Moderate Coupling Research Articles

Room-temperature polarization of individual nuclear spins in diamond via anisotropic hyperfine coupling and coherent population trapping

We employ the electronic spin of a single nitrogen-vacancy (NV) defect in diamond to detect and control the quantum state of remote nuclear spins coupled by hyperfine interaction. More precisely, our work focuses on individual ^{13}text {C} nuclei featuring a moderate hyperfine coupling strength (sim 1 MHz) with the NV’s electron spin. Two different methods providing an efficient room-temperature polarization of these peculiar ^{13}text {C} nuclear spins are described. The first one is based on a polarization transfer from the NV electron spin to the ^{13}text {C} nucleus, which is mediated by the anisotropic component of the hyperfine interaction. The second one relies on coherent population trapping (CPT) within a Lambda -type energy-level configuration in the microwave domain, which enables to initialize the ^{13}text {C} nuclear spin in any quantum state superposition on the Bloch sphere. This CPT protocol is performed in an unusual regime for which relaxation from the excited level of the Lambda -scheme is externally triggered by optical pumping and separated in time from coherent microwave excitations. For these two polarization techniques, we investigate the impact of optical illumination on the nuclear spin polarization efficiency. This work adds new methods to the quantum toolbox used for coherent control of individual nuclear spins in diamond, which might find applications in quantum metrology.Graphical

Designing high-efficiency metal and semimetal contacts to two-dimensional semiconductor γ-GeSe

Forming a low-resistance semiconductor–metal contact is a critical step to achieve a high-performance two-dimensional (2D) semiconductor nanoelectronic device. Motivated by the recent discovery of monolayer γ-GeSe with exceptional high electrical conductivity reaching 105 S/m, we computationally investigate the interface contact properties of γ-GeSe with four representative classes of metallic systems, including 2D semimetal (graphene), 2D metal (NbS2), 3D semimetal (Bi), and 3D metal (Au) using first-principle density functional theory simulations. We found that these metals exhibit rich contact formation physics with 2D γ-GeSe, yielding contacts of heterostructures with weak and moderate couplings. Importantly, γ-GeSe/NbS2 is an Ohmic contact while γ-GeSe/Bi is an n-type Schottky contact with an ultralow barrier height of 0.07 eV. For γ-GeSe/graphene contact, the electronic properties can be adjusted via the interlayer distance or via an external electric field. Finally, we show that the contact properties can also be further controlled using layer-number engineering of γ-GeSe. Our findings provide a useful guideline for designing high-performance 2D nanoelectronics based on 2D γ-GeSe.

Computational Evaluation of Potential Molecular Catalysts for Nitrous Oxide Decomposition.

Nitrous oxide (N2O) is a potent greenhouse gas (GHG) with limited use as a mild anesthetic and underdeveloped reactivity. Nitrous oxide splitting (decomposition) is critical to its mitigation as a GHG. Although heterogeneous catalysts for N2O decomposition have been developed, highly efficient, long-lived solid catalysts are still needed, and the details of the catalytic pathways are not well understood. Reported herein is a computational evaluation of three potential molecular (homogeneous) catalysts for N2O splitting, which could aid in the development of more active and robust catalysts and provide deeper mechanistic insights: one Cu(I)-based, [(CF3O)4Al]Cu (A-1), and two Ru(III)-based, Cl(POR)Ru (B-1) and (NTA)Ru (C-1) (POR = porphyrin, NTA = nitrilotriacetate). The structures and energetic viability of potential intermediates and key transition states are evaluated according to a two-stage reaction pathway: (A) deoxygenation (DO), during which a metal-N2O complex undergoes N-O bond cleavage to produce N2 and a metal-oxo species and (B) (di)oxygen evolution (OER), in which the metal-oxo species dimerizes to a dimetal-peroxo complex, followed by conversion to a metal-dioxygen species from which dioxygen dissociates. For the (F-L)Cu(I) activator (A-1), deoxygenation of N2O is facilitated by an O-bound (F-L)Cu-O-N2 or better by a bimetallic N,O-bonded, (F-L)Cu-NNO-Cu(F-L) complex; the resulting copper-oxyl (F-L)Cu-O is converted exergonically to (F-L)Cu-(η2,η2-O2)-Cu(F-L), which leads to dioxygen species (F-L)Cu(η2-O2), that favorably dissociates O2. Key features of the DO/OER process for (POR)ClRu (B-1) include endergonic N2O coordination, facile N2 evolution from LR'u-N2O-RuL to Cl(POR)RuO, moderate barrier coupling of Cl(POR)RuO to peroxo Cl(POR)Ru(O2)Ru(POR)Cl, and eventual O2 dissociation from Cl(POR)Ru(η1-O2), which is nearly thermoneutral. N2O decomposition promoted by (NTA)Ru(III) (C-1) can proceed with exergonic N2O coordination, facile N2 dissociation from (NTA)Ru-ON2 or (NTA)Ru-N2O-Ru(NTA) to form (NTA)Ru-O; dimerization of the (NTA)Ru-oxo species is facile to produce (NTA)Ru-O-O-Ru(NTA), and subsequent OE from the peroxo species is moderately endergonic. Considering the overall energetics, (F-L)Cu and Cl(POR)Ru derivatives are deemed the best candidates for promoting facile N2O decomposition.

Spatially Coupled PLDPC-Hadamard Convolutional Codes

We propose a new type of ultimate-Shannon-limit-approaching codes called spatially coupled protograph-based low-density parity-check Hadamard convolutional codes (SC-PLDPCH-CCs), which are constructed by spatially coupling PLDPC-Hadamard block codes. We develop an efficient decoding algorithm that combines pipeline decoding and layered scheduling for the decoding of SC-PLDPCH-CCs, and analyze the latency and complexity of the decoder. To estimate the decoding thresholds of SC-PLDPCH-CCs, we first propose a layered protograph extrinsic information transfer (PEXIT) algorithm to evaluate the thresholds of spatially coupled PLDPC-Hadamard terminated codes (SC-PLDPCH-TDCs) with a moderate coupling length. With the use of the proposed layered PEXIT method, we develop a genetic algorithm to find good SC-PLDPCH-TDCs in a systematic way. Then we extend the coupling length of these SC-PLDPCH-TDCs to form good SC-PLDPCH-CCs. Results show that our constructed SC-PLDPCH-CCs can achieve comparable thresholds to the block code counterparts. Simulations illustrate the superiority of the SC-PLDPCH-CCs over the block code counterparts and other state-of-the-art low-rate codes in terms of error performance. For the rate-0.00295 SC-PLDPCH-CC, a bit error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−5</sup> is achieved at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$E_{b}/N_{0} = -1.465$ </tex-math></inline-formula> dB, which is only 0.125 dB from the ultimate Shannon limit.

Spatial–Temporal Characteristics and Driving Factors of the Coupling Coordination between Population Health and Economic Development in China

Promoting the coordinated development of population health and the economy is an important part of building a “Healthy China” and promoting high-quality economic development. Based on the systematic construction of the population health and economic development evaluation index system, this paper uses the coupled coordination model, geodetector, and geographically weighted regression (GWR) to comprehensively measure the population health level and economic development level at the provincial scale in China in 2000 and 2015, and reveals the spatial and temporal evolution characteristics of the coupled coordination relationship between the population health level and economic development level at the provincial scale in China from 2000 to 2015 and its driving factors. The results show the following: (1) China’s population health and economic development are in a high-level coupling stage, and the coupling level increases slightly with time; spatially, two types of running-in coupling and high-level coupling coexist; the coupling degree in the eastern and central regions tends to increase, while the coupling degree in the western region tends to weaken. (2) China’s population health and economic development are in a good coupling coordination stage as a whole, and the coupling coordination degree has an increasing trend; spatially, the coupling coordination degree shows high spatial differentiation characteristics in the east and low in the west; the good and high-quality coupling coordination type area tends to expand to the west, while the moderate coupling coordination type area tends to shrink to the west; there is also positive spatial agglomeration of coupling coordination degree, and the spatial agglomeration is gradually enhanced. (3) The coupling coordination of China’s population health and economic development is driven by multiple factors such as natural conditions, health resources, culture quality, and urbanization level; the interaction between factors is stronger than that of a single factor, and the driving effect of each factor also shows significant spatial heterogeneity. This study is intended to provide a scientific basis for promoting harmonious population health and economic development.

Effects of thermostats/barostats on physical properties of liquids by molecular dynamics simulations

Thermostats/barostats that maintain temperature/pressure as constant (on average) in molecular dynamics (MD) simulations are essential for simulatingisothermal-isochoric (NVT) or isothermal-isobaric (NPT) systems. In order to simulate accurate physical properties, thermostats/barostats need not only sample a correct NVT/NPT ensemble but also minimally disturb the particles’ (Newtonian) dynamics. In other words, the NVT/NPT simulations need to yield accurate physical properties (dynamics properties included, e.g. diffusivity and viscosity) and also their fluctuations. However, few studies have studied in-depth the effects of thermostats/barostats on a comprehensively wide range of simulated properties. In this work, commonly used thermostats, e.g. Andersen, stochastic dynamics (SD), Berendsen, V-rescale and Nosé-Hoover thermostats, and barostats, e.g. Berendsen and Parrinello-Rahman barostats, with different coupling strengths were studied to see whether they can yield accurate simulated properties and fluctuations. The theoretical values of physical properties’ fluctuations were calculated via statistical mechanics to provide a comparison benchmark for MD simulations. Particularly, the accuracy of thermostats was studied in the context of NPT or non-equilibrium molecular dynamics (NEMD) simulations, which has long been overlooked. Berendsen thermostat/barostat suppresses the fluctuations of energy/volume (due to the exponential decay of deviation of the simulated temperature/pressure from the target value) and yields inaccurate simulated properties. In addition, Andersen and SD thermostats perturb the particles’ dynamics so violently (due to the random component in the algorithm) that they fail to accurately simulate dynamics properties. Overall, Nosé-Hoover/V-rescale thermostat, and Parrinello-Rahman barostat with moderate coupling strength are recommended for common NVT/NPT production simulations. NPT or NEMD simulations would require more efficient (or stronger coupling) thermostats than NVT equilibrium molecular dynamics (EMD) to maintain the simulated temperature close to the target.

Study on the Coupling Coordination between New-Type Urbanization and Water Ecological Environment and Its Driving Factors: Evidence from Jiangxi Province, China.

With the rapid development of urbanization, problems such as the degradation of water ecological environment have emerged. How to improve the water ecological environment in the process of urbanization has become one of the urgent problems facing policy makers. This paper studies the coupling coordination relationship between new-type urbanization and water ecological environment, with the purpose of using insights gained from the study to help improve the quality of water ecological environment and promote sustainable development of water ecological environment. We take 11 cities in China’s Jiangxi Province as the research object, and construct the coupling coordination evaluation indicator system of new-type urbanization and water ecological environment, then using the coupling coordination degree model to examine the state of coupling coordination between new-type urbanization and water ecological environment from 2009 to 2019. We further explore its driving factors employing random effect panel Tobit model. The results show that: (1) The level of new-type urbanization in Jiangxi Province shows a steady upward trend, and the water ecological environment level tends to rise steadily and slowly, although the comprehensive score of water ecological environment in most cities is lower than 0.1, indicating that the situation of water ecological environment is not optimal yet and there is room for improvement. (2) In 2009, 2014 and 2019, the coupling coordination development level between new-type urbanization and water ecological environment in Jiangxi Province showed an upward trend, from moderate maladjustment recession to mild maladjustment recession, and from low coupling coordination to moderate coupling coordination, although the overall coupling coordination degree was low. (3) The investment in scientific and technological innovation, degree of opening-up and government capacity are positively correlated with the coupling coordination degree, while economic development level, resource agglomeration ability, education level and industrialization level are negatively correlated with the coupling coordination degree. These results can provide insights to support new-type urbanization and water ecological environment in the future, and hold great significance for urban sustainable development.

Gravitational waves from first-order electroweak phase transition in a model with light sgoldstinos

We study previously unexplored possibility of triggering the first order electroweak phase transition (EWPT) by interactions of the Standard Model (SM) particles with the sector responsible for low scale supersymmetry breaking. The low-energy theory apart from the SM particles contains additional scalar degrees of freedom — sgoldstinos — which contribute to the effective scalar potential and thus can trigger the first order EWPT. Remarkably, the latter requires only moderate couplings in the scalar sector. The perturbative description in terms of the effective theory seems formally to be applicable upto the scale of supersymmetry breaking: the Landau pole in the scalar sector is above 108-109 GeV. We calculate the gravitational wave signal generated at this transition (it can be tested, e.g. by LISA, BBO and DECIGO) and briefly discuss the collider phenomenology of this scenario.

Effective electronic forces and potentials from ab initio path integral Monte Carlo simulations.

The rigorous description of correlated quantum many-body systems constitutes one of the most challenging tasks in contemporary physics and related disciplines. In this context, a particularly useful tool is the concept of effective pair potentials that take into account the effects of the complex many-body medium consistently. In this work, we present extensive, highly accurate ab initio path integral Monte Carlo (PIMC) results for the effective interaction and the effective force between two electrons in the presence of the uniform electron gas. This gives us a direct insight into finite-size effects, thereby, opening up the possibility for novel domain decompositions and methodological advances. In addition, we present unassailable numerical proof for an effective attraction between two electrons under moderate coupling conditions, without the mediation of an underlying ionic structure. Finally, we compare our exact PIMC results to effective potentials from linear-response theory, and we demonstrate their usefulness for the description of the dynamic structure factor. All PIMC results are made freely available online and can be used as a thorough benchmark for new developments and approximations.

Superconductivity in Kagome Metal YRu3Si2 with Strong Electron Correlations

We report the detailed physical properties of YRu3Si2 with the Ru kagome lattice at normal and superconducting states. The results of resistivity and magnetization show that YRu3Si2 is a type-II bulk superconductor with T c ∼ 3.0 K. The specific heat measurement further suggests that this superconductivity could originate from the weak or moderate electron-phonon coupling. On the other hand, both large Kadawaki–Woods ratio and Wilson ratio indicate that there is a strong electron correlation effect in this system, which may have a connection with the featured flat band of kagome lattice.

Interfacial electronic properties of metal/CsSnBr3 heterojunctions

All-inorganic lead-free perovskite CsSnBr3, has been proved good stability and optoelectronic properties in theory and experiment. However, the interfacial electronic properties of metal/CsSnBr3 are still unclear in electronic devices. Herein, we systematically investigate the interfacial properties of metal electrodes (Al, Ag and Au) and CsSnBr3 with different atomic terminals (SnBr2-T and CsBr-T) through the first-principles calculation. SnBr2-T and CsBr-T have various contact types and Schottky barriers due to their different interaction strengths with metals. In particular, the moderate interlayer coupling strength with Al leads to the ultra-low Schottky barrier and tunneling barrier, which makes Al possess the best contact performance among the studied metals. Furthermore, the external electric field can be effective in regulating the Schottky barrier and realizing the Ohmic contact. These findings provide useful guidance for the design of perovskite-based nanoelectronic devices with high performance.

Efficient Adversarial Generation of Thermally Activated Delayed Fluorescence Molecules.

Adversarial generative models are becoming an essential tool in molecular design and discovery due to their efficiency in exploring the desired chemical space with the assistance of deep learning. In this article, we introduce an integrated framework by combining the modules of algorithmic synthesis, deep prediction, adversarial generation, and fine screening for the purpose of effective design of the thermally activated delayed fluorescence (TADF) molecules that can be used in the organic light-emitting diode devices. The retrosynthetic rules are employed to algorithmically synthesize the D–A complex based on the empirically defined donor and acceptor moieties, which is followed by the high-throughput labeling and prediction with the deep neural network. The new D–A molecules are subsequently generated via the adversarial autoencoder, with the excited-state property distributions perfectly matching those of the original samples. Fine screening of the generated molecules, including the spin–orbital coupling calculation and the excited-state optimization, is eventually implemented to select the qualified TADF candidates within the novel chemical space. Further investigation shows that the created structures fully mimic the original D–A samples by maintaining a significant charge transfer characteristic, a minimal adiabatic singlet–triplet gap, and a moderate spin–orbital coupling that are desirable for the delayed fluorescence.

Bayesian Inference of Fault Slip and Coupling Along the Tuosuo Lake Segment of the Kunlun Fault, China

AbstractIntegrating newly‐processed Interferometric Synthetic Aperture Radar (InSAR) and available Global Positioning System (GPS) data, we obtained a dense interseismic velocity field across the Tuosuo Lake segment of the Kunlun fault. Using a Bayesian approach, we obtained the spatial distribution of fault slip and coupling. The slip rate below the locking depth, which represents contemporary steady‐state faulting, demonstrates an overall decrease, and decreases non‐monotonically toward the eastern fault tip in the range of ∼5–11 mm/a. Two strongly locked zones are identified, extending 100–120 km along‐strike and being separated by narrow sections of moderate coupling. These locked zones were likely ruptured during the 1937 M 7.5 Huashixia earthquake. The estimated moment accumulation rate provides a recurrence time of 428 ± 57 years for an earthquake similar to the 1937 event. The Maduo Mw 7.3 earthquake on 21 May 2021 imposed stress loading on the locked zones, equivalent to tectonic loading of ∼10 years, implying increased seismic hazard along the Kunlun fault.

A frequency domain approach for analyzing motion responses of integrated offshore fish cage and wind turbine under wind and wave actions

This paper presents a frequency domain (FD) approach for analyzing wind and wave coupling effect on motion responses of an integrated offshore fish cage and wind turbine named COSPAR. FD analysis was performed by ANSYS AQWA to obtain response amplitude operators (RAOs) for surge, pitch and heave motions as well as mooring tension RAOs. Wind thrust was estimated by using a linearization method based on the pre-defined thrust coefficients determined from a fixed-bottom wind turbine. In AQWA, wind force coefficients were specified to incorporate the thrust force and moment in the hydrodynamic response analysis of a floating structure. The present FD method was applied to a DeepCwind floating wind turbine problem for its validation. It was found that the AQWA-FD approach can furnish wind and wave induced motion responses that are comparable with published experimental and numerical results. Thereby, this approach can be used to readily obtain a preliminary design for subsequent refinements for an optimal design. Based on the FD analysis results, DeepCwind and COSPAR have a pronounced the coupling effect only in the pitch motion. Moreover, COSPAR shows not only moderate coupling effect, but steadier and less RAOs in surge, heave, pitch and mooring tension than those of DeepCwind. In addition, it is evident that COSPAR can provide a stable working platform for offshore fish farms from wind, wave and current loads, and has acceptable pitch angles for a steady wind turbine operation.

Quantum Rajeev–Ranken model as an anharmonic oscillator

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves of wavenumber k in a scalar field theory pseudodual to the 1 + 1D SU(2) principal chiral model. Classically, the RR model is Liouville integrable. Here, we interpret the model as a novel 3D cylindrically symmetric quartic oscillator with an additional rotational energy. The quantum theory has two dimensionless parameters. Upon separating variables in the Schrödinger equation, we find that the radial equation has a four-term recurrence relation. It is of type [0, 1, 16] and lies beyond the ellipsoidal Lamé and Heun equations in Ince’s classification. At strong coupling λ, the energies of highly excited states are shown to depend on the scaling variable λk. The energy spectrum at weak coupling and its dependence on k in a double-scaling strong coupling limit are obtained. The semi-classical Wentzel-Kramers-Brillouin (WKB) quantization condition is expressed in terms of elliptic integrals. Numerical inversion enables us to establish a (λk)2/3 dispersion relation for highly energetic quantized “screwons” at moderate and strong coupling. We also suggest a mapping between our radial equation and the one of Zinn-Justin and Jentschura that could facilitate a resurgent WKB expansion for energy levels. In another direction, we show that the equations of motion of the RR model can also be viewed as Euler equations for a step-3 nilpotent Lie algebra. We use our canonical quantization to uncover an infinite dimensional reducible unitary representation of this nilpotent algebra, which is then decomposed using its Casimir operators.

The dark Z′ and sterile neutrinos behind current anomalies

We show how, in the B−L extension of the SM (BLSM) with an Inverse Seesaw (IS) mechanism for neutrino mass generation, a light Z′ state with moderate couplings to SM objects, hence ‘dark’ in its nature, can be associated, in conjunction with light sterile neutrinos, to some present day data anomalies, such as the anomalous magnetic moment of the muon as well as a possible signal indicating the existence of sterile neutrinos in neutrino beam experiments.

Simultaneous control of plasmon–exciton and plasmon–trion couplings in an Au nanosphere and monolayer WS2 hybrid system

Simultaneous control of plasmon–exciton and plasmon–trion couplings is fundamentally interesting for tailoring the strong light–matter interaction at the nanoscale and is intriguing for developing high-efficiency optoelectronic and nonlinear photonic devices. Here, we integrate the monolayer WS2 with the Au nanosphere to take full advantages of both the strong excitonic effect and local field enhancement effect to realize strong resonance couplings between the dipolar plasmon mode and the exciton, as well as the trion, at room temperature. Interestingly, from the dark-field scattering spectrum, a transition from the dominated plasmon–exciton coupling to the plasmon–exciton–trion coupling in the hybrid system by simply increasing the radius of the nanosphere is revealed. This evolution of the scattering spectrum is further analyzed using the coupled-oscillator model to extract Rabi splittings of 89 and 48 meV for plasmon–exciton and plasmon–trion couplings, implying that the hybrid system enters the moderate coupling region. The moderate coupling imparts the hybrid system with a remarkable light-emitting capacity, rendering 1265- and 680-fold photoluminescence (PL) enhancement for the exciton and trion emissions, respectively. Our findings provide a facile way for the manipulation of excitonic quasiparticles in semiconductors at room temperature.

Investigation of field-controlled magnetocaloric switching and magnetodielectric phenomena in spin-chain compound Er2BaNiO5

The Haldane spin-chain compound Er2BaNiO5 has been known to possess magnetoelectric coupling below the magnetic ordering temperature. Here we report various low-temperature magnetic and magnetocaloric properties, and magnetodielectric (MD) effect above magnetic ordering temperature in this compound. The present compound displays a coexistence of conventional and inverse magnetocaloric effects with a large entropy change of 5.9 and −2.5 J kg−1 K−1, respectively. Further, it exhibits a remarkable switching between them, which can be tuned with temperature and magnetic field. In addition, evolution of two magnetic field-dependent metamagnetic transitions at 19.7 and 27.7 kOe, and their correlation with magnetocaloric switching effect, make this compound effective for potential applications. On the other hand, demonstration of intrinsic MD effect (1.9%) near and above antiferromagnetic ordering temperature, through a moderate coupling between electric dipoles and magnetic spins, establishes this compound as a useful candidate for future research. A detailed analysis of these findings, in a framework of different magnetic interactions and magnetocrystalline anisotropies, is discussed here. Overall, these results may provide a future pathway to tune the magnetic, MD, and magnetocaloric properties in this compound toward better application potential.

Type-II Dirac cones and electron-phonon interaction in monolayer biphenylene from first-principles calculations

We report a first-principles investigation of electronic structure, topological bands, and electron-phonon interactions in metallic biphenylene sheets. Biphenylene is a recently synthesized $s{p}^{2}$-bonded carbon allotrope. We find coupling of electrons at the Fermi surface to very high frequency carbon-derived phonons, analogous to superconducting ${\mathrm{MgB}}_{2}$. This leads to low-temperature weak coupling superconductivity due to an unusual combination of exceptionally large logarithmically averaged phonon frequency ${\ensuremath{\omega}}_{\text{log}}$=1369 K and moderate electron-phonon coupling. The electronic structure shows a two-band Fermi surface dominated by C ${p}_{z}$ orbitals and a pair of type-II tilted Dirac cones along the $\mathrm{\ensuremath{\Gamma}}$-Y line at the Brillouin zone boundary. Berry curvature and edge-state calculations show that monolayer biphenylene is a two-dimensional ${\mathbb{Z}}_{2}$ topological material. Thus, monolayer biphenylene is predicted to be a topological superconductor based on C $p$ orbitals and high-frequency phonons.

Two mechanisms of remote synchronization in a chain of Stuart-Landau oscillators.

Remote synchronization implies that oscillators interacting not directly but via an additional unit (hub) adjust their frequencies and exhibit frequency locking while the hub remains asynchronous. In this paper, we analyze the mechanisms of remote synchrony in a small network of three coupled Stuart-Landau oscillators using recent results on higher-order phase reduction. We analytically demonstrate the role of two factors promoting remote synchrony. These factors are the nonisochronicity of oscillators and the coupling terms appearing in the second-order phase approximation. We show a good correspondence between our theory and numerical results for small and moderate coupling strengths.