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Journal of Physics A Mathematical and Theoretical

Casimir effect for impurity in periodic background in one dimension

Michael Bordag B

Published May 19, 2020• 0 citations
Publisher

Abstract

Abstract We consider a Bose gas in a one-dimensional periodic background formed by generalized delta function potentials with one and two impurities. We investigate the scattering off these impurities and their bound state levels. Besides expected features, we observe a kind of long-range correlation between the bound state levels of two impurities. Further, we define and calculate the vacuum energy of the impurity. It causes a force acting on the impurity relative to the background. We define the vacuum energy as a mode sum. In order to get a discrete spectrum we start from a finite lattice and use Chebychev polynomials to get a general expression. These allow also for quite easy investigation of impurities in finite lattices.

DOI
10.1088/1751-8121/ab9463

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Abstract We consider a Bose gas in a one-dimensional periodic background formed by generalized delta function potentials with one and two impurities. We investigate the scattering off these impurities and their bound state levels. Besides expected features, we observe a kind of long-range correlation between the bound state levels of two impurities. Further, we define and calculate the vacuum energy of the impurity. It causes a force acting on the impurity relative to the background. We define the vacuum energy as a mode sum. In order to get a discrete spectrum we start from a finite lattice and use Chebychev polynomials to get a general expression. These allow also for quite easy investigation of impurities in finite lattices.
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.
Abstract We investigate Bose–Einstein condensation of a noninteracting gas of Bose particles moving in the background of a periodic lattice of delta functions. In the one-dimensional case, where one has no condensation in the free case, this property persists also in the presence of the lattice for all examples which are considered in the present paper and we could only formulate some conditions which are necessary for condensation. We also considered the three-dimensional case and showed that the lattice does not destroy condensation. We calculated, for small coupling, the change in the critical temperature, which is lowered by the lattice. Finally, we took another, more general view on the problem using heat kernel expansion, and discuss BEC for Casimir effect related configurations.
We study the fluctuation-induced interaction between two impurities in a weakly interacting one-dimensional Bose gas using the field-theoretical approach. At separations between impurities shorter and of the order of the healing length of the system, the induced interaction has a classical origin and behaves exponentially. At separations longer than the healing length, the interaction is of a quantum origin and scales as the third power of the inverse distance. Finite temperature destroys the quasi-long-range order of the Bose gas and, accordingly, the induced interaction becomes exponentially suppressed beyond the thermal length. We obtain analytical expressions for the induced interaction at zero and finite temperature that are valid at arbitrary distances. We discuss experimental realizations as well as possible formation of bound states of two impurities, known as bipolarons.
Abstract The electromagnetic Casimir effect manifests as the interaction between uncharged conducting objects that are placed in a vacuum. More generally, the Casimir-like effect denotes an induced interaction between external bodies in a fluctuating medium. We study the Casimir-like interaction between two impurities embedded in a weakly interacting one-dimensional Bose gas. We develop a theory based on the Gross–Pitaevskii equation that accounts for the effect of quantum fluctuations. At small separations, the induced interaction between the impurities decays exponentially with the distance. This is a classical result that can be understood using the mean-field Gross–Pitaevskii equation. We find that at larger distances, the induced interaction crosses over into a power law dependence due to the quantum fluctuations. We obtain an analytic expression for the interaction that interpolates between the two limiting behaviors. The obtained result does not require any regularization.
Motivated by a recent experiment by Meinert et al. [arXiv:1608.08200], we study the dynamics of an impurity moving in the background of a harmonically trapped one-dimensional Bose gas in the hard-core limit. We show that due to the hidden ``lattice'' structure of background bosons, the impurity effectively feels a quasiperiodic potential via impurity-boson interactions that can drive the Bloch oscillation under an external force, even in the absence of real lattice potentials. Meanwhile, the inhomogeneous density of trapped bosons imposes an additional harmonic potential to the impurity, resulting in a similar oscillation dynamics but with a different period and amplitude. We show that the sign and strength of the impurity-boson coupling can significantly affect the above two potentials in determining the impurity dynamics.
We study the ground state of a one-dimensional (1D) trapped Bose gas with two mobile impurity particles. To investigate this setup, we develop a variational procedure in which the coordinates of the impurity particles are slowlike variables. We validate our method using the exact results obtained for small systems. Then, we discuss energies and pair densities for systems that contain of the order of 100 atoms. We show that bosonic noninteracting impurities cluster. To explain this clustering, we calculate and discuss induced impurity-impurity potentials in a harmonic trap. Further, we compute the force between static impurities in a ring (in the manner of the Casimir force), and contrast the two effective potentials: the one obtained from the mean-field approximation, and the one due to the one-phonon exchange. Our formalism and findings are important for understanding (beyond the polaron model) the physics of modern 1D cold-atom systems with more than one impurity.