Structure Of Eigenstates Research Articles

Assigning temperatures to eigenstates.

In the study of thermalization in finite isolated quantum systems, an inescapable issue is the definition of temperature. We examine and compare different possible ways of assigning temperatures to energies or equivalently to eigenstates in such systems. A commonly used assignment of temperature in the context of thermalization is based on the canonical energy-temperature relationship, which depends only on energy eigenvalues and not on the structure of eigenstates. For eigenstates, we consider defining temperature by minimizing the distance between (full or reduced) eigenstate density matrices and canonical density matrices. We show that for full eigenstates, the minimizing temperature depends on the distance measure chosen and matches the canonical temperature for the trace distance; however, the two matrices are not close. With reduced density matrices, the minimizing temperature has fluctuations that scale with subsystem and system size but appears to be independent of distance measure. In particular limits, the two matrices become equivalent while the temperature tends to the canonical temperature.

Generalized Survival Probability.

Survival probability measures the probability that a system taken out of equilibrium has not yet transitioned from its initial state. Inspired by the generalized entropies used to analyze nonergodic states, we introduce a generalized version of the survival probability and discuss how it can assist in studies of the structure of eigenstates and ergodicity.

Quantum Chaos in the Extended Dicke Model.

We systematically study the chaotic signatures in a quantum many-body system consisting of an ensemble of interacting two-level atoms coupled to a single-mode bosonic field, the so-called extended Dicke model. The presence of the atom–atom interaction also leads us to explore how the atomic interaction affects the chaotic characters of the model. By analyzing the energy spectral statistics and the structure of eigenstates, we reveal the quantum signatures of chaos in the model and discuss the effect of the atomic interaction. We also investigate the dependence of the boundary of chaos extracted from both eigenvalue-based and eigenstate-based indicators on the atomic interaction. We show that the impact of the atomic interaction on the spectral statistics is stronger than on the structure of eigenstates. Qualitatively, the integrablity-to-chaos transition found in the Dicke model is amplified when the interatomic interaction in the extended Dicke model is switched on.

Single-photon Transport in a Waveguide-cavity-emitter System

We investigate the single-photon transport properties in a hybrid waveguide quantum electrodynamics system, in which a one-dimensional waveguide is simultaneously coupled to a cavity and a driven Λ-type three-level atom. The cavity and the atom are also coupled to each other. We show that when the waveguide is coupled to the cavity and the atom at the same point, double electromagnetically induced transparency (EIT) can be observed from the system. The physical mechanism of the double EIT effect has been given by setting up the eigenstate structure for the whole system. When the waveguide is coupled to the cavity and the atom at different points, we demonstrate that the controllable nonreciprocal scattering effect can be achieved by modulating the non-zero cavity-emitter separation and the phase associated with the cavity-emitter coupling strength. These results have potential applications in realization of photonic coherent control.

Non-ergodic delocalized phase with Poisson level statistics

Motivated by the many-body localization (MBL) phase in generic interacting disordered quantum systems, we develop a model simulating the same eigenstate structure like in MBL, but in the random-matrix setting. Demonstrating the absence of energy level repulsion (Poisson statistics), this model carries non-ergodic eigenstates, delocalized over the extensive number of configurations in the Hilbert space. On the above example, we formulate general conditions to a single-particle and random-matrix models in order to carry such states, based on the transparent generalization of the Anderson localization of single-particle states and multiple resonances.

Fock-space anatomy of eigenstates across the many-body localization transition

We explore the Fock-space structure of eigenstates across the many-body localisation (MBL) transition in a disordered, interacting quantum spin-1/2 chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition; and is intimately connected to the discontinuous jump in the multifractal exponents characterising the Fock-space wavefunctions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the lengthscale also implies discontinuous behaviour of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume-scale in the ergodic phase and a length-scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless-like scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.

Rare thermal bubbles at the many-body localization transition from the Fock space point of view

In this work we study the many-body localization (MBL) transition and relate it to the eigenstate structure in the Fock space. Besides the standard entanglement and multifractal probes, we introduce the radial probability distribution of eigenstate coefficients with respect to the Hamming distance in the Fock space from the wave function maximum and relate the cumulants of this distribution to the properties of the quasi-local integrals of motion in the MBL phase. We demonstrate non-self-averaging property of the many-body fractal dimension $D_q$ and directly relate it to the jump of $D_q$ as well as of the localization length of the integrals of motion at the MBL transition. We provide an example of the continuous many-body transition confirming the above relation via the self-averaging of $D_q$ in the whole range of parameters. Introducing a simple toy-model, which hosts ergodic thermal bubbles, we give analytical evidences both in standard probes and in terms of newly introduced radial probability distribution that the MBL transition in the Fock space is consistent with the avalanche mechanism for delocalization, i.e., the Kosterlitz-Thouless scenario. Thus, we show that the MBL transition can been seen as a transition between ergodic states to non-ergodic extended states and put the upper bound for the disorder scaling for the genuine Anderson localization transition with respect to the non-interacting case.

Open quantum systems in thermal nonergodic environments

The dynamics of an open system crucially depends on the correlation function of its environment, $C_B(t)$. We show that for thermal non-Harmonic environments $C_B(t)$ may not decay to zero but to an offset, $C_0>0$. The presence of such offset is determined by the environment eigenstate structure, and whether it fulfills or not the eigenstate thermalization hypothesis. Moreover, we show that a $C_0>0$ could render the weak coupling approximation inaccurate and prevent the open system to thermalize. Finally, for a realistic environment of dye molecules, we show the emergence of the offset by using matrix product states (MPS), and discuss its link to a 1/f noise spectrum that, in contrast to previous models, extends to zero frequencies. Thus, our results may be relevant in describing dissipation in quantum technological devices like superconducting qubits, which are known to be affected by such noise.

Quantum coherence in ergodic and many-body localized systems

Quantum coherence quantifies the amount of superposition a quantum state can have in a given basis. Since there is a difference in the structure of eigenstates of the ergodic and many-body localized systems, we expect them also to differ in terms of their coherences in a given basis. Here, we numerically calculate different measures of quantum coherence in the excited eigenstates of an interacting disordered Hamiltonian as a function of the disorder. We show that quantum coherence can be used as an order parameter to detect the well-studied ergodic to many-body-localized phase transition. We also perform quantum quench studies to distinguish the behavior of coherence in thermalized and localized phases. We then present a protocol to calculate measurement-based localizable coherence to investigate the thermal and many-body localized phases. The protocol allows one to investigate quantum correlations experimentally in a non-destructive way, in contrast to measures that require tracing out a subsystem, which always destroys coherence and correlation.

Magic-angle semimetals with chiral symmetry

We construct and solve a two-dimensional, chirally symmetric model of Dirac cones subjected to a quasiperiodic modulation. In real space, this is realized with a quasiperiodic hopping term. This hopping model, as we show, at the Dirac node energy has a rich phase diagram with a semimetal-to-metal phase transition at intermediate amplitude of the quasiperiodic modulation, and a transition to a phase with a diverging density of states and sub-diffusive transport when the quasiperiodic hopping is strongest. We further demonstrate that the semimetal-to-metal phase transition can be characterized by the multifractal structure of eigenstates in momentum space and can be considered as a unique "unfreezing" transition. This unfreezing transition in momentum space generates flat bands with a dramatically renormalized bandwidth in the metallic phase similar to the phenomena of the band structure of twisted bilayer graphene at the magic angle. We characterize the nature of this transition numerically as well as analytically in terms of the formation of a band of topological zero modes. For pure quasiperiodic hopping, we provide strong numerical evidence that the low-energy density of states develops a divergence and the eigenstates exhibit Chalker (quantum-critical) scaling despite the model not being random. At particular commensurate limits the model realizes higher-order topological insulating phases. We discuss how these systems can be realized in experiments on ultracold atoms and metamaterials.

Subradiance of multilevel fermionic atoms in arrays with filling n≥2

We investigate the subradiance properties of $n\geq 2$ multilevel fermionic atoms loaded into the lowest motional level of a single trap (e.g.~a single optical lattice site or an optical tweezer). As pointed out in our previous work [arXiv:1907.05541], perfectly dark subradiant states emerge from the interplay between fermionic statistics and dipolar interactions. While in [arXiv:1907.05541] we focused on the $n=2$ case, here we provide an in-depth analysis of the single-site dark states for generic filling $n$, and show a tight connection between generic dark states and total angular momentum eigenstates. We show how the latter can also be used to understand the full eigenstate structure of the single-site problem, which we analyze numerically. Apart from this, we discuss two possible schemes to coherently prepare dark states using either a Raman transition or an external magnetic field to lift the Zeeman degeneracy. Although the analysis focuses on the single-site problem, we show that multi-site dark states can be trivially constructed in any geometry out of product states of single-site dark states. Finally, we discuss some possible implementations with alkaline-earth(-like) atoms such as $^{171}$Yb or $^{87}$Sr loaded into optical lattices, where they could be used for potential applications in quantum metrology and quantum information.

Electromagnetically and optomechanically induced transparency and amplification in an atom-assisted cavity optomechanical system

We propose an atom-assisted cavity optomechanical system consisting of a single $\mathrm{\ensuremath{\Lambda}}$-type three-level atom, a mechanical resonator, and a sideband-driven cavity. The analytic expression of the system's evolution is constructed under the consideration of sideband-driven fields, then steady-state solution of electromagnetically and optomechanically induced transparency and amplification is presented. In the resolved red sideband case, we demonstrate the appearance of electromagnetically and optomechanically induced transparency when the two sets of two-photon resonances are satisfied. Moreover, we set up the eigenstate structure for this atom-assisted cavity optomechanical system, which gives an explicit explanation for the mechanism of the transparency. In the resolved blue sideband case, we can achieve the controllable amplification of the probe light, which is induced by the constructive interactions of photon, atom, and phonon. The proposed scheme offers insight for the tripartite coherent interaction among photons, atoms, and phonons, which will provide great flexibility for phonon storage and motivate the development of quantum information processing.

Weyl Semimetal to Metal Phase Transitions Driven by Quasiperiodic Potentials.

We explore the stability of three-dimensional Weyl and Dirac semimetals subject to quasiperiodic potentials. We present numerical evidence that the semimetal is stable for weak quasiperiodic potentials, despite being unstable for weak random potentials. As the quasiperiodic potential strength increases, the semimetal transitions to a metal, then to an "inverted" semimetal, and then finally to a metal again. The semimetal and metal are distinguished by the density of states at the Weyl point, as well as by level statistics, transport, and the momentum-space structure of eigenstates near the Weyl point. The critical properties of the transitions in quasiperiodic systems differ from those in random systems: we do not find a clear critical scaling regime in energy; instead, at the quasiperiodic transitions, the density of states appears to jump abruptly (and discontinuously to within our resolution).

Algebraic properties and spectral collapse in nonlinear quantum Rabi models

We investigate the origin of spectral collapse occurring in nonlinear Rabi Hamiltonians with an su(1,1) coupling scheme, showing how the collapse can be triggered by the competition between the Rabi parameter g and the field frequency W. The collapse already appears in the model Hamiltonian where the atomic-energy term is absent. After showing that su(1,1) is the dynamical algebra of the Hamiltonian, we demonstrate how the occurrence of spectral collapse can be directly related to the three types of equivalence classes characterizing the structure of this algebra. We highlight how the dramatic change of the spectrum significantly affects the structure of eigenstates represented in a suitable momentum–coordinate picture.

Control of a single-particle localization in open quantum systems

We investigate the possibility to control localization properties of the asymptotic state of an open quantum system with a tunable synthetic dissipation. The control mechanism relies on the matching between properties of dissipative operators, acting on neighboring sites and specified by a single control parameter, and the spatial phase structure of eigenstates of the system Hamiltonian. As a result, the latter coincide (or near coincide) with the dark states of the operators. In a disorder-free Hamiltonian with a flat band, one can either obtain a dominating localized asymptotic state or populate whole flat and/or dispersive bands, depending on the value of the control parameter. In a disordered Anderson system, the asymptotic state can be localized anywhere in the spectrum of the Hamiltonian. The dissipative control is robust with respect to an additional local dephasing.

Non-local currents and the structure of eigenstates in planar discrete systems with local symmetries

Local symmetries are spatial symmetries present in a subdomain of a complex system. By using and extending a framework of so-called non-local currents that has been established recently, we show that one can gain knowledge about the structure of eigenstates in locally symmetric setups through a Kirchhoff-type law for the non-local currents. The framework is applicable to all discrete planar Schrödinger setups, including those with non-uniform connectivity. Conditions for spatially constant non-local currents are derived and we explore two types of locally symmetric subsystems in detail, closed-loops and one-dimensional open ended chains. We find these systems to support locally similar or even locally symmetric eigenstates.

Structure of eigenstates and quench dynamics at an excited-state quantum phase transition

We study the structure of the eigenstates and the dynamics of a system that undergoes an excited state quantum phase transition (ESQPT). The analysis is performed for two-level pairing models characterized by a U(n+1) algebraic structure. They exhibit a second order phase transition between two limiting dynamical symmetries represented by the U(n) and SO(n+1) subalgebras. They are, or can be mapped onto, models of interacting bosons. We show that the eigenstates with energies very close to the ESQPT critical point, E_{ESQPT}, are highly localized in the U(n)-basis. Consequently, the dynamics of a system initially prepared in a U(n)-basis vector with energy close to E_{ESQPT} may be extremely slow. Signatures of an ESQPT can therefore be found in the structures of the eigenstates and in the speed of the system evolution after a sudden quench. Our findings can be tested experimentally with trapped ions.

Entanglement and the sign structure of quantum states

Many body quantum eigenstates of generic Hamiltonians at finite energy density typically satisfy "volume law" of entanglement entropy: the von Neumann entanglement entropy and the Renyi entropies for a subregion scale in proportion to its volume. Here we provide a connection between the volume law and the sign structure of eigenstates. In particular, we ask the question: can a positive wavefunction support a volume law entanglement? Remarkably, we find that a typical random positive wavefunction, exhibits a constant law for Renyi entanglement entropies $S_n$ for $n>1$, despite arbitrary large amplitude fluctuations. We also provide evidence that the modulus of the finite energy density eigenstates of generic local Hamiltonians show similar behavior.

Gain- or loss-induced localization in one-dimensionalPT-symmetric tight-binding models

We investigate the properties of PT-symmetric tight-binding models by considering both bounded and unbounded models. For the bounded case, we obtain closed form expressions for the corresponding energy spectra and we analyze the structure of eigenstates as well as their dependence on the gain/loss contrast parameter. For unbounded PT-lattices, we explore their scattering properties through the development of analytical models. Based on our approach we identify a mechanism that is responsible to the emergence of localized states that are entirely due to the presence of gain and loss. The derived expressions for the transmission and reflection coefficients allow one to better understand the role of PT-symmetry in energy transport problems occurring in such PT-symmetric tight-binding settings. Our analytical results are further exemplified via pertinent examples.

Quantum effect of inductance on geometric Cooper-pair transport

We introduce a model for a flux-assisted Cooper-pair pump, the sluice, which is used to study geometric charge transport. Our model allows for a nonvanishing loop inductance going beyond the usual treatment with an exact phase bias. We derive the device Hamiltonian and current operators for different elements of the system and calculate the pumped charge carried by the ground state in the adiabatic limit. We show that extending the model beyond the exact phase bias has a weak but potentially non-negligible effect on the charge transport. This effect is observed to depend on the external flux bias. The adiabatic energy level and eigenstate structures are studied and the unusual features are explained. Finally, we discuss the requirements of adiabaticity.