Table of contents

The realization of microscopic heat engines has gained a surge of research interest in statistical physics, soft matter, and biological physics. A typical microscopic heat engine employs a colloidal particle trapped in a confining potential, which is modulated in time to mimic the cycle operations. Here, we use a lanthanide-doped upconverting particle (UCP) suspended in a passive aqueous bath, which is highly absorptive at 975 nm and converts near infra red (NIR) photons to visible, as the working substance of the engine. When a single UCP is optically trapped with a 975 nm laser, it behaves like an active particle by executing motion subjected to an asymmetric temperature profile along the direction of propagation of the laser. The strong absorption of 975 nm light by the particle introduces a temperature gradient and results in significant thermophoretic diffusion along the temperature gradient. However, the activity of the particle vanishes when the trapping wavelength is switched to 1064 nm. We carefully regulate the wavelength-dependent activity of the particle to engineer all four cycles of a Stirling engine by using a combination of 1064 nm and 975 nm wavelengths. Since the motion of the particle is stochastic, the work done on the particle due to the stiffness modulation per cycle is random. We provide statistical estimation for this work averaged over five cycles which can be extended towards several cycles to make a Stirling engine. Our experiment proposes a robust set-up to systematically harness temperature which is a crucial factor behind building microscopic engines.

An outstanding challenge involves understanding the many-particle entanglement of liquid states of quantum matter that arise in systems of interacting electrons. The Fermi liquid (FL) shows a violation of the area-law in real-space entanglement entropy of a subsystem, believed to be a signature of the ground state of a gapless quantum critical system of interacting fermions. Here, we apply a T = 0 renormalization group approach to the FL, unveiling the long-wavelength quantum fluctuations from which long-range entanglement arises. A similar analysis of non-Fermi liquids such as the 2D marginal Fermi liquid (MFL) and the 1D Tomonaga–Luttinger liquid reveals a universal logarithmic violation of the area-law in gapless electronic liquids, with a proportionality constant that depends on the nature of the underlying Fermi surface. We extend this analysis to classify the gapped quantum liquids emergent from the destabilisation of the Fermi surface by renormalisation group relevant quantum fluctuations arising from backscattering processes.

We consider the overdamped dynamics of different stochastic processes, including Brownian motion and autoregressive processes, continuous time random walks, fractional Brownian motion, and scaled Brownian motion, confined by an harmonic potential. We discuss the effect of both static and dynamic noise representing two kinds of localisation error prevalent in experimental single-particle tracking data. To characterise how such noise affects the dynamics of the pure, noise-free processes we investigate the ensemble-averaged and time-averaged mean squared displacements as well as the associated ergodicity breaking parameter. Process inference in the presence of noise is demonstrated to become more challenging, as typically the noise dominates the short-time behaviour of statistical measures, while the long time behaviour is dominated by the external confinement. In particular, we see that while static noise generally leads to a more subdiffusive apparent behaviour, dynamic noise makes the signal seem more superdiffusive. Our detailed study complements tools for analysing noisy time series and will be useful in data assimilation of stochastic data.

The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system's properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.

At temperatures well below the Fermi temperature T_F, the coupling of magnetic fluctuations to particle-hole excitations in a two-component Fermi gas leads to non-analytic corrections in the renormalized free energy and makes the transition to itinerant ferromagnetism first order. On the other hand, despite that larger symmetry often introduces larger degeneracies in the low-lying states, here we show that for a Fermi gas with SU(N > 2)-symmetry in three space dimensions the ferromagnetic phase transition is first order in agreement with the predictions of Landau's mean-field theory in its minimal formulation, which contains a cubic term in the free-energy (Cazalilla et al 2009 New J. Phys.11 103033). By performing unrestricted Hartree–Fock calculations for an SU(N > 2)-symmetric Fermi gas with short range interactions, we find the order parameter undergoes a finite jump across the transition. In addition, for SU(N > 2) we do not observe any tri-critical point up to temperatures $T \simeq 0.5\: T_F$, for which the thermal smearing of the Fermi surface and thermal fluctuations are substantial. Going beyond mean-field theory, we find that the coupling of magnetic fluctuations to particle-hole excitations makes the transition more abrupt and further enhances the tendency of the gas to become fully polarized for smaller values of N and the gas parameter $k_F a_s$. In our study, we also clarify the role of time reversal symmetry in the microscopic Hamiltonian and obtain the temperature dependence of Tan's contact. For the latter, the presence of the tri-critical point for N = 2 leads to a more pronounced temperature dependence around the transition than for SU(N > 2)-symmetric gases. Although the results are obtained for a microscopic model relevant to ultracold atomic gases, they may also apply to models of solid state systems with SU(N > 2) symmetry provided Coulomb interactions are sufficiently well screened.

We investigate theoretically the superfluidity of a one-dimensional boson system whose hopping energy is periodically modulated with a zero time average, which results in the suppression of first-order single-particle hopping processes. The dynamics of this Floquet-engineered flat-band system is entirely driven by correlations and described by exotic Hamiltonian and current operators. We employ exact diagonalization and compare our results with those of the conventional, undriven Bose–Hubbard system. We focus on the two main manifestations of superfluidity, the Hess-Fairbank effect and the metastability of supercurrents, with explicit inclusion of an impurity when relevant. Among the novel superfluid features, we highlight the presence of a cat-like ground state, with branches that have opposite crystal momentum but carry the same flux-dependent current, and the essential role of the interference between the collective components of the ground-state wave function. Calculation of the dynamic form factor reveals the presence of an acoustic mode that guarantees superfluidity in the thermodynamic limit.

Electronic structures and magnetotransport properties of topological Dirac semimetal (TDSM) nanoribbons are studied by adopting the tight-binding lattice model and the Landauer–Büttiker formula based on the non-equilibrium Green's function. For concreteness, the TDSM material Cd₃As₂ grown along the experimentally accessible [110] crystallographic direction is taken as an example. We found that the electronic structures of the TDSM nanoribbon depend on both the strength and direction of the magnetic field (MF). The transversal local charge density (LCD) distribution of the electronic states in the TDSM nanoribbon is moved gradually from the center toward the hinge of each surface as a [010] direction MF strength is increased, forming the two-sided hinge states. However, one-sided surface states are generated in the TDSM nanoribbon when a [001] direction MF is applied. As a result, one-sided hinge states can be achieved once a tilted MF is placed to the TDSM nanoribbon. The underlying physical mechanism of the desired one-sided hinge states is attributed to both the orbital and Zeeman effects of the MF, which is given by analytical analyses. In addition, typical Aharonov–Bohm interference patterns are observed in the charge conductance of the two-terminal TDSM nanoribbon with a tilted MF. This conductance behaviour originates from the unique interfering loop shaped by the one-sided hinge states. These findings may not only further our understanding on the external-field-induced higher-order (HO) topological phases but also provide an alternative method to probe the HO boundary states.

Dirac cones (DCs) are an important band structure in topological insulators (TIs) for realizing topological phase transition, and they provide unique ways to artificially regulate wave transport. Herein, we proposed a simple method to achieve Dirac hierarchy in three-dimensional (3D) acoustic TIs with rich and controllable topological phase transitions. The split of multifold DCs in each bulk Dirac hierarchy induced boundary Dirac hierarchy, including topological surface states and topological hinge states. We successfully realized 3D higher-order topological insulators (HOTIs) that exhibited two-fold boundary Dirac hierarchy with hinge states and achieved energy transport along three independent directions based on hinge-to-hinge channels. The proposed method is not limited to single hinges, and it provides a new design idea for multidimensional sound transport, serving as the basis for controllable acoustic functional devices.

We analyse the transport of diffusive particles that switch between mobile and immobile states with finite rates. We focus on the effect of advection on the density functions and mean squared displacements (MSDs). At relevant intermediate time scales we find strong anomalous diffusion with cubic scaling in time of the MSD for high Péclet numbers. The cubic scaling exists for short and long mean residence times in the immobile state $\tau_\mathrm{im}$. For long $\tau_\mathrm{im}$ the plateau in the MSD at intermediate times, previously found in the absence of advection, also exists for high Péclet numbers. Initially immobile tracers are subject to the newly observed regime of advection induced subdiffusion for short immobilisations and high Péclet numbers. In the long-time limit the effective advection velocity is reduced compared to advection in the mobile phase. In contrast, the MSD is enhanced by advection. We explore physical mechanisms behind the emerging non-Gaussian density functions and the features of the MSD.

Defects play significant roles in spin-current-related physical processes in intrinsic ferromagnetic semiconductors (FMSs), which are great promise for spintronics applications. However, current defect calculation methods cannot be used to investigate charged defects in FMSs due to the spin polarization of both the charged defect states and ionized carriers, which is not well treated in current defect calculation methods. In order to solve this problem, we propose a spin-distinguishable charge correction (SDCC) method that uses spin-polarized band edge charge density instead of spin-unpolarized uniform background charge density as the compensating charge for charged defects. We apply our method to study the defect properties of CrI₃ monolayer and find it can be doped n-type under the Cr-rich growth condition but difficult to be doped p-type. The SDCC method proposed here is generally suitable for all FMSs, which will be useful for the studies of defect properties of magnetic semiconductors.

We consider collective excitations in the superfluid state of Fermi condensed charged gases. The dispersion and damping of collective excitations at nonzero temperatures are examined, and the coexistence and interaction of different branches of collective excitations: plasma oscillations, pair-breaking Higgs modes, and Carlson–Goldman phonon-like excitations are taken into account. The path integral methods for superfluid Fermi gases and for Coulomb gas are combined into a unified formalism that extends the Gaussian fluctuation approximation to account for plasmonic modes. This approximation of Gaussian pair and density fluctuations is able to describe all branches of collective excitations existing in a charged superfluid. The spectra of collective excitations are determined in two ways: from the spectral functions and from the complex poles of the fluctuation propagator. A resonant avoided crossing of different modes is shown. It is accompanied by resonant enhancement of the response provided by the pair-breaking modes due to their interaction with plasma oscillations. This may facilitate the experimental observation of the pair-breaking modes.

There are two types of port-based teleportation (PBT) protocols: deterministic—when the state always arrives to the receiver but is imperfectly transmitted and probabilistic—when the state reaches the receiver intact with high probability. We introduce the minimal set of requirements that define a feasible PBT protocol and construct a simple PBT protocol that satisfies these requirements: it teleports an unknown state of a qubit with success probability $p_{\textrm {succ}} = 1-\frac{N+2}{2^{N+1}}$ and fidelity $1-O(\frac{1}{N})$ with the resource state consisting of N maximally entangled states. This protocol is not reducible from either the deterministic or probabilistic PBT protocol. We define the corresponding efficient superdense coding protocols which transmit more classical bits with fewer maximally entangled states. Furthermore, we introduce rigorous methods for comparing and converting between different PBT protocols.

We present a scheme of positron acceleration by intense terahertz (THz) wave together with the driving large-charge electron beam in a plasma channel. The THz wave rapidly evolves into a transversely uniform acceleration field and a weakly focusing/defocusing lateral field in the channel. The THz wave is partially formed with the scheme of coherent transition radiation when the electron beam goes through a metal foil and partially because of the wakefield in the plasma channel. The electron beam continuously supplies energy to the THz wave. Such a field structure offers the feasibility of long-distance positron acceleration while preserving beam quality. By two-dimensional simulations, we demonstrate the acceleration of positrons from initial 1 GeV to 126.8 GeV with a charge of ∼10 pC over a distance of 1 m. The energy spread of accelerated positrons is 2.2%. This scheme can utilize the electron beam either from laser-driven or conventional accelerators, showing prospects towards high-quality and flexible THz-driven relativistic positron sources of ∼100 GeV.

Coherent emission coming from relativistic charged bunches is of great interest in a wide range of user-oriented applications and high-resolution diagnostics. The complete characterization of such emission is therefore important in view of a complete understanding of its potential. Here we present a complete temporally-resolved characterization of the radiation emitted by ultra-short relativistic electron bunches using a temporal diagnostic based on electro-optical sampling with a few tens fs of temporal resolution. We have characterized, for the first time to our knowledge, the evolution of the radiation (in THz range) both in amplitude and direction of propagation by varying the detection (i.e. the observer) position from the near to the far field (FF) range. Results show that in the near-field regime the emitted radiation propagates collinearly with the electron beam; while, approaching the FF regime, the radiation behaves as the classical Cherenkov radiation.

A key aspect of ultracold bosonic quantum gases in deep optical lattice potential wells is the realization of the strongly interacting Mott insulating phase. Many characteristics of this phase are well understood, however little is known about the effects of a random external potential on its gapped quasiparticle and quasihole low-energy excitations. In the present study we investigate the effect of disorder upon the excitations of the Mott insulating state at zero temperature described by the Bose–Hubbard model. Using a field-theoretical approach we obtain a resummed expression for the disorder ensemble average of the spectral function. Its analysis shows that disorder leads to an increase of the effective mass of both quasiparticle and quasihole excitations. Furthermore, it yields the emergence of damped states, which exponentially decay during propagation in space and dominate the whole band when disorder becomes comparable to interactions. We argue that such damped-localized states correspond to single-particle excitations of the Bose-glass phase.

We present a spin two-axis-twisting mechanism via coherent population trapping (CPT) based atom–photon interactions. CPT happens and the atoms are trapped in the dark state (coherent superposition of two ground states) when the ground states are resonantly coupled to a common excited state. Close to CPT, the atoms behave as two dark-state based spins, which interact with the common cavity vacuum fields. The otherwise nonexistent interaction is created between them and is identified to be responsible for the two-axis-twisting of the ground state spin. The essential difference from the previous schemes is the compatibility of the twisting spin squeezing with the resonant atom-light interaction. The CPT resonant unit serves as a kind of new ingredients for the quantum networks.

We study with a 3D particle-in-cell simulation discontinuities between an electron–positron pair plasma and magnetized electrons and protons. A pair plasma is injected at one simulation boundary with a speed 0.6c along its normal. It expands into an electron-proton plasma and a magnetic field that points orthogonally to the injection direction. Diamagnetic currents expel the magnetic field from within the pair plasma and pile it up in front of it. It pushes electrons, which induces an electric field pulse ahead of the magnetic one. This initial electromagnetic pulse (EMP) confines the pair plasma magnetically and accelerates protons electrically. The fast flow of the injected pair plasma across the protons behind the initial EMP triggers the filamentation instability. Some electrons and positrons cross the injection boundary and build up a second EMP. Electron-cyclotron drift instabilities perturb the plasma ahead of both EMPs seeding a Rayleigh–Taylor (RT)-type instability. Despite equally strong perturbations ahead of both EMPs, the second EMP is much more stable than the initial one. We attribute the rapid collapse of the initial EMP to the filamentation instability, which perturbed the plasma behind it. The RT-type instability transforms the planar EMPs into transition layers, in which magnetic flux ropes and electrostatic forces due to uneven numbers of electrons and positrons slow down and compress the pair plasma and accelerate protons. In our simulation, the expansion speed of the pair cloud decreased by about an order of magnitude and its density increased by the same factor. Its small thickness implies that it is capable of separating a relativistic pair outflow from an electron-proton plasma, which is essential for collimating relativistic jets of pair plasma in collisionless astrophysical plasma.

Bell nonlocality—the existence of quantum correlations that cannot be explained by classical means—is certainly one of the most striking features of quantum mechanics. Its range of applications in device-independent protocols is constantly growing. Many relevant quantum features can be inferred from violations of Bell inequalities, including entanglement detection and quantification, and state certification applicable to systems of arbitrary number of particles. A complete characterisation of nonlocal correlations for many-body systems is, however, a computationally intractable problem. Even if one restricts the analysis to specific classes of states, no general method to tailor Bell inequalities to be violated by a given state is known. In this work we provide a general construction of Bell expressions tailored to the graph states of any prime local dimension. These form a broad class of multipartite quantum states that have many applications in quantum information, including quantum error correction. We analytically determine their maximal quantum values, a number of high relevance for device-independent applications of Bell inequalities. Importantly, the number of expectation values to determine in order to test the violation of our inequalities scales only linearly with the system size, which we expect to be the optimal scaling one can hope for in this case. Finally, we show that these inequalities can be used for self-testing of multi-qutrit graph states such as the well-known four-qutrit absolutely maximally entangled state AME(4,3).

On the basis of the generalized Poisson–Boltzmann equation derived from the Bogolyubov chain of equations for the equilibrium distribution functions in the pair correlation approximation, a general expression is proposed for the Helmholtz free energy of a system that contains any number of components and whose particles interact via arbitrary potentials. This opens up an extraordinary opportunity to simultaneously treat a whole range of physical effects including partial ionization, quantum effects of diffraction and electron degeneracy, short- and long-range interactions of charged particles with neutrals, finite size effects, etc. It is shown that all medium constituents are tied together in a single screening matrix, whose determinant and trace determine the excess contribution to the free energy. The approach developed is then applied to the problem of the ionization potential depression (IPD) leading to quite simple analytical expressions, which turn out to be useful for various practical purposes. In particular, for a single ionization from the neutral state the IPD is shown to significantly depend on the ionization degree such that it consists of the difference of charged and neutral contributions for a fully ionized plasma and turns non-zero for an almost neutral medium. On the other hand, for a multiple ionization process finite size effects of atoms and ions are demonstrated to be of great importance and accounted for in order to achieve good agreement with experimental data on the IPD under warm dense matter conditions.

We study nonequilibrium steady states of a one-dimensional stochastic model, originally introduced as an approximation of the discrete nonlinear Schrödinger equation. This model is characterized by two conserved quantities, namely mass and energy; it displays a 'normal', homogeneous phase, separated by a condensed (negative-temperature) phase, where a macroscopic fraction of energy is localized on a single lattice site. When steadily maintained out of equilibrium by external reservoirs, the system exhibits coupled transport herein studied within the framework of linear response theory. We find that the Onsager coefficients satisfy an exact scaling relationship, which allows reducing their dependence on the thermodynamic variables to that on the energy density for unitary mass density. We also determine the structure of the nonequilibrium steady states in proximity of the critical line, proving the existence of paths which partially enter the condensed region. This phenomenon is a consequence of the Joule effect: the temperature increase induced by the mass current is so strong as to drive the system to negative temperatures. Finally, since the model attains a diverging temperature at finite energy, in such a limit the energy–mass conversion efficiency reaches the ideal Carnot value.

The beneficial combination of micro- and nano-patterned surfaces with magneto-optical materials was investigated over the recent years. Due to their resonant behavior, these structures are commonly used to enhance the non-reciprocal magneto-optical effects. In this paper, a novel kind of magneto-optical intensity effect is enhanced with an all-dielectric grating patterned on a magnetic nanocomposite layer. This nanocomposite is made of CoFe₂O₄ nanoparticles (NPs) embedded in a silica matrix by sol–gel technique. The demonstrated magneto-optical intensity effect is reciprocal and it is observed with transverse magnetic field, for both polarization (TE and TM) and small angles of incidence. Such effect is not explained by the classical appearance of off-diagonal elements in the permittivity tensor of the magneto-optical material under magnetic field. However, it can be attributed to a magneto-induced reciprocal modification of the diagonal elements. Furthermore, this effect strongly depends on the NPs orientation inside the magneto-optical film and can originate from the magnetostrictive property of the magnetic CoFe₂O₄ NPs.

The all-versus-nothing proof of Bell nonlocality is a prominent demonstration of Bell's theorem without inequalities. There are two kinds of such proofs: the deterministic all-versus-nothing proof and the probabilistic all-versus-nothing proof, which have received extensive research attention. Traditionally, all previous deterministic all-versus-nothing proofs are constructed based on stabilizer states. However, this work presents new deterministic proofs derived from non-stabilizer states, thereby breaking away from this conventional approach. These novel results not only significantly broaden the range of demonstrations of Bell nonlocality without inequalities but also offer valuable resources for certain quantum information processing applications.

We present the experimental implementation of a two-qubit phase gate, using a radio frequency (RF) controlled trapped-ion quantum processor. The RF-driven gate is generated by a pulsed dynamical decoupling sequence applied to the ions' carrier transitions only. It allows for a tunable phase shift with high-fidelity results. The conditional phase shift is measured using a Ramsey-type measurement with an inferred fringe contrast of up to $99_{-2}^{+1}\%$. We also prepare a Bell state using this laser-free gate. The phase gate is robust against common sources of error. We investigate the effect of the excitation of the center-of-mass (COM) mode, errors in the axial trap frequency, pulse area errors and errors in sequence timing. The contrast of the phase gate is not significantly reduced up to a COM mode excitation $\lt$20 phonons, trap frequency errors of +10%, and pulse area errors of −8%. The phase shift is not significantly affected up to $\lt$10 phonons and pulse area errors of −2%. Both, contrast and phase shift are robust to timing errors up to −30% and +15%. The gate implementation is resource efficient, since only a single driving field is required per ion. Furthermore, it holds the potential for fast gate speeds (gate times on the order of 100 µs) by using two axial motional modes of a two-ion crystal through improved setups.

Neodymium spherical magnets are inexpensive objects that demonstrate how dipolar particles self-assemble into various structures ranging from 1D chains to 3D crystals. Assemblies of these magnets are nicknamed magnetostructures and this paper focuses on a variety called magnetotubes, which are some curved square lattices forming cylinders. We experimentally and numerically observe that such magnetotubes can self-buckle, above a critical aspect ratio. In fact, the underlying dipolar ordering of such structures is found to exhibit a collective reorganization, altering the mechanical stability of the entire system. We identify the conditions in which these phenomena occur, and we emphasize that metastable states coexist. This suggests that a wide variety of magnetostructures, including chains and magnetocrystals, may collapse due to the coexistence of multiple ground states and global reorientation of dipoles.

The demixing and sorting strategies for chiral active mixtures are crucial to the biochemical and pharmaceutical industries. However, it remains uncertain whether chiral mixed particles can spontaneously demix without the aid of specific strategies. In this paper, we investigate the demixing behaviors of binary mixtures in a model of chiral active particles to understand the demixing mechanism of chiral active mixtures. We demonstrate that chiral mixed particles can spontaneously demix in motility-induced phase separation (MIPS). The hidden velocity alignment in MIPS allows particles of different types to accumulate in different clusters, thereby facilitating separation. There exists an optimal angular velocity or packing fraction at which this separation is optimal. Noise (translational or rotational diffusion) can promote mixture separation in certain cases, rather than always being detrimental to the process. Since the order caused by the hidden velocity alignment in this process is not global, the separation behavior is strongly dependent on the system size. Furthermore, we also discovered that the mixture separation caused by MIPS is different from that resulting from explicit velocity alignment. Our findings are crucial for understanding the demixing mechanism of chiral active mixtures and can be applied to experiments attempting to separate various active mixtures in the future.

We propose in this article a quantum square ring that conveniently generates, annihilates and distills the Aharonov Casher phase with the aid of entanglement. The non-Abelian phase is carried by a pair of spin-entangled particles traversing the square ring. At maximal entanglement, dynamic phases are eliminated from the ring and geometric phases are generated in discrete values. By contrast, at partial to no entanglement, both geometric and dynamic phases take on discrete or locally continuous values depending only on the wavelength and the ring size. We have shown that entanglement in a non-Abelian system could greatly simplify future experimental efforts revolving around the studies of geometric phases.

Multipartite entanglement plays a central role in optical quantum technologies. One way to entangle two photons is to prepare them in orthogonal internal states, for example, in two polarisations, and then send them through a balanced beam splitter. Post-selecting on the cases where there is one photon in each output port results in a maximally entangled state. This idea can be extended to schemes for the post-selected generation of larger entangled states. Typically, switching between different types of entangled states requires different arrangements of beam splitters and so a new experimental setup. Here, we demonstrate a simple and versatile scheme to generate different types of genuine tripartite entangled states with only one experimental setup. We send three photons through a three-port splitter and vary their internal states before post-selecting on certain output distributions. This results in the generation of tripartite W, G and GHZ states. We obtain fidelities of up to $(87.3\pm 1.1)\%$ with regard to the respective ideal states, confirming a successful generation of genuine tripartite entanglement.

The world faces Covid-19 waves, and the overall pattern of confirmed cases shows periodic oscillations. In this paper, we investigate the spatiotemporal spread of Covid-19 in the network-organized SIR model with an extrinsic incubation period of the driving factors. Firstly, Our analysis shows the occurrences of Hopf bifurcation and periodic outbreaks consistent with the actual spread of Covid-19. And we investigate periodic waves on spatial scales using Turing instability, and the spread of infected individuals increases the localized hot spots. We study the effect of the incubation period, and more incubation periods generate Turing instability resulting in periodic outbreaks. There is an occurrence of bursting states at peaks of periodic waves due to small diffusion of infected and susceptible, which means stable and unstable areas try to convert each other due to high competition among nodes. Also, We note the disappearance of these bursts when infected and susceptible individuals' movements are easier; thus, the dominance of infected individuals prevails everywhere. Effective policy interventions and seasonality can cause periodic perturbations in the model, and therefore we study the impact of these perturbations on the spread of Covid-19. Periodic perturbations on the driving factors, infected individuals show co-existing spatial patterns. Chaotic outbreak becomes periodic outbreaks through alternating periodic or period-2 outbreaks as we regulate the amplitude and frequency of infected individuals. In short, regulations can erase period-2 and chaotic spread through policy interventions.