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:nlin/0605006 · Journal of Physics A Mathematical and Theoretical

Nonlinear dynamics of quantum systems and soliton theory

Eldad Bettelheim, Alexander G. Abanov, Paul B. Wiegmann

Published Feb 6, 2007• 41 citations
arXivHTMLPDF

Abstract

We show that spacetime evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a spacetime dependence of a matrix element of fermionic systems related to the orthogonality catastrophe or boundary states with the τ-function of the modified KP-hierarchy. The established relation allows us to apply the apparatus of soliton theory to the study of nonlinear aspects of quantum dynamics. We also describe a bosonization in momentum space —a representation of a fermion operator by a Bose field in the presence of a boundary state.

DOI
10.1088/1751-8113/40/8/f02

Related papers

Nonlinear quantum-dynamical system based on the Kadomtsev-Petviashvili II equation

Yair Zarmi

The structure of soliton solutions of classical integrable nonlinear evolution equations, which can be solved through the Hirota transformation, suggests a new way for the construction of nonlinear quantum-dynamical systems that are based on the classical equations. In the new approach, the classical soliton solution is mapped into an operator, U, which is a nonlinear functional of the particle-number operators over a Fock space of quantum particles. U obeys the evolution equation; the classical soliton solutions are the eigenvalues of U in multi-particle states in the Fock space. The construction easily allows for the incorporation of particle interactions, which generate soliton effects that do not have a classical analog. In this paper, this new approach is applied to the case of the Kadomtsev-Petviashvili II equation. The nonlinear quantum-dynamical system describes a set of M = (2S + 1) particles with intrinsic spin S, which interact in clusters of 1 ≤ N ≤ (M − 1) particles.

Journal of Mathematical Physics•Jun 1, 2013•3 citations

The Quantum Nonlinear Schrödinger Model; Gelfand-Levitan Equation and Classical Soliton

Miki Wadati, Atsuo Kuniba, Tetsuro Konishi

Through the quantum inverse scattering method the nonlinear Schrödinger model is studied for the attractive case. Making use of the bound state operators and the quantum Gelfand-Levitan equation, the matrix elements of the field operator φ( x ) and the density operator φ + ( x )φ( x ) are explicitly calculated. Each matrix element is found to be expressed in a product form. Based on the results, the connection between classical soliton and the quantum field theory is investigated. It is shown that the wave form of the classical soliton is related to the matrix element of the field operator in the limit n →∞, where n is the particle number making the bound state. The powerfulness of the quantum Gelfand-Levitan equation is stressed.

Journal of the Physical Society of Japan•May 15, 1985•24 citations

Quantum states of dark solitons in the 1D Bose gas

Jun Sato, Rina Kanamoto, Eriko Kaminishi +1 more

Abstract We present a series of quantum states that are characterized by dark solitons of the nonlinear Schrödinger equation (i.e. the Gross–Pitaevskii equation) for the one-dimensional Bose gas interacting through the repulsive delta-function potentials. The classical solutions satisfy the periodic boundary conditions and we simply call them classical dark solitons. Through exact solutions we show corresponding aspects between the states and the solitons in the weak coupling case: the quantum and classical density profiles completely overlap with each other not only at an initial time but also at later times over a long period of time, and they move together with the same speed in time; the matrix element of the bosonic field operator between the quantum states has exactly the same profiles of the square amplitude and the phase as the classical complex scalar field of a classical dark soliton not only at the initial time but also at later times, and the corresponding profiles move together for a long period of time. We suggest that the corresponding properties hold rigorously in the weak coupling limit. Furthermore, we argue that the lifetime of the dark soliton-like density profile in the quantum state becomes infinitely long as the coupling constant approaches zero, by comparing it with the quantum speed limit time. Thus, we call the quantum states quantum dark soliton states.

arXiv:1602.08329•New Journal of Physics•Jul 1, 2016•36 citations

Quantum dynamics

In this chapter, we derived some elements of quantum mechanics, which are essential for the further development of our theory: the momentum of a system of Fermions in the second quantization, the coordinate and momentum of a harmonic oscillator as a unique operator at two different moments of time, Boson and Fermion operator algebra, coherent states, the electron-field interaction, the quantization of the electromagnetic field, Boson and Fermion distributions, and densities of states in a degenerate, or a non-degenerate system of Fermions. Our starting point is the wave nature of a quantum particle, the Hamiltonian equations were obtained as group velocities in the two conjugate spaces of the wave, of the coordinates and of the momentum. In this way, the Schr¨odinger equation and the electron-field potential of interaction are obtained from quantum equations generated by the particle wave function.

BENTHAM SCIENCE PUBLISHERS eBooks•Dec 17, 2014•0 citations

Quantum dark solitons in the 1D Bose gas and the superfluid velocity

Jun Sato, Rina Kanamoto, Eriko Kaminishi +1 more

We give explicit connections of quantum one-hole excited states to classical solitons for the one-dimensional Bose gas with repulsive short-range interactions. We call the quantum states connected to classical solitons the quantum soliton states. We show that the matrix element of the canonical field operator between quantum soliton states with $N-1$ and $N$ particles is given by a dark soliton of the Gross-Pitaevskii equation in the weak coupling case. We suggest that the matrix element corresponds to the order parameter of BEC in the quantum soliton state. The result should be useful in the study of many-body effects in Bose-Einstein condensation and superfluids. For instance, we derive the superfluid velocity for a quantum soliton state.

arXiv:1204.3960•Apr 18, 2012•6 citations

Classical Soliton as a Limit of the Quantum Field Theory

Miki Wadachi, Masa‐aki Sakagami

In the Nonlinear Schrödinger model, the correspondence between classical soliton and its quantum field theory is investigated. It is shown explicitly that moving soliton as well as soliton at rest arise as the matrix elements of a field operator in the limit n →∞, where n is the particle number making the bound state.

Journal of the Physical Society of Japan•Jun 15, 1984•51 citations
We show that spacetime evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a spacetime dependence of a matrix element of fermionic systems related to the orthogonality catastrophe or boundary states with the τ-function of the modified KP-hierarchy. The established relation allows us to apply the apparatus of soliton theory to the study of nonlinear aspects of quantum dynamics. We also describe a bosonization in momentum space —a representation of a fermion operator by a Bose field in the presence of a boundary state.
The structure of soliton solutions of classical integrable nonlinear evolution equations, which can be solved through the Hirota transformation, suggests a new way for the construction of nonlinear quantum-dynamical systems that are based on the classical equations. In the new approach, the classical soliton solution is mapped into an operator, U, which is a nonlinear functional of the particle-number operators over a Fock space of quantum particles. U obeys the evolution equation; the classical soliton solutions are the eigenvalues of U in multi-particle states in the Fock space. The construction easily allows for the incorporation of particle interactions, which generate soliton effects that do not have a classical analog. In this paper, this new approach is applied to the case of the Kadomtsev-Petviashvili II equation. The nonlinear quantum-dynamical system describes a set of M = (2S + 1) particles with intrinsic spin S, which interact in clusters of 1 ≤ N ≤ (M − 1) particles.
Through the quantum inverse scattering method the nonlinear Schrödinger model is studied for the attractive case. Making use of the bound state operators and the quantum Gelfand-Levitan equation, the matrix elements of the field operator φ( x ) and the density operator φ + ( x )φ( x ) are explicitly calculated. Each matrix element is found to be expressed in a product form. Based on the results, the connection between classical soliton and the quantum field theory is investigated. It is shown that the wave form of the classical soliton is related to the matrix element of the field operator in the limit n →∞, where n is the particle number making the bound state. The powerfulness of the quantum Gelfand-Levitan equation is stressed.
Abstract We present a series of quantum states that are characterized by dark solitons of the nonlinear Schrödinger equation (i.e. the Gross–Pitaevskii equation) for the one-dimensional Bose gas interacting through the repulsive delta-function potentials. The classical solutions satisfy the periodic boundary conditions and we simply call them classical dark solitons. Through exact solutions we show corresponding aspects between the states and the solitons in the weak coupling case: the quantum and classical density profiles completely overlap with each other not only at an initial time but also at later times over a long period of time, and they move together with the same speed in time; the matrix element of the bosonic field operator between the quantum states has exactly the same profiles of the square amplitude and the phase as the classical complex scalar field of a classical dark soliton not only at the initial time but also at later times, and the corresponding profiles move together for a long period of time. We suggest that the corresponding properties hold rigorously in the weak coupling limit. Furthermore, we argue that the lifetime of the dark soliton-like density profile in the quantum state becomes infinitely long as the coupling constant approaches zero, by comparing it with the quantum speed limit time. Thus, we call the quantum states quantum dark soliton states.
In this chapter, we derived some elements of quantum mechanics, which are essential for the further development of our theory: the momentum of a system of Fermions in the second quantization, the coordinate and momentum of a harmonic oscillator as a unique operator at two different moments of time, Boson and Fermion operator algebra, coherent states, the electron-field interaction, the quantization of the electromagnetic field, Boson and Fermion distributions, and densities of states in a degenerate, or a non-degenerate system of Fermions. Our starting point is the wave nature of a quantum particle, the Hamiltonian equations were obtained as group velocities in the two conjugate spaces of the wave, of the coordinates and of the momentum. In this way, the Schr¨odinger equation and the electron-field potential of interaction are obtained from quantum equations generated by the particle wave function.
We give explicit connections of quantum one-hole excited states to classical solitons for the one-dimensional Bose gas with repulsive short-range interactions. We call the quantum states connected to classical solitons the quantum soliton states. We show that the matrix element of the canonical field operator between quantum soliton states with N−1N-1N−1 and NNN particles is given by a dark soliton of the Gross-Pitaevskii equation in the weak coupling case. We suggest that the matrix element corresponds to the order parameter of BEC in the quantum soliton state. The result should be useful in the study of many-body effects in Bose-Einstein condensation and superfluids. For instance, we derive the superfluid velocity for a quantum soliton state.