Abstract
We study the effect of the boundary on a system of weakly interacting bosons in one dimension. It strongly influences the boson density which is completely suppressed at the boundary position. Away from it, the density is depleted over the distances on the order of the healing length at the mean-field level. Quantum fluctuations modify the density profile considerably. The local density approaches the average one as an inverse square of the distance from the boundary. We calculate an analytic expression for the density profile at arbitrary separations from the boundary. We then consider the problem of localization of a foreign quantum particle (impurity) in the potential created by the inhomogeneous boson density. At the mean-field level, we find exact results for the energy spectrum of the bound states, the corresponding wave functions, and the condition for interaction-induced localization. The quantum contribution to the boson density gives rise to small corrections of the bound state energy levels. However, it is fundamentally important for the existence of a long-range Casimir-like interaction between the impurity and the boundary.
Highlights
Boundaries play an important role in one-dimensional quantum liquids affecting, e.g., correlation and response functions, as well as the ground-state energy [1,2,3,4,5,6,7,8]
We find the condition for the appearance of the bound states, which is a threshold for a single dimensionless parameter that involves the masses of the impurity and of the particles of the Bose gas as well as the interaction strengths
In the case of repulsion, the density profile of the Bose gas forms an attractive potential for the impurity, which can lead to the bound states in the spectrum
Summary
Boundaries play an important role in one-dimensional quantum liquids affecting, e.g., correlation and response functions, as well as the ground-state energy [1,2,3,4,5,6,7,8]. Away from it, the particle density in fermionic systems shows socalled Friedel oscillations [9]. Friedel oscillations in fermionic systems are studied within the harmonic Tomonaga-Luttinger liquid description [3,10]. This is the low-energy theory of both interacting fermions and bosons in one dimension [11]. We solve the Schrödinger equation and characterize the impurity by the energy spectrum of the bound states, their wave functions, and the mean position. We find the mean-field solution and the first two quantum corrections In the Appendix, we present a simplified procedure that leads to the density in the regime of large separations from the boundary
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