Superradiant phase transitions (SPTs) are a fascinating area of study in quantum physics, characterized by the singularity of quantum fluctuation at the critical point. The Dicke model, describing the collective interaction between N two-level systems and a quantum field, theoretically predicts SPTs in the thermodynamic limit (N→∞). However, the Rabi model (N=1) also exhibits SPTs under specific conditions, like the classical oscillator limit (Ω/ω → ∞, where Ω and ω are the frequencies of the spin and field, respectively). SPTs represent a unique type of quantum phase transition occurring with parameter changes at zero temperature. Increasing the coupling strength beyond a critical point leads to an abrupt ground state change from a normal phase to a superradiant phase, where the ground state becomes macroscopically occupied and twofold degenerate, indicating Z2 symmetry breaking. This results in significant quantum effects like spin-field entanglement and large-amplitude quantum superposition, crucial for quantum metrology and computation. Cavity or circuit quantum electrodynamics (QED) systems provide a promising platform to realize the Dicke and Rabi models. However, achieving the critical parameter regime and preparing the ultralow-temperature ground state for equilibrium SPT remain experimentally challenging. Furthermore, a no-go theorem challenges the existence of equilibrium SPT in cavity QED systems. This theorem arises from neglecting the A² term (the squared term of the electromagnetic vector potential) in standard Dicke and Rabi Hamiltonians, which eliminates the singularity of quantum fluctuation. The debate on the A² term's inclusion in effective light-matter interaction models persists. While recent work proposed SPT schemes immune to the no-go theorem using hybrid circuit QED with an auxiliary squeezing term, current technology limitations hinder practical implementation. Quantum simulation offers an alternative approach to experimentally investigate SPTs. While dynamics of the quantum Rabi model (QRM) and nonequilibrium SPTs of the Dicke model have been simulated using various platforms (Bose-Einstein condensates, trapped ions), the effect of the no-go theorem remained experimentally unexplored. This research employs nuclear magnetic resonance (NMR) quantum simulation to address these challenges.

The theoretical prediction of superradiant phase transitions (SPTs) has a rich history, beginning with the Dicke model [2,3] and extending to the Rabi model [4–6], where the thermodynamic limit is replaced with the classical oscillator limit. Previous work highlights the entanglement and quantum superposition phenomena in the superradiant phase [7,8], underscoring its potential for quantum technologies. The experimental exploration has largely been limited by the difficulties in cavity QED systems, with the A² term posing a significant theoretical barrier [12–20]. Nataf and Ciuti’s no-go theorem [14] demonstrated the challenges in observing equilibrium SPTs due to the coupling-dependent potential introduced by the A² term. This spurred further investigations into alternative approaches, including hybrid systems [21,22], but experimental limitations in achieving strong quadratic optomechanical coupling persisted [23]. Existing quantum simulations mainly focused on dynamical aspects or nonequilibrium SPTs [24–30], leaving the experimental exploration of the no-go theorem’s impact largely untouched. This study builds upon this foundation by utilizing NMR quantum simulation to overcome these experimental hurdles and directly investigate the SPT in the presence of the A² term.

This research leverages nuclear magnetic resonance (NMR) as a quantum simulator to investigate the equilibrium superradiant phase transition (SPT) in the presence of the A² term, typically considered a significant obstacle to observing the phenomenon. The experimental methodology involves several key steps: 1. **Physical Model and Mapping:** The researchers utilize the Rabi model with the A² term and an antisqueezing term (H = HR + HA + HAS). A spin-to-oscillator mapping scheme (equation 2) is employed, where N spins simulate an oscillator and one spin simulates a two-level system (Fig. 1a). This mapping, similar to the Holstein-Primakoff transformation, is exact in the limit N → ∞. ¹³C-iodotriuroethylene dissolved in d-chloroform serves as a 4-qubit simulator (Fig. 1c), with experiments conducted on a 400 MHz spectrometer. The Hamiltonian for this model includes terms for spin interactions. 2. **State Preparation:** The system is initialized to a pseudo-pure state (PPS) using selective-transition approach [47], reaching a fidelity of approximately 0.991. The antisqueezing effect is implemented through a gradient ascent pulse engineering (GRAPE) pulse [48], transforming the oscillator's ground state from a vacuum state to a squeezed vacuum state. The adiabatic method [50] is used to prepare the ground state of the transformed Hamiltonian (Hs), which is a crucial step in observing the phase transition. 3. **Measurement:** A three-step measurement process is used to obtain the zero-point fluctuation (ZPF) (Fig. 2): measuring diagonal elements for ⟨a†a + âa†⟩, measuring the fourth qubit with/without a readout operator Û₁ for ⟨a + a†⟩, and measuring the third qubit with/without a readout operator Û₂ for ⟨a² + a†²⟩. These measurements are performed via the ¹³C channel using SWAP gates, accounting for the low natural abundance of ¹³C in the sample. The order parameter Φ for the SPT is obtained by measuring corresponding expectation values in the prepared ground state, utilizing a post-processing step outlined in the methods section to address the limitations of the finite-dimensional Hilbert space (Eqs. 10-14). 4. **Quantum State Tomography:** This technique is used to reconstruct the experimentally prepared ground states, allowing for the characterization of entanglement and the verification of the generation of Schrödinger cat states by analyzing the Wigner function. The von Neumann entropy is calculated from the reduced density matrix of the oscillator to quantify the entanglement. The methodology meticulously addresses the challenges of simulating a system with a limited number of qubits and incorporates advanced techniques for pulse engineering, state preparation, and measurement.

The study's key findings demonstrate the experimental realization of an equilibrium superradiant phase transition (SPT) beyond the constraints imposed by the no-go theorem, achieved through the introduction of an antisqueezing effect. Specifically: 1. **Antisqueezing-Enhanced ZPF:** The experiment demonstrates an exponential enhancement of the zero-point fluctuation (ZPF) of the oscillator with increasing antisqueezing strength (Fig. 2), validating the theoretical prediction and showcasing the key mechanism behind the recovery of the SPT. The measured ZPF values closely match the theoretical expectations. 2. **Recovering of SPT:** The experiments successfully simulate the equilibrium SPT in the presence of the A² term. Without antisqueezing (ξ = 0), the phase transition is absent (Fig. 3a), confirming the no-go theorem. However, with antisqueezing (ξ/ω = 0.26), a reversed SPT is observed (Fig. 3b), where the order parameter (Φ) increases from near zero to a finite value as the coupling strength (λ) decreases. This reversed transition arises from the interplay between the antisqueezing and A² terms, differing significantly from typical SPT behavior. The results show good agreement with theoretical predictions, especially in the classical oscillator limit (Ω/ω → ∞). 3. **Entangled and Squeezed Schrödinger Cat States:** The experimental results reveal the generation of strongly entangled states in the superradiant phase (Fig. 3b), quantified by the von Neumann entropy. The Wigner functions (Fig. 3c) clearly exhibit negative values and distinct interference fringes, demonstrating the creation of squeezed Schrödinger cat states, crucial resources for quantum metrology and fault-tolerant quantum computing. These states display large amplitude separation and distinct interference fringes. 4. **Ground-State Phase Diagram:** The experimentally determined ground-state phase diagram (Fig. 4) confirms the existence of the reversed SPT. The critical point scaling exponent (γ = -0.676) is close to the universal value (-2/3) predicted for the Rabi and Dicke models, further supporting the experimental realization of SPT. The diagram also delineates the regions of normal phase (NP), superradiant phase (SP), and unstable phase (UP), consistent with the theoretical boundaries. The results demonstrate that the system transitions from SP to NP with decreasing λ. These findings collectively demonstrate the successful experimental simulation of the SPT beyond the no-go theorem, highlighting the crucial role of antisqueezing in recovering the ground state singularity.

This research presents compelling evidence of an equilibrium SPT beyond the no-go theorem, achieved by incorporating an antisqueezing effect. The findings do not contradict the original no-go theorem; instead, they demonstrate how antisqueezing modifies the system, effectively relaxing the limitations imposed by the A² term. The experimental observation of enhanced ZPF directly supports the mechanism responsible for recovering the singularity of the ground state. The experimental realization of strongly entangled states and squeezed Schrödinger cat states in the superradiant phase opens up new avenues for quantum information processing and metrology. The meticulously characterized phase diagram further validates the experimental observation of SPT, showing consistency with theoretical predictions, particularly in the classical oscillator limit. The research significantly advances our understanding of SPTs and offers a novel pathway for investigating quantum optical effects using NMR and other spin systems.

This study provides the first experimental demonstration of equilibrium SPT beyond the no-go theorem using an antisqueezing effect. The findings show that the A² term is not an absolute barrier to observing equilibrium SPTs in cavity QED systems. The antisqueezing-enhanced ZPF plays a critical role in this process. The experimental generation of entangled states and Schrödinger cat states opens exciting possibilities for quantum technologies. The methodology presented is adaptable to other platforms like trapped ions and NV centers, promising further advancements in the field.

The experiment is conducted using a 4-qubit NMR system, limiting the simulation to a truncated Hilbert space. While the adiabatic method is used for ground state preparation, the finite-time evolution might introduce some non-adiabatic effects. The post-processing step used to obtain the order parameter relies on a theoretical model and could introduce some approximation errors. Future research could explore larger-scale simulations and investigate the effects of decoherence and noise in more realistic experimental settings.