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Classical and quantum metrology of the Lieb-Liniger model

We study the classical and quantum Fisher information for the Lieb-Liniger model. The 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. Taking advantage of the fact that the Lieb-Liniger model is exactly solvable, the Fisher information is determined for periodic and hard-wall boundary conditions, varying number of particles, and for 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 wavefunctions is largest when the Lieb-Liniger parameter is at the crossover between the Bose-Einstein condensate and Tonks-Girardeau limits. The saturability of this Fisher information by the absorption imaging method is assessed by a specific modeling of the latter.