n-Phase Dicke Models: Multiplicity in Quantum Phases

• n-Phase Dicke models are generalized quantum systems that exhibit more than the conventional normal–superradiant phases by incorporating extra ordered regimes or attractor states.
• They achieve rich phase diagrams through methods like periodic driving, open-system dynamics, and multi-ensemble or multi-level atomic constructions, leading to multistability and multicritical behavior.
• These models offer practical insights for quantum optics and condensed matter physics, enabling experimental access to novel quantum phase transitions and dynamic attractor phenomena.

Searching arXiv for recent and foundational papers on multi-phase and generalized Dicke models. n-Phase Dicke models are not a single canonical Hamiltonian class in the literature; rather, the term is most usefully understood as an umbrella for Dicke-type systems whose phase structure exceeds the standard normal–superradiant dichotomy of the one-mode, two-level, equilibrium Dicke model. In the strict standard model, the thermodynamic phase structure is the familiar normal and superradiant pair, with a single critical line and finite-$N$ smoothing [(Casta~nos et al., 2011); (Hirsch et al., 2012)]. The literature broadens this picture in several distinct ways: by periodic driving that generates an infinite hierarchy of sideband-assisted critical channels and multistable Floquet quasienergy minima (Bastidas et al., 2011); by open-system dynamics that supports multiple stable attractors, including two distinct superradiant branches, coexistence sectors, and limit cycles (Bhaseen et al., 2011); by multi-ensemble constructions in which separately conserved collective spins produce more than one superradiant ordering (Mivehvar, 2023, Liu et al., 2023); and by multi-level atomic generalizations that realize arbitrary-order multicriticality (Xu et al., 2020). In that sense, “$n$-phase Dicke models” denotes not one microscopic generalization, but a family of generalized Dicke settings in which multiple ordered sectors, multiple critical manifolds, or multiple asymptotic phases arise.

1. Baseline: the standard Dicke transition and finite-$N$ structure

The reference point is the standard Dicke Hamiltonian for $N$ identical two-level atoms collectively coupled to a single cavity mode. In dimensionless or rescaled forms used across the literature, its equilibrium structure is the normal-to-superradiant quantum phase transition at the critical coupling

$g_c=\frac{1}{2}\sqrt{\omega\omega_0},$

or equivalent notations such as $\gamma_c=\sqrt{\omega_A}/2$ depending on conventions [(Bastidas et al., 2011); (Hirsch et al., 2012)]. In this baseline problem, the phase content is not intrinsically “$n$-phase”: there is one normal phase and one superradiant phase (Casta~nos et al., 2011).

Finite-$N$ analyses refine this statement without changing the basic two-phase structure. Symmetry-adapted coherent-state treatments show that parity must be respected explicitly, and that the apparent singularities of photon number, excited-atom number, and fluctuations at the transition are thermodynamic-limit effects rather than true finite-$N$ divergences (Casta~nos et al., 2011). Variation-after-projection with symmetry-adapted coherent states further shows that finite-$N$ transition behavior is governed by competition between local minima of the projected energy surface; for $n$0 at resonance, the corresponding critical coupling is shifted upward to $n$1, rather than the thermodynamic $n$2 (Hirsch et al., 2012). A related finite-size scaling analysis gives

$n$3

for the ground state and

$n$4

for the first excited state, with $n$5 (Nahmad-Achar et al., 2012).

This baseline matters because much of the “$n$6-phase” literature should be read as an enrichment of this standard picture rather than a replacement for it. The key distinction is whether the generalization produces multiple thermodynamic phases, multiple nonequilibrium attractors, or multiple critical structures.

2. Floquet-engineered multi-phase behavior

A major route to n-phase Dicke behavior is periodic driving. In the monochromatically driven Dicke model,

$n$7

the periodically modulated atom-field coupling produces a ladder of sideband-assisted instabilities and, at stronger drive, a multistable quasienergy landscape (Bastidas et al., 2011).

In the thermodynamic-limit normal-phase analysis, the fluctuation coordinates satisfy Mathieu equations, and the phase boundaries are determined by Floquet-parametric instability tongues around

$n$8

with

$n$9

for the resonant case $N$0 (Bastidas et al., 2011). The $N$1-th instability region has width scaling as

$N$2

so the weak-drive regime realizes an infinite hierarchy of progressively narrower sideband transition channels (Bastidas et al., 2011). This is one precise sense in which a driven Dicke model becomes “$N$3-phase-like”: there is not one normal–superradiant critical line, but infinitely many Floquet-dressed critical lines indexed by $N$4.

The Floquet construction is made explicit by transforming to the rotating frame with

$N$5

and retaining the zero Fourier component of the transformed Hamiltonian, $N$6 (Bastidas et al., 2011). Each $N$7-sector is then an effective time-independent Floquet Dicke Hamiltonian. For example, in the $N$8 sector the critical line shifts to

$N$9

on resonance, while the $N$0 and $N$1 sectors acquire distinct effective couplings and transition lines (Bastidas et al., 2011).

At strong drive, the focus shifts from sideband multiplicity to multistability within a given Floquet sector. Using Holstein–Primakoff variables and macroscopic displacements, the effective $N$2 Hamiltonian yields a quasienergy surface

$N$3

with order parameters $N$4 and $N$5 for the photon and atomic displacements (Bastidas et al., 2011). The normal phase corresponds to $N$6, while superradiant-like phases have $N$7. As $N$8 increases, the quasienergy surface develops multiple local extrema; the number of minima can be $N$9, and the phase structure contains both second-order and first-order nonequilibrium quantum phase transitions (Bastidas et al., 2011).

This Floquet scenario is therefore a paradigmatic n-phase Dicke mechanism: periodic modulation creates both an infinite ladder of sideband critical channels and, in the strong-drive regime, multiple coexisting quasienergy minima and metastable macroscopic states.

3. Open-system Dicke models: attractor multiplicity, coexistence, and limit cycles

A second major notion of n-phase Dicke physics is dynamical rather than equilibrium-based. In the open generalized Dicke model with cavity loss,

$g_c=\frac{1}{2}\sqrt{\omega\omega_0},$0

and

$g_c=\frac{1}{2}\sqrt{\omega\omega_0},$1

the long-time phase is defined by the set of stable semiclassical attractors rather than by an equilibrium free-energy minimum (Bhaseen et al., 2011). This produces a genuinely multi-phase nonequilibrium phase diagram.

The semiclassical equations

$g_c=\frac{1}{2}\sqrt{\omega\omega_0},$2

admit several distinct stable asymptotic states (Bhaseen et al., 2011). The paper identifies the following attractor classes: the normal non-superradiant state $g_c=\frac{1}{2}\sqrt{\omega\omega_0},$3, the inverted non-superradiant state $g_c=\frac{1}{2}\sqrt{\omega\omega_0},$4, the superradiant A branch (SRA), the superradiant B branch (SRB), coexistence sectors such as $g_c=\frac{1}{2}\sqrt{\omega\omega_0},$5, $g_c=\frac{1}{2}\sqrt{\omega\omega_0},$6, $g_c=\frac{1}{2}\sqrt{\omega\omega_0},$7, and narrow $g_c=\frac{1}{2}\sqrt{\omega\omega_0},$8 regions, together with persistent oscillation or limit-cycle phases where no fixed point is stable (Bhaseen et al., 2011).

SRA is the nonequilibrium continuation of the standard Dicke superradiant state and satisfies

$g_c=\frac{1}{2}\sqrt{\omega\omega_0},$9

with the branch $\gamma_c=\sqrt{\omega_A}/2$0 selected (Bhaseen et al., 2011). SRB is distinct and exists only for $\gamma_c=\sqrt{\omega_A}/2$1, determined by

$\gamma_c=\sqrt{\omega_A}/2$2

which gives

$\gamma_c=\sqrt{\omega_A}/2$3

It is therefore a separate superradiant phase rather than a trivial deformation of SRA (Bhaseen et al., 2011).

The critical couplings for instability of the normal and inverted states are

$\gamma_c=\sqrt{\omega_A}/2$4

while SRB exists for $\gamma_c=\sqrt{\omega_A}/2$5, $\gamma_c=\sqrt{\omega_A}/2$6, and $\gamma_c=\sqrt{\omega_A}/2$7 with

$\gamma_c=\sqrt{\omega_A}/2$8

Hopf bifurcations occur at $\gamma_c=\sqrt{\omega_A}/2$9, opening regions of oscillatory instability and persistent oscillations (Bhaseen et al., 2011).

This open-system formulation is one of the clearest operational realizations of an n-phase Dicke model. If phases are counted by qualitative stable-attractor sets, there are at least five basic classes—normal, inverted, SRA, SRB, and limit cycle—and more if coexistence sectors are counted separately (Bhaseen et al., 2011). Importantly, these phases are not organized by energy minimization. The inverted state may be dynamically stable even though it is not an equilibrium minimum-energy state, and finite-time experiments can miss asymptotic attractors because some growth and relaxation rates are slow, with sweep durations of order $n$0 required in certain regimes (Bhaseen et al., 2011).

The same basic theme reappears in an open, spin-1 generalized Dicke model with independent co- and counter-rotating couplings and quadratic Zeeman shift $n$1, where semiclassical analysis yields evidence of transitions to steady-state and oscillatory superradiance as well as chaotic dynamics (Adiv et al., 2024). In that model the total Hamiltonian is

$n$2

with cavity loss described by

$n$3

(Adiv et al., 2024). Here the multi-level internal structure and openness together generate normal, steady superradiant, oscillatory superradiant, multistable, and chaotic regimes. This suggests a broader principle: once the Dicke model is both open and internally structured, “phase” naturally generalizes from equilibrium order to attractor taxonomy.

4. Multi-ensemble constructions and multiple ordered superradiant sectors

A third route to n-phase Dicke behavior is to split the matter sector into multiple independently conserved ensembles. In the two-ensemble Dicke model,

$n$4

with $n$5 and $n$6 separately conserved, the special case $n$7 yields two distinct superradiant orderings: an $n$8-ferromagnetic branch and an $n$9-ferrimagnetic branch (Mivehvar, 2023).

In the $N$0 frame,

$N$1

the steady-state cavity amplitude is proportional to the staggered $N$2-spin,

$N$3

so the natural order parameter is

$N$4

Two nontrivial steady-state classes arise (Mivehvar, 2023).

The $N$5-ferromagnetic superradiant branch satisfies

$N$6

while the $N$7-ferrimagnetic branch satisfies

$N$8

Their critical couplings are

$N$9

so the ferrimagnetic branch always appears first, while the ferromagnetic threshold diverges as $N$0 (Mivehvar, 2023).

The important structural point is that these two superradiant branches belong to different conserved sectors of

$N$1

Thus they are independent superradiant instabilities rather than two parametrizations of the same ordered phase (Mivehvar, 2023). In the large-coupling coexistence regime, both parity doublets $N$2xFi-SR and $N$3xFo-SR are stable simultaneously, and the cavity Husimi function in finite-size quantum calculations develops four partially disjoint lobes, indicating coexistence of all four superradiant states (Mivehvar, 2023).

A related but distinct two-ensemble generalization with direct inter-ensemble interaction

$N$4

realizes three equilibrium phases: paramagnetic-normal, ferromagnetic-superradiant, and antiferromagnetic-normal (Liu et al., 2023). The mean-field energy density

$N$5

has three corresponding minima (Liu et al., 2023).

The paramagnetic-normal phase has $N$6, $N$7. The ferromagnetic-superradiant phase has $N$8 and $N$9, with boundary

$N$0

The antiferromagnetic-normal phase has $N$1 and $N$2, with boundary $N$3 (Liu et al., 2023). The direct boundary between the two ordered phases,

$N$4

is first order (Liu et al., 2023). This model is therefore a particularly clean realization of a genuinely multi-phase Dicke system in the equilibrium sense.

Together, these two-ensemble constructions show that splitting the matter sector into independently structured collective components is a robust mechanism for generating more than one ordered Dicke phase. A plausible implication is that further multiplication of ensembles or interaction channels should produce richer phase hierarchies, although those broader generalizations are not developed explicitly in these works.

5. Multi-level atoms, arbitrary-order multicriticality, and generalized n-phase structure

A different, more formal notion of n-phase Dicke behavior arises when the atoms themselves are promoted from two-level to $N$5-level systems. In the generalized Dicke Hamiltonian

$N$6

the single-atom operators $N$7 and $N$8 act on an $N$9-dimensional internal space and satisfy the Dicke $n$00 symmetry via

$n$01

(Xu et al., 2020). For $n$02, the number of independent tunable parameters is $n$03, and the model does not support multicriticality. For $n$04, the atomic-level structure and dipole matrix elements create enough parameter freedom to tune multiple Landau coefficients to zero (Xu et al., 2020).

Mean-field theory introduces a real cavity amplitude $n$05 through

$n$06

leading to the single-atom mean-field Hamiltonian

$n$07

Because of the $n$08 symmetry, the mean-field ground-state energy has the Landau expansion

$n$09

Ordinary Dicke criticality is given by $n$10, $n$11, while the $n$12-order critical manifold is defined by

$n$13

(Xu et al., 2020). This is a direct analytical realization of arbitrary-order multicriticality in a generalized Dicke model.

For the experimentally relevant T-class, where the dipole operator is tridiagonal,

$n$14

the multicritical conditions simplify drastically: $n$15 Thus an $n$16-level system can realize criticality up to order $n$17 by tuning adjacent dipole couplings against the level spacings (Xu et al., 2020). The paper gives an explicit $n$18 Raman implementation with a five-level T-class Hamiltonian. Setting $n$19, the fifth-order multicritical point occurs at

$n$20

(Xu et al., 2020).

In this multi-level framework, the phase content remains, at the most basic level, normal and superradiant, but the geometry of the phase boundary becomes far richer: second-order critical manifolds, first-order surfaces, tricritical lines, tetracritical points, and, in principle, arbitrary-order multicritical loci (Xu et al., 2020). This is therefore an n-phase Dicke construction in a different sense from multistability or multiple ordered phases: here the “$n$21” is encoded in the order of multicriticality supported by the internal atomic Hilbert space.

6. Related directions: finite-$n$22 spectral richness, lattice generalizations, and limits of the term

Several additional strands of the literature illuminate what should and should not be called an n-phase Dicke model. Exact finite-$n$23 studies of the $n$24 Dicke model show a much richer spectral organization than the single-qubit Rabi problem—four regular singular points, determinant-valued $n$25-functions, two exceptional baseline families, and possible same-parity degeneracies—but this is spectral complexity rather than a many-phase thermodynamic classification (Braak, 2013). Likewise, numerical diagonalization of the standard Dicke Hamiltonian clarifies how finite-size phase diagnostics sharpen toward the thermodynamic normal–superradiant transition and how displaced/coherent bases become essential in the superradiant regime, but it does not produce extra phases by itself (Bastarrachea-Magnani et al., 2011).

The full Dicke model with independent rotating and counter-rotating couplings at finite temperature also remains fundamentally a two-phase normal–superradiant system, although the symmetry class of the transition depends on whether one is in the $n$26, $n$27, or $n$28 regime (Alcalde et al., 2010). In the first two cases the broken symmetry is continuous $n$29 and the superradiant phase carries a Goldstone mode, whereas in the generic full model it is discrete $n$30 with no Goldstone mode (Alcalde et al., 2010). This is an important caution: distinct symmetry realizations do not automatically imply distinct thermodynamic phases.

On the other hand, generalized Dicke lattice models in driven-dissipative cavity arrays do move beyond the simple two-phase picture. In the Dicke lattice implementation with NV-center ensembles coupled to superconducting cavities, the normal phase, a homogeneous superradiant phase, a finite-wavevector superradiant phase with spontaneously broken translation symmetry, and an unstable regime for $n$31 all appear in the phase diagram (Zou et al., 2014). The critical coupling for a given momentum mode is

$n$32

and in the finite-$n$33 region the ordering wavevector is

$n$34

(Zou et al., 2014). This is a distinct multi-phase mechanism again: momentum-space structure and dissipation produce multiple superradiant states distinguished by ordering wavevector.

Finally, some generalized Dicke models enrich the phase structure by changing the order of the transition rather than multiplying the number of stable phases. An extended Dicke Hamiltonian with an all-to-all frustrating term $n$35, derived from gauge-invariant circuit-QED coupling, supports a first-order quantum phase transition into a dipolar superradiant phase, with a metastable ordered branch already present for $n$36 and an actual transition at

$n$37

(Mukhin et al., 2017). This is not an n-phase hierarchy in the literal sense, but it shows how competing minima, metastability, and first-order switching enter generalized Dicke physics.

Taken together, these results suggest a useful editorial classification. “n-Phase Dicke models” can denote at least four non-equivalent phenomena: multiple equilibrium phases distinguished by different matter and/or photonic order parameters (Liu et al., 2023); multiple superradiant sectors protected by separate conservation laws (Mivehvar, 2023); multiple nonequilibrium attractor classes in open or driven settings [(Bhaseen et al., 2011); (Adiv et al., 2024); (Bastidas et al., 2011)]; and multi-level constructions supporting arbitrary-order multicriticality rather than merely additional ordered states (Xu et al., 2020). What unifies them is the breakdown of the ordinary single-transition Dicke paradigm. What separates them is the mechanism—Floquet dressing, openness, multi-ensemble structure, multi-level internal Hilbert space, spatial lattice structure, or frustration—and the corresponding definition of “phase.”