We review analyses of open quantum systems. We show how non-Hermiticity arises in an open quantum system with an infinite environment, focusing on the one-body problem. One of the reasons for taking the present approach is that we can solve the problem completely, making it easier to see the structures of problems involving open quantum systems. We show that this results in the discovery of a new complete set, which is one of the main topics of the present article. Another reason for focusing on the one-body problem is that the theory permits the strong coupling between the system and the environment. In the current research landscape, it is valuable to revisit the one-body problem for open quantum systems, which can be solved accurately for arbitrary strengths of the system-environment couplings. A rigorous understanding of the problem structures in the present approach will be helpful when we tackle problems with many-body interactions. First, we consider potential scattering and directly define the resonant state as an eigenstate of the Schrödinger equation under the Siegert outgoing boundary condition. We show that the resonant eigenstate can have a complex energy eigenvalue, even though the Hamiltonian is seemingly Hermitian. Second, we introduce the Feshbach formalism, which eliminates the infinite degrees of freedom of the environment and represents its effect as a complex potential. The resulting effective Hamiltonian is explicitly non-Hermitian. By unifying these two ways of defining resonant states, we obtain a new complete set of bases for the scattering problem that contains all discrete eigenstates, including resonant states. We finally mention the non-Markovian dynamics of open quantum systems. We emphasize the time-reversal symmetry of the dynamics that continuously connects the past and the future. We can capture it using the new complete set that we develop here.
We review analyses of open quantum systems. We show how non-Hermiticity arises in an open quantum system with an infinite environment, focusing on the one-body problem. One of the reasons for taking the present approach is that we can solve the problem completely, making it easier to see the structures of problems involving open quantum systems. We show that this results in the discovery of a new complete set, which is one of the main topics of the present article. Another reason for focusing on the one-body problem is that the theory permits the strong coupling between the system and the environment. In the current research landscape, it is valuable to revisit the one-body problem for open quantum systems, which can be solved accurately for arbitrary strengths of the system-environment couplings. A rigorous understanding of the problem structures in the present approach will be helpful when we tackle problems with many-body interactions. First, we consider potential scattering and directly define the resonant state as an eigenstate of the Schrödinger equation under the Siegert outgoing boundary condition. We show that the resonant eigenstate can have a complex energy eigenvalue, even though the Hamiltonian is seemingly Hermitian. Second, we introduce the Feshbach formalism, which eliminates the infinite degrees of freedom of the environment and represents its effect as a complex potential. The resulting effective Hamiltonian is explicitly non-Hermitian. By unifying these two ways of defining resonant states, we obtain a new complete set of bases for the scattering problem that contains all discrete eigenstates, including resonant states. We finally mention the non-Markovian dynamics of open quantum systems. We emphasize the time-reversal symmetry of the dynamics that continuously connects the past and the future. We can capture it using the new complete set that we develop here.
We review the methodology to theoretically treat parity-time- (PT-) symmetric, non-Hermitian quantum many-body systems. They are realized as open quantum systems withPTsymmetry and couplings to the environment which are compatible.PT-symmetric non-Hermitian quantum systems show a variety of fascinating properties which single them out among generic open systems. The study of the latter has a long history in quantum theory. These studies are based on the Hermiticity of the combined system-reservoir setup and were developed by the atomic, molecular, and optical physics as well as the condensed matter physics communities. The interest of the mathematical physics community inPT-symmetric, non-Hermitian systems led to a new perspective and the development of the elegant mathematical formalisms ofPT-symmetric and biorthogonal quantum mechanics, which do not make any reference to the environment. In the mathematical physics research, the focus is mainly on the remarkable spectral properties of the Hamiltonians and the characteristics of the corresponding single-particle eigenstates. Despite being non-Hermitian, the Hamiltonians can show parameter regimes, in which all eigenvalues are real. To investigate emergent quantum many-body phenomena in condensed matter physics and to make contact to experiments one, however, needs to study expectation values of observables and correlation functions. One furthermore, has to investigate statistical ensembles and not only eigenstates. The adoption of the concepts ofPT-symmetric and biorthogonal quantum mechanics by parts of the condensed matter community led to a controversial status of the methodology. There is no consensus on fundamental issues, such as, what a proper observable is, how expectation values are supposed to be computed, and what adequate equilibrium statistical ensembles and their corresponding density matrices are. With the technological progress in engineering and controlling open quantum many-body systems it is high time to reconcile the Hermitian with thePT-symmetric and biorthogonal perspectives. We comprehensively review the different approaches, including the overreaching idea of pseudo-Hermiticity. To motivate the Hermitian perspective, which we propagate here, we mainly focus on the ancilla approach. It allows to embed a non-Hermitian system into a larger, Hermitian one. In contrast to other techniques, e.g. master equations, it does not rely on any approximations. We discuss the peculiarities ofPT-symmetric and biorthogonal quantum mechanics. In these, what is considered to be an observable depends on the Hamiltonian or the selected (biorthonormal) basis. Crucially in addition, what is denoted as an 'expectation value' lacks a direct probabilistic interpretation, and what is viewed as the canonical density matrix is non-stationary and non-Hermitian. Furthermore, the non-unitarity of the time evolution is hidden within the formalism. We pick up several model Hamiltonians, which so far were either investigated from the Hermitian perspective or from thePT-symmetric and biorthogonal one, and study them within the respective alternative framework. This includes a simple two-level, single-particle problem but also a many-body lattice model showing quantum critical behavior. Comparing the outcome of the two types of computations shows that the Hermitian approach, which, admittedly, is in parts clumsy, always leads to results which are physically sensible. In the rare cases, in which a comparison to experimental data is possible, they furthermore agree to these. In contrast, the mathematically elegantPT-symmetric and biorthogonal approaches lead to results which, are partly difficult to interpret physically. We thus conclude that the Hermitian methodology should be employed. However, to fully appreciate the physics ofPT-symmetric, non-Hermitian quantum many-body systems, it is also important to be aware of the main concepts ofPT-symmetric and biorthogonal quantum mechanics. Our conclusion has far reaching consequences for the application of Green function methods, functional integrals, and generating functionals, which are at the heart of a large number of many-body methods. They cannot be transferred in their established forms to treatPT-symmetric, non-Hermitian quantum systems. It can be considered as an irony of fate that these methods are available only within the mathematical formalisms ofPT-symmetric and biorthogonal quantum mechanics.
Abstract Every realistic quantum system is coupled to an environment. Including the environment restores a closed, Hermitian description, but at the cost of an intractably large Hilbert space. The Feshbach–Löwdin projection eliminates the unwanted degrees of freedom exactly, yielding an effective Hamiltonian H eff on the relevant subspace alone—which generically becomes non-Hermitian when the eliminated subspace contains an energetically accessible continuum and outgoing boundary conditions are imposed. We illustrate this through a minimal model: a particle in a box with a delta barrier at its center. Direct elimination of one half-well reduces all information about the discarded sector to a single boundary parameter β ; the same result is then rederived via the Green’s function and Dyson equation, offering a pedagogical illustration of how boundary data enter the Green’s-function formalism and relating β to a self-energy term. Whether β is real or complex determines whether H eff is Hermitian or not, and the full range of behavior—from Hermitian dynamics through finite-width resonances to the counterintuitive quantum Zeno effect—follows from this single parameter.
The study of open quantum systems forms one of the main problems of modern physics. This thesis aims at contributing to this field, and focuses on the investigation of non-Markovian dynamics of open quantum systems. Here, as a special case, quantum many-body systems provide for a rich field of research. With the ability to control them being of essential importance for the development of quantum technologies, the characterization of their dynamical properties has also drawn a lot of interest lately. One of the main methodological tasks in the study of open quantum systems is to reduce the infinite degrees of freedom of the total system. In order to achieve this, the thesis makes use of the tensor network method matrix product states and combines it with the picture of the quantum stochastic Schrödinger equation. However, when computing the dynamics of many-body systems of system size N, even the reduced system dynamics scale with 2N, making larger system sizes difficult to access. Therefore, especially when computing non-Markovian time-dynamics where the information backflow from the reservoir has to be taken into account, efficiently constructed algorithms are required. This thesis proposes these type of algorithms and demonstrates their application. They enable an access to larger system sizes, thus making it possible to investigate non-equilibrium steady-states of quantum many-body systems in a non-Markovian interaction for system sizes up to N=30. This thesis consists of three parts. The first part provides an introduction to the theory of open quantum systems and to the methodology for modeling them. The second part examines many-body system-reservoir interaction in the non-Markovian regime by investigating the case of fully coherent self-feedback. It uses the one-dimensional Heisenberg spin-1/2 chain as a paradigmatic and recently intensely studied many-body system. It is demonstrated that this gives rise to new conditions for the feedback phase and to highly non-trivial states within the system as well as to the possibility of partially characterizing the system non-invasively and storing excitation within it. Next, a novel approach to the boundary-driven Heisenberg chain is presented, where the chain is exposed to a coherent driving field, thus enabling a comparison to the well-studied incoherently driven model. The third part demonstrates the use of an external pump to control the feedback phase of a Lambda-type three-level system, where the external pump pulse is shown to be a new control parameter. The thesis concludes with the proposition to enhance the indistinguishability of two photons by feedback control.
Calculations for open quantum systems are performed usually by taking into account their embedding into one common environment, which is mostly the common continuum of scattering wave functions. Realistic quantum systems are coupled however often to more than one continuum. For example, the conductance of an open cavity needs at least two environments, namely the input and the output channel. In the present paper, we study generic features of an open quantum system coupled to two channels. We compare the results with those characteristic of a one-channel system. Of special interest is the parameter range which is influenced by singular (exceptional) points. Here, the states of the system are mixed via the environment and the dynamics is determined, in any case, by nonlinear processes. These processes restore the resonance structure of the cross section in the one-channel case, such that there is no need to consider them in calculations for concrete systems. In the two-channel case, however, they cause new effects. Most interesting is the enhanced conductance of an open cavity in this parameter range. This enhancement follows directly from the properties of the eigenstates of a non-Hermitian Hamilton operator (and is not related to the existence of a superradiant state). The study of the conductance allows, among others, an experimental test of the non-Hermitian quantum physics.
Non-Hermiticity plays a fundamental role in open quantum systems and describes a wide variety of effects of interactions with environments, including quantum measurement. However, understanding its consequences in strongly interacting systems is still elusive due to the interplay between non-perturbative strong correlations and non-Hermiticity. While the Wilsonian renormalization group (RG) method has been applied to tackle this problem, its foundation, based on the existence of the partition function, is ill-defined. In this paper, we establish a microscopic foundation of the non-Hermitian RG method from a few-body perspective. We show that the invariance of the scattering amplitude under RG transformations enables us to rigorously derive the non-Hermitian RG equation, giving a physically transparent interpretation of RG flows. We discuss a detailed structure of such RG flows in a non-relativistic two-body system with inelastic two-body loss, and show its relation to a non-Hermitian quantum scale anomaly. Our analysis suggests that non-Hermitian complex potentials often used in high-energy physics can be interpreted as being caused by quantum measurement, where the detection of elastically scattered particles updates the observer's knowledge, resulting in a nonunitary state change of the system. We apply our formalism to nuclear physics, find the emergence of a critical semicircle, and show that several nuclei are located near the critical semicircle in the coherent neutron-nucleus scattering. We also propose that the localized dineutron in two-neutron halo nuclei can be interpreted as the quantum measurement effect on the imaginary potential associated with absorption into the core nucleus. Our result bridges different contexts of non-Hermitian systems in high-energy and atomic, molecular, and optical physics, opening an interdisciplinary playground of non-Hermitian few-body physics.
It is one of the most important and long-standing issues of physics to derive the irreversibility out of a time-reversal symmetric equation of motion. The present paper considers the breaking of the time-reversal symmetry in open quantum systems and the emergence of an arrow of time. We claim that the time-reversal symmetric Schrödinger equation can have eigenstates that break the time-reversal symmetry if the system is open in the sense that it has at least a countably infinite number of states. Such eigenstates, namely the resonant and anti-resonant states, have complex eigenvalues. We show that, although these states are often called "unphysical", they observe the probability conservation in a particular way. We also comment that the seemingly Hermitian Hamiltonian is non-Hermitian in the functional space of the resonant and anti-resonant states, and hence there is no contradiction in the fact that it has complex eigenvalues. We finally show how the existence of the states that break the time-reversal symmetry affects the quantum dynamics. The dynamics that starts from a time-reversal symmetric initial state is dominated by the resonant states for t > 0 ; this explains the phenomenon of the arrow of time, in which the decay excels the growth. The time-reversal symmetry holds in that the dynamic ending at a time-reversal symmetric final state is dominated by the anti-resonant states for t < 0 .