Finite-Size Dicke-Stark Model

• The finite-size Dicke-Stark model is a framework describing a finite ensemble of two-level systems interacting with a bosonic mode and nonlinear Stark terms, leading to unique entangled states like the W-state.
• It combines collective spin-boson coupling with Stark interactions, resulting in modified phase transitions and critical couplings that differ from the mean-field Dicke paradigm through measurable finite-size scaling.
• This model informs experimental platforms such as circuit-QED and trapped ions by elucidating measurement-induced symmetry breaking, dynamical transitions, and enhanced quantum fluctuations for realistic mesoscopic ensembles.

The finite-size Dicke-Stark model describes a finite ensemble of interacting two-level systems (qubits or spins), subject to both collective coupling to a quantized bosonic mode (light field) and, in general, additional nonlinear Stark-like interactions. The model extends the Dicke paradigm by explicitly incorporating finite-$N$ (non-thermodynamic) effects, which strongly influence entanglement, phase transitions, and the quantum-to-classical crossover dynamics. Its analysis is essential for understanding contemporary quantum optical platforms (circuit-QED, trapped ions, cold atoms in cavities) operating in the mesoscopic regime.

1. Model Definition and Core Hamiltonian Structure

The finite-size Dicke-Stark (often called "generalized Dicke") Hamiltonian incorporates $N_q$ interacting qubits spin-coupled to a quantized mode and modified by Stark interactions. In its minimal form,

$$H = \omega_f a^\dagger a + \omega J_z + \lambda (a^\dagger + a) J_x + \eta J_x^2 + U a^\dagger a J_z + \ldots$$

where:

• $a^\dagger$ ($a$): bosonic mode operators (field frequency $\omega_f$).
• $J_{x,z}$: collective spin operators.
• $\lambda$: light–matter coupling.
• $\eta$: inter-qubit coupling (exchange/Ising-like).
• $U$: nonlinear Stark coupling (Stark shift proportional to photon number).
• The model may include further nonlinearities, e.g., $(a^\dagger a)^2$ terms, “$A^2$-type” terms, or explicit anisotropies in the light–matter interaction.

Crucially, for finite $N_q$, combinatorial effects in the collective spin sector and Hilbert space discretization generate effects absent in the thermodynamic (mean-field) limit.

2. Ground-State Structure, Entanglement, and the W-State Region

A distinctive feature of the finite model (Robles et al., 2010) is that for moderate $N_q$, and for couplings above a critical threshold, the ground state of the qubit ensemble hosts a maximally entangled $W$-state component:

$$|\psi_g^{(1)}\rangle = c_0^{(1)}|0\rangle|N_q/2,\,1-N_q/2\rangle + c_1^{(1)}|1\rangle|N_q/2,\,-N_q/2\rangle,$$

with $|N_q/2, m\rangle$ collective spin basis states and coefficients $c_{0,1}^{(1)}$ (functions of $h(\lambda, \eta)$). In the limiting case $h \to \infty$, field and ensemble become separable and the qubits realize the $W$-state,

$|W\rangle \equiv |N_q/2, 1-N_q/2\rangle,$

which exhibits maximal pairwise entanglement and is robust to particle loss. This multipartite entangled configuration is unique to the finite model; it disappears in the infinite-$N_q$ (classical) limit.

The area in the $(\lambda, \eta)$ parameter space supporting the $W$-state shrinks as $N_q$ increases, collapsing onto the critical phase boundary in the thermodynamic limit. Thus, finite-size effects guarantee a finite regime of strong long-range entanglement \textbf{not} present in conventional mean-field Dicke models.

3. Quantum Phase Transition, Critical Coupling, and Scaling

The model features a quantum phase transition from a vacuum (normal) phase to an entangled, one-excitation phase at a critical coupling. Under the rotating-wave approximation and weak $\eta$ (Robles et al., 2010), the critical field–ensemble coupling is:

$$\lambda_c = 2 \sqrt{\left[\omega + \frac{\eta}{N_q - 1}\right]\omega_f},$$

so increasing inter-qubit interaction shifts the threshold for the transition. Beyond $\lambda_c$, the system occupies a sector with a single shared excitation between the field and the spin ensemble, where the $W$-state is realized under matching conditions. As $N_q$ grows, this critical surface narrows; in the classical limit ($N_q \rightarrow \infty$), it becomes a sharp phase boundary.

Finite-size scaling analyses (in the context of more general open/closed Dicke and Dicke–Stark models) provide critical exponents for quantum Fisher information, photon fluctuations, and order parameters, typically distinct from those of the single-particle (quantum Rabi, $N_q=1$) or standard Dicke universality classes (Zhang et al., 2015, Chen et al., 2017, Chen et al., 2020). For instance, the QFI at criticality scales as $F_{Q,N}(\lambda_c) \sim N^{-0.65}$ (Zhang et al., 2015), while in the Rabi–Stark model, the gap exponent is $z\nu_x=2$—substantially different from the Dicke class (Chen et al., 2020).

4. Classical Limit, Dynamical Transitions, and Onset of Chaos

Semiclassical analysis and classical mappings (Robles et al., 2010, Bakemeier et al., 2013) clarify how collective variables map to phase-space structures:

• For large $N_q$, expectation values such as $\langle \hat{J}_z \rangle$ and conjugate variables $(j_z, \varphi$) describe effective classical dynamics.
• The lowest possible value of $j_z$ (the minimal projection) sets the energetic floor and the boundary between different dynamical regimes.
• Regular (ordered) Rabi oscillations arise for energies below a threshold set by the minimum $j_z$. As the energy crosses this threshold, Poincaré sections reveal a transition to chaotic (disordered) dynamics—quantum imprints to the emergence of classical chaos.

Specifically, the critical energy for the order/disorder transition aligns with the minimal angular momentum projection: $E/(\omega j) \lesssim -1$ corresponds to localized oscillations, while higher energy density reaches the chaotic regime.

5. Finite-Size and Quantum Fluctuations: Fluctuation Statistics and Measurement-Induced Dynamics

Quantum fluctuation statistics deviate sharply from mean-field predictions in the finite-size regime (Gammelmark et al., 2012). Near criticality:

• Photon number variances become super-Poissonian, scaling nontrivially with system size (e.g., $\mathrm{Var}\, n / n \propto N_q^{0.19}$).
• Enhanced quantum fluctuations are associated with the ground-state wavefunction splitting into coherent superpositions.
• Approaching second-order transitions, these enhanced fluctuations serve as experimental and theoretical signatures distinct to finite $N_q$.

Measurement-induced dynamics, driven by stochastic processes such as cavity homodyne detection, produce rapid selection of symmetry-broken states from parity-symmetric initial conditions. Stochastic Schrödinger equations incorporating cavity loss (Gammelmark et al., 2012, Imai et al., 2018), e.g.

$$d|\psi\rangle = \left[-i H\,dt - \frac{\kappa}{2}\left(a^\dagger a - \langle a + a^\dagger \rangle a + \frac{1}{4} \langle a + a^\dagger \rangle^2\right)dt + \sqrt{\kappa} \left(a - \frac{1}{2} \langle a + a^\dagger \rangle\right)dW\right]|\psi\rangle,$$

cause the system, within measurement quantum trajectories, to localize in one symmetry sector on a timescale governed by the photon loss rate and the separation between coherent components: $\tau \approx 1 / (\kappa\, \Delta\alpha^2)$.

Loss mechanisms thus not only stabilize the symmetry-broken (superradiant) phase for finite $N_q$ but are essential for the emergence of observable superradiance and macroscopic order in experiments (Imai et al., 2018).

6. Universality, Critical Exponents, and Comparison with Other Models

Critical exponents in the finite-size Dicke–Stark model depend on the interplay of the system size, interaction structure, and nonlinearity. Results show:

• The finite-size scaling exponent for the lowest excitation energy (gap), variance of field quadrature, and QFI deviate from both the standard Dicke and quantum Rabi models (Chen et al., 2020), evidencing nontrivial universality classes in the presence of Stark nonlinearities.
• Universal parametric curves for observables such as the field quadrature and atomic population (Castaños et al., 2012) survive both in mean-field, symmetry-adapted, and exact diagonalization approaches, with convergence in the thermodynamic limit.
• Finite-size corrections regularize divergences and modify the locus and rounding of the singularities near transition points, with photon number variance and Fano factor maximum scaling as $N_q^{0.4}$ and the soft mode damping vanishing as $N_q^{-0.44}$ in open-system variants (Konya et al., 2012).

The scaling behavior of entanglement, fluctuation observables, and the bifurcation structure in phase diagrams all exhibit $N_q$-dependent corrections, making the finite-$N$ regime qualitatively distinct from the mean-field (thermodynamic) scenario.

7. Experimental Relevance and Control Applications

The finite-size Dicke–Stark model is directly relevant for circuit-QED, cold atoms in optical cavities, and related quantum simulation platforms operating at mesoscopic $N_q$. Experimentally accessible consequences include:

• Regimes enabling the deterministic production and stabilization of $W$-states and GHZ-like superpositions via frequency tuning and pulse control, exploiting selective Rabi oscillations dictated by state-dependent detuning (Mu et al., 2020).
• Observation of measurement-induced symmetry breaking on experimentally relevant timescales, even though the exact eigenstates of the closed Hamiltonian conserve parity.
• Manifestations of genuine multipartite entanglement, accessible via modern quantum tomography and entanglement monotones (He et al., 2014).
• Distinct finite-size induced fluctuations, such as super-Poissonian photon statistics and altered spin squeezing, distinguishable from mean-field predictions even at $N_q \sim 10^2$ (Müller et al., 5 Mar 2025).

These properties provide both stringent tests and control handles for quantum devices intended for information processing, entanglement generation, and quantum metrology.

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In summary, the finite-size Dicke–Stark model provides a framework where quantum entanglement, critical phenomena, and dynamical transitions at the few- to mesoscopic-ensemble scale can be precisely studied and utilized. It captures the breakdown of mean-field universality for finite system sizes, the emergence of robust multipartite entangled states, the interplay of quantum and classical dynamics, and the essential role of measurement backaction and dissipation in stabilizing macroscopic order and symmetry breaking (Robles et al., 2010, Gammelmark et al., 2012, Castaños et al., 2012, Konya et al., 2012, Bakemeier et al., 2013, He et al., 2014, Müller et al., 5 Mar 2025). These features are germane both for foundational studies of quantum phase transitions and for practical quantum engineering in state-of-the-art laboratory platforms.