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arXiv:2408.05270 · Physical review. B./Physical review. B

Topological transitions in quantum jump dynamics: Hidden exceptional points

Andrei I. Pavlov, Yuval Gefen, Alexander Shnirman

Published Mar 5, 2025• 15 citations
arXivHTMLPDF

Abstract

Complex spectra of dissipative quantum systems may exhibit degeneracies known as exceptional points (EPs). At these points the systems' dynamics may undergo drastic changes. Phenomena associated with EPs and their applications have been extensively studied in relation to various experimental platforms, including, i.e., the superconducting circuits. While most of the studies focus on EPs appearing due to the variation of the system's physical parameters, we focus on EPs emerging in the full counting statistics of the system. We consider a monitored three-level system and find multiple EPs in the Lindbladian eigenvalues considered as functions of a counting field. These “hidden” EPs are not accessible without the insertion of the counting field into the Linbladian, i.e., if only the density matrix of the system is studied. Nevertheless, we show that the hidden EPs are accessible experimentally. We demonstrate that these EPs signify transitions between different topological classes which are rigorously characterized in terms of the braid theory. Furthermore, we identify dynamical observables affected by these transitions and demonstrate how experimentally measured quantum jump distributions can be used to spot transitions between different topological regimes. Additionally, we establish a duality between the conventional Lindbladian EPs (zero counting field) and some of the “hidden” ones. Our findings allow for easier experimental observations of the EP transitions, normally concealed by the Lindbladian steady state, without applying postselection schemes. These results can be directly generalized to any monitored open system as long as it is described within the Lindbladian formalism.

DOI
10.1103/physrevb.111.104301

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Complex spectra of dissipative quantum systems may exhibit degeneracies known as exceptional points (EPs). At these points the systems' dynamics may undergo drastic changes. Phenomena associated with EPs and their applications have been extensively studied in relation to various experimental platforms, including, i.e., the superconducting circuits. While most of the studies focus on EPs appearing due to the variation of the system's physical parameters, we focus on EPs emerging in the full counting statistics of the system. We consider a monitored three-level system and find multiple EPs in the Lindbladian eigenvalues considered as functions of a counting field. These “hidden” EPs are not accessible without the insertion of the counting field into the Linbladian, i.e., if only the density matrix of the system is studied. Nevertheless, we show that the hidden EPs are accessible experimentally. We demonstrate that these EPs signify transitions between different topological classes which are rigorously characterized in terms of the braid theory. Furthermore, we identify dynamical observables affected by these transitions and demonstrate how experimentally measured quantum jump distributions can be used to spot transitions between different topological regimes. Additionally, we establish a duality between the conventional Lindbladian EPs (zero counting field) and some of the “hidden” ones. Our findings allow for easier experimental observations of the EP transitions, normally concealed by the Lindbladian steady state, without applying postselection schemes. These results can be directly generalized to any monitored open system as long as it is described within the Lindbladian formalism.
Non-Hermitian systems exhibit a variety of unique features rooted in the presence of exceptional points (EPs). The distinct topological structure in the proximity of an EP gives rise to counterintuitive behaviors absent in Hermitian systems, which emerge after encircling the EP either quasistatically or dynamically. However, experimental exploration of EP encirclement in quantum systems, particularly those involving high-order EPs, remains challenging due to the difficulty of coherently controlling more degrees of freedom. In this work, we experimentally investigate the eigenvalue braiding and state transfer arising from the encirclement of EP in a three-dimensional non-Hermitian quantum system using superconducting circuits. We characterize the second- and third-order EPs through the coalescence of eigenvalues. Then we reveal the topological structure near the EP3 by quasistatically encircling it along various paths with three independent parameters, which yields the eigenvalue braiding described by the braid group B 3 . Additionally, we observe chiral state transfer between three eigenstates under a fast driving scheme when no EPs are enclosed, while time-symmetric behavior occurs when at least one EP is encircled. Our findings offer insights into understanding non-Hermitian topological structures and the manipulation of quantum states through dynamic operations.
Exceptional points (EPs) are non-Hermitian degeneracies at which two or more eigenvalues and their corresponding eigenvectors coalesce. In open quantum systems, exceptional points can arise in the Lindbladian governing the dissipative dynamics. Their signatures have so far been mainly identified in finite-time observables, such as transient currents, while steady-state average currents generally provide no direct evidence of the underlying exceptional-point structure. In this work, we demonstrate that signatures of Lindbladian EPs can nevertheless be accessed in the steady-state regime through current noise. We derive general expressions for current correlation functions within a Lindblad master-equation framework and show, in particular, how exceptional points affect their behaviour as a function of the time delay. We illustrate these results with the paradigmatic example of two interacting qubits coupled to two reservoirs, where the steady-state noise clearly distinguishes overdamped, underdamped, and critical regimes. Our results establish current correlation functions as a steady-state probe of Lindbladian EPs in open quantum systems.
Exceptional points (EPs) are degeneracies of classical and quantum open systems, which are studied in many areas of physics including optics, optoelectronics, plasmonics, and condensed matter physics. In the semiclassical regime, open systems can be described by phenomenological effective non-Hermitian Hamiltonians (NHHs) capturing the effects of gain and loss in terms of imaginary fields. The EPs that characterize the spectra of such Hamiltonians (HEPs) describe the time evolution of a system without quantum jumps. It is well known that a full quantum treatment describing more generic dynamics must crucially take into account such quantum jumps. In a recent paper [F. Minganti et al., Phys. Rev. A 100, 062131 (2019)], we generalized the notion of EPs to the spectra of Liouvillian superoperators governing open system dynamics described by Lindblad master equations. Intriguingly, we found that in situations where a classical-to-quantum correspondence exists, the two types of dynamics can yield different EPs. In a recent experimental work [M. Naghiloo et al., Nat. Phys. 15, 1232 (2019)], it was shown that one can engineer a non-Hermitian Hamiltonian in the quantum limit by postselecting on certain quantum jump trajectories. This raises an interesting question concerning the relation between Hamiltonian and Lindbladian EPs, and quantum trajectories. We discuss these connections by introducing a hybrid-Liouvillian superoperator, capable of describing the passage from an NHH (when one postselects only those trajectories without quantum jumps) to a true Liouvillian including quantum jumps (without postselection). Beyond its fundamental interest, our approach allows to intuitively relate the effects of postselection and finite-efficiency detectors.
Open quantum systems can be approximately described by non-Hermitian Hamiltonians (NHHs) and Liouvillian superoperators. The two approaches differ by quantum jump terms corresponding to a measurement of the system by its environment. We analyze the emergence of exceptional points (EPs) in NHHs and Liouvillian superoperators. In particular, we show how EPs in NHHs relate to a novel type of EPs -- multi-block EPs -- in the no-jump Liouvillian, i.e. the Liouvillian superoperator in absence of quantum jump terms. We further analyze how quantum jump terms modify the multi-block structure. To illustrate our general findings, we present two prime examples: qubits and qutrits coupled to additional ground state levels that serve as sinks of the population. In those examples, we can navigate through the EP block structure by a variation of physical parameters. We analyze how the dynamics of the population of the states is affected by the order of the EPs. Additionally, we demonstrate that the quantum geometric tensor serves as a sensitive indicator of EPs of different kinds.
The appearance of topological singularities, namely exceptional points (EPs) is an intriguing feature of parameter-dependent open quantum or wave systems. EPs are the special type of nonHermitian degeneracies where two (or more) eigenstates of the underlying system coalesce. In this paper, we present a three level non-Hermitian Hamiltonian which hosts three interacting eigenstates. The matrix elements are optimized in such a way that the intermediate eigenstate interacts with both the other states and the underlying system hosts at least two different second order EPs. The impact of quasi-static parameter variation along a cyclic contour around the embedded EPs on the dynamics of interacting eigenvalues is well investigated in the context of cascaded state conversion. Such dynamics of the eigenvalues shows a clear signature of the third order EP with a combined effect of both the second order EPs. Moreover, we examine the accumulation of phases around the identified EPs and study the hallmark of phase exchange during cascaded state conversions accompanied by the parametric encirclement of the third order EP.
Exceptional points (EPs) are the spectral singularities and one of the central concepts of non-Hermitian physics, originating from the inevitable energy exchange with the surrounding environment. EPs exist in diverse physical systems and give rise to many counterintuitive effects, offering rich opportunities to control the dynamics and alter the properties of optical, electronic, acoustic, and mechanical states. The last two decades have witnessed the flourishing of non-Hermitian physics and associated applications related to coalesced eigenstates at EPs in a plethora of classical systems. While stemming from the quantum mechanism, the implementation of EPs in real quantum systems still faces challenges of tuning and stabilizing the systems at EPs, as well as the additional noises that hinder the observation of relevant phenomena. This review mainly focuses on summarizing the current efforts and opportunities offered by quantum EPs that result from or produce observable quantum effects. We introduce the concepts of Hamiltonian and Liouvillian EPs in the quantum regime and focus on their different properties in connection with quantum jumps and decoherence. We then provide a comprehensive discussion covering the theoretical and experimental advances in accessing EPs in diverse quantum systems and platforms. Special attention is paid to EP-based quantum-optics applications with state-of-art technologies. Finally, we present a discussion on the existing challenges of constructing quantum EPs at the nanoscale and an outlook on the fundamental science and applied technologies of quantum EPs, aiming to provide valuable insights for future research and building quantum devices with high performance and advanced functionalities.