The atom-loss functional $F[\rho](k)$ of the one-dimensional Bose gas: evaluation beyond the asymptotic regimes and loss-driven rapidity dynamics

In the Lieb-Liniger gas (one-dimensional bosons with repulsive contact interactions, the paradigmatic integrable cold-atom system), out-of-equilibrium states are encoded in the rapidity distribution $\rho(k)$. Atom losses, which are unavoidable in experiments, drive a slow evolution of $\rho(k)$ governed by a loss functional $F[\rho](k)$ that Bouchoule, Doyon and Dubail expressed exactly as a double sum over Bethe eigenstates involving finite-volume form factors of the $K$-body annihilation operator. Closed results exist only in the two asymptotic regimes: the ideal Bose gas, where $F=KK!\,\rho(k)$, and the hard-core Tonks-Girardeau limit, where $F$ is a nonlinear, nonlocal functional that admits an exact solution; generic couplings are accessible only through a computationally costly Monte Carlo evaluation. The problem asked here is a controlled evaluation of $F$ at generic interaction strength and temperature: a thermodynamic-limit or resummed form of the Bethe-state double sum, low-temperature and near-zero-entropy treatments, the resulting quantitative intermediate-coupling dynamics including the loss-generated $1/k^4$ rapidity tail (established exactly so far only in the Tonks-Girardeau and weakly interacting quasicondensate regimes, with physical arguments at arbitrary coupling), efficient evaluation enabling the solution of the inhomogeneous lossy generalized-hydrodynamics equation for trapped gases, and evolutions for higher $K$. No resolution of this core is known in the literature to date.

Background

The one-dimensional (1D) Bose gas with repulsive contact interactions is described by the Lieb-Liniger Hamiltonian

$$H=\int_0^L dx\;\Big[\frac{\hbar^2}{2m}\,\partial_x\Psi^\dagger\,\partial_x\Psi+\frac{g}{2}\,\Psi^{\dagger}(x)^2\Psi(x)^2\Big],$$

where $\Psi(x)$ annihilates a boson at position $x$ and $g>0$ is the interaction strength. It is a quantum integrable system: its exact eigenstates are constructed by the Bethe ansatz and labeled by real numbers $\{\lambda_i\}$, called rapidities, which act as generalized quasi-momenta. Integrability provides an extensive family of conserved quantities, one for each function of the rapidity. Consequently a homogeneous gas prepared far from equilibrium does not relax to an ordinary Gibbs ensemble but to a generalized Gibbs ensemble (GGE), a density matrix fully parameterized by the rapidity distribution $\rho(k)$ (the density of rapidities near $k$; the line density is $n=\int\rho(k)\,dk$).

Atoms are progressively lost in every cold-atom experiment: one at a time by collisions with the background gas, two at a time by inelastic two-body processes, or three at a time by three-body recombination. In a Markovian description, $K$-body losses are governed by a Lindblad master equation (the standard quantum master equation for dissipation) with jump operator $\Psi(x)^K$, annihilating $K$ atoms at the same point, and rate constant $G$. Bouchoule, Doyon and Dubail considered the slow-loss regime, where the loss rate $Gn^{K-1}$ is much smaller than the inverse intrinsic relaxation time $1/\tau$ of the gas: between two loss events the gas re-relaxes to a GGE, so its state is at all times captured by $\rho(k)$, which now evolves slowly. To lowest order in $Gn^{K-1}\tau$ the evolution is the nonlinear kinetic equation

$$\frac{d}{dt}\rho(k)=-G\,n^{K-1}\,F[\rho](k),$$

whose central object, the loss functional $F$, has the exact representation $F[\rho](k)=-n^{1-K}\langle\Psi^{\dagger K}(0)\,[\,Q_k,\Psi^K(0)]\rangle_{[\rho]}$, an expectation value in the GGE labeled by $\rho$, where $Q_k$ measures the density of rapidities in a window around $k$. Its exact finite-volume form is a double sum over pairs of Bethe eigenstates,

$$F[\rho](k)=n^{1-K}\sum_{\{\lambda_i\},\{\mu_j\}}p(\{\lambda_i\})\,\big|\langle\{\mu_j\}|\Psi(0)^K|\{\lambda_i\}\rangle\big|^2\Big(\sum_i\delta_\sigma(k-\lambda_i)-\sum_j\delta_\sigma(k-\mu_j)\Big),$$

where $p(\{\lambda_i\})$ is the GGE weight of the initial Bethe state and $\langle\{\mu_j\}|\Psi(0)^K|\{\lambda_i\}\rangle$ are finite-volume form factors (matrix elements of local operators between exact eigenstates), known exactly from the work of Piroli and Calabrese; the bracket is the change in rapidity content induced by a loss event, so the squared form factor plays the role of a conditional probability for the rapidity transition $\{\lambda_i\}\to\{\mu_j\}$.

Closed analytic expressions for $F$ are known only in two asymptotic regimes. For $g\to 0$ (ideal Bose gas) one finds the linear law $F[\rho](k)=K\,K!\,\rho(k)$, and the rapidity distribution is simply rescaled in time. For $g\to\infty$ (the hard-core Tonks-Girardeau regime, where bosons become impenetrable and map to free fermions) only one-body losses are relevant, and $F$ is a nonlinear, nonlocal functional,

$$F[\rho](k)=\rho(k)-2\pi\Big(\rho(k)^2-\big(\tfrac{1}{\pi}\,\mathrm{PV}\!\int\tfrac{\rho(\lambda)\,d\lambda}{k-\lambda}\big)^2\Big)+\tfrac{2n}{\pi}\,\mathrm{PV}\!\int\tfrac{\rho(k)-\rho(\lambda)}{(k-\lambda)^2}\,d\lambda,$$

with PV denoting the Cauchy principal value; remarkably, the resulting evolution admits an exact closed-form solution for one-body losses. In that regime the functional behaves as $F[\rho](k)\sim C/k^4$ at large $k$: losses populate algebraic $1/k^4$ tails of the rapidity distribution, in sharp contrast with the Gaussian decay of a thermal state. These tails add to the Tan contact (the universal amplitude of the $1/|p|^4$ large-momentum tail of the momentum distribution, related by Tan's relation to the adiabatic derivative of the energy with respect to the coupling), so Tan's relation breaks down in lossy gases; Bouchoule and Dubail established this breakdown and the loss-driven growth of the tail amplitude $C_r(t)$ exactly in both asymptotic regimes (Tonks-Girardeau and weakly interacting quasicondensate) and gave a physical argument at arbitrary coupling.

At large scales an inhomogeneous gas is described by generalized hydrodynamics (GHD), the hydrodynamic theory of integrable systems built on a position-dependent rapidity distribution $\rho(x,k)$ transported with an effective velocity $v^{\mathrm{eff}}[\rho]$ and accelerated in rapidity space by a trapping potential $V(x)$. Losses enter the GHD equation as the source term $-G\,n^{K-1}F[\rho](k)$, and solving this lossy GHD equation for a trapped gas requires evaluating $F$ for a great many rapidity distributions, one for each space-time point. On the weakly interacting side sits the quasicondensate (quasi-BEC) regime, where density fluctuations are suppressed and the gas is described by Bogoliubov theory (a linearization of the phase and density variables into independent phonon-like modes); there, uniform losses were shown theoretically and experimentally to act as a cooling mechanism, and the Bogoliubov analysis gives the evolution of the mode occupations under $K$-body losses. The rapidity-based and quasicondensate descriptions have been matched in the weak-repulsion limit, for the tail amplitude, but their precise general connection remains to be made.

What is missing, and what this problem targets, is the regime of generic interaction strength and temperature: the double sum over Bethe states is accessible only through a nested Monte Carlo sampling that costs of order days of laptop time for a single evolution; no thermodynamic-limit or resummed expression for $F$ (or for the form factors of $\Psi(0)^K$) exists; no low-temperature or near-zero-entropy treatment is available; and the quantitative evolution of the rapidity distribution and of its $1/k^4$ tail at intermediate coupling is unknown, although it is precisely there that cold-atom experiments typically operate.

Problem Statement

Obtain a controlled evaluation of the loss functional $F[\rho](k)$ of the Lieb-Liniger gas at generic interaction strength and temperature, and thereby make the slow-loss theory of the rapidity distribution quantitatively predictive beyond the asymptotic regimes, aiming at the quantitative comparison with realistic, spatially inhomogeneous cold-atom experiments identified in Ref. 1 as requiring further analytical and numerical developments. The following results are linked by their common dependence on evaluating $F$ beyond the asymptotic regimes; items (1), (3) and (4) express the core requirement, while items (2) and (5) are optional extensions that support the objective without being individually required for completion:

(1) Analytic evaluation of $F$: a closed thermodynamic-limit expression, or a convergent resummation, of the finite-volume double sum over Bethe states defining $F[\rho](k)$, equivalently the thermodynamic-limit form factors of the $K$-body annihilation operator $\Psi(0)^K$ and of the conditional rapidity transition probabilities entering that sum, valid for generic repulsion $g$ and generic $K$, together with the demonstration that it reduces to the known ideal-Bose-gas law $F=KK!\rho$ and to the Tonks-Girardeau functional in the hard-core limit.

(2) Low-temperature and near-zero-entropy evaluation (optional): a controlled computation of $F$ for low-temperature states, for example through an effective Luttinger-liquid expansion (the universal low-energy field theory of a 1D Bose fluid), and for states close to zero-entropy states such as split Fermi seas (rapidity distributions supported on several intervals), where standard finite-temperature form-factor expansions degenerate.

(3) Intermediate-coupling dynamics: from (1), (2) or an equivalent controlled scheme, the quantitative time evolution of $\rho(k,t)$ and of the loss-generated $1/k^4$ rapidity-tail amplitude $C_r(t)$ at intermediate interaction strengths, where the tail's existence is established but its evolution is known exactly only in the Tonks-Girardeau and weakly interacting quasicondensate regimes.

(4) Inhomogeneous gases: an efficient evaluation of $F$, by analytic simplification, resummation or a numerical scheme of demonstrated efficiency, making it feasible to solve the lossy generalized-hydrodynamics equation

$$\partial_t\rho+\partial_x\big(v^{\mathrm{eff}}\rho\big)-(\partial_x V)\,\partial_k\rho=-G\,n^{K-1}\,F[\rho](k)$$

for a trapped gas, enabling quantitative comparison with inhomogeneous lossy cold-atom experiments.

(5) Higher-$K$ evolutions (optional): analytic time evolutions, or controlled approximations, for higher $K$ in interacting regimes, in the spirit of the known exact Tonks-Girardeau one-body-loss solution. When several loss channels act simultaneously, their contributions add linearly within the slow-loss regime; this is an exact property of the framework, not a remaining open requirement.

Completion means that the loss-driven rapidity dynamics of the Lieb-Liniger gas is under quantitative, expert-verifiable control at generic coupling and temperature, as expressed by items (1), (3) and (4); any single component alone (for example only the thermodynamic-limit form factors without the hydrodynamic application, or an efficient numerical scheme without a closed form), and likewise the optional directions (2) and (5), is valuable but constitutes partial progress toward this objective.

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