Superradiant Phase Transition (SRPT)

• SRPT is a symmetry-breaking quantum phase transition where macroscopic bosonic occupations emerge from collective light–matter interactions.
• It is modeled using generalized Dicke models with multi-level systems and multi-mode bosonic fields to capture complex coupling dynamics.
• TRK sum rule compliant diamagnetic terms and cross-mode couplings uniquely influence the transition order and symmetry properties.

A superradiant phase transition (SRPT) is a symmetry-breaking quantum phase transition in which a macroscopic number of bosonic excitations—most commonly, photons or polaritons—develop in a mode that is collectively coupled to an ensemble of two-level (or multi-level) systems. The transition signifies a qualitative change in the ground state from a “normal phase,” where both matter and field remain unexcited, to a “superradiant phase” characterized by nonzero mean-field amplitudes of the bosonic modes and macroscopic collective polarization of the matter subsystem. The SRPT has been studied extensively within variants of the Dicke model, including multi-level and multi-mode cases, and is realized when the collective coupling exceeds a critical threshold. In the generalized context of three-level lambda systems interacting with two bosonic modes, the SRPT displays unique features concerning its order, symmetry structure, and the interplay with microscopic constraints such as the Thomas–Reiche–Kuhn sum rule.

1. Generalized Dicke Model with Three-Level Lambda Systems and Two Bosonic Modes

The system consists of $N$ identical three-level constituents arranged in a lambda configuration, where two ground states ($E_1$, $E_2$, detuning $\delta = E_2-E_1$) and one excited state ($E_3$, with $\Delta = E_3-E_1$) are coupled via two distinct bosonic modes of frequencies $\omega_1$, $\omega_2$. The Hamiltonian includes collective dipole interactions, cross-mode couplings, and bosonic self-interaction (diamagnetic) terms:

$$H = \sum_{n=1}^3 E_n I + \sum_{n=1}^2 \omega_n a_n^\dagger a_n + \frac{g_1}{\sqrt{N}} (I_{3,1} + I_{1,3})[a_1^\dagger + a_1 + \chi_1 (a_2^\dagger + a_2)] + \frac{g_2}{\sqrt{N}} (I_{3,2} + I_{2,3})[a_2^\dagger + a_2 + \chi_2 (a_1^\dagger + a_1)]$$

$$+ \sum_{n=1}^2 \frac{\kappa_n^2}{\omega_n} (a_n^\dagger + a_n)^2 + 2\frac{\kappa_3^2}{\sqrt{\omega_1 \omega_2}} (a_1^\dagger + a_1)(a_2^\dagger + a_2)$$

where $I_{i,j}$ are collective lowering/raising operators for the atoms, $a_n$ are bosonic mode operators, $g_1$, $g_2$ are light–matter coupling strengths, $\chi_1$, $\chi_2$ mix the two transitions, and $\kappa_n$ parameterize bosonic self-interactions (diamagnetic contributions). The effective atomic-cavity couplings and diamagnetic terms are derived from the minimal-coupling Hamiltonian and are subject to the Thomas–Reiche–Kuhn (TRK) sum rule.

2. Characterization and Order of the Superradiant Phase Transition

In the thermodynamic limit ($N \rightarrow \infty$), the system supports a SRPT with the following features:

• Normal phase: All mean fields vanish; atomic population remains in one ground state, and bosonic modes remain in vacuum.
• Superradiant phase: Mean fields ($\Psi_2$, $\Psi_3$ for atomic levels; $\varphi_1$, $\varphi_2$ for bosonic modes) become nonzero, signaling macroscopic occupation of both atomic and photonic degrees of freedom.

Depending on the Hamiltonian parameters:

• The transition may be second order (continuous change of order parameters as the coupling crosses the critical value) or first order (order parameters jump discontinuously at the transition point).
• The phase boundary is a composite of both first- and second-order segments, meeting at a multicritical point.

The condition for the onset of the SRPT can be formulated, e.g., for the $g_1$ channel, as $g_1 \leq g_{1,c}$, where the critical coupling $g_{1,c}$ depends on the detunings, bosonic frequencies, diamagnetic terms, and mixing parameters. When equality is reached, local stability of the normal phase is lost and a superradiant phase emerges.

Numerical minimization of the mean-field ground-state energy reveals discontinuous jumps in certain regions of parameter space, distinguishing first-order transitions from the continuous (second-order) regime.

3. Symmetry Properties and Phase Diagram Structure

The symmetry structure crucially influences the phase diagram:

• For $\kappa_n = \chi_n = 0$, the system has two independent parity symmetries (associated with each bosonic mode and its coupled atomic transition): $\Pi'_n = \exp[-i\pi(-I + a_n^\dagger a_n)]$.
  • In this regime, superradiant phases may selectively break one or the other parity, creating two distinct symmetry-broken “blue” or “red” phases.
• Inclusion of diamagnetic ($\kappa_n$) and cross-mode mixing ($\chi_n$) terms causes the separate parities to fail to commute with the Hamiltonian; instead, a combined global parity emerges:

$$\Pi = \exp[-i\pi(-I + a_1^\dagger a_1 + a_2^\dagger a_2)]$$

• In the full model, both bosonic modes participate equally in the symmetry breaking, yielding a unique superradiant phase with simultaneous macroscopic excitation of both fields.

The destruction of separate channel parities by diamagnetic and mixing terms collapses the phase diagram into a single superradiant domain, with the order and location of phase boundaries determined by the interplay of $g_{1,2}$, $\kappa_{1,2,3}$, and $\chi_{1,2}$.

4. Influence of Microscopics, Diamagnetic Terms, and the Thomas–Reiche–Kuhn Sum Rule

The effective model is strictly derived from the minimal-coupling Hamiltonian for atoms in a cavity:

• The diamagnetic term $(a_n^\dagger + a_n)^2$ arises naturally and increases the energetic cost for exciting the bosonic field.
• The TRK sum rule, a consequence of gauge invariance, imposes upper bounds on the coupling strengths, e.g., $g_1 \leq \sqrt{\Delta/\omega_1}\kappa$, $g_2 \leq \sqrt{(\Delta - \delta)/\omega_2}\kappa$.

In conventional (Hopfield) models and for two-level systems, enforcing the physical value of the diamagnetic term (TRK compliant) prohibits the SRPT (no-go theorem). However, in the present multi-level, multi-mode lambda configuration:

• Even under the TRK sum rule, a SRPT is permitted.
• In atomic realizations where the TRK-imposed bound is active, the SRPT is found to be strictly first order—i.e., the transition involves a macroscopic jump in the order parameters and no continuous softening of the excitation spectrum.
• This finding demonstrates that a “no-go theorem” does not universally apply to systems with a richer level structure and multiple electromagnetic modes, provided the microscopic coupling topology is sufficiently complex.

5. Experimental and Theoretical Implications

This analysis yields several implications:

• Atomic and artificial cavity QED: Real three-level atomic systems (with suitable dipole moments and nonorthogonal polarizations) operating in multimode cavities or with engineered polaritonic spectra can access a superradiant phase transition, potentially of first order, even in the presence of physically realistic diamagnetic couplings and under TRK constraints.
• Criticality and experimental observables: The order of the transition (first versus second) is vital when analyzing quantum critical behavior and fluctuations, and provides guidance for the interpretation of experimental data near and above threshold.
• Design of quantum simulators and condensed matter analogs: The findings motivate the use of three-level systems, photonic crystals, or engineered circuit QED platforms to realize and manipulate superradiant phases, bypassing restrictions inherent to conventional two-level setups.
• Theoretical significance: The existence of a SRPT under TRK-compliant couplings in generalized Dicke models provides an explicit counterexample to the widely interpreted “no-go theorem” for equilibrium superradiance, specifically in multi-level, multi-mode systems.

6. Summary Table: Key Model Features

Aspect	Feature	SRPT Impact
Level structure	Three-level lambda (two ground + one excited)	Allows two independent light–matter channels
Bosonic modes	Two modes (frequencies $\omega_1$, $\omega_2$)	Enable multimode collective effects and parity mixing
Diamagnetic/self-interaction	Terms $\kappa_n$, cross-terms (e.g., $\kappa_3$) from minimal coupling	Raise excitation cost but do not universally forbid SRPT
Cross-mode coupling	Parameters $\chi_1$, $\chi_2$	Mix transition channels; influence symmetry structure
Phase transition order	First or second order, depending on parameters (typically first order under TRK bounds)	Determines observables: jump/discontinuity vs. continuous onset
Symmetry breaking	Single global parity broken (for nonzero $\kappa_n,\chi_n$); two parities in their absence	Determines nature and number of superradiant phases
Applicability (TRK sum rule)	Enforced in atomic settings; criticality determined by $\kappa_n$, $g_n$, etc.	SRPT survives in multi-level, multi-mode; first order if TRK

7. Relevance to Broader SRPT Research

This work constitutes a foundational extension of the Dicke phase transition paradigm to more complex matter–field coupled systems, highlighting:

• The critical role of system symmetry and Hamiltonian topology in determining phase diagram structure and transition order.
• The interplay between realistic microscopic constraints (such as the TRK sum rule) and emergent collective phenomena.
• The necessity of moving beyond two-level, single-mode models to fully capture the diversity and potential of equilibrium superradiant phase transitions in quantum optical and condensed matter systems.

Such findings guide both the theoretical classification of quantum phase transitions in light–matter systems and the engineering of platforms for observing and utilizing superradiant phases in quantum technologies (Hayn et al., 2012).