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arXiv:2207.01383

Algebraic approach for investigation of a multi-mode quantum system dynamics

Andrei Gaidash, Козубов Антон Владимирович, Alexei D. Kiselev, George P. Miroshnichenko

Published Jul 4, 2022• 0 citations
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Abstract

We introduce algebraic approach for superoperators that might be useful tool for investigation of quantum (bosonic) multi-mode systems and its dynamics. In order to demonstrate potential of proposed method we consider multi-mode Liouvillian superoperator that describes relaxation dynamics of a quantum system (including thermalization and intermode coupling). Considered algebraic structure of superoperators that form Liouvillian and their algebraic properties allows us to diagonilize multi-mode Liouvillian to find its spectrum. Also it allows to derive linear by mean number of thermal (environmental) photons approximation for time-evolution superoperator that keeps amount of considered dimensions in Fock space finite (assuming initial amount of dimensions finite) that might be helpful regarding entanglement dynamics problems. Conjugate Liouvillian is considered as well in order to perform analysis in Heisenberg picture, it can be implemented for multi-time correlation functions derivation.

DOI
10.48550/arxiv.2207.01383

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We introduce algebraic approach for superoperators that might be useful tool for investigation of quantum (bosonic) multi-mode systems and its dynamics. In order to demonstrate potential of proposed method we consider multi-mode Liouvillian superoperator that describes relaxation dynamics of a quantum system (including thermalization and intermode coupling). Considered algebraic structure of superoperators that form Liouvillian and their algebraic properties allows us to diagonilize multi-mode Liouvillian to find its spectrum. Also it allows to derive linear by mean number of thermal (environmental) photons approximation for time-evolution superoperator that keeps amount of considered dimensions in Fock space finite (assuming initial amount of dimensions finite) that might be helpful regarding entanglement dynamics problems. Conjugate Liouvillian is considered as well in order to perform analysis in Heisenberg picture, it can be implemented for multi-time correlation functions derivation.
We develop the algebraic method based on the Lie algebra of quadratic combinations of left and right superoperators associated with matrices to study the Lindblad dynamics of multimode bosonic systems coupled a thermal bath and described by the Liouvillian superoperator that takes into account both dynamical (coherent) and environment mediated (incoherent) interactions between the modes. Our algebraic technique is applied to transform the Liouvillian into the diagonalized form by eliminating jump superoperators and solve the spectral problem. The temperature independent effective non-Hermitian Hamiltonian, H^eff\hat{H}_{eff}H^eff​, is found to govern both the diagonalized Liouvillian and the spectral properties. It is shown that the Liouvillian exceptional points are represented by the points in the parameter space where the matrix, HHH, associated with H^eff\hat{H}_{eff}H^eff​ is non-diagonalizable. We use our method to derive the low-temperature approximation for the superpropagator and to study the special case of a two mode system representing the photonic polarization modes. For this system, we describe the geometry of exceptional points in the space of frequency and relaxation vectors parameterizing the intermode couplings and, for a single-photon state, evaluate the time dependence of the speed of evolution as a function of the angles characterizing the couplings and the initial state.
For an open quantum system described by the Lindblad equation, full characterization of its dynamics typically needs the knowledge of the Liouvillian spectrum and correlation functions. Solving the Liouvillian spectrum and correlation functions are usually formidable tasks, and most previous studies are constrained to simple models and lower-order correlations. In this work, we derive a closed form of dynamical recursion equations for all even-order correlation functions associated with the quadratic Liouvillian superoperator, thus extending Wick's theorem and enabling the reconstruction of the corresponding Liouvillian spectrum. Furthermore, we study the quartic Liouvillian superoperator, establishing the necessary and sufficient conditions for the closure of second-order correlation functions. Building on this, the dynamical expressions for even-order correlation functions under quadratic dissipation are derived, culminating in a method for constructing the Liouvillian spectrum in this extended framework.
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.
We employ an exact solution of the thermal bath Lindblad master equation with the Liouvillian superoperator that takes into account both dynamic and environment-induced intermode couplings to study the speed of evolution and quantum speed limit (QSL) times of a open multi-mode bosonic system. The time-dependent QSL times are defined from quantum speed limits, giving upper bounds on the rate of change of two different measures of distinguishability: the fidelity of evolution and the Hilbert-Schmidt distance. For Gaussian states, we derive explicit expressions for the evolution speed and the QSL times. General analytical results are applied to the special case of a two-mode system where the intermode couplings can be characterized by two intermode coupling vectors: the frequency vector and the relaxation rate vector. For the system initially prepared in a two-mode squeezed state, dynamical regimes are generally determined by the intermode coupling vectors, the squeezing parameter and temperature. When the vectors are parallel, different regimes may be associated with the disentanglement time, which is found to be an increasing (a decreasing) function of the length of the relaxation vector when the squeezing parameter is below (above) its temperature-dependent critical value. Alternatively, we study dynamical regimes related to the long-time asymptotic behavior of the QSL times, which is characterized by linear time dependence with the proportionality coefficients defined as the long-time asymptotic ratios. These coefficients are evaluated as a function of the squeezing parameter at varying temperatures and relaxation vector lengths. We also discuss how the magnitude and orientation of the intermode coupling vectors influence the maximum speed of evolution and dynamics of the entropy and the mutual information.
Abstract Open quantum systems that interact with a Markovian environment can be described by a Lindblad master equation. The generator of time-translation is given by a Liouvillian superoperator acting on the density matrix of the system. As the Fock space for a single bosonic mode is already infinite-dimensional, the diagonalization of the Liouvillian has to be done on the creation- and annihilation-superoperators, a process called ‘third quantization’. We propose a method to solve the Liouvillian for quadratic systems using a single symplectic transformation. We show that the non-Hermitian effective Hamiltonian of the system, next to incorporating the dynamics of the system, is a tool to analyze its symmetries. As an example, we use the effective Hamiltonian to formulate a -‘symmetry’ of an open system. We describe how the inclusion of source terms allows us to obtain the cumulant generating function for observables such as the photon current.
We present an exact Liouvillian spectral reconstruction for a finite-temperature bosonic mode, including eigenvalues, physical eigenoperators, and expansion coefficients for arbitrary initial conditions. The eigenvalues are independent of the thermal occupation, whereas the physical eigenoperators retain the reservoir-temperature dependence, with the thermal stationary state corresponding to the zero Liouvillian mode. Photon-number statistics and two-time correlations connect the spectral structure with relaxation, fluctuations, and temporal coherence by revealing the Liouvillian sectors selected by different observables. The zero-temperature dynamics are recovered in the limit ̅ ̅ ̅ ̅ ̅ 𝑛 → 0 .