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arXiv:2406.09134 · Optics Express

Entanglement, squeezing and non-locality limits in filtered two-mode squeezed mixed states

Souvik Agasti

Published Apr 9, 2025• 4 citations
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Abstract

We investigate the entanglement and non-locality between specific spectral components of continuous variable two-mode squeezed mixed states, identifying their limits. These spectral components are selected from output modes using filters commonly employed in optomechanical systems. Both entanglement and non-locality reach their peak when the filters are identical. However, increasing the degree of input squeezing while applying non-identical filters disrupts both entanglement and non-locality, leading to a bell-shaped pattern. Additionally, we provide precise boundaries for entanglement and non-locality. Furthermore, we also evaluate the squeezing of two-mode hybrid quadrature as a measure of entanglement, thereby demonstrating how it remains analogous to logarithmic negativity. Combined with the filter, the population of two-mode squeezed thermal light influences the angle of a maximally squeezed hybrid quadrature.

DOI
10.1364/oe.557971

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We investigate the entanglement and non-locality between specific spectral components of continuous variable two-mode squeezed mixed states, identifying their limits. These spectral components are selected from output modes using filters commonly employed in optomechanical systems. Both entanglement and non-locality reach their peak when the filters are identical. However, increasing the degree of input squeezing while applying non-identical filters disrupts both entanglement and non-locality, leading to a bell-shaped pattern. Additionally, we provide precise boundaries for entanglement and non-locality. Furthermore, we also evaluate the squeezing of two-mode hybrid quadrature as a measure of entanglement, thereby demonstrating how it remains analogous to logarithmic negativity. Combined with the filter, the population of two-mode squeezed thermal light influences the angle of a maximally squeezed hybrid quadrature.
We explore how entanglement and non-locality evolve between specific spectral components of two-mode squeezed states in thermal environments. These spectral components are extracted from output modes using filters that are frequently utilized in optomechanical systems. We consider two distinct thermalization scenarios: one occurring in the vacuum state prior to entering the nonlinear crystal for squeezing and another after the generation of the two-mode squeezed vacuum but before passing through filters and detectors. Entanglement and non-locality generally remain at their peak when identical filters are applied throughout. In the first scenario, higher initial squeezing causes the entanglement dissipation to slow down at the beginning of the time evolution, followed by a progressive acceleration of entanglement dissipation over time. However, the dissipation rate of non-locality, even though it changes over time it moreover remains the same irrespective of the initial degree of squeezing. In the second scenario, greater squeezing results in a more rapid loss of both entanglement and non-locality. We identify the evolution of specific boundaries for entanglement and non-locality and the conditions for their optimization. Finally, for all the cases, increasing the thermal population of the environment enhances the rate of dissipation, whereas stronger interaction slows dissipation in a normalized dimensionless time scale.
Spectral components of continuous squeezed fields are entangled. In this article we review and clarify this phenomenon by analyzing systematically the relations between the correlations of modes filtered from stationary continuous fields and the cross-power spectrum between the operators of the corresponding spectral components. Moreover, we study the specific spectral components that are filtered in homodyne or heterodyne detections and their entanglement properties. In particular, we establish the equivalence between two-mode squeezing variance and logarithmic negativity for the spectral components of continuous stationary fields, thereby demonstrating that the measurement of the homodyne or heterodyne spectrum is, in fact, a direct measurement of the logarithmic negativity between specific spectral modes. As an illustrative example, we apply these concepts to the analysis of entanglement in ponderomotive squeezing.
We investigate how to generate continuous-variable entanglement between distant optomechanical and spin systems, by transferring input two-mode squeezed vacuum state to the system. Such a setup has been proposed for backaction evading gravitational-wave measurement, squeezing the output noise below the standard quantum limit. We find that the spin cavity entanglement saturates to a particular value when no mechanics are involved even though the entanglement of the input beam increases steadily, and drops down when the mechanical oscillator interacts with the cavity. Our study also reveals that the spin optical readout rate enables the robustness of the spin-cavity entanglement with input squeezing whereas the optomechanical coupling strength disables it. The entanglement reaches its maximum when the effective resonance frequency and bandwidth of the cavity match the spin system. Determining collective quadrature fluctuations, our analysis also shows that even though the entanglement between spin and cavity, and cavity and mechanics is significantly present; it is still impossible to obtain entanglement between spin and mechanical oscillator.
We explore here how to generate a two-mode squeezed output using reservoir engineering in a double-cavity optomechanical system coupled to a common mechanical resonator. Such hybrid platforms are experimentally accessible in electro-optomechanical interfaces and are relevant for high-fidelity state transfer, quantum communication, and metrological applications. By examining violations of the CHSH Bell inequality, we demonstrate that maximal squeezing does not necessarily imply nonlocality; instead, nonlocal correlations can emerge in states with lower squeezing. Furthermore, by analyzing the CHSH inequality across different cavity finesse values, we find that the parameter region supporting nonlocality can broaden even as the squeezing region shrinks. Across all regimes considered, our results emphasize the crucial influence of the mixedness of the state in determining the relationship between squeezing and nonlocality.
We study theoretically a two-mode optomechanical system where two cavity modes are coupled to a single mechanical mode. It is shown that the whole system can yield good squeezing, which is comparable to that produced by dispersive coupling, the numerical results which we obtained showed periodic oscillations with a maximum peaks in one of the two quadrature component with a value less than 0.5. We also showed that entanglement can be achieved by calculating the variance in quadrature component of the field operators. Our results are useful in non- degenerate parametric oscillation.
We derive a general condition for nonlocality using the Banaszek-Wódkiewicz phase-space Wigner representation of Bell's function in continuous-variable Gaussian states, showing rigorously that while entanglement is necessary, it is not sufficient for nonlocality, whereas nonlocality always implies entanglement. Furthermore, to demonstrate the applicability of the framework, we analyze the two-mode squeezed output generated by a double-cavity optomechanical system coupled to a common mechanical resonator-an experimentally accessible hybrid electro/optomechanical interface. We show that maximal entanglement does not necessarily lead to Bell violation, whereas nonlocality may arise in states with lower entanglement, highlighting the crucial role of mixedness. The developed formalism is universally applicable to all bipartite Gaussian states and provides a general tool for investigating foundational and operational aspects of nonlocality in continuous-variable quantum platforms.