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Dark-mode theory for quadratic bosonic networks

Jie‐Qiao Liao

Published Jul 29, 2026• 0 citations
Publisher

Abstract

Abstract Determining dark modes in quadratic bosonic networks is both fundamental and imperative, since the dark modes are critical to coherent network manipulation and quantum resource transfer, and such networks are applicable to wide parameter regimes and various physical areas. Here, we develop a symplectic singular value decomposition (SSVD) method for identifying the number and form of the dark modes in quadratic bosonic networks. This method maps the emergence of dark modes onto a computationally tractable symplectic null-space problem, enabling their systematic identification in large-scale quadratic bosonic networks.We adopt the SSVD method to study both the collective cooling of multiple mechanical modes and optomechanical entanglement generation in linearized optomechanical networks. We find that the appearance of the dark modes suppresses the cooling performance and entanglement generation. Our work paves the way for engineering dark modes in quadratic bosonic networks, and the SSVD method has wide applicability in quantum physics, quantum information science, and many-body physics.

DOI
10.21203/rs.3.rs-10256816/v1

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Abstract Determining dark modes in quadratic bosonic networks is both fundamental and imperative, since the dark modes are critical to coherent network manipulation and quantum resource transfer, and such networks are applicable to wide parameter regimes and various physical areas. Here, we develop a symplectic singular value decomposition (SSVD) method for identifying the number and form of the dark modes in quadratic bosonic networks. This method maps the emergence of dark modes onto a computationally tractable symplectic null-space problem, enabling their systematic identification in large-scale quadratic bosonic networks.We adopt the SSVD method to study both the collective cooling of multiple mechanical modes and optomechanical entanglement generation in linearized optomechanical networks. We find that the appearance of the dark modes suppresses the cooling performance and entanglement generation. Our work paves the way for engineering dark modes in quadratic bosonic networks, and the SSVD method has wide applicability in quantum physics, quantum information science, and many-body physics.
We propose and prove two theorems for determining the number of dark modes in linear two-component quantum networks composed of two types of bosonic modes. This is achieved by diagonalizing the two sub-networks of the same type of modes, mapping the networks to either a standard or a thick arrowhead matrix, and analyzing the linear dependence and independence between the column vectors associated with degenerate normal modes in the coupling matrix. We confirm the two theorems by checking the simultaneous ground-state cooling of the mechanical modes in linearized optomechanical networks. These results also work for linear fermionic networks and other networks described by quadratic coupled-mode Hamiltonian. The present method can be extended to study the dark-state effect in driven atom systems and to construct large decoherence-free subspaces for processing quantum information. This work will initiate the studies on dynamical, transport, and statistical properties of linear networks with decoupled subspaces.
We theoretically investigate a quadratic optomechanical system comprising a single-mode optical cavity linearly coupled to one mechanical resonator and quadratically coupled to a second resonator. By tuning the cavity detuning and optomechanical coupling strengths, we demonstrate the transition from optical bistability to multistability with up to seven steady-state solutions. Notably, simultaneous ground-state cooling of both mechanical resonators occurs on the dynamically stable branch of the nonlinear steady-state solutions, offering new opportunities for combined nonlinear optical and quantum cooling functionalities. Beyond the multistable regime, we systematically study dual-mode ground-state cooling and find that robust simultaneous cooling can be achieved over a broad parameter range, except when the linear and quadratic couplings become comparable, where a dark-mode effect arises. In this case, tuning the second-order optomechanical-induced frequency shifts effectively suppresses dark-mode interference, enabling controllable and simultaneous ground-state cooling. Our results provide a versatile framework for engineering multimode quantum states in optomechanical systems and open new avenues for the development of multifunctional quantum devices, including ultra-sensitive sensors, scalable quantum memories, and integrated quantum networks.
Dark modes arise when destructive interference prevents selected bosonic degrees of freedom from coupling to environmental channels. We formulate a Hamiltonian criterion for identifying such modes in multimode bosonic systems by separating two requirements: the candidate mode must be invisible to the direct system--environment coupling, and its generated operator space must remain invariant under the intrinsic system dynamics. For linear environment coupling and quadratic system Hamiltonians, the criterion is reduced to the the familiar null-space and invariant-subspace conditions of passive linear dark-mode theory. We then extend the analysis to nonlinear scenarios. For a two-photon conversion channel coupled to an auxiliary environmental mode, interference among nonlinear conversion pathways can reduce the environment coupling to a single collective two-photon channel, leaving a complementary bosonic mode decoupled from the environment. We show that preserving this mode under nonlinear intrinsic dynamics generally requires more than conventional Kerr-type quartic interactions: correlated four-boson conversion processes are needed to cancel mixed nonlinear conversion between the dark and environment-coupled collective modes. Finally, we show that the Bogoliubov dark mode of a parametrically driven optomechanical satisfies the same Hamiltonian criterion through an active canonical transformation. These results provide a unified Hamiltonian framework for identifying and engineering dark modes in linear, nonlinear, and driven bosonic systems.
Highly efficient transfer of quantum resources including quantum excitations, states, and information on coupled bosonic networks is an important task in quantum physics and quantum information science. Here we propose a bipartite-graph framework to characterize quantum excitation transfer in coupled bosonic networks. This is achieved by diagonalizing the intermediate subnetwork between the sender and the receiver to construct a bipartite-graph configuration, and hence this treatment can be understood from the viewpoint of network deformation. We examine the covariance matrix of the coupled bosonic networks in both the original and bipartite-graph representations. In particular, we investigate quantum excitation transfer in both the finite and infinite intermediate-normal-mode cases and show the dependence of the transfer efficiency on the network configurations and system parameters. We find the bounds of maximally transferred excitations for various network configurations and reveal the underlying physical mechanisms. We also discover that the dark-mode effect will degrade the excitation transfer efficiency. Our findings provide a new insight for the design and optimization of networks in physics, information theory, and complex system science.
Quantum entanglement not only plays an important role in the study of the fundamentals of quantum theory, but also is considered as a crucial resource in quantum information science. The generation of macroscopic entanglement involving multiple optical and mechanical modes is a desired task in cavity optomechanics. However, the dark-mode effect is a critical obstacle against the generation of quantum entanglement in multimode optomechanical systems consisting of multiple degenerate or near-degenerate mechanical modes coupled to a common cavity mode. Here we propose an auxiliary-cavity-mode method to enhance optomechanical entanglement in a multimode optomechanical system by breaking the dark-mode effect. We find that the introduction of the auxiliary cavity mode not only assists the entanglement creation between the cavity mode and the mechanical modes, but also improves the immunity of the optomechanical entanglement to the thermal excitations by about three orders of magnitude. We also study the optomechanical entanglement in the network-coupled optomechanical system consisting of two mechanical modes and two cavity modes. By analyzing the correspondence between the optomechanical entanglement and the dark-mode effect, we find that optomechanical entanglement can be largely enhanced once the dark mode is broken. In addition, we study the mechanical entanglement and find that it is negligibly small. We also present some discussions on the experimental implementation with a microwave optomechanical setup, on the relationship between the dark-mode-breaking mechanism and the center-of-mass and relative coordinates, and on the explanation of the important role of the dark-mode breaking in the enhancement of optomechanical entanglement. Our results pave the way towards the preparation of entangled optomechanical networks and noise-resistant quantum resources.