Energetics and structural properties of two- and three-boson systems in the presence of 1D spin-orbit coupling

Doerte Blume; Qingze Guan

REVIEW 2 major objections 5 minor 73 references

Energetics and structural properties of two- and three-boson systems in the presence of 1D spin-orbit coupling

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Each Efimov state becomes four states under 1D spin-orbit coupling, and the ground state binds more deeply.

desk verdict A solid numerical follow-up that maps SOC-modified two- and three-boson binding, but the unquantified equal-scattering-length assumption keeps the headline critical values from being ready for direct experimental comparison. read the letter →

arxiv 1908.06918 v1 pith:HV434IYX submitted 2019-08-19 cond-mat.quant-gas

Qingze Guan , Doerte Blume This is my paper · ORCID

classification cond-mat.quant-gas

keywords Efimoveffectspin-orbitcouplingthree-bosonsystembindingenergyRashba-Dresselhausgeneralizeddetuningmomentumdistributioncoldatoms

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what happens to the Efimov effect—the discrete scaling symmetry that produces an infinite ladder of three-boson states—when each boson carries a one-dimensional spin-orbit coupling of the equal Rashba-Dresselhaus form. It establishes that every ordinary Efimov energy level becomes a manifold of four levels, and that the lowest level of the lowest manifold binds more strongly while the three excited levels bind more weakly than in the absence of spin-orbit coupling. The paper maps the binding-energy surfaces of the two- and three-boson systems over the scattering length and detuning parameters, showing that the binding enhancement tracks the degeneracy of the lowest non-interacting dispersion curve. This is relevant because it gives cold-atom experiments a concrete few-body signature of spin-orbit coupling and provides the two-body threshold input needed for many-body studies of spin-orbit-coupled Bose gases.

What carries the argument

The load-bearing object is the lowest non-interacting relative dispersion curve $E^{(0)}_{N,{\rm rel,ni}}(q_{1,z}, \ldots, \tilde\delta)$ of the $N$-boson relative Hamiltonian, with the generalized detuning $\tilde\delta = \delta/2 + \hbar k_{\rm so} q_{N,z}/\mu_N$ combining the bare detuning with the center-of-mass quasi-momentum. The threshold energy is the minimum of this curve, and the paper shows that the ground-state binding energy is maximized at parameter values where this minimum has maximal degeneracy: three degenerate minima for the two-boson system near $\tilde\delta = 1.7E_{\rm so}$, and four degenerate minima for the three-boson system near $\tilde\delta = 0$ with a secondary enhancement near $\tilde\delta = 2.27E_{\rm so}$. The momentum distributions of weakly-bound states are concentrated at the same quasi-momenta, linking the single-particle dispersion structure to the few-body wave function.

What would settle it

A loss-spectroscopy experiment with Raman-dressed bosons at $\Omega = 2E_{\rm so}$ and $\tilde\delta = 0$ could check whether the three-body loss resonance marking the ground Efimov trimer appears near $(a_s k_{\rm so})^{-1} = -1.304$ rather than near $-0.504$; observing the resonance at the latter would indicate the predicted binding enhancement is absent.

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Extended reading notes

Core claim

For a system of three identical bosons with short-range two-body interactions and 1D spin-orbit coupling of equal Rashba-Dresselhaus type, the discrete scale invariance of the Efimov effect survives, and each Efimov state is replaced by a four-state manifold. For Raman coupling $\Omega = 2E_{\rm so}$, the ground state of the lowest manifold is bound for $(a_s k_{\rm so})^{-1} \ge -1.304$, whereas with infinitesimally small Raman coupling it is bound only for $(a_s k_{\rm so})^{-1} \ge -0.504$; the excited states of the manifold are less bound than their counterparts without spin-orbit coupling. The two-boson system likewise gains a bound state on the negative scattering-length side that does not exist without spin-orbit coupling, with binding strongest near a generalized detuning of about $1.7E_{\rm so}$. In both systems, the binding energy of the ground state is largest when the global minimum of the lowest non-interacting relative dispersion curve has the largest degeneracy, and the momentum distributions of weakly-bound states mirror the locations of those minima.

Load-bearing premise

The paper assumes the two-body interaction is identical in all spin channels, whereas real Raman-dressed cold-atom experiments generally have different scattering lengths in different spin channels.

Editorial extensions

If this is right

At $\Omega = 2E_{\rm so}$, the lowest three-boson Efimov state remains bound on the negative scattering-length side down to $(a_s k_{\rm so})^{-1} = -1.304$, while the excited states of the manifold become bound only for larger $(a_s k_{\rm so})^{-1}$ than in the spin-orbit-free case.
The two-boson system acquires a dimer on the negative scattering-length side when spin-orbit coupling is finite, with the most favorable generalized detuning near $1.7E_{\rm so}$.
The critical generalized total momentum of the total ground state jumps discontinuously when the ground state switches from bound to scattering character, yielding phase diagrams with distinct finite-momentum phases for both two and three bosons.
The generalized radial scaling law from the authors' prior work implies that the mapped lowest-manifold energy surfaces also describe all higher Efimov manifolds after discrete rescaling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the equal-scattering-length assumption is relaxed, the four-state manifold will split asymmetrically and the phase boundaries will shift; the qualitative binding enhancement may survive, but the quantitative contours will not. A dedicated calculation with spin-dependent scattering lengths could test this.
The correlation between binding enhancement and dispersion-curve degeneracy suggests a mechanism generalizable to other single-particle dispersions: any modification that increases the degeneracy of the non-interacting threshold should enhance the few-body ground-state binding. Other synthetic gauge-field geometries could be probed this way.
The predicted zero average mechanical momentum of the total ground state offers a direct experimental check: time-of-flight imaging of the dressed-state cloud should show no net mechanical momentum even when the generalized momentum is finite.
The enhanced two-body binding on the negative scattering-length side implies that spin-orbit-coupled Bose gases may support dimer formation where ordinary s-wave Bose gases do not, with potential consequences for pairing and droplet physics.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: this is a competent follow-up to the same group's PRX paper. The fourfold-manifold structure and generalized scaling law were already there; what's new here is the detailed numerical mapping of the lowest manifold energy surfaces for Omega=2Eso, the two-boson binding-energy surfaces and phase diagram, the three-boson phase diagram, and the momentum distributions. The analytic Omega=0 construction in Appendix A is a genuine addition: it shows how the manifold arises from a single short-range energy for all four Mz channels, and it gives independent support for the numerical results.

The paper does several things well. The Hellmann-Feynman argument that the critical total momentum has zero average lab-frame mechanical momentum is clean and useful. The comparison between the binding-energy maxima and the degeneracy of the non-interacting dispersion minima is a nice structural insight, and the momentum distributions back it up. The computations are done from the explicit Hamiltonian with no fitting to a scaling law, so the circularity burden is low.

The soft spots are real but not fatal. First, no convergence checks or error bars are reported, and the contour plots lack numeric labels, so the quantitative claims (e.g., the critical (askso)^-1 = -1.304 for the three-boson ground state) cannot be independently checked from the preprint. Second, the spin-independent two-body interaction is load-bearing. The fourfold manifold in Appendix A relies on one short-range energy E3,sr for all four Mz states. In a real Raman-coupled system the parallel and antiparallel scattering lengths generally differ; that asymmetry will split the manifold, move the atom-dimer threshold, and shift the phase boundaries in Figs. 4 and 9. The paper acknowledges this in Sec. III.A but gives no estimate of the size of the effect. For a paper that quotes critical scattering lengths for atom-loss experiments, this is the largest unquantified gap between model and experiment. Third, the results are for a single Omega=2Eso and a single kso/kappa* ratio, so the parameter space covered is a slice.

Despite that, the central qualitative claim—that finite SOC enhances the ground-state dimer and trimer binding and weakens excited states—is well supported and likely robust, even if unequal scattering lengths move the numbers. The paper is honest about its assumptions, and the analytic Omega=0 construction gives it independent grounding. I'd send it to peer review; the referee should push for error bars, labeled contours, and ideally a quantitative statement about unequal scattering lengths before publication. I would cite it for the phase diagrams and the momentum-distribution analysis.

Referee Report

2 major / 5 minor

Summary. This paper presents a detailed numerical and analytical study of two- and three-boson systems with 1D spin-orbit coupling of the equal Rashba-Dresselhaus type. The authors map out the binding energy surfaces of the lowest two- and three-boson states as functions of the s-wave scattering length, the generalized detuning, and the Raman coupling, focusing on Omega = 2 Eso. They show that each ordinary Efimov state splits into a four-state manifold when Omega = 0, and that for finite Omega the ground state of the lowest manifold can be more strongly bound than in the absence of spin-orbit coupling, while the excited states are less bound. The binding enhancement is correlated with the degeneracy of the lowest non-interacting relative dispersion curves. The paper also computes momentum distributions of weakly bound states, total ground state energies as functions of the center-of-mass momentum, and phase diagrams that distinguish scattering and bound states with different total momenta. Analytical results for Omega = 0 are provided in Appendix A, and the Hellman-Feynman argument in Sec. II.C shows that the critical total momentum of the ground state corresponds to vanishing mechanical momentum in the lab frame.

Significance. If the results hold, this is a valuable extension of few-body Efimov physics to systems with synthetic spin-orbit coupling. The paper provides concrete quantitative predictions, such as the critical scattering length (askso)^-1 >= -1.304 for the enhanced binding of the three-boson ground state at Omega = 2 Eso, and a physical picture (degeneracy of the non-interacting dispersion) that explains the binding trends. The analytic Omega = 0 construction in Appendix A is a clean and useful result, and the systematic mapping of energy surfaces and phase diagrams goes significantly beyond the authors' earlier work. The Hellman-Feynman argument for the vanishing lab-frame momentum at the critical total momentum is elegant. The manuscript is generally well organized and should be of interest to the ultracold-atom and few-body communities, provided the scope limitations and numerical convergence issues are addressed.

major comments (2)

[Abstract, Secs. II.D, IV.A, V] The abstract and the concluding section state the fourfold-manifold result and the quantitative critical value (askso)^-1 >= -1.304 without the qualification that the two-body interactions are assumed to be identical in all spin channels. This assumption, introduced in Sec. II.D, is essential: the Omega = 0 construction in Appendix A (Eqs. A7-A10) uses a single short-range energy E3,sr for all M_z states, and any spin dependence in the interactions would split the fourfold degeneracy and modify the thresholds and phase boundaries. Since the abstract advertises the fourfold manifold as a general consequence of 1D spin-orbit coupling, the authors should either qualify the claims explicitly or add a discussion of the expected modifications when the parallel and antiparallel scattering lengths differ.
[Secs. III and IV (Figs. 1, 4, 5, 9)] The quantitative predictions, including the critical values -1.304 and -1.016 and the phase boundaries in Figs. 4 and 9, are obtained with explicitly correlated Gaussian calculations, but no convergence checks, basis sizes, or numerical uncertainties are reported. A statement about the convergence with respect to basis size and range parameters (or a specific reference to convergence tests in Ref. [41]) is needed to support the precision of these numbers, which are central to the paper's claims.

minor comments (5)

[Figs. 1, 4, 5, 9] The contour plots and phase diagrams do not have numeric labels on the contours, which makes quantitative reading of the energy surfaces and phase boundaries difficult; please add contour labels or a color bar with values.
[Sec. III.A] The phrase 'infinitesimally small but finite Omega' should be defined explicitly at first use, e.g., as the Omega -> 0+ limit in which the threshold is taken as the absolute minimum of all non-interacting relative dispersion curves.
[Sec. III.A] The statement that 'the qualitative behavior is expected to be similar' for unequal scattering lengths is unsupported; please either soften this remark or provide a heuristic argument for why the enhancement would persist.
[Secs. III and IV, Eq. (31)-(32)] Please specify whether the momentum distributions n(q) are normalized and to what value, since they are used to compare peak positions with dispersion minima.
[Sec. II.C] The discussion of the Hellman-Feynman result would benefit from a brief caveat that the argument assumes differentiability of the ground-state energy as a function of q_N; the phase boundaries in Figs. 4 and 9 involve degeneracies, and a note on how the conclusion applies there would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the binding-energy surfaces and phase boundaries follow from direct Hamiltonian diagonalization; the only self-citation (generalized scaling law) is used to extend, not to construct, the results.

full rationale

The paper's main results — two- and three-boson binding energies, critical scattering lengths, momentum distributions, and phase diagrams — are obtained by numerically diagonalizing the relative Hamiltonian (Eqs. (10)-(13)) with explicitly correlated Gaussians for fixed kso, Omega, delta-tilde, and kappa*/kso, and by the Omega=0 analytical construction in Appendix A. The Omega=0 construction (Eqs. A7-A14) starts from known Efimov eigenstates at Omega=delta=kso=0, applies the momentum-dependent spin transformation, and verifies the resulting states are eigenstates; it is a genuine derivation, not a restatement of the target result. Threshold energies in Eqs. (15)-(18) are computed from the same Hamiltonian. No parameter is fitted to the binding-energy surfaces being predicted; the three-body regulator kappa* is fixed once at unitarity in the absence of SOC, and the quoted critical value (askso)^-1 >= -1.304 is an output of the diagonalization. The paper also checks that successive Efimov binding energies reproduce the zero-range scaling factor to within 0.1% (Sec. IV), providing an external consistency check. The only self-citation is Ref. [41]'s generalized radial scaling law, invoked to state that the lowest-manifold plots also describe higher manifolds; this extends the results but is not the basis for the computed surfaces, so it is not load-bearing in a circular sense. The equal-scattering-length assumption is explicitly flagged as a limitation requiring separate calculations, which is a correctness caveat rather than circularity. No equation in the paper is equivalent to its own input by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard few-body physics assumptions: the two-body Gaussian potential tuned to a scattering length, a three-body Gaussian regulator tuned to kappa*, universality, and the validity of the explicitly correlated Gaussian variational approach. The spin-orbit coupled Hamiltonian is taken from the standard Raman scheme. No hidden fitted parameters or invented entities are introduced; the only tunable quantities are physical parameters (as, kappa*, Omega, delta) and regulator parameters whose effects are supposed to drop out in the universal regime.

free parameters (2)

v0 (Gaussian two-body depth) = tuned to desired s-wave scattering length as
The depth of the model potential in Eq. (27) is adjusted to reproduce the physical scattering length as; in the universal regime the results depend on as, not v0.
V0 (three-body Gaussian strength) = tuned to kappa* = 0.0152/r0
The height of the three-body regulator in Eq. (28) is set so that the lowest three-boson state at unitarity has the desired Efimov parameter kappa*; results depend on kappa*, not V0.

assumptions (5)

domain assumption The generalized radial scaling law for the three-boson spectrum in the presence of 1D spin-orbit coupling, established in Ref. [41], holds for the parameter regime considered.
Invoked in Sec. IV.A and Sec. V to assert that the mapped lowest manifold also determines all higher-lying Efimov manifolds by discrete rescaling.
standard math The explicitly correlated Gaussian variational approach converges to the exact eigenstates of the model Hamiltonian for the reported results.
The method is standard (Refs. [58,59]) and used as the numerical workhorse, but the paper does not present convergence checks.
domain assumption The low-energy properties are universal: the finite-range Gaussian potentials with ranges r0 and R0 much smaller than all other length scales reproduce zero-range universal results.
Stated in Sec. II.D; the ratio R0/r0 = sqrt(8) and kappa*r0 = 0.0152 are chosen to be in the universal regime, and universality is checked through the 515.29 scaling ratio.
standard math The scattering threshold for the three-boson system is the minimum of the three-atom threshold and the atom-dimer threshold, as defined in Eqs. (16)-(18).
Standard few-body threshold condition; the atom-dimer threshold requires mapping the two-body binding energy dependence on generalized detuning.
standard math The Hellman-Feynman theorem applies to the q_N dependence of the ground state energy, justifying Eq. (21).
Used in Sec. II.C to derive the zero lab-frame mechanical momentum of the total ground state.

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Cite this review

Pith. "Pith review of Energetics and structural properties of two- and three-boson systems in the presence of 1D spin-orbit coupling." pith.science (2026). https://pith.science/paper/HV434IYX

@misc{pith2026190806918,
  author       = {Pith},
  title        = {Pith review of: Energetics and structural properties of two- and three-boson systems in the presence of 1D spin-orbit coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HV434IYX}},
  note         = {Machine review of arXiv:1908.06918}
}

read the original abstract

It was shown recently that the discrete scaling symmetry, which underlies the Efimov effect in the three identical boson system with two-body short-range interactions, survives when single-particle 1D spin-orbit coupling terms are added to the Hamiltonian. Each three-body energy level in the ordinary Efimov scenario turns into an energy manifold that contains four energy levels in the presence of 1D spin-orbit coupling (equal mixture of Rashba-Dresselhaus coupling). This work provides a detailed characterization of the energy levels in these manifolds. The two-boson energies, which enter into the three-boson scattering threshold, are analyzed in detail. Moreover, the structural properties, e.g., momentum distributions of the two- and three-boson systems, are analyzed for various parameter combinations.

Figures

Figures reproduced from arXiv: 1908.06918 by the authors.

**Figure 1.** FIG. 1: (Color online) The contours show the negative of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 3.** FIG. 3: (Color online) The ground state energy [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 6.** Figure 6: FIG. 6: (Color online) Correlations between the lowest non [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 5.** Figure 5: FIG. 5: (Color online) The contours show the negative of the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 7.** Figure 7: FIG. 7: (Color online) The ground state energy [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]

**Figure 8.** Figure 8: FIG. 8: The critical center-of-mass momentum [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9: (Color online) “Phase diagram” for the total ground [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

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Reference graph

Works this paper leans on

73 extracted references · 69 canonical work pages

[41]

Z.-Y. Shi, H. Zhai, and X. Cui, Eﬁmov physics and uni- versal trimers in spin-orbit-coupled ultracold atomic mix- tures, Phys. Rev. A 91, 023618 (2015)

work page 2015
[1]

Eﬁmov, Energy levels arising from resonant two-body forces in a three-body system, Phys

V. Eﬁmov, Energy levels arising from resonant two-body forces in a three-body system, Phys. Lett. B 33, 563 (1970). 15

work page 1970
[2]

far- away particles

(12) The shift introduces a non-trivial dependence of the eigenenergy on the total generalized momentum # »qN. No such dependence exists for the system without spin-orbit coupling. Using Eqs. (10) and (11), the total N-particle Hamiltonian for ﬁxed # »qN is given by ˆHN( # »qN) = # »q 2 N 2µN I1,···,N +ˆHN,rel. (13) Equations (10) and (11) can be viewed a...

work page
[3]

Eﬁmov, Weakly-bound states of three resonantly- interacting particles, Yad

V. Eﬁmov, Weakly-bound states of three resonantly- interacting particles, Yad. Fiz. 12, 1080-91 (1970)

work page 1970
[4]

Eﬁmov, Level spectrum of three resonantly interacting particles, Sov

V. Eﬁmov, Level spectrum of three resonantly interacting particles, Sov. Phys. JETP Lett. 16, 34 (1972)

work page 1972
[5]

Eﬁmov, Energy levels of three resonantly interacting particles, Nucl

V. Eﬁmov, Energy levels of three resonantly interacting particles, Nucl. Phys. A 210, 157 (1973)

work page 1973
[6]

Braaten and H.-W

E. Braaten and H.-W. Hammer, Universality in few-body systems with large scattering length, Phys. Rep.428, 259 (2006)

2006
[7]

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, and R. Grimm, Evidence for Eﬁmov quan- tum states in an ultracold gas of caesium atoms, Nature (London) 440, 315 (2006)

work page 2006

Show all 73 references

[8]

Zaccanti, B

M. Zaccanti, B. Deissler, C. D’Errico, M. Fattori, M. Jona-Lasinio, S. M¨ uller, G. Roati, M. Inguscio, G. Mod- ugno, Observation of an Eﬁmov spectrum in an atomic system, Nat. Phys. 5, 586 (2009)

2009
[9]

Ferlaino and R

F. Ferlaino and R. Grimm, Trend: Forty years of Eﬁmov physics: How a bizarre prediction turned into a hot topic, Physics 3, 9 (2010)

2010
[10]

Ferlaino, A

F. Ferlaino, A. Zenesini, M. Berninger, B. Huang, H.-C. N¨ agerl, and R. Grimm, Eﬁmov Resonances in Ultracold Quantum Gases, Few-Body Syst. 51, 113 (2011)

2011
[11]

Berninger, A

M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. N¨ agerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M. Hutson, Universality of the Three-Body Parameter for Eﬁmov States in Ultracold Cesium, Phys. Rev. Lett. 107, 120401 (2011)

2011
[12]

Huang, L

B. Huang, L. A. Sidorenkov, R. Grimm, and J. M. Hut- son, Observation of the Second Triatomic Resonance in Eﬁmov’s Scenario, Phys. Rev. Lett. 112, 190401 (2014)

2014
[13]

Kunitski, S

M. Kunitski, S. Zeller, J. Voigtsberger, A. Kalinin, L. P. H. Schmidt, M. Sch¨ oﬄer, A. Czasch, W. Sch¨ ollkopf, R. E. Grisenti, T. Jahnke, D. Blume, R. D¨ orner, Observation of the Eﬁmov state of the helium trimer, Science 348, 551 (2015)

2015
[14]

Lompe, T

T. Lompe, T. B. Ottenstein, F. Serwane, A. N. Wenz, G Z¨ urn, S. Jochim, Radio-Frequency Association of Eﬁmov Trimers, Science 330, 940 (2010)

2010
[15]

Nakajima, M

S. Nakajima, M. Horikoshi, T. Mukaiyama, P. Naidon, and M. Ueda, Measurement of an Eﬁmov Trimer Binding Energy in a Three-Component Mixture of6Li, Phys. Rev. Lett. 106, 143201 (2011)

2011
[16]

R. D. Amado and F. C. Greenwood, There Is No Eﬁmov Eﬀect for Four or More Particles, Phys. Rev. D 7, 2517 (1973)

1973
[17]

S. K. Adhikari and A. C. Fonseca, Four-body Eﬁmov eﬀect in a Born-Oppenheimer model, Phys. Rev. D 24, 416 (1981)

1981
[18]

Platter, H.-W

L. Platter, H.-W. Hammer, and U.-G. Meißner, Four- boson system with short-range interactions, Phys. Rev. A 70, 052101 (2004)

2004
[19]

von Stecher, J

J. von Stecher, J. P. D’Incao, and C. H. Greene, Signa- tures of universal four-body phenomena and their rela- tion to the Eﬁmov eﬀect, Nat. Phys. 5, 417 (2009)

2009
[20]

von Stecher, Weakly bound cluster states of Eﬁmov character, J

J. von Stecher, Weakly bound cluster states of Eﬁmov character, J. Phys. B 43, 101002 (2010)

2010
[21]

A. N. Nicholson, N-Body Eﬁmov States from Two- Particle Noise, Phys. Rev. Lett. 109, 073003 (2012)

2012
[22]

Gattobigio and A

M. Gattobigio and A. Kievsky, Universality and scaling in the N-body sector of Eﬁmov physics, Phys. Rev. A 90, 012502 (2014)

2014
[23]

Yan and D

Y. Yan and D. Blume, Energy and structural proper- ties of N-boson clusters attached to three-body Eﬁmov states: Two-body zero-range interactions and the role of the three-body regulator, Phys. Rev. A 92, 033626 (2015)

2015
[24]

Naidon and S

P. Naidon and S. Endo, Eﬁmov Physics: a review, Rep. Prog. Phys. 80, 056001 (2017)

2017
[25]

C. H. Greene, P. Giannakeas, and J. P´ erez-R´ ıos, Univer- sal few-body physics and cluster formation, Rev. Mod. Phys. 89, 035006 (2017)

2017
[26]

Levinsen, M

J. Levinsen, M. M. Parish, and G. M. Bruun, Impurity in a Bose-Einstein Condensate and the Eﬁmov Eﬀect, Phys. Rev. Lett. 115, 125302 (2015)

2015
[27]

M. Sun, H. Zhai, and X. Cui, Visualizing the Eﬁmov Cor- relation in Bose Polarons, Phys. Rev. Lett. 119, 013401 (2017)

2017
[28]

M. J. Gullans, S. Diehl, S. T. Rittenhouse, B. P. Ruzic, J. P. D’Incao, P. Julienne, A. V. Gorshkov, and J. M. Taylor, Eﬁmov States of Strongly Interacting Photons, Phys. Rev. Lett. 119, 233601 (2017)

2017
[29]

V. E. Colussi, J. P. Corson, and J. P. D’Incao, Dynamics of Three-Body Correlations in Quenched Unitary Bose Gases, Phys. Rev. Lett. 120, 100401 (2018)

2018
[30]

S. M. Yoshida, S. Endo, J. Levinsen, and M. M. Parish, Universality of an Impurity in a Bose-Einstein Conden- sate, Phys. Rev. X 8, 011024 (2018)

2018
[31]

Sun and X

M. Sun and X. Cui, Eﬁmov physics in the presence of a Fermi sea, Phys. Rev. A 99, 060701(R) (2019)

2019
[32]

Jonsell, H

S. Jonsell, H. Heiselberg, and C. J. Pethick, Universal Be- havior of the Energy of Trapped Few-Boson Systems with Large Scattering Length, Phys. Rev. Lett. 89, 250401 (2002)

2002
[33]

Stoll and T

M. Stoll and T. K¨ ohler, Production of three-body Eﬁmov molecules in an optical lattice, Phys. Rev. A 72, 022714 (2005)

2005
[34]

Werner and Y

F. Werner and Y. Castin, Unitary Quantum Three- Body Problem in a Harmonic Trap, Phys. Rev. Lett. 97, 150401 (2006)

2006
[35]

Portegies and S

J. Portegies and S. Kokkelmans, Eﬁmov Trimers in a Harmonic Potential, Few-Body Syst. 51, 219 (2011)

2011
[36]

Blume, Few-body physics with ultracold atomic and molecular systems in traps, Rep

D. Blume, Few-body physics with ultracold atomic and molecular systems in traps, Rep. Prog. Phys. 75, 4 (2012)

2012
[37]

T¨ olle, H.-W

S. T¨ olle, H.-W. Hammer and B. Ch. Metsch, Convergence properties of the eﬀective theory for trapped bosons, J. Phys. G: Nuclear and Particle Physics40, 055004 (2013)

2013
[38]

Levinsen, P

J. Levinsen, P. Massignan, and M. M. Parish, Eﬁmov Trimers under Strong Conﬁnement, Phys. Rev. X 4, 031020 (2014)

2014
[39]

Z.-Y. Shi, X. Cui, and H. Zhai, Universal Trimers In- duced by Spin-Orbit Coupling in Ultracold Fermi Gases, Phys. Rev. Lett. 112, 013201 (2014)

2014
[40]

Cui and W

X. Cui and W. Yi, Universal Borromean Binding in Spin- Orbit-Coupled Ultracold Fermi Gases, Phys. Rev. X 4, 031026 (2014)

2014
[42]

Guan and D

Q. Guan and D. Blume, Three-Boson Spectrum in the Presence of 1D Spin-Orbit Coupling: Eﬁmovs General- ized Radial Scaling Law, Phys. Rev. X 8, 021057 (2018)

2018
[43]

J. P. Vyasanakere and V. B. Shenoy, Bound states of two spin-1/2 fermions in a synthetic non-Abelian gauge ﬁeld, Phys. Rev. B 83, 094515 (2011)

2011
[44]

J. P. Vyasanakere, S. Zhang, and V. B. Shenoy, BCS- 16 BEC crossover induced by a synthetic non-Abelian gauge ﬁeld, Phys. Rev. B 84, 014512 (2011)

2011
[45]

J. P. Vyasanakere and V. B. Shenoy, Rashbons: prop- erties and their signiﬁcance, New J. Phys. 14, 043041 (2012)

2012
[46]

Yu, Short-range correlations in dilute atomic Fermi gases with spin-orbit coupling, Phys

Z. Yu, Short-range correlations in dilute atomic Fermi gases with spin-orbit coupling, Phys. Rev. A 85, 042711 (2012)

2012
[47]

Cui, Mixed-partial-wave scattering with spin-orbit coupling and validity of pseudopotentials, Phys

X. Cui, Mixed-partial-wave scattering with spin-orbit coupling and validity of pseudopotentials, Phys. Rev. A 85, 022705 (2012)

2012
[48]

J. P. Vyasanakere and V. B. Shenoy, Flow-enhanced pair- ing and other unusual eﬀects in Fermi gases in synthetic gauge ﬁelds, Phys. Rev. A 88, 033609 (2013)

2013
[49]

L. Dong, L. Jiang, H. Hu, and H. Pu, Finite-momentum dimer bound state in a spin-orbit-coupled Fermi gas, Phys. Rev. A 87, 043616 (2013)

2013
[50]

Wu and Z

Y. Wu and Z. Yu, Short-range asymptotic behavior of the wave functions of interacting spin-1/2 fermionic atoms with spin-orbit coupling: A model study, Phys. Rev. A 87, 032703 (2013)

2013
[51]

Guan and D

Q. Guan and D. Blume, Scattering framework for two particles with isotropic spin-orbit coupling applicable to all energies, Phys. Rev. A 94, 022706 (2016)

2016
[52]

Cui, Multichannel molecular state and rectiﬁed short- range boundary condition for spin-orbit-coupled ultra- cold fermions near p-wave resonances, Phys

X. Cui, Multichannel molecular state and rectiﬁed short- range boundary condition for spin-orbit-coupled ultra- cold fermions near p-wave resonances, Phys. Rev. A 95, 030701(R) (2017)

2017
[53]

Wang and C

S.-J. Wang and C. H. Greene, General formalism for ultracold scattering with isotropic spin-orbit coupling, Phys. Rev. A 91, 022706 (2015)

2015
[54]

Li and L

R. Li and L. Yin, Pair condensation in a dilute Bose gas with Rashba spin-orbit coupling, New J. Phys. 16, 053013 (2014)

2014
[55]

Z. Xu, Z. Yu, and S. Zhang, Evidence for correlated states in a cluster of bosons with Rashba spin-orbit coupling, New J. Phys. 18, 025002 (2016)

2016
[56]

Luo and L

D. Luo and L. Yin, BCS-pairing state of a dilute Bose gas with spin-orbit coupling, Phys. Rev. A 96, 013609 (2017)

2017
[57]

Gu and L

Q. Gu and L. Yin, Spin-orbit-couplinginduced resonance in an ultracold Bose gas, Phys. Rev. A98, 013617 (2018)

2018
[58]

Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I. B. Spielman, Spin- orbit-coupled Bose-Einstein condensates, Nature (Lon- don) 471, 83 (2011)

2011
[59]

Suzuki and K

Y. Suzuki and K. Varga, Stochastic Variational Ap- proach to Quantum-Mechanical Few-Body Problems , 1st ed. (Springer-Verlag Berlin Heidelberg, 1998)

1998
[60]

Mitroy, S

J. Mitroy, S. Bubin, W. Horiuchi, Y. Suzuki, L. Adamow- icz, W. Cencek, K. Szalewicz, J. Komasa, D. Blume, and K. Varga, Theory and application of explicitly correlated Gaussians, Rev. Mod. Phys. 85, 693 (2013)

2013
[61]

Guan, One-, two-, and three-body systems with spin- orbit coupling, Ph.D

Q. Guan, One-, two-, and three-body systems with spin- orbit coupling, Ph.D. thesis, Washington State Univer- sity (2017)

2017
[62]

Zhai, Degenerate quantum gases with spin-orbit cou- pling: a review, Rep

H. Zhai, Degenerate quantum gases with spin-orbit cou- pling: a review, Rep. Prog. Phys. 78, 026001 (2015)

2015
[63]

Feynman, Forces in Molecules, Phys

R. Feynman, Forces in Molecules, Phys. Rev. 56, 340 (1939)

1939
[64]

Kievsky and M

A. Kievsky and M. Gattobigio, Universal nature and ﬁnite-range corrections in elastic atom-dimer scattering below the dimer breakup threshold, Phys. Rev. A 87, 052719 (2013)

2013
[65]

Yu and H

Z.-Q. Yu and H. Zhai, Spin-Orbit Coupled Fermi Gases across a Feshbach Resonance, Phys. Rev. Lett. 107, 195305 (2011)

2011
[66]

Zhai, Spin-Orbit Coupled Quantum Gases, Int

H. Zhai, Spin-Orbit Coupled Quantum Gases, Int. J. Mod. Phys. B 26, 1230001 (2012)

2012
[67]

Peng, X.-J

S.-G. Peng, X.-J. Liu, H. Hui, and K. Jiang, Momentum- resolved radio-frequency spectroscopy of a spin-orbit- coupled atomic Fermi gas near a Feshbach resonance in harmonic traps, Phys. Rev. A 86, 063610 (2012)

2012
[68]

Liu and H

X.-J. Liu and H. Hu, Inhomogeneous Fulde-Ferrell super- ﬂuidity in spin-orbit-coupled atomic Fermi gases, Phys. Rev. A 87, 051608(R) (2013)

2013
[69]

Liu and H

X.-J. Liu and H. Hu, Topological Fulde-Ferrell superﬂuid in spin-orbit-coupled atomic Fermi gases, Phys. Rev. A 88, 023622 (2013)

2013
[70]

Jiang, E

L. Jiang, E. Tiesinga, X.-J. Liu, H. Hu, and H. Pu, Spin- orbit-coupled topological Fulde-Ferrell states of fermions in a harmonic trap, Phys. Rev. A 90, 053606 (2014)

2014
[71]

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

2010
[72]

Galitski and I

V. Galitski and I. B. Spielman, Spin-orbit coupling in quantum gases, Nature (London) 494, 49 (2013)

2013
[73]

Goldman, G

N. Goldman, G. Juzeli¯ unas, P. ¨Ohberg, and I. B. Spiel- man, Light-induced gauge ﬁelds for ultracold atoms, Rep. Prog. Phys. 77, 126401 (2014)

2014

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [41]

Z.-Y. Shi, H. Zhai, and X. Cui, Eﬁmov physics and uni- versal trimers in spin-orbit-coupled ultracold atomic mix- tures, Phys. Rev. A 91, 023618 (2015)

work page 2015

[2] [1]

Eﬁmov, Energy levels arising from resonant two-body forces in a three-body system, Phys

V. Eﬁmov, Energy levels arising from resonant two-body forces in a three-body system, Phys. Lett. B 33, 563 (1970). 15

work page 1970

[3] [2]

far- away particles

(12) The shift introduces a non-trivial dependence of the eigenenergy on the total generalized momentum # »qN. No such dependence exists for the system without spin-orbit coupling. Using Eqs. (10) and (11), the total N-particle Hamiltonian for ﬁxed # »qN is given by ˆHN( # »qN) = # »q 2 N 2µN I1,···,N +ˆHN,rel. (13) Equations (10) and (11) can be viewed a...

work page

[4] [3]

Eﬁmov, Weakly-bound states of three resonantly- interacting particles, Yad

V. Eﬁmov, Weakly-bound states of three resonantly- interacting particles, Yad. Fiz. 12, 1080-91 (1970)

work page 1970

[5] [4]

Eﬁmov, Level spectrum of three resonantly interacting particles, Sov

V. Eﬁmov, Level spectrum of three resonantly interacting particles, Sov. Phys. JETP Lett. 16, 34 (1972)

work page 1972

[6] [5]

Eﬁmov, Energy levels of three resonantly interacting particles, Nucl

V. Eﬁmov, Energy levels of three resonantly interacting particles, Nucl. Phys. A 210, 157 (1973)

work page 1973

[7] [6]

Braaten and H.-W

E. Braaten and H.-W. Hammer, Universality in few-body systems with large scattering length, Phys. Rep.428, 259 (2006)

2006

[8] [7]

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, and R. Grimm, Evidence for Eﬁmov quan- tum states in an ultracold gas of caesium atoms, Nature (London) 440, 315 (2006)

work page 2006