Liouvillian EPs in Open Quantum Systems

Updated 17 October 2025

Liouvillian Exceptional Points (EPs) are non-Hermitian degeneracies in open quantum systems where eigenvalues and eigenmatrices coalesce into a Jordan block structure.
They induce critical phenomena such as non-exponential relaxation, chiral state transfer, and accelerated dissipation, which are pivotal for quantum sensing and control.
Analytical tools like the Newton polygon and tropical geometry enable precise determination of EP order and scaling, bridging theory with experimental observations.

Liouvillian Exceptional Points (EPs) are non-Hermitian degeneracies that arise in the quantum dynamical generators—the Liouvillian superoperators—governing open quantum systems. Distinguished from exceptional points in non-Hermitian Hamiltonians, Liouvillian EPs originate from the interplay of coherent evolution and dissipative quantum jumps encoded in the Lindblad framework. Their paper has elucidated dynamical phase transitions, critical phenomena, and universal features unique to systems evolving under irreversible quantum processes.

1. Mathematical Definition and Core Properties

A Liouvillian exceptional point occurs at parameter values where two or more eigenvalues of the Liouvillian superoperator coincide and, crucially, their corresponding right and left eigenmatrices coalesce, rendering the operator non-diagonalizable and necessitating a Jordan block structure for its spectral decomposition (Minganti et al., 2019). Consider the general Lindblad master equation for the quantum density matrix $\rho$ : $\frac{d\rho}{dt} = \mathcal{L}[\rho] = -i[\hat{H}, \rho] + \sum_{\mu} \left( \hat{L}_\mu \rho \hat{L}_\mu^{\dagger} - \frac{1}{2} \{\hat{L}_\mu^{\dagger} \hat{L}_\mu, \rho\} \right)$ where $\hat{H}$ is the Hamiltonian, $\hat{L}_\mu$ are Lindblad jump operators, and $\mathcal{L}$ is the Liouvillian.

The eigenproblem is

$\mathcal{L}(\rho_n) = \lambda_n \rho_n\,,\qquad \mathcal{L}^\dagger(\sigma_n) = \lambda_n^* \sigma_n$

At an EP, not only do two (or more) eigenvalues $\lambda_n$ merge, but also the eigenmatrices $\rho_n$ and $\sigma_n$ become linearly dependent; generalized eigenmatrices must be constructed to form a complete basis. The algebraic multiplicity at the EP exceeds the geometric multiplicity, and the dynamics exhibit non-trivial polynomial prefactors atop the usual exponential behavior.

Key EP signatures in Liouvillians include:

Jordan block structure in the superoperator spectrum
Polynomial (secular) corrections to dissipative decay
Heightened sensitivity in transient dynamics
Absence of EPs at steady state due to trace constraints—defective zero eigenvalues are forbidden (Minganti et al., 2019)

2. Liouvillian vs. Hamiltonian Exceptional Points

While Hamiltonian EPs (HEPs) are defined for non-Hermitian effective Hamiltonians—often arising in “no-jump” or semiclassical approximations—the full quantum Liouvillian always incorporates quantum jump processes. The fundamental distinction is that HEPs may be realized by postselecting on quantum trajectories without jumps (or in the limit of negligible jump rates), whereas LEPs reflect the complete stochastic/irreversible evolution (Minganti et al., 2020).

Theoretical analysis demonstrates:

The spectrum of the full Liouvillian often hosts EPs at different parameter values than those predicted by the non-Hermitian Hamiltonian.
LEPs and HEPs can become equivalent only in semiclassical (quantum-jump-free or “dark state”) limits (Minganti et al., 2019, Minganti et al., 2020).
The order of the Liouvillian EP generally exceeds that of the Hamiltonian EP. For instance, a second-order HEP frequently yields a third-order LEP (Wiersig, 2020).

A central theorem is that LEPs and HEPs generically differ in the quantum regime; only for dark (jump-annihilated) states or under perfect postselection can a mapping be constructed (Minganti et al., 2019).

3. Spectral Structure, Higher-Order EPs, and Probes

Owing to the enlarged space of density matrices for the Liouvillian, LEPs of any finite or even infinite order (in continuous-variable, bosonic spaces) are possible (Arkhipov et al., 2020). The order and scaling behavior of these EPs are accessible using two analytical tools (P et al., 9 Oct 2025):

Newton Polygon Method: Construction of the Newton polygon from the coefficients in the characteristic polynomial for $\mathcal{L}$ identifies the Puiseux expansion and the fractional scaling exponents $(1/N)$ indicative of an order- $N$ EP. Each nonvertical segment in the convex hull designates a cluster of eigenvalues whose leading dependence on the perturbation scales with a specific fractional power.
Tropical Geometry (Amoebas): The amoeba construction maps the zero locus of the characteristic polynomial under logarithmic projection, with “tentacles” in the amoeba geometry corresponding to the scaling behaviors and exchange of eigenvalues upon parameter encirclement.

In dissipative bosonic systems, EPs associated with higher-order moments are revealed via correlation functions. For example, if the non-Hermitian Hamiltonian admits an EP of order $n$ , the Liouvillian can exhibit an EP of order $m\geq 2k(n-1)+1$ for $k\in \{\frac{1}{2},1,2,3,\dots\}$ (Arkhipov et al., 2020). The scaling of eigenvalue splitting $\Delta\lambda \sim \epsilon^{1/(2k)}$ is encoded in spectral and coherence measurements.

4. Physical Consequences: Dynamics, Chiral State Transfer, and Universal Behavior

LEPs influence open-system dynamics markedly:

Non-exponential Relaxation: At an EP, dynamics exhibit polynomial (e.g., linear or quadratic) temporal prefactors in $\rho(t)$ : $[1 + \alpha t + \beta t^2 + \dots]e^{\lambda t}$ , as seen in critically damped regimes (Khandelwal et al., 2021, Tay, 2023).
Critical Damping and Fastest Relaxation: At the EP, systems demonstrate the fastest possible relaxation (critical damping) to the steady state, a phenomenon robust even beyond the Markovian approximation (Khandelwal et al., 2021, Tay, 2023, Khandelwal et al., 12 Sep 2024).
Transient Chirality: Encircling an EP parametrically leads to non-reciprocal (chiral) state transfer, where the final outcome depends on the encircling direction (Sun et al., 21 Aug 2024, Gao et al., 17 Jan 2025). The chirality is preserved at intermediate times but ultimately erased by steady-state relaxation, revealing the uniquely transient nature of LEP-induced non-Hermitian topology.

A table summarizes some distinctive dynamic signatures:

Feature	Hamiltonian EP	Liouvillian EP (LEP)
Spectrum	Non-Hermitian matrix	Superoperator on density matrices
Quantum jumps?	Neglected	Included
Jordan structure	Yes (eigenvector coalescence)	Yes (eigenmatrix coalescence)
Steady state	Can be defective	Never defective (trace constraint)
Experimental probe	Post-selection, no jumps	Tomography, full evolution, quantum trajectories

LEPs govern the scaling and sensitivity characteristics relevant for quantum sensing—the eigenvalue splitting near an order- $N$ EP follows a $\propto \epsilon^{1/N}$ law, supporting applications in enhanced sensing and control (Wiersig, 2020, Gao et al., 17 Jan 2025).

5. Impact of Quantum Jumps, Hybrid Formalisms, and Environmental Effects

Quantum jumps—dissipative terms restoring population, decohering off-diagonal elements, or causing state repopulation—are the principal distinction between LEPs and HEPs. Inclusion of these terms significantly alters both the spectral structure and physical observables:

Hybrid Liouvillian Superoperators: By interpolating between the NHH limit ( $q=0$ , no jumps) and the full Liouvillian ( $q=1$ , all jumps included), the hybrid Liouvillian framework models the effects of imperfect detector efficiency or partial postselection (Minganti et al., 2020, Kopciuch et al., 3 Jun 2025). Even weak quantum jump contributions can split or shift predicted spectral degeneracies and EP orders.
Effect of Thermal Photons: For example, in circuit QED devices, thermal photons modify effective loss rates and shift HEPs, but LEPs remain set by the quantum-jump rates irrespective of thermal occupation, evidencing breakdown of correspondence in the presence of noise (Chimczak et al., 2023).
Markovian vs. Non-Markovian Dynamics: The strict identification of EPs in the Liouvillian spectrum holds in the Markovian approximation; non-Markovian memory effects destroy the simple exponential or polynomial decay structure needed to define conventional LEPs (Seshadri et al., 8 Jan 2024).

6. Experimental Observation and Practical Applications

Quantum process tomography is a powerful tool for reconstructing the Liouvillian and observing LEPs in real devices (Abo et al., 26 Jan 2024). Experiments on IBM quantum processors demonstrate the appearance of LEPs (coalescence of Liouvillian eigenvalues and eigenoperators) without any corresponding Hamiltonian EPs. Single-photon interferometric platforms have realized transient chiral state transfer by dynamically encircling a Liouvillian EP, and atomic platform experiments confirm LEP-induced state switching (Gao et al., 17 Jan 2025, Sun et al., 21 Aug 2024).

Practical implications include:

Enhanced and Accelerated Relaxation: LEPs can maximize the Liouvillian gap, speeding up dissipation—a feature exploited in optimal atomic cooling protocols and quantum state preparation (Zhou et al., 2023).
Topological and Sensing Phenomena: LEPs underlie chiral state transfer, topological protection of dynamical processes, and heightened sensitivity, especially relevant for quantum metrology and robust quantum control strategies (Wiersig, 2020, Sun et al., 21 Aug 2024).
Quantum Heat Engine Performance: Encircling a LEP produces a topological transition in the eigenenergy landscape, manifesting in work enhancement for quantum heat engines (Bu et al., 2023).

7. Theoretical Insights, Classification, and Design Principles

Recent advances leverage Newton polygon and tropical geometry techniques for rigorous order and scaling classification:

Scaling Behavior: The order and anisotropy of LEPs are encoded in the slopes of the non-vertical segments of the Newton polygon constructed from the characteristic polynomial of the “flattened” Liouvillian superoperator. For instance, a segment slope $-1/2$ signals a second-order (square-root) EP, while $-1/3$ indicates a third-order (cube-root) branch point (P et al., 9 Oct 2025).
Hybrid and Discrete Dynamics: LEPs are neither exclusive to continuous systems nor to Lindblad dynamics—they have been analytically identified in stroboscopic (discrete-time, brickwork) quantum circuits, and the sensing advantages persist under CPTP-mapped evolution (Popkov et al., 12 Oct 2025).
Robustness Beyond Markovian Approximations: Fundamental signatures of critical damping and non-diagonalizability at LEPs can be recovered even in the exact Heisenberg equation formalism, extending the concept well beyond conventional approximate treatments (Khandelwal et al., 12 Sep 2024).

A plausible implication is that engineering and exploiting specific orders and locations of LEPs—by the appropriate choice of Hamiltonian, dissipative channels, and perturbations—can be systematically realized via these algebraic and geometric methods.

In summary, Liouvillian Exceptional Points provide a universal framework for understanding non-Hermitian dynamics in open quantum systems, encompassing environmental effects, quantum jumps, and both continuous and discrete architectures. Their hierarchical and tunable spectral structure underpins a spectrum of phenomenology ranging from critical damping and dynamical phase transitions, through chiral transfer and enhanced sensing, to foundational consequences for dissipative many-body systems and quantum technological applications.