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arXiv:2208.12006 · Physical Review Letters

Lie Algebraic Quantum Phase Reduction

Wataru Setoyama, Yoshihiko Hasegawa

Published Feb 27, 2024• 7 citations
arXivHTMLPDF

Abstract

We introduce a general framework of phase reduction theory for quantum nonlinear oscillators. By employing the quantum trajectory theory, we define the limit-cycle trajectory and the phase according to a stochastic Schrödinger equation. Because a perturbation is represented by unitary transformation in quantum dynamics, we calculate phase response curves with respect to generators of a Lie algebra. Our method shows that the continuous measurement yields phase clusters and alters the phase response curves. The observable clusters capture the phase dynamics of individual quantum oscillators, unlike indirect indicators obtained from density operators. Furthermore, our method can be applied to finite-level systems that lack classical counterparts.

DOI
10.1103/physrevlett.132.093602

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We introduce a general framework of phase reduction theory for quantum nonlinear oscillators. By employing the quantum trajectory theory, we define the limit-cycle trajectory and the phase according to a stochastic Schrödinger equation. Because a perturbation is represented by unitary transformation in quantum dynamics, we calculate phase response curves with respect to generators of a Lie algebra. Our method shows that the continuous measurement yields phase clusters and alters the phase response curves. The observable clusters capture the phase dynamics of individual quantum oscillators, unlike indirect indicators obtained from density operators. Furthermore, our method can be applied to finite-level systems that lack classical counterparts.
Measurement backaction inherently alters observed dynamics in quantum physics. In the realm of quantum synchronization, this backaction induces a phase bias, making the assessment of synchronization critically dependent on the choice of the observables. In this study we extend the quantum phase reduction approach [W. Setoyama and Y. Hasegawa, Phys. Rev. Lett. 132, 093602 (2024)] into heterodyne detection, offering a comprehensive theoretical framework for analyzing quantum synchronization dynamics through uniform continuous measurement over all possible quadrature observables. This method averages out the backaction, allowing for unbiased evaluation of synchronization between quantum oscillators while avoiding measurement-induced phase bias. Furthermore, by defining the phase and limit-cycle solution independently of specific observables, our proposed method consistently adapts to the scenario where the observables are freely modified during the time evolution. Through simulations of noise-induced synchronization, our method reveals that the number of phase clusters between oscillators is restricted by their bosonic levels.
Quantum technologies rely on the control of quantum systems at the level of individual quanta. Mathematically, this control is described by Hamiltonian or Liouvillian evolution, requiring the application of various techniques to solve the resulting dynamic equations. Here, we present a tutorial for how the quantum dynamics of systems can be solved using a Lie-algebra decoupling method. The approach involves identifying a Lie algebra that governs the dynamics of the system, enabling the derivation of differential equations to solve the Schrödinger equation. As background, we include an overview of Lie groups and Lie algebras aimed at a general-physicist audience. We then prove the Lie-algebra decoupling theorem and apply it to both closed and open dynamics. The results represent a broad methodology to find the dynamics of quantum systems with applications across many fields of modern quantum research.
Beyond their use as numerical tools, quantum trajectories can be ascribed a degree of reality in terms of quantum measurement theory. In fact, they arise naturally from considering continuous observation of a damped quantum system. A particularly useful form of quantum trajectories is as linear (but non-unitary) stochastic Schrödinger equations. In the limit where a strong local oscillator is used in the detection, and where the system is not driven, these quantum trajectories can be solved. This gives an alternative derivation of the probability distributions for completed homodyne and heterodyne detection schemes. It also allows the previously intractable problem of real-time adaptive measurements to be treated. The results for an analytically soluble example of adaptive phase measurements are presented, and future developments discussed.
Beyond their use as numerical tools, quantum trajectories can be ascribed a degree of reality in terms of quantum measurement theory. In fact, they arise naturally from considering continuous observation of a damped quantum system. A particularly useful form of quantum trajectories is as linear (but non-unitary) stochastic Schrodinger equations. In the limit where a strong local oscillator is used in the detection, and where the system is not driven, these quantum trajectories can be solved. This gives an alternate derivation of the probability distributions for completed homodyne and heterodyne detection schemes. It also allows the previously intractable problem of real-time adaptive measurements to be treated. The results for an analytically soluble example of adaptive phase measurements are presented, and future developments discussed.
We study the time evolution of driven quantum systems using analytic, algebraic, and numerical methods. First, we obtain analytic solutions for driven free and oscillator systems by shifting the coordinate and phase of the undriven wave function. We also factorize the quantum evolution operator using the generators of the Lie algebra comprising the Hamiltonian. We obtain coupled ODE's for the time evolution of the Lie algebra parameters. These parameters allow us to find physical properties of oscillator dynamics. In particular we find phase-space trajectories and transition probabilities. We then search for chaotic behavior in the Lie algebra parameters as a signature for dynamical chaos in the quantum system. We plot the trajectories, transition probabilities, and Lyapunov exponents for a wide range of the following physical parameters: strength and duration of the driving force, frequency difference, and anharmonicity of the oscillator. We identify conditions for the appearance of chaos in the system.
The algebra of generalized linear quantum canonical transformations is examined in the prespective of Schwinger's unitary-canonical basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the quantum canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with the dynamical symmetry are examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.