Photon Squeezing and Its Signatures of Quantum Phase Transitions in the Open Quantum Rabi-Stark Model

Abstract: As a hallmark of nonclassical light, squeezed light is of profound theoretical interest and holds broad practical promise for emerging quantum technologies. In this work, we investigate steady-state optical quadrature squeezing in the open quantum Rabi-Stark model by employing the quantum dressed master equation. Both numerically and analytically, we find that positive (negative) Stark coupling tends to enhance (suppress) the squeezing effect. The quadrature squeezing exhibits distinct signatures associated with both first- and second-order quantum phase transitions (QPTs). Notably, a sharp vanishing of squeezing is observed across the first-order QPT, suggesting its potential as a sensitive probe of such transitions. In the vicinity of the second-order QPT, we further demonstrate that the squeezing factor displays finite-size scaling behavior, indicating a promising route toward the realization of near-perfect squeezing. Moreover, we establish a quantitative criterion for the disruption of quantum criticality induced by thermal fluctuations, which may offer valuable guidance for future experiments. These findings contribute to a deep understanding of nonclassical light in light-matter interacting systems and provide useful insights for the design of strong optical squeezing states.

• The paper demonstrates that positive Stark coupling enhances photon squeezing, showing abrupt changes at first-order quantum phase transitions.
• It employs a quantum dressed master equation to capture dissipative dynamics and critical scaling near both first- and second-order transitions.
• The detailed analytical and numerical insights provide actionable guidance for optimizing nonclassical light generation in tunable cavity and circuit-QED systems.

Photon Squeezing and Quantum Phase Transition Signatures in the Open Quantum Rabi-Stark Model

Introduction

The study explores quadrature photon squeezing in the open quantum Rabi-Stark model (QRSM), probing both theoretical underpinnings and observable signatures associated with quantum phase transitions (QPTs). The QRSM extends the quantum Rabi model by introducing a tunable nonlinear Stark term, $U$, generating a Hamiltonian of the form $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$. The interplay between the nonlinear Stark coupling, quantum dissipation, and critical phenomena in ultra-strong and deep-strong coupling regimes is addressed using the quantum dressed master equation (DME) formalism, providing a robust treatment beyond the limitations of the standard optical master equation.

Theoretical Model and Dissipation Formalism

The QRSM introduces the Stark term $U a^\dagger a \sigma_z$, generalizing light-matter interactions to regimes where the dispersive nonlinearity and the parity symmetry crucially affect the energy spectrum and quantum states. The model undergoes both first- and second-order QPTs, with analytical expressions available for the corresponding critical points: the first-order QPT at

$g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$

for positive $U$, and a second-order superradiant phase transition (SRPT) for $U \rightarrow \pm 1$, associated with spectral collapse.

To incorporate dissipation under realistic conditions, the authors employ the quantum DME, constructed in the eigenbasis of $H_{\text{RS}}$. This resolves unphysical ground-state excitation in the standard master equation, especially near the ultrastrong-coupling regime. Qubit and photon fields are each coupled to independent Ohmic bosonic thermal baths, yielding a time evolution for the reduced system density matrix $\rho_s$ containing generalized Lindblad dissipators in the dressed basis. This approach enables analysis of steady-state properties, including nonclassical squeezing, under weak but nonzero system-environment coupling.

Quadrature Squeezing and Analytical Formalism

Photon squeezing is characterized by the principal quadrature variance:

$$\xi_B^2 = \min_\theta \{ \langle (\Delta X_\theta)^2 \rangle \}$$

where $X_\theta = a e^{-i\theta} + a^\dagger e^{i\theta}$ and $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$0 indicates squeezing. The authors present numerically exact and analytical results for $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$1 by expressing it in terms of photon number and two-photon coherence terms. In the steady state at low temperature, the underlying ground state is expanded in the Fock basis, and $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$2 is decomposed into contributions from two-photon processes, specifically transitions $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$3.

The analysis identifies that the dominant low-excitation terms—$$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$4 (from $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$5) which enhances squeezing, and $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$6 (from $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$7) which suppresses it—largely determine the steady-state squeezing in experimentally relevant parameter regimes. Higher-excitation processes become relevant only when the linear coupling $$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$8 is sufficiently large.

Stark-Induced Squeezing and QPT Signatures

Numerical calculations reveal parameter domains where positive Stark coupling ($$H_{\text{RS}} = a^\dagger a + \frac{\Delta}{2} \sigma_z + U a^\dagger a \sigma_z + g \sigma_x(a + a^\dagger)$$9) markedly enhances photon squeezing, evidenced by strong reduction of $U a^\dagger a \sigma_z$0 (down to $U a^\dagger a \sigma_z$1 for $U a^\dagger a \sigma_z$2 at large $U a^\dagger a \sigma_z$3), while negative $U a^\dagger a \sigma_z$4 suppresses squeezing. The enhancement arises because a positive $U a^\dagger a \sigma_z$5 favors population in even-photon-number manifolds ($U a^\dagger a \sigma_z$6 relative to $U a^\dagger a \sigma_z$7), thus reinforcing the beneficial two-photon coherence on squeezing.

In proximity to a first-order QPT, squeezing exhibits an abrupt, step-like transition as $U a^\dagger a \sigma_z$8 crosses the critical value. The sharp vanishing of photon squeezing directly corresponds to a parity switch in the ground state, driven by the energy-level crossing intrinsic to the first-order transition. This abrupt behavior provides a strong, experimentally accessible signature for the nonequilibrium probing of first-order QPTs via homodyne detection.

Critical Scaling at the Superradiant Phase Transition

The analysis demonstrates that near the second-order SRPT ($U a^\dagger a \sigma_z$9), $g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$0 exhibits critical enhancement, following a finite-size scaling law:

$g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$1

with system size $g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$2. The scaling exponent converges to $g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$3, in congruence with the QRM ground-state exponents, and is verified via numerical fits to the DME steady state. This implies the feasibility of achieving arbitrarily strong steady-state squeezing by approaching the critical regime with increasing system size and sufficiently low temperature.

The analysis further introduces a universal finite-size scaling function for $g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$4, collapsing numerical data across system sizes and temperatures onto a single curve when plotted as a function of the quantum scaling variable $g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$5 with $g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$6. Breakdown of this scaling, and thus destruction of quantum criticality, occurs once the energy gap $g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$7 at the SRPT satisfies $g_c^{(1)} = \sqrt{(1 - U^2)\Delta} / (2U)$8, providing a practical criterion for the persistence of critical squeezing in dissipative environments.

Implications and Outlook

The study establishes strong theoretical and computational evidence that the QRSM, under weak thermal dissipation, enables scalable control over nonclassical photon squeezing, with tunability via the nonlinear Stark term. The connection between squeezing and the structure of quantum phase transitions, specifically the step-like change at first-order and critical divergence at second-order transitions, suggests new protocols for quantum metrology, quantum information processing, and experimental studies of light-matter models in the ultrastrong coupling regime.

Practically, the findings motivate the engineering of cavity and circuit-QED systems with tunable Stark couplings to access new regimes of nonclassical light. The scaling relations and finite-size analysis provide guidance for optimizing squeezing generation under realistic constraints. Theoretically, the approach strengthens connections between non-equilibrium quantum optics, QPT critical theory, and open quantum system dynamics.

Conclusion

This work characterizes photon quadrature squeezing in the open quantum Rabi-Stark model, revealing enhancements and signatures tightly linked to both first- and second-order quantum phase transitions. The precise analytical decomposition of squeezing into two-photon processes, combined with numerical DME solutions, elucidates the mechanism by which Stark coupling polarity controls nonclassicality. Critical scaling of squeezing at the SRPT is robust to thermal dissipation up to a temperature determined by the system’s finite-size gap, offering valuable benchmarks for future experimental realization. The results contribute to the broader understanding of nonclassical state generation and the role of critical phenomena in driven-dissipative light-matter systems.