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arXiv:1611.04643 · IEEE Transactions on Automatic Control

Dark Modes of Quantum Linear Systems

Yu Pan, Daoyi Dong, Ian R. Petersen

Published Mar 7, 2017• 15 citations
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Abstract

In this paper, we develop a direct method for the characterization of dark modes. The results can be used to construct a transformation that separates dark and bright modes, through the decomposition of system dynamics. We also study a synthesis problem by engineering the system-environment coupling and Hamiltonian engineering. We apply the theory to investigate an optomechanical dark mode.

DOI
10.1109/tac.2017.2677878

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In this paper, we develop a direct method for the characterization of dark modes. The results can be used to construct a transformation that separates dark and bright modes, through the decomposition of system dynamics. We also study a synthesis problem by engineering the system-environment coupling and Hamiltonian engineering. We apply the theory to investigate an optomechanical dark mode.
In this note, we are concerned with dark modes in a class of non-Markovian open quantum systems. Based on a microscopic model, a time-convoluted linear quantum stochastic differential equation and an output equation are derived to describe the system dynamics. The definition of dark modes is given building on the input-output structure of the system. Then, we present a necessary and sufficient condition for the existence of dark modes. Also, the problem of dark mode synthesis via Hamiltonian engineering is constructively solved and an example is presented to illustrate our results.
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.
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.
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Optomechanical dark mode plays a central role in effective mechanically-mediated conversion of two different optical fields. In this paper, two approaches are proposed to generate pure optomechanical dark mode, in which the optomechanical bright mode is absolutely eliminated even with small cooperativity and different losses of the two optical cavities. Only the optomechanical dark mode is left to facilitate quantum state transfer. This result provides a new perspective to utilize the dark state or dark mode for quantum information processing.
We investigate the dissipative dynamics of multilevel quantum systems strongly coupled either to a lossy cavity mode or directly to a bosonic environment. By deriving spectral localization bounds, we establish conditions under which strong system-bath coupling gives rise to a hierarchy of population relaxation timescales. Our approach builds on the reaction-coordinate polaron-transform framework. By mapping the original strong-coupling problem onto an effective weakly dissipative model, we analyze the spectrum of the resulting Liouvillian superoperator through localization bounds. For the generalized V model, we find that strong system-bath coupling gives rise to a bright-dark structure in the effective system-bath coupling operator: a single collective mode remains strongly coupled to the environment, while the remaining modes become progressively dark. Consequently, the dynamics separate into fast and slow sectors and, at finite coupling strengths, develop a hierarchy of population relaxation timescales. Numerical simulations based on both secular and non-secular quantum master equations corroborate the emergence of timescale separation and the pronounced slowing down of dissipative dynamics at strong coupling. Our results reveal a general mechanism underlying anomalously slow relaxation in strongly coupled open quantum systems and provide a route for engineering long-lived states through system-environment interactions.