Excitation Of Collective States Research Articles

Isoscalar giant resonances of <inline-formula><tex-math id="M1">\begin{document}$^{{\bf{18}}}_{{\boldsymbol{\Lambda\Lambda}}}{\bf{O}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M1.png"/></alternatives></inline-formula> in relativistic approach

The interactions between hyperon-nucleon and hyperon-hyperon have been an important topic in strangeness nuclear physics, which play an important role in understanding the properties of hypernuclei and equation of state of strangeness nuclear matter. It is very difficult to perform a direct scattering experiment of the nucleon and hyperon because the short lifetime of the hyperon. Therefore, the hyperon-nucleon interaction and the hyperon-hyperon interaction have been mainly investigated experimentally by <inline-formula><tex-math id="M4">\begin{document}$\gamma$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M4.png"/></alternatives></inline-formula> spectroscopy of single-<inline-formula><tex-math id="M5">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M5.png"/></alternatives></inline-formula> hypernuclei or double-<inline-formula><tex-math id="M6">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M6.png"/></alternatives></inline-formula> hypernuclei. There are also many theoretical methods developed to describe the properties of hypernuclei. Most of these models focus mostly on the ground state properties of hypernuclei, and have given exciting results in producing the banding energy, the energy of single-particle levels, deformations, and other properties of hypernuclei. Only a few researches adopting Skyrme energy density functionals is devoted to the study of the collective excitation properties of hypernuclei. In present work, we have extended the relativistic mean field and relativistic random phase approximation theories to study the collective excitation properties of hypernuclei, and use the methods to study the isoscalar collective excited state properties of double <inline-formula><tex-math id="M7">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M7.png"/></alternatives></inline-formula> hypernuclei. First, the effect of <inline-formula><tex-math id="M8">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M8.png"/></alternatives></inline-formula> hyperons on the single-particle energy of <sup>16</sup>O and <inline-formula><tex-math id="M9">\begin{document}$^{18}_{\Lambda\Lambda}{\rm{O}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M9.png"/></alternatives></inline-formula> are discussed in the relativistic mean field theory, the calculations are performed within TM1 parameter set and related hyperon-nucleon interaction, and hyperon-hyperon interaction. We find that it gives a larger attractive effect on the <inline-formula><tex-math id="M10">\begin{document}${{\mathrm{s}}}_{1/2}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M10.png"/></alternatives></inline-formula> state of proton and neutron, while gives a weaker attractive effect on the state around Fermi surface. The self-consistent relativistic random phase approximation is used to study the collectively excited state properties of hypernucleus <inline-formula><tex-math id="M11">\begin{document}$^{18}_{\Lambda\Lambda}{\rm{O}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M11.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M11.png"/></alternatives></inline-formula>. The isoscalar giant monopole resonance and quadrupole resonance are calculated and analysed in detail, we pay more attention to the effect of the inclusion of <inline-formula><tex-math id="M12">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M12.png"/></alternatives></inline-formula> hyperons on the properties of giant resonances. Comparing with the strength distributions of <sup>16</sup>O, changes of response function of <inline-formula><tex-math id="M13">\begin{document}$^{18}_{\Lambda\Lambda}{\rm{O}}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M13.png"/></alternatives></inline-formula> are evidently found both on the isoscalar giant monopole resonance and quadrupole resonance. It is shown that the difference comes mainly from the change of Hartree energy of particle-hole configuration and the contribution of the excitations of <inline-formula><tex-math id="M14">\begin{document}$\Lambda$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20231531_M14.png"/></alternatives></inline-formula> hyperons. We find that the hyperon-hyperon residual interactions have small effect on the monopole resonance function and quadrupole response function in the low-energy region, and have almost no effect on the response functions in the high-energy region.

Semi-microscopic optical-model with channel coupling analysis to describe neutron-nucleus scattering of light and heavy nuclei

Many studies utilizing various potential models have been conducted on the scattering of nucleons from nuclei. In our study, we added channel coupling to a semi-microscopic optical model. With incident energies between 10 and 30 MeV, we sought to compare the predicted reaction observables for the neutron scattering of the following nuclei: 12C, 16O, 54Fe, 58Ni, 120Sn, and 208Pb. We did this twice, once using the density-dependent M3Y-Reid nucleon-nucleon effective bare interaction and once using the density-independent M3Y-Reid effective bare interaction. The remaining terms of the optical potential model take the form of Woods-Saxon and its derivatives, and the real volume term is obtained by folding the NN effective bare potential over the density of the nuclear target using the density-dependent M3Y-Reid NN effective bare interaction and density-independent M3Y-Reid NN effective bare interaction. To investigate how the potential characteristics depend on energy, the ground state is connected to a select few low-lying collective excited states. The density-dependent M3Y-Reid NN effective bare interaction-based semi-microscopic model reproduces the reaction features and forecasts the experimental total cross sections. The parameters of potential depth exhibit a linear energy dependence. We contrast the outcomes of the M3Y-Reid density-dependent model with those attained using the M3Y-Reid nucleon-nucleon effective bare interaction, which is density-independent. The M3Y Reid density-dependent model performs well when the two models are compared.

Collective States in Molecular Monolayers on 2D Materials.

Collective excited states form in organic two-dimensional layers through Coulomb coupling of the molecular transition dipole moments. They manifest as characteristic strong and narrow peaks in the excitation and emission spectra that are shifted to lower energies compared with the monomer transition. We study experimentally and theoretically how robust the collective states are against homogeneous and inhomogeneous broadening, as well as spatial disorder that occurs in real molecular monolayers. Using a microscopic model for a two-dimensional dipole lattice in real space, we calculate the properties of collective states and their extinction spectra. We find that the collective states persist even for 1-10% random variation in the molecular position and in the transition frequency, with a peak position and integrated intensity similar to those for the perfectly ordered system. We measured the optical response of a monolayer of the perylene derivative MePTCDI on two-dimensional materials. On the wide-band-gap insulator hexagonal boron nitride, it shows strong emission from the collective state with a line width that is dominated by the inhomogeneous broadening of the molecular state. When the semimetal graphene is used as a substrate, however, the luminescence is completely quenched. By combining optical absorption, luminescence, and multiwavelength Raman scattering, we verify that the MePTCDI molecules form very similar collective monolayer states on hexagonal boron nitride and graphene substrates, but on graphene the line width is dominated by nonradiative excitation transfer from the molecules to the substrate. Our study highlights the transition from the localized molecular state of the monomer to a delocalized collective state in the two-dimensional molecular lattice that is entirely based on Coulomb coupling between optically active excitations of the electrons and molecular vibrations. The excellent properties of organic monolayers make them promising candidates for components of soft-matter optoelectronic devices.

One-step implementation of multiqubit controlled–controlled-Z gates with Rydberg atoms

In this paper, we propose a one-step scheme for generating a multiqubit controlled–controlled-Z (CCZ) gate based on Rydberg atoms where an amplitude-modulated field is employed to induce Rydberg antiblockade. The Rydberg atoms can form a Rabi oscillation between the ground state and the collective excited state effectively, so a one-step three-qubit CCZ gate can be easily achieved through a Rabi cycle. Numerical simulation results show that the scheme has a high fidelity and robustness against errors which are caused by parameter errors, intrinsic errors, and atomic decays. Therefore, we hope that this scheme will enable fast and robust quantum computing in the near future.

A Multiconfigurational Wave Function Implementation of the Frenkel Exciton Model for Molecular Aggregates.

We present an implementation of the Frenkel exciton model into the OpenMolcas program package enabling calculations of collective electronic excited states of molecular aggregates based on a multiconfigurational wave function description of the individual monomers. The computational protocol avoids using diabatization schemes and, thus, supermolecule calculations. Additionally, the use of the Cholesky decomposition of the two-electron integrals entering pair interactions enhances the efficiency of the computational scheme. The application of the method is exemplified for two test systems, that is, a formaldehyde oxime and a bacteriochlorophyll-like dimer. For the sake of comparison with the dipole approximation, we restrict our considerations to situations where intermonomer exchange can be neglected. The protocol is expected to be beneficial for aggregates composed of molecules with extended π systems, unpaired electrons such as radicals or transition metal centers, where it should outperform widely used methods based on time-dependent density functional theory.

Nuclear Equation of State from Nuclear Collective Excited State Properties

Nuclear Equation of State from Nuclear Collective Excited State Properties

The exciton size. Where are the limits?

The concept of exciton implies a collective excited state able to travel in a particle-like fashion. Its size is determined by the radius of excited electron-hole pair and, although it may vary by two orders of magnitude, it is always spatially restricted, while its delocalization length owing to the exciton wavefunction spatial dynamics may provide even a larger scale of changes. In this work, the limitations of exciton sizes are discussed by analysis where the exciton concept is still applicable. It is shown that the exciton size can be as small as few angstroms, but even smaller sizes can be, probably, justified. At the same time, coupling of exciton to polariton mode can enlarge the exciton-polariton coherence length to values as high as 20 µm, thus extending the scale of possible exciton sizes up to five orders of magnitude.

The Kuramoto–Lohe model and collective absorption of a photon

Light absorption by molecular exciton states in disordered networks is studied. The main purpose of this paper is to look at how phases of the intermediate ground–excited state superposition interfere during the absorption process. How does this phase average enable, or suppress, absorption to a delocalized state? To address this question, a theory for phase oscillators is used to predict the purity of the collective excited state of the network. The results of the study suggest that collective absorption by molecular exciton states requires a sufficiently large electronic coupling between molecules in the network compared to the random distribution of transition energies at the sites, even when the molecular network is completely isolated from the environment degrees of freedom. The ‘dividing line’ between absorption to a mixture of, essentially, localized excited states and coherent excitation of a pure delocalized exciton state is suggested to be predicted by the threshold of phase synchronization.

Quantum dynamics of disordered arrays of interacting superconducting qubits: Signatures of quantum collective states

We study theoretically the collective quantum dynamics occurring in various interacting superconducting qubits arrays (SQAs) in the presence of a spread of individual qubit frequencies. The interaction is provided by mutual inductive coupling between adjacent qubits (short-range Ising interaction) or inductive coupling to a low-dissipative resonator (long-range exchange interaction). In the absence of interaction the Fourier transform of temporal correlation function of the total polarization ($z$-projection of the total spin), i.e. the dynamic susceptibility $C(\omega)$, demonstrates a set of sharp small magnitude resonances corresponding to the transitions of individual superconducting qubits. We show that even a weak interaction between qubits can overcome the disorder with a simultaneous formation of the collective excited states. This collective behavior manifests itself by a single large resonance in $C(\omega)$. In the presence of a weak non-resonant microwave photon field in the low-dissipative resonator, the positions of dominant resonances depend on the number of photons, i.e. the collective ac Stark effect. Coupling of an SQA to the transmission line allows a straightforward experimental access of the collective states in microwave transmission experiments and, at the same time, to employ SQAs as sensitive single-photon detectors.

Shape and structure for the low-lying states of the Ge80 nucleus

The ground-state shape of the $^{80}\mathrm{Ge}$ nucleus and its evolution for the lowest collective excited states have been investigated in the framework of the Bohr model in order to clarify if the shape coexistence phenomenon is present. The obtained results, which are largely in agreement with the most recent experimental data, indicate a prolate shape for the ground state, respectively a sudden switch to a more deformed prolate one for certain excited states. Also, the first-excited ${0}_{2}^{+}$ state is found to be very high in energy, namely, at 2208 keV, being close to recent predictions made with shell-model calculations. Additionally, using the parameters fit for the few available experimental data for the ground band, the structure of the $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ bands has been built to support future experiments.

The Seamless Connection of Local and Collective Excited States in Subsystem Time-Dependent Density Functional Theory.

The theoretical understanding of photoinduced processes in multichromophoric systems requires, as an essential ingredient, the possibility of accurately describing their electronically excited states. However, the size of these systems often prohibits the usage of conventional electronic-structure methods, so that often multiscale approaches based on phenomenologically motivated models are employed. In contrast, subsystem time-dependent density functional theory (sTDDFT) allows for a subsystem-based ab initio description of multichromophoric systems and therefore allows for, in principle, an exact description of photoinduced processes. This Perspective aims to outline the theoretical foundations and commonly used practical realizations as well as to illustrate benefits of recent developments and open issues in the field of sTDDFT. Prospective, potential future applications and possible methodological developments are discussed.

Size-reduction of Rydberg collective excited states in cold atomic system

The collective effect of large amounts of atoms exhibit an enhanced interaction between light and atoms. This holds great interest in quantum optics, and quantum information. When a collective excited state of a group of atoms during Rabi oscillation is varying, the oscillation exhibits rich dynamics. Here, we experimentally observe a size-reduction effect of the Rydberg collective state during Rabi oscillation in cold atomic dilute gases. The Rydberg collective state was first created by the Rydberg quantum memory, and we observed a decreased oscillation frequency effect by measuring the time traces of the retrieved light field amplitude, which exhibited chirped characteristics. This is caused by the simultaneous decay to the overall ground state and the overall loss of atoms. The observed oscillations are dependent on the effective Rabi frequency and detuning of the coupling laser, and the dephasing from inhomogeneous broadening. The reported results show the potential prospects of studying the dynamics of the collective effect of a large amount of atoms and manipulating a single-photon wave-packet based on the interaction between light and Rydberg atoms.

Flat-Band Localization in Creutz Superradiance Lattices.

Flat bands play an important role in diffraction-free photonics and attract fundamental interest in many-body physics. Here we report the engineering of flat-band localization of collective excited states of atoms in Creutz superradiance lattices with tunable synthetic gauge fields. Magnitudes and phases of the lattice hopping coefficients can be independently tuned to control the state components of the flat band and the Aharonov-Bohm phases. We can selectively excite the flat band and control the flat-band localization with the synthetic gauge field. Our study provides a room-temperature platform for flat bands of atoms and holds promising applications in exploring correlated topological materials.

Radiative process of two entangled uniformly accelerated atoms in a thermal bath: a possible case of anti-Unruh event

We study the radiative process of two entangled two-level atoms uniformly accelerated in a thermal bath, coupled to a massless scalar field. First, by using the positive frequency Wightman function from the Minkowski modes with a Rindler transformation we provide the transition probabilities for the transitions from maximally entangled symmetric and anti-symmetric Bell states to the collective excited or ground state in (1 + 1) and (1 + 3) dimensions. We observe a possible case of anti-Unruh-like event in these transition probabilities, though the (1+1) and (1+3) dimensional results are not completely equivalent. We infer that thermal bath plays a major role in the occurrence of the anti-Unruh-like effect, as it is also present in the transition probabilities corresponding to a single detector in this case. Second, we have considered the Green’s functions in terms of the Rindler modes with the vacuum of Unruh modes for estimating the same. Here the anti-Unruh effect appears only for the transition from the anti-symmetric state to the collective excited or ground state. It is noticed that here the (1 + 1) and (1 + 3) dimensional results are equivalent, and for a single detector, we do not observe any anti-Unruh effect. This suggests that the entanglement between the states of the atoms is the main cause for the observed anti-Unruh effect in this case. In going through the investigation, we find that the transition probability for a single detector case is symmetric under the interchange between the thermal bath’s temperature and the Unruh temperature for Rindler mode analysis; whereas this is not the case for Minkowski mode. We further comment on whether this observation may shed light on the analogy between an accelerated observer and a real thermal bath. An elaborate investigation for the classifications of our observed anti-Unruh effects, i.e., either weak or strong anti-Unruh effect, is also thoroughly demonstrated.

Effects of velocity-dependent and spin-orbit terms of the Skyrme interaction on neutron elastic scattering observables

Elastic scattering of a series of doubly closed-shell nuclei at low energy has been described in the framework of optical potential model generated from the particle-vibration coupling approach on top of the collective excited states obtained from the random-phase approximation. We focus on the effects of the spin-orbit and velocity-dependent interaction on the angular distributions and analyzing powers by comparing these observables with experimental data. It has been found that the contribution of the two-body velocity-dependent and two-body spin-orbit terms are important to improve results for $^{16}\mathrm{O},\phantom{\rule{0.16em}{0ex}}^{40}\mathrm{Ca},\phantom{\rule{0.16em}{0ex}}^{48}\mathrm{Ca}$, and $^{208}\mathrm{Pb}$. The velocity-dependent interactions strongly reduce the absorption on the surface while the spin-orbit interactions sightly increase the absorption in the interior.

Describing the Spin Dependence of p-Wave Neutron Scattering on Spherical Nuclei in a Two-Phonon Approximation of the Coupled Channel Technique

The spin dependence of p-wave neutron scattering by spherical nuclei in region A ≈ 60–130 is analyzed. The spin-orbit splitting of the 3p-maximum of the neutron force function is described using a two-phonon approximation of the coupled channel thechnique. The cross section of inelastic neutron scattering through a composite nucleus is comparable to that of the direct excitation of collective states at an energy of ∼1 MeV.

Shape evolution of neutron-rich Mo106,108,110 isotopes in the triaxial degree of freedom

Background: Neutron-rich nuclei with mass number between 100 and 110 attract much attention, since several kinds of shapes, such as spherical, prolate, oblate, and triaxial shapes, are predicted. In particular, for neutron-rich Mo isotopes, different models predict different magnitudes and rigidity of triaxial deformation. Previous interpretations of experimental results based solely on low-lying 22+ states are insufficient to distinguish between the rigid triaxial shape, γ vibration, or γ-soft rotor. Purpose: The shape evolution of Mo106, Mo108, and Mo110 is investigated through their 21+-state lifetimes, decay-branching ratios of the 22+ state, and energies of the low-lying collective excited states with Kπ=0+,2+, and 4+. Method: β-delayed γ-ray spectroscopy was employed for neutron-rich Nb and Zr isotopes produced at the RIKEN RI Beam Factory to populate excited states in Mo106, Mo108, and Mo110. The EUroball-RIKEN Cluster Array was used for high-resolution γ-ray detection and lifetimes of the 21+ states were determined using the UK fast-timing array of LaBr3(Ce) detectors. Results: New γ-ray transitions and levels are reported, including newly assigned 02+ states in Mo108,110. Quadrupole deformations were obtained for Mo106,108,110 from their 21+ energies and lifetimes. The β-delayed neutron-emission probabilities of Nb108 and Nb110 were determined by examining the γ rays of their respective daughter decays. Conclusions: The even-odd energy staggering in the 22+ band was compared with typical patterns of the γ-vibrational band, rigid triaxial rotor, and γ-soft rotor. The very small even-odd staggering of Mo106, Mo108, and Mo110 favors a γ-vibrational band assignment. The kinematic moment of inertia for the 22+ band showed a trend similar to the ground-state band, which is as expected for the γ-vibrational band. Beyond-mean-field calculations employing the constrained Hartree-Fock-Bogoliubov and local quasiparticle-random-phase approximation method using the SLy5+T interaction reproduced the ground and 22+ bands in Mo106 and Mo108. The collective wave functions are consistent with the interpretation of the 22+ band as the γ-vibrational band of the prolate shape. However, the staggering pattern observed in Mo110 differs from the one suggested in the calculations which predict a γ-soft rotor. There was no experimental indication of the oblate shape or the γ-soft rotor predicted in these Mo isotopes.10 MoreReceived 28 August 2019Revised 19 December 2019Accepted 17 January 2020DOI:https://doi.org/10.1103/PhysRevC.101.044311©2020 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasBeta decayCollective levelsNuclear structure & decaysNucleon distributionProperties90 ≤ A ≤ 149Nuclear Physics

Membrane Environment Enables Ultrafast Isomerization of Amphiphilic Azobenzene.

The non‐covalent affinity of photoresponsive molecules to biotargets represents an attractive tool for achieving effective cell photo‐stimulation. Here, an amphiphilic azobenzene that preferentially dwells within the plasma membrane is studied. In particular, its isomerization dynamics in different media is investigated. It is found that in molecular aggregates formed in water, the isomerization reaction is hindered, while radiative deactivation is favored. However, once protected by a lipid shell, the photochromic molecule reacquires its ultrafast photoisomerization capacity. This behavior is explained considering collective excited states that may form in aggregates, locking the conformational dynamics and redistributing the oscillator strength. By applying the pump probe technique in different media, an isomerization time in the order of 10 ps is identified and the deactivation in the aggregate in water is also characterized. Finally, it is demonstrated that the reversible modulation of membrane potential of HEK293 cells via illumination with visible light can be indeed related to the recovered trans→cis photoreaction in lipid membrane. These data fully account for the recently reported experiments in neurons, showing that the amphiphilic azobenzenes, once partitioned in the cell membrane, are effective light actuators for the modification of the electrical state of the membrane.

Phase transition and thermodynamic properties of <i>N</i> two-level atoms in an optomechanical cavity at finite temperature

Optomechanical cavity is a powerful connection between a nanomechanical oscillator and a quantized electromagnetic field. In this system, a novel photon-phonon nonlinear interaction arising from the nanomechanical oscillation is produced through the radiation pressure. Now this nonlinear photon-phonon interaction has become an important resource for implementing high-precision measurements and processing quantum information. Motivated by T. Esslinger group’s experiment, it is very meaningful to explore the exotic quantum phenomena when a ultra-cold BEC is trapped in an optomechanical cavity. In this paper, we mainly investigate phase transition and the finite-temperature thermodynamic properties of a Bose-Einstein condensate in an optomechanical cavity. It’s worth mentioning that at zero temperature many different mean-field approximate methods have been used to analyze the ground state properties of a Bose-Einstein condensate in an optomechanical cavity. Two common methods are Holstein-Primakoff transformation and spin coherent state variation. In this paper, an interesting imaginary-time path integral approach has been introduced to study finite temperature thermodynamic properties and phase transition of a Bose-Einstein condensate in an optomechanical cavity. First, we obtained system's partition function by taking imaginary-time path integration. Meanwhile, an effective action has been obtained by means of this method, which is the basic of the variation to get the numerical solution of photon number and the expression of the atomic number. At zero temperature, these results are consistent with what we have obtained by Holstein-Primakoff transformation or spin coherent state variational method. By adjusting the atom-field coupling strength and other parameters the second-order phase transition from the normal phase to the superradiant phase has been revealed. Meanwhile, a new unstable superradiant state was also found. And we found that in addition to the normal phase and superradiation phase, there exists an un-solution region of the mean photon number. Meanwhile, we find that the nonlinear photon-phonon interaction does not affect the normal phase. However, in the superradiant phase, the nonlinear photon-phonon interaction can enhance the macroscopic collective excitations. At the same time, the thermodynamic properties of the system are also discussed. According to the obtained distribution function, we can derive the analytical expression of the average energy and the free energy. Furthermore, the expression of entropy at finite temperature can also be obtained. we find the nonlinear photon-phonon interaction does not affect the average energy in the normal phase, but the average energy in the superradiant phase can deeply deviate in the large nonlinear photon-phonon interaction. It’s worth mentioning that the mean photon number and average energy in the finite-temperature tend to be consistent with the case in absolute zero temperature in the strong coupling region, while the entropy in the superradiant phase is rapidly reduced to zero as the atom-field coupling strength increases. In other words, strongly coupled collective excited states are highly ordered and are not affected by thermal fluctuations in the temperature range we are considering. The thermodynamic properties, such as the entropy and corresponding specific heat, characterize the Dicke phase transition.

Low-lying collective excited states in nonintegrable pairing models based on the stationary-phase approximation to the path integral

For a description of large-amplitude collective motion associated with nuclear pairing, requantization of time-dependent mean-field dynamics is performed using the stationary-phase approximation (SPA) to the path integral. We overcome the difficulty of the SPA, which is known to be applicable to integrable systems only, by developing a requantization approach combining the SPA with the adiabatic self-consistent collective coordinate method (ASCC+SPA). We apply the ASCC+SPA to multi-level pairing models, which are non-integrable systems, to study the nuclear pairing dynamics. The ASCC+SPA gives a reasonable description of low-lying excited $0^+$ states in non-integrable pairing systems.