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We propose an $\alpha$-separable graph Hamiltonian network ($\alpha$-SGHN) that reveals complex interaction patterns between particles in lattice systems. Utilizing trajectory data, $\alpha$-SGHN infers potential interactions without prior knowledge about particle coupling, overcoming the limitations of traditional graph neural networks that require predefined links. Furthermore, $\alpha$-SGHN preserves all conservation laws during trajectory prediction. Experimental results demonstrate that our model, incorporating structural information, outperforms baseline models based on conventional neural networks in predicting lattice systems. We anticipate that the results presented will be applicable beyond the specific onsite and inter-site interaction lattices studied, including the Frenkel-Kontorova model, the rotator lattice, and the Toda lattice., Comment: 17pages

Light-matter interactions provide an ideal testground for interplay of critical phenomena, topological transitions, quantum metrology and non-Hermitian physics. We consider two fundamental non-Hermitian Jaynes-Cummings models which possess real energy spectra in parity-time (PT) symmetry and anti-PT symmetry. We show that the quantum Fisher information is critical around the transitions at the exceptional points and exhibits a super universality with respect to different parameters, all energy levels, both models, symmetric phases and symmetry-broken phases. The transitions are found to be both symmetry-breaking Landau-class transitions (LCTs) and symmetry-protected topological-class of transitions (TCTs), thus realizing a simultaneous occurrence of critical LCTs and TCTs which are conventionally incompatible due to contrary symmetry requirements., Comment: 7 pages, 3 figures, 1 table

The quantum Rabi model (QRM) is a fundamental model for light-matter interactions. A fascinating feature of the QRM is that it manifests a quantum phase transition which is applicable for critical quantum metrology (CQM). Effective application for CQM needs the exact location of the transition point, however the conventional expression for the transition coupling is only valid in the extreme limit of low frequency, while apart from zero frequency an accurate location is still lacking. In the present work we conversely use the quantum Fisher information (QFI) in the CQM to identify the transition coupling, which finds out that transition coupling indeed much deviates from the conventional one once a finite frequency is turned on. Polaron picture is applied to analytically reproduce the numeric QFI. An accurate expression for the transition coupling is obtained by the inspiration from the fractional-power-law effect of polaron frequency renormalization. From the QFI in the polaron picture we find that the transition occurs around a point where the effective velocity and the susceptibility of the single-photon absorption rate reach maximum. Our result provides an accurate reference of transition couplings for quantum metrology at non-zero frequencies. The formulation of the QFI in the polaron picture also prepares an analytic method with an accurate compensation for the parameter regime difficult to access for the numerics. Besides the integer/fractional power law analysis to extract the underlying physics of transition, the QFI/velocity relation may also add some insight in bridging the QFI and transition observables., Comment: 14 pages, 8 figures

The Jaynes-Cummings Model (JCM) is a fundamental model and building block for light-matter interactions, quantum information and quantum computation. We analytically analyze the topological feature manifested by the JCM in the presence of non-Hermiticity which may be effectively induced by dissipation and decay rates. Indeed, the eigenstates of the JCM are topologically characterized by spin windings in two-dimensional plane. The non-Hermiticity tilts the spin winding plane and induces out-of-plane component, while the topological feature is maintained. In particular, besides the invariant spin texture nodes, we find a non-Hermiticity-induced reversal transition of the tilting angle and spin winding direction with a fractional phase gain at gap closing, a partially level-independent reversal transition without gap closing, and a completely level-independent super invariant point with untilted angle and also without gap closing. Our result demonstrates that the topological feature is robust against non-Hermiticity, which would be favorable in practical applications. On the other hand, one may conversely make use of the disadvantageous dissipation and decay rates to reverse the spin winding direction, which might add a control way for topological manipulation of quantum systems in light-matter interactions., Comment: 15 pages, 5 figures

Besides exploring novel transition patterns, acquiring a full understanding of the transition nature is an ultimate pursuit in studies of phase transitions. The fundamental models of light-matter interactions manifest single-qubit topological phase transitions, which is calling for an analytical demonstration apart from numerical studies. We present a rigorous study for topological transitions in Jaynes-Cummings Model generally with Stark non-linear Coupling. In terms of the properties of Hermite polynomials, we show that the topological structure of the eigen wave function has an exact correspondence to the spin winding by nodes, which yields a full spin winding without anti-winding nodes. The spurious fractional contribution to the winding number of the winding angle at infinity is found to be actually integer. Thus, the phase transitions in the model have a nature of topological phase transitions and the excitation number is endowed as a topological quantum number. The principal transition establishes a paradigmatic case that a transition is both symmetry-breaking Landau class of transition and symmetry-protected topological class of transition simultaneously, while conventionally these two classes of transitions are incompatible due to the contrary symmetry requirements. We also give an understanding for the origin of unconventional topological transitions in the presence of counter-rotating terms. Our results may provide a deeper insight for the few-body phase transitions in light-matter interactions., Comment: 14 pages, 7 figures

The superradiant phase transition (SPT) has been widely studied in cavity quantum electrodynamics (CQED). However, this SPT is still subject of ongoing debates due to the no-go theorem induced by the so-called ${\bf A}^2$ term (AT). We propose a hybrid quantum system, consisting of a single-mode cavity simultaneously coupled to both a two-level system and yttrium-iron-garnet sphere supporting magnons with Kerr nonlinearity, to restore the SPT against the AT. The Kerr magnons here can effectively introduce an additional strong and tunable AT to counteract the intrinsic AT, via adiabatically eliminating the degrees of freedom of the magnons. We show that the Kerr magnons induced SPT can exist in both cases of ignoring and including the intrinsic AT. Without the intrinsic AT, the critical coupling strength can be dramatically reduced by introducing the Kerr magnons, which greatly relaxes the experimental conditions for observing the SPT. With the intrinsic AT, the forbidden SPT can be recovered with the Kerr magnons in a reversed way. Our work paves a potential way to manipulate the SPT against the AT in hybrid systems combining CQED and nonlinear magnonics.

By extracting different levels of topological information a new light is shed on the energy spectrum of the anisotropic quantum Rabi model (QRM) which is the fundamental model of light-matter interactions with indispensable counter-rotating terms in ultra-strong couplings. Besides conventional topological transitions (TTs) at gap closing, abundant unconventional TTs including a particular one universal for different energy levels are unveiled underlying level anticrossings without gap closing by tracking the wave-function nodes. On the other hand, it is found that the nodes have a correspondence to spin windings, which not only endows the nodes a more explicit topological character in supporting single-qubit TTs but also turns the topological information physically detectable. Furthermore, hidden small-spin-knot transitions are exposed for the ground state, while more kinds of spin-knot transitions emerge in excited states including unmatched node numbers and spin winding numbers. As a surprise, frequently the spin windings produce portraits in high spiritual similarity with abstract artistic works, which demonstrates that the anisotropic QRM may be the Picasso of physical models. This signifies that art is joining the dialogue between mathematics and physics which was triggered by the milestone work of revealing integrability of the QRM., Comment: 17 pages, 12 figures

The rapid spread of COVID-19 disease has had a significant impact on the world. In this paper, we study COVID-19 data interpretation and visualization using open-data sources for 351 cities and towns in Massachusetts from December 6, 2020 to September 25, 2021. Because cities are embedded in rather complex transportation networks, we construct the spatio-temporal dynamic graph model, in which the graph attention neural network is utilized as a deep learning method to learn the pandemic transition probability among major cities in Massachusetts. Using the spectral graph wavelet transform (SGWT), we process the COVID-19 data on the dynamic graph, which enables us to design effective tools to analyze and detect spatio-temporal patterns in the pandemic spreading. We design a new node classification method, which effectively identifies the anomaly cities based on spectral graph wavelet coefficients. It can assist administrations or public health organizations in monitoring the spread of the pandemic and developing preventive measures. Unlike most work focusing on the evolution of confirmed cases over time, we focus on the spatio-temporal patterns of pandemic evolution among cities. Through the data analysis and visualization, a better understanding of the epidemiological development at the city level is obtained and can be helpful with city-specific surveillance., Comment: Accepted by IEEE Transactions on Signal and Information Processing over Networks

Research into electronic nanomaterials has recently seen a growing focus into the synthesis of structures with unconventional curved geometries including bent wires in planar systems and three-dimensional architectures obtained by rolling up nanomembranes. The inclusion of these geometries has led to the prediction and observation of a series of novel effects that either result from shape-driven modifications of the electronic motion or from an intrinsic change of electronic and magnetic properties due to peculiar confinement effects. Moreover, local strains often generated by curvature also trigger the appearance of new phenomena due to the essential role played by electromechanical coupling in solids. Here we review the recent developments in the discovery of these shape-, confinement- and strain-induced curvature effects at the nanoscale, and discuss their potential use in electronic and spintronic devices., Comment: 17 pages, 4 figures. Review article

Universality is a common quality in different physical parameters that is rooted in the deep nature of physical systems. Scaling relation is a typical universality for critical phenomena around a quantum phase transition, while topological classification provides another type of universality essentially different from the critical universality. Both classes of universalities can be present in a single-qubit system with light-matter interactions, as exhibiting generally in the fundamental quantum Rabi model with anisotropy not only for linear coupling but also for nonlinear Stark coupling (NSC). In low frequencies different levels of scaling relations are demonstrated, holding for anisotropic or/and NSCs, locally or globally. At finite frequencies such a critical universality breaks down and diversity is dominant. However, common topological feature of the ground state can be extracted from the node number, which yields a topological class of universality amidst the critical diversity. Both conventional and unconventional topological transitions emerge, with their meeting, which never occurs in linear interaction, enabled by the nonlinear coupling to form topological quadruple points which are found to be spin-invariant points. Sensitivity analysis indicates that the NSC can be another approach to manipulate topological transitions in addition to coupling anisotropy., Comment: 17 pages, 9 figures

The quantum Rabi model (QRM) with linear coupling between light mode and qubit exhibits the analog of a second order phase transition for vanishing mode frequency which allows for criticality-enhanced quantum metrology in a few-body system. We show that the QRM including a non-linear coupling term exhibits much higher measurement precisions due to its first order like phase transition at \emph{finite} frequency, avoiding the detrimental slowing-down effect close to the critical point of the linear QRM. When a bias term is added to the Hamiltonian, the system can be used as a fluxmeter or magnetometer if implemented in circuit QED platforms., Comment: 7 pages, 5 figures

Conventionally the occurrence of topological phase transitions (TPTs) requires gap closing, whereas there are also unconventional cases without need of gap closing. Although traditionally TPTs lie in many-body systems in condensed matter, both cases of TPTs may find analogs in few-body systems. Indeed, the ground-state node number provides a topological classification for single-qubit systems. While the no-node theorem of spinless systems is shown to also restrict the fundamental quantum Rabi model in light-matter interactions, it is demonstrated that the limitation of the no-node theorem can be broken not only in a small counter-rotating term (CRT) but also in the large-CRT regime, which striates a rich phase diagram with different TPTs. While these transitions are mostly accompanied with gap closing and parity reversal, a hidden node-phase transition is revealed that has neither gap closing nor parity change, which turns out to be an analog of the unconventional TPT in condensed matter. A hysteresis sign for the unconventional TPT is unveiled via the transition from amplitude squeezing to phase squeezing in the gapped phase. The imprints in the Wigner function are also addressed. The clarified mechanisms provide some special insights for the subtle role of the CRT., Comment: 13 pages, 6 figures

We study the ground state (GS) and excitation gap of anisotropic quantum Rabi model (QRM) which connects the fundamental QRM and the Jaynes-Cummings model (JCM). While the GS has a second-order quantum phase transition (QPT) in the low frequency limit, turning on finite frequencies we shed a novel light on the phase diagram to illuminate a fine structure of first-order transition series. We find the QPT is accompanied with a hidden symmetry breaking, whereas the emerging series transitions are topological transitions without symmetry breaking. The topological structure of the wave function provides a novel universality classification in bridging the QRM and the JCM. We show that the conventionally established triple point is actually a quintuple or sextuple point and following the penta-/hexa-criticality emerge a series of tetra-criticalities., Comment: 5 pages, 5 figures

Quantum Rabi model (QRM) is fascinating not only because of its broad relevance and but also due to its few-body quantum phase transition. In practice both the bias and the nonlinear coupling in QRM are important controlling parameters in experimental setups. We study the interplay of the bias and the nonlinear interaction with the linear coupling in the ground state which exhibits various patterns of symmetry breaking and different orders of transitions. Several situations of tricriticality are unveiled in the low frequency limit and at finite frequencies. We find that the full quantum-mechanical effect leads to novel transitions, tricriticalities and quadruple points, which are much beyond the semiclassical picture. We clarify the underlying mechanisms by analyzing the energy competitions and the essential changeovers of the quantum states, which enables us to extract most analytic phase boundaries., Comment: 19 pages, 15 figures

The polaron picture is employed to investigate and elucidate the physics of the two-qubit quantum Rabi model, which describes two identical qubits coupled to a common harmonic oscillator. This approach enables us to obtain the ground-state energy and some other simpler physical observables with high accuracy in all regimes of the coupling strengths $g$, while there is no constraint to the ration of tunneling frequency $\Omega$ and field frequency $\omega$, which is not simultaneously possible using previous methods. Besides, we also discover a new phenomenon which is not present in the one-qubit Rabi model: with the increase of coupling strength $g$, there is a transition of the ground state of the system from a multipolaron state to a bipolaron state. However, the tunneling frequency $\Omega$ counteracts this process. Specifically, when tunneling frequency $\Omega=0$, the system always stays in the bipolaron state.

We investigate a class of cyclic evolutions for %the cyclic evolution of driven two-level quantum systems (effective spin-1/2) with a particular focus on the geometric characteristics of the driving and their specific imprints on the quantum dynamics. By introducing the concept of geometric field curvature for any field trajectory in the parameter space we are able to unveil underlying patterns in the overall quantum behavior: the knowledge of the field curvature provides a non-standard and fresh access to the interrelation between field and spin trajectories, and the corresponding quantum phases acquired in non-adiabatic cyclic evolutions. In this context, we single out setups in which the driving field curvature can be employed to demonstrate a pure geometric control of the quantum phases. Furthermore, the driving field curvature can be naturally exploited to introduce the geometrical torque and derive a general expression for the total quantum phase acquired in a cycle. Remarkably, such relation allows to access the mechanisms controlling the changeover of the quantum phase across a topological transition and to disentangle the role of the spin and field topological windings. As for implementations, we discuss a series of physical systems and platforms to demonstrate how the geometric control of the quantum phases can be realized for pendular field drivings. This includes setups based on superconducting islands coupled to a Josephson junction and inversion asymmetric nanochannels with suitably tailored geometric shapes., Comment: 13 pages, 5 figures

An analytical variational method for the ground state of the biased quantum Rabi model in the ultra-strong coupling regime is presented. This analytical variational method can be obtained by a unitary transformation or alternatively by assuming the form of ground state wave function. The key point of the method is to introduce a variational parameter $\lambda$, which can be determined by minimizing the energy functional. Using this method, we calculate physical observables with high accuracy in comparison with the numerical exact one. Our method evidently improves the results from the widely used general rotating-wave approximation (GRWA) in both qualitative and quantitative ways.

Consider the time-domain multiple cavity scattering problem, which arises in diverse scientific areas and has significant industrial and military applications. The multiple cavity embedded in an infinite ground plane, is filled with inhomogeneous media characterized by variable dielectric permittivities and magnetic permeabilities. Corresponding to the transverse electric or magnetic polarization, the scattering problem can be studied for the Helmholtz equation in frequency domain and wave equation in time-domain, respectively. A novel transparent boundary condition in time-domain is developed to reformulate the cavity scattering problem into an initial-boundary value problem in a bounded domain. The well-posedness and stability are established for the reduced problem. Moreover, a priori estimates for the electric field is obtained with a minimum requirement for the data by directly studying the wave equation., Comment: 29 pages,3 figures. arXiv admin note: text overlap with arXiv:1604.00944

We employ a polaron picture to investigate the properties of the two-photon quantum Rabi model (QRM), which describes a two-level or spin-half system coupled with a single bosonic mode by a two-photon process. In the polaron picture, the coupling in the two-photon process leads to spin-related asymmetry so that the original single bosonic mode splits into two separated frequency modes for the opposite spins, which correspond to two \textit{bare} polarons. Furthermore, the tunneling causes these two bare polarons to exchange their components with each other, thus leading to additional \textit{induced} polarons. According to this picture, the variational ground-state wave function of the two-photon QRM can be correctly constructed, with the ground-state energy and other physical observables in good agreement with the exact numerics in all the coupling regimes. Furthermore, generalization to multiple induced polarons involving higher orders in the tunneling effect provides a systematic way to yield a rapid convergence in accuracy even around the difficult spectral collapse point. In addition, the polaron picture provides a distinctive understanding of the spectral collapse behavior, that is about the existence of discrete energy levels apart from the collapsed spectrum at the spectral collapse point. This work illustrates that the polaron picture is helpful to capture the key physics in this nonlinear light-matter interaction model and indicates that this method can be applicable to more complicated QRM-related models.

The experimental advance on light-matter interaction into strong couplings has invalidated Jaynes-Cummings model and brought quantum Rabi model (QRM) to more relevance. The QRM only involves linear coupling via a single-photon process (SPP), while nonlinear two-photon process (TPP) is weaker and conventionally neglected. However, we find a contrary trend that enhancing the linear coupling might not suppress more the nonlinear effect but backfire to trigger some collapse of linear characters. Indeed, in strong SPP couplings a tiny strength of TPP may dramatically change properties of the system, like a symmetry spontaneous breaking. By extracting the ground-state phase diagram including both SPP and TPP, we find TPP in low frequency limit induces a quantum phase transition with continuity-discontinuity double faces, which split into two distinct transitions at finite frequencies and yields a triple point. Our analysis unveils a subtle SPP-TPP entanglement., Comment: 6 pages, 4 figures, model extended, new references added

We investigate the properties of a weak link between two Rashba-based superconducting nanowires with geometric misalignment. By applying an external magnetic field the system can be driven into a topological non-trivial regime. We demonstrate that the Josephson current can be modulated in amplitude and sign through the variation of the applied field and, remarkably, via the angle controlling the spin-orbit locking mismatch at the interface of the nanowires. The proposed setup with misaligned coplanar nanowires provides the building block configuration for the manipulation of coherent transport via geometric-controlled mixing/splitting of interface states.

We investigate the nature of the superconducting state in curved nanostructures with Rashba spin-orbit coupling (RSOC). In bent nanostructures with inhomogeneous curvature we find a local enhancement or suppression of the superconducting order parameter, with the effect that can be tailored by tuning either the RSOC strength or the carrier density. Apart from the local superconducting spin-singlet amplitude control, the geometric curvature generates non-trivial textures of the spin-triplet pairs through a spatial variation of the d-vector. By employing the representative case of an elliptically deformed quantum ring, we demonstrate that the amplitude of the d-vector strongly depends on the strength of the local curvature and it generally exhibits a three-dimensional profile whose winding is tied to that of the single electron spin in the normal state. Our findings unveil novel paths to manipulate the quantum structure of the superconducting state in RSOC nanostructures through their geometry., Comment: 5 pages, 4 figures

We present a frequency-renormalized multipolaron expansion method to explore the ground state of quantum Rabi model (QRM). The main idea is to take polaron as starting point to expand the ground state of QRM. The polarons are deformed and displaced oscillator states with variationally determined frequency-renormalization and displacement parameters. This method is an extension of the previously proposed polaron concept and the coherent state expansion used in the literature, which shows high efficiency in describing the physics of the QRM. The proposed method is expected to be useful for solving other more complicated light-matter interaction models., Comment: 7 pages, 6 figures

We present a mean-photon-number dependent variational method, which works well in whole coupling regime if the photon energy is dominant over the spin-flipping, to evaluate the properties of the Rabi model for both the ground state and the excited states. For the ground state, it is shown that the previous approximate methods, the generalized rotating-wave approximation (only working well in the strong coupling limit) and the generalized variational method (only working well in the weak coupling limit), can be recovered in the corresponding coupling limits. The key point of our method is to tailor the merits of these two existing methods by introducing a mean-photon-number dependent variational parameter. For the excited states,our method yields considerable improvements over the generalized rotating-wave approximation. The variational method proposed could be readily applied to the more complex models, for which an analytic formula is difficult to be formulated., Comment: 13 pages, 5 figures

The concept of the polaron in condensed matter physics has been extended to the Rabi model, where polarons resulting from the coupling between a two-level system and single-mode photons represent two oppositely displaced oscillators. Interestingly, tunneling between these two displaced oscillators can induce an anti-polaron, which has not been systematically explored in the literature, especially in the presence of an asymmetric term. In this paper, we present a systematic analysis of the competition between the polaron and anti-polaron under the interplay of the coupling strength and the asymmetric term. While intuitively the anti-polaron should be secondary owing to its higher potential energy, we find that, under certain conditions, the minor anti-polaron may gain a reversal in the weight over the major polaron. If the asymmetric amplitude $\epsilon$ is smaller than the harmonic frequency $\omega$, such an overweighted anti-polaron can occur beyond a critical value of the coupling strength $g$; if $\epsilon$ is larger, the anti-polaron can even be always overweighted at any $g$. We propose that the explicit occurrence of the overweighted anti-polaron can be monitored by a displacement transition from negative to positive values. This displacement is an experimentally accessible observable, which can be measured by quantum optical methods, such as balanced Homodyne detection., Comment: 14 pages, 8 figures

We investigate first- and second-order quantum phase transitions of the anisotropic quantum Rabi model, in which the rotating- and counter-rotating terms are allowed to have different coupling strength. The model interpolates between two known limits with distinct universal properties. Through a combination of analytic and numerical approaches we extract the phase diagram, scaling functions, and critical exponents, which allows us to establish that the universality class at finite? anisotropy is the same as the isotropic limit. We also reveal other interesting features, including a superradiance-induced freezing of the effective mass and discontinuous scaling functions in the Jaynes-Cummings limit. Our findings are relevant in a variety of systems able to realize strong coupling between light and matter, such as circuit QED setups where a finite anisotropy appears quite naturally., Comment: 12 pages, 4 figures