Quantum Dynamics in Noisy Backgrounds: from Sampling to Dissipation and Fluctuations

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.

In the previous chapter we studied the kinematics of many-body systems, and the form of operators representing the physical properties of a system, all of which were embodied by the quantum field. In this chapter we shall study the quantum dynamics of many-body systems, which can also be embodied by the quantum fields. We shall employ the fact that the quantum dynamics of a system, instead of being described in terms of the dynamics of the states or the evolution operator, i.e. as previously done through the Schrodinger equation, can instead be carried by the quantum fields. The quantum dynamics is therefore expressed in terms of the correlation functions or Green's functions of the quantum fields evaluated with respect to some state of the system. In particular we shall consider the general case of quantum dynamics for arbitrary non-equilibrium states. After introducing various types of Green's functions and relating them to measurable quantities, we will discuss the simplifications reigning for the special case of equilibrium states. Quantum dynamics Quantum dynamics can be described in different ways since quantum mechanics is a linear theory and the dynamics described by a unitary transformation of states. This will come in handy in the next chapter when we study a quantum theory in terms of its perturbative expansion using the so-called interaction picture. Here we first discuss the Schrodinger and Heisenberg pictures. The Schrodinger picture Having the Hamiltonian on the multi-particle space at hand we can consider the dynamics described in the multi-particle space.

We consider evolution of dynamical systems described by non-Hermitian Hamiltonians, using the density operator approach. The latter is formulated both at the level of the Hilbert space and the phase space, and adapted for applications to open quantum systems. We illustrate the formalism using a family of non-Hermitian system, which generators are quadratic with respect to both momentum and position. Despite the initial simplicity of a Hamiltonian, the structure of its solutions and spectral characteristics are nontrivial, and they can drastically change depending on parameters of the model and its symmetry in phase space. We present analytical solutions in L2(ℝ) and in phase space, and an explicit form of the similarity transformation changing these generators into the corresponding normal operators.

Searching for Anderson localization of light in three dimensions has challenged experimental and theoretical research for the last decades. Here the problem is analyzed through large-scale numerical simulations, using a radiative Hamiltonian, i.e., a non-Hermitian long-range hopping Hamiltonian, well suited to model light-matter interaction in cold atomic clouds. Light interaction in atomic clouds is considered in the presence of positional and diagonal disorder. Due to the interplay of disorder and cooperative effects (sub- and super-radiance) a novel type of localization transition is shown to emerge, differing in several aspects from standard localization transitions which occur along the real energy axis. The localization transition discussed here is characterized by a mobility edge along the imaginary energy axis of the eigenvalues which is mostly independent of the real energy value of the eigenmodes. Differently from usual mobility edges it separates extended states from hybrid localized states and it manifests itself in the large moments of the participation ratio of the eigenstates. Our prediction of a mobility edge in the imaginary axis, i.e., depending on the eigenmode lifetime, paves the way to achieve control both in the time and space domains of open quantum systems.

Motivated by recent progress on non-Hermitian topological band theories, we study the energy spectrum of a generic two-band non-Hermitian Hamiltonian. We prove rigorously that the complex energy spectrum of such a non-Hermitian Hamiltonian is restricted to the lower complex plane, provided that the parameters of the Hamiltonian satisfy a certain constraint. Furthermore, we consider one specific scenario where such a non-Hermitian Hamiltonian can arise, namely a two-band model coupled to an environment, and show that this aforementioned constraint originates from very general physical considerations. Using this construction we extract the real-space behaviour of the non-hermitian terms. Our findings are relevant in the definition of the energy gap in non-Hermitian topological band theories and also have implications on simulations of such theories using quantum systems.

In the present work, we suggest an approach for describing dynamics of finite-dimensional quantum systems in terms of pseudostochastic maps acting on probability distributions, which are obtained via minimal informationally complete quantum measurements. The suggested method for probability representation of quantum dynamics preserves the tensor product structure, which makes it favourable for the analysis of multi-qubit systems. A key advantage of the suggested approach is that minimal informationally complete positive operator-valued measures (MIC-POVMs) are easier to construct in comparison with their symmetric versions (SIC-POVMs). We establish a correspondence between the standard quantum-mechanical formalism and the MIC-POVM-based probability formalism. Within the latter approach, we derive equations for the unitary von-Neumann evolution and the Markovian dissipative evolution, which is governed by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) generator. We apply the MIC-POVM-based probability representation to the digital quantum computing model. In particular, for the case of spin-1/2 evolution, we demonstrate identifying a transition of a dissipative quantum dynamics to a completely classical-like stochastic dynamics. One of the most important findings is that the MIC-POVM-based probability representation gives more strict requirements for revealing the non-classical character of dissipative quantum dynamics in comparison with the SIC-POVM-based approach. Our results give a physical interpretation of quantum computations and pave a way for exploring the resources of noisy intermediate-scale quantum devices.

Krein-Hermitian Hamiltonians, i.e., Hamiltonians Hermitian with respect to an indefinite inner product, have emerged as an important class of non-Hermitian Hamiltonians in physics, encompassing both single-particle bosonic Bogoliubov-de Gennes (BdG) Hamiltonians and so-called "$PT$-symmetric" non-Hermitian Hamiltonians. In particular, they have attracted considerable scrutiny owing to the recent surge in interest for boson topology. Motivated by these developments, we formulate a perturbative Krein-unitary Schrieffer-Wolff transformation for finite-size dynamically stable Krein-Hermitian Hamiltonians, yielding an effective Hamiltonian for a subspace of interest. The effective Hamiltonian is Krein Hermitian and, for sufficiently small perturbations, also dynamically stable. As an application, we use this transformation to justify codimension-based analyses of band touchings in bosonic BdG Hamiltonians, which complement topological characterization. We use this simple approach based on symmetry and codimension to revisit known topological magnon band touchings in several materials of recent interest.

One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namely, that the eigenstates are exponentially localized at the boundary of the system. For open quantum systems, a short-time evolution can often be well described by the effective non-Hermitian Hamiltonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvillian superoperators generate time evolution. In this Letter, we find that Liouvillian superoperators can exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It is shown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such that the damping in a class of open quantum systems is algebraic under periodic boundary conditions but exponential under open boundary conditions. Moreover, the non-Hermitian skin effect and non-Bloch bands cause a chiral damping with a sharp wave front. These phenomena are beyond the effective non-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged open quantum dynamics.

In this paper we review an approach to the dynamics of open quantum systems based of non-Hermitian Hamiltonians. Non-Hermitian Hamiltonians arise naturally when one wish to study a subsystem interacting with a continuum of states. Moreover, quantum subsystems with probability sinks or sources are naturally described by non-Hermitian Hamiltonians. Herein, we discuss a non-Hermitian formalism based on the density matrix. We show both how to derive the equations of motion of the density matrix and how to define statistical averages properly. It turns out that the laws of evolution of the normalized density matrix are intrinsically non-linear. We also show how to define correlation functions and a non-Hermitian entropy with a non zero production rate. The formalism has been generalized to the case of hybrid quantum-classical systems using a partial Wigner representation. The equations of motion and the statistical averages are defined analogously to the pure quantum case. However, the definition of the entropy requires to introduce a non-Hermitian linear entropy functional.

Open systems, governed by non-Hermitian Hamiltonians, evolve fundamentally differently from their Hermitian counterparts and facilitate many unusual applications. Though non-Hermitian but parity-time (PT) symmetric dynamics has been realized in various classical or semiclassical systems, its fully quantum demonstration is still lacking. Here we engineer a highly controllable anti-Hermitian spin-boson model in a circuit QED structure composed of a decaying artificial atom (pseudospin) interacting with a bosonic mode stored in a microwave resonator. Besides observing abrupt changes in the spin-boson entanglement evolution and bifurcation transition in quantum Rabi splitting, we demonstrate super-sensitive quantum sensing by mapping the observable of interest to a hitherto unobserved PT-manifested entanglement evolution. These results pave the way for exploring non-Hermitian entanglement dynamics and PT-enhanced quantum sensing empowered by nonclassical correlations.

Digital quantum simulators provide a diversified tool for solving the evolution of quantum systems with complicated Hamiltonians and hold great potential for a wide range of applications. Although much attention is paid to the unitary evolution of closed quantum systems, dissipation and noise are vital in understanding the dynamics of practical quantum systems. In this work, we experimentally demonstrate a digital simulation of an open quantum system in a controllable Markovian environment with the assistance of a single ancillary qubit. By Trotterizing the quantum Liouvillians, the continuous evolution of an open quantum system is effectively realized, and its application in error mitigation is demonstrated by adjusting the simulated noise intensities. High-order Trotter for open quantum dynamics is also experimentally investigated and shows higher accuracy. Our results represent a significant step toward hardware-efficient simulation of open quantum systems and error mitigation in quantum algorithms in noisy intermediate-scale quantum systems.

Trace preserving quantum dynamics using a novel reparametrization-neutral summation-by-parts difference operator

Differential Equations with Applications to Mathematical Physics

We consider a quantum particle, moving on a lattice with a tight-binding Hamiltonian, which is subjected to measurements to detect its arrival at a particular chosen set of sites. The projective measurements are made at regular time intervals $\ensuremath{\tau}$, and we consider the evolution of the wave function until the time a detection occurs. We study the probabilities of its first detection at some time and, conversely, the probability of it not being detected (i.e., surviving) up to that time. We propose a general perturbative approach for understanding the dynamics which maps the evolution operator, which consists of unitary transformations followed by projections, to one described by a non-Hermitian Hamiltonian. For some examples of a particle moving on one- and two-dimensional lattices with one or more detection sites, we use this approach to find exact expressions for the survival probability and find excellent agreement with direct numerical results. A mean-field model with hopping between all pairs of sites and detection at one site is solved exactly. For the one- and two-dimensional systems, the survival probability is shown to have a power-law decay with time, where the power depends on the initial position of the particle. Finally, we show an interesting and nontrivial connection between the dynamics of the particle in our model and the evolution of a particle under a non-Hermitian Hamiltonian with a large absorbing potential at some sites.

There are many formalisms to describe quantum decoherence. However, many of them give a non-general and ad hoc definition of ‘pointer basis’ or ‘moving preferred basis’ , and this fact is a problem for the decoherence program. In this paper, we will consider quantum systems under a general theoretical framework for decoherence and we will present a tentative definition of the moving preferred basis. These ideas are implemented in a well-known open system model. The obtained decoherence and the relaxation times are defined and compared with those of the literature for the Lee–Friedrichs model.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.