Lie algebraic techniques are powerful and widely used tools for studying dynamics and metrology in quantum optics. When the Hamiltonian generates a Lie algebra with finite dimension, the unitary evolution can be expressed as a finite product of exponentials using the Wei-Norman expansion. The system is then exactly described by a finite set of scalar differential equations, even if the Hilbert space is infinite. However, the differential equations provided by the Wei-Norman expansion are nonlinear and often have singularities that prevent both analytic and numerical evaluation. We derive a new Lie algebra expansion for the quantum Fisher information, which results in linear differential equations. Together with existing Lie algebra techniques this allows many metrology problems to be analysed entirely in the Heisenberg picture. This substantially reduces the calculations involved in many metrology problems, and provides analytical solutions for problems that cannot even be solved numerically using the Wei-Norman expansion. It also allows us to study general features of metrology problems, valid for all quantum states. We provide detailed examples of these methods applied to problems in quantum optics and nonlinear optomechanics. Published by the American Physical Society 2024
Lie algebraic techniques are powerful and widely used tools for studying dynamics and metrology in quantum optics. When the Hamiltonian generates a Lie algebra with finite dimension, the unitary evolution can be expressed as a finite product of exponentials using the Wei-Norman expansion. The system is then exactly described by a finite set of scalar differential equations, even if the Hilbert space is infinite. However, the differential equations provided by the Wei-Norman expansion are nonlinear and often have singularities that prevent both analytic and numerical evaluation. We derive a new Lie algebra expansion for the quantum Fisher information, which results in linear differential equations. Together with existing Lie algebra techniques this allows many metrology problems to be analysed entirely in the Heisenberg picture. This substantially reduces the calculations involved in many metrology problems, and provides analytical solutions for problems that cannot even be solved numerically using the Wei-Norman expansion. It also allows us to study general features of metrology problems, valid for all quantum states. We provide detailed examples of these methods applied to problems in quantum optics and nonlinear optomechanics. Published by the American Physical Society 2024
In this thesis, I focus on the quantum Fisher information (QFI) - a central quantity in parameter estimation theory for quantum systems, which provides a lower bound on the precision one can achieve. In the first part of the thesis, I explore a Lie algebraic approach to evaluate the quantum Fisher information for unitary dynamics. I show how the Lie algebra associated with the unitary evolution, together with the initial state, allows one to compute various quantities of interest, circumventing the computation of the time-evolved state. In particular, I apply this shortcut approach to calculate the QFI and use it for the QFI evaluation in various prototype models, both finiteand infinite-dimensional. In the second part of the thesis, I shift focus to non-unitary dynamics, such as those described by a quantum master equation. In this setting, I theoretically study the description of open system dynamics in the Liouville space of vectorized density matrices and investigate the so-called dissipative quantum Fisher information (DQFI) within that framework. Specifically, I demonstrate how the metric defined in Liouville space differs from its counterpart in Hilbert space. I establish a map between the Hilbert space QFI and the Liouville dissipative quantum Fisher information for qubit and qudit systems, expressed via the quantum state’s purity. Furthermore, I discuss the physical meaning of the DQFI by studying its behavior under completely positive trace-preserving (CPTP) maps. The present results demonstrate that the Lie algebraic approach provides a computationally efficient framework for the QFI evaluation. This approach allows to avoid the explicit solution of quantum dynamics, and remains well-defined in cases when conventional approaches like WeiNorman expansion exhibit singularities.
This thesis focuses on the mathematical description and application of nonlinear cavity optomechanical systems. The first part is concerned with solving the dynamics of the standard nonlinear optomechanical Hamiltonian with an additional time-dependent mechanical displacement and single-mode squeezing term. The solution is based on identifying a Lie algebra that generates the time-evolution of the system, which reduces the problem to considering a finite set of coupled ordinary differential equations of real functions. The second part applies the solutions of the extended optomechanical Hamiltonian to the study of non-Gaussianity. We compute the non-Gaussian character of an optomechanical state as a function of the parameters in the Hamiltonian, and investigate the interplay between the non-Gaussianity, the strength of the nonlinear coupling and the strength of the single-mode mechanical squeezing term. We find that the strength and form of the nonlinear coupling strongly impacts the non-Gaussianity, and that its relationship with the squeezing term is highly complex. The third part concerns the use of nonlinear optomechanical systems as quantum sensors. We derive a general expression of the quantum Fisher information given the extended optomechanical Hamiltonian and demonstrate its applicability through three concrete examples: estimating the strength of a nonlinear light--matter coupling, the strength of a time-modulated mechanical displacement, and the strength of a single-mode mechanical squeezing parameter, all of which are modulated at resonance. In the last Chapter of the thesis, we consider the estimation of a constant gravitational acceleration with an optomechanical system. Our results suggest that optomechanical systems could, in principle, be used as powerful quantum sensors.
This chapter aims to develop an appropriate notation and to facilitate the subsequent discussion of the role played by the Lie algebras. It is unlikely that one manages to express a sufficiently realistic model only in terms of the operators that span a finite-dimensional Lie algebra, but it may be possible to simulate most of the relevant features of many physical phenomena by means of such simple models. The chapter discusses well-known approximate methods for the treatment of the time-evolution equation. If the Hamiltonian operator belongs to a finite-dimensional Lie algebra, then the differential equation for the time-evolution operator is easier to solve, and even the calculation of transition probabilities and expectation values of observables may be straightforward. The most relevant feature of this application of the Lie algebraic method is the occurrence of a matrix equation of finite dimension in contrast with the expansion in a basis set of state vectors, which typically leads to matrices of infinite dimension.
Abstract Quantum Fisher information (QFI) is a key concept of quantum metrology, which is widely used in quantum parameter estimation. We introduce a general approach to analytically get the QFI of pure states for unitary parametrization processes governed by arbitrary finite dimensional Lie algebras. We show the universality of this approach by comparing the result of algebra with the previous work and applying it to and algebras. For case, there are some interesting findings. The phase of the initial coherent state has an optimal value for maximum QFI, and the QFI can vanish and revive during the time evolution.
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.
Many important quantum systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a finite Lie algebra, H ̂ ( t ) = ∑ l = 1 L η l ( t ) g ̂ l . The Wei-Norman framework provides a powerful analytical method for computing the exact time-evolution operator in a factorized form, U ̂ ( t ) = ∏ l = 1 L e Λ l ( t ) g ̂ l . By exploiting the underlying algebraic structure, the method yields a system of differential equations—the Wei-Norman equations—which determine the time-evolution operator coefficients Λ l ( t ) from the Hamiltonian ones η l ( t ) . Although robust and formally extendable, applying the method by hand becomes increasingly difficult or even intractable for high-order systems ( L ≥ 6 ). In this work, we introduce , a Python library that automates the application of this method. Given the commutators between generators, the library symbolically derives the Wei-Norman equations and efficiently computes nested commutators, similarity transformations, and the differential equations that yield Baker-Campbell-Hausdorff-like relations. We validate the library by recovering known results for quantum systems associated with the Lie algebras su ( 1 , 1 ) , su ( 2 ) , sl ( 2 ) , and so ( 2 , 1 ) , and further demonstrate its capabilities by deriving the Wei-Norman equations for a system of two coupled time-dependent harmonic oscillators involving 11 generators. Additionally, we specialize the library to handle the Lie group SU ( N ) using a generic Cartan-Weyl basis for its algebra, and demonstrate its versatility by deriving the Wei-Norman equations for su ( 2 ) , su ( 3 ) , and su ( 4 ) , including the explicit construction of a set of quantum gates for universal quantum computation.