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arXiv:2303.15137 · Physical Review A

Production of genuine multimode entanglement in circular waveguides with long-range coupling

Tonipe Anuradha, Ayan Patra, Rivu Gupta, Amit Rai, Aditi Sen

Published Mar 11, 2024• 7 citations
arXivHTMLPDF

Abstract

Starting with a product initial state, squeezed (squeezed coherent) state in one of the modes, and vacuum in the rest, we report that a circular waveguide comprising modes coupled with varying coupling strength is capable of producing genuine multimode entanglement (GME), quantified via the generalized geometric measure (GGM). We demonstrate that, for a fixed coupling and squeezing strength, the GME content of the resulting state increases as the range of couplings between the waveguides increases, although the GGM collapses and revives with the variation of coupling strength and time. The advantage of long-range coupling can be emphasized by measuring the area under the GGM curve, which clearly illustrates growing trends of GME with the increasing range of couplings. Moreover, long-range couplings help in generating a higher GGM for a fixed coupling strength. We analytically determine the exact expression of GGM for systems involving an arbitrary number of modes, when all the modes interact with each other equally. The entire analysis is performed in the phase-space formalism. We manifest the constructive effect of disorder in the coupling parameter, which promises a steady production of GME, independent of the coupling strength.

DOI
10.1103/physreva.109.032411

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Starting with a product initial state, squeezed (squeezed coherent) state in one of the modes, and vacuum in the rest, we report that a circular waveguide comprising modes coupled with varying coupling strength is capable of producing genuine multimode entanglement (GME), quantified via the generalized geometric measure (GGM). We demonstrate that, for a fixed coupling and squeezing strength, the GME content of the resulting state increases as the range of couplings between the waveguides increases, although the GGM collapses and revives with the variation of coupling strength and time. The advantage of long-range coupling can be emphasized by measuring the area under the GGM curve, which clearly illustrates growing trends of GME with the increasing range of couplings. Moreover, long-range couplings help in generating a higher GGM for a fixed coupling strength. We analytically determine the exact expression of GGM for systems involving an arbitrary number of modes, when all the modes interact with each other equally. The entire analysis is performed in the phase-space formalism. We manifest the constructive effect of disorder in the coupling parameter, which promises a steady production of GME, independent of the coupling strength.
Genuine multimode entanglement in continuous variable systems can be quantified by exploring the geometry of the state space, namely, via the generalized geometric measure (GGM). It is defined by the shortest distance of a given multimode state from a nongenuinely multimode entangled state. For the multimode Gaussian states, we derive a closed-form expression of GGM in terms of the symplectic eigenvalues of the reduced states. Following that prescription, the characteristics of GGM for typical three- and four-mode Gaussian states are investigated. In the non-Gaussian paradigm, we compute GGM for photon-added as well as -subtracted states having three and four modes and find that both addition and subtraction of photons enhance the genuine multimode entanglement of the state compared to its Gaussian counterpart. Our analysis reveals that, when an initial three-mode vacuum state is evolved according to an interacting Hamiltonian, photon addition is more beneficial in increasing GGM compared to photon subtraction, while the scenario reverses when one considers the four mode non-Gaussian states. Specifically, subtracting photons from four-mode squeezed vacuum states almost always results in higher multimode entanglement content than that of photon addition to both single as well as multimode and constrained as well as unconstrained operations. Furthermore, we observe that GGM freezes under subtraction of photons involving multiple modes, in some specific cases. This feature does not appear while adding photons. Finally, we relate the enhancements of GGM with the distance-based non-Gaussianity measure.
Trends of genuine entanglement in Haar uniformly generated multimode pure Gaussian states with fixed average energy per mode are explored. A distance-based metric known as the generalized geometric measure (GGM) is used to quantify genuine entanglement. The GGM of a state is defined as its minimum distance from the set of all non-genuinely entangled states. To begin with, we derive an expression for the Haar averaged value of any function defined on the set of energy-constrained states. Subsequently, we investigate states with a large number of modes and provide a closed-form expression for the Haar averaged GGM in terms of the average energy per mode. Furthermore, we demonstrate that typical states closely approximate their Haar averaged GGM value, with deviation probabilities bounded by an exponentially suppressed limit. We then analyze the GGM content of typical states with a finite number of modes and present the distribution of GGM. Our findings indicate that as the number of modes increases, the distribution shifts towards higher entanglement values and becomes more concentrated. We quantify these features by computing the Haar averaged GGM and the standard deviation of the GGM distribution, revealing that the former increases while the latter decreases with the number of modes.
We propose a deterministic scheme of generating genuine multiparty entangled states in quantum networks of arbitrary size having various geometric structures---we refer to it as entanglement circulation. The procedure involves optimization over a set of two-qubit arbitrary unitary operators and the entanglement of the initial resource state. We report that the set of unitary operators that maximize the genuine multipartite entanglement quantified via generalized geometric measure (GGM) is not unique. We prove that the GGM of the resulting state of arbitrary qubits coincides with the minimum GGM of the initial resource states. By fixing the output state as the six-qubit one, we find the optimal way to create such states according to the available resource. Moreover, we show that the method proposed here can be implemented by using logic gates, or by using the time dynamics of realizable spin Hamiltonians. In case of an ordered system, GGM varies periodically with time while the evolution via disordered models lead to a low but constant multipartite entanglement in outputs at a critical time, which decreases exponentially with the increase of the strength of the disorder.
Measurement-based quantum correlation mimics several characteristics of multipartite quantum correlations and at the same time, it reduces the parent system to a smaller subsystem. On the other hand, genuine multipartite entanglement measures can capture certain features of a multisite composite system that are inaccessible via bipartite quantum correlation quantifiers. We merge these two concepts by introducing localizable genuine multimode entanglement for continuous variable systems, both for Gaussian and non-Gaussian multimode parent states. We report a compact form of localizable generalized geometric measure for multimode Gaussian states when Gaussian measurements are performed in some of the modes. We show that non-Gaussian measurements can concentrate more genuine multimode entanglement compared to the Gaussian ones. For non-Gaussian states with non-Gaussian measurements, we find that although four-mode squeezed vacuum state has permutation symmetry with respect to the exchange of first and third modes as well as the second and the fourth modes, the symmetry can be broken by performing measurements in one of the modes in case of addition while for subtraction, such symmetry is preserved, thereby providing a method for distinguishing multimode photon-added and -subtracted states via localizations.
We present a scheme for generating and manipulating three-mode squeezed states with genuine tripartite entanglement by injecting single-mode squeezed light into an array of coupled optical waveguides. We explore the possibility to selectively generate single-mode squeezing or multimode squeezing at the output of an elliptical waveguide array, determined solely by the input light polarization. We study the effect of losses in the waveguide array and show that quantum correlations and squeezing are preserved for realistic parameters. Our results show that arrays of optical waveguides are suitable platforms for generating multimode quantum light, which could lead to novel applications in quantum metrology.
Multimode squeezing is a key resource for continuous-variable quantum technologies, but its spatial range in bosonic lattices is usually tied to dispersive propagation. Here we show that parametric pairing can create an exactly flat Bogoliubov band spanned by compact Bogoliubov generators, while producing phase-sensitive anomalous correlations and sub-vacuum collective-mode squeezing that extend far beyond their finite support. Each compact generator mixes annihilation and creation operators, thereby encoding the Bogoliubov squeezing structure, and neighboring translated generators can have nonzero commutator overlap. Enforcing canonical bosonic commutation relations therefore requires spatially extended linear combinations of these translated compact generators, which define the canonical Bogoliubov modes. The squeezing transformation of these modes varies with momentum and is quantified by the squeezing-sector symplectic quantum metric. A complex-momentum singularity of the analytically continued canonical Bogoliubov modes sets both the anomalous-correlation decay length and the momentum-space width of this metric, defining an intrinsic quantum-geometric length scale. This singularity can be tuned continuously while preserving exact flatness, thereby controlling the spatial range of correlations and squeezing. Away from exact flatness, long-distance correlations persist through multiple decay channels, while modes localized by defects and dimerized boundaries provide complementary probes of the underlying quantum geometry. In the weak-damping limit, the same quantum-geometric length scale can be extracted from frequency-filtered two-port correlations. Our results establish the quantum geometry of canonical Bogoliubov modes as a mechanism for spatially extended quantum resources arising from compact Bogoliubov generators.