Using the Wigner function in phase space, we study quantum steering and entanglement between two coupled harmonic oscillators. We derive expressions for purity and quantum steering in both directions and identify several important selection rules. Our results extend the work reported in {\color{blue} [Phys. Rev. E 97, 042203 (2018)]} focused on the weak coupling regime, revealing significant deviations in the ultra-strong coupling regime. In particular, Makarov's prediction of a separable ground state contrasts with our exact calculations, highlighting the limitations of his approach under strong coupling conditions. We show that quantum steering between excited oscillators is completely absent even in the ultra-strong coupling regime. Similarly, resonant oscillators have no steering, and ground states cannot steer any receiver state. We find that quantum steering becomes notably more pronounced as the system approaches resonance and within specific ranges of ultra-strong coupling. This behavior is marked by a clear asymmetry, where steering is present in only one direction, highlighting the delicate balance of interaction strengths that govern the emergence of quantum correlations. These results advance our understanding of how excitation levels and coupling strengths influence quantum steering and entanglement in coupled harmonic oscillators.
Using the Wigner function in phase space, we study quantum steering and entanglement between two coupled harmonic oscillators. We derive expressions for purity and quantum steering in both directions and identify several important selection rules. Our results extend the work reported in {\color{blue} [Phys. Rev. E 97, 042203 (2018)]} focused on the weak coupling regime, revealing significant deviations in the ultra-strong coupling regime. In particular, Makarov's prediction of a separable ground state contrasts with our exact calculations, highlighting the limitations of his approach under strong coupling conditions. We show that quantum steering between excited oscillators is completely absent even in the ultra-strong coupling regime. Similarly, resonant oscillators have no steering, and ground states cannot steer any receiver state. We find that quantum steering becomes notably more pronounced as the system approaches resonance and within specific ranges of ultra-strong coupling. This behavior is marked by a clear asymmetry, where steering is present in only one direction, highlighting the delicate balance of interaction strengths that govern the emergence of quantum correlations. These results advance our understanding of how excitation levels and coupling strengths influence quantum steering and entanglement in coupled harmonic oscillators.
Quantum steering is one of the most intriguing phenomena in quantum mechanics and is essential for understanding correlations in multi-body systems. Despite its importance, analytical results for coupled three-body oscillators remain scarce. In this work, we investigate this phenomenon through a geometrical diagonalization approach, which reduces the degrees of freedom associated with the system's steering properties. Specifically, we derive analytical expressions for quantum steering in all possible directions using the Wigner function framework, as it provides a complete description of the system's quantum state. Our results indicate that excitations significantly enhance quantum steering across the system; this stands in contrast to the ground state (0,0,0), which exhibits no steerable correlations. Furthermore, both the directionality and topology of these correlations are governed by the spatial distribution of the excitations rather than their magnitude. We also observe symmetric steering behavior between oscillators x, y, and z under equivalent excitation conditions, which can be formalized as Sx→z(n,m,l)(θ)=Sx→y(n,m,l)(−θ),Sz→x(n,m,l)(θ)=Sy→x(n,m,l)(−θ), and Sy→z(n,m,l)(θ)=Sz→y(n,m,l)(−θ). Therefore, we elucidate how excitation levels and mixing angles generate and enhance steering in three coupled harmonic oscillators.
Abstract Quantum steering is one of the most intriguing phenomena in quantum mechanics and is essential for understanding correlations in multipartite systems. Despite its importance, analytical results for coupled three-body oscillators remain scarce. In this work, we investigate this phenomenon using a geometrical diagonalization approach, which reduces the degrees of freedom associated with the system’s steering properties to a single effective mixing parameter, making the problem fully tractable within the Wigner function framework. We derive closed-form expressions for quantum steering in all possible directions. Our analysis shows that steering vanishes in the ground state and increases with the quantum number of the excited mode, while excitations significantly enhance quantum steering across the system. However, steering disappears for multi-excitation states involving more than one nonzero quantum number. Furthermore, both the directionality and topology of these correlations are governed by the spatial distribution of the excitations rather than their magnitude. In particular, exciting one mode produces a hub-and-spoke structure in which that mode influences the other two, whereas exciting a different mode leads to a distinct configuration in which no steering is observed between the remaining modes and the excited one. We further uncover a cyclic balance relation representing the equivalence between clockwise and counterclockwise steering flows, as well as a sign-sensitive directionality in which steering in a given direction appears only for specific signs of the effective mixing parameter. We also observe symmetric steering behavior between oscillators under equivalent excitation conditions, reflecting underlying algebraic symmetries consistent with the root structure of the associated Lie algebra. These findings provide an analytical description of quantum steering in tripartite quantum states of coupled oscillators and offer guidance for designing new protocols based on quantum steering in continuous-variable systems.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−P for arbitrary excitation numbers (n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θ and ϑ2, the frequency parameter β0 and the coupling strength ε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1 and strong couplings ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.
Quantum entanglement serves as a key phenomenon in understanding correlations in many-body systems, but analytical results remain scarce for coupled three-body oscillators. In this work, we address this gap by introducing a geometrical diagonalization approach that constrains Euler angles, thereby reducing the degrees of freedom in the entanglement analysis. It consists of deriving analytical expressions for linear entropy and purity under the bipartitions (x∣yz), (y∣xz), and (xy∣z) using the Wigner function framework. Our results indicate that excitations in any oscillator basically enhance the redistribution of correlations across the system. The mixing angle θ governs entanglement intensity, ranging from separability to maximal correlation. Moreover, we reveal the symmetry relations, notably SLy[(n,m,l),θ]=SLz[(n,m,l),−θ] and an intrinsic symmetry within (x∣yz). Hence, we clarify how excitation levels and mixing angles create and enhance entanglement in the three coupled harmonic oscillators.
We study a system of two quadratically-coupled harmonic oscillators, prepared respectively in the ground state and in a coherent state. This toy model can describe for instance the dynamics of one mode of a quantum field evolving in a harmonically-oscillating classical background. In such situations, it is well-known that, depending on the background oscillation amplitude and coupling strength, the system may exhibit parametric resonance. Using the Feynman-Vernon influence functional, we analyze the interplay between parametric resonance and quantum entanglement by perturbatively computing the purity of the mixed state resulting from tracing over one of the oscillators. We find a runaway behavior of the quantum entanglement between the two oscillators, as well as a rapid breakdown of the validity of perturbation theory, in regions of parametric instability. This provides a quantitative confirmation of various statements in the literature and is highly relevant to questions pertaining to the validity of the semiclassical approximation in coupled quantum-classical systems.
We consider two systems A and B that share Einstein–Podolsky–Rosen (EPR) steering correlations and study how these correlations will decay when each of the systems is independently coupled to a reservoir. EPR steering is a directional form of entanglement, and the measure of steering can change depending on whether system A is steered by B, or vice versa. First, we examine the decay of the steering correlations of the two-mode squeezed state. We find that if system B is coupled to a reservoir, then the decoherence of the steering of A by B is particularly marked, to the extent that there is a sudden death of steering after a finite time. We find a different directional effect if the reservoirs are thermally excited. Second, we study the decoherence of the steering of a Schrödinger cat state, modeled as the entangled state of a spin and harmonic oscillator, when the macroscopic system (the cat) is coupled to a reservoir.