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

Physical Review A

Energy eigenvalues and squeezing properties of general systems of coupled quantum anharmonic oscillators

Ning Ning Chung, Lock Yue Chew

Published Sep 18, 2007• 20 citations
Publisher

Abstract

We have generalized the two-step approach to the solution of systems of $N$ coupled quantum anharmonic oscillators. By using the squeezed vacuum state of each individual oscillator, we construct the tensor product state, and obtain the optimal squeezed vacuum product state through energy minimization. We then employ this optimal state and its associated bosonic operators to define a basis set to construct the Heisenberg matrix. The diagonalization of the matrix enables us to obtain the energy eigenvalues of the coupled oscillators. In particular, we have applied our formalism to determine the eigenenergies of systems of two coupled quantum anharmonic oscillators perturbed by a general polynomial potential, as well as three and four coupled systems. Furthermore, by performing a first-order perturbation analysis about the optimal squeezed vacuum product state, we have also examined into the squeezing properties of two coupled oscillator systems.

DOI
10.1103/physreva.76.032113

Related papers

Two-step approach to the dynamics of coupled anharmonic oscillators

Ning Ning Chung, Lock Yue Chew

We have further extended the two-step approach developed by Chung and Chew [N. N. Chung and L. Y. Chew, Phys. Rev. A 76, 032113 (2007)] to the solution of the quantum dynamics of general systems of $N$-coupled anharmonic oscillators. The idea is to employ an optimized basis set to represent the dynamical quantum states of these oscillator systems. The set is generated via the action of the optimized Bogoliubov transformed bosonic operators on the optimal squeezed vacuum product state. The procedure requires (i) applying the two-step approach to the eigendecomposition of the time evolution operator and (ii) transforming the representation of the initial state from the original to the optimal bases. We have applied the formalism to examine the dynamics of squeezing and entanglement of several anharmonic oscillator systems.

Physical Review A•Jul 1, 2009•14 citations

Calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential

Mojtaba Jafarpour, Davood Afshar

We choose the squeezed vacuum state as a one-parameter trial wavefunction, to minimize the energy of an anharmonic oscillator, with a polynomial perturbation potential. Using the optimal vacuum state obtained, we generate the optimal generalized number state basis, and set up the Hamiltonian in the normal-ordered form. Then, truncating the basis, we produce finite-dimensional matrices in a simple manner, and find their eigenvalues by standard methods. We will apply our method to the Morse potential, whose exact eigenvalues are known, as a check on the accuracy of the results that we have derived. Two models of the double well and the quartic potential are considered as examples.

Journal of Physics A Mathematical and General•Dec 21, 2001•61 citations

Study of continuous variable entanglement in coupled oscillator systems by means of the dynamical two-step approach

Ning Ning Chung

We have extended the applicability of the two-step approach to higher-dimensional problem, and have further developed it to evaluate the quantum dynamics and stationary states of a general class of coupled anharmonic oscillator system. This new approach has enabled us to investigate the topic of continuous variable entanglement within these coupled oscillator systems with good numerical efficiency. In particular, we have applied it to study the relation between entanglement, squeezing, and quantum uncertainties, of the systems’ ground states. Interestingly, while the relation between quadrature squeezing and entanglement depends sensitively on the nonlinear potential, the influence of nonlinear perturbation on the entropy-uncertainty relation is minimal, especially when the entanglement between the two oscillators is large. We have also explored into the dynamics of entanglement where we have found that entanglement dynamics can depend completely on the global classical dynamical regime, without being influenced by the local classical behavior.

•Jan 1, 2009•0 citations

First and Second-Order Energy Eigenvalues of One-Dimensional Quantum Harmonic and Anharmonic Oscillator with Linear, Quadratic, Cubic and Polynomial Perturbation Potential

B. I. Madububa, N. I. Achuko, JP. C. Mbagwu +2 more

This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).

ASM Science Journal•May 18, 2022•0 citations

Optimized Perturbation Theory for Wave Functions of Quantum Systems

Tetsuo Hatsuda, Teiji Kunihiro, Toshiaki Tanaka

The notion of the optimized perturbation, which has been successfully applied to energy eigenvalues, is generalized to treat wave functions of quantum systems. The key ingredient is to construct an envelope of a set of perturbative wave functions. This leads to a condition similar to that obtained from the principle of minimal sensitivity. Applications of the method to the quantum anharmonic oscillator and the double well potential show that uniformly valid wave functions with correct asymptotic behavior are obtained in the first-order optimized perturbation even for strong couplings.

arXiv:hep-th/9612097•Physical Review Letters•Apr 28, 1997•35 citations

Relation of the entanglement entropy and uncertainty product in ground states of coupled anharmonic oscillators

Ning Ning Chung, Chen Er, Yong Siah Teo +1 more

We explore quantitatively the fundamental connection between the entropy and local uncertainty product as an entanglement measure within the lowest energy eigenstate of a class of two-coupled anharmonic oscillator systems. We find that these two quantum aspects approach a unique relation as the entanglement between the oscillators becomes large if the potential of the first collective mode displays a generic single-well potential structure. This implies that the uncertainty product can be a good choice for the characterization of entanglement when the entropy is sufficiently large. Interestingly, although this relation is robust against the anharmonic perturbation, we find that it can be altered effectively by a quantum catastrophe.

Physical Review A•Jul 16, 2010•12 citations
We have generalized the two-step approach to the solution of systems of NNN coupled quantum anharmonic oscillators. By using the squeezed vacuum state of each individual oscillator, we construct the tensor product state, and obtain the optimal squeezed vacuum product state through energy minimization. We then employ this optimal state and its associated bosonic operators to define a basis set to construct the Heisenberg matrix. The diagonalization of the matrix enables us to obtain the energy eigenvalues of the coupled oscillators. In particular, we have applied our formalism to determine the eigenenergies of systems of two coupled quantum anharmonic oscillators perturbed by a general polynomial potential, as well as three and four coupled systems. Furthermore, by performing a first-order perturbation analysis about the optimal squeezed vacuum product state, we have also examined into the squeezing properties of two coupled oscillator systems.
We have further extended the two-step approach developed by Chung and Chew [N. N. Chung and L. Y. Chew, Phys. Rev. A 76, 032113 (2007)] to the solution of the quantum dynamics of general systems of NNN-coupled anharmonic oscillators. The idea is to employ an optimized basis set to represent the dynamical quantum states of these oscillator systems. The set is generated via the action of the optimized Bogoliubov transformed bosonic operators on the optimal squeezed vacuum product state. The procedure requires (i) applying the two-step approach to the eigendecomposition of the time evolution operator and (ii) transforming the representation of the initial state from the original to the optimal bases. We have applied the formalism to examine the dynamics of squeezing and entanglement of several anharmonic oscillator systems.
We choose the squeezed vacuum state as a one-parameter trial wavefunction, to minimize the energy of an anharmonic oscillator, with a polynomial perturbation potential. Using the optimal vacuum state obtained, we generate the optimal generalized number state basis, and set up the Hamiltonian in the normal-ordered form. Then, truncating the basis, we produce finite-dimensional matrices in a simple manner, and find their eigenvalues by standard methods. We will apply our method to the Morse potential, whose exact eigenvalues are known, as a check on the accuracy of the results that we have derived. Two models of the double well and the quartic potential are considered as examples.
We have extended the applicability of the two-step approach to higher-dimensional problem, and have further developed it to evaluate the quantum dynamics and stationary states of a general class of coupled anharmonic oscillator system. This new approach has enabled us to investigate the topic of continuous variable entanglement within these coupled oscillator systems with good numerical efficiency. In particular, we have applied it to study the relation between entanglement, squeezing, and quantum uncertainties, of the systems’ ground states. Interestingly, while the relation between quadrature squeezing and entanglement depends sensitively on the nonlinear potential, the influence of nonlinear perturbation on the entropy-uncertainty relation is minimal, especially when the entanglement between the two oscillators is large. We have also explored into the dynamics of entanglement where we have found that entanglement dynamics can depend completely on the global classical dynamical regime, without being influenced by the local classical behavior.
This work is aimed at obtaining the energy eigenvalues for one-dimensional quantum harmonic and anharmonic oscillators perturbed by linear, quadratic, cubic and polynomial potentials. To obtain the solutions of the energy eigenvalues, we employed the time-independent perturbation theory to calculate the first and the second-order energy correction, which we used to obtain the complete generalised energy eigenvalues of the quantum harmonic oscillators with linear, quadratic, cubic and polynomial perturbation potential of the same unperturbed Hamiltonian (H).
The notion of the optimized perturbation, which has been successfully applied to energy eigenvalues, is generalized to treat wave functions of quantum systems. The key ingredient is to construct an envelope of a set of perturbative wave functions. This leads to a condition similar to that obtained from the principle of minimal sensitivity. Applications of the method to the quantum anharmonic oscillator and the double well potential show that uniformly valid wave functions with correct asymptotic behavior are obtained in the first-order optimized perturbation even for strong couplings.
We explore quantitatively the fundamental connection between the entropy and local uncertainty product as an entanglement measure within the lowest energy eigenstate of a class of two-coupled anharmonic oscillator systems. We find that these two quantum aspects approach a unique relation as the entanglement between the oscillators becomes large if the potential of the first collective mode displays a generic single-well potential structure. This implies that the uncertainty product can be a good choice for the characterization of entanglement when the entropy is sufficiently large. Interestingly, although this relation is robust against the anharmonic perturbation, we find that it can be altered effectively by a quantum catastrophe.