Dark-State Phase Transition

• Dark-state phase transition is a nonequilibrium phenomenon in open quantum systems marked by transitions between pure stationary states arising from competing dissipative decay and quantum coherence.
• It is realized in diverse platforms such as cavity-QED setups, Rydberg ensembles, and Bose-Hubbard chains, where interactions and interference create distinct dark and mixed phases.
• Control parameters like coherent rates and Liouvillian spectra act as order observables, revealing critical behavior and distinguishing first-order from second-order transition phenomenology.

A dark-state phase transition is a nonequilibrium phase transition in an open quantum system whose stationary structure is governed by dark states, i.e. stationary states of a dissipative, Lindblad-type time evolution with zero von Neumann entropy and therefore pure steady quantum states (Roscher et al., 2018). In the most specific formulation, it denotes a transition within a dark manifold itself: irreversible decay, interactions, and quantum interference compete so that the system switches between distinct non-fluctuating stationary states rather than between a fluctuating active phase and a classical absorbing state (Carollo et al., 2021). Related usages occur in driven-dissipative light-matter systems, monitored fermionic chains, cavity-QED dark subspaces, and dissipative superfluids, where the transition can be encoded in the Liouvillian spectrum, in the dynamical response, or in the dimension of the dark sector (Soriente et al., 2019).

1. Conceptual definition

In open many-body quantum systems, a dark state is a pure steady state annihilated by the dissipative dynamics. The notion becomes nontrivial when the set of dark states is not a single trivial absorbing configuration but a structured dark space with competing stationary states. The paper introducing the term “dark space phase transition” identifies precisely this situation: two competing dark states, a trivial one corresponding to a classical absorbing state and an emergent one which is quantum coherent, together with a nonequilibrium phase transition within this dark space (Carollo et al., 2021).

This formulation differs from the classical absorbing-state paradigm. In classical models such as epidemic spreading, the transition is from a fluctuating active phase to an absorbing configuration state. In the quantum setting of the dark space phase transition, both phases are dark and non-fluctuating, but the nature of the dark state changes. The emergent phase is collective and quantum coherent, and its existence relies on interference between different dynamical paths, a phenomenology that cannot be encountered in classical systems (Carollo et al., 2021).

The literature also uses the expression in a broader sense for transitions involving a dark state on one side and a mixed state on the other. In a driven open Rydberg spin ensemble, for example, the transition connects a zero entropy dark state with a finite entropy mixed state and thus goes beyond the field of equilibrium statistical mechanics and becomes of genuine nonequilibrium character (Roscher et al., 2018). A plausible implication is that “dark-state phase transition” now names a family of nonequilibrium critical phenomena unified by the central role of dissipation-protected stationary states rather than a single universality class.

2. Microscopic framework of the dark space transition

A concrete many-body realization is an array of $N$ three-level units, in one dimension or higher dimension, with local basis states $\ket{\bullet}$ (contagious infected), $\ket{*}$ (non-contagious infected), and $\ket{\circ}$ (healthy) (Carollo et al., 2021). The deterministic evolution of the density matrix is governed by the Lindblad quantum master equation

$$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$

with dissipator

$$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$

and Hamiltonian

$$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$

Here $\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$, $\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$, and $\Pi_\bullet^k$ is a facilitation operator indicating the presence of contagious neighbors (Carollo et al., 2021).

The three ingredients have sharply differentiated roles. Irreversible decay $\ket{\bullet}$0 drives the system toward the healthy absorbing configuration. Interactions enter through facilitation: $\ket{\bullet}$1 is only allowed if at least one neighbor is in $\ket{\bullet}$2. Quantum coherence and interference arise from the coherent transitions $\ket{\bullet}$3 and from the facilitated channel. It is this interplay of irreversible decay, interactions, and quantum interference that produces the dark space phase transition (Carollo et al., 2021).

In the thermodynamic limit, the model yields two fundamentally different dark states. The trivial, classical absorbing state is

$\ket{\bullet}$4

The emergent quantum-coherent dark state is

$\ket{\bullet}$5

a collective state with finite coherence between contagious and healthy states, measurable through off-diagonal observables such as $\ket{\bullet}$6, where $\ket{\bullet}$7 (Carollo et al., 2021). The emergent state has no classical analog and only appears collectively in the thermodynamic limit.

3. Order parameters, criticality, and first-order phenomenology

For the nonequilibrium dark space transition, the control parameter is the coherent infection rate $\ket{\bullet}$8. At fixed $\ket{\bullet}$9 and $\ket{*}$0, the threshold is

$\ket{*}$1

For $\ket{*}$2, the system ends in the trivial dark state $\ket{*}$3; for $\ket{*}$4, it ends in the emergent dark state $\ket{*}$5 (Carollo et al., 2021).

The order parameter is the density of contagious sites,

$\ket{*}$6

It vanishes in $\ket{*}$7 and is nonzero in $\ket{*}$8. For fully connected or infinite-dimensional lattices, the stationary solutions are

$\ket{*}$9

At $\ket{\circ}$0, the gap in the imaginary part of the effective non-Hermitian Hamiltonian $\ket{\circ}$1 closes as $\ket{\circ}$2, indicating a phase transition (Carollo et al., 2021).

A distinct but related first-order dark-state phenomenology appears in a driven open Rydberg spin ensemble on a $\ket{\circ}$3-dimensional square lattice (Roscher et al., 2018). There the order parameter is the local Rydberg density $\ket{\circ}$4, with $\ket{\circ}$5 in the dark state and $\ket{\circ}$6 in the mixed phase. The coarse-grained dynamics is described by

$\ket{\circ}$7

with Landau potential

$\ket{\circ}$8

and multiplicative non-thermal noise

$\ket{\circ}$9

The multiplicative character is crucial: the noise vanishes when $$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$0, so fluctuations vanish strictly in the dark state (Roscher et al., 2018).

The functional RG analysis of that first-order dark-state transition shows strong dimensional dependence. In $$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$1, fluctuations smooth out the potential barrier during the RG flow and the first-order transition is replaced by a second-order transition. For $$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$2, the barrier survives the RG flow and the first-order transition persists (Roscher et al., 2018). At coexistence, the long-wavelength theory develops a flat deterministic potential together with a vanishing noise kernel over an extended domain $$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$3, supporting an extensive set of noiseless steady configurations. This suggests a sharp distinction between equilibrium first-order transitions and absorbing or dark-state coexistence.

4. Driven-dissipative realizations in light-matter and superfluid systems

In the interpolating Dicke-Tavis-Cummings model, infinitesimal dissipation stabilizes new steady-state solutions and extends the region in which the normal phase exists into domains that would be occupied by the superradiant phase in the closed system (Soriente et al., 2019). The resulting dissipation-stabilized regime, termed the excited Normal Phase (e-NP), is statically identical to the usual normal phase: in both cases the cavity is empty and all spins are unexcited. The transition between them is therefore invisible in static or equal-time observables and appears only in the dynamical response. The cavity response function

$$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$4

shows a peak inversion across the transition, and the dominant fluctuations flip from being particlelike to holelike. The Liouvillian eigenvalues exhibit behavior akin to exceptional points (Soriente et al., 2019).

A different driven open Dicke model exhibits a transition from a predominantly superradiant to a predominantly subradiant state (Gegg et al., 2017). The transition is controlled by the interplay of resonant driving, cavity decay, spontaneous emission, and dephasing. Reported signatures include a kink in the mean TLS excitation as a function of driving, a local minimum in the mean cavity photon number, a peak in the second-order photon correlation function $$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$5, and a switch in a collectivity measure $$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$6 from enhancement of the maximal-spin Dicke sector to enhancement of low-$$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$7 subradiant sectors (Gegg et al., 2017). After the drive is switched off, relaxation generates a cascade of dark Dicke states, with dark state populations up to unity.

In a Bose-Hubbard chain with a single lossy site, the identified transition is first-order and occurs between a discrete dark soliton and a uniform superfluid (Ceulemans et al., 30 Sep 2025). The Lindblad dynamics is

$$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$8

with jump operator $$\frac{d}{dt}{\rho}_t = \mathcal{L}[\rho_t] = -i[H,\rho_t] + \mathcal{D}[\rho_t],$$9. The dark soliton satisfies $$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$0, or equivalently $$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$1 in the discrete Gross-Pitaevskii description, so the lossy site is empty and the state is dark. Classical-field simulations and Bogoliubov stability analysis identify a bistable region and an upper bistability boundary $$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$2 in the ideal theory (Ceulemans et al., 30 Sep 2025).

5. Dark subspaces, order parameters from nullity, and stability of degenerate manifolds

In the Tavis-Cummings model, dark states are many-body quantum states with nonzero excitation in the collection of qubits but from which no photon emission is possible (Theerthagiri et al., 17 Sep 2025). With no photons in the cavity and emission operator

$$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$3

a dark state satisfies $$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$4 and $$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$5. The number of linearly independent dark states in the $$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$6-excitation sector is

$$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$7

This formula holds even when the couplings $$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$8 are completely non-uniform and disordered, provided they are all nonzero (Theerthagiri et al., 17 Sep 2025).

The same work identifies a dark-bright phase transition in which the number of independent dark states plays the role of an order parameter. With tuning parameter $$\mathcal{D}[\rho]=\gamma\sum_{k=1}^N\left(J^{(k)}_-\rho J^{ (k)}_+-\frac{1}{2}\left\{J^{ (k)}_+ J^{(k)}_-,\rho \right\}\right), \qquad J_- = \ket{\circ}\bra{*},$$9 and order parameter

$$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$0

the thermodynamic-limit behavior is

$$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$1

The critical point is $$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$2, and the transition is second-order (Theerthagiri et al., 17 Sep 2025). Experimentally, the dark-state number can be accessed by a no-photon protocol: if no photons are detected from a lossy cavity, the qubits collapse onto a dark state, and summing over product-state preparations yields the dimension of the dark subspace (Theerthagiri et al., 17 Sep 2025).

The dark-space transition of the facilitated three-level model has a more compact dark manifold: the two dark states $$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$3 together span a two-dimensional steady-state dark subspace,

$$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$4

which can encode a qubit (Carollo et al., 2021). The proposed significance is collective encoding of quantum information based on non-fluctuating, collectively robust steady states.

A contrasting many-body setting is the $$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$5-Dicke model with two bosonic modes (Hayn et al., 2011). There a family of dark states exists only when the two ground states are degenerate, $$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$6, and is characterized by $$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$7 and $$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$8. In that limit, $$H=\sum_k [ \Omega_1 \lambda^{(k)}_1 + \Omega_2 \Pi^k_{\bullet} \lambda^{(k)}_6 ].$$9 is arbitrary within normalization, so the dark manifold reflects a degeneracy rather than a robust ordered phase. The spectrum contains a zero mode,

$\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$0

and the dark state is stable only for $\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$1 and below the critical couplings, with stability criterion

$\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$2

The authors emphasize that the many-body dark state is meta-stable or even unstable and is not robust against perturbations (Hayn et al., 2011).

6. Measurement-induced dark-state transitions

Measurement-induced phase transitions can also be framed as quantum phase transitions in the dark state of an effective non-Hermitian Hamiltonian (Müller et al., 2021). In a one-dimensional ring of free fermions with algebraically decaying hopping $\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$3, continuous local density measurements at rate $\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$4, and $\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$5, the competition between long-range coherent hopping and disentangling local measurements produces three phases: an area law phase, a critical phase with logarithmic entanglement growth, and a new algebraic scaling phase (Müller et al., 2021).

For $\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$6, independently of the measurement rate, the algebraic phase is characterized by

$\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$7

and by connected density-density correlations

$\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$8

This phase is slower than volume law but faster than logarithmic scaling (Müller et al., 2021). For $\lambda_1 = \ket{\bullet}\bra{*} + \text{h.c.}$9, the transition between the area law and critical phases is Berezinskii-Kosterlitz-Thouless-like and depends on $\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$0.

The effective field theory is a non-Hermitian sine-Gordon type theory with an additional long-range purely imaginary hopping term. The first-order RG equations for the long-range coupling $\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$1 and stiffness $\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$2 are

$\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$3

For $\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$4, the long-range term is always relevant, which explains why the algebraic phase is independent of the measurement strength $\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$5 (Müller et al., 2021). Exact numerical simulations of the monitored wave functions and analytical predictions from replica field theory show excellent quantitative agreement, reinforcing the interpretation of monitored criticality as a dark-state transition of an effective non-Hermitian generator.

7. Terminological scope and adjacent usages

The phrase “dark-state phase transition” should be distinguished from phase transitions in a dark sector. Several recent arXiv works use “dark” to denote dark matter or dark-energy sectors rather than dissipation-protected pure stationary states. One study rescues overabundant fermionic dark matter through a strongly first order phase transition after freeze-out, where the relic abundance is diluted by entropy injection,

$\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$6

and the resulting stochastic gravitational-wave signal can lie within the sensitivity of future experiments (Huang et al., 18 Feb 2026). Another considers a nearly-conformal dark sector in which a confining first-order phase transition can generate the NANOGrav signal of a stochastic gravitational wave background (Fujikura et al., 2023).

Other examples include the population of approximately Weyl invariant secluded dark sectors through a phase transition controlled by the Hubble parameter, with the condition $\lambda_6 = \ket{*}\bra{\circ} + \text{h.c.}$7 and heavy dark matter generically predicted (Redi et al., 2022); a conformal freeze-in scenario where a dark CFT confines and the cosmological phase transition completes promptly without significant supercooling (Luo et al., 10 Feb 2025); and a galactic model positing a phase transition from radially emitted dark matter to dark energy, with Bose-Einstein condensation proposed as one mechanism (Nikitin, 2021). These usages are conceptually separate from dark-state phase transitions in open quantum dynamics.

Within quantum many-body physics proper, the term therefore refers most precisely to transitions in which dark states determine the stationary critical structure: transitions inside dark space, transitions from dark to mixed steady states, or transitions whose order parameter is the dimension or dynamical response of a dark sector. This suggests a unifying viewpoint in which the relevant object is neither an equilibrium free energy nor a classical absorbing configuration, but the dark manifold of a Lindbladian or effective non-Hermitian evolution.