Discrete Eigenstates Research Articles

Role of losses and coupling in bright-dark metasurfaces mimicking non-Hermitian parity-time symmetric systems

All real systems are essentially non-conservative in nature, which is non-Hermitian in the fundamental premises of the quantum realm. Such open systems obeying parity-time (PT) symmetry can offer intriguing physics, which has sparked much attention in recent years. The thoughtful interplay of loss and gain in PT-symmetric systems can tune the eigenstates from real to complex space passing through a singular point known as an exceptional point (EP) with concurrently coalescing the discrete eigenstates. In this regard, artificially engineered near-field coupled metasurfaces provide a remarkable podium to introduce as well as control loss and coupling strength simultaneously, manifesting a practical channel to attain EP. Motivated by this notion, we have theoretically explored near-field coupled dark-bright-mode assisted terahertz metasurfaces imitating a typical non-Hermitian PT-symmetric system. Structurally, such metasurfaces are realized by utilizing a pair of orthogonally twisted near-field coupled split-ring resonators (bright-dark-resonators). We have meticulously examined such a system with two distinctive approaches, i.e., the loss parameter and inter-resonator coupling strength. Eventually, our exploration demonstrates that judicious modulation of the loss parameter can invoke an undemanding route to approach the other wisely elusive exceptional point. We believe that this comprehensive study of non-Hermitian quantum systems in a photonic context can lead to the development of accurate and precise ultra-sensitive sensors besides other futuristic photonic devices.

Confined-state physics and signs of fermionization of moiré excitons in WSe2/MoSe2 heterobilayers

We revisit and extend the standard bosonic interpretation of interlayer excitons (ILX) in the moiré potential of twisted heterostructures of transition-metal dichalcogenides. In our experiments, we probe a high quality MoSe2/WSe2 van der Waals bilayer heterostructure via density-dependent photoluminescence spectroscopy and reveal strongly developed, unconventional spectral shifts of the emergent moiré exciton resonances. The observation of saturating blueshifts of successive exciton resonances allow us to explain their physics in terms of a model utilizing fermionic saturable absorbers. This approach is strongly inspired by established quantum-dot models, which underlines the close analogy of ILX trapped in pockets of the moiré potential, and quantum emitters with discrete eigenstates.

Reservoir-assisted symmetry breaking and coalesced zero-energy modes in an open PT -symmetric Su-Schrieffer-Heeger model

We study a model consisting of a central $\mathcal{PT}$-symmetric trimer with non-Hermitian strength parameter $\gamma$ coupled to two semi-infinite Su-Schrieffer-Heeger (SSH) leads. We show the existence of two zero-energy modes, one of which is localized while the other is anti-localized. For the remaining eigenvalues, we demonstrate two qualitatively distinct types of $\mathcal{PT}$-symmetry breaking. Within a subset of the parameter space corresponding to the topologically non-trivial phase of the SSH chains, a gap opens within the broken $\mathcal{PT}$ regime of the discrete eigenvalue spectrum. For relatively smaller values of $\gamma$, the eigenvalues are embedded in the two SSH bands and hence become destabilized primarily due to the resonance interaction with the continuum. We refer to this as reservoir-assisted $\mathcal{PT}$-symmetry breaking. As the value of $\gamma$ is increased, the eigenvalues exit the SSH bands and the discrete eigenstates become more strongly localized in the central trimer region. This approximate decoupling results in the discrete spectrum behaving more like the independent trimer, including both a region in which the $\mathcal{PT}$ symmetry is restored (the gap) and a second region in which it is broken again. At the exceptional point (EP) marking the boundary between the gap and the second $\mathcal{PT}$-broken region, two of the eigenstates coalesce with the localized zero-energy mode, resulting in a third-order exceptional point (EP3). At the other boundaries of the parameter space at which the gap vanishes, similar higher-order EPs can form as pairs of the discrete eigenstates coalesce with either of the two zero-energy states. The EPs of order $N$ formed of the localized zero-energy state give rise to a characteristic $\sim t^{2N-2}$ evolution in the survival probability dynamics, which we propose to measure in a photonic lattice experiment.

Time-Dependent Unitary Transformation Method in the Strong-Field-Ionization Regime with the Kramers-Henneberger Picture.

Time evolution operators of a strongly ionizing medium are calculated by a time-dependent unitary transformation (TDUT) method. The TDUT method has been employed in a quantum mechanical system composed of discrete states. This method is especially helpful for solving molecular rotational dynamics in quasi-adiabatic regimes because the strict unitary nature of the propagation operator allows us to set the temporal step size to large; a tight limitation on the temporal step size () can be circumvented by the strict unitary nature. On the other hand, in a strongly ionizing system where the Hamiltonian is not Hermitian, the same approach cannot be directly applied because it is demanding to define a set of field-dressed eigenstates. In this study, the TDUT method was applied to the ionizing regime using the Kramers-Henneberger frame, in which the strong-field-dressed discrete eigenstates are given by the field-free discrete eigenstates in a moving frame. Although the present work verifies the method for a one-dimensional atom as a prototype, the method can be applied to three-dimensional atoms, and molecules exposed to strong laser fields.

Exact Classical and Quantum Mechanics of a Generalized Singular Equation of Quadratic Liénard Type

Authors introduce a generalized singular differential equation of quadratic Liénard type for study of exact classical and quantum mechanical solutions. The equation is shown to exhibit periodic solutions and to include the linear harmonic oscillator equation and the Painlevé-Gambier XVII equation as special cases. It is also shown that the equation may exhibit discrete eigenstates as quantum behavior under Nikiforov-Uvarov approach after several point transformations.

The arrow of time in open quantum systems and dynamical breaking of the resonance–anti-resonance symmetry

Open quantum systems are often represented by non-Hermitian effective Hamiltonians that have complex eigenvalues associated with resonances. In previous work we showed that the evolution of tight-binding open systems can be represented by an explicitly time-reversal symmetric expansion involving all the discrete eigenstates of the effective Hamiltonian. These eigenstates include complex-conjugate pairs of resonant and anti-resonant states. An initially time-reversal-symmetric state contains equal contributions from the resonant and anti-resonant states. Here we show that as the state evolves in time, the symmetry between the resonant and anti-resonant states is automatically broken, with resonant states becoming dominant for and anti-resonant states becoming dominant for . Further, we show that there is a time-scale for this symmetry-breaking, which we associate with the ‘Zeno time’. We also compare the time-reversal symmetric expansion with an asymmetric expansion used previously by several researchers. We show how the present time-reversal symmetric expansion bypasses the non-Hilbert nature of the resonant and anti-resonant states, which previously introduced exponential divergences into the asymmetric expansion.

Partial wave decomposition in Friedrichs model with self-interacting continua

We consider the nonrelativistic model of coupling bare discrete states with continuum states in which the continuum states can have interactions among themselves. By partial-wave decomposition and constraint to the conserved angular momentum eigenstates, the model can be reduced to Friedrichs-like model with additional interactions between the continua. If a kind of factorizable form factor is chosen, the model can be exactly solvable, that is, the generalized discrete eigenstates including bound states, virtual states, and resonances can all be represented using the original bare states, and so do the in-state and out-state. The exact S matrix is thus obtained. We then discuss the behaviors of the dynamically generated S-wave and P-wave discrete states as the coupling is varying when there is only one self-interacting bare continuum state. We find that even when the potential is repulsive there could also be resonances and virtual states. In the P-wave cases with attractive interaction, we find that when there is a near-threshold bound state, there will always be an accompanying virtual state and we also give a more general argument of this effect.

Extensions of solvable potentials with finitely many discrete eigenstates

We address the problem of rational extensions of six examples of shape-invariant potentials having finitely many discrete eigenstates. The overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions, which are an essential ingredient for multi-indexed and iso-spectral extensions of these potentials. They have exactly the same form as the eigenfunctions but their degrees are much higher than nmax so that their energies are lower than the groundstate energy.

ON THE SUPERSYMMETRY OF THE DIRAC–KEPLER PROBLEM PLUS A COULOMB-TYPE SCALAR POTENTIAL IN (D+1) DIMENSIONS AND THE GENERALIZED LIPPMANN–JOHNSON OPERATOR

We will study the Dirac–Kepler problem plus a Coulomb-type scalar potential by generalizing the Lippmann–Johnson operator to D spatial dimensions. From this operator, we construct the supersymmetric generators to obtain the energy spectrum for discrete excited eigenstates and the radial spinor for the SUSY ground state.

Resonant Spectrum Analysis of the Conductance of an Open Quantum System and Three Types of Fano Parameter

We explain the Fano peak (an asymmetric resonance peak) as an interference effect involving resonant states. We reveal that there are three types of Fano asymmetry according to their origins: the interference between a resonant state and an anti-resonant state, that between a resonant state and a bound state, and that between two resonant states. We show that the last two show the asymmetric energy dependence given by Fano, but the first one shows a slightly different form. In order to show the above, we analytically and microscopically derive a formula where we express the conductance purely in terms of the summation over all discrete eigenstates including resonant states and anti-resonant states, without any background integrals. We thereby obtain microscopic expressions of the Fano parameters that describe the three types of the Fano asymmetry. One of the expressions indicate that the corresponding Fano parameter becomes complex under an external magnetic field.

Resonant-state Expansion of the Green’s Function of Open Quantum Systems

Our series of recent work on the transmission coefficient of open quantum systems in one dimension will be reviewed. The transmission coefficient is equivalent to the conductance of a quantum dot connected to leads of quantum wires. We will show that the transmission coefficient is given by a sum over all discrete eigenstates without a background integral. An apparent "background" is in fact not a background but generated by tails of various resonance peaks. By using the expression, we will show that the Fano asymmetry of a resonance peak is caused by the interference between various discrete eigenstates. In particular, an unstable resonance can strongly skew the peak of a nearby resonance.

Correlation-induced single-flux-quantum penetration in quantum rings

Conventionally, the states of a two-dimensional quantum ring in a high magnetic field have a well-defined spatial structure. But Coulomb repulsion between individual orbits causes oscillations in the size of this structure each time a magnetic flux-quantum enters or leaves the system. This effect has now been measured experimentally in semiconducting quantum rings. The confined electronic states of mesoscopic structures in a magnetic field are arranged in Landau levels consisting of spatially discrete eigenstates. These Landau orbits are the quantum mechanical analogue of classical cyclotron orbits. Here we present magnetoconductance oscillations in semiconductor rings, which visualize the spatial discreteness of the Landau orbits in high magnetic fields (typically B>2 T). We will show that these oscillations are caused by the flux-quantized, discrete electronic size of the ring leading to a corresponding modulation of its two-point conductance. The oscillation period is given by the number of flux quanta penetrating the conducting area of the structure. These high-field oscillations are distinctively different from the well-known Aharonov–Bohm effect1, where, most generally, the penetration of individual flux quanta h/e through a nanostructure causes periodic crossings of field-dependent energy levels, which give rise to magneto-quantum oscillations in its conductance2,3,4.

Resonant States of Open Quantum Systems

We first show that quantum resonant states observe particle number conservation and hence are consistent with the probabilistic interpretation of quantum mechanics. We then present for a class of quantum open systems, a resonant-state expansion of the sum of the retarded and advanced Green's functions. The expansion is given purely in terms of all discrete eigenstates and does not contain any background integrals. Using the expansion, we argue that the Fano asymmetry of resonance peaks is interpreted as interference between discrete eigenstates. We microscopically derive the Fano parameters for several cases.

Quantum-confinement effect on recombination dynamics and carrier localization in cubic InN and In xGa 1 − xN quantum boxes

In this study, the quantum confinement effect on recombination dynamics and carrier localization in cubic InN ( c-InN) and cubic In x Ga 1 − x N ( c-In x Ga 1 − x N) low dimensional structures are theoretically examined. The small InN and In-rich In x Ga 1 − x N low dimensional structures show a strong quantum confinement effect, which results in ground states away from the band edge and discrete eigen-states. Depending on composition, temperature, and size of the InN and In x Ga 1 − x N low dimensional structures, quantum confinement effect can affect the exciton dimensions (D). In InN quantum cubes, the strong quantum confinement effect leads to temperature-dependent radiative lifetimes showing a large size variation. The nearly-temperature-independent and shorter radiative lifetimes in small InN and In-rich In x Ga 1 − x N low dimensional structures suggest that the strong quantum confinement leads to 0 D carrier confinement, stronger carrier localization, and high recombination efficiency. Reported radiative lifetimes were found to match very well with our simulation results of In-rich quantum cubes, which indicates that spontaneous emissions come from the radiative recombination of localized excitons in In-rich In x Ga 1 − x N clusters. The simulation results could provide important information for the designs and interpretations of c-InN and c-In x Ga 1 − x N devices.

A simple model of molecular electronic devices and its analytical solution

I present an analytically solvable model for molecular electronic devices (MEDs). The model contains all the essential components: semi-infinite contacts, molecule-contact interface, and of course, the molecule. I obtain explicit expression for the reflection coefficient r(E), as well as for the transmission probability T(E) [T(E)=1-mid R:r(E)mid R:(2)]. r(E) exhibits a surprisingly simple structure if studied as a function of a complex energy variable E. In this case, r(E) can be expressed in terms of a finite number of eigenvalues of a non-Hermitian Hamiltonian. This Hamiltonian also yields the molecular part of the MED wave function. Considering various MEDs, it is illustrated that the theory presented allows for a transparent interpretation of molecular conductance in terms of discrete eigenstates.

Quantum regime of free electron lasers starting from noise

We investigate the regime of a high-gain free-electron laser starting from noise. In the first part, we neglect the radiation propagation and we formulate a linear theory of the N-particle free-electron laser Hamiltonian model, quantizing both the radiation field and the electron motion. Quantum effects such as frequency shift, line narrowing, limitation for bunching and energy spread, and minimum uncertainty states are described. Using a second-quantization formalism, we demonstrate entanglement between the recoiling electrons and the radiation field. In the second part, we describe the field classically but we include propagation effects (i.e. slippage) and we demonstrate the novel regime of SASE with high temporal coherence and discrete spectrum. Furthermore, we describe quantum purification'' of SASE: the classical chaotic spiking behavior disappears and the spectrum becomes a series of discrete very narrow lines which correspond to transitions between discrete momentum eigenstates ( which originate high temporal coherence).

Exact quantum motions of two ions in a two-dimensional paul trap

We investigate the exact quantum motions of two ions in a two-dimensional Paul trap. When the Coulomb-correlation of the system is considered, we obtain a set of exact solutions of the system and the corresponding perturbed solutions, containing the exact discrete eigenstates and eigenenergies, and the approximate piecewise continuous spectra and the band structure. Taking several quantum states with lower quantum numbers as examples, we compute the average distances between two ions and give the plots of probability distribution of the two ions.

EFFECTS OF THE MAGNETIC FLUX AND OF THE ELECTRON MOMENTUM ON THE TRANSMISSION AMPLITUDE IN THE AHARONOV–BOHM INTERFEROMETER

A characterization of the two-terminal open-ring Aharonov–Bohm interferometer is made by analyzing the phase space plots in the complex transmission amplitude plane. Two types of plots are considered: type 1 plot uses the magnetic flux as the variable parameter and type 2 plot which uses the electron momentum as the variable parameter. In type 1 plot, the trajectory closes upon itself only when the ratio R of the arm lengths (of the interferometer) is a rational fraction, and the shape and the type of the generated flower-like pattern is sensitive to the electron momentum. For momenta corresponding to discrete eigenstates of the perfect ring (i.e., the ring without the leads), the trajectory passes through the origin a certain fixed number of times before closing upon itself, whereas for arbitrary momenta it never passes through the origin. Although the transmission coefficient is periodic in the flux with the elementary flux quantum as the basic period, the phenomenon of electron transmission is shown not to be so when analyzed via the present technique. The periodicity is seen to spread over several flux units whenever R is a rational fraction whereas there is absolutely no periodicity present when R is an irrational number. In type 2 plot, closed trajectories passing through the origin a number of times are seen for R being a rational fraction. The case R = 1 (i.e., a symmetric ring) with zero flux is rather pathological — it presents a closed loop surrounding the origin. For irrational R values, the trajectories never close.

Density operators that extremize Tsallis entropy and thermal stability effects

Quite general, analytical (both exact and approximate) forms for discrete probability distributions (PDs) that maximize Tsallis entropy for a fixed variance are here investigated. They apply, for instance, in a wide variety of scenarios in which the system is characterized by a series of discrete eigenstates of the Hamiltonian. Using these discrete PDs as “weights” leads to density operators of a rather general character. The present study allows one to vividly exhibit the effects of non-extensivity. Varying Tsallis’ non-extensivity index q one is seen to pass from unstable to stable systems and even to unphysical situations of infinite energy.

Effective Hamiltonian and unitarity of the S matrix.

The properties of open quantum systems are described well by an effective Hamiltonian H that consists of two parts: the Hamiltonian H of the closed system with discrete eigenstates and the coupling matrix W between discrete states and continuum. The eigenvalues of H determine the poles of the S matrix. The coupling matrix elements W(cc')(k) between the eigenstates k of H and the continuum may be very different from the coupling matrix elements W(cc')(k) between the eigenstates of H and the continuum. Due to the unitarity of the S matrix, the W(cc')(k) depend on energy in a nontrivial manner. This conflicts with the assumptions of some approaches to reactions in the overlapping regime. Explicit expressions for the wave functions of the resonance states and for their phases in the neighborhood of, respectively, avoided level crossings in the complex plane and double poles of the S matrix are given.