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:quant-ph/0310162 · Journal of Physics A Mathematical and General

Coarse grained and fine dynamics in trapped ion Raman schemes

B Militello, P Aniello, A Messina

Published Aug 5, 2004• 3 citations
arXivHTMLPDF

Abstract

A novel result concerning Raman coupling schemes implemented using trapped ions is obtained. By means of an operator perturbative approach, it is shown that the complete time evolution of these systems can be expressed, with a high degree of accuracy, as the product of two unitary evolutions. The first one describes the time evolution related to an effective coarse grained dynamic. The second is a suitable correction restoring the fine dynamics suppressed by the coarse graining performed to adiabatically eliminate the nonresonantly coupled atomic level. The case where a decoherence source is present is also studied.

DOI
10.1088/0305-4470/37/33/014

Related papers

Coarse grained and fine dynamics in trapped ion Raman schemes

Benedetto Militello, Paolo Aniello, ANTONINO MESSINA

A novel result concerning Raman coupling schemes implemented using trapped ions is obtained. By means of an operator perturbative approach, it is shown that the complete time evolution of these systems can be expressed, with a high degree of accuracy, as the product of two unitary evolutions. The first one describes the time evolution related to an effective coarse grained dynamics. The second is a suitable correction restoring the fine dynamics suppressed by the coarse graining performed to adiabatically eliminate the nonresonantly coupled atomic level.

•Jan 1, 2008•5 citations

Time-dependent perturbation treatment of independent Raman schemes

Benedetto Militello, Paolo Aniello, ANTONINO MESSINA

The problem of a trapped ion subjected to the action of two or more independent Raman schemes is analysed through a suitable time-dependent perturbative approach based on the factorization of the evolution operator in terms of other unitary operators. We show that the dynamics of the system may be traced back to an effective Hamiltonian up to a suitable dressing. Moreover, we give the method to write the master equation corresponding to the case wherein spontaneous decays occur

Journal of Physics A Mathematical and Theoretical•Apr 17, 2007•2 citations

Perturbative Treatment of the Evolution Operator Associated with Raman Couplings

Benedetto Militello

A novel perturbative treatment of the time evolution operator of a quantum system is applied to the model describing a Raman-driven trapped ion in order to obtain a suitable 'effective model'.It is shown that the associated effective Hamiltonian describes the system dynamics up to a certain transformation which may be interpreted as a 'dynamical dressing' of the effective model.

arXiv:quant-ph/0605091•Symmetry Integrability and Geometry Methods and Applications•May 10, 2006•1 citation

Systematic time-coarse graining for driven quantum systems

Leon Bello, Fan, Wentao, Aditya Gandotra +1 more

Many recent advancements in quantum computing leverage strong drives on nonlinear systems for state preparation, signal amplification, or gate operation. However, the interplay within such strongly driven system introduces multi-scale dynamics that affects the long-time behavior of the system in non-trivial ways that are very difficult to model. Therefore, the analysis of these systems often relies on effective Hamiltonian models that introduce additional nonlinear processes which approximate the long-time dynamics so that highly oscillatory terms may be ignored. However, the removal of such high frequency transitions can only be performed rigorously within a systematic framework of time-coarse graining, which is a fundamentally irreversible operation. This implies that standard approaches with unitary effective models cannot accurately capture the long-time behavior of strongly driven nonlinear quantum systems in general. We introduce a systematic perturbation theory for obtaining the complete non-unitary effective model of the time-coarse grained (TCG) dynamics of a driven quantum system to any order in the coupling strengths. We derive a closed-form analytical formula for both unitary and non-unitary contributions, in the form of an effective Hamiltonian and non-unitary (pseudo-)dissipators. Remarkably, even though the effective theory presumes unitary time evolution at the microscopic level, the time-coarse grained dynamics is found to follow a non-unitary time evolution in general. This occurs even when there is no open heat reservoir for the system to become entangled with or dissipate into. We demonstrate the effectiveness of the new method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.

arXiv:2407.06068•Jul 8, 2024•0 citations

Systematic perturbation theory for dynamical coarse-graining

Gernot Schaller, Philipp Zedler, Tobias Brandes

We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant, we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix---even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics; i.e., it is suitable for analyzing non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.

arXiv:0812.1507•Physical Review A•Mar 20, 2009•41 citations

Coarse-Grained Effective Hamiltonian via the Magnus Expansion for a Three-Level System

Nicola Macrì, Luigi Giannelli, Elisabetta Paladino +1 more

Quantum state processing is one of the main tools of quantum technologies. While real systems are complicated and/or may be driven by non-ideal control, they may nevertheless exhibit simple dynamics approximately confined to a low-energy Hilbert subspace. Adiabatic elimination is the simplest approximation scheme allowing us to derive in certain cases an effective Hamiltonian operating in a low-dimensional Hilbert subspace. However, these approximations may present ambiguities and difficulties, hindering a systematic improvement of their accuracy in larger and larger systems. Here, we use the Magnus expansion as a systematic tool to derive ambiguity-free effective Hamiltonians. We show that the validity of the approximations ultimately leverages only on a proper coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations.

arXiv:2212.08508•Entropy•Jan 27, 2023•3 citations
A novel result concerning Raman coupling schemes implemented using trapped ions is obtained. By means of an operator perturbative approach, it is shown that the complete time evolution of these systems can be expressed, with a high degree of accuracy, as the product of two unitary evolutions. The first one describes the time evolution related to an effective coarse grained dynamic. The second is a suitable correction restoring the fine dynamics suppressed by the coarse graining performed to adiabatically eliminate the nonresonantly coupled atomic level. The case where a decoherence source is present is also studied.
A novel result concerning Raman coupling schemes implemented using trapped ions is obtained. By means of an operator perturbative approach, it is shown that the complete time evolution of these systems can be expressed, with a high degree of accuracy, as the product of two unitary evolutions. The first one describes the time evolution related to an effective coarse grained dynamics. The second is a suitable correction restoring the fine dynamics suppressed by the coarse graining performed to adiabatically eliminate the nonresonantly coupled atomic level.
Many recent advancements in quantum computing leverage strong drives on nonlinear systems for state preparation, signal amplification, or gate operation. However, the interplay within such strongly driven system introduces multi-scale dynamics that affects the long-time behavior of the system in non-trivial ways that are very difficult to model. Therefore, the analysis of these systems often relies on effective Hamiltonian models that introduce additional nonlinear processes which approximate the long-time dynamics so that highly oscillatory terms may be ignored. However, the removal of such high frequency transitions can only be performed rigorously within a systematic framework of time-coarse graining, which is a fundamentally irreversible operation. This implies that standard approaches with unitary effective models cannot accurately capture the long-time behavior of strongly driven nonlinear quantum systems in general. We introduce a systematic perturbation theory for obtaining the complete non-unitary effective model of the time-coarse grained (TCG) dynamics of a driven quantum system to any order in the coupling strengths. We derive a closed-form analytical formula for both unitary and non-unitary contributions, in the form of an effective Hamiltonian and non-unitary (pseudo-)dissipators. Remarkably, even though the effective theory presumes unitary time evolution at the microscopic level, the time-coarse grained dynamics is found to follow a non-unitary time evolution in general. This occurs even when there is no open heat reservoir for the system to become entangled with or dissipate into. We demonstrate the effectiveness of the new method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.
We demonstrate how the dynamical coarse-graining approach can be systematically extended to higher orders in the coupling between system and reservoir. Up to second order in the coupling constant, we explicitly show that dynamical coarse-graining unconditionally preserves positivity of the density matrix---even for bath density matrices that are not in equilibrium and also for time-dependent system Hamiltonians. By construction, the approach correctly captures the short-time dynamics; i.e., it is suitable for analyzing non-Markovian effects. We compare the dynamics with the exact solution for highly non-Markovian systems and find a remarkable quality of the coarse-graining approach. The extension to higher orders is straightforward but rather tedious. The approach is especially useful for bath correlation functions of simple structure and for small system dimensions.
Quantum state processing is one of the main tools of quantum technologies. While real systems are complicated and/or may be driven by non-ideal control, they may nevertheless exhibit simple dynamics approximately confined to a low-energy Hilbert subspace. Adiabatic elimination is the simplest approximation scheme allowing us to derive in certain cases an effective Hamiltonian operating in a low-dimensional Hilbert subspace. However, these approximations may present ambiguities and difficulties, hindering a systematic improvement of their accuracy in larger and larger systems. Here, we use the Magnus expansion as a systematic tool to derive ambiguity-free effective Hamiltonians. We show that the validity of the approximations ultimately leverages only on a proper coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations.