Experimental quantum simulation of superradiant phase transition beyond no-go theorem via antisqueezing

Superradiant phase transitions (SPT) emerge in the Dicke/Rabi models when light-matter coupling crosses a critical point, yielding a transition from a normal phase to a superradiant phase with macroscopic ground-state occupation and spontaneous Z2 symmetry breaking. While theory predicts ground-state SPT in the classical oscillator limit of the Rabi model (Ω/ω → ∞), its existence in cavity QED is challenged by a no-go theorem arising from the A2 term, which removes the singularity in quantum fluctuations. Additionally, achieving the required parameters and ultralow temperatures is experimentally difficult. This work asks whether and how equilibrium SPT can be realized beyond the no-go theorem by engineering an antisqueezing term that exponentially enhances zero-point fluctuations (ZPF) to recover the ground-state singularity. Using an NMR quantum simulator, the study aims to demonstrate equilibrium SPT recovery, characterize the associated phase diagram, and realize entangled and squeezed Schrödinger cat states relevant to quantum metrology and computation.

The Dicke model SPT in the thermodynamic limit and its Rabi-model counterpart in the classical oscillator limit have been theoretically established (Hepp & Lieb 1973; Wang & Hioe 1973; Hwang et al. 2015). The no-go theorem, due to the A2 term in cavity QED, forbids equilibrium SPT by eliminating the critical singularity (Rzażewski et al. 1975; Knight et al. 1978; Nataf & Ciuti 2010; Viehmann et al. 2011; Vukics et al. 2014; Jaako et al. 2016; Andolina et al. 2019, 2020). Hybrid circuit QED proposals suggested auxiliary squeezing to bypass the no-go theorem, but practical realization is limited by weak quadratic optomechanical coupling at single-photon levels (Aspelmeyer et al. 2014; Lü et al. 2018). Quantum simulation platforms have explored Rabi/Dicke dynamics and nonequilibrium transitions (BECs, trapped ions, superconducting circuits), and a trapped-ion experiment observed a QRM quantum phase transition, yet the explicit effect of the no-go theorem had not been experimentally addressed. Recent theory and experiments have highlighted antisqueezing-enhanced light-matter interactions via parametric amplification, motivating this study to test antisqueezing-enabled equilibrium SPT.

Physical model: The effective Hamiltonian is H = H_R + H_A + H_AS, with H_R = Ωσ_x + ω a†a + λ(a† + a)σ_z, A2 term H_A = (α λ^2/2)(a† + a)^2 (α ≥ 1), and an antisqueezing term H_AS = ξ (a† + a)^2. In the classical oscillator limit Ω/ω → ∞, the A2 term removes SPT; adding antisqueezing can recover it by enhancing ZPF. Theoretical criticality occurs when λ̃ ≡ 2λ/√(Ωω) satisfies λ̃ = √(1 + αλ̃^2 − 4ξ/ω), with instability when 1 + αλ̃^2 − 4ξ/ω < 0. Spin-to-oscillator mapping and platform: The bosonic mode is mapped to N nuclear spins via a binary encoding of Fock states using a spin-to-oscillator mapping akin to Holstein-Primakoff (exact as N→∞). In this experiment, a 4-qubit NMR quantum simulator is used: one 13C spin encodes the two-level system and three 19F spins encode the truncated boson (2^3 levels). The molecule is 13C-iodotriuroethylene in d-chloroform, run on a Bruker Avance III 400 MHz spectrometer at room temperature. The natural NMR Hamiltonian is H_NMR = Σ_i ν_i I_i^z + Σ_{i<j} J_ij I_i^z I_j^z with chemical shifts and J-couplings provided. The system is initialized from thermal equilibrium to a pseudo-pure state ρ_PPS with polarization ε≈10^-5 using line-selective pulses; PPS fidelity ≈0.991. Ground-state preparation: An exact squeezing transformation maps the ground state of H to S(r)|G⟩_s, where H_s = S(r) H S(r) is a transformed Hamiltonian with renormalized parameters and r = (1/4) ln(1 + αλ̃^2 − 4ξ/ω). The ground state of H_s is prepared by adiabatic evolution: starting from the ground state of H_0 = Σ_i σ_i^z (prepared by simultaneous π/2 y-pulses), the system evolves under Ĥ(t) = [1 − s(t)] H_0 + s(t) H_s, with s(t) slowly ramped from 0 to 1 over L steps. Control sequences are implemented via GRAPE pulses (adiabatic evolution pulse duration ~26 ms). Measurement protocols: Zero-point fluctuations under antisqueezing are characterized by preparing truncated squeezed vacuum states with a GRAPE-implemented squeezing operator (duration ~15 ms) and measuring ⟨a†a + aa†⟩, ⟨a + a†⟩, and ⟨a^2 + a†2⟩ via tailored readout operations (π/2 readouts on each qubit; custom GRAPE readouts Û1 = Σ_n |n⟩⟨n+1| + |5⟩⟨0| and Û2 = Σ_n |n⟩⟨n+2| + |4⟩⟨0| + |5⟩⟨1|) and SWAPs to the 13C channel (necessary due to 1% natural abundance of 13C). Order parameter and postprocessing: The SPT order parameter is Φ = (ω/2)⟨a†a⟩ in the original frame. Using postprocessing with the known squeezing relation S†(γ) a†a S(γ) = cosh(2γ) a†a − (1/2) sinh(2γ) (a^2 + a†2) + sinh^2 γ, Φ is computed from expectations measured in the prepared ground state |G⟩_s (or equivalently |G⟩_0 in the postprocessed scheme). Quantum state tomography is performed to reconstruct reduced density matrices and Wigner functions for selected parameter points.

• Antisqueezing-enhanced ZPF: Measured zero-point fluctuation (ZPF) of the oscillator increases exponentially with antisqueezing strength ξ, consistent with theory. Error bars shrink with increasing ξ due to larger signal amplitude. Wigner functions of reconstructed squeezed states confirm squeezing.
• Recovery of equilibrium SPT beyond the no-go theorem: With A2 term present and no antisqueezing (ξ = 0), no phase transition is observed (order parameter Φ remains ≈0), confirming the no-go theorem. Introducing antisqueezing (ξ/ω = 0.26) recovers SPT: Φ exhibits a sharp increase at a critical coupling consistent with theory. A reversed transition is observed as λ is decreased, with Φ changing from ≈0 to finite around λ ≈ 0.63 (for finite Ω/ω), reflecting competition between A2 and antisqueezing terms.
• Classical oscillator limit approach: Increasing Ω/ω moves the observed transition toward the analytical critical point λ̃ = √(1 + αλ̃^2 − 4ξ/ω), with faster growth of Φ near criticality. Finite truncation limits Φ deep in SP, mitigated by postprocessing.
• Entanglement and Schrödinger cat states: In the superradiant phase (e.g., λ = 0.2, 0.3 with Ω/ω = 25 and ξ/ω = 0.26), reconstructed ground states show strong spin-oscillator entanglement (increased von Neumann entropy) and squeezed cat states with negative Wigner distributions, large peak separation, and interference fringes. In the normal phase (e.g., λ = 1), entanglement is low and Wigner function lacks interference.
• Phase diagram and scaling: The measured phase diagram φ(λ, Ω/ω) for ξ/ω = 0.26 matches theoretical NP/SP boundaries increasingly well as Ω/ω grows. Finite-frequency scaling at criticality yields an exponent γ ≈ 0.676, close to the universal 2/3 for Rabi/Dicke models. A necessary condition Ω/ω > 1/4 (for fixed ξ/ω) is observed for SPT. An unstable phase (UP) appears for too large ξ (when 1 + αλ̃^2 − 4ξ/ω < 0), indicating the Hamiltonian becomes unbounded from below.
• Practical parameters and measurements: Demonstrations include Ω/ω values such as 5, 25, 50; antisqueezing ξ/ω = 0.26; critical coupling near λ ≈ 0.63; GRAPE pulse durations of 15 ms (squeezing) and 26 ms (adiabatic evolution).

The experiments directly address whether equilibrium SPT can be realized despite the A2 no-go theorem by introducing antisqueezing. By exponentially enhancing zero-point fluctuations, antisqueezing restores the ground-state singularity necessary for SPT, leading to a measurable order parameter transition even with the A2 term included. The observed reversed transition as λ decreases stems from the competition between the λ-dependent A2 contribution and the λ-independent antisqueezing-induced modification of the oscillator potential. The phase boundaries and critical scaling align with theoretical predictions and universal behavior of the Rabi/Dicke universality class, supporting that the NMR quantum simulator faithfully captures the equilibrium quantum criticality in the finite-frequency regime. The generation of strong spin-oscillator entanglement and squeezed cat states in the SP underlines the resourcefulness of the phase for metrology and fault-tolerant computation. These results demonstrate that λ^2-induced no-go constraints are not absolute and can be relaxed by engineered antisqueezing, suggesting broader applicability across platforms where parametric amplification is available.

This work provides a proof-of-principle experimental quantum simulation of an equilibrium superradiant phase transition beyond the no-go theorem by introducing antisqueezing. Using an NMR platform with a spin-to-oscillator mapping and adiabatic ground-state preparation, the study shows: (i) antisqueezing exponentially enhances ZPF; (ii) SPT reappears and can reverse with decreasing coupling in the presence of the A2 term; (iii) strong entanglement and squeezed Schrödinger cat states are realized in the superradiant phase; and (iv) phase boundaries and finite-frequency scaling (γ ≈ 0.676) are consistent with theory. These findings suggest new routes to realize equilibrium SPT and associated quantum resources in diverse platforms, including trapped ions and NV centers. Future work could scale the bosonic Hilbert space, implement stronger and tunable antisqueezing, explore dynamics and critical exponents with higher precision, and investigate robustness and applications in quantum metrology and fault-tolerant bosonic encoding.

• Finite truncation of the bosonic Hilbert space (three 19F qubits → 2^3 levels) limits accuracy deep in the superradiant phase and reduces the observed order parameter magnitude; truncated squeezing loses validity for large r.
• The approach relies on postprocessing via known squeezing transformations; direct implementation of large squeezing in small Hilbert spaces is challenging.
• Experiments are in the finite-frequency regime; exact classical oscillator limit (Ω/ω → ∞) is approached but not reached.
• Natural abundance sample leads to only ~1% 13C molecules, necessitating SWAP-based readout and potentially reducing signal-to-noise.
• Excessive antisqueezing drives the system to an unstable phase where the effective Hamiltonian is unbounded from below; coupling cannot be reduced arbitrarily.
• The study does not resolve the fundamental theoretical debate on including the A2 term in effective models; it demonstrates a simulation-based route to bypass its constraints.