Aleek Maity | Searcharxiv

arXiv2025

The Dirac--Bergmann approach to optimal control theory

We present a novel framework for optimal control in both classical and quantum systems. Our approach leverages the Dirac--Bergmann algorithm: a systematic method for formulating and solving constrained dynamical systems. In contrast to the standard Pontryagin Principle, which is used in control theory, our approach bypasses the need to perform a variation to obtain the optimal solution. Instead, the Dirac--Bergmann algorithm generates the optimal solution automatically, through the closure of the Poission Bracket algebra of the full set of constraints and the Hamiltonian. The efficacy of our framework is demonstrated through two quintessential examples: (1) the classical brachistochrone problem and (2) the time-optimal control of a generic quantum system, relevant for quantum technological applications. In the latter example, both closed and open quantum systems are discussed.

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arXiv2024

Dirac Brackets $\leftrightarrow$ Lindblad Equation: A Correspondence

The time evolution of an open quantum system is governed by the Gorini-Kossakowski-Sudarshan-Lindlad equation for the reduced density operator of the system. This operator is obtained from the full density operator of the composite system involving the system itself, the bath, and the interactions between them, by performing a partial trace over the bath degrees of freedom. The entanglement between the system and the bath leads to a generalized Liouville evolution that involves, amongst other things, dissipation and decoherence of the system. In a similar fashion, the time evolution of a physical observable in a classically constrained dynamical system is governed by a generalization of the Liouville equation, in which the usual Poisson bracket is replaced by the so-called Dirac bracket. The generalization takes into account the reduction in the phase space of the system because of constraints which arise either because they are introduced by hand, or because of singular Lagrangians. We derive an intriguing, but precise classical-quantum correspondence between the aforementioned situations which connects the Lindblad operators to the constraints. The correspondence is illustrated in the ubiquitous example of coupled simple harmonic oscillators studied earlier in various contexts like quantum optics, dissipative quantum systems, Brownian motion, and the area law for the entropy of black holes.

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