Psi35,0,0(R)|2. The molecular potential for the state 3Sigma(5s–35s) (green) is modelled for a polarizability alpha = 319 a.u. and a scattering length aRb = -18.5a0. Not shown is the repulsive part of the potential for R < 500a0 that results from a zero crossing in the scattering length a(k(R)) at approximately 500a0. The potential supports two vibrational bound states (wavefunctions given in blue) in the outermost potential wells at R = 1,900a0 with binding energies (in frequency units) of EB(v = 0) = -23.4 MHz and EB(v = 1) = -10.6 MHz.

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To observe the 3Sigma ultralong-range molecules formed from s-state Rydberg atoms, we prepare a spin-polarized, magnetically trapped sample of ultracold 87Rb atoms in the state 5s1/2, mF = 2 (mF denoting the projection of the angular momentum on the magnetic field axis), and excite them via the 5p3/2 level to the Rydberg state ns1/2 (Methods). Polarizations are chosen to conserve the Rydberg electron spin with respect to the ground-state atoms, which should ensure that only triplet bound states form. Spectra obtained for 87Rb atoms that differ essentially in only their principal quantum numbers are presented in Fig. 2. The peak at the origin of each spectrum is the atomic Rydberg line 87Rb(ns1/2mS = 1/2) (mS denoting the orientation of the spin with respect to the magnetic field), and the slightly broadened secondary peak, or 'shoulder', on the lower-frequency, or red, side corresponds to the mS = –1/2 atomic Rydberg state. These two atomic levels are shifted by plusminusmuBB0 owing to the Zeeman effect of the magnetic offset field, B0 (muB denoting the Bohr magneton), with the mS = -1/2 state being present because the residual inhomogeneous magnetic field direction of the trap can cause a spin flip during Rydberg excitation. The smaller peaks appearing farther to the red side are assigned to the ultralong-range Rydberg molecules. The Zeeman effect shifts atomic as well as molecular lines, and the measured binding energies, EB, therefore correspond to the differences between the centres of multiplet lines (inset, Fig. 3). The energy positions of these centres are given by the magnetic offset field B0, which was measured independently for each spectrum.

Figure 2: Spectra of the Rydberg states 35s, 36s and 37s.
Figure 2 : Spectra of the Rydberg states 35s, 36s and 37s. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

The overview spectra (right) are centred around the atomic Rydberg lines (5smF = 2) right arrow (nsmS = 1/2). The additional shoulder at approx-3 MHz corresponds to the magnetic-field-dependent transitions (5smF = 2) right arrow (nsmS = -1/2). On the left are shown the observed molecular lines at higher resolution. We assign the leftmost line of each spectrum to the 3Sigma(5sns)(v = 0) bound state. The highest-lying lines in the 35s and 36s spectra are in good agreement with the modelled excited states 3Sigma(5sns)(v = 1). Peaks not yet assigned are marked with diamonds. The error bars (2sigma) are determined from 15 (right) and 30 (left) independent spectra.

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Figure 3: Measured and calculated binding energies, EB.
Figure 3 : Measured and calculated binding energies, EB. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

The solid lines are the calculated binding energies for the v = 0, 1 states, assuming a scattering length aRb = -18.5a0; shaded areas show the theory for aRb = -(18.5 plusminus 0.5)a0. Symbols represent the measured line centres for the molecular v = 0, 1 states (red) and the unassigned states (green). For the strong spectral lines (v = 0, n = 34–37), errors are given by the laser linewidth of plusminus0.5 MHz; for the other lines, errors are given by the 95% confidence bounds of the fit. The inset illustrates the definition of EB as the energy difference between the line centres of atomic doublet n2s lines and molecular triplet 3Sigma(5sns) lines in a magnetic field B0 leading to a Zeeman splitting DeltaB = 2muBB0.

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We assign the lines at the highest binding energies to the vibrational ground state v = 0 of the triplet molecule 3Sigma(5sns) (v denoting the vibrational quantum number). Considering our peak atomic densities, nG, we expect the Franck–Condon factor for excitation of two free ground-state atoms to the bound molecular state to be on the order of 10-2. However, Franck–Condon factors cannot be directly derived from the relative line intensities in the measured spectra because the excitation of the atomic Rydberg states is strongly suppressed by van der Waals blockade (ref. 20 and references therein).

We assign the observed spectral lines corresponding to different principal quantum numbers using the previously described theory based on the Fermi–Greene model. With an accurately known ground-state polarizability for rubidium of alpha = 319(6) atomic units (a.u.) (ref. 21), the only free parameter in equations (1) and (2) is the triplet scattering length, aRb. We find that the binding energies measured for the 3Sigma(5sns)(v = 0) states and obtained with the model agree best when using aRb = -18.5a0. This value is close to theoretical predictions for aRb, which range between -13a0 and -17a0 for the triplet case and +0.6a0 and +2.0a0 for the singlet case16. Figure 3 shows the eigenenergies of the modelled potentials and the measured binding energies, EB. In view of the approximate nature of the model, the description of the dependence of the eigenenergies and binding energies on the principal quantum numbers, n, of the ground states 3Sigma(5sns)(v = 0) is surprisingly good. Having experiment-based determination of the electron–atom scattering length in the low-energy regime (kinetic energy, <100 meV) is useful because this parameter is important in the context of, for example, electron solvation22 and structure calculations of negative ions23. We note, however, that our result depends upon the model used to calculate the mean-field potential; a precision analysis might therefore benefit from an extension of the Fermi–Greene model24, 25.

The modelled molecular potentials (Fig. 1) also support an excited bound state, yielding eigenenergies for the first vibrational states (v = 1) that are in good agreement with the corresponding lines seen in the measured 35s and 36s spectra (Figs 2 and 3). Although this good agreement between model and experiment is reassuring, we note that the additional unassigned lines in the spectra clearly indicate that p-wave contributions to the molecular potential cannot be neglected. In this system, the p-wave contributions leave the outermost potential well of the pure s-wave potential, and therefore also the v = 0 state, nearly unchanged. However, the barriers between the potential wells are lowered, leading to a higher number of excited states and affecting the assignment of the v = 1 states given in Figs 2 and 3. Recent calculations based on the Fermi–Greene model including p-wave scattering9 (I. Liu and J. M. Rost, personal communication) confirmed this qualitative explanation and the experimental results for the excited molecular states.

In addition to their vibrational modes, the 3Sigma(5sns) molecules also exhibit rotational spectral features determined only by their masses and bond lengths. The rotational constants range from 11.5 kHz for the 3Sigma(5s–35s) molecule to 9.0 kHz for the 3Sigma(5s–37s) molecule and are thus far below the resolution of the present measurements.

We further characterize the molecular states by Stark-effect measurements of the molecular ground state (v = 0; see Methods). The Stark spectra of the atomic 35s state and the molecular 3Sigma(5s–35s)(v = 0) state in Fig. 4 both show, as expected, a quadratic Stark shift with the electric field. The relative polarizabilities of the atomic and the molecular v = 0 states are alpha = 1,542(7) times 107 a.u. and alpha = 1,524(4) times 107 a.u., respectively. (We note that absolute polarizability values are affected by a systematic error of 12% arising from the calibration of the electric field.) The fact that the values are very similar to each other supports the model assumption that the bound ground-state atom does not perturb the Rydberg wavefunction significantly.

Figure 4: Stark map of the atomic 35s state and the molecular 3Sigma(5s–35s)(v = 0) state.
Figure 4 : Stark map of the atomic 35s state and the molecular 3|[Sgr]|(5s|[ndash]|35s)(v = 0) state. Unfortunately we are unable to provide accessible alternative text for this. If you require assistance to access this image, or to obtain a text description, please contact npg@nature.com

The line centres of both the atomic state (right) and the molecular state (left) (symbols) show a quadratic Stark effect. Their polarizabilities are determined to be alpha = 1,542(7) times 107 a.u. and alpha = 1,524(4) times 107 a.u. for the atomic 35s state and the 3Sigma(5s–35s)(v = 0) state, respectively (white lines). The error bars represent the finite laser linewidth of plusminus0.5 MHz.

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To investigate the lifetime of the molecular v = 0 state as well as that of the atomic Rydberg state, we apply an excitation pulse of fixed length to either the molecular or the atomic resonance and change the time between excitation and field ionization. From the exponential decrease of detected ions, we determine the respective lifetimes, tau, of the molecules and Rydberg atoms (Methods). In Table 1, we compare the measured molecular lifetimes with those of the atomic Rydberg states. The lifetimes of the Rydberg atoms are slightly longer than the lifetimes of the corresponding atoms in free space. We attribute this effect to suppression of the decay of black-body radiation by the metallic environment of the chamber, which changes the spectral mode density. More importantly, however, the data show that the molecular lines decay faster than the atomic states by a factor of typically 3–4. Considering that the atomic and molecular polarizabilities are almost identical, such a reduction in the molecular lifetime is noteworthy. A possible additional decay channel for the molecular states could be ion-pair formation (that is, Rb(ns) + Rb(5sright arrow Rb+ + Rb-), as observed with rubidium for low principal quantum numbers26, but full clarification of the additional molecular decay process will be the subject of further study.


The molecular bound states observed here with rubidium are expected to form with all species exhibiting a negative scattering length for electron–atom interactions, for example the other alkali atoms16. Successful observation of such states requires sufficiently high densities, to provide a reasonable number of atom pairs with distances on the order of 100 nm, and low temperatures, to avoid collisions during the excitation. Future theoretical investigations are needed to clarify the nature of the additional bound states evident in the spectra, and further development of our experimental approach may soon reveal trimer states10, p-wave bound states9 and high-l Rydberg states—so-called trilobite molecules5. In conclusion, we note that the molecular state described here might find use in pump–probe experiments as a reference in studying collective Rydberg excitation (ref. 20 and references therein) or in determining correlation functions in ultracold gases. In view of the recent significant progress (ref. 20, references therein and refs 27, 28) in coherent excitation of Rydberg atoms, it might also be possible to create coherent superposition states between free and bound atoms.

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Methods Summary

The ultracold sample of 87Rb atoms was produced in a Ioffe–Pritchard trap with a magnetic offset field B0 = 0.8 G, where the atoms are trapped in the ground state 5s1/2, mF = 2. At a temperature T = 3.5 muK and peak density nG = 1.5 times 1013 cm-3, the cigar-shaped cloud had a radial diameter of 28 mum (1/e2, where e denotes the Euler number). The Rydberg state ns1/2 was addressed by a two-photon excitation 5s1/2 right arrow 5p3/2 right arrow ns1/2 using continuous-wave lasers at 780 nm and 480 nm with a combined linewidth of <1 MHz. The laser power of the red laser was 800 nW in a 1/e2 diameter of 1 mm and the power of the blue laser was 50 mW in a 1/e2 diameter of 80 mum, both of which diameters were larger than the sample. To avoid resonant scattering, the 780-nm laser was blue-detuned (>400 MHz) from the intermediate level 5p3/2. The excitation pulse had a duration of 3 mus and was directly followed by field ionization of the Rydberg atoms in an electric field of up to 440 V cm-1. The ions were detected using a microchannel plate. For the lifetime measurements, the time between excitation and field ionization of the Rydberg atoms was varied from 0 mus to up to 150 mus. The temperature and density of the remaining ground-state atoms were measured by absorption imaging. Further details regarding the setup can be found in ref. 29 and Methods, which also details in full the measurements of the lifetime and the Stark effect and the data analysis.

Full methods accompany this paper.

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References

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Acknowledgements

We would like to thank C. Greene and J. M. Rost for discussions and P. Kollmann for his contribution in the early stage of the experiment. This work is supported by the Deutsche Forschungsgemeinschaft as part of the SFB/TRR21 and under contract PF 381/4-1, and by the Landesstiftung Baden-Württemberg. B.B. acknowledges support from the Carl Zeiss foundation and J.P.S. thanks the Alexander von Humboldt foundation for financial support.

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Online Methods

In general, the experimental sequence was divided in three parts: preparation of the cold sample of 87Rb atoms, Rydberg excitation and detection, and absorption imaging of the cloud.

The rubidium atoms were trapped magnetically in the hyperfine state F = 2, mF = 2 of the 5s1/2 state. The cloverleaf trap had a magnetic offset field B0 = 0.8 G and produced a cigar-shaped cloud. At a temperature T = 3.5 muK, the sample had an extension (1/e2 diameter) of 28 mum in the radial direction and one of 380 mum in the axial direction with a peak atomic density nG = 1.5 times 1013 cm-3.

In this ultracold sample of 87Rb, atoms were excited to the ns Rydberg state through a two-photon excitation, 5s1/2 right arrow 5p3/2 right arrow ns1/2 (34 less than or equal to n less than or equal to 40), using two continuous-wave lasers with respective wavelengths of 780 nm and 480 nm. The 780-nm laser was blue-detuned (Delta > 400 MHz) from the intermediate 5p3/2 level to avoid resonant scattering and heating of the atoms. The collinear laser beams propagated along the axial direction of the atomic cloud, which was also the quantization axis. The polarizations of the beams were sigma+ (780 nm) and sigma- (480 nm) in this configuration. By choosing the polarizations as described, only the magnetic sublevel mS = +1/2 of the Rydberg state was addressed. Both laser beams were significantly larger in diameter than the cloud, to ensure a constant Rabi frequency. The red laser had a 1/e2 diameter of 1 mm and a power of 0.8 muW and the blue laser was focused to 80 mum (1/e2 diameter) at a power of 50 mW. In all experiments described in this Letter, the excitation pulses had a fixed duration of 3 mus. The pulses were generated with acousto-optical modulators with rise times of 12 ns; that is, the pulses had a rectangular shape. Directly after the excitation, the created Rydberg atoms were ionized in an electric field of up to 440 V cm-1. The same field accelerated the ions towards a microchannel-plate detector, the current from which was recorded as a measure of the total number of Rydberg atoms that were excited. Details of the laser system, excitation scheme and detection of the Rydberg atoms are given in ref. 29.

Finally, the ground-state atoms were imaged in absorption after a time of flight of 20 ms, to determine the temperature and density of the cloud.

Spectroscopic data

The spectra presented in Fig. 2 are the averages of 15 (overview spectra) and 30 (high-resolution spectra) frequency scans each measured in a single atomic sample. After preparing the cloud, the Rydberg excitation and detection was carried out up to 60 times while stepwise changes were made to the frequency of the red laser. The repetition rate of the Rydberg excitation was 330 Hz. As every spectrum is the average of scans taken in single clouds, the 2sigma errors in Fig. 2 are a measure of the shot-to-shot variation in the Rydberg signal. Furthermore, the overview spectra are composed of two frequency ranges red- and blue-detuned from the atomic ns resonances.

For the Stark spectra, the experimental procedure was the same except for the additional d.c. electric field that was applied. The spectra shown in Fig. 4 are averages of five independent measurements. For each electric field, the line positions were determined by a Gaussian fit (white data) and finally fitted to a parabola to extract the polarizabilities that describe the quadratic Stark effect for the levels.

Lifetime

As the laser linewidth was larger than the natural linewidths of the Rydberg states studied, the lifetime was measured by recording the exponentially decreasing Rydberg signal over time after the excitation had occurred. The field-ionization pulse was applied at a variable time delay, t, referenced to the optical excitation in the experiment. Each measurement was performed for twelve different values of t ranging from 0 mus to 150 mus. To compensate for variations in the Rydberg signal from shot to shot, the experiment was repeated 90 times for every t value. The statistical 2sigma errors in the means of this data are 10% for the atomic ns resonances and 13% for the molecular 3Sigma(5sns)(v = 0) ground states. The lifetimes given in Table 1 are the results of exponential fits to the mean Rydberg signal versus t for the tabulated atomic and molecular states.