Felicetti and Colleagues Introduce Nonlocal Quantum Fluctuations for Remote System Control

Scientists have identified a new mechanism driving phase transitions through nonlocal quantum fluctuations by Alessandro Coppo of  National Research Council (ISC-CNR), and colleagues fromPurdue University, Aalto University, Universita degli Studi di Pavia and Sapienza University. The team reveal that entanglement between environmental modes can induce correlated symmetry breaking in spatially separated systems, offering a fresh perspective beyond traditional thermal and quantum critical phenomena. Their theoretical investigation of coupled nonlinear quantum resonators shows that these emergent nonlocal phase transitions, governed by quantum correlations in the environment, manifest as spontaneous symmetry breaking of a collective mode shared between remote systems near a critical point. The findings highlight the key role of environmental entanglement in shaping the critical behaviour of complex quantum systems and expands our understanding of emergent phenomena.

Entanglement drives symmetry breaking via nonlocal quantum fluctuations

Phase transitions and quantum fluctuations in quantum critical phenomena are currently under investigation. Nonlocal quantum fluctuations represent a new fundamental mechanism to drive phase transitions, demonstrating that entanglement between environmental modes can induce correlated symmetry breaking in remote systems, irrespective of spatial separation. A theoretical investigation using the framework of driven, dissipative phase transitions focuses on a system comprising two nonlinear quantum resonators placed at arbitrarily large spatial separations, each coupled to independent local Markovian baths.

The study considers a regime involving remote environmental modes prepared in broadband entangled states. Quantum correlations in the environments govern the system’s critical behaviour near the critical point, where susceptibility to weak perturbations diverges. These correlations manifest locally as effective thermal fluctuations, yet give rise to an emergent nonlocal phase transition at the global level, marked by the spontaneous symmetry breaking of a collective mode shared by the two remote systems.

The study of phase transitions is relevant across diverse research areas including condensed matter physics, statistical mechanics, complex systems, cosmology, and high-energy physics. Critical phenomena are considered both a conceptual framework and a practical resource for quantum computing and metrology in quantum information. Typically, phase transitions are studied in systems at thermal equilibrium with their environment, with classical transitions driven by thermal fluctuations and quantum transitions persisting at zero temperature.

Driven-dissipative systems can exhibit critical phenomena, where the non-equilibrium steady-state manifold undergoes a nonanalytic change in response to an infinitesimal variation of a control parameter. Experimental observation of this behaviour has occurred in various atomic and solid-state systems, with applications in proof-of-concept sensing experiments. Such systems extend the concept of thermal and quantum phase transitions to those driven out of thermal equilibrium by an external source.

Critical phenomena are typically studied in the thermodynamic limit of many-body systems, where the number of constituents tends to infinity. However, non-analyticities can also emerge in finite-component systems, provided they can explore an infinite-dimensional Hilbert space. Nonlinear quantum resonators provide a paradigmatic and physically relevant example, with first- and second-order dissipative phase transitions recently observed with trapped ions and superconducting circuits, enabling the study of critical phenomena in a highly controllable way and offering high interest for quantum-computing applications.

Both atomic and solid-state systems can interact with entangled signals supported by optical or telecom photons or propagating microwaves. This work introduces a novel conceptual framework in which phase transitions are induced by nonlocal quantum fluctuations. A driven-dissipative quantum nonlinear resonator embedded in a Markovian bath serves as a minimal model, effectively reproducing the critical phenomena of a broad class of fully-connected models, such as the infinite-range Ising, Rabi, and Dicke models.

Theoretical study focuses on a system with two identical copies of such critical systems placed at an arbitrarily large separation, each interacting with its own bath. The baths are not directly coupled, but are assumed to be in an entangled state. Modelling the dynamics using the Caldeira-Leggett model of bosonic environments leads to the derivation of the Lindblad master equation for a broadband two-mode squeezed state of the baths, allowing characterisation of the global system’s critical behaviour using complementary analytical and numerical methods.

A Gaussian theory, including second-order quantum fluctuations on top of mean-field solutions, describes the driven-dissipative phase diagram of a single critical resonator, far from the critical region, and captures the two-mode squeezing of Bogoliubov excitations due to the nonlocal bath fluctuations. Interestingly, exact full-quantum numerical simulations show nontrivial properties of the global-system steady state near the critical point, despite the Gaussian model not predicting any modification to the symmetry breaking. A local observer would witness a standard driven-dissipative phase transition in the presence of a thermal bath, yet an observer with access to the global system could resolve the emergence of delocalized collective modes, for which the phase transition is enhanced or suppressed.

Numerical evidence indicates that, owing to the nonlocal character of the transition, the two critical resonators undergo correlated symmetry breaking despite being uncoupled and separated by arbitrarily large distances. To understand the emergence of this nonlocal symmetry breaking, a non-Gaussian theory valid in the critical region is developed. Phase-space methods are used, building on the slaving principle, which uses the separation of time scales induced by critical slowing down.

This analytical approach unveils the emergence of nonlocal critical modes induced by nonlocal quantum fluctuations and provides a clear interpretation of their physical origin. These analytical and numerical results open a new research direction centred on critical phenomena induced by nonlocal quantum fluctuations, including their experimental realisation and application in quantum information science. A general framework for understanding symmetry breaking is a main contribution of this work.

When criticality arises from Z2 spontaneous symmetry breaking, the Hamiltonian of a single, closed critical quantum system commutes with the parity operator Π. As a control parameter λ is varied across a critical value λc, the system undergoes a phase transition. For λ > λc, the low-energy sector develops two quasi-degenerate parity eigenstates, resulting from the superposition of the symmetry-broken states |R⟩ and |L⟩, related by Π|R⟩= |L⟩. In the thermodynamic limit, the degeneracy becomes exact, and the system spontaneously settles into either |R⟩ or |L⟩. These two phases can be characterised by an order parameter, defined as the expectation value of an observable O anticommuting with Π, vanishing in the normal phase and acquiring a finite value with opposite sign in the states |L⟩ and |R⟩. Similar critical phenomena arise in non-equilibrium open quantum systems, where phase transitions occur in the steady state ρ reached asymptotically under the combined effect of external driving and dissipation. In the normal phase, the steady state ρ0 preserves the symmetry, yielding a vanishing order parameter.

Beyond the critical point, in the thermodynamic limit, the steady state becomes a classical mixture of two symmetry-broken states ρL and ρR, with the global steady state reading ρ= (ρL + ρR)/2 ⊗(ρL + ρR)/2. The central setup of this work introduces a configuration where the two critical systems and their respective environments remain spatially remote and strictly uncoupled, while the environments are prepared in an entangled state. This raises fundamental questions: can quantum correlations between the environmental states actively modify the structure of the symmetry-broken phase and, more broadly, the very nature of the phase transition itself. Can they give rise to a non-locally ordered phase where the two systems become either correlated or anti-correlated?

The model focuses on driven Kerr resonators, which are minimal critical systems whose dissipative phase transitions have recently been experimentally characterised. The Hamiltonian is written in the frame rotating at the drive frequency: Ha= ωa† a+ λ 2 a† a† + aa + εa† a† aa, where ω is the drive-to-cavity detuning, λ is the two-photon drive intensity, and ε is the Kerr nonlinearity. This model becomes critical in the limit ε→0, which plays the role of an effective thermodynamic limit, reproducing the critical behaviour of the fully-connected Ising model up to corrections of order O(1/N2), under a re-parametrization in which ε∼1/N, where N is the number of spins. The resonator is assumed to be coupled to a 3-dimensional system and environment.

Entanglement’s role as a long-range driver of material phase changes

Researchers have established a new mechanism for phase transitions, those shifts in a material’s properties, showing that entanglement, a uniquely quantum connection between particles, can induce changes even when systems are far apart. Professors acknowledges a key hurdle: their current theoretical model relies on creating highly specific, broadband entangled states, a challenging feat with existing technology. Despite this, identifying a new driver for phase transitions remains significant. This theoretical advance offers a novel perspective for designing materials with tailored responses, potentially impacting fields like quantum sensing and information processing, even with incremental improvements in entanglement generation technology.

Researchers demonstrated that nonlocal quantum fluctuations, arising from entanglement shared between environmental modes, can drive phase transitions in systems even when spatially separated. This finding establishes entanglement as a fundamental mechanism for symmetry breaking, alongside thermal and quantum fluctuations. The study, using a theoretical model of two nonlinear quantum resonators, reveals that correlations in the environment govern the critical behaviour of the system near a transition point. The authors note the need for advancements in generating the specific broadband entangled states used in their model to further explore these phenomena.