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arxiv: 2501.09497 · v1 · submitted 2025-01-16 · ❄️ cond-mat.quant-gas

Atom-Molecule Superradiance and Entanglement with Cavity-Mediated Three-Body Interactions

Yun Chen , Yuqi Wang , Jingjun You , Yingqi Liu , Su Yi , Yuangang Deng This is my paper

Pith reviewed 2026-05-23 05:28 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas
keywords superradiancecavity QEDultracold atomsmoleculesthree-body interactionsquantum phase transitionentanglementphotoassociation
0
0 comments X

The pith

Cavity-enhanced photoassociation creates long-range three-body interactions that drive hybrid atom-molecule superradiance with cubic photon scaling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a scheme in which an optical cavity boosts photoassociation in an atomic Bose condensate to form biatomic molecules. The resulting tripartite coupling generates effective long-range three-body interactions among the particles. Above a critical pump strength these interactions stabilize a self-organized square lattice of molecules and produce a hybrid superradiant phase in which both atoms and molecules couple coherently to the cavity field, spontaneously breaking U(1) symmetry. The steady-state photon number in this phase scales with the cube of the total atom number, a signature of bosonic enhancement that differs from the scaling seen in purely atomic superradiance. Photon-matter entanglement is shown to serve as an order parameter for the transition.

Core claim

Cavity-mediated tripartite atom-molecule-photon coupling realizes dominant long-range three-body interactions that, beyond a pump threshold, stabilize a self-organized square-lattice molecular condensate and induce hybrid atom-molecule superradiance accompanied by spontaneous U(1) symmetry breaking; the resulting steady-state intracavity photon number scales cubically with total atom number.

What carries the argument

Tripartite cavity-atom-molecule coupling that mediates long-range three-body interactions and drives self-organization of the molecular lattice.

If this is right

  • A molecular square-lattice phase appears above a critical pump strength.
  • Hybrid atom-molecule superradiance emerges with spontaneous U(1) symmetry breaking.
  • Steady-state photon number scales cubically with total atom number.
  • Photon-matter entanglement serves as a diagnostic of the superradiant transition.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The scheme offers a route to engineer effective higher-order interactions that are otherwise hard to realize in ultracold gases.
  • Cubic scaling may amplify collective signals in larger ensembles, potentially aiding precision measurements.
  • Analogous cavity couplings could be explored with other molecular species or different lattice geometries.
  • The nonequilibrium dynamics may connect to studies of quantum superchemistry in driven-dissipative settings.

Load-bearing premise

Cavity-enhanced photoassociation can be arranged to produce stable biatomic molecules while keeping three-body interactions dominant over losses and competing two-body processes.

What would settle it

Direct measurement of the steady-state intracavity photon number as a function of total atom number, showing whether the scaling exponent is three or lower.

Figures

Figures reproduced from arXiv: 2501.09497 by Jingjun You, Su Yi, Yingqi Liu, Yuangang Deng, Yun Chen, Yuqi Wang.

Figure 1
Figure 1. Figure 1: (a) Scheme for creating atom-molecule superradi [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Ground state phase diagram for N = 1×104 and ~δ/EL = 1. The insets show α dependence of Vg for N and SQL phases. Ω dependence of ˜ |α| (b), Nm (c), and Θ (d) for different ∆˜ c. The density (e), phase (f), and momentum distribution for SQL phase with ~Ω˜/EL = 2 and ~∆˜ c/EL = 4 × 103 . ˆb †ˆb+2 ˆm†mˆ , δ ′ = δ+EL/~, and Ω is light-matter coupling. ˜ This nonlinear tripartite interaction, involving cavi… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Distributions of entropy S2 on Ω- ˜ ∆˜ c parameter plane. (b) Ω dependence of ˜ Ns (solid line) and Nm (dashed line). (c) Distribution of g (2) aa and g (2) bb as a function of Ω for ˜ ~∆˜ c/EL = 8 × 103 . Ω˜ cr ∼ N −1/2 . The slight deviation between analytic threshold (red dashed line) and numerical result (blue solid line) for large N is ascribe to PA field induced strong spatially dependent two-bod… view at source ↗
read the original abstract

Ultracold atoms coupled to optical cavities offer a powerful platform for studying strongly correlated many-body physics. Here, we propose an experimental scheme for creating biatomic molecules via cavity-enhanced photoassociation from an atomic condensate. This setup realizes long-range three-body interactions mediated by tripartite cavity-atom-molecule coupling. Beyond a critical pump strength, a self-organized square lattice phase for molecular condensate emerges, resulting in hybrid atom-molecule superradiance with spontaneous $U(1)$ symmetry breaking. Distinct from previously observed ultracold bosonic (fermionic) atomic superradiance, our findings demonstrate bosonic enhancement characterized by a cubic scaling of steady-state photon number with total atom number. Additionally, strong photon-matter entanglement is shown to effectively characterize superradiant quantum phase transition. Our findings deepen the understanding of quantum superchemistry and exotic many-body nonequilibrium dynamics in cavity-coupled quantum gases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes an experimental scheme for cavity-enhanced photoassociation to create biatomic molecules from an atomic Bose condensate, thereby engineering long-range three-body interactions through tripartite cavity-atom-molecule coupling. It predicts that above a critical pump strength a self-organized square-lattice phase of the molecular condensate appears, producing hybrid atom-molecule superradiance accompanied by spontaneous U(1) symmetry breaking. A distinctive bosonic-enhancement signature is claimed: the steady-state intracavity photon number scales cubically with total atom number. Photon-matter entanglement is further proposed as an order parameter for the superradiant quantum phase transition.

Significance. If the three-body interaction can be shown to dominate, the work would open a route to cavity-mediated quantum superchemistry and to nonequilibrium phases with tunable higher-order interactions. The N³ photon scaling constitutes a falsifiable, parameter-free prediction that differs qualitatively from conventional Dicke superradiance and could be tested in existing cavity-QED setups with ultracold bosons.

major comments (2)
  1. [Sections describing the effective Hamiltonian and steady-state analysis] The central claim that cavity-mediated three-body interactions dominate and produce the reported square-lattice phase and N³ scaling rests on the assumption that two-body inelastic losses and spontaneous-emission decoherence remain sub-dominant throughout the relevant parameter window. No quantitative comparison of the respective rates (photoassociation loss versus cavity-mediated three-body strength) is supplied for the self-organized regime.
  2. [Section on steady-state photon number and scaling analysis] The derivation of the cubic photon-number scaling is presented as a direct consequence of bosonic enhancement in the tripartite coupling, yet the manuscript does not demonstrate that this scaling survives when the molecular condensate fraction and the cavity detuning are varied self-consistently with the pump strength.
minor comments (2)
  1. [Model Hamiltonian section] Notation for the tripartite coupling strength and the effective three-body interaction coefficient should be introduced with explicit definitions and units in the main text rather than only in the supplementary material.
  2. [Figures showing phase diagram and photon number] Figure captions for the phase diagram should explicitly state the range of pump strengths and atom numbers over which the N³ scaling is observed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Sections describing the effective Hamiltonian and steady-state analysis] The central claim that cavity-mediated three-body interactions dominate and produce the reported square-lattice phase and N³ scaling rests on the assumption that two-body inelastic losses and spontaneous-emission decoherence remain sub-dominant throughout the relevant parameter window. No quantitative comparison of the respective rates (photoassociation loss versus cavity-mediated three-body strength) is supplied for the self-organized regime.

    Authors: We agree that a quantitative comparison of rates is necessary to substantiate the dominance of the three-body interactions. In the revised manuscript we will add an analysis (likely in an appendix) that compares the cavity-mediated three-body coupling strength to two-body photoassociation losses and spontaneous-emission decoherence, using typical experimental parameters for cavity-QED setups with ultracold bosons. This will identify the parameter window in which the coherent three-body dynamics remain dominant. revision: yes

  2. Referee: [Section on steady-state photon number and scaling analysis] The derivation of the cubic photon-number scaling is presented as a direct consequence of bosonic enhancement in the tripartite coupling, yet the manuscript does not demonstrate that this scaling survives when the molecular condensate fraction and the cavity detuning are varied self-consistently with the pump strength.

    Authors: The cubic scaling is obtained from the steady-state solution of the mean-field equations in which the molecular condensate fraction is determined self-consistently. In the revision we will augment the steady-state analysis with explicit expressions and numerical results showing how the molecular fraction and effective detuning evolve with pump strength, and we will verify that the N³ scaling persists throughout the self-organized phase. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from proposed model

full rationale

The paper proposes a cavity-enhanced photoassociation scheme to realize an effective Hamiltonian with tripartite coupling, then analyzes its mean-field or steady-state behavior to obtain the square-lattice molecular phase, U(1) breaking, and N^3 photon scaling. These outcomes follow from solving the model equations rather than from any fitted parameter renamed as a prediction, self-definitional loop, or load-bearing self-citation. No quoted step reduces a claimed result to its own input by construction. The dominance of three-body terms over losses is an assumption whose validity is external to the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no specific free parameters, axioms, or invented entities can be extracted from the provided text.

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Works this paper leans on

77 extracted references · 77 canonical work pages

  1. [1]

    Bloch, J

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

    work page 2008

  2. [2]

    Saffman, T

    M. Saffman, T. G. Walker, and K. Mølmer, Quantum information with rydberg atoms, Rev. Mod. Phys. 82, 2313 (2010)

    work page 2010

  3. [3]

    Ritsch, P

    H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Cold atoms in cavity-generated dynamical optical poten- tials, Rev. Mod. Phys. 85, 553 (2013)

    work page 2013

  4. [4]

    Qu´ em´ ener and P

    G. Qu´ em´ ener and P. S. Julienne, Ultracold molecules un- der control!, Chem. Rev. 112, 4949 (2012)

    work page 2012

  5. [5]

    J. J. Park, Y.-K. Lu, A. O. Jamison, T. V. Tscherbul, and W. Ketterle, A feshbach resonance in collisions between triplet ground-state molecules, Nature (London) 614, 54 (2023)

    work page 2023

  6. [6]

    Finelli, A

    S. Finelli, A. Ciamei, B. Restivo, M. Schem- mer, A. Cosco, M. Inguscio, A. Trenkwalder, K. Zaremba-Kopczyk, M. Gronowski, M. Tomza, and M. Zaccanti, Ultracold LiCr: A new pathway to quantum gases of paramagnetic polar molecules, PRX Quantum 5, 020358 (2024)

    work page 2024

  7. [7]

    Micheli, G

    A. Micheli, G. K. Brennen, and P. Zoller, A tool- box for lattice-spin models with polar molecules, Nat. Phys. 2, 341 (2006)

    work page 2006

  8. [8]

    J. P. Covey, S. A. Moses, M. G¨ arttner, A. Safavi-Naini, M. T. Miecnikowski, Z.-K. Fu, J. Schachenmayer, P. S. Julienne, A. M. Rey, D. S. Jin, and J. Ye, Doublon dy- namics and polar molecule production in an optical lat- tice, Nat. Commun. 7, 11279 (2016)

    work page 2016

  9. [9]

    Altman, K

    E. Altman, K. R. Brown, G. Carleo, L. D. Carr, E. Dem- ler, B. Chin, C.and DeMarco, S. E. Economou, M. A. Eriksson, K.-M. C. Fu, M. Greiner, K. R. Hazzard, R. G. Hulet, A. J. Koll´ ar, B. L. Lev, M. D. Lukin, R.-C. Ma, X. Mi, S. Misra, C. Monroe, K. Murch, Z. Nazario, K.-K. Ni, A. C. Potter, P. Roushan, M. Saffman, M. Schleier- Smith, I. Siddiqi, R. Simmon...

    work page 2021

  10. [10]

    J. A. Blackmore, L. Caldwell, P. D. Gregory, E. M. Bridge, R. Sawant, J. Aldegunde, J. Mur-Petit, D. Jaksch, J. M. Hutson, B. E. Sauer, M. R. Tar- butt, and S. L. Cornish, Ultracold molecules for quan- tum simulation: rotational coherences in caf and rbcs, Quantum Sci. Technol. 4, 014010 (2018)

    work page 2018

  11. [11]

    P. D. Gregory, J. A. Blackmore, S. L. Bromley, J. M. Hutson, and S. L. Cornish, Robust storage qubits in ul- tracold polar molecules, Nat. Phys. 17, 1149 (2021)

    work page 2021

  12. [12]

    Wang, Quantum phase transitions of polar molecules in bilayer systems, Phys

    D.-W. Wang, Quantum phase transitions of polar molecules in bilayer systems, Phys. Rev. Lett. 98, 060403 (2007)

    work page 2007

  13. [13]

    H. P. B¨ uchler, E. Demler, M. Lukin, A. Micheli, N. Prokof’ev, G. Pupillo, and P. Zoller, Strongly cor- related 2d quantum phases with cold polar molecules: Controlling the shape of the interaction potential, Phys. Rev. Lett. 98, 060404 (2007)

    work page 2007

  14. [14]

    N. R. Cooper and G. V. Shlyapnikov, Stable topological superfluid phase of ultracold polar fermionic molecules, Phys. Rev. Lett. 103, 155302 (2009)

    work page 2009

  15. [15]

    Capogrosso-Sansone, C

    B. Capogrosso-Sansone, C. Trefzger, M. Lewen- stein, P. Zoller, and G. Pupillo, Quantum phases of cold polar molecules in 2d optical lattices, Phys. Rev. Lett. 104, 125301 (2010)

    work page 2010

  16. [16]

    DeMille, Quantum computation with trapped polar molecules, Phys

    D. DeMille, Quantum computation with trapped polar molecules, Phys. Rev. Lett. 88, 067901 (2002)

    work page 2002

  17. [17]

    Andr´ e, D

    A. Andr´ e, D. DeMille, J. M. Doyle, M. D. Lukin, S. E. Maxwell, P. Rabl, R. J. Schoelkopf, and P. Zoller, A coherent all-electrical interface between pol ar molecules and mesoscopic superconducting resonators, Nat. Phys. 2, 636 (2006)

    work page 2006

  18. [18]

    Sawant, J

    R. Sawant, J. A. Blackmore, P. D. Gregory, J. Mur-Petit, D. Jaksch, J. Aldegunde, J. M. Hutson, M. R. Tarbutt, and S. L. Cornish, Ultracold polar molecules as qudits, New J. Phys. 22, 013027 (2020)

    work page 2020

  19. [19]

    Cornish, M

    S. Cornish, M. Tarbutt, and K. Hazzard, Quantum computation and quantum simulation with ultracold molecules, Nat. Phys. 20, 730 (2024)

    work page 2024

  20. [20]

    L. R. B. Picard, A. J. Park, G. E. Patenotte, S. Gebret- sadkan, D. Wellnitz, A. M. Rey, and K.-K. Ni, Entangle- ment and iswap gate between molecular qubits, Nature (London) 10.1038/s41586-024-08177-3 (2024)

    work page doi:10.1038/s41586-024-08177-3 2024

  21. [21]

    L. D. Carr, D. DeMille, R. V. Krems, and J. Ye, Cold and ultracold molecules: science, technology and appli- cations, New J. Phys. 11, 055049 (2009)

    work page 2009

  22. [22]

    C. Chin, V. V. Flambaum, and M. G. Kozlov, Ultracold molecules: new probes on the variation of fundamental constants, New J. Phys. 11, 055048 (2009)

    work page 2009

  23. [23]

    T. S. Roussy, L. Caldwell, T. Wright, W. B. Cairn- 12 cross, Y. Shagam, K. B. Ng, N. Schlossberger, S. Y. Park, A. Wang, J. Ye, and E. A. Cornell, An im- proved bound on the electron’s electric dipole moment, Science 381, 46 (2023)

    work page 2023

  24. [24]

    L. D. Marco, G. Valtolina, K. Matsuda, W. G. Tobias, J. P. Covey, and J. Ye, A degenerate fermi gas of polar molecules, Science 363, 853 (2019)

    work page 2019

  25. [25]

    W. G. Tobias, K. Matsuda, G. Valtolina, L. De Marco, J.-R. Li, and J. Ye, Thermalization and sub-poissonian density fluctuations in a degenerate molecular fermi gas, Phys. Rev. Lett. 124, 033401 (2020)

    work page 2020

  26. [26]

    Schindewolf, R

    A. Schindewolf, R. Bause, X.-Y. Chen, M. Duda, T. Karman, I. Bloch, and X.-Y. Luo, Evaporation of microwave-shielded polar molecules to quantum degen- eracy, Nature (London) 607, 677 (2022)

    work page 2022

  27. [27]

    Duda, X.-Y

    M. Duda, X.-Y. Chen, A. Schindewolf, R. Bause, J. von Milczewski, R. Schmidt, I. Bloch, and X.- Y. Luo, Transition from a polaronic condensate to a degenerate fermi gas of heteronuclear molecules, Nat. Phys. 19, 720 (2023)

    work page 2023

  28. [28]

    Prehn, A.and Ibr¨ ugger, R

    M. Prehn, A.and Ibr¨ ugger, R. Gl¨ ockner, G. Rempe, and M. Zeppenfeld, Optoelectrical cooling of po- lar molecules to submillikelvin temperatures, Phys. Rev. Lett. 116, 063005 (2016)

    work page 2016

  29. [29]

    Zeppenfeld, R

    M. Zeppenfeld, R. G. B. G. U. Englert, A. Prehn, M. Mie- lenz, C. Sommer, L. D. van Buuren, M. Motsch, and G. Rempe, Sisyphus cooling of electrically trapped poly- atomic molecules, Nature (London) 491, 570 (2012)

    work page 2012

  30. [30]

    Ding, Y.-W

    S.-Q. Ding, Y.-W. Wu, I. A. Finneran, J. J. Burau, and J. Ye, Sub-doppler cooling and com- pressed trapping of yo molecules at µ K temperatures, Phys. Rev. X 10, 021049 (2020)

    work page 2020

  31. [31]

    Mitra, N

    D. Mitra, N. B. Vilas, C. Hallas, L. Anderegg, B. L. Augenbraun, L. Baum, C. Miller, S. Raval, and J. M. Doyle, Direct laser cooling of a symmetric top molecule, Science 369, 1366 (2020)

    work page 2020

  32. [32]

    O. S. d. M. M. H. P. A. N. B. Z. J. J. K. S. J. P. S. J. D. S. Ni, K. K. and J. Ye, A high phase-space-density gas of polar molecules, Science 322, 231 (2008)

    work page 2008

  33. [33]

    C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010)

    work page 2010

  34. [34]

    S. A. Moses, J. P. Covey, M. T. Miecnikowski, B. Yan, B. Gadway, J. Ye, and D. S. Jin, Creation of a low- entropy quantum gas of polar molecules in an optical lattice, Science 350, 659 (2015)

    work page 2015

  35. [35]

    J. T. Zhang, Y. Yu, W. B. Cairncross, K. Wang, L. R. B. Picard, J. D. Hood, Y.-W. Lin, J. M. Hutson, and K.-K. Ni, Forming a single molecule by magnetoassociation in an optical tweezer, Phys. Rev. Lett. 124, 253401 (2020)

    work page 2020

  36. [36]

    K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, Ultracold photoassociation spectroscopy: Long-range molecules and atomic scattering, Rev. Mod. Phys. 78, 483 (2006)

    work page 2006

  37. [37]

    Winkler, G

    K. Winkler, G. Thalhammer, M. Theis, H. Ritsch, R. Grimm, and J. H. Denschlag, Atom-molecule dark states in a bose-einstein condensate, Phys. Rev. Lett. 95, 063202 (2005)

    work page 2005

  38. [38]

    L. R. Liu, J. D. Hood, Y. Yu, J. T. Zhang, N. R. Hutzler, T. Rosenband, and K.-K. Ni, Building one molecule from a reservoir of two atoms, Science 360, 900 (2018)

    work page 2018

  39. [39]

    Cao, B.-Y

    J. Cao, B.-Y. Wang, H. Yang, Z.-J. Fan, Z. Su, J. Rui, B. Zhao, and J.-W. Pan, Observation of photoassoci- ation resonances in ultracold atom-molecule collisions, Phys. Rev. Lett. 132, 093403 (2024)

    work page 2024

  40. [40]

    S. A. Will, J. W. Park, Z. Z. Yan, H. Loh, and M. W. Zwierlein, Coherent microwave control of ultracold 23Na40K molecules, Phys. Rev. Lett. 116, 225306 (2016)

    work page 2016

  41. [41]

    Cappellini, L

    G. Cappellini, L. F. Livi, L. Franchi, D. Tusi, D. Bene- dicto Orenes, M. Inguscio, J. Catani, and L. Fallani, Co- herent manipulation of orbital feshbach molecules of two- electron atoms, Phys. Rev. X 9, 011028 (2019)

    work page 2019

  42. [42]

    J. W. Park, S. A. Will, and M. W. Zwierlein, Ultracold dipolar gas of fermionic 23Na40K molecules in their abso- lute ground state, Phys. Rev. Lett. 114, 205302 (2015)

    work page 2015

  43. [43]

    M. Guo, B. Zhu, B. Lu, X. Ye, F. Wang, R. Vexiau, N. Bouloufa-Maafa, G. Qu´ em´ ener, O. Dulieu, and D. Wang, Creation of an ultracold gas of ground-state dipolar 23Na87Rb molecules, Phys. Rev. Lett. 116, 205303 (2016)

    work page 2016

  44. [44]

    L. R. B. Picard, G. E. Patenotte, A. J. Park, S. F. Gebretsadkan, and K.-K. Ni, Site-selective preparation and multistate readout of molecules in optical tweezers, PRX Quantum 5, 020344 (2024)

    work page 2024

  45. [45]

    J.-R. Li, K. Matsuda, C. Miller, A. N. Carroll, W. G. Tobias, J. S. Higgins, and J. Ye, Tun- able itinerant spin dynamics with polar molecules, Nature (London) 614, 70 (2023)

    work page 2023

  46. [46]

    Christakis, J

    L. Christakis, J. S. Rosenberg, R. Raj, S. Chi, A. Morn- ingstar, D. A. Huse, Z. Z. Yan, and W. S. Bakr, Probing site-resolved correlations in a spin system of ultracold molecules, Nature (London) 614, 64 (2023)

    work page 2023

  47. [47]

    Karman and J

    T. Karman and J. M. Hutson, Microwave shielding of ultracold polar molecules, Phys. Rev. Lett. 121, 163401 (2018)

    work page 2018

  48. [48]

    Anderegg, S

    L. Anderegg, S. Burchesky, Y.-C. Bao, S. S. Yu, T. Kar- man, E. Chae, K.-K. Ni, W. Ketterle, and J. M. Doyle, Observation of microwave shielding of ultracold molecules, Science 373, 779 (2021)

    work page 2021

  49. [49]

    X.-Y. Chen, A. Schindewolf, S. Eppelt, R. Bause, M. Duda, S. Biswas, T. Karman, T. Hilker, I. Bloch, and X.-Y. Luo, Field-linked resonances of polar molecules, Nature (London) 614, 59 (2023)

    work page 2023

  50. [50]

    Bigagli, W

    N. Bigagli, W. Yuan, S. Zhang, B. Bulatovic, T. Karman, I. Stevenson, and S. Will, Observation of bose-einstein condensation of dipolar molecules, Nature (London) 631, 289 (2024)

    work page 2024

  51. [51]

    Mivehvar, F

    F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, Cavity qed with quantum gases: new paradigms in many-body physics, Adv. Phys 70, 1 (2021)

    work page 2021

  52. [52]

    Zhang, Y

    X.-T. Zhang, Y. Chen, Z.-M. Wu, J. Wang, J.-J. Fan, S. j. Deng, and H.-B. Wu, Observation of a superradiant quantum phase transition in an intracavity degenerate fermi gas, Science 373, 1359 (2021)

    work page 2021

  53. [53]

    Baumann, C

    K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, Dicke quantum phase transition with a superfluid gas in an optical cavity, Nature (London) 464, 1301 (2010)

    work page 2010

  54. [55]

    L´ eonard, A

    J. L´ eonard, A. Morales, P. Zupancic, T. Esslinger, and T. Donner, Supersolid formation in a quan- tum gas breaking a continuous translational symmetry, Nature (London) 543, 87 (2017)

    work page 2017

  55. [56]

    L´ eonard, A

    J. L´ eonard, A. Morales, P. Zupancic, T. Donner, and T. Esslinger, Monitoring and manipulating higgs 13 and goldstone modes in a supersolid quantum gas, Science 358, 1415 (2017)

    work page 2017

  56. [57]

    Deng and S

    Y.-G. Deng and S. Yi, Self-ordered supersolid phase beyond dicke superradiance in a ring cavity, Phys. Rev. Res. 5, 013002 (2023)

    work page 2023

  57. [58]

    R. M. Kroeze, Y. Guo, V. D. Vaidya, J. Keeling, and B. L. Lev, Spinor self-ordering of a quantum gas in a cavity, Phys. Rev. Lett. 121, 163601 (2018)

    work page 2018

  58. [59]

    R. M. Kroeze, Y. Guo, and B. L. Lev, Dy- namical spin-orbit coupling of a quantum gas, Phys. Rev. Lett. 123, 160404 (2019)

    work page 2019

  59. [60]

    Bilitewski, L

    T. Bilitewski, L. De Marco, J.-R. Li, K. Matsuda, W. G. Tobias, G. Valtolina, J. Ye, and A. M. Rey, Dynam- ical generation of spin squeezing in ultracold dipolar molecules, Phys. Rev. Lett. 126, 113401 (2021)

    work page 2021

  60. [61]

    Y.-C. Bao, S. S. Yu, L. Anderegg, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, Dipolar spin-exchange and entanglement between molecules in an optical tweezer ar- ray, Science 382, 1138 (2023)

    work page 2023

  61. [62]

    C. M. Holland, Y.-K. Lu, and L. W. Cheuk, On-demand entanglement of molecules in a reconfigurable optical tweezer array, Science 382, 1143 (2023)

    work page 2023

  62. [63]

    Finger, R

    F. Finger, R. Rosa-Medina, N. Reiter, P. Christodoulou, T. Donner, and T. Esslinger, Spin- and momentum- correlated atom pairs mediated by photon ex- change and seeded by vacuum fluctuations, Phys. Rev. Lett. 132, 093402 (2024)

    work page 2024

  63. [64]

    T. Rom, T. Best, O. Mandel, A. Widera, M. Greiner, T. W. H¨ ansch, and I. Bloch, State se- lective production of molecules in optical lattices, Phys. Rev. Lett. 93, 073002 (2004)

    work page 2004

  64. [65]

    S. Will, T. Best, U. Schneider, L. Hackerm¨ uler, D.-S. L¨ uhmann, and I. Bloch, Time-resolved observation of coherent multi-body interactions in quantum phase re- vivals, Nature (London) 465, 197 (2010)

    work page 2010

  65. [66]

    Kraemer, M

    T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. N¨ agerl,et al. , Evidence for efimov quantum states in an ultracold gas of caesium atoms, Nature 440, 315 (2006)

    work page 2006

  66. [67]

    Sekino and Y

    Y. Sekino and Y. Nishida, Quantum droplet of one-dimensional bosons with a three-body attraction, Phys. Rev. A 97, 011602 (2018)

    work page 2018

  67. [68]

    P. R. Johnson, E. Tiesinga, J. V. Porto, and C. J. Williams, Effective three-body inter- actions of neutral bosons in optical lattices, New Journal of Physics 11, 093022 (2009)

    work page 2009

  68. [69]

    See supplemental material for a derivation and discus- sion.,

    work page

  69. [70]

    Chotia, B

    A. Chotia, B. Neyenhuis, S. A. Moses, B. Yan, J. P. Covey, M. Foss-Feig, A. M. Rey, D. S. Jin, and J. Ye, Long-lived dipolar molecules and feshbach molecules in a 3d optical lattice, Phys. Rev. Lett. 108, 080405 (2012)

    work page 2012

  70. [71]

    J. L. Bohn, A. M. Rey, and J. Ye, Cold molecules: Progress in quantum engineering of chemistry and quan- tum matter, Science 357, 1002 (2017)

    work page 2017

  71. [72]

    H. Son, J. J. Park, Y.-K. Lu, A. O. Jamison, T. Kar- man, and W. Ketterle, Control of reactive collisions by quantum interference, Science 375, 1006 (2022)

    work page 2022

  72. [73]

    W. G. Tobias, K. Matsuda, J.-R. Li, C. Miller, A. N. Car- roll, T. Bilitewski, A. M. Rey, and J. Ye, Reactions be- tween layer-resolved molecules mediated by dipolar spin exchange, Science 375, 1299 (2022)

    work page 2022

  73. [74]

    Luo, Y.-Q

    X.-Y. Luo, Y.-Q. Zou, L.-N. Wu, Q. Liu, M.-F. Han, M. K. Tey, and L. You, Deterministic entanglement gen- eration from driving through quantum phase transitions, Science 355, 620 (2017)

    work page 2017

  74. [75]

    Mivehvar, F

    F. Mivehvar, F. Piazza, T. Donner, and H. Ritsch, Cavity qed with quantum gases: new paradigms in many-body physics, Advances in Physics 70, 1 (2021)

    work page 2021

  75. [76]

    Mottl, F

    R. Mottl, F. Brennecke, K. Baumann, R. Landig, T. Don- ner, and T. Esslinger, Roton-type mode softening in a quantum gas with cavity-mediated long-range interac- tions, Science 336, 1570 (2012)

    work page 2012

  76. [77]

    P. B. Blakie, D. Baillie, L. Chomaz, and F. Fer- laino, Supersolidity in an elongated dipolar condensate, Phys. Rev. Res. 2, 043318 (2020)

    work page 2020

  77. [78]

    Baksic and C

    A. Baksic and C. Ciuti, Controlling discrete and continuous symmetries in ”superradiant” phase transitions with circuit qed systems, Phys. Rev. Lett. 112, 173601 (2014)

    work page 2014