# Observing the emergence of a quantum phase transition shell by shell

## Abstract

Many-body physics describes phenomena that cannot be understood by looking only at the constituents of a system1. Striking examples are broken symmetry, phase transitions and collective excitations2. To understand how such collective behaviour emerges as a system is gradually assembled from individual particles has been a goal in atomic, nuclear and solid-state physics for decades3,4,5,6. Here we observe the few-body precursor of a quantum phase transition from a normal to a superfluid phase. The transition is signalled by the softening of the mode associated with amplitude vibrations of the order parameter, usually referred to as a Higgs mode7. We achieve fine control over ultracold fermions confined to two-dimensional harmonic potentials and prepare closed-shell configurations of 2, 6 and 12 fermionic atoms in the ground state with high fidelity. Spectroscopy is then performed on our mesoscopic system while tuning the pair energy from zero to a value larger than the shell spacing. Using full atom counting statistics, we find the lowest resonance to consist of coherently excited pairs only. The distinct non-monotonic interaction dependence of this many-body excitation, combined with comparison with numerical calculations allows us to identify it as the precursor of the Higgs mode. Our atomic simulator provides a way to study the emergence of collective phenomena and the thermodynamic limit, particle by particle.

## Access options

Rent or Buy article

from\$8.99

All prices are NET prices.

## Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

## References

1. 1.

Anderson, P. W. More is different. Science 177, 393–396 (1972).

2. 2.

Sachdev, S. Quantum Phase Transitions 2nd edn (Cambridge Univ. Press, 2011).

3. 3.

Bohr, A. & Mottelson, B. R. Nuclear Structure Vols. I, II (Benjamin, 1975).

4. 4.

Grebenev, S., Toennies, J. P. & Vilesov, A. F. Superfluidity within a small helium-4 cluster: the microscopic Andronikashvili experiment. Science 279, 2083–2086 (1998).

5. 5.

Wenz, A. N. et al. From few to many: observing the formation of a Fermi sea one atom at a time. Science 342, 457–460 (2013).

6. 6.

Launey, K. D. Emergent Phenomena in Atomic Nuclei from Large-Scale Modeling (World Scientific, 2017).

7. 7.

Bruun, G. M. Long-lived Higgs mode in a two-dimensional confined Fermi system. Phys. Rev. A 90, 023621 (2014).

8. 8.

Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. 80, 885–964 (2008).

9. 9.

Bjerlin, J., Reimann, S. M. & Bruun, G. M. Few-body precursor of the Higgs mode in a Fermi gas. Phys. Rev. Lett. 116, 155302 (2016).

10. 10.

Sooryakumar, R. & Klein, M. V. Raman scattering by superconducting-gap excitations and their coupling to charge-density waves. Phys. Rev. Lett. 45, 660–662 (1980).

11. 11.

Rüegg, C. et al. Quantum magnets under pressure: controlling elementary excitations in TlCuCl3. Phys. Rev. Lett. 100, 205701 (2008).

12. 12.

Bissbort, U. et al. Detecting the amplitude mode of strongly interacting lattice bosons by Bragg scattering. Phys. Rev. Lett. 106, 205303 (2011).

13. 13.

Endres, M. et al. The ‘Higgs’ amplitude mode at the two-dimensional superfluid/Mott insulator transition. Nature 487, 454–458 (2012).

14. 14.

Matsunaga, R. et al. Higgs amplitude mode in the BCS superconductors Nb1−xTixN induced by terahertz pulse excitation. Phys. Rev. Lett. 111, 057002 (2013).

15. 15.

Léonard, J., Morales, A., Zupancic, P., Donner, T. & Esslinger, T. Monitoring and manipulating Higgs and Goldstone modes in a supersolid quantum gas. Science 358, 1415–1418 (2017).

16. 16.

Katsumi, K. et al. Higgs mode in the d-wave superconductor Bi2Sr2CaCu2O8+x driven by an intense terahertz pulse. Phys. Rev. Lett. 120, 117001 (2018).

17. 17.

Behrle, A. et al. Higgs mode in a strongly interacting fermionic superfluid. Nat. Phys. 14, 781–785 (2018).

18. 18.

Serwane, F. et al. Deterministic preparation of a tunable few-fermion system. Science 332, 336–338 (2011).

19. 19.

Kohmoto, M. & Takada, Y. Superconductivity from an insulator. J. Phys. Soc. Jpn 59, 1541–1544 (1990).

20. 20.

Nozières, P. & Pistolesi, F. From semiconductors to superconductors: a simple model for pseudogaps. Eur. Phys. J. B 10, 649–662 (1999).

21. 21.

Heiselberg, H. & Mottelson, B. Shell structure and pairing for interacting fermions in a trap. Phys. Rev. Lett. 88, 190401 (2002).

22. 22.

Bruun, G. M. Low-energy monopole modes of a trapped atomic Fermi gas. Phys. Rev. Lett. 89, 263002 (2002).

23. 23.

Rontani, M., Eriksson, G., Åberg, S. & Reimann, S. M. On the renormalization of contact interactions for the configuration-interaction method in two-dimensions. J. Phys. At. Mol. Opt. Phys. 50, 065301 (2017).

24. 24.

Zürn, G. et al. Precise characterization of 6Li Feshbach resonances using trap-sideband-resolved RF spectroscopy of weakly bound molecules. Phys. Rev. Lett. 110, 135301 (2013).

25. 25.

Randeria, M., Duan, J. M. & Shieh, L. Y. Superconductivity in a two-dimensional Fermi gas: evolution from Cooper pairing to Bose condensation. Phys. Rev. B 41, 327–343 (1990).

26. 26.

Idziaszek, Z. & Calarco, T. Analytical solutions for the dynamics of two trapped interacting ultracold atoms. Phys. Rev. A 74, 022712 (2006).

27. 27.

Bruun, G. M. & Mottelson, B. R. Low energy collective modes of a superfluid trapped atomic Fermi gas. Phys. Rev. Lett. 87, 270403 (2001).

28. 28.

D’Alessio, L., Kafri, Y., Polkovnikov, A. & Rigol, M. From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics. Adv. Phys. 65, 239–362 (2016).

29. 29.

Cremon, J. Quantum Few-Body Physics with the Configuration Interaction Approach: Method Development and Application to Physical Systems. Ph.D. thesis, Lund University (2010).

30. 30.

Hu, H., Mulkerin, B. C., Toniolo, U., He, L. & Liu, X.-J. Reduced quantum anomaly in a quasi-two-dimensional fermi superfluid: significance of the confinement-induced effective range of interactions. Phys. Rev. Lett. 122, 070401 (2019).

## Acknowledgements

The experimental work has been supported by the ERC consolidator grant 725636, the Heidelberg Center for Quantum Dynamics, the DFG Collaborative Research Centre SFB 1225 (ISOQUANT) and the European Union’s Horizon 2020 research and innovation programme under grant agreement number 817482 PASQuanS. K.S. acknowledges support by the Landesgraduiertenförderung Baden-Württemberg. P.M.P. acknowledges funding from the Daimler and Benz Foundation. S.M.R. and J.B. acknowledge financial support by the Swedish Research Council, the Knut and Alice Wallenberg Foundation and NanoLund. G.M.B. acknowledges financial support from the Independent Research Fund Denmark—Natural Sciences via grant number DFF-8021-00233B and the Danish National Research Foundation through the Center of Excellence “CCQ” (grant agreement number DNRF156).

## Author information

Authors

### Contributions

L.B. and M.H. contributed equally to this work. L.B., M.H. and K.S. performed the measurements and analysed the data. J.B., S.M.R. and G.M.B. developed the theoretical framework. J.B. performed the numerical calculations. P.M.P. and S.J. supervised the experimental part of the project. All authors contributed to the discussion of the results and the writing of the manuscript.

### Corresponding authors

Correspondence to Luca Bayha or Marvin Holten.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks Hui Hu and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Experimental protocol.

The sequence can be separated into three parts. First, several evaporation and spilling stages are combined with a transfer from a quasi-1D to a quasi-2D trap geometry. This is needed to prepare closed-shell ground state configurations of up to 12 atoms. Second, we excite the system at some defined frequency fexc and magnetic offset field B using a sinusoidal modulation of either the radial or axial confinement. Third, detection is implemented by spilling to the ground state a second time and a transfer of all remaining atoms to the magneto-optical trap, where we count them.

### Extended Data Fig. 2 Excitation spectrum for two particles.

We define the two-body excitation energy Eexc as the energy difference between the ground state and the lowest monopole excitation of the two atom system. It is measured using the same modulation scheme as for the Higgs mode. The system is initialized with one filled shell, that is, two particles. The analytical solution of the two-body problem (solid line) shows good agreement with the measurement (blue points). The systematic uncertainty of around 2% on the measured radial and axial trap frequencies that enters into the analytical solution is indicated by the grey error band. Residual systematic deviations can be explained by the trap anharmonicity. For the measurement (blue points) error bars are extracted from the fit to the spectrum and are smaller than the data points.

### Extended Data Fig. 3 Comparison of different modulation schemes.

A modulation of the radial trap frequency leads to similar transition probabilities for the pair excitation mode and the higher excited states (top). In contrast, a modulation of the axial confinement effectively only modulates the interaction strength and couples predominantly to the pair excitation mode. Excitations to higher states are suppressed by this modulation scheme (bottom). This qualitative observation agrees with the coupling elements that were predicted in ref. 9. The two modulation amplitudes have been chosen such that they lead to a similar response of the pair excitation mode. The data are taken for EB = 0.09hfr. For this measurement the radial trap frequency was 2fr = 1,660 Hz. Error bars show the standard error of the mean. Each data point is the average of at least 24 measurements.

### Extended Data Fig. 4 Probabilities of different atom numbers remaining in the lowest two shells for the N = 6 initial state.

af, The probabilities for different remaining atom numbers after modulating the 6-atom ground state with a defined frequency given on the y axis and subsequent removal of excited atoms. All possible excitations manifest themselves by a reduced probability of remaining in the ground state of 6 atoms (a). We find that the lowest excitation, or Higgs mode, mostly consists of excitations to four atoms (c), while the higher excited peaks are predominantly generated by the loss of a single atom (b). For each setting the experiment is repeated between 42 and 47 times.

### Extended Data Fig. 5 Probabilities of different atom numbers remaining in the lowest three shells for the N = 12 initial state.

af, The probabilities for different remaining atom numbers after modulating the 12-atom ground state with a defined frequency given on the y axis and subsequent removal of excited atoms. All possible excitations manifest themselves by a reduced probability of remaining in the ground state of 12 atoms (a). We find that lowest excitation, or Higgs mode, mainly consists of excitations to ten atoms (c), while the higher excited peaks are predominantly generated by the loss of even more atoms (df). For each setting the experiment is repeated between 19 and 63 times.

### Extended Data Fig. 6 Numerically calculated excitation spectrum for 6 particles.

a, The level spectrum obtained by exact diagonalization with parameters A = 20 and γ = 0.99 for the potential as well as the experimental results. The calculation includes states up to $${E}_{{\rm{sp}}}^{{\rm{cut}}}=10\hbar {\omega }_{{\rm{r}}}$$ and up to a many-body energy of 28ħωr. For comparison the experimental data are shown by green diamonds. Values and errors bars are obtained as in Fig. 3c. b, The numerically calculated excitation spectrum for a modulation of the interaction strength. As in the experiment we observe that this modulation couples to two non-monotonous modes. We note that the calculations performed for b employ a smaller cut-off ($${E}_{{\rm{sp}}}^{{\rm{cut}}}=6\hbar {\omega }_{{\rm{r}}}$$ and a maximal many-body energy of 24ħωr) than in a owing to the computational demand in calculating the matrix element (equation (4)).

## Rights and permissions

Reprints and Permissions

Bayha, L., Holten, M., Klemt, R. et al. Observing the emergence of a quantum phase transition shell by shell. Nature 587, 583–587 (2020). https://doi.org/10.1038/s41586-020-2936-y

• Accepted:

• Published:

• Issue Date:

• ### Observation of Pauli Crystals

• Marvin Holten
• , Luca Bayha
• , Keerthan Subramanian
• , Carl Heintze
• , Philipp M. Preiss
•  & Selim Jochim

Physical Review Letters (2021)