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Strict Time-Resolved Steady States via Affine-Eigenstate Mapping: A Robust Framework for Ultracold Atom–Molecule Dynamics

Yuehua Chen, Gaoyang Du, Chenglong Yang, Shuyu Dai, Bo Cui

Published Jul 1, 2026• 0 citations

Abstract

We propose a theoretical framework based on an affine-eigenstate transformation for analyzing ultracold atom-molecule conversion dynamics with particle loss. The transformation maps the mean-field dynamics to an effective two-mode representation in which fixed points, Bloch-sphere trajectories, and linear stability can be examined in a common set of variables. We give the derivation of the transformed Hamiltonian and specify the invertibility and conjugate-condition requirements under which the mapping is used. Within this representation, we distinguish ordinary, pseudo, and strict self-trapping regimes. The strict regime is associated with the balanced condition S=0 in the transformed variables; in the corresponding linearized dissipative flow, the leading attractor/repeller bifurcation term controlled by SΓ- vanishes, explaining the observed robustness against atom- and molecule-loss imbalance. We also introduce von Neumann and linear-entropy diagnostics for future mixed-state or ensemble descriptions in the transformed two-level representation, and we provide an inverse reconstruction procedure for preparing initial states that realize strict self-trapping. Finally, we discuss the limits of the mean-field and Markovian approximations and outline how finite-particle simulations and phase-modulated control protocols could connect this mechanism to decoherence-resilient quantum simulations and information-processing architectures.

DOI: 10.3390/e28070752

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We propose a theoretical framework based on an affine-eigenstate transformation for analyzing ultracold atom-molecule conversion dynamics with particle loss. The transformation maps the mean-field dynamics to an effective two-mode representation in which fixed points, Bloch-sphere trajectories, and linear stability can be examined in a common set of variables. We give the derivation of the transformed Hamiltonian and specify the invertibility and conjugate-condition requirements under which the mapping is used. Within this representation, we distinguish ordinary, pseudo, and strict self-trapping regimes. The strict regime is associated with the balanced condition S=0 in the transformed variables; in the corresponding linearized dissipative flow, the leading attractor/repeller bifurcation term controlled by SΓ- vanishes, explaining the observed robustness against atom- and molecule-loss imbalance. We also introduce von Neumann and linear-entropy diagnostics for future mixed-state or ensemble descriptions in the transformed two-level representation, and we provide an inverse reconstruction procedure for preparing initial states that realize strict self-trapping. Finally, we discuss the limits of the mean-field and Markovian approximations and outline how finite-particle simulations and phase-modulated control protocols could connect this mechanism to decoherence-resilient quantum simulations and information-processing architectures.

Abstract Based on the mean-field approximation and the two-mode Bose–Hubbard model, we investigate the dynamical properties of atom-polyatomic molecule conversion systems under dissipation and dephasing. For such a many-body system problem, we partition phase space based on the states of fixed points, study the movement of fixed points and the transition to attractors in an open system. Additionally, for all the atoms that constitute a molecule, depending on whether the species of these atoms are identical or not, we introduce the innovative concepts of unbiased and biased systems. We discover that in biased conversion systems under particle losses, the evolution trajectories are distributed on a three-dimensional Bloch sphere with an internal structure. There exists a significant difference from the two-dimensional Hilbert space of an unbiased system. Finally, we discuss the sweeping rate of the time-varying magnetic field applied externally to the conversion system, and analyze a feasible scheme for the quantum coherent synthesis of triatomic molecules.

Based on the mean-field approximation and the phase-space analysis, we study the dynamics of atom-molecule conversion systems subject to particle losses. Starting from the many-body dynamics described by a master equation, an effective nonlinear Schr\"odinger equation is introduced. The classical phase space is then specified and classified by fixed points. The boundaries, which separate different dynamical regimes, have been established and discussed. The effect of particle loss on the conversion efficiency and the self-trapping is explored. By numerically solving the master equation, we show that the mean-field approximation is a good approach to study the dynamics of this atom-molecule conversion system.

We numerically demonstrate the dynamic stabilization of a strongly interacting many-body bosonic system which can be realized by coupled ultracold atom-molecule gases. The system is initialized to an unstable equilibrium state corresponding to a saddle point in the classical phase space, where subsequent free evolution gives rise to atom-molecule conversion. To control and stabilize the system, periodic modulation is applied that suddenly shifts the relative phase between the atomic and the molecular modes and limits their further interconversion. The stability diagram for the range of modulation amplitudes and periods that stabilize the dynamics is given. The validity of the phase diagram obtained from the time-average calculation is discussed by using the orbit tracking method, and the difference in contrast with the maximum absolute deviation analysis is shown as well. A brief quantum analysis shows that quantum fluctuations can put serious limitations on the applicability of the mean-field results.

We investigate the conversion dynamics of ultracold bosonic atoms to molecules in an external field with a Gaussian temporal shape and a linear frequency sweep. Under the mean-field theory, we study the atom–molecule conversion efficiency and the dynamics in the conversion process. By using the Fock-state basis, we further research the quantum many-body dynamics, wherein the many-body effect on the conversion is discussed. We find that the atom–molecule conversion dynamics depend on the particle interaction, peak coupled strength, sweep rate, and total particle number. Notably, some analytical results are derived in the regime without particle interactions and in the sudden limit, respectively. Our theory indicates that the ultracold atom–molecule conversion can be achieved stably and efficiently in the Gaussian external field model.

We consider a simple quantum model of atom-molecule conversion where bosonic atoms can combine into diatomic molecules and vice versa. The many-particle system can be expressed in terms of the generators of a deformed

\text{su}(2)

algebra, and the mean-field dynamics takes place on a deformed version of the Bloch sphere, a teardrop shaped surface with a cusp singularity. We analyze the mean-field and many-particle correspondence, which shows typical features of quantum-classical correspondence. We demonstrate that semiclassical methods can be employed to recover full many-particle features from the mean-field description in cold atom systems with atom-molecule conversion, and we derive an analytic expression for the many-particle density of states in the limit of large particle numbers.

We study nonlinear mean-field dynamics of ultracold molecule formation in the case when the external field configuration is defined by the level-crossing Demkov-Kunike model, characterized by a bell-shaped coupling and finite variation of the detuning. Analyzing the fast sweep rate regime of the strong interaction limit, which models a situation when the peak value of the coupling is large enough and the resonance crossing is sufficiently fast, we construct a highly accurate ansatz to describe the temporal dynamics of the molecule formation in the mentioned interaction regime. The absolute error of the constructed approximation is less than 3 × 10 −6 for the final transition probability while at certain time points it might increase up to 10 −3 . Examining the role of the different terms in the constructed approximation, we prove that in the fast sweep rate regime of the strong interaction limit the temporal dynamics of the atom-molecule conversion effectively consists of the process of resonance crossing, which is governed by a nonlinear equation, followed by atom-molecular coherent oscillations which are basically described by a solution of the linear problem, associated with the considered nonlinear one.