Resonant Bose Matter and Metastable Branches

A resonant Bose gas is intrinsically metastable in present cold-atom realizations because the same short-range three-body configurations that generate Efimov physics also enable recombination into deeply bound molecules. A useful state can still form after a quench, but its name and equation of state are meaningful only relative to the preparation protocol and the ordering of many-body, equilibration, range, and loss times.

Required background. Efimov Physics and the Three-Body Parameter supplies $\kappa_*$ and inelasticity, and Universal Relations and Tan Contact supplies short-distance observables.

Helpful background. Lindblad Field Dynamics describes reduced evolution when loss can be treated as Markovian.

Density and lifetime scales

For a homogeneous gas define

$$k_n=(6\pi^2n)^{1/3}, \qquad E_n=\frac{k_n^2}{2m}, \qquad t_n=\frac{1}{E_n}.$$

The numerical convention for $k_n$ is chosen for comparison with a one-component ideal Fermi density; all reported dimensionless coefficients must use the same definition. At two-body unitarity, a density-only transient would scale in $k_n$, $E_n$, and $t_n$. Resonance width adds $k_nR^*$, range adds $k_nR$, and Efimov physics adds $k_n/\kappa_*$. None is removed by setting $a^{-1}=0$.

If local three-body loss obeys

$$\frac{\mathrm dn}{\mathrm dt}=-L_3n^3,$$

the instantaneous per-particle loss time is

$$\tau_3=\frac{1}{L_3n^2}.$$

Heating and preferential loss make $L_3$ and $n$ time dependent in a real protocol. A single initial $\tau_3$ is therefore only a first diagnostic.

Three distinct state claims

Short-time correlated state. For $t_{\rm range}\ll t\lesssim t_n$, two-body correlations and a high-momentum tail can build before global equilibration. The observable is explicitly time dependent.

Prethermal or quasi-steady state. If local observables become nearly stationary over a window

$$t_n\lesssim t\ll\tau_3,$$

while slow variables such as density continue to drift, a prethermal description may be justified. Collapse of momentum distributions after instantaneous density rescaling tests this statement; it does not establish Gibbs equilibrium.

Equilibrium metastable branch. This stronger term requires internal equilibration time $\tau_{\rm eq}\ll\tau_3$, protocol independence within uncertainty, thermodynamic relations among energy, pressure, and response, and a specified continuation that excludes deeply bound clusters. A stationary-looking momentum distribution alone is insufficient.

Experiments on quenched homogeneous gases observed density-scaled prethermal dynamics Eigen et al. 2018, building on earlier time-resolved observations Makotyn et al. 2014. These results support transient universality in their measured windows, not a stable ground state.

Contacts, Efimov scale, and loss

The two-body contact controls the leading $k^{-4}$ momentum tail. For identical bosons, a three-body contact and log-periodic dependence on $\kappa_*$ also enter subleading short-distance observables. Measurements of both contacts at unitarity found explicitly developing two- and three-body correlations Fletcher et al. 2017.

Loss is not merely an experimental nuisance that can be removed from the Hamiltonian prediction. It changes density, injects correlation-dependent selection, and can heat the remaining gas. A non-Hermitian three-body term or Lindblad jump $L(\mathbf x)\propto\psi^3(\mathbf x)$ may reproduce inclusive number decay, but coherent deep-channel memory, molecule production, and finite-range dynamics can invalidate a local Markov model.

Evidence boundary through 10 August 2026

The current-source check for this page was completed on 10 August 2026. By that cutoff, the strongest experimental evidence supports universal or approximately universal post-quench dynamics over finite observation windows, with measured three-body loss and developing contacts. A 2025 narrow-resonance experiment instead found coherent atom–molecule oscillations governed by the Feshbach coupling van de Kraats et al. 2025, directly demonstrating that density-only scaling is not generic when resonance width is resolved. A 2024 conserving triplet theory describes departure from the prethermal stage but remains a model calculation rather than evidence of an equilibrium phase van de Kraats et al. 2024.

Consequently, no protocol-independent, long-lived equilibrium equation of state for a broad-resonance unitary Bose gas is treated here as experimentally established. Thermodynamically stable model gases with a three-body regulator can possess Efimov-liquid phases, but the regulator and stability mechanism are physical parts of those models, as made explicit by Piatecki and Krauth 2014.

New experimental phase claims, updated resonance records, and evidence assessments after this cutoff belong in Quantum Matter and Emergent Phenomena Research; they should not be silently promoted into this teaching account.

A validity assessment for a proposed branch

Report at least:

1. $a$, $R$ or $R^*$, $\kappa_*$, and the loss calibration;
2. $k_nR$, $k_nR^*$, and $k_n/\kappa_*$ over the evolving density range;
3. ramp time, hold time, $t_n$, $\tau_{\rm eq}$ estimate, and $\tau_3$;
4. which observables plateau and whether they collapse under instantaneous scaling;
5. sensitivity to initial temperature, ramp shape, and density; and
6. whether thermodynamic, contact, and number-decay relations agree.

If $\tau_{\rm eq}$ is not separated from $\tau_3$, restrict the conclusion to transient dynamics. If $k_nR^*$ is not small, retain explicit molecular dynamics.

Common pitfalls

Calling a plateau equilibrium. A slowly varying prethermal observable can coexist with loss and memory of the quench.

Dropping the Efimov input at unitarity. Two-body scale invariance does not eliminate $\kappa_*$ or inelasticity.

Rescaling by the initial density only. Loss changes $k_n$ and $t_n$ during the observation window.

Exercises

Compare many-body and loss times

A gas has $E_n/h=5\,\mathrm{kHz}$ and a per-particle loss rate $1/\tau_3=500\,\mathrm{s}^{-1}$. Estimate $\tau_3/t_n$ using $t_n=\hbar/E_n$.

Solution

$E_n/\hbar=2\pi\times5\,\mathrm{kHz}$, so $t_n\simeq31.8\,\mu\mathrm s$. The loss time is $2\,\mathrm{ms}$, giving $\tau_3/t_n\simeq63$. This permits many density times before substantial loss, but equilibrium still requires an independent estimate of $\tau_{\rm eq}$.

Classify a narrow-resonance quench

Suppose $k_nR^*=1.5$ and coherent molecular oscillations are observed. Can a density-only, one-channel prethermal description be inferred?

Solution

No. The width parameter is leading because $k_nR^*$ exceeds unity, and the observed molecular frequency supplies an additional scale. A two-channel dynamical model and probe-calibrated molecular population are required.

Continue

From Few-Body Inputs to Many-Body Predictions turns these times and scales into an error budget. Few-Body Data in the Virial Expansion gives the controlled high-temperature equilibrium limit. Two-Channel Resonance Models treats width-driven molecular dynamics.

References

• Eigen, Christoph, Jake A. P. Glidden, Raphael Lopes, Eric A. Cornell, Robert P. Smith, and Zoran Hadzibabic. “Universal Prethermal Dynamics of Bose Gases Quenched to Unitarity.” Nature 563 (2018): 221–224. DOI.
• Fletcher, Richard J., Jay Man, Raphael Lopes, Petar Christodoulou, Julian Schmitt, Maximilian Sohmen, Nir Navon, Robert P. Smith, and Zoran Hadzibabic. “Two- and Three-Body Contacts in the Unitary Bose Gas.” Science 355 (2017): 377–380. DOI.
• Makotyn, P., C. E. Klauss, D. L. Goldberger, E. A. Cornell, and D. S. Jin. “Universal Dynamics of a Degenerate Unitary Bose Gas.” Nature Physics 10 (2014): 116–119. DOI.
• Piatecki, Swann, and Werner Krauth. “Efimov-Driven Phase Transitions of the Unitary Bose Gas.” Nature Communications 5 (2014): 3503. DOI.
• van de Kraats, J., D. J. M. Ahmed-Braun, V. E. Colussi, and S. J. J. M. F. Kokkelmans. “Resonance Triplet Dynamics in the Quenched Unitary Bose Gas.” Physical Review Research 6 (2024): L012056. DOI.
• van de Kraats, J., D. J. M. Ahmed-Braun, B. K. Yuen, C. Robens, V. E. Colussi, S. J. J. M. F. Kokkelmans, and M. W. Mitchell. “Universal Coherent Atom–Molecule Oscillations in the Dynamics of the Unitary Bose Gas near a Narrow Feshbach Resonance.” Physical Review Research 7 (2025): L012025. DOI.