Excited Eigenstates Research Articles

Embedding Quantum Many-Body Scars into Decoherence-Free Subspaces.

Quantum many-body scars are nonthermal excited eigenstates of nonintegrable Hamiltonians, which could support coherent revival dynamics from special initial states when scars form an equally spaced tower in the energy spectrum. For open quantum systems, engineering many-body scarred dynamics by a controlled coupling to the environment remains largely unexplored. Here, we provide a general framework to exactly embed quantum many-body scars into the decoherence-free subspaces of Lindblad master equations. The dissipative scarred dynamics manifest persistent periodic oscillations for generic initial states, and can be practically utilized to prepare scar states with potential quantum metrology applications. We construct the Liouvillian dissipators with the local projectors that annihilate the whole scar towers, and utilize the Hamiltonian part to rotate the undesired states out of the null space of dissipators. We demonstrate our protocol through several typical models hosting many-body scar towers and propose an experimental scheme to observe the dissipative scarred dynamics based on digital quantum simulations and resetting ancilla qubits.

Prethermal Stability of Eigenstates under High Frequency Floquet Driving.

Systems subject to high-frequency driving exhibit Floquet prethermalization, that is, they heat exponentially slowly on a timescale that is large in the drive frequency, τ_{h}∼exp(ω). Nonetheless, local observables can decay much faster via energy conserving processes, which are expected to cause a rapid decay in the fidelity of an initial state. Here we show instead that the fidelities of eigenstates of the time-averaged Hamiltonian, H_{0}, display an exponentially long lifetime over a wide range of frequencies-even as generic initial states decay rapidly. When H_{0} has quantum scars, or highly excited eigenstates of low entanglement, this leads to long-lived nonthermal behavior of local observables in certain initial states. We present a two-channel theory describing the fidelity decay time τ_{f}: the interzone channel causes fidelity decay through energy absorption, i.e., coupling across Floquet zones, and ties τ_{f} to the slow heating timescale, while the intrazone channel causes hybridization between states in the same Floquet zone. Our work informs the robustness of experimental approaches for using Floquet engineering to generate interesting many-body Hamiltonians, with and without scars.

Eigenstate entanglement entropy in the integrable spin-1/2XYZ model.

We study the average and the standard deviation of the entanglement entropy of highly excited eigenstates of the integrable interacting spin-1/2XYZ chain away from and at special lines with U(1) symmetry and supersymmetry. We universally find that the average eigenstate entanglement entropy exhibits a volume-law coefficient that is smaller than that of quantum-chaotic interacting models. At the supersymmetric point, we resolve the effect that degeneracies have on the computed averages. We further find that the normalized standard deviation of the eigenstate entanglement entropy decays polynomially with increasing system size, which we contrast with the exponential decay in quantum-chaotic interacting models. Our results provide state-of-the art numerical evidence that integrability in spin-1/2 chains reduces the average and increases the standard deviation of the entanglement entropy of highly excited energy eigenstates when compared with those in quantum-chaotic interacting models.

Preparing quantum many-body scar states on quantum computers

Quantum many-body scar states are highly excited eigenstates of many-body systems that exhibit atypical entanglement and correlation properties relative to typical eigenstates at the same energy density. Scar states also give rise to infinitely long-lived coherent dynamics when the system is prepared in a special initial state having finite overlap with them. Many models with exact scar states have been constructed, but the fate of scarred eigenstates and dynamics when these models are perturbed is difficult to study with classical computational techniques. In this work, we propose state preparation protocols that enable the use of quantum computers to study this question. We present protocols both for individual scar states in a particular model, as well as superpositions of them that give rise to coherent dynamics. For superpositions of scar states, we present both a system-size-linear depth unitary and a finite-depth nonunitary state preparation protocol, the latter of which uses measurement and postselection to reduce the circuit depth. For individual scarred eigenstates, we formulate an exact state preparation approach based on matrix product states that yields quasipolynomial-depth circuits, as well as a variational approach with a polynomial-depth ansatz circuit. We also provide proof of principle state-preparation demonstrations on superconducting quantum hardware.

Scale-Invariant Survival Probability at Eigenstate Transitions.

Understanding quantum phase transitions in highly excited Hamiltonian eigenstates is currently far from being complete. It is particularly important to establish tools for their characterization in time domain. Here, we argue that a scaled survival probability, where time is measured in units of a typical Heisenberg time, exhibits a scale-invariant behavior at eigenstate transitions. We first demonstrate this property in two paradigmatic quadratic models, the one-dimensional Aubry-Andre model and three-dimensional Anderson model. Surprisingly, we then show that similar phenomenology emerges in the interacting avalanche model of ergodicity breaking phase transitions. This establishes an intriguing similarity between localization transition in quadratic systems and ergodicity breaking phase transition in interacting systems.

Extracting Quantum Many-Body Scarred Eigenstates with Matrix Product States.

Quantum many-body scarred systems host nonthermal excited eigenstates immersed in a sea of thermal ones. In cases where exact expressions for these special eigenstates are not known, it is computationally demanding to distinguish them from their exponentially many thermal neighbors. We propose a matrix-product-state (MPS) algorithm, dubbed DMRG-S, to extract such states at system sizes far beyond the scope of exact diagonalization. Using this technique, we obtain scarred eigenstates in Rydberg-blockaded chains of up to 80 sites and perform a finite-size scaling study to address the lingering question of the stability for the Néel state revivals in the thermodynamic limit. Our method also provides a systematic way to obtain exact MPS representations for scarred eigenstates near the target energy without apriori knowledge. In particular, we find several new scarred eigenstates with exact MPS representations in kinetically constrained spin and clock models. The combination of numerical and analytical investigations in our work provides a new methodology for future studies of quantum many-body scars.

Correspondence between excited energy eigenstates and local minima of the energy landscape in quantum spin systems

The quantum-classical correspondence between local minima on the classical energy landscape and excited eigenstates in the energy spectrum is studied within the context of many-body quantum spin systems. In mean-field approximations of a quantum problem, one usually focuses on attaining the global minimum of the resulting energy function, while other minimum solutions are usually ignored. For frustrated systems, a strict distinction between global and local minimum is often not tenable since first-order-type transitions can interchange the roles played by two different minima. This begs the question of whether there is any physical interpretation for the local minima encountered in mean-field approximations of quantum systems. We look at the problem from the perspective of quantum spin systems. Two models are studied, a frustrated model with quenched disorder and a pure system without frustration. Accurate classical energies of the minima are compared with the full spectrum of energy levels, allowing us to search for signs of correspondence between them. It is found that the local minima can generally be interpreted as excited energy eigenstates. Instances of spurious minima are also reported.

Hidden Bethe states in a partially integrable model

We present a one-dimensional multi-component model, known to be partially integrable when restricted to the subspaces made of only two components. By constructing fully anti-symmetrized bases, we find integrable excited eigenstates corresponding to the totally anti-symmetric irreducible representation of the permutation operator in the otherwise non-integrable subspaces. We establish rigorously the breakdown of integrability in those subspaces by showing explicitly the violation of the Yang-Baxter's equation. We further solve the constraints from Yang-Baxter's equation to find exceptional momenta that allows Bethe Ansatz solutions of solitonic bound states. These integrable eigenstates have distinct dynamical consequence from the embedded integrable subspaces previously known, as they do not span their separate Krylov subspaces, and a generic initial state can partly overlap with them and therefore have slow thermalization. However, this novel form of weak ergodicity breaking contrasts that of quantum many-body scars in that the integrable eigenstates involved do not have necessarily low entanglement. Our approach provides a complementary route to arrive at quantum many-body scars since, instead of solving towers of single mode excited states based on a solvable ground state in a non-integrable model, we identify the integrable eigenstates that survive in a deformation of the Hamiltonian away from its integrable point.

Many-body excitations in trapped Bose gas: A non-Hermitian approach

We study a physically motivated model for a trapped dilute gas of Bosons with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states of this system by accounting for the scattering of atoms in pairs from the macroscopic state. We start with an approximate many-body Hamiltonian, H a p p \mathcal {H}_{\mathrm {app}} , in the Bosonic Fock space. This H a p p \mathcal {H}_{\mathrm {app}} conserves the total number of atoms. Inspired by Wu [J. Math. Phys. 2 (1961), 105–123], we apply a non-unitary transformation to H a p p \mathcal {H}_{\mathrm {app}} . Key in this procedure is the pair-excitation kernel, which obeys a nonlinear integro-partial differential equation. In the stationary case, we develop an existence theory for solutions to this equation by a variational principle. We connect this theory to a system of partial differential equations for one-particle excitation (“quasiparticle”-) wave functions derived by Fetter [Ann. Phys. 70 (1972), 67–101], and prove existence of solutions for this system. These wave functions solve an eigenvalue problem for a J J -self-adjoint operator. From the non-Hermitian Hamiltonian, we derive a one-particle nonlocal equation for low-lying excitations, describe its solutions, and recover Fetter’s energy spectrum. We also analytically provide an explicit construction of the excited eigenstates of the reduced Hamiltonian in the N N -particle sector of Fock space.

Excited-eigenstate entanglement properties of XX spin chains with random long-range interactions

Quantum information theoretical measures are useful tools for characterizing quantum dynamical phases. However, employing them to study excited states of random spin systems is a challenging problem. Here, we report results for the entanglement entropy (EE) scaling of excited eigenstates of random XX antiferromagnetic spin chains with long-range (LR) interactions decaying as a power law with distance with exponent $\alpha$. To this end, we extend the real-space renormalization group technique for excited states (RSRG-X) to solve this problem with LR interaction. For comparison, we perform numerical exact diagonalization (ED) calculations. From the distribution of energy level spacings, as obtained by ED for up to $N\sim 18$ spins, we find indications of a delocalization transition at $\alpha_c \approx 1$ in the middle of the energy spectrum. With RSRG-X and ED, we show that for $\alpha>\alpha^*$ the entanglement entropy (EE) of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for $\alpha<\alpha^*$ EE displays an algebraic growth with the subsystem size $l$, $S_l\sim l^{\beta}$, with $0<\beta<1$. We find that $\alpha^* \approx 1$ coincides with the delocalization transition $\alpha_c$ in the middle of the many-body spectrum. An interpretation of these results based on the structure of the RG rules is proposed, which is due to {\it rainbow} proliferation for very long-range interactions $\alpha\ll 1$. We also investigate the effective temperature dependence of the EE allowing us to study the half-chain entanglement entropy of eigenstates at different energy densities, where we find that the crossover in EE occurs at $\alpha^* < 1$.

Analyzing Time‐Average Excitation Rate Among Quantum Dot Eigenstates Triggered by Time‐Dependent Noise Strength

Present investigation rigorously explores the time‐average excitation rate (TAER) of ground‐state electronic population of GaAs quantum dot (QD) to the excited eigenstates. The QD confinement potential contains Gaussian impurity (the dopant) and Gaussian white noise (GWN). GWN can be categorized as either additive or multiplicative depending upon its way of attachment to the doped QD. The excitation occurs as soon as the noise strength begins fluctuating with time in periodic or random manner. The study highlights the coupled influence of the varying physical quantity, presence/absence of noise, the mode of attachment of noise to QD, and the nature of time‐fluctuation of the noise strength (periodic/random), and finally harnesses and tailors the features of the TAER profiles. The TAER plots unveil steady rise, steady fall, maximization (linked with the emergence of large nonlinear optical response), minimization, and saturation (linked with dynamic freezing). The findings appear to be significant in the practical usage of QD‐based devices.

Localization of a mobile impurity interacting with an Anderson insulator

Thermalizing and localized many-body quantum systems present two distinct dynamical phases of matter. Recently, the fate of a localized system coupled to a thermalizing system viewed as a quantum bath received significant theoretical and experimental attention. In this work, we study a mobile impurity, representing a small quantum bath, that interacts locally with an Anderson insulator with a finite density of localized particles. Using static Hartree approximation to obtain an effective disorder strength, we formulate an analytic criterion for the perturbative stability of the localization. Next, we use an approximate dynamical Hartree method and the quasi-exact time-evolved block decimation (TEBD) algorithm to study the dynamics of the system. We find that the dynamical Hartree approach which completely ignores entanglement between the impurity and localized particles predicts the delocalization of the system. In contrast, the full numerical simulation of the unitary dynamics with TEBD suggests the stability of localization on numerically accessible timescales. Finally, using an extension of the density matrix renormalization group algorithm to excited states (DMRG-X), we approximate the highly excited eigenstates of the system. We find that the impurity remains localized in the eigenstates and entanglement is enhanced in a finite region around the position of the impurity, confirming the dynamical predictions. Dynamics and the DMRG-X results provide compelling evidence for the stability of localization.

Unsupervised detection of decoupled subspaces: Many-body scars and beyond

Highly excited eigenstates of quantum many-body systems are typically featureless thermal states. Some systems, however, possess a small number of special, low-entanglement eigenstates known as quantum scars. We introduce a quantum-inspired machine-learning platform based on a quantum variational autoencoder (QVAE) that detects families of scar states in spectra of many-body systems. Unlike a classical autoencoder, QVAE performs a parametrized unitary operation, allowing us to compress a single eigenstate into a smaller number of qubits. We demonstrate that the autoencoder trained on a scar state is able to detect the whole family of scar states sharing common features with the input state. We identify families of quantum many-body scars in the PXP model beyond the ${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{3}$ families and find dynamically decoupled subspaces in the Hilbert space of disordered, interacting spin-ladder model. The possibility of an automatic detection of subspaces of scar states opens new pathways in studies of models with a weak breakdown of ergodicity and fragmented Hilbert spaces.

Cooling and state preparation in an optical lattice via Markovian feedback control

We propose and investigate a scheme based on Markovian feedback control that allows for the preparation of single targeted eigenstates of a system of bosonic atoms in a one-dimensional optical lattice with high fidelity. It can be used for in-situ cooling the interacting system without particle loss, both for weak and strong interactions, and for experimentally preparing and probing individual excited eigenstates. For that purpose the system is assumed to be probed weakly via homodyne detection of photons that are scattered off-resonantly by the atoms from a structured probe beam into a cavity mode. By applying an inertial force to the system that is proportional to the measured signal, the system is then guided into a pure target state. The scheme is found to be robust against reduced measurement efficiencies.

Free-Fermion Multiply Excited Eigenstates and Their Experimental Signatures in 1D Arrays of Two-Level Atoms.

One-dimensional (1D) subwavelength atom arrays display multiply excited subradiant eigenstates which are reminiscent of free fermions. So far, these states have been associated with subradiant states with decay rates ∝N^{-3}, with N the number of atoms, which fundamentally prevents detection of their fermionic features by optical means. In this Letter, we show that free-fermion states generally appear whenever the band of singly excited states has a quadratic dispersion relation at the band edge and, hence, may also be obtained with radiant and even superradiant states. 1D arrays have free-fermion multiply excited eigenstates that are typically either subradiant or (super)radiant, and we show that a simple transformation acts between the two families. Based on this correspondence, we propose different means for their preparation and analyze their experimental signature in optical detection.

Tuning nonequilibrium heat current and two-photon statistics via composite qubit-resonator interaction

Quantum thermal transport and two-photon statistics serve as two representative nonequilibrium features in circuit quantum electrodynamics (cQED) systems. Here we investigate quantum heat flow and two-photon correlation function at steady state in a composite qubit-resonator model, where one qubit shows both transverse and longitudinal couplings to a single-mode optical resonator. With weak qubit-resonator interaction, we unravel two microscopic transport pictures, i.e., cotunneling and cyclic heat exchange processes corresponding to transverse and longitudinal couplings, respectively. The nonmonotonic behavior of the heat current is exhibited by tuning the temperature bias with the weak longitudinal coupling. At strong qubit-resonator coupling, the heat current also exhibits a nonmonotonic feature by increasing qubit-resonator coupling strength, which tightly relies on the scattering processes between the qubit and the corresponding thermal bath. Furthermore, the longitudinal coupling is preferred to enhance heat current in the strong qubit-resonator coupling regime. For two-photon correlation function, it exhibits an antibunching-to-bunching transition by tuning the composite angle, which is mainly dominated by the modulation of the energy gap between the first and the second excited eigenstates. Our results are expected to deepen the understanding of nonequilibrium thermal transport and nonclassical photon radiation based on the cQED platform.

Conserved quantities from entanglement Hamiltonian

We show that the subregion entanglement Hamiltonians of excited eigenstates of a quantum many-body system are approximately linear combinations of subregionally (quasi)local approximate conserved quantities, with relative commutation errors $\mathcal{O}\left(\frac{\text{subregion boundary area}}{\text{subregion volume}}\right)$. By diagonalizing an entanglement Hamiltonian superdensity matrix (EHSM) for an ensemble of eigenstates, we can obtain these conserved quantities as the EHSM eigen-operators with nonzero eigenvalues. For free fermions, we find the number of nonzero EHSM eigenvalues is cut off around the order of subregion volume, and some of their EHSM eigen-operators can be rather nonlocal, although subregionally quasilocal. In the interacting XYZ model, we numerically find the nonzero EHSM eigenvalues decay roughly in power law if the system is integrable, with the exponent $s\approx 1$ ($s\approx 1.5\sim 2$) if the eigenstates are extended (many-body localized). For fully chaotic systems, only two EHSM eigenvalues are significantly nonzero, the eigen-operators of which correspond to the identity and the subregion Hamiltonian.

Extended quasiparticle approach to non-resonant and resonant X-ray emission spectroscopy.

The initial states of the secondary processes of X-ray emission spectroscopy (XES) and resonant inelastic X-ray scattering (RIXS) are highly excited eigenstates with a deep core hole after a X-ray photoelectron spectroscopy (XPS) process and a X-ray photoabsorption spectroscopy (XAS) process, respectively, so that the XES and RIXS calculation offers a good example of extended quasiparticle theory (EQPT) (K. Ohno, S. Ono and T. Isobe, J. Chem. Phys., 2017, 146, 084108) which is applicable to any initial exited eigenstate. We apply the standard one-shot GW + Bethe-Salpeter equation (BSE) approach in MBPT to this problem on the basis of EQPT and analyze XES and RIXS spectra for CH4, NH3, H2O, and CH3OH molecules. We also suggest a simpler approach only using the GW calculation without solving the BSE to compute the XES and RIXS energies, although it cannot give the spectral intensity. Moreover, according to extended Kohn-Sham theory (T. Nakashima, H. Raebiger and K. Ohno, Phys. Rev. B, 2021, 104, L201116), we give a justification and comment of applying the method relying on time-dependent density functional theory as well as the one-shot GW + BSE approach to this problem.

Topologically protected two-fluid edge states

Edge states reveal the nontrivial topology of energy band in the bulk. As localized states at boundaries, many-body edge states may obey a special symmetry that is broken in the bulk. When local particle-particle interaction is induced, they may support a particular property. We consider an extended two-dimensional Su-Schrieffer-Heeger Hubbard model and examine the appearance of $\ensuremath{\eta}$-pairing states, which are excited eigenstates related to superconductivity. In the absence of Hubbard interaction, the energy band is characterized by topologically invariant polarization in association with edge states. In the presence of on-site Hubbard interaction, $\ensuremath{\eta}$-pairing edge states appear in the topologically nontrivial phase, resulting in the condensation of pairs at the boundary. In addition, as Hamiltonian eigenstates, the edge states contain paired fermions and unpaired fermions. Neither affects the other; they act as two-fluid states. From numerical simulations of many-body scattering processes, a clear manifestation and experimental detection scheme of topologically protected two-fluid edge states are provided.

Emergent Ergodicity at the Transition between Many-Body Localized Phases.

Strongly disordered systems in the many-body localized (MBL) phase can exhibit ground state order in highly excited eigenstates. The interplay between localization, symmetry, and topology has led to the characterization of a broad landscape of MBL phases ranging from spin glasses and time crystals to symmetry protected topological phases. Understanding the nature of phase transitions between these different forms of eigenstate order remains an essential open question. Here, we conjecture that no direct transition between distinct MBL orders can occur in one dimension; rather, an ergodic phase always intervenes. Motivated by recent advances in Rydberg-atom-based quantum simulation, we propose an experimental protocol where the intervening ergodic phase can be diagnosed via the dynamics of local observables.