Liouvillian Exceptional Point

• Liouvillian exceptional points are non-diagonalizable degeneracies in the Liouvillian superoperator, marked by coalescing eigenvalues and eigenoperators into Jordan blocks.
• They differ from Hamiltonian exceptional points by including quantum-jump recycling, which leads to richer spectral structures and altered dynamical responses.
• Experimental diagnostics such as process tomography and resolvent spectroscopy reveal their impact on relaxation rates, critical damping, and topological mode conversion.

Searching arXiv for papers on Liouvillian exceptional points to ground the article in published work. arXiv search query: "Liouvillian exceptional point" A Liouvillian exceptional point is a non-diagonalizable spectral degeneracy of the Liouvillian superoperator that generates open-system dynamics. In operator form, the density matrix obeys a master equation $\dot\rho=\mathcal L[\rho]$, and an LEP occurs when two or more Liouvillian eigenvalues coalesce simultaneously with their corresponding eigenoperators, so that the generator develops a Jordan block. Because $\mathcal L$ includes both coherent evolution and quantum-jump recycling terms, LEPs are generally inequivalent to exceptional points of an effective non-Hermitian Hamiltonian, and can differ in existence, location, order, and topology. Recent work places LEPs at the intersection of non-Hermitian spectral theory, dissipative quantum dynamics, open-system thermodynamics, and topological mode conversion, with extensions from standard Lindblad generators to non-Markovian pseudomode constructions and even discrete completely positive trace-preserving circuits (Minganti et al., 2020, Zhang et al., 6 Dec 2025, Popkov et al., 12 Oct 2025).

1. Formal definition and algebraic structure

For a time-independent Markovian open quantum system, the Liouvillian is defined through the Lindblad equation

$$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$

After vectorization, $\rho\mapsto |\rho\rangle\rangle$, the Liouvillian becomes a non-Hermitian matrix acting on Liouville space. One then solves

$$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$

with biorthogonal left and right eigenmodes. The complex eigenvalues determine decay rates and oscillation frequencies, while the zero mode is the steady state when it is unique (Molina, 1 Feb 2026, Minganti et al., 2020).

An LEP of order two occurs when $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$ and the corresponding eigenoperators coalesce, so that the algebraic multiplicity exceeds the geometric multiplicity. Equivalently, the relevant invariant subspace is described by a Jordan block

$$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$

or, in rank language, the kernel dimension drops below the algebraic multiplicity. In matrix formulations this is detected by simultaneous eigenvalue coincidence and defectiveness, often through a vanishing discriminant together with a rank condition (Zhang et al., 6 Dec 2025, Abo et al., 2024, Zhou et al., 13 May 2026).

The dynamical hallmark of defectiveness is the appearance of polynomial prefactors multiplying exponentials. At a second-order Jordan block one obtains terms of the form $t\,e^{\lambda_{\rm EP} t}$; at higher order, higher-degree polynomials appear. This structure is the Liouvillian analogue of critical damping in non-Hermitian dynamics and underlies the anomalous relaxation signatures associated with LEPs (Khandelwal et al., 2021, Khandelwal et al., 2024).

2. Relation to non-Hermitian Hamiltonian exceptional points

The effective non-Hermitian Hamiltonian

$$H_{\rm eff}=H-\frac{i}{2}\sum_\mu L_\mu^\dagger L_\mu$$

describes no-jump evolution. Its exceptional points are Hamiltonian exceptional points, defined purely at the conditional-wavefunction level. LEPs instead belong to the full Liouvillian and therefore retain the recycling terms $L_\mu \rho L_\mu^\dagger$. This enlargement from Hilbert space to operator space is not a minor technicality: the Liouvillian spectrum is generically richer, and the corresponding singularities need not track those of $\mathcal L$0 (Minganti et al., 2020, Sun et al., 2024).

A useful interpolation between the two descriptions is provided by the hybrid-Liouvillian formalism,

$\mathcal L$1

or equivalently by the jump-weight parameter $\mathcal L$2 in $\mathcal L$3. At $\mathcal L$4 one recovers the no-jump non-Hermitian evolution; at $\mathcal L$5 one obtains the full Liouvillian. Varying $\mathcal L$6 or $\mathcal L$7 shows explicitly that quantum jumps can move, split, remove, or create spectral degeneracies, and can also change their order (Minganti et al., 2020, Kopciuch et al., 3 Jun 2025).

Concrete models illustrate all of these possibilities. A driven-lossy qubit with $\mathcal L$8, $\mathcal L$9, and $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$0 has a diagonal effective Hamiltonian whose eigenvalues never coalesce, hence no HEPs, yet its Liouvillian exhibits second-order LEPs at $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$1 (Abo et al., 2024). In an effective atomic-vapor model, the no-jump sector can host an EP3, while turning on repopulation reduces the degeneracy to an EP2 or removes it entirely, thereby showing that the physical Liouvillian singularity can have a lower order than its NHH precursor (Kopciuch et al., 3 Jun 2025). These results establish that LEPs are not merely re-labeled HEPs, but singularities of the full quantum dissipative generator.

3. Dynamical consequences: relaxation, critical damping, and thermodynamic speedup

The long-time approach to stationarity is governed by the Liouvillian spectral gap. In a standard ordering,

$$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$2

depending on convention. Several exactly solvable or reduced models show that tuning to an LEP can maximize this gap and therefore minimize the characteristic relaxation time. In a three-level atomic model with coherent drive $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$3 and decay $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$4, the LEP occurs at $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$5, where the static gap reaches $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$6, and Floquet modulation of the dissipation can enlarge the effective gap beyond the static limit (Zhou et al., 2023). In a trapped-ion quantum-battery model, the reduced slow Liouvillian block develops an EP when the cubic discriminant vanishes, and the spectral gap $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$7 reaches a local maximum exactly at that defective degeneracy, minimizing the charging time $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$8 (Zhou et al., 13 May 2026).

Higher-order LEPs generate critical-damping-type dynamics. In the two-qubit thermal machine analyzed by Khandelwal, Brunner, and Haack, a third-order LEP occurs when $$\frac{d\rho}{dt} = \mathcal L[\rho] = -\,i[H,\rho] + \sum_\mu \Bigl( L_\mu \rho L_\mu^\dagger - \frac12\{L_\mu^\dagger L_\mu,\rho\} \Bigr).$$9. At that point three eigenvalues merge at $\rho\mapsto |\rho\rangle\rangle$0, the corresponding eigenmatrices coalesce, and the exact solution acquires both $\rho\mapsto |\rho\rangle\rangle$1 and $\rho\mapsto |\rho\rangle\rangle$2 terms. The resulting relaxation is the open-system analogue of a critically damped oscillator and is asymptotically faster than the overdamped nonoscillatory alternatives considered in that model (Khandelwal et al., 2021). Closely related behavior survives beyond the conventional master-equation regime: in a dissipative double quantum dot, the exact Heisenberg-equation generator and the corresponding master-equation Liouvillian share the same square-root structure and the same critical coupling $\rho\mapsto |\rho\rangle\rangle$3, and the critical-damping signature persists beyond Born-Markov validity (Khandelwal et al., 2024).

These spectral effects have direct thermodynamic and control implications. Encircling an LEP in a single-ion quantum heat engine yields positive net work, with larger net work obtained closer to the LEP, while LEP conditions in sideband and EIT cooling reproduce analytically optimal detuning and Rabi-frequency conditions for fast cooling (Bu et al., 2023, Zhou et al., 2023). The common mechanism is dynamical rather than static: spectral reorganization of slow Liouvillian modes changes the route to the steady state.

4. Encircling, chirality, and non-Markovian topology

Parameter-space encircling of LEPs generalizes the branch-point physics of non-Hermitian Hamiltonians to open-system superoperators. For a time-dependent Liouvillian $\rho\mapsto |\rho\rangle\rangle$4, the propagator is

$\rho\mapsto |\rho\rangle\rangle$5

Near a second-order LEP, the instantaneous spectral sheets are joined by a square-root branch cut, and slow encircling can induce directional mode conversion among decay modes. In Liouvillian settings, however, this chirality is typically transient because the steady state eventually dominates the dynamics (Sun et al., 2024).

This transient nature was made explicit in single-photon interferometric simulations of open-system dynamics. In that work, clockwise and counterclockwise encircling of a Liouvillian EP produce different final states only for intermediate encircling times. When dephasing opens a nonzero Liouvillian gap, sufficiently long protocols drive both directions back to the same instantaneous steady-state sheet, and the chirality vanishes. The experiment further reported a scaling collapse of the chirality $\rho\mapsto |\rho\rangle\rangle$6 as a function of $\rho\mapsto |\rho\rangle\rangle$7, with fitted exponent $\rho\mapsto |\rho\rangle\rangle$8 (Gao et al., 17 Jan 2025). This result sharpens an important distinction: LEP encircling can control transient decay-mode populations even when it does not alter the ultimate steady state.

A qualitatively different topological structure arises in the non-Markovian pseudomode construction of a qubit coupled to a reservoir with memory. There the extended Liouvillian

$\rho\mapsto |\rho\rangle\rangle$9

hosts two coinciding LEP2s at $$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$0. For a closed loop

$$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$1

the generalized winding number

$$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$2

takes two distinct values on a single encircling path: $$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$3 This “hybrid” invariant was measured in a superconducting circuit where a transmon was sideband-coupled to a decaying resonator acting as a Lorentzian reservoir. Repeating the experiment at large $$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$4 removed the distinction, so the dual half-integer topology was identified as a purely non-Markovian effect (Zhang et al., 6 Dec 2025).

5. Higher-order LEPs, asymptotic exceptional steady states, and geometric methods

LEPs are not restricted to simple EP2 scenarios. In a driven two-level Liouvillian with decay and dephasing,

$$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$5

the $$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$6 Liouvillian matrix has one steady eigenvalue and three nontrivial modes. A third-order LEP occurs when

$$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$7

so that all three nonzero modes coalesce. Because the pure-decay and pure-dephasing Liouvillians do not commute, the LEP position moves with the mixing parameter $$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$8, and at $$\mathcal L |r_\mu\rangle\rangle = \lambda_\mu |r_\mu\rangle\rangle, \qquad \langle\langle \ell_\mu|\mathcal L = \lambda_\mu \langle\langle \ell_\mu|,$$9 the EP is pushed to infinity (Wu et al., 1 Dec 2025). This is a direct illustration of the sensitivity of high-order LEPs to the structure, not just the magnitude, of dissipation.

In linear bosonic and continuous-variable settings, the possible order can be much larger. For dissipative linear bosonic systems, an $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$0th-order HEP of the effective non-Hermitian Hamiltonian implies LEPs of order $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$1 in the Liouvillian, and steady-state coherence functions diagnose them through non-exponential time dependence and higher-power Lorentzian lineshapes: squared Lorentzians in power spectra and cubic Lorentzians in intensity-fluctuation spectra were identified for second- and third-order signatures in a bimodal cavity with incoherent coupling (Arkhipov et al., 2020). For a damped harmonic oscillator in a generic environment, the LEP occurs when the modified oscillator frequency vanishes, $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$2, corresponding to critical damping in Caldeira-Leggett and Markovian Hu-Paz-Zhang equations or to an infinite-effective-mass limit in a modified Kossakowski-Lindblad equation. At the EP, each fixed-$\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$3 sector exhibits an $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$4-fold degeneracy and a single Jordan block of order $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$5 (Tay, 2023).

Exactly solvable fermionic models show a complementary many-body route to high-order LEPs. In a class of quadratic open fermion models treated by third quantization, the Liouvillian develops an EP whose order approaches system size and appears as a quasisteady state with gapless spectrum. Adding many-body quantum-jump perturbations lifts that degeneracy and opens a finite Liouvillian gap with fractional power-law scaling whose exponent depends sensitively on the perturbation form (Xu et al., 31 Jan 2026). Discrete-time open dynamics can also host LEPs: in two-qubit brickwork CPTP circuits, the stroboscopic transfer matrix develops a Jordan block at a surface defined by the vanishing of an explicit square root $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$6, and the resulting time-domain signal acquires an $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$7 factor in the defective subspace (Popkov et al., 12 Oct 2025).

The geometry of LEP splitting has also been systematized algebraically. Newton polygons and tropical geometry encode the Puiseux exponents of eigenvalue branches near a defective point. A polygon edge of slope $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$8 signals the leading $\lambda_1=\lambda_2\equiv\lambda_{\rm EP}$9 splitting of an $$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$0th-order EP, while multiple lower faces reveal anisotropic or hybrid splittings, such as coexistence of linear and cube-root scaling or square-root splitting with frozen modes, depending on the perturbation direction (P et al., 9 Oct 2025). At the opposite end of the spectrum, zero modes obey a no-go theorem in finite dimension: steady-state eigenspaces at physical jump strength are semisimple. Even so, dissipative many-body systems can approach an “asymptotic exceptional steady state” in the thermodynamic limit, with $$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$1 and normalized overlap $$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$2 between the steady state and first metastable mode, exhibiting exponential scaling in a dissipative 3SAT solver and polynomial scaling in AKLT-type symmetry-protected topological state preparation (Hu et al., 3 Apr 2025).

6. Diagnostics, experimental platforms, and conceptual boundaries

LEPs are diagnosed by reconstructing or probing the Liouvillian rather than only the conditional Hamiltonian. Quantum process tomography reconstructs the short-time transfer matrix $$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$3 and hence the experimental Liouvillian; resolvent spectroscopy identifies higher-order poles and super-Lorentzian lineshapes; coherence functions reveal Jordan-block-induced polynomial decay; and full state tomography of reduced or extended systems can track the complex Liouvillian eigenvalues along parameter loops (Abo et al., 2024, Molina, 1 Feb 2026).

Platform	Diagnostic	Key LEP feature
IBM superconducting processor	Quantum process tomography	LEPs without any HEP (Abo et al., 2024)
Superconducting qubit + decaying resonator	Joint-state tomography in extended space	Non-Markovian twofold LEP2 with $$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$4 winding (Zhang et al., 6 Dec 2025)
Single trapped ion	Time-resolved tomography and parameter scans	Heat-engine enhancement and third-order LEPs (Bu et al., 2023, Wu et al., 1 Dec 2025)
Collective spin with polarized bath	Frequency-resolved resolvent spectroscopy	Super-Lorentzian response and state-dependent visibility (Molina, 1 Feb 2026)

Spectroscopy introduces additional subtleties. In the collective-spin model, the emission spectrum

$$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$5

contains both simple and second-order poles near an EP, leading to a super-Lorentzian correction

$$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$6

However, weak $$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$7 symmetry sectors impose selection rules, and the overlap of the source state with the Jordan chain can strongly suppress the $$\mathcal J_2(\lambda_{\rm EP}) = \begin{pmatrix} \lambda_{\rm EP} & 1\ 0 & \lambda_{\rm EP} \end{pmatrix},$$8 term. As a result, steady-state fluorescence may conceal an EP that becomes unambiguous for infinite-temperature or random initial states (Molina, 1 Feb 2026). This state dependence is a recurrent theme in LEP diagnostics.

The conceptual status of LEPs outside time-local descriptions remains unsettled. One line of work argues that exact non-Markovian dynamics, as described by NEGF Dyson equations or non-Markovian master equations with memory kernels, does not admit a fixed finite-dimensional Liouvillian matrix and therefore does not allow the introduction of exceptional points in the strict spectral sense (Seshadri et al., 2024). Another line constructs non-Markovian LEPs in an extended Liouvillian acting on system-plus-pseudomode space and reports experimentally accessible non-Markovian topology in that enlarged generator (Zhang et al., 6 Dec 2025). The literature therefore contains two distinct viewpoints: LEPs as singularities of a time-local generator, and LEPs as singularities of an extended effective generator that embeds memory degrees of freedom. This suggests that the meaning of “non-Markovian LEP” depends on what is taken to be the fundamental dynamical operator.

Across these settings, the central theme is stable: LEPs are spectral singularities of dissipative quantum generators, not merely of no-jump Hamiltonians. They reorganize relaxation channels, modify spectral gaps, produce polynomial decay, reshape topological encircling phenomena, and, in higher-order or many-body settings, connect to asymptotic criticality and computational complexity. Their modern study now spans Markovian and pseudomode descriptions, bosonic and fermionic many-body systems, continuous and discrete time, and both tomographic and spectroscopic diagnostics (Sun et al., 2024).