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

Journal of Modern Optics

Symbolic Computation Approach to Nonlinear Dynamics

J. Bajer, Petr Lisoněk

Published Apr 1, 1991• 9 citations
Publisher

Abstract

The description and demonstration of a short-time approximation method are presented for solution of the Heisenberg equation using a computer for operator algebra symbolic computation to obtain an operator series. Multimode nonlinear processes can be studied by this method in a fully quantum way; the symbolic operator or standard numerical results as well as the graphical dependences can be computed. Although a use for the method can be seen in its ability to improve numerical predictions rather than to discover new states or effects, we have obtained squeezing parameters and variances of photon number for initially squeezed states (and some other cases) in degenerate parametric processes.

DOI
10.1080/09500349114550711

Related papers

Heisenberg Equations of Motion

Z. Ficek, Mohamed Ridza Wahiddin

In the master equation method, we have already illustrated a powerful technique for the calculation of the dynamics of an atomic system interacting with the vacuum field. Another technique for calculating the dynamics of an atomic system coupled to the elec- tromagnetic (EM) field involves Heisenberg equations of motion for the system’s operators. A difference between the master equation and the Heisenberg equations is that the later involves dynamics of the operators, which allows to analyse the evolution of an atomic system in terms of the field and atomic operators. This creates some problems with handling the Heisenberg equations as, in general, operators do not commute and then in the course of solution of the equations we may face the problem of ordering of the operators. It is usually resolved by putting the operators in the normal order. We have gained some experience with the Heisenberg equations of motion in Chapter 6, Example 6.3, where we studied squeezing generation in the nonlinear degenerate parametric amplifier (DPA) process. Here, we illustrate the technique on the standard model of a two-level atom interacting with a multi-mode field. We then generalize the technique to some specific models such as Lorenz- Maxwell and Langevin equations, the derivation of which involves some approximations that can be applied only in some limited cases. We also present in detail the Floquet approach, which is usually applied to problems determined by differential equations with time- dependent coefficients.

•Apr 19, 2016•0 citations

Tensor-network approach to quantum optical state evolution beyond the Fock basis

Nikolay Kapridov, Egor Tiunov, Dmitry Chermoshentsev

Understanding the quantum evolution of light in nonlinear media is central to the development of next-generation quantum technologies. Yet, modeling these processes remains computationally demanding, as the required resources grow rapidly with photon number and phase-space resolution. Here, we introduce a tensor-network approach that efficiently captures the dynamics of nonlinear optical systems in a continuous-variable representation. Using the matrix product state (MPS) formalism, both quantum states and operators are encoded in a highly compressed form, enabling direct numerical integration of the Schrödinger equation. We demonstrate the method by simulating degenerate spontaneous parametric down-conversion (SPDC) and show that it accurately reproduces established theoretical benchmarks - energy conservation, pump depletion, and quadrature squeezing - even in regimes where conventional Fock-basis simulations become infeasible. For high-intensity pump fields $(α=100)$, the MPS representation achieves compression ratios below $3\times10^{-4}$ while preserving physical fidelity. This framework opens a scalable route to modeling multimode quantum light and nonlinear optical phenomena beyond the reach of traditional methods.

quant-pharXiv:2511.15295•Nov 20, 2025•0 citations

Dynamics of parametric processes with a trilinear hamiltonian

Govind Prasad Agrawal, C. L. Mehta

The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.

Journal of Physics A Mathematical Nuclear and General•Mar 21, 1974•50 citations

Quantum properties of kerr nonlinear directional couplers via phase-space representation / Rafael Julius

Rafael Julius

Quantum statistical properties of interacting optical fields in Kerr nonlinear couplers may be regarded as one of the most fundamental problems yet to be solved in quantum optics. As an analytical solution to this system is not always possible, a semi-analytic or direct numerical solution must be employed in order to obtain a more complete description of their evolution. This study investigates the quantum properties of four-mode couplers composed of two and four Kerr nonlinear waveguides, both with and without cavity set up via phase space representation. In these systems, the electromagnetic fields are described by their Hamiltonians, while the time-evolution of the systems are described by the Von-Neumann equation. Following standard techniques, the master equations for the density matrices of the systems are obtained and further converted to the corresponding classical Fokker- Planck equations using both positive P and Wigner representations. The corresponding set of Langevin Stochastic equations is then obtained from the Fokker-Planck equations by means of Ito rules. Finally, the systems are integrated numerically and averaged over many trajectories to get the relevant information. The effects of self-action nonlinearity, cross-action coupling and the initial energy of the coherent light on the evolution of the field quadrature variances are investigated. Furthermore, the influences of multimode interaction on the dynamics of the squeezed states are discussed. In this research, we show that the system may produce squeezed states in both quadrature variances for all calculated parameters. The multimode two channel model can provide better squeezing in contrast to the conventional device. The amplitude oscillation of maximal squeezing may be amplified by manipulation of the initial state of the coherent light in the first mode, and the range may be extended through multimode interaction. When the model is contained in a cavity, the system is able to generate better amplitude oscillations and squeezing range with leaf-revival-collapse-like behaviour; with the whole oscillation able to evolute below the standard quantum limit Regular photon transition and self trapping effects of the interacting modes in the four channels model are simulated, and squeezing is predicted for various initial intensities of the coherent field and nonlinear coupling parameter g. The cavity setup caused the system to undergo fluctuation amplification and the potential for different types of squeezing to be generated is observed. Depending on how we choose the interaction parameter values, the system may generate better squeezing in comparison to the cavity-less model.

•Jan 1, 2014•0 citations

Non-perturbative solution of nonlinear Heisenberg equations

Ladislav Mišta, Radim Filip

A new non-perturbative method of solution of the nonlinear Heisenberg equations in a finite-dimensional subspace is illustrated. The method, being a counterpart of the traditional Schrödinger picture method, is based on a finite operator expansion into the elementary processes. It provides us with insight into the nonlinear quantal interaction from a different point of view. Thus, one can investigate the nonlinear system in both pictures of quantum mechanics.

arXiv:quant-ph/0012023•Journal of Physics A Mathematical and General•Jun 27, 2001•5 citations

Non-perturbative solution of nonlinear Heisenberg equations

Radim Filip

A new non-perturbative method of solution of the nonlinear Heisenberg equations in finite-dimensional subspace is illustrated. The method, being a counterpart of the traditional Schrödinger picture method, is based on a finite operator expansion into the elementary processes. It provides us with the insight into the nonlinear quantal interaction from the different point of view. Thus one can investigate the nonlinear system in both pictures of quantum mechanics. 1

•Jan 1, 2000•5 citations
The description and demonstration of a short-time approximation method are presented for solution of the Heisenberg equation using a computer for operator algebra symbolic computation to obtain an operator series. Multimode nonlinear processes can be studied by this method in a fully quantum way; the symbolic operator or standard numerical results as well as the graphical dependences can be computed. Although a use for the method can be seen in its ability to improve numerical predictions rather than to discover new states or effects, we have obtained squeezing parameters and variances of photon number for initially squeezed states (and some other cases) in degenerate parametric processes.
In the master equation method, we have already illustrated a powerful technique for the calculation of the dynamics of an atomic system interacting with the vacuum field. Another technique for calculating the dynamics of an atomic system coupled to the elec- tromagnetic (EM) field involves Heisenberg equations of motion for the system’s operators. A difference between the master equation and the Heisenberg equations is that the later involves dynamics of the operators, which allows to analyse the evolution of an atomic system in terms of the field and atomic operators. This creates some problems with handling the Heisenberg equations as, in general, operators do not commute and then in the course of solution of the equations we may face the problem of ordering of the operators. It is usually resolved by putting the operators in the normal order. We have gained some experience with the Heisenberg equations of motion in Chapter 6, Example 6.3, where we studied squeezing generation in the nonlinear degenerate parametric amplifier (DPA) process. Here, we illustrate the technique on the standard model of a two-level atom interacting with a multi-mode field. We then generalize the technique to some specific models such as Lorenz- Maxwell and Langevin equations, the derivation of which involves some approximations that can be applied only in some limited cases. We also present in detail the Floquet approach, which is usually applied to problems determined by differential equations with time- dependent coefficients.
Understanding the quantum evolution of light in nonlinear media is central to the development of next-generation quantum technologies. Yet, modeling these processes remains computationally demanding, as the required resources grow rapidly with photon number and phase-space resolution. Here, we introduce a tensor-network approach that efficiently captures the dynamics of nonlinear optical systems in a continuous-variable representation. Using the matrix product state (MPS) formalism, both quantum states and operators are encoded in a highly compressed form, enabling direct numerical integration of the Schrödinger equation. We demonstrate the method by simulating degenerate spontaneous parametric down-conversion (SPDC) and show that it accurately reproduces established theoretical benchmarks - energy conservation, pump depletion, and quadrature squeezing - even in regimes where conventional Fock-basis simulations become infeasible. For high-intensity pump fields (α=100)(α=100)(α=100), the MPS representation achieves compression ratios below 3×10−43\times10^{-4}3×10−4 while preserving physical fidelity. This framework opens a scalable route to modeling multimode quantum light and nonlinear optical phenomena beyond the reach of traditional methods.
The quantum dynamics of a parametric interaction of the electromagnetic field with a nonlinear medium is considered. The three nonlinear coupled Heisenberg equations of motion are solved under the short-time approximation. The characteristic function for the normal ordering rule of association is evaluated. This function is then used to obtain the time dependence of the density operator. Assuming that the initial state of the system is a coherent state, explicit expressions for the diagonal coherent state representation of the reduced density operators for the pump as well as for the signal mode are obtained. For the pump mode, the initially coherent state remains coherent, whereas for the signal mode the diagonal coherent state representation is found to be a gaussian distribution whose variance is proportional to the average number of photons initially present in the pump mode.
Quantum statistical properties of interacting optical fields in Kerr nonlinear couplers may be regarded as one of the most fundamental problems yet to be solved in quantum optics. As an analytical solution to this system is not always possible, a semi-analytic or direct numerical solution must be employed in order to obtain a more complete description of their evolution. This study investigates the quantum properties of four-mode couplers composed of two and four Kerr nonlinear waveguides, both with and without cavity set up via phase space representation. In these systems, the electromagnetic fields are described by their Hamiltonians, while the time-evolution of the systems are described by the Von-Neumann equation. Following standard techniques, the master equations for the density matrices of the systems are obtained and further converted to the corresponding classical Fokker- Planck equations using both positive P and Wigner representations. The corresponding set of Langevin Stochastic equations is then obtained from the Fokker-Planck equations by means of Ito rules. Finally, the systems are integrated numerically and averaged over many trajectories to get the relevant information. The effects of self-action nonlinearity, cross-action coupling and the initial energy of the coherent light on the evolution of the field quadrature variances are investigated. Furthermore, the influences of multimode interaction on the dynamics of the squeezed states are discussed. In this research, we show that the system may produce squeezed states in both quadrature variances for all calculated parameters. The multimode two channel model can provide better squeezing in contrast to the conventional device. The amplitude oscillation of maximal squeezing may be amplified by manipulation of the initial state of the coherent light in the first mode, and the range may be extended through multimode interaction. When the model is contained in a cavity, the system is able to generate better amplitude oscillations and squeezing range with leaf-revival-collapse-like behaviour; with the whole oscillation able to evolute below the standard quantum limit Regular photon transition and self trapping effects of the interacting modes in the four channels model are simulated, and squeezing is predicted for various initial intensities of the coherent field and nonlinear coupling parameter g. The cavity setup caused the system to undergo fluctuation amplification and the potential for different types of squeezing to be generated is observed. Depending on how we choose the interaction parameter values, the system may generate better squeezing in comparison to the cavity-less model.