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Related Concept Videos

State Space Representation01:27

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Transfer Function to State Space01:23

Transfer Function to State Space

State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...
State Space to Transfer Function01:21

State Space to Transfer Function

The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:
Classical Mechanics01:12

Classical Mechanics

Classical mechanics provides a mathematical description of the motion of bodies under the influence of forces. A key principle within this field is the work-energy theorem, which establishes a bridge between the net work done on an object and its kinetic energy.The work-energy theorem states that the net work done on a particle by all the forces acting on it equals the change in its kinetic energy.In simple terms, the work-energy theorem is a method to analyze the effects of forces on an...
The Quantum-Mechanical Model of an Atom02:45

The Quantum-Mechanical Model of an Atom

Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra. Schrödinger...

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Related Experiment Video

Updated: Jun 5, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Phase-space algorithm for simulating quantum nonlinear response functions of bosons using stochastic classical

Benoit Palmieri1, Yuki Nagata, Shaul Mukamel

  • 1Department of Chemistry, University of California, Irvine, California 92697-2025, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|January 15, 2011
PubMed
Summary
This summary is machine-generated.

This study presents a new algorithm for calculating quantum many-body nonlinear response functions in boson systems. It translates complex quantum dynamics into classical stochastic processes for efficient computation.

Related Experiment Videos

Last Updated: Jun 5, 2026

Generation and Coherent Control of Pulsed Quantum Frequency Combs
06:42

Generation and Coherent Control of Pulsed Quantum Frequency Combs

Published on: June 8, 2018

Area of Science:

  • Quantum Many-Body Physics
  • Nonlinear Dynamics
  • Statistical Mechanics

Background:

  • Calculating quantum many-body nonlinear response functions is computationally challenging.
  • Existing methods often struggle with the complexity of driven boson systems.
  • The positive P-representation offers a potential pathway for simplification.

Purpose of the Study:

  • To develop an efficient algorithm for computing quantum many-body nonlinear response functions.
  • To adapt the positive P-representation for driven boson systems.
  • To provide classical simulation methods for quantum dynamics.

Main Methods:

  • Utilizing the positive P-representation of the density matrix.
  • Mapping N-boson quantum dynamics to 4N classical stochastic degrees of freedom.
  • Employing coupled Langevin equations with multiplicative noise to propagate P-representation parameters.

Main Results:

  • Successfully calculated first- and third-order quantum many-body nonlinear response functions.
  • Developed two classical computational approaches: nonequilibrium and equilibrium simulations.
  • Demonstrated the formalism's applicability to boson systems coupled to a harmonic bath.

Conclusions:

  • The developed algorithm provides an efficient classical simulation method for quantum many-body nonlinear response.
  • The positive P-representation formalism is effective for driven boson systems.
  • The approach is generalizable to open quantum systems.