Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:gr-qc/9302015 · Physical Review A

Quantum behavior of general time-dependent quadratic systems linearly coupled to a bath

Jason Twamley

Published Oct 1, 1993• 16 citations
arXivHTMLPDF

Abstract

In this paper we solve for the quantum propagator of a general time-dependent system quadratic in both position and momentum, linearly coupled to an infinite bath of harmonic oscillators. We work in the regime where the quantum optical master equation is valid. We map this master equation to a Schr\"odinger equation on super-Hilbert space and utilize Lie-algebraic techniques to solve for the dynamics in this space. We then map back to the original Hilbert space to obtain the solution of the quantum dynamics. The Lie-algebraic techniques used are preferable to the standard Wei-Norman methods in that only coupled systems of first-order ordinary differential equations and purely algebraic equations need only be solved. We look at two examples.

DOI
10.1103/physreva.48.2627

Related papers

A quadratic time-dependent quantum harmonic oscillator

F. E. Onah, E. García Herrera, J. A. Ruelas-Galván +3 more

We present a Lie algebraic approach to a Hamiltonian class covering driven, parametric quantum harmonic oscillators where the parameter set-mass, frequency, driving strength, and parametric pumping-is time-dependent. Our unitary-transformation-based approach provides a solution to our general quadratic time-dependent quantum harmonic model. As an example, we show an analytic solution to the periodically driven quantum harmonic oscillator without the rotating wave approximation; it works for any given detuning and coupling strength regime. For the sake of validation, we provide an analytic solution to the historical Caldirola-Kanai quantum harmonic oscillator and show that there exists a unitary transformation within our framework that takes a generalized version of it onto the Paul trap Hamiltonian. In addition, we show how our approach provides the dynamics of generalized models whose Schrödinger equation becomes numerically unstable in the laboratory frame.

arXiv:2211.13281•Scientific Reports•May 23, 2023•18 citations

Integrability of quadratic non-autonomous quantum linear systems

Sergeĭ K. Suslov, Raquel M. López

The Quantum Harmonic Oscillator is one of the most important models in Quantum Mechanics. Analogous to the classical mass vibrating back and forth on a spring, the quantum oscillator system has attracted substantial attention over the years because of its importance in many advanced and difficult quantum problems. This dissertation deals with solving generalized models of the time-dependent Schrodinger equation which are called generalized quantum harmonic oscillators, and these are characterized by an arbitrary quadratic Hamiltonian of linear momentum and position operators. The primary challenge in this work is that most quantum models with timedependence are not solvable explicitly, yet this challenge became the driving motivation for this work. In this dissertation, the methods used to solve the time-dependent Schrodinger equation are the fundamental singularity (or Green's function) and the Fourier (eigenfunction expansion) methods. Certain Riccati- and Ermakov-type systems arise, and these systems are highlighted and investigated. The overall aims of this dissertation are to show that quadratic Hamiltonian systems are completely integrable systems, and to provide explicit approaches to solving the time-dependent SchrDodinger equation governed by an arbitrary quadratic Hamiltonian operator. The methods and results established in the dissertation are not yet well recognized in the literature, yet hold for high promise for further future research. Finally, the most recent results in the dissertation correspond to the harmonic oscillator group and its symmetries. A simple derivation of the maximum kinematical invariance groups of the free particle and quantum harmonic oscillator is constructed from the view point of the Riccati- and Ermakov-type systems, which shows an alternative to the traditional Lie Algebra approach. To conclude, a missing class of solutions of the time-dependent Schrodinger equation for the simple harmonic oscillator in one dimension is constructed. Probability distributions of the particle linear position and momentum, are emphasized with Mathematica animations. The eigenfunctions qualitatively differ from the traditional standing waves of the one-dimensional Schrodinger equation. The physical relevance of these dynamic states is still questionable, and in order to investigate their physical meaning, animations could also be created for the squeezed coherent states. This will be addressed in future work.

•Jan 1, 2012•1 citation

Lindblad dynamics of the damped and forced quantum harmonic oscillator: General solution

H. J. Korsch

The quantum dynamics of a damped and forced harmonic oscillator described by a Lindblad master equation is analyzed. The master equation is converted into a matrix-vector representation and the resulting non-Hermitian Schrödinger equation is solved by Lie-algebraic techniques allowing the construction of the general solution for the density operator.

arXiv:1909.03206•Sep 7, 2019•0 citations

Solving the quantum master equation of coupled harmonic oscillators with Lie-algebra methods

Lucas Teuber, Stefan Scheel

Based on a Liouville-space formulation of open systems, we present two methods to solve the quantum dynamics of coupled harmonic oscillators experiencing Markovian loss. The starting point is the quantum master equation in Liouville space which is generated by a Liouvillian that induces a Lie algebra. We show how this Lie algebra allows to define ladder operators that construct Fock-like eigenstates of the Liouvillian. These eigenstates are used to decompose the time-evolved density matrix and, together with the accompanying eigenvalues, provide insight into the transport properties of the lossy system. Additionally, a Wei-Norman expansion of the generated time evolution can be found by a structure analysis of the algebra. This structure analysis yields a construction principle to implement effective non-Hermitian Hamiltonians in lossy systems.

arXiv:1912.07320•Physical Review A•Apr 27, 2020•24 citations

Dynamics of a harmonic oscillator coupled with a Glauber amplifier

R Grimaudo, V I Man’ko, M A Man’ko +1 more

Abstract A system of a quantum harmonic oscillator bi-linearly coupled with a Glauber amplifier is analysed considering a time-dependent Hamiltonian model. The Hilbert space of this system may be exactly subdivided into invariant finite dimensional subspaces. Resorting to the Jordan–Schwinger map, the dynamical problem within each invariant subspace may be traced back to an effective SU(2) Hamiltonian model expressed in terms of spin variables only. This circumstance allows to analytically solve the dynamical problem and thus to study the exact dynamics of the oscillator-amplifier system under specific time-dependent scenarios. Peculiar physical effects are brought to light by comparing the dynamics of such a system with that of two interacting standard oscillators.

arXiv:1910.02703•Physica Scripta•Sep 10, 2019•17 citations

Quantum propagator for a general time-dependent quadratic Hamiltonian: Application to interacting oscillators in external fields

Shohreh Janjan, Fardin Kheirandish

Abstract In this paper, we find the quantum propagator for a general time-dependent quadratic Hamiltonian. The method is based on the properties of the propagator and the fact that the quantum propagator fulfills two independent partial differential equations originating from Heisenberg equations for positions and momenta. As an application of the method, we find the quantum propagator for a linear chain of interacting oscillators for both periodic and Dirichlet boundary conditions. The state and excitation propagation along the harmonic chain in the absence and presence of an external classical source is studied and discussed. The location of the first maxima of the probability amplitude P(n, τ) is a straight line in the (n, τ)-plane, indicating a constant speed of excitation propagation along the chain.

arXiv:2305.19052•Physica Scripta•Apr 20, 2023•0 citations
In this paper we solve for the quantum propagator of a general time-dependent system quadratic in both position and momentum, linearly coupled to an infinite bath of harmonic oscillators. We work in the regime where the quantum optical master equation is valid. We map this master equation to a Schr"odinger equation on super-Hilbert space and utilize Lie-algebraic techniques to solve for the dynamics in this space. We then map back to the original Hilbert space to obtain the solution of the quantum dynamics. The Lie-algebraic techniques used are preferable to the standard Wei-Norman methods in that only coupled systems of first-order ordinary differential equations and purely algebraic equations need only be solved. We look at two examples.
We present a Lie algebraic approach to a Hamiltonian class covering driven, parametric quantum harmonic oscillators where the parameter set-mass, frequency, driving strength, and parametric pumping-is time-dependent. Our unitary-transformation-based approach provides a solution to our general quadratic time-dependent quantum harmonic model. As an example, we show an analytic solution to the periodically driven quantum harmonic oscillator without the rotating wave approximation; it works for any given detuning and coupling strength regime. For the sake of validation, we provide an analytic solution to the historical Caldirola-Kanai quantum harmonic oscillator and show that there exists a unitary transformation within our framework that takes a generalized version of it onto the Paul trap Hamiltonian. In addition, we show how our approach provides the dynamics of generalized models whose Schrödinger equation becomes numerically unstable in the laboratory frame.
The Quantum Harmonic Oscillator is one of the most important models in Quantum Mechanics. Analogous to the classical mass vibrating back and forth on a spring, the quantum oscillator system has attracted substantial attention over the years because of its importance in many advanced and difficult quantum problems. This dissertation deals with solving generalized models of the time-dependent Schrodinger equation which are called generalized quantum harmonic oscillators, and these are characterized by an arbitrary quadratic Hamiltonian of linear momentum and position operators. The primary challenge in this work is that most quantum models with timedependence are not solvable explicitly, yet this challenge became the driving motivation for this work. In this dissertation, the methods used to solve the time-dependent Schrodinger equation are the fundamental singularity (or Green's function) and the Fourier (eigenfunction expansion) methods. Certain Riccati- and Ermakov-type systems arise, and these systems are highlighted and investigated. The overall aims of this dissertation are to show that quadratic Hamiltonian systems are completely integrable systems, and to provide explicit approaches to solving the time-dependent SchrDodinger equation governed by an arbitrary quadratic Hamiltonian operator. The methods and results established in the dissertation are not yet well recognized in the literature, yet hold for high promise for further future research. Finally, the most recent results in the dissertation correspond to the harmonic oscillator group and its symmetries. A simple derivation of the maximum kinematical invariance groups of the free particle and quantum harmonic oscillator is constructed from the view point of the Riccati- and Ermakov-type systems, which shows an alternative to the traditional Lie Algebra approach. To conclude, a missing class of solutions of the time-dependent Schrodinger equation for the simple harmonic oscillator in one dimension is constructed. Probability distributions of the particle linear position and momentum, are emphasized with Mathematica animations. The eigenfunctions qualitatively differ from the traditional standing waves of the one-dimensional Schrodinger equation. The physical relevance of these dynamic states is still questionable, and in order to investigate their physical meaning, animations could also be created for the squeezed coherent states. This will be addressed in future work.
Based on a Liouville-space formulation of open systems, we present two methods to solve the quantum dynamics of coupled harmonic oscillators experiencing Markovian loss. The starting point is the quantum master equation in Liouville space which is generated by a Liouvillian that induces a Lie algebra. We show how this Lie algebra allows to define ladder operators that construct Fock-like eigenstates of the Liouvillian. These eigenstates are used to decompose the time-evolved density matrix and, together with the accompanying eigenvalues, provide insight into the transport properties of the lossy system. Additionally, a Wei-Norman expansion of the generated time evolution can be found by a structure analysis of the algebra. This structure analysis yields a construction principle to implement effective non-Hermitian Hamiltonians in lossy systems.
Abstract A system of a quantum harmonic oscillator bi-linearly coupled with a Glauber amplifier is analysed considering a time-dependent Hamiltonian model. The Hilbert space of this system may be exactly subdivided into invariant finite dimensional subspaces. Resorting to the Jordan–Schwinger map, the dynamical problem within each invariant subspace may be traced back to an effective SU(2) Hamiltonian model expressed in terms of spin variables only. This circumstance allows to analytically solve the dynamical problem and thus to study the exact dynamics of the oscillator-amplifier system under specific time-dependent scenarios. Peculiar physical effects are brought to light by comparing the dynamics of such a system with that of two interacting standard oscillators.
Abstract In this paper, we find the quantum propagator for a general time-dependent quadratic Hamiltonian. The method is based on the properties of the propagator and the fact that the quantum propagator fulfills two independent partial differential equations originating from Heisenberg equations for positions and momenta. As an application of the method, we find the quantum propagator for a linear chain of interacting oscillators for both periodic and Dirichlet boundary conditions. The state and excitation propagation along the harmonic chain in the absence and presence of an external classical source is studied and discussed. The location of the first maxima of the probability amplitude P(n, τ) is a straight line in the (n, τ)-plane, indicating a constant speed of excitation propagation along the chain.