Eigenenergy Spectrum Research Articles

Spectral collapse in multiqubit two-photon Rabi model

We have shown that the smallest possible singel-qubit critical coupling strength of the N-qubit two-photon Rabi model is only 1/N times that of the two-photon Rabi model. The spectral collapse can thus occur at a more attainable value of the critical coupling. For both of the two-qubit and three-qubit cases, we have also rigorously demonstrated that at the critical coupling the system not only has a set of discrete eigenenergies but also a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in the presence of a finite potential well and a nonlocal potential. The energy difference of each qubit, which specifies both the depth of the finite potential well and the strength of the nonlocal potential, determines the number of bound states available, implying that the extent of the incomplete spectral collapse can be monitored in a straightforward manner.

Hybrid Quantum-Classical Eigensolver without Variation or Parametric Gates

The use of near-term quantum devices that lack quantum error correction, for addressing quantum chemistry and physics problems, requires hybrid quantum-classical algorithms and techniques. Here, we present a process for obtaining the eigenenergy spectrum of electronic quantum systems. This is achieved by projecting the Hamiltonian of a quantum system onto a limited effective Hilbert space specified by a set of computational bases. From this projection, an effective Hamiltonian is obtained. Furthermore, a process for preparing short depth quantum circuits to measure the corresponding diagonal and off-diagonal terms of the effective Hamiltonian is given, whereby quantum entanglement and ancilla qubits are used. The effective Hamiltonian is then diagonalized on a classical computer using numerical algorithms to obtain the eigenvalues. The use case of this approach is demonstrated for ground state and excited states of BeH2 and LiH molecules, and the density of states, which agrees well with exact solutions. Additionally, hardware demonstration is presented using IBM quantum devices for H2 molecule.

Exceptional non-Hermitian topological edge mode and its application to active matter

Topological materials exhibit edge-localized scattering-free modes protected by their nontrivial bulk topology through the bulk-edge correspondence in Hermitian systems. While topological phenomena have recently been much investigated in non-Hermitian systems with dissipations and injections, the fundamental principle of their edge modes has not fully been established. Here, we reveal that, in non-Hermitian systems, robust gapless edge modes can ubiquitously appear owing to a mechanism that is distinct from bulk topology, thus indicating the breakdown of the bulk-edge correspondence. The robustness of these edge modes originates from yet another topological structure accompanying the branchpoint singularity around an exceptional point, at which eigenvectors coalesce and the Hamiltonian becomes nondiagonalizable. Their characteristic complex eigenenergy spectra are applicable to realize lasing wave packets that propagate along the edge of the sample. We numerically confirm the emergence and the robustness of the proposed edge modes in the prototypical lattice models. Furthermore, we show that these edge modes appear in a model of chiral active matter based on the hydrodynamic description, demonstrating that active matter can exhibit an inherently non-Hermitian topological feature. The proposed general mechanism would serve as an alternative designing principle to realize scattering-free edge current in non-Hermitian devices, going beyond the existing frameworks of non-Hermitian topological phases.

Manipulating the spectral collapse in two-photon Rabi model

We have investigated the eigenenergy spectrum of the two-photon Rabi model with a full quadratic coupling, particularly the special feature “spectral collapse”. The critical coupling strength is reduced by half from that of the two-photon Rabi model, implying that the spectral collapse can now occur at a more attainable value of the critical coupling. At the critical coupling some discrete eigenenergy levels still survive below the continuous energy spectrum, i.e. an incomplete spectral collapse, and the set of discrete eigenenergies has a one-to-one mapping with that of a particle of variable effective mass in a finite potential well. Since the energy difference between the two atomic levels specifies the depth of the potential well, the number of bound states available (or the extent of the “spectral collapse”) can be straightforwardly monitored. Obviously, this bears a great resemblance to the spectral collapse of the two-photon Rabi model, at least qualitatively. Moreover, since the full quadratic coupling includes an extra term proportional to the photon number operator only, our analysis indicates that one may manipulate the critical coupling of the two-photon Rabi model by incorporating an adjustable proportionality constant to this extra term.

Enhanced photon blockade via driving a trapped Λ -type atom in a hybrid optomechanical system

A scheme is proposed to enhance the photon blockade effect in a hybrid optomechanical system with a $\mathrm{\ensuremath{\Lambda}}$-type atom driven by the microwave. Through analyzing the conventional and unconventional blockade mechanisms, we find that the enhanced photon blockade effect can be attributed to two aspects: (i) The nonresonant coupled $\mathrm{\ensuremath{\Lambda}}$-type atom reconstructs the anharmonic eigenenergy spectrum and (ii) the microwave driving field promotes the destructive quantum interference for two-photon excitation. By means of the joint enhancement effect, the perfect photon blockade, i.e., the second-order correlation function ${g}^{(2)}(0)\ensuremath{\simeq}0$, can be achieved without the strong single-photon optomechanical coupling as reported in the standard optomechanical system. All the analyses and derivations are further verified via simulating numerically the quantum master equation of the initial Hamiltonian, showing good agreement between analytical and numerical results. Moreover, the optimal parameter relation is given to optimize the photon blockade and maximize the occupancy probability of single-photon excitation at the same time. Our scheme provides a feasible method to engineer a high-quality and efficient single-photon source.

Demystifying the spectral collapse in two-photon Rabi model

We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, particularly the special feature “spectral collapse”, by means of an elementary quantum mechanics approach. The eigenenergy spectrum is found to consist of both a set of discrete energy levels and a continuous energy spectrum. Each of these eigenenergies has a two-fold degeneracy corresponding to the spin degree of freedom. The discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a “Lorentzian function” potential well, and the continuous energy spectrum can be derived from the scattering problem associated with a potential barrier. The number of bound states available at the critical coupling is determined by the energy difference between the two atomic levels so that the extent of the “spectral collapse” can be monitored in a straightforward manner.

Bound states of spin–orbit coupled cold atoms in a Dirac delta-function potential

Dirac delta-function potential is widely studied in quantum mechanics because it usually can be exactly solved and at the same time is useful in modeling various physical systems. Here we study a system of delta-potential trapped spin–orbit coupled cold atoms. The spin–orbit coupled atomic matter wave has two kinds of evanescent modes, one of which has a pure imaginary wavevector and is an ordinary evanescent wave; while the other with a complex number wave vector is recognized as an oscillating evanescent wave. We identified the eigenenergy spectra and the existence of bound states in this system. The bound states can be constructed analytically using the two kinds of evanescent modes and we found that they exhibit typical features of stripe phase, separated phase or zero-momentum phase. In addition to that, the properties of semi-bound states are also discussed, which is a localized wave packet on a plane wave background.

Enhancement of photon blockade effect via quantum interference.

We study the photon blockade effect in a coupled cavity system, which is formed by a linear cavity coupled to a Kerr-type nonlinear cavity via a photon-hopping interaction. We explain the physical phenomenon from the viewpoint of the conventional and unconventional photon blockade effects. The corresponding physical mechanisms of the two kinds of photon blockade effects are based on the anharmonicity in the eigenenergy spectrum and the destructive quantum interference between two different transition paths, respectively. In particular, we find that the photon blockade via destructive quantum interference also exists in the conventional photon blockade regime and that the unconventional photon blockade occurs in both the weak- and strong-Kerr nonlinearity cases. The photon blockade effect can be observed by calculating the second-order correlation function of the cavity field. This model is general and hence it can be implemented in various experimental setups such as coupled optical-cavity systems, coupled photon-magnon systems, and coupled superconducting-resonator systems. We present some discussions on the experimental feasibility.

Splitting of conductance resonance through a magnetic quantum dot in graphene

We report a dual resonance feature in ballistic conductance through a quantum Hall graphene nanoribbon with a magnetic quantum dot. Such a magnetic quantum dot localizes Dirac fermions exhibiting anisotropic eigenenergy spectra with broken time-reversal symmetry. Interplay between the localized states and quantum Hall edge states is found to be two-fold, showing Breit-Wigner and Fano resonances, which is reminiscent of a double quantum dot system. By fitting the numerical results with the Fano-Breit-Wigner lineshape from the double quantum dot model, we demonstrate that the two-fold resonance is due to the valley mixing that comes from the coupling of the magnetic quantum dot with quantum Hall edge channels; an effective double quantum dot system emerges from a single magnetic quantum dot in virtue of the valley degree of freedom. It is further confirmed that the coupling is weaker for the Fano resonance and stronger for the Breit-Wigner resonace.

Spectral features of the tunneling-induced transparency and the Autler-Townes doublet and triplet in a triple quantum dot

We theoretically investigate the spectral features of tunneling-induced transparency (TIT) and Autler-Townes (AT) doublet and triplet in a triple-quantum-dot system. By analyzing the eigenenergy spectrum of the system Hamiltonian, we can discriminate TIT and double TIT from AT doublet and triplet, respectively. For the resonant case, the presence of the TIT does not exhibit distinguishable anticrossing in the eigenenergy spectrum in the weak-tunneling regime, while the occurrence of double anticrossings in the strong-tunneling regime shows that the TIT evolves to the AT doublet. For the off-resonance case, the appearance of a new detuning-dependent dip in the absorption spectrum leads to double TIT behavior in the weak-tunneling regime due to no distinguished anticrossing occurring in the eigenenergy spectrum. However, in the strong-tunneling regime, a new detuning-dependent dip in the absorption spectrum results in AT triplet owing to the presence of triple anticrossings in the eigenenergy spectrum. Our results can be applied to quantum measurement and quantum-optics devices in solid systems.

Lowest energy Frenkel and charge transfer exciton intermixing in one-dimensional copper phthalocyanine molecular lattice

We report the results of the combined experimental and theoretical studies of the low-lying exciton states in crystalline copper phthalocyanine. We derive the eigen energy spectrum for the two lowest intramolecular Frenkel excitons coupled to the intermolecular charge transfer exciton state and compare it with temperature dependent optical absorption spectra measured experimentally, to obtain the parameters of the Frenkel-charge-transfer exciton intermixing. The two Frenkel exciton states are spaced apart by 0.26 eV, and the charge transfer exciton state is 50 meV above the lowest Frenkel exciton. Both Frenkel excitons are strongly mixed with the charge transfer exciton, showing the coupling constant 0.17 eV which agrees with earlier experimental measurements. These results can be used for the proper interpretation of the physical properties of crystalline phthalocyanines.

Saddle point localization of molecular wavefunctions.

The quantum mechanical description of isomerization is based on bound eigenstates of the molecular potential energy surface. For the near-minimum regions there is a textbook-based relationship between the potential and eigenenergies. Here we show how the saddle point region that connects the two minima is encoded in the eigenstates of the model quartic potential and in the energy levels of the [H, C, N] potential energy surface. We model the spacing of the eigenenergies with the energy dependent classical oscillation frequency decreasing to zero at the saddle point. The eigenstates with the smallest spacing are localized at the saddle point. The analysis of the HCN ↔ HNC isomerization states shows that the eigenstates with small energy spacing relative to the effective (v1, v3, ℓ) bending potentials are highly localized in the bending coordinate at the transition state. These spectroscopically detectable states represent a chemical marker of the transition state in the eigenenergy spectrum. The method developed here provides a basis for modeling characteristic patterns in the eigenenergy spectrum of bound states.

Quantum Levy Flights and Multifractality of Dipolar Excitations in a Random System.

We consider dipolar excitations propagating via dipole-induced exchange among immobile molecules randomly spaced in a lattice. The character of the propagation is determined by long-range hops (Levy flights). We analyze the eigenenergy spectra and the multifractal structure of the wave functions. In 1D and 2D, all states are localized, although in 2D the localization length can be extremely large leading to an effective localization-delocalization crossover in realistic systems. In 3D, all eigenstates are extended but not always ergodic, and we identify the energy intervals of ergodic and nonergodic states. The reduction of the lattice filling induces an ergodic to nonergodic transition, and the excitations are mostly nonergodic at low filling.

Spectral domains for bosonic pair creation in static electromagnetic fields

We study the emission spectrum of bosons created from the vacuum by combined static electric and magnetic fields. Depending on the spatial extension of the magnetic field, we find four regimes of pair creation, characterized by different growth behaviors of the number of the produced particles. We show that these regimes manifest themselves in the eigenenergy spectrum of the Klein–Gordon Hamiltonian. The regimes also lead to rather different kinetic energy spectra of the emitted bosons, whose peak positions can be obtained from a generalized Fano-like perturbative approach.

Analysis of two and three dipolar bosons in a spherical harmonic trap

As dipolar gases become more readily accessible in experiment there is a need to develop a comprehensive theoretical framework of the few-body physics of these systems. Here, we extend the coupled-pair approach developed for the unitary two-component Fermi gas to a few-body system of dipolar bosons in a spherical harmonic trap. The long range and anisotropy of the dipole-dipole interaction is handled by a flexible and efficient correlated gaussian basis with stochastically variational optimisation. We calculate the eigenenergy spectrum and structural properties of two and three trapped bosonic dipoles. This demonstrates the efficiency and flexibility of the coupled-pair approach at dealing with systems with complex interactions.

Transient evolution of eigenmodes in dynamic cavities and time-varying media

In this paper, we investigate the perturbation of natural eigenmodes of dynamic cavities with boundaries moving at quasi-static speeds relative to the wave velocity. For an arbitrarily shaped source-free cavity, the amplitude of the irrotational mode is modeled as a damped harmonic oscillator with time-varying eigenfrequency, i.e., a parametric oscillator. It is found that the effect of the pure Doppler shift of the resonance frequencies of the eigenmodes is small at nonrelativistic speeds. However, it is known that any spectrum of eigenenergies that is perturbed by a space- and/or time-fluctuating medium can develop frequency shifts of arbitrary magnitude. By using a linear dynamic (time-dependent) shift for the cavity broad resonances, we find that Doppler-like large shifts result in a mere frequency modulation of the total (resultant) field amplitude, while nonuniform red or blue shift can create a hybrid amplitude and frequency modulation. Interestingly, the combined action of red and blue shifts of uniform magnitude can also create a hybrid modulation. If the angle between modal wave vector and stirrer speed is accounted for in the static (time-independent) shift, the resulting red and blue shifts lead to irregular hybrid modulations. This can occur even for regular perturbations in regular cavities. In addition, owing to the stochastic nature of mode-stirred cavities, the effect of random Doppler-like shifts is also investigated, leading to a Fokker-Planck equation whose diffusion coefficient shows quadratic dependence on the mode amplitude. Thus, the analysis of random perturbations offers an effective framework for observed instantaneous Doppler effects in closed electromagnetic environments. The mathematical framework obtained in terms of stochastic differential equations is useful to predict the nonstationary response of dynamic cavities with complicated or unknown boundary geometry.

Comment on: “Multi-photon Rabi model: Generalized parity and its applications” by B. Gardas and J. Dajka [Phys. Lett. A 377 (2013) 3205

Abstract Recently Gardas and Dajka examined the multi-photon Rabi model and showed that it had a discrete eigenenergy spectrum with normalizable eigenstates. However, their analysis is actually invalid because it has been rigorously proven that the model does not have eigenstates in the Hilbert space spanned by the photon number states.

Higgs Algebraic Symmetry of Screened System in a Spherical Geometry

The orbits and the dynamical symmetries for the screened Coulomb potentials and isotropic harmonic oscillators have been studied by Wu and Zeng [Z.B. Wu and J.Y. Zeng, Phys. Rev. A 62 (2000) 032509]. We find similar properties in the corresponding systems in a spherical space, whose dynamical symmetries are described by Higgs algebra. There exist extended Runge—Lenz vector for screened Coulomb potentials and extended quadruple tensor for screened harmonic oscillators. They, together with angular momentum, constitute the generators of the geometrical symmetry group. Moreover, there exist an infinite number of closed orbits for suitable angular momentum values, and we give the equations of the classical orbits. The eigenenergy spectrum and corresponding eigenstates in these systems are derived.

The role of power-law correlated disorder in the Anderson metal-insulator transition

We study the influence of scale-free correlated disorder on the metal-insulator transition in the Anderson model of localization. We use standard transfer matrix calculations and perform finite-size scaling of the largest inverse Lyapunov exponent to obtain the localization length for respective 3D tight-binding systems. The density of states is obtained from the full spectrum of eigenenergies of the Anderson Hamiltonian. We discuss the phase diagram of the metal-insulator transition and the influence of the correlated disorder on the critical exponents.

Dynamical localization of two electrons in triple-quantum-dot shuttles

The dynamical localization phenomena in two-electron quantum-dot shuttles driven by an ac field have been investigated and analyzed by the Floquet theory. The dynamical localization occurs near the anti-crossings in Floquet eigenenergy spectrum. The oscillation of the quantum-dot shuttles may increase the possibility of the dynamical localization. Especially, even if the two electrons are initialized in two neighbor dots, they can be localized there for appropriate intensity of the driven field. The studies may help the understanding of dynamical localization in electron shuttles and expand the application potential of nanoelectromechanical devices.