Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Quantum chaos in ultracold collisions of gas-phase erbium atoms


Atomic and molecular samples reduced to temperatures below one microkelvin, yet still in the gas phase, afford unprecedented energy resolution in probing and manipulating the interactions between their constituent particles. As a result of this resolution, atoms can be made to scatter resonantly on demand, through the precise control of a magnetic field1. For simple atoms, such as alkalis, scattering resonances are extremely well characterized2. However, ultracold physics is now poised to enter a new regime, where much more complex species can be cooled and studied, including magnetic lanthanide atoms and even molecules. For molecules, it has been speculated3,4 that a dense set of resonances in ultracold collision cross-sections will probably exhibit essentially random fluctuations, much as the observed energy spectra of nuclear scattering do5. According to the Bohigas–Giannoni–Schmit conjecture, such fluctuations would imply chaotic dynamics of the underlying classical motion driving the collision6,7,8. This would necessitate new ways of looking at the fundamental interactions in ultracold atomic and molecular systems, as well as perhaps new chaos-driven states of ultracold matter. Here we describe the experimental demonstration that random spectra are indeed found at ultralow temperatures. In the experiment, an ultracold gas of erbium atoms is shown to exhibit many Fano–Feshbach resonances, of the order of three per gauss for bosons. Analysis of their statistics verifies that their distribution of nearest-neighbour spacings is what one would expect from random matrix theory9. The density and statistics of these resonances are explained by fully quantum mechanical scattering calculations that locate their origin in the anisotropy of the atoms’ potential energy surface. Our results therefore reveal chaotic behaviour in the native interaction between ultracold atoms.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Prices vary by article type



Prices may be subject to local taxes which are calculated during checkout

Figure 1: Fano–Feshbach spectrum of 168Er and 166Er from 0 to 70 G.
Figure 2: Mean resonance density for bosonic Er as a function of largest included partial-wave Lmax.
Figure 3: Loss-maxima position and staircase function for 168Er.
Figure 4: NNS distribution and number variance.

Similar content being viewed by others


  1. Inouye, S. et al. Observation of Feshbach resonances in a Bose–Einstein condensate. Nature 392, 151–154 (1998)

    Article  CAS  ADS  Google Scholar 

  2. Chin, C., Grimm, R., Julienne, P. S. & Tiesinga, E. Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225–1286 (2010)

    Article  CAS  ADS  Google Scholar 

  3. Mayle, M., Ruzic, B. P. & Bohn, J. L. Statistical aspects of ultracold resonant scattering. Phys. Rev. A 85, 062712 (2012)

    Article  ADS  Google Scholar 

  4. Mayle, M., Quéméner, G., Ruzic, B. P. & Bohn, J. L. Scattering of ultracold molecules in the highly resonant regime. Phys. Rev. A 87, 012709 (2013)

    Article  ADS  Google Scholar 

  5. Guhr, T., Müller-Groeling, A. & Weidenmüller, H. A. Random-matrix theories in quantum physics: common concepts. Phys. Rep. 299, 189–425 (1998)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  6. Bohigas, O., Giannoni, M. J. & Schmit, C. Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1–4 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  7. Weidenmüller, H. A. & Mitchell, G. E. Random matrices and chaos in nuclear physics: nuclear structure. Rev. Mod. Phys. 81, 539–589 (2009)

    Article  ADS  Google Scholar 

  8. Mitchell, G. E., Richter, A. & Weidenmüller, H. A. Random matrices and chaos in nuclear physics: nuclear reactions. Rev. Mod. Phys. 82, 2845–2901 (2010)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  9. Brody, T. A. A statistical measure for the repulsion of energy levels. Lett. N. Cim. Ser. 2 7, 482–484 (1973)

    Article  Google Scholar 

  10. Lu, M., Burdick, N. Q., Youn, S. H. & Lev, B. L. Strongly dipolar Bose-Einstein condensate of dysprosium. Phys. Rev. Lett. 107, 190401 (2011)

    Article  ADS  Google Scholar 

  11. Lu, M., Burdick, N. Q. & Lev, B. L. Quantum degenerate dipolar Fermi gas. Phys. Rev. Lett. 108, 215301 (2012)

    Article  ADS  Google Scholar 

  12. Aikawa, K. et al. Bose-Einstein condensation of erbium. Phys. Rev. Lett. 108, 210401 (2012)

    Article  CAS  ADS  Google Scholar 

  13. Aikawa, K. et al. Reaching Fermi degeneracy via universal dipolar scattering. Phys. Rev. Lett. 112, 010404 (2014)

    Article  CAS  ADS  Google Scholar 

  14. Hancox, C. I., Doret, S. C., Hummon, M. T., Luo, L. & Doyle, J. M. Magnetic trapping of rare-earth atoms at millikelvin temperatures. Nature 431, 281–284 (2004)

    Article  CAS  ADS  Google Scholar 

  15. Connolly, C. B., Au, Y. S., Doret, S. C., Ketterle, W. & Doyle, J. M. Large spin relaxation rates in trapped submerged-shell atoms. Phys. Rev. A 81, 010702 (2010)

    Article  ADS  Google Scholar 

  16. Kokoouline, V., Santra, R. & Greene, C. H. Multichannel cold collisions between metastable Sr atoms. Phys. Rev. Lett. 90, 253201 (2003)

    Article  ADS  Google Scholar 

  17. Krems, R. V., Groenenboom, G. C. & Dalgarno, A. Electronic interaction anisotropy between atoms in arbitrary angular momentum states. J. Phys. Chem. A 108, 8941–8948 (2004)

    Article  CAS  Google Scholar 

  18. Petrov, A., Tiesinga, E. & Kotochigova, S. Anisotropy-induced Feshbach resonances in a quantum dipolar gas of highly magnetic atoms. Phys. Rev. Lett. 109, 103002 (2012)

    Article  ADS  Google Scholar 

  19. Berninger, M. et al. Feshbach resonances, weakly bound molecular states, and coupled-channel potentials for cesium at high magnetic fields. Phys. Rev. A 87, 032517 (2013)

    Article  ADS  Google Scholar 

  20. Takekoshi, T. et al. Towards the production of ultracold ground-state RbCs molecules: Feshbach resonances, weakly bound states, and the coupled-channel model. Phys. Rev. A 85, 032506 (2012)

    Article  ADS  Google Scholar 

  21. Kotochigova, S., Levine, H. & Tupitsyn, I. Correlated relativistic calculation of the giant resonance in the Gd3+ absorption spectrum. Int. J. Quantum Chem. 65, 575–584 (1997)

    Article  CAS  Google Scholar 

  22. Kramida, A. et al. NIST Atomic Spectra Database Version 5.0. (2013)

  23. Lawler, J. E., Wyart, J. & Den Hartog, E. A. Atomic transition probabilities of Er I. J. Phys. At. Mol. Opt. Phys. 43, 235001 (2010)

    Article  ADS  Google Scholar 

  24. Gao, B. Zero-energy bound or quasibound states and their implications for diatomic systems with an asymptotic van der Waals interaction. Phys. Rev. A 62, 050702(R) (2000)

    Article  ADS  Google Scholar 

  25. Wigner, E. P. On a class of analytic functions from the quantum theory of collisions. Ann. Math. 53, 36–67 (1951)

    Article  MathSciNet  Google Scholar 

  26. Dyson, F. J. & Mehta, M. L. Statistical theory of the energy levels of complex systems. IV. J. Math. Phys. 4, 701–712 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  27. Alt, H. et al. Gaussian orthogonal ensemble statistics in a microwave stadium billiard with chaotic dynamics: Porter-Thomas distribution and algebraic decay of time correlations. Phys. Rev. Lett. 74, 62–65 (1995)

    Article  CAS  ADS  Google Scholar 

  28. Taylor, J. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements 272 (Univ. Science Books, 1997)

    Google Scholar 

  29. Brody, T. A. et al. Random-matrix physics: spectrum and strength fluctuations. Rev. Mod. Phys. 53, 385–479 (1981)

    Article  MathSciNet  CAS  ADS  Google Scholar 

  30. Gutzwiller, M. Chaos in Classical and Quantum Mechanics (Springer, 1990)

    Book  Google Scholar 

  31. Frisch, A. et al. Narrow-line magneto-optical trap for erbium. Phys. Rev. A 85, 051401 (2012)

    Article  ADS  Google Scholar 

  32. Watts, J. D., Gauss, J. & Bartlett, R. J. Coupled-cluster methods with noniterative triple excitations for restricted open-shell Hartree-Fock and other general single determinant reference functions: energies and analytical gradients. J. Chem. Phys. 98, 8718 (1993)

    Article  CAS  ADS  Google Scholar 

  33. Oganesyan, V. & Huse, D. A. Localization of interacting fermions at high temperature. Phys. Rev. B 75, 155111 (2007)

    Article  ADS  Google Scholar 

  34. Kollath, C., Roux, G., Biroli, G. & Läuchli, A. M. Statistical properties of the spectrum of the extended Bose-Hubbard model. J. Stat. Mech. 2010, P08011 (2010)

    Article  MathSciNet  Google Scholar 

Download references


The Innsbruck group thanks R. Grimm for discussions and S. Baier, C. Ravensbergen and M. Brownnutt for reading the manuscript. S.K. and A.P. thank E. Tiesinga for discussions. J.L.B. is supported by an ARO MURI. The Innsbruck team is supported by the Austrian Science Fund (FWF) through a START grant under project no. Y479-N20 and by the European Research Council under project no. 259435. K.A. is supported within the Lise-Meitner program of the FWF. Research at Temple University is supported by AFOSR and NSF PHY-1308573.

Author information

Authors and Affiliations



A.F., M.M., K.A. and F.F. did the experimental work and statistical analysis of the data. C.M., A.P. and S.K. did the theoretical work on coupled-channel calculations and RQDT. J.L.B. did the theoretical work on RMT. The manuscript was written with substantial contributions from all authors.

Corresponding author

Correspondence to Francesca Ferlaino.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Extended data figures and tables

Extended Data Figure 1 Fano–Feshbach spectrum of fermionic 167Er from 0 to 4.5 G.

The trap-loss spectroscopy is performed in an optically trapped sample of fermionic Er atoms at a temperature of 0.4TF, where TF = 1.0(1) μK is the Fermi temperature. The atoms are spin-polarized in the lowest Zeeman sublevel, mF = −19/2. We keep the atomic sample at the magnetic probing field for a holding time of 100 ms. We observe 115 resonances up to 4.5 G, which we take to be Fano–Feshbach resonances between identical fermions. The corresponding mean density is about 26 resonances per gauss.

Extended Data Figure 2 Elastic rate coefficient of mJ = −6 168Er collisions.

The s-wave elastic rate coefficient as a function of magnetic field assuming a collision energy of E = kB × (360 nK). Partial waves up to = 20 are included. A divergence of the elastic rate coefficient, that is, the position of a Fano–Feshbach resonance, is marked with squares.

Extended Data Figure 3 Statistical analysis of high-density Fano–Feshbach resonances of isotope 166Er.

a, Positions of the resonances are marked with vertical lines. b, The staircase function shows a similar behaviour to 168Er (Fig. 3). A linear fit to the data above 30 G is plotted in a lighter colour. From the staircase function, we calculate a mean density of resonances of , which corresponds to a mean distance between resonances of .

Extended Data Figure 4 NNS distribution and number variance.

a, 168Er NNS distribution above 30 G with a bin size of 140 mG. For the error bars we assume a Poisson counting error. The plot shows the experimental data (circles) with the corresponding Brody distribution (solid line). The parameter-free distributions PP (dashed line) and PWD (dash–dot line) are shown, and for the Poisson distribution and for the W–D distribution. b, Number variance, Σ2, for the same experimental data (solid line) with a 2σ confidence band (shaded area). The number variance of the experimental data shows a clear deviation from the number variance of a Poisson distribution.

Extended Data Figure 5 Density of resonances as a function of the magnetic field.

The densities of resonances for 168Er (a) and 166Er (b) are given as an averaged derivative of the staircase function (dark solid lines) using an averaging region 10 G in size. At small magnetic field values, the density of resonances is about 1.5 G−1, and it increases with the magnetic field up to about 30 G. For larger magnetic fields, the density is roughly constant. The staircase function also suggests this behaviour. From the data above 30 G, we calculate the mean value (light solid lines) and the standard deviation (light dotted lines) of the density of resonances.

Extended Data Table 1 Fano–Feshbach resonances data for isotope 168Er
Extended Data Table 2 Fano–Feshbach resonances data for isotope 166Er

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Frisch, A., Mark, M., Aikawa, K. et al. Quantum chaos in ultracold collisions of gas-phase erbium atoms. Nature 507, 475–479 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing