Analytical approach to the Bose-polaron problem in one dimension

Abstract

We discuss the ground state properties of a one-dimensional bosonic system doped with an impurity (the so-called Bose polaron problem). We introduce a formalism that allows us to calculate analytically the thermodynamic zero-temperature properties of this system with weak and moderate boson-boson interaction strengths for any boson-impurity interaction. Our approach is validated by comparing to exact quantum Monte Carlo calculations. In addition, we test the method in finite size systems using numerical results based upon the similarity renormalization group. We argue that the introduced approach provides a simple analytical tool for studies of strongly interacting impurity problems in one dimension.The weakly-interacting Bose gas is a beautiful model system [1], which is often used to study emergent manybody phenomena such as superfluidity, Bose-Einstein condensation, and topologically non-trivial many-body excitations like solitons and vortices. The basic properties of this model are well understood theoretically and tested experimentally (see, e.g., Refs. [2,3]). However, some important questions still remain open. One of them concerns the reaction of a Bose gas to a mobile impurity particle, which is usually referred to as the Bose polaron problem, in analogy to the polaron studied by Landau and Pekar [4]. Polaron problems are among the simplest problems exhibiting non-trivial many-body effects that shed light on the interplay of one-and many-body physics. However, the fate of the impurity in a gas is not of only formal interest. Properties of many systems in condensed matter physics can be understood by studying a single particle interacting with a reservoir. Prominent examples are given by a single 3 He atom in liquid 4 He [5] and an electron in an ionic crystal (often described as a particle interacting with a Bose field of ion vibrations), see [6] and references therein. The apparent simplicity of these problems is misleading, as to date they resist a full theoretical solution. Fortunately, experiments with cold atomic gases, realizing the idea of a quantum simulator [7], open up the possibility to create and study the Bose polaron [8][9][10][11][12][13][14][15] in a laboratory. This intriguing possibility motivated a flurry of recent theoretical works on this problem [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].One-dimensional (1D) systems are of special interest in this context, because strongly interacting bosons in 1D fermionize [36]. This phenomenon simplifies the analysis. For example, if all the masses in the system are identical, the Bose polaron problem is exactly solvable [37]. This is also true if the impurity is infinitely heavy [30]. Therefore, a solution of the problem for weak and moderate boson-boson interaction is enough to complete the picture for all interaction strengths. However, theoretical approaches face challenges in describing these parameter regimes if the boson-impurity interactions are strong [38]. In this case accurate results can be obtai...

Many-body quantum dynamics and induced correlations of Bose polarons

“…Experimentally Bose [20][21][22][23][24] and Fermi [12,13,17] polarons have been observed and these experiments confirmed the importance of higher-order correlations for the description of the polaronic properties. The experiments in turn have spurred additional several theoretical investigations which have aimed at describing different polaronic aspects [25,26] by operating e.g.within the Fröhlich model [27][28][29][30][31], effective Hamiltonian approximations [8,[32][33][34], variational approaches [7,9,22,[35][36][37], renormalization group methods [25,38,39] and the path integral formalism [40,41].…”

Dissipative correlated dynamics of a moving impurity immersed in a Bose–Einstein condensate

“…These experiments triggered an intense theoretical activity in order to describe the polaron characteristics by operating within different frameworks [31,32]. These include, but are not restricted to, the mean-field approximation [33][34][35][36], the Fröhlich model [37][38][39][40][41][42], variational methods [14,17,24,25,27], effective Hamiltonian approaches [16,21,43,44] and renormalization group techniques [20,31,45]. While the majority of these investigations have been mainly focused on the equilibrium properties of the emergent quasiparticles, the dynamics of impurities is far less explored.…”

Entanglement-assisted tunneling dynamics of impurities in a double well immersed in a bath of lattice trapped bosons

“…In the extreme case such systems consist of a single impurity immersed in a majority species. These setups have been studied theoretically [25][26][27][28][29][30][31][32] and experimentally [33][34][35][36], for a single impurity, serving as a simulator of polaron physics, as well as for many impurities [37][38][39][40][41][42] and are indeed a subject of ongoing research. While the ground state properties of a single impurity in a bath are to a certain extent well understood, less focus has been placed on the transport properties and the emergent collisions of the impurity through the bath [43][44][45][46].…”