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arXiv:1210.4183 · Europhysics Letters (EPL)

Effective potential and polaronic mass shift in a trapped dynamical impurity–Luttinger liquid system

Julius Bonart, Leticia F. Cugliandolo

Published Jan 1, 2013• 22 citations
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Abstract

Recent experiments with cold atoms on the impurity motion in one-dimensional liquids of interacting bosons have revealed an interesting interplay between the polaronic impurity mass shift and the renormalization of the optical potential. We show that the influence of the external trap on the Bose gas leads to a steeper effective potential for the impurity. We propose a framework in which this potential renormalization and the mass shift can be quantitatively understood by combining a semi-classical theory of density wave excitations in the Luttinger liquid with the non-equilibrium formalism of a quantum Brownian particle. The obtained theoretical results reproduce well recent experimental data.

DOI
10.1209/0295-5075/101/16003

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Recent experiments with cold atoms on the impurity motion in one-dimensional liquids of interacting bosons have revealed an interesting interplay between the polaronic impurity mass shift and the renormalization of the optical potential. We show that the influence of the external trap on the Bose gas leads to a steeper effective potential for the impurity. We propose a framework in which this potential renormalization and the mass shift can be quantitatively understood by combining a semi-classical theory of density wave excitations in the Luttinger liquid with the non-equilibrium formalism of a quantum Brownian particle. The obtained theoretical results reproduce well recent experimental data.
Impurity motion in one-dimensional ultracold quantum liquids confined in an optical trap has attracted much interest recently. As a step towards its full understanding, we construct a generating functional from which we derive the position nonequilibrium correlation function of a quantum Brownian particle with general Gaussian nonfactorizing initial conditions. We investigate the slow dynamics of a particle confined in a harmonic potential after a position measurement; the rapid relaxation of a particle trapped in a harmonic potential after a quantum quench realized as a sudden change in the potential parameters; and the evolution of an impurity in contact with a one-dimensional bosonic quantum gas. We argue that such an impurity--Luttinger-liquid system, which has been recently realized experimentally, admits a simple modeling as quantum Brownian motion in a super-Ohmic bath.
One-dimensional Bose gases present an interesting setting to study the physics of Bose polarons, as density fluctuations play an enhanced role due to reduced dimensionality. Theoretical descriptions of this system have predominantly relied on contact pseudopotentials to model the impurity-bath interaction, leading to unphysical results in the strongly coupled limit. In this work, we analytically solve the Gross-Pitaevskii equation, using a square-well potential instead of a zero-range potential, for the ground-state wave function of a static impurity. We compute perturbative corrections arising from infinitesimally slow impurity motion. The polaron energy and effective mass remain finite in the strongly coupled regime, in contrast to the divergent behavior obtained using a contact potential. In this limit, we characterize the polaron properties in terms of the dimensionless ratio w ̃ ≡ w / ξ between the interaction range w of the impurity-bath potential and the coherence length ξ of the Bose gas. The effective mass exhibits a 1 / w ̃ scaling. The energy of the attractive polaron scales as − 1 / w ̃ 3 , whereas the repulsive polaron features subleading corrections to the dark soliton energy at the order w ̃ 3 .
We investigate a temporal evolution of an impurity atom in a one-dimensional trapped Bose gas following a sudden change of the boson-impurity interaction strength. Our focus is on the effects of inhomogeneity due to the harmonic confinement. These effects can be described by an effective one-body model where both the mass and the spring constant are renormalized. This is in contrast to the classic renormalization, which addresses only the mass. We propose an effective single-particle Hamiltonian and apply the multilayer multiconfiguration time-dependent Hartree method for bosons to explore its validity. Numerical results suggest that the effective mass is smaller than the impurity mass, which means that it cannot straightforwardly be extracted from translationally invariant models.
We study the dynamics of an impurity embedded in a trapped Bose-Einstein condensate, i.e., the Bose polaron problem. This problem is treated by recalling open quantum systems techniques: the impurity corresponds to a particle performing quantum Brownian motion, while the excitation modes of the gas play the role of the environment. It is crucial that the model considers a parabolic trapping potential to resemble the experimental conditions. Thus, we detail here how the formal derivation changes due to the gas trap, in comparison to the homogeneous gas. More importantly, we elucidate all aspects in which the gas trap plays a relevant role, with an emphasis on the enhancement of the non-Markovian character of the dynamics. We first find that the presence of a gas trap leads to a new form of the bath-impurity coupling constant and a larger degree in the super-Ohmicity of the spectral density. We then solve the quantum Langevin equation to derive the position and momentum variances of the impurity, where the former is a measurable quantity. For the particular case of an untrapped impurity, the asymptotic behavior of this quantity is found to be motion superdiffusive. When the impurity is trapped, we find position squeezing, casting the system suitable for implementing quantum metrology and sensing protocols. We detail how both superdiffusion and squeezing can be enhanced or inhibited by tuning the Bose-Einstein condensate trap frequency. Compared to the homogeneous gas case, the form of the bath-impurity coupling constant changes, and this is manifested as a different dependence of the system dynamics on the past history. To quantify this, we introduce several techniques to compare the different amount of memory effects arising in the homogeneous and inhomogeneous gas. We find that it is higher in the second case. This analysis paves the way to the study of non-Markovianity in ultracold gases, and the possibility to exploit such a property in the realization of new quantum devices.
We study the dynamics of a heavy particle of mass M moving in a one-dimensional repulsively interacting Fermi gas. The Fermi gas is described using the Luttinger model and bosonization. By transforming to a frame comoving with the heavy particle, we map the model onto a generalized ``quantum impurity problem.'' A renormalization-group calculation reveals a crossover from strong to weak coupling upon scaling down in temperature. Above the crossover temperature scale T∗{\mathit{T}}^{\mathrm{*}}T∗=(m/M)EF{\mathit{E}}_{\mathit{F}}EF​, the particle's mobility \ensuremath{\mu} is found to be (roughly) temperature independent and proportional to the dimensionless conductance g, characterizing the one-dimensional Luttinger liquid. Here m(\ensuremath{\ll}M) is the fermion mass, and EF{\mathit{E}}_{\mathit{F}}EF​ the Fermi energy. Below T∗{\mathit{T}}^{\mathrm{*}}T∗, in the weak-coupling regime, the mobility grows and diverges as \ensuremath{\mu}(T)\ensuremath{\sim}${\mathit{T}}^{\mathrm{\ensuremath{-}}4}$ in the T\ensuremath{\rightarrow}0 limit. \textcopyright{} 1996 The American Physical Society.
Mobile impurity atoms immersed in Bose–Einstein condensates provide a new platform for exploring Bose polarons. Recent experimental advances in the field of ultracold atoms make it possible to realize such systems with highly tunable microscopic parameters and to explore equilibrium and dynamical properties of polarons using a rich toolbox of atomic physics. In this paper we present a detailed theoretical analysis of Bose polarons in one-dimensional systems of ultracold atoms. By combining a non-perturbative renormalization group approach with numerically exact diffusion Monte Carlo calculations we obtain not only detailed numerical results over a broad range of parameters but also qualitative understanding of different regimes of the system. We find that an accurate description of Bose polarons requires the inclusion of two-phonon scattering terms which go beyond the commonly used Fröhlich model. Furthermore we show that when the Bose gas is in the strongly interacting regime, one needs to include interactions between the phonon modes. We use several theoretical approaches to calculate the polaron energy and its effective mass. The former can be measured using radio-frequency spectroscopy and the latter can be studied experimentally using impurity oscillations in a harmonic trapping potential. We compare our theoretical results for the effective mass to the experiments by Catani et al (2012 Phys. Rev. A 85 023623). In the weak-to-intermediate coupling regimes we obtain excellent quantitative agreement between theory and experiment, without any free fitting parameter. We supplement our analysis by full dynamical simulations of polaron oscillations in a shallow trapping potential. We also use our renormalization group approach to analyze the full phase diagram and identify regions that support repulsive and attractive polarons, as well as multi-particle bound states.