Classical and quantum metrology of the Lieb-Liniger model

This is the authors' abstract. We don't add key points for this paper.

We study the classical and quantum Fisher information of the interaction coupling in the Lieb-Liniger model. Fisher information has been studied extensively when the parameter is inscribed on a quantum state by a unitary process, e.g., Mach-Zehnder or Ramsey interferometry. Here we investigate the case that a Hamiltonian parameter to be estimated is imprinted on eigenstates of that Hamiltonian and thus is not necessarily encoded by a unitary operator. We take advantage of the fact that the Lieb-Liniger model is exactly soluble for both periodic and hard-wall boundary conditions. For the latter case, we provide a derivation of the explicit expression for the norm of the Lieb-Liniger wave function. The Fisher information is determined for various small numbers of particles, for both ground state and excited states of type I and type II in the Lieb-Liniger terminology. We discuss the dependence of the Fisher information on interaction strength and system size, to further evaluate the metrological aspects of the model. Particularly noteworthy is the fact that the Fisher information displays a maximum when we vary the system size, indicating that the distinguishability of the wave functions is largest when the Lieb-Liniger parameter is at the crossover between Bose-Einstein condensate and Tonks-Girardeau limits. The saturability of this Fisher information by employing standard absorption imaging as the measurement method is assessed by a specific modeling of the latter.