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arXiv:2412.13674

Manifolds of exceptional points and effective Zeno limit of an open two-qubit system

Vladislav Popkov, Carlo Presilla A, Mario Salerno

Published Dec 18, 2024• 0 citations
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Abstract

We analytically investigate the Liouvillian exceptional point manifolds (LEPMs) of a two-qubit open system, where one qubit is coupled to a dissipative polarization bath. Exploiting a Z_2 symmetry, we block-diagonalize the Liouvillian and show that one symmetry block yields two planar LEPMs while the other one exhibits a more intricate, multi-sheet topology. The intersection curves of these manifolds provide a phase diagram for effective Zeno transitions at small dissipation. These results are consistent with a perturbative extrapolation from the strong Zeno regime. Interestingly, we find that the fastest relaxation to the non-equilibrium steady state occurs on LEPMs associated with the transition to the effective Zeno regime.

DOI
10.48550/arxiv.2412.13674

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We analytically investigate the Liouvillian exceptional point manifolds (LEPMs) of a two-qubit open system, where one qubit is coupled to a dissipative polarization bath. Exploiting a Z_2 symmetry, we block-diagonalize the Liouvillian and show that one symmetry block yields two planar LEPMs while the other one exhibits a more intricate, multi-sheet topology. The intersection curves of these manifolds provide a phase diagram for effective Zeno transitions at small dissipation. These results are consistent with a perturbative extrapolation from the strong Zeno regime. Interestingly, we find that the fastest relaxation to the non-equilibrium steady state occurs on LEPMs associated with the transition to the effective Zeno regime.
Exceptional points in non-Hermitian quantum systems give rise to novel genuine quantum phenomena. Recent explorations of exceptional-point-induced quantum phase transitions have extended from discrete-variable to continuous-variable-encoded quantum systems. However, quantum phase transitions driven by Liouvillian exceptional points (LEPs) in continuous-variable platforms remain largely unexplored. Here, we construct and investigate a Liouvillian exceptional structure based on a driven-dissipative Kerr-cat qubit. Through numerical simulations, we reveal a quantum phase transition occurring at the LEP characterized by a sudden change in dynamical behavior from underdamped oscillations to overdamped relaxations as visualized via Wigner functions and Bloch sphere trajectories. Notably the negativity of the Wigner function serves as a direct signature of genuine quantum coherence unattainable in conventional single-qubit non-Hermitian systems. Furthermore, we introduce the phase difference between the off-diagonal elements of the Liouvillian eigenmatrices as a novel parameter to quantify the transition. Our results establish the Kerr-cat qubit as a novel continuous-variable setting for exploring dissipative quantum criticality and intrinsic non-Hermitian physics.
Abstract Exceptional points in non-Hermitian quantum systems give rise to genuine quantum phenomena. Recent explorations of exceptional-point-induced phase transitions have extended from discrete-variable to continuous-variable-encoded quantum systems. However, phase transitions driven by Liouvillian exceptional points (LEPs) in continuous-variable platforms remain largely unexplored. Here, we construct and investigate a Liouvillian exceptional structure based on a driven-dissipative Kerr-cat qubit. Through numerical simulations, we numerically reveal a non-Hermitian phase transition occurring at the LEP characterized by a sudden change in dynamical behavior from underdamped oscillations to overdamped relaxations as visualized via Wigner functions and Bloch sphere trajectories. The dynamical pattern of Wigner interference fringes provides a phase-space signature of the LEP-induced phase transition, while Wigner negativity certifies the genuine quantum nonclassicality of the system. Our results establish the Kerr-cat qubit as a continuous-variable setting for exploring dissipative quantum criticality and intrinsic non-Hermitian physics.
Non-Hermitian (NH) systems can display exotic topological phenomena without Hermitian counterparts, enabled by exceptional points (EPs). So far, investigations of NH topology have been restricted to EPs of the NH Hamiltonian, which governs the system dynamics conditional upon no quantum jumps occurring. The Liouvillian superoperator, which combines the effects of quantum jumps with NH Hamiltonian dynamics, possesses EPs (LEPs) that are significantly different from those of the corresponding NH Hamiltonian. We here study the topological features of the LEPs in the system consisting of a qubit coupled to a non-Markovian reservoir. We find that two distinct winding numbers can be simultaneously produced by executing a single closed path encircling the twofold LEP2, formed by two coinciding LEP2s, each involving a pair of coalescing eigenvectors of the extended Liouvillian superoperator. We experimentally demonstrate this purely non-Markovian phenomenon with a circuit, where a superconducting qubit is coupled to a decaying resonator which acts as a reservoir with memory effects. The results push the exploration of exceptional topology from the Markovian to non-Markovian regime.
We consider an open quantum system with dissipation, described by a Lindblad Master equation (LME). For dissipation locally acting and sufficiently strong, a separation of the relaxation timescales occurs, which, in terms of the eigenvalues of the Liouvillian, implies a grouping of the latter in distinct vertical stripes in the complex plane at positions determined by the eigenvalues of the dissipator. We derive effective LME equations describing the modes within each stripe separately, and solve them perturbatively, obtaining for the full set of eigenvalues and eigenstates of the Liouvillian explicit expressions correct at order 1/Γ included, where Γ is the strength of the dissipation. As an example, we apply our general results to quantum XYZ spin chains coupled, at one boundary, to a dissipative bath of polarization.
The coupling between self-sustained oscillators, such as lasers or polariton condensates, leads to different steady-states in different regimes. Weak coupling allows the oscillators to behave independently, with different frequencies and uncorrelated phases, while stronger coupling can produce synchronization. In classical or semiclassical theories transitions between these steady-states correspond to changes in the topology of the phase space attractors and occur at generalized exceptional points. We investigate the corresponding changes in a quantum theory of coupled lasers or condensates. We show how the different steady-state regimes of the classical limit give rise to distinct forms for the spectra and eigenmatrices of the slow modes of the Liouvillian. By following the spectral flow between the different regimes we show that the synchronization transition is controlled by cascades of exceptional points in these slow modes, protected by a generalized PT symmetry. Our results show how singularities of the quantum Liouvillian control the different dynamical regimes, and give rise to experimental signatures of the synchronization transition which remain well-defined in the few-particle regime dominated by quantum effects.