non-Hermitian Quantum Dynamics Research Articles

Energetic Cost for Speedy Synchronization in Non-Hermitian Quantum Dynamics.

Quantum synchronization is crucial for understanding complex dynamics and holds potential applications in quantum computing and communication. Therefore, assessing the thermodynamic resources required for finite-time synchronization in continuous-variable systems is a critical challenge. In the present work, we find these resources to be extensive for large systems. We also bound the speed of quantum and classical synchronization in coupled damped oscillators with non-Hermitian anti-PT-symmetric interactions, and show that the speed of synchronization is limited by the interaction strength relative to the damping. Compared to the classical limit, we find that quantum synchronization is slowed by the noncommutativity of the Hermitian and anti-Hermitian terms. Our general results could be tested experimentally, and we suggest an implementation in photonic systems.

Quantum simulation of the bosonic Kitaev chain

Superconducting quantum circuits are a natural platform for quantum simulations of a wide variety of important lattice models describing topological phenomena, spanning condensed matter and high-energy physics. One such model is the bosonic analog of the well-known fermionic Kitaev chain, a 1D tight-binding model with both nearest-neighbor hopping and pairing terms. Despite being fully Hermitian, the bosonic Kitaev chain exhibits a number of striking features associated with non-Hermitian systems, including chiral transport and a dramatic sensitivity to boundary conditions known as the non-Hermitian skin effect. Here, using a multimode superconducting parametric cavity, we implement the bosonic Kitaev chain in synthetic dimensions. The lattice sites are mapped to frequency modes of the cavity, and the in situ tunable complex hopping and pairing terms are created by parametric pumping at the mode-difference and mode-sum frequencies, respectively. We experimentally demonstrate important precursors of nontrivial topology and the non-Hermitian skin effect in the bosonic Kitaev chain, including chiral transport, quadrature wavefunction localization, and sensitivity to boundary conditions. Our experiment is an important first step towards exploring genuine many-body non-Hermitian quantum dynamics.

Experimental quantum simulation of non-Hermitian dynamical topological states using stochastic Schrödinger equation

Noise is ubiquitous in real quantum systems, leading to non-Hermitian quantum dynamics, and may affect the fundamental states of matter. Here we report in an experiment a quantum simulation of the two-dimensional non-Hermitian quantum anomalous Hall (QAH) model using the nuclear magnetic resonance processor. Unlike the usual experiments using auxiliary qubits, we develop a stochastic average approach based on the stochastic Schrödinger equation to realize the non-Hermitian dissipative quantum dynamics, which has advantages in saving the quantum simulation sources and simplifying the implementation of quantum gates. We demonstrate the stability of dynamical topology against weak noise and observe two types of dynamical topological transitions driven by strong noise. Moreover, a region where the emergent topology is always robust regardless of the noise strength is observed. Our work shows a feasible quantum simulation approach for dissipative quantum dynamics with stochastic Schrödinger equation and opens a route to investigate non-Hermitian dynamical topological physics.

Non-Hermitian Edge Burst.

We unveil an unexpected non-Hermitian phenomenon, dubbed edge burst, in non-Hermitian quantum dynamics. Specifically, in a class of non-Hermitian quantum walk in periodic lattices with open boundary condition, an exceptionally large portion of loss occurs at the system boundary. The physical origin of this edge burst is found to be an interplay between two unique non-Hermitian phenomena: non-Hermitian skin effect and imaginary gap closing. Furthermore, we establish a universal bulk-edge scaling relation underlying the non-Hermitian edge burst. Our predictions are experimentally accessible in various non-Hermitian systems including quantum-optical and cold-atom platforms.

Emergent Mott insulators and non-Hermitian conservation laws in an interacting bosonic chain with noninteger filling and nonreciprocal hopping

We investigate the ground state and quantum dynamics of an interacting bosonic chain with nonreciprocal hopping. In sharp contrast to its Hermitian counterpart, the ground state can support Mott insulators in systems with noninteger filling due to the competition between nonreciprocal hopping and the on-site interaction. For the quantum dynamics, conservation laws for non-Hermitian systems manifest a stark difference from their Hermitian counterpart. In particular, for any Hermitian operator that commutes with the Hamiltonian operator, its expectation value is guaranteed to be nonconserved in the non-Hermitian quantum dynamics. To systematically identify the non-Hermitian conservation law, we establish a generic approach for constructing the conserved quantities in non-Hermitian many-body quantum systems with completely real spectra, and illustrate it concretely by the system under study. The direct experimental observation of Mott insulators in systems with noninteger filling and non-Hermitian conservation laws can be performed by ultracold atoms in optical lattices with the engineered nonreciprocal hopping.

Rephasing Invariant for Three-Neutrino Oscillations Governed by a Non-Hermitian Hamiltonian

Time-reversal symmetry is broken for mixed and possibly unstable Dirac neutrino propagation through absorbing media. This implies that interplay between the neutrino mixing, refraction, absorption and/or decay can be described by non-Hermitian quantum dynamics. We derive an identity which sets up direct connection between the fundamental neutrino parameters (mixing angles, CP-violating phase, mass-squared splittings) in vacuum and their effective counterparts in matter.

Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard.

Exciton-polaritons are hybrid light-matter quasiparticles formed by strongly interacting photons and excitons (electron-hole pairs) in semiconductor microcavities. They have emerged as a robust solid-state platform for next-generation optoelectronic applications as well as for fundamental studies of quantum many-body physics. Importantly, exciton-polaritons are a profoundly open (that is, non-Hermitian) quantum system, which requires constant pumping of energy and continuously decays, releasing coherent radiation. Thus, the exciton-polaritons always exist in a balanced potential landscape of gain and loss. However, the inherent non-Hermitian nature of this potential has so far been largely ignored in exciton-polariton physics. Here we demonstrate that non-Hermiticity dramatically modifies the structure of modes and spectral degeneracies in exciton-polariton systems, and, therefore, will affect their quantum transport, localization and dynamical properties. Using a spatially structured optical pump, we create a chaotic exciton-polariton billiard--a two-dimensional area enclosed by a curved potential barrier. Eigenmodes of this billiard exhibit multiple non-Hermitian spectral degeneracies, known as exceptional points. Such points can cause remarkable wave phenomena, such as unidirectional transport, anomalous lasing/absorption and chiral modes. By varying parameters of the billiard, we observe crossing and anti-crossing of energy levels and reveal the non-trivial topological modal structure exclusive to non-Hermitian systems. We also observe mode switching and a topological Berry phase for a parameter loop encircling the exceptional point. Our findings pave the way to studies of non-Hermitian quantum dynamics of exciton-polaritons, which may uncover novel operating principles for polariton-based devices.

Non-Hermitian quantum dynamics and entanglement of coupled nonlinear resonators

We consider a generalization of a recently proposed non-Hermitian model for resonant cavities coupled by a chiral mirror by taking into account number non-conservation and nonlinear interactions. We analyze non-Hermitian quantum dynamics of populations and entanglement of the cavity modes. We find that the interplay between initial coherence and non-Hermitian coupling leads to a counterintuitive population transfer. While an initially coherent cavity mode is depleted, the other empty cavity can be populated more than or less than the initially filled one. Moreover, the presence of nonlinearity yields population collapse and revival as well as bipartite entanglement of the cavity modes. In addition to coupled cavities, we point out that similar models can be found in -symmetric Bose-Hubbard dimers of Bose-Einstein condensates or in coupled soliton-plasmon waveguides. We specifically illustrate the quantum dynamics of populations and entanglement in a heuristic model that we propose for a soliton-plasmon system with soliton amplitude-dependent asymmetric interaction. Degree of asymmetry, nonlinearity and coherence are examined to control plasmon excitations and soliton-plasmon entanglement. Relations to -symmetric lasers and Jahn-Teller systems are pointed out.

Non-Hermitian exciton dynamics in a photosynthetic unit system

The non-Hermitian quantum dynamics of excitonic energy transfer in photosynthetic systems is investigated using a dissipative two-level dimer model. The approach is based on Green's function formalism which permits consideration of decoherence and intersite transfer processes on comparable terms. The results indicate a combination of coherent and incoherent behavior at higher temperatures with the possibility of exceptional points occurring at the coherent-incoherent crossover regime at critical temperatures. When each dimer site is coupled equally to the environmental sources of dissipation, the excitonic wavepacket evolves with time with a coherent component, which can be attributed to the indistinguishability of the sources of dissipation. The time evolution characteristics of the B850 Bchls dimer system is analysed using typical parameter estimates in photosynthetic systems, and the quantum brachistochrone passage times are obtained for a range of parameters.

Classical limit of non-Hermitian quantum dynamics—a generalized canonical structure

We investigate the classical limit of non-Hermitian quantum dynamics arising from a coherent state approximation, and show that the resulting classical phase space dynamics can be described by generalized ‘canonical’ equations of motion, for both conservative and dissipative motion. The dynamical equations combine a symplectic flow associated with the Hermitian part of the Hamiltonian with a metric gradient flow associated with the anti-Hermitian part of the Hamiltonian. We derive this structure of the classical limit of quantum systems in the case of a Euclidean phase space geometry. As an example we show that the classical dynamics of a damped and driven oscillator can be linked to a non-Hermitian quantum system, and investigate the quantum classical correspondence. Furthermore, we present an example of an angular momentum system whose classical phase space is spherical and show that the generalized canonical structure persists for this nontrivial phase space geometry.

Adiabatic approximation in open systems: an alternative approach

The adiabatic approximation in open systems is formulated through the effective Hamiltonian approach. By introducing an ancilla, we embed the open system dynamics into a non-Hermitian quantum dynamics of a composite system composed of the open system and ancilla, the adiabatic evolution of the open system is then defined as the adiabatic dynamics of the composite system. Validity and invalidity conditions for this approximation as well as the relation to the other definition are established and discussed. A high-order adiabatic approximation for open systems is introduced. As an example, the adiabatic condition for an open spin- particle in time-dependent magnetic fields is analysed.

Non-Hermitian quantum dynamics.

An algorithm previously obtained for complex-energy modal probabilities in non-Hermitian quantum dynamics is generalized to determine expectation values for arbitrary observables. When cast in conventional form, the formalism including the generalized algorithms appears as a natural generalization of conventional quantum theory, in that conventional quantum theory obtains in the continuous Hermitian limit.

Non-Hermitian quantum theory of multiphoton ionization

The construction of physical occupation and transition probabilities in non-Hermitian, effective-Hamiltonian models of coupled near-resonant discrete states which decay to a continuum is analyzed. We consider for definiteness the particular physical example of atomic multiphoton ionization. The possibility of finite duration and arbitrary modulation of this interaction, and of the subsequent ionization and/or decay, in such a system allows us to invoke switching considerations in conventional $S$-matrix theory. The effective Hamiltonian $\stackrel{^}{H}(t)$ is derived for both stationary and time-dependent Schr\odinger equations. These derivations yield the same effective $\stackrel{^}{H}(t)$, and further reveal that this operator is associated with discrete-state-projected scattering states. The Hermitian conjugate operator ${\stackrel{^}{H}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)$ is similarly shown to be associated with discrete-state scattering states. This shows that effective-Hamiltonian theories are intrinsically $S$-matrix theories. This fact, in turn, is employed to construct transition amplitudes. The possibility of resonance in the atom-field interaction requires that both the projected incoming states and projected outgoing states be employed in this construction. The projected incoming and outgoing states in the (discrete) bound space are quite conveniently described by effective time-dependent Schr\odinger equations, supplemented by initial- and final-state boundary conditions. The fact of independent-exponential decay from each mode throughout the history of the interaction, for an arbitrary initial superposition state, in the adiabatic limit, suggests a practical construction for intermediate-time bound-state probabilities. This construction permits the formal definition of individual-state probabilities, which satisfy a generalized adiabatic theorem, and leads to the satisfying result that the total bound-state probability at all intermediate times is the sum of complex-mode probabilities. In marked contrast to the conventional norm-of-state definition of nonionization probability, this sum does not have oscillations at intermediate times, in the limit of adiabatic modulation of the interaction. Superposition-state probabilities, however, exhibit oscillations at intermediate times. The resulting non-Hermitian quantum dynamics is especially suited, and even essential, for an accurate description of near-resonance ionization. Precise matching of the time-dependent $\stackrel{^}{H}(t)$ and ${\stackrel{^}{H}}^{\ifmmode\dagger\else\textdagger\fi{}}(t)$ with bases of their instantaneous eigenstates is required by approximate unitarity in the discrete decaying space, for a consistent theory. We illustrate these considerations in a general way in the context of resolvent-operator techniques. The implementation of the theory in both its time-dependent and stationary-state formulations is presented in an Appendix, for the completely general two-state non-Hermitian Hamiltonian. The results of these formulations are found to agree, in the adiabatic limit, to all orders in the non-Hermitian interaction. The utility of direct diagonalization of the effective $\stackrel{^}{H}(t)$, advocated by Armstrong and Baker, is transparent. The overall context of interpretation of their earlier work is, however, significantly altered. Similar considerations apply for nonswitchable interactions, such as for $K$-meson decays. These results represent a practical generalization of quantum mechanics to non-Hermitian systems. The utility, and the necessity, of such a generalization has considerable theoretical interest, and direct experimental implications.