We study a one-dimensional Bose gas with two internal states described by the Yang-Gaudin model and calculate analytically the dispersion relation of a polaron quasiparticle, which is the lowest excitation branch. We study the dispersion in the thermodynamic limit in the regimes of weak and strong interaction without limitations on the momentum. At weak interaction the polaron dispersion is in the vicinity of the dark soliton one; we calculate the leading deviation in the parametric form. At strong interaction we find an ansatz for the explicit form of the polaron dispersion. It has the form of a power series of the sine function of the momentum with interaction-dependent coefficients. By increasing the power of the series, the corresponding coefficients show faster decay and thus one practically needs only a few of them; we give the results for the first three. The coefficients of the series are connected to the Maclaurin series of the polaron dispersion and thus it is sufficient to calculate latter quantity to automatically find the power series result for the dispersion at all momenta. The derived results for the dispersion can be used to obtain explicit expressions for the exponents of the power-law singularities in the response functions at the spectral edge.
We study a one-dimensional Bose gas with two internal states described by the Yang-Gaudin model and calculate analytically the dispersion relation of a polaron quasiparticle, which is the lowest excitation branch. We study the dispersion in the thermodynamic limit in the regimes of weak and strong interaction without limitations on the momentum. At weak interaction the polaron dispersion is in the vicinity of the dark soliton one; we calculate the leading deviation in the parametric form. At strong interaction we find an ansatz for the explicit form of the polaron dispersion. It has the form of a power series of the sine function of the momentum with interaction-dependent coefficients. By increasing the power of the series, the corresponding coefficients show faster decay and thus one practically needs only a few of them; we give the results for the first three. The coefficients of the series are connected to the Maclaurin series of the polaron dispersion and thus it is sufficient to calculate latter quantity to automatically find the power series result for the dispersion at all momenta. The derived results for the dispersion can be used to obtain explicit expressions for the exponents of the power-law singularities in the response functions at the spectral edge.
Bose polarons, quasiparticles composed of mobile impurities surrounded by cold Bose gas, can experience strong interactions mediated by the many-body environment and form bipolaron bound states. Here we present a detailed study of heavy polarons in a one-dimensional Bose gas by formulating a nonperturbative theory and complementing it with exact numerical simulations. We develop an analytic approach for weak boson-boson interactions and arbitrarily strong impurity-boson couplings. Our approach is based on a mean-field theory that accounts for deformations of the superfluid by the impurities and in this way minimizes quantum fluctuations. The mean-field equations are solved exactly in the Born-Oppenheimer approximation, leading to an analytic expression for the interaction potential of heavy polarons, which is found to be in excellent agreement with quantum Monte Carlo (QMC) results. In the strong coupling limit, the potential substantially deviates from the exponential form valid for weak coupling and has a linear shape at short distances. Taking into account the leading-order Born-Huang corrections, we calculate bipolaron binding energies for impurity-boson mass ratios as low as 3 and find excellent agreement with QMC results.
Abstract We study the excitation spectrum of two-component delta-function interacting bosons confined to a single spatial dimension, the Yang–Gaudin Bose gas. We show that there are pronounced finite-size effects in the dispersion relations of excitations, perhaps best illustrated by the spinon single particle dispersion which exhibits a gap at 2k F and a finite-momentum roton-like minimum. Such features occur at energies far above the finite volume excitation gap, vanish slowly as 1/ L for fixed spinon number, and can persist to the thermodynamic limit at fixed spinon density. Features such as the 2k F gap also persist to multi-particle excitation continua. Our results show that excitations in the finite system can behave in a qualitatively different manner to analogous excitations in the thermodynamic limit . The Yang–Gaudin Bose gas is also host to multi-spinon bound states, known as Λ -strings. We study these excitations both in the thermodynamic limit under the string hypothesis and in finite size systems where string deviations are taken into account. In the zero-temperature limit we present a simple relation between the length n Λ -string dressed energies ϵ n ( λ ) and the dressed energy ϵ ( k ) . We solve the Yang–Yang–Takahashi equations numerically and compare to the analytical solution obtained under the strong couple expansion, revealing that the length n Λ -string dressed energy is Lorentzian over a wide range of real string centers λ in the vicinity of λ = 0 . We then examine the finite size effects present in the dispersion of the two-spinon bound states by numerically solving the Bethe ansatz equations with string deviations.
An impurity particle interacting with a Bose-Einstein condensate (BEC) leads to the formation of a quasiparticle known as the Bose polaron. We investigate the properties of the two-dimensional Bose polaron, applying a variational ansatz that contains up to three Bogoliubov excitations of the BEC. Similar to its three-dimensional counterpart, we observe the existence of two quasiparticle branches, namely, the attractive and the repulsive polarons, at different coupling strengths. We find that their energies agree well with recent quantum Monte Carlo calculations. In particular, we observe that the inclusion of three excitations is crucial to capture the attractive polaron energy towards the regime of strong attraction, where the quasiparticle properties are dominated by few-body correlations. We also calculate the attractive polaron effective mass and residue, where we find significant differences between considering a weakly interacting Bose medium and taking the noninteracting limit, signaling enhanced impurity dressing by excitations in the latter case. By contrast, the spectral weight of the metastable repulsive polaron is largely insensitive to the interactions in the BEC and the number of Bogoliubov excitations. Our model may be experimentally realized in dilute atomic vapors and atomically thin semiconductors.
We use variational methods to study a spin impurity in a one-dimensional Bose lattice gas. Both in the strongly interacting superfluid regime and in the Mott regime we find that the impurity binds with a hole, forming a polaron. Our calculations for the dispersion of the polaron are consistent with recent experiments by Fukuhara et al. [Nat. Phys. 9, 235 (2013)] and give a better understanding of their numerical simulations. We find that for sufficiently weak interactions there are ranges of momentum for which the polaron is unstable. We propose experimentally studying the stability of the polaron by measuring the correlation between the impurity and the hole. We also study two interacting impurities, finding stable bipolarons for sufficiently strong interactions.
The polaron model is a basic model of quantum field theory describing a single particle interacting with a bosonic field. It arises in many physical contexts. We are mostly concerned with models applicable in the context of an impurity atom in a Bose-Einstein condensate as well as the problem of electrons moving in polar crystals. The model has a simple structure in which the interaction of the particle with the field is given by a term linear in the field’s creation and annihilation operators. In this work, we investigate the properties of this model by providing rigorous estimates on various energies relevant to the problem. The estimates are obtained, for the most part, by suitable operator techniques which constitute the principal mathematical substance of the thesis. The first application of these techniques is to derive the polaron model rigorously from first principles, i.e., from a full microscopic quantum-mechanical many-body problem involving an impurity in an otherwise homogeneous system. We accomplish this for the N + 1 Bose gas in the mean-field regime by showing that a suitable polaron-type Hamiltonian arises at weak interactions as a low-energy effective theory for this problem. In the second part, we investigate rigorously the ground state of the model at fixed momentum and for large values of the coupling constant. Qualitatively, the system is expected to display a transition from the quasi-particle behavior at small momenta, where the dispersion relation is parabolic and the particle moves through the medium dragging along a cloud of phonons, to the radiative behavior at larger momenta where the polaron decelerates and emits free phonons. At the same time, in the strong coupling regime, the bosonic field is expected to behave purely classically. Accordingly, the effective mass of the polaron at strong coupling is conjectured to be asymptotically equal to the one obtained from the semiclassical counterpart of the problem, first studied by Landau and Pekar in the 1940s. For polaron models with regularized form factors and phonon dispersion relations of superfluid type, i.e., bounded below by a linear function of the wavenumbers for all phonon momenta as in the interacting Bose gas, we prove that for a large window of momenta below the radiation threshold, the energy-momentum relation at strong coupling is indeed essentially a parabola with semi-latus rectum equal to the Landau–Pekar effective mass, as expected. For the Fröhlich polaron describing electrons in polar crystals where the dispersion relation is of the optical type and the form factor is formally UV–singular due to the nature of the point charge-dipole interaction, we are able to give the corresponding upper bound. In contrast to the regular case, this requires the inclusion of the quantum fluctuations of the phonon field, which makes the problem considerably more difficult. The results are supplemented by studies on the absolute ground-state energy at strong coupling, a proof of the divergence of the effective mass with the coupling constant for a wide class of polaron models, as well as the discussion of the apparent UV singularity of the Fröhlich model and the application of the techniques used for its removal for the energy estimates.