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 Mathematical Physics

Cranking approach for a complex quantum system

V. M. Kolomietz, S. V. Radionov

Published Jun 1, 2010• 5 citations
Publisher

Abstract

Within random matrix model, the response of a complex quantum system Ĥ(q[t]) on time variations in an external parameter q. In the limit of weak coupling of the external parameter to the quantum system, the different dynamical regimes of the diffusion in a space of the occupancies of eigenstates of Ĥ(q) are considered. The main focus is made on measuring the role of memory effects in the quantum diffusive dynamics. Possible macroscopic manifestations of the quantum mechanical diffusion are discussed in the context of the cranking approach, where the time variations in the classical parameter q provides the constancy of energy of the total system.

DOI
10.1063/1.3436579

Related papers

Non-Markovian response of complex quantum systems

S. V. Radionov

We study the perturbative response of a complex quantum system on time changes of an external parameter $X$. The driven dynamics is treated in adiabatic basis of the system's Hamiltonian $\hat{H}[X]$. Within a random matrix approach we obtained non--Markovian Fokker--Planck equation for the occupancy of given adiabatic state. We observed normal diffusion regime of the driven quantum dynamics at quite small values of the memory time defined by the time scales of the $X$--correlations and energy--distribution of the coupling matrix elements $(\partial \hat{H}/\partial X)_{nm}$. Here the normal energy diffusion was found to drop out with the width of the matrix elements' energy--distribution and the diffusion may be significantly suppressed with the decrease of the correlations between the matrix elements. In the opposite limit of relatively large memory times we obtained ballistic regime of the dynamics.

arXiv:1911.01820•Nov 4, 2019•0 citations

Collective motion in a quantum diffusive environment

V. M. Kolomietz, Sven Åberg, S. V. Radionov

The general problem of dissipation in macroscopic large-amplitude collective motion and its relation to energy diffusion of intrinsic degrees of freedom of a nucleus is studied. By applying the cranking approach to the nuclear many-body system, a set of coupled dynamical equations for the collective classical variable and the quantum mechanical occupancies of the intrinsic nuclear states is derived in the limit of weak coupling of the collective and intrinsic subsystems. Different dynamical regimes of the intrinsic nuclear motion and its consequences on time properties of collective dissipation are discussed.

arXiv:0706.2406•Physical Review C•Jan 9, 2008•6 citations

Diffusion and dissipation in complex quantum systems

Michael Wilkinson

The Hamiltonian of a quantum-mechanical system contains a time-dependent parameter X. The system is prepared in a highly excited eigenstate of the instantaneous Hamiltonian at t=0, and we compute the amplitudes 〈n(t)\ensuremath{\Vert}\ensuremath{\psi}(t)〉 to be in the eigenstates \ensuremath{\Vert}n〉 of the instantaneous Hamiltonian H^(X(t)) at later times. It is found that for generic systems, with no symmetries yielding additional quantum numbers, the occupation probability spreads diffusively away from the initial state. This diffusion is an irreversible process, and can be related to a model for dissipation. The diffusion constant D is investigated numerically in a random matrix model as a function of X\ifmmode \dot{}\else \.{}\fi{}, the rate of change of the parameter. Good agreement with theoretical predictions is found in two limiting regimes. When X\ifmmode \dot{}\else \.{}\fi{} is large, D is proportional to X\ifmmode \dot{}\else \.{}\fi{} $^{2}$, corresponding to the Ohmic dissipation predicted by the Kubo formula. When X\ifmmode \dot{}\else \.{}\fi{} is small, Landau-Zener transitions are the mechanism for diffusion, and D is proportional to X\ifmmode \dot{}\else \.{}\fi{} $^{3/2}$.

Physical Review A•May 1, 1990•69 citations

A general and solvable random matrix model for spin decoherence

François David

We propose and solve a simple but very general quantum model of a SU (2) spin interacting with a large external system with N states. The coupling is described by a random Hamiltonian in a new general Gaussian SU (2) × U ( N ) random matrix ensemble, that we introduce in this paper. We solve the model in the large N limit, for any value of the spin j and for any choice of the coupling matrix element distributions in the different possible angular momentum channels l (and provided that the internal dynamics of the spin is slow). Besides its mathematical interest as a non-trivial random matrix model, it allows us to study and illustrate in a simple framework various phenomena: the decoherence dynamics, the conditions of emergence of the classical phase space for the spin, and the properties of quantum diffusion in phase space. The large time evolution for the spin is shown to be non-Markovian in general, the Markov property emerging in some specific cases for the dynamics and the initial conditions.

arXiv:1009.1282•Journal of Statistical Mechanics Theory and Experiment•Jan 6, 2011•2 citations

Random-matrix reduction in projective quantum mechanics: Numerical simulations

Alexey A. Kryukov

We present numerical simulations supporting the random-matrix state-reduction framework developed in the companion theoretical paper. The simulations test the main derived features of the model: isotropic diffusion generated by Gaussian Unitary Ensemble Hamiltonians in projective state space, the restriction of this diffusion to Brownian motion on the classical submanifold, Born-rule frequencies for detector-defined outcome classes, and stroboscopic Newtonian motion for macroscopic systems under repeated environmental monitoring. We also compare GUE and GOE random Hamiltonians and show that GOE fails to produce the required isotropic complex projective diffusion. Further simulations examine finite-resolution detector records in the double-slit experiment, Zeno stability of recorded equivalence classes, effective irreversibility from high-dimensional state-space dynamics and loss of path information, and tensor-product particle-device dynamics in the device limit. The results show that microscopic state reduction, stable measurement records, effective irreversibility, and macroscopic classicality can be described as different coarse-grained manifestations of the same stochastic unitary mechanism.

quant-pharXiv:2606.19273•Jun 18, 2026•0 citations

Stochastic aspects of large amplitude collective motion

Aurel Bulgac, G. Do Dang, Dimitri Kusnezov

We consider externally driven many-body systems with complex spectra of intrinsic states. The effects of the coupling to the external world are analyzed by assuming time-dependent random matrix properties for the intrinsic system. We derive and solve evolution equations for intrinsic state population probabilities, average excitation energy and its fluctuations. The diffusive process is likely to be dominated by memory effects and an unexpected type of fluctuations.

Physics Reports•Jan 1, 1996•8 citations
Within random matrix model, the response of a complex quantum system Ĥ(q[t]) on time variations in an external parameter q. In the limit of weak coupling of the external parameter to the quantum system, the different dynamical regimes of the diffusion in a space of the occupancies of eigenstates of Ĥ(q) are considered. The main focus is made on measuring the role of memory effects in the quantum diffusive dynamics. Possible macroscopic manifestations of the quantum mechanical diffusion are discussed in the context of the cranking approach, where the time variations in the classical parameter q provides the constancy of energy of the total system.
We study the perturbative response of a complex quantum system on time changes of an external parameter XXX. The driven dynamics is treated in adiabatic basis of the system's Hamiltonian H^[X]\hat{H}[X]H^[X]. Within a random matrix approach we obtained non--Markovian Fokker--Planck equation for the occupancy of given adiabatic state. We observed normal diffusion regime of the driven quantum dynamics at quite small values of the memory time defined by the time scales of the XXX--correlations and energy--distribution of the coupling matrix elements (∂H^/∂X)nm(\partial \hat{H}/\partial X)_{nm}(∂H^/∂X)nm​. Here the normal energy diffusion was found to drop out with the width of the matrix elements' energy--distribution and the diffusion may be significantly suppressed with the decrease of the correlations between the matrix elements. In the opposite limit of relatively large memory times we obtained ballistic regime of the dynamics.
The general problem of dissipation in macroscopic large-amplitude collective motion and its relation to energy diffusion of intrinsic degrees of freedom of a nucleus is studied. By applying the cranking approach to the nuclear many-body system, a set of coupled dynamical equations for the collective classical variable and the quantum mechanical occupancies of the intrinsic nuclear states is derived in the limit of weak coupling of the collective and intrinsic subsystems. Different dynamical regimes of the intrinsic nuclear motion and its consequences on time properties of collective dissipation are discussed.
The Hamiltonian of a quantum-mechanical system contains a time-dependent parameter X. The system is prepared in a highly excited eigenstate of the instantaneous Hamiltonian at t=0, and we compute the amplitudes 〈n(t)\ensuremath{\Vert}\ensuremath{\psi}(t)〉 to be in the eigenstates \ensuremath{\Vert}n〉 of the instantaneous Hamiltonian H^(X(t)) at later times. It is found that for generic systems, with no symmetries yielding additional quantum numbers, the occupation probability spreads diffusively away from the initial state. This diffusion is an irreversible process, and can be related to a model for dissipation. The diffusion constant D is investigated numerically in a random matrix model as a function of X\ifmmode \dot{}\else \.{}\fi{}, the rate of change of the parameter. Good agreement with theoretical predictions is found in two limiting regimes. When X\ifmmode \dot{}\else \.{}\fi{} is large, D is proportional to X\ifmmode \dot{}\else \.{}\fi{} 2^{2}2, corresponding to the Ohmic dissipation predicted by the Kubo formula. When X\ifmmode \dot{}\else \.{}\fi{} is small, Landau-Zener transitions are the mechanism for diffusion, and D is proportional to X\ifmmode \dot{}\else \.{}\fi{} 3/2^{3/2}3/2.
We propose and solve a simple but very general quantum model of a SU (2) spin interacting with a large external system with N states. The coupling is described by a random Hamiltonian in a new general Gaussian SU (2) × U ( N ) random matrix ensemble, that we introduce in this paper. We solve the model in the large N limit, for any value of the spin j and for any choice of the coupling matrix element distributions in the different possible angular momentum channels l (and provided that the internal dynamics of the spin is slow). Besides its mathematical interest as a non-trivial random matrix model, it allows us to study and illustrate in a simple framework various phenomena: the decoherence dynamics, the conditions of emergence of the classical phase space for the spin, and the properties of quantum diffusion in phase space. The large time evolution for the spin is shown to be non-Markovian in general, the Markov property emerging in some specific cases for the dynamics and the initial conditions.
We present numerical simulations supporting the random-matrix state-reduction framework developed in the companion theoretical paper. The simulations test the main derived features of the model: isotropic diffusion generated by Gaussian Unitary Ensemble Hamiltonians in projective state space, the restriction of this diffusion to Brownian motion on the classical submanifold, Born-rule frequencies for detector-defined outcome classes, and stroboscopic Newtonian motion for macroscopic systems under repeated environmental monitoring. We also compare GUE and GOE random Hamiltonians and show that GOE fails to produce the required isotropic complex projective diffusion. Further simulations examine finite-resolution detector records in the double-slit experiment, Zeno stability of recorded equivalence classes, effective irreversibility from high-dimensional state-space dynamics and loss of path information, and tensor-product particle-device dynamics in the device limit. The results show that microscopic state reduction, stable measurement records, effective irreversibility, and macroscopic classicality can be described as different coarse-grained manifestations of the same stochastic unitary mechanism.