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arXiv:2206.13478 · Physical review. B./Physical review. B

Full view on the dynamics of an impurity coupled to two one-dimensional baths

Martino Stefanini, Massimo Capone, Alessandro Silva

Published May 31, 2023• 6 citations
arXivHTMLPDF

Abstract

We consider a model for the motion of an impurity interacting with two parallel, one-dimensional baths, described as two Tomonaga-Luttinger liquid systems. The impurity is able to move along the baths, and to jump from one to the other. We provide a perturbative expression for the evolution of the system when the impurity is injected in one of the baths, with a given wave packet. We obtain an approximation formally of infinite order in the impurity-bath coupling, which allows us to reproduce the orthogonality catastrophe. We monitor and discuss the dynamics of the impurity and its effect on the baths, in particular for a Gaussian wave packet. Besides the motion of the impurity, we also analyze the dynamics of the bath density and momentum density (i.e., the particle current), and show that it fits an intuitive semiclassical interpretation. We also quantify the correlation that is established between the baths by calculating the interbath, equal-time spatial correlation functions of both bath density and momentum, finding a complex pattern. We show that this pattern contains information on both the impurity motion and on the baths, and that these can be unveiled by taking appropriate ``slices'' of the time evolution.

DOI
10.1103/physrevb.107.184316

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In this thesis, we have explored various aspects of the dynamics of an impurity moving in more than one 1D bath. A recurring theme has been the investigation of the orthogonality catastrophe (OC) that follows the injection of the impurity in the system. This phenomenon has been studied by calculating the Green’s function of the impurity, which also described the time evolution and spectral features of the latter. The distinctive signature of the OC is that the Green’s function shows a power-law tail at long times. This function has been calculated using suitable perturbative expansions in the impurity-bath coupling, namely the Linked Cluster Expansion (LCE) and a time-dependent perturbation theory around a nontrivial dynamics. All of our results are nonperturbative in the inter-bath hopping. 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We consider a model for the motion of an impurity interacting with two parallel, one-dimensional baths, described as two Tomonaga-Luttinger liquid systems. The impurity is able to move along the baths, and to jump from one to the other. We provide a perturbative expression for the evolution of the system when the impurity is injected in one of the baths, with a given wave packet. We obtain an approximation formally of infinite order in the impurity-bath coupling, which allows us to reproduce the orthogonality catastrophe. We monitor and discuss the dynamics of the impurity and its effect on the baths, in particular for a Gaussian wave packet. Besides the motion of the impurity, we also analyze the dynamics of the bath density and momentum density (i.e., the particle current), and show that it fits an intuitive semiclassical interpretation. We also quantify the correlation that is established between the baths by calculating the interbath, equal-time spatial correlation functions of both bath density and momentum, finding a complex pattern. We show that this pattern contains information on both the impurity motion and on the baths, and that these can be unveiled by taking appropriate ``slices'' of the time evolution.
In this thesis, we have explored various aspects of the dynamics of an impurity moving in more than one 1D bath. A recurring theme has been the investigation of the orthogonality catastrophe (OC) that follows the injection of the impurity in the system. This phenomenon has been studied by calculating the Green’s function of the impurity, which also described the time evolution and spectral features of the latter. The distinctive signature of the OC is that the Green’s function shows a power-law tail at long times. This function has been calculated using suitable perturbative expansions in the impurity-bath coupling, namely the Linked Cluster Expansion (LCE) and a time-dependent perturbation theory around a nontrivial dynamics. All of our results are nonperturbative in the inter-bath hopping. In the two-bath scenario we have performed a detailed asymptotic expansion of the LCE Green's function at long times, which turned out to be very accurate in comparison with its numerical evaluation. The expansion has allowed us to obtain the renormalisation of the dispersion of the impurity bands, as well as the exponent of the power-law decay and the lifetime of the odd mode. One of our main results is that the OC, leading to the breakdown of the quasiparticle picture, survives the inclusion of a second 1D bath and dominates the long-time behaviour of all the components of the Green's function. In particular, the exponent characterising the long-time behaviour of the Green's function is given by half of the average of the exponents of the individual baths and, notably, is the same for the intra-bath Green's functions and for that connecting the two baths. In the case of two asymmetric baths, the Green's function is nonuniversal, acquiring a high-frequency component at short times and exhibiting persistent oscillations at longer times. In real experiments, the temperature of the baths is always finite, so we have extended the LCE treatment to this scenario. The effect of the temperature is to suppress the Green's function, limiting the possibility to observe its coherent oscillations and the power-law tail. At sufficiently long times, we found analytically that this suppression is exponential in time, with a different decay constant for the even and odd modes. For low temperatures, the two decay constants are approximately coinciding, and are proportional to the temperature. Using a perturbation theory in the inter-band part of the interaction we have been able to reproduce the LCE Green’s function and the OC with a method which also allows us to access the evolution of the whole impurity-bath system. The advantage of this approach is that it provides an analytic expression for the time evolution of the state of the whole impurity-baths system. This has allowed us to compute the time evolution of observables beyond the impurity Green's function, including the often-neglected properties of the baths. Moreover, the numerical effort required by the approach is sufficiently low so that we have been able to treat the case in which the impurity is initialised in a wave packet of an (almost) arbitrary shape. On the impurity side, we have analysed the time evolution of its population within each bath, observing how the persistent oscillations of the free impurity are damped and slowed down by the interaction with the baths. The impurity momentum is subjected to damping, as well. We have observed that momentum is transferred to the baths in two steps: a short transient, connected to the bath relaxation and independent of the inter-bath hopping, and a much slower decay caused by the emission of phonons during the deexcitation of the odd mode. Finally, we have also examined the time evolution of the probability density of finding the impurity in a given position and bath, for various Gaussian wave packets. We have studied the time evolution of observables describing the dynamics of the baths, which are rarely discussed in the literature on mobile impurities, but are nonetheless accessible to experiments. We have looked at the number of excited phonons, which shows a slow logarithmic divergence in time, related to the OC, superimposed with the faster growth caused by the emission of phonons from the odd mode decay. The particle density of the baths shows a semiclassical behaviour, intuitively similar to that of a pond in which a stone has been thrown. After the impurity has been injected into one of the baths, a localised density depletion forms and follows its motion. At the same time, two wave fronts are generated and propagate away. Moreover, each time the impurity oscillates between the baths, a new pair of ripples is emitted. The emission of ripples can be suppressed by employing wider wave packets. We have also found that the bath momentum density displays a behaviour analogous to that of the density. When the initial impurity wave packet has a markedly non-Gaussian shape, we find complex interference phenomena, both in the ripples and within the central trough. We examined the inter-bath, equal-times connected density and momentum density correlations, which revealed a rich structure in real space. This structure is best understood by taking “slices” of the correlation function along the relative and centre-of-mass coordinate, which show both the light-cone propagation of correlations and the motion of the impurity. We have made the first steps towards a many-baths system, in which the impurity moves in a 1D or 2D lattice of 1D baths. We have obtained the time evolution of the state of the system with the perturbative technique developed before. The impurity Green's function has revealed qualitative differences from the two-baths setup. First of all, the Green's function shows a complex short-time behaviour, caused by interference effects between the various paths of propagation within the lattice of baths. More importantly, we have found that each band of the noninteracting impurity is characterised by its own OC exponent, which is proportional to the degeneracy of the band. We have then speculated that it may be possible to tune the OC exponent either by properly designing the lattice of baths, or by changing the degeneracies by means of external magnetic fields. We also shown that for generic lattices of baths, the OC exponents vanish in the limit of an infinite number of lattice sites. Lastly, we showed that in the case of two identical baths there exists a unitary transformation that diagonalises the impurity degrees of freedom, thus achieving a complete decoupling of the latter from the baths. On the basis of this transformation, we have sketched two variational approximations for the ground-state and dynamics of the system.
We consider a linearly dispersing quantum impurity interacting through a contact density-density term with a one-dimensional (1D) superfluid described by the Tomonaga-Luttinger liquid theory. Using a linked-cluster expansion we characterize the impurity dynamics by calculating approximate expressions for the single-particle Green's function and for the time evolution of the density profile. We show the existence of two different dynamical regimes: (i) a quasiparticle regime, dominated by the presence of a finite lifetime for the host impurity, and (ii) an infrared-dominated regime, where the impurity causes a Anderson's orthogonality catastrophe of the 1D1\text{D}1D bath. We discuss the possible experimental consequences of these findings for cold-atom experiments.
We analyze the behavior of a mobile quantum impurity, restricted to a one-dimensional motion, in a bath formed by a two-leg bosonic ladder through a combination of field theory [Tomonaga-Luttinger liquid (TLL)] and numerical (density-matrix renormalization group) techniques. We compute the Green's function of the impurity as a function of time at different momenta. We find a power-law decay at zero momentum, which signals the breakdown of a quasiparticle description of the impurity motion. We compute the exponent both for the limits of weak and strong impurity-bath interactions. At small impurity-bath interaction, the impurity experiences the ladder as a single channel one-dimensional bath, but with an effective coupling reduced by a factor of 2\sqrt{2}2​. We compare the numerical results for the exponent at zero momentum with a semianalytical expression, initially established for the chain, and find excellent agreement without adjustable parameters. We find an increase of the exponent with increasing transverse tunneling in the bath. At small tunneling, the exponent is compatible with the TLL prediction, while larger tunneling shows strong deviations. Finally, we show that, as a function of the momentum of the impurity, two different regimes of decay of the Green's function exist, similar to the single chain case. The power-law regime occurs for small momentum, while at large momentum the Green's function shows a faster decay, corresponding to the one expected in a polaronic regime. In this last regime, we compute the lifetime of the polaron numerically. We compute the critical momentum marking the transition between these two regimes. We compare with analytical predictions based on the structure factor of the bath and find good agreement with the numerical results. Finally, we discuss the consequences of our results for cold atomic experiments.
We study the dynamics of a mobile impurity in a two-leg bosonic ladder. The impurity moves both along and across the legs and interacts with a bath of interacting bosonic particles present in the ladder. We use both analytical (Tomonaga-Luttinger liquid) and numerical [Density Matrix Renormalization Group (DMRG)] methods to compute the Green's function of the impurity. We find that for a small impurity-bath interaction, the symmetric mode of the impurity effectively couples only to the gapless mode of the bath while the antisymmetric mode of the impurity couples to both gapped and gapless modes of the bath. We compute the time dependence of the Green's function of the impurity, for impurity created in either the antisymmetric or symmetric mode with a given momentum. The latter case leads to a decay as a power law below a critical momentum and exponential above, while the former case exhibits both power-law and exponential decay depending on the transverse tunneling of the impurity. We compare the DMRG results with analytical results using the linked cluster expansion and find good agreement. In addition, we use DMRG to extract the lifetime of the quasiparticle, when the Green's function decays exponentially. We also treat the case of an infinite bath-impurity coupling for which both the symmetric and antisymmetric modes are systematically affected. For this case, the impurity Green's function in the symmetric mode decays as a power law at zero momentum. The corresponding exponent increases with increasing transverse tunneling of the impurity. We compare our results with other impurity problems for which the motion of either the impurity or the bath is limited to a single chain. Finally, we comment on the consequences of our findings for experiments with the ultracold gases.
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.
By means of the time-dependent density-matrix renormalization-group (TDMRG) method we are able to follow the real-time dynamics of a single impurity embedded in a one-dimensional bath of interacting bosons. We focus on the impurity breathing mode, which is found to be well described by a single oscillation frequency and a damping rate. If the impurity is very weakly coupled to the bath, a Luttinger-liquid description is valid and the impurity suffers an Abraham-Lorentz radiation-reaction friction. For a large portion of the explored parameter space, the TDMRG results fall well beyond the Luttinger-liquid paradigm.