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arXiv:2007.10771 · Physical Review Research

Density profile of a semi-infinite one-dimensional Bose gas and bound states of the impurity

Aleksandra Petković, Benjamin Reichert, Zoran Ristivojević

Published Oct 20, 2020• 10 citations
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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.

DOI
10.1103/physrevresearch.2.043104

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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.
We consider an impurity in a semi-infinite one-dimensional system of weakly-interacting bosons. We calculate the interaction potential for the impurity due to the end of the system, i.e., the wall. For local repulsive (attractive) interaction between the impurity and the Bose gas, the interaction potential is attractive (repulsive). At short distances from the wall it decays exponentially crossing over into a universal 1/r21/{r}^{2}1/r2 behavior at separations rrr above the healing length. Our results can also be interpreted as a Casimir-like interaction between two impurities, where one of them is infinitely strongly coupled to the Bose gas. We discuss various scenarios for the induced interaction between the impurities using the scattering approach. We finally address the phenomenon of localization of the impurity near the wall. In the paper we mainly study the case of a static impurity, however the universal 1/r21/{r}^{2}1/r2 interaction also holds for a slowly moving impurity.
We consider an impurity in a one-dimensional weakly interacting Bose gas and analytically calculate the density profile of the Bose gas. Within the mean-field approximation, by increasing the distance from the impurity, the Bose gas density saturates exponentially fast to its mean thermodynamic-limit value at distances beyond the healing length. The effect of quantum fluctuations drastically changes this behavior, leading to a power-law decay of the density deviation from the mean density. At distances longer than the healing length and shorter than a new length scale proportional to the impurity coupling strength, the power-law exponent is 2, while at the longest distances the corresponding exponent becomes 3. The latter crossover does not exist in two special cases. The first one is realized for infinitely strongly coupled impurity; then the density deviation always decays with the exponent 2. The second special case occurs when the new length scale is smaller than the healing length, i.e., at weak impurity coupling; then the density deviation always decays with the exponent 3. The obtained results are exact in the impurity coupling strength and account for the leading order in the interaction between the particles of the Bose gas.
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.
We study a weakly interacting one-dimensional Bose gas with two impurities coupled locally to the boson density. We derive analytical results for the induced interaction between the impurities at arbitrary positive coupling and separation rrr. At $r\ensuremath{\lesssim}\ensuremath{\xi}$, where $\ensuremath{\xi}$ denotes the healing length of the Bose gas, the interaction is well described by the mean-field contribution. Its form changes as the coupling is increased, approaching a linear function of rrr at short distances in the regime of strong coupling. The mean-field contribution decays exponentially at arbitrary coupling for $r\ensuremath{\gg}\ensuremath{\xi}$. At such long distances, however, the effect of quantum fluctuations becomes important, giving rise to a long-ranged quantum contribution to the induced interaction. At longest distances it behaves as 1/r31/{r}^{3}1/r3, while at strong coupling we find an intermediate distance regime with a slower decay, 1/r1/r1/r. The quantum contribution in the crossover regime is also calculated. The induced interaction between impurities (i.e., polarons) is attractive and leads to the formation of their bound state, known as bipolaron. We discuss its binding energy.
We discuss impurities in a one-dimensional Bose gas with arbitrary boson-boson and boson-impurity interactions. To fully account for quantum effects, we employ numerical simulations based on the density-matrix renormalization group (DMRG) and—in the regime of strong boson-boson interactions—the mapping to weakly interacting fermions. A mean-field description of the Bose polaron based on coupled Gross–Pitaevskii Schrödinger equations predicts the existence of a self-localized polaron. We here show that such a solution does not exist and is an artifact of the underlying decoupling approximation. To this end we consider a mobile impurity in a box potential. Our work demonstrates that correlations between the impurity position and the bosons are important even in the limit where mean-field approaches are expected to work well. Furthermore, we derive analytical approximations for the energy of a single polaron formed by a heavy impurity for arbitrary interaction strengths and large but finite boson-boson couplings which accurately reproduce DMRG results. This demonstrates that the polaron problem of a heavy impurity in a one-dimensional Bose gas can be accurately approximated by a proper mean-field description plus a linearized treatment of quantum fluctuations for arbitrary boson-boson and impurity-boson couplings. Finally, we determine the polaron-polaron interaction potential V ( r ) in the Born-Oppenheimer approximation for small and intermediate distances r , which in the Tonks gas limit is oscillatory due to Friedel oscillations in the Bose gas.