© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Bulletin of the American Physical Society

Quantum Control of the Squeezing Operator with Dynamics using Wei-Norman Factorization and the Time Evolution Operator*

Riley Martell, Manuel Berrondo, Ty Beus, Ray Hagimoto, Jean-François S. Van Huele

Published Oct 1, 2018• 0 citations

Abstract

Control of quantum phenomena would allow for expanding control theory from classical systems to microscopic ones whose behavior is dictated by quantum mechanics. A current goal of quantum control is to develop a systematic methodology for the manipulation of systems. The approach typically used to solve dynamic quantum systems is useful to analyze characteristics of a system represented by a defined operator. The squeeze operator's actions are characterized by finding the time evolution operator using the Wei-Norman method on the associated Hamiltonian and applying this to number (Fock) states, coherent states, and Schrodinger cat states. This specific case analyzing the Squeeze operator shows that the Wei-Norman method to find time-evolution operator can reveal the dynamics of any system with an associated Lie Algebra basis. Documenting a variety of initial states and initial parameters in a library of cases provides a foundation to achieving greater control in experimental applications as well. *I would like to thank Brigham Young University for hosting this Research Experience for Undergraduates program and the National Science Foundation for funding this research through grant #1757998. To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2018.4CS.L04.6

Related papers

Exploring the Solvability of the Jaynes-Cummings and Jaynes-Cummings-like Models: Implementing Quantum Control

Austen Couvertier

In this paper we aim to explore the dynamics and overall solvability of the Jaynes-Cummings \& Jaynes-Cummings-Like models. As a lens to understand these dynamics, we focused on cases where the parameters of the system were made time-dependent. All previous work on solving the dynamics of the Jaynes-Cummings models has relied heavily on the use of differential methods and setting the parameters as time-independent constants which were zero or one. To account for this, we utilized the Wei-Norman method which allowed us to analytical solve the time-dependent Hamiltonian. Through the use of this method, we can understand the more general characteristics of the dynamics associated with the Jaynes-Cummings model. This research related back to the field of Quantum information and control as the model highlights interactions with qubits and our time-dependent parameters can force certain transitions.

•Jan 1, 2017•0 citations

Time-evolution of squeezed coherent states of a generalized quantum parametric oscillator

Şirin A. Büyükaşık, Zehra Çayiç

Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.

Journal of Mathematical Physics•Jun 1, 2019•11 citations

Symdyn: An automated algebraic solution for high-order quantum systems

D. Martínez-Tibaduiza, Vladimir Vargas-Calderón, J. G. Dueñas +2 more

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.

arXiv:2503.22061•Physical Review Research•Dec 15, 2025•0 citations

Solving Quantum Dynamics with a Lie-Algebra Decoupling Method

Sofia Qvarfort, Igor Pikovski

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.

arXiv:2210.11894•PRX Quantum•Jan 27, 2025•9 citations

Dynamics of squeezed states of a generalized quantum parametric oscillator

Şirin A. Büyükaşık, Zehra Çayiç

Time-evolution of squeezed coherent states of a generalized Caldirola-Kanai type quantum parametric oscillator is found explicitly using the exact evolution operator obtained by the Wei-Norman algebraic approach. Properties of these states are investigated according to the parameters of the unitary squeeze operator and the time-variable parameters of the generalized quadratic Hamiltonian. As an application, we consider exactly solvable quantum models with specific frequency modification for which the corresponding classical oscillator is in underdamping case and driving forces are of sinusoidal type. For each model we explicitly provide the evolution of the squeezed coherent states and discuss their behavior.

Journal of Physics Conference Series•Apr 1, 2019•0 citations

New Families of Single—Qubit Control Fields: An Algorithm

Marco Enríquez, Alfonso Jaimes-Nájera

Abstract The dynamics of a two-level system interacting with a new class of analytically solvable driving fields is analyzed. Such fields are obtained using an inverse-engineering approach, which allows to exactly factorize the correspondent time-evolution operator via the Wei-Norman theorem. This technique is presented as an algorithm and using an example we show that the free parameters involved can be used to tune some aspects of the dynamics on demand such as the atomic population inversion.

Journal of Physics Conference Series•Apr 1, 2020•0 citations

Control of quantum phenomena would allow for expanding control theory from classical systems to microscopic ones whose behavior is dictated by quantum mechanics. A current goal of quantum control is to develop a systematic methodology for the manipulation of systems. The approach typically used to solve dynamic quantum systems is useful to analyze characteristics of a system represented by a defined operator. The squeeze operator's actions are characterized by finding the time evolution operator using the Wei-Norman method on the associated Hamiltonian and applying this to number (Fock) states, coherent states, and Schrodinger cat states. This specific case analyzing the Squeeze operator shows that the Wei-Norman method to find time-evolution operator can reveal the dynamics of any system with an associated Lie Algebra basis. Documenting a variety of initial states and initial parameters in a library of cases provides a foundation to achieving greater control in experimental applications as well. *I would like to thank Brigham Young University for hosting this Research Experience for Undergraduates program and the National Science Foundation for funding this research through grant #1757998. To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2018.4CS.L04.6

In this paper we aim to explore the dynamics and overall solvability of the Jaynes-Cummings \& Jaynes-Cummings-Like models. As a lens to understand these dynamics, we focused on cases where the parameters of the system were made time-dependent. All previous work on solving the dynamics of the Jaynes-Cummings models has relied heavily on the use of differential methods and setting the parameters as time-independent constants which were zero or one. To account for this, we utilized the Wei-Norman method which allowed us to analytical solve the time-dependent Hamiltonian. Through the use of this method, we can understand the more general characteristics of the dynamics associated with the Jaynes-Cummings model. This research related back to the field of Quantum information and control as the model highlights interactions with qubits and our time-dependent parameters can force certain transitions.

Time evolution of squeezed coherent states for a quantum parametric oscillator with the most general self-adjoint quadratic Hamiltonian is found explicitly. For this, we use the unitary displacement and squeeze operators in coordinate representation and the evolution operator obtained by the Wei-Norman Lie algebraic approach. Then, we analyze squeezing properties of the wave packets according to the complex parameter of the squeeze operator and the time-variable parameters of the Hamiltonian. As an application, we construct all exactly solvable generalized quantum oscillator models classically corresponding to a driven simple harmonic oscillator. For each model, defined according to the frequency modification in position space, we describe explicitly the squeezing and displacement properties of the wave packets. This allows us to see the exact influence of all parameters and make a basic comparison between the different models.

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.

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.

Time-evolution of squeezed coherent states of a generalized Caldirola-Kanai type quantum parametric oscillator is found explicitly using the exact evolution operator obtained by the Wei-Norman algebraic approach. Properties of these states are investigated according to the parameters of the unitary squeeze operator and the time-variable parameters of the generalized quadratic Hamiltonian. As an application, we consider exactly solvable quantum models with specific frequency modification for which the corresponding classical oscillator is in underdamping case and driving forces are of sinusoidal type. For each model we explicitly provide the evolution of the squeezed coherent states and discuss their behavior.