Extracting the physical content of Liouvillian eigenmodes: Semiclassical quantization

Abstract: Unlike in closed quantum systems where individual energy eigenstates are understood as physical excitations, open quantum systems have distinct right and left eigenstates of the Liouvillian that decay with time and are difficult to interpret. Here we introduce a physically motivated quasiprobability measure combining the two types of eigenstates that interprets a Liouville eigenmode as a set of coherences. This coherence measure is intimately connected to the return probability and allows one to visualize the modes as quasiprobability distributions in a "doubled" phase space. Using this measure we show that, remarkably, an oscillator retains its quantized "orbits" in phase space for a large class of linear and nonlinear damping, thus providing a formulation of semiclassical quantization for open systems. The orbits have measurable dynamical signatures and are broadened in the presence of a thermal bath, similar to energy levels. For quadratic systems, our results yield an extension of the concept of invariant tori, which play a central role in Hamiltonian systems.

• The paper introduces a novel coherence-based quasiprobability measure to interpret Liouvillian eigenmodes in dissipative quantum systems.
• It demonstrates that semiclassical quantization survives in open systems via a clear correspondence between Liouvillian eigenmodes and quantized phase-space orbits.
• The method enables experimental predictions of decay rates and thermal broadening in practical platforms like cavity QED and circuit QED.

Extracting Physical Content from Liouvillian Eigenmodes in Open Quantum Systems

Introduction and Context

This work addresses the challenge of interpreting eigenmodes of the Liouvillian superoperator in open quantum systems, particularly in settings where dissipation and decoherence preclude the conventional physical intuition attached to Hamiltonian eigenstates in closed systems. While semiclassical quantization connects quantum eigenstates to invariant tori and classical orbits (via, e.g., Einstein–Brillouin–Keller quantization), open systems complicate this picture due to the non-Hermitian nature of the Liouvillian, manifesting in distinct right and left eigenstates associated with complex eigenvalues. The main technical contribution is the introduction of a quasiprobability measure enabling a physical interpretation of Liouvillian eigenmodes as sets of coherences, thereby extending the semiclassical quantization program to the domain of Markovian open quantum dynamics.

Formulation of a Coherence-Based Quasiprobability Measure

The proposed measure builds on the spectral projector $F_p = |R_p\rangle\langle L_p|$, constructed from the biorthonormal right and left eigenstates of the Liouvillian for eigenvalue $\lambda_p$. This measure is invariant under rescaling ("gauge" transformations) of the eigenstates—a crucial property given the inherent ambiguities in the definition of non-Hermitian eigenvectors. When expressed in a tensor-product basis via the Choi isomorphism, the diagonal entries define a "coherence matrix" $C_p(i,j)$, interpreted as the weight representing coherence between basis elements $|i\rangle$ and $|j\rangle$. This construction specializes to the standard probability distribution in the unitary (Hamiltonian) limit, where $|R_p\rangle = |L_p\rangle$.

This measure is physically operational: it captures contributions to the return probability of a generic initial state and allows direct visualization as a joint quasiprobability distribution over products of states in a "doubled" phase space. Notably, this approach is fully compatible with the requirements of operator normalization and positivity where appropriate and is robust for application across arbitrary completely positive trace-preserving (CPTP) dynamics, including many-body and nonlinear cases.

Semiclassical Quantization in Open Systems

The work demonstrates that, for an isotropically damped quantum oscillator (including generalized nonlinear damping), the quantization of phase-space orbits survives the introduction of arbitrary damping. Specifically, eigenspaces of the Liouvillian are in one-to-one correspondence with pairs of Fock states, and the coherence matrix robustly isolates the corresponding quantized orbits in phase space. For a harmonic oscillator with single-particle loss, an eigenmode labeled by $(k,l)$ corresponds to coherence between $|m\rangle$ and $|n\rangle$ with $l = m-n$ and $\lambda_p$0. The decay rate of these coherences is given by the real part of the associated eigenvalue and is experimentally relevant—governing physically measurable survival and coherence decay probabilities. This robust structure extends to nonlinear models such as the lossy Kerr oscillator, underlining its generality for isotropic, unidirectional Lindblad flows in Fock space.

Crucially, in these settings, the spectral broadening due to thermal noise is closely analogous to level broadening in closed systems, but with unique non-Lorentzian lineshapes and experimentally distinguishable relaxation signatures, including multistage and nonexponential spatial decay.

Visualization in Doubled Phase Space and Extension to Quadratic Systems

The proposed framework naturally leads to a phase-space visualization by constructing a super Husimi function in the product of two (doubled) phase spaces. This allows for both the marginalization over "ket" and "bra" spaces—yielding meaningful reduced distributions—and a full characterization of coherence between different Fock states or coherent states. In the quadratic Liouvillian scenario (e.g., multimode coupled oscillators), this construction identifies quantized tori in a $\lambda_p$1-dimensional phase space, precisely extending the concept of classical invariant tori in the presence of Markovian dissipation. The left-right Husimi function localizes on these tori, confirming analytically that area quantization persists for the dissipative normal modes.

Thermal Effects and Broadening

Introducing a thermal bath modifies the Liouvillian by enabling both injection and loss processes, captured in the formalism via modified Lindblad operators. The unidirectional spectral support of the eigenmodes is replaced by a distribution that overlaps many Fock states, computed exactly in terms of Meixner polynomials. The resulting thermal broadening obeys well-defined scaling with temperature (and excitation number), with traces observable in the late-time behavior of return probabilities and in the structure of the coherence matrix.

Generality and Implications

The coherence-based interpretation applies broadly to Markovian models satisfying CPTP conditions, including nonlinear and many-body systems, and extends seamlessly to quadratic fermionic systems by virtue of the Choi isomorphism and appropriate phase-space representations.

Practically, the results provide direct predictions for experimentally observable quantities, such as the decay of population and coherences in cavity QED, circuit QED, and related platforms, where the quantized structure of dissipative spectra is accessible via e.g., Wigner tomography or Loschmidt echo measurements. Theoretically, the persistence of semiclassical quantization structures in open quantum systems provides new tools for spectral analysis and supports extensions of semiclassical eigenfunction theory to the master equation setting.

An immediate open direction is the fate of quantized tori under integrability-breaking perturbations, potentially leading to a generalization of EBK quantization in open systems, and the investigation of coherence structures in settings lacking a simple classical limit, such as in open fermionic or spin systems, or systems with limit-cycle attractors.

Conclusion

This work establishes a robust, physically meaningful correspondence between Liouvillian eigenmodes and quantum coherences, enabling a form of semiclassical quantization in dissipative quantum systems. The quasiprobability measures introduced here generalize the concept of classical orbits and quantized tori to the Lindblad dynamics of open quantum systems, bridging an interpretive gap and supporting both theoretical extensions and experimental applications in the characterization of non-Hermitian quantum spectra and dynamics.