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

Piecewise time-independent procedure to control two-level systems

J. Kuhn, M. G. E. da Luz

Published May 17, 2007• 11 citations
Publisher

Abstract

We present a general and mathematically simple method to control two-level systems. It is based on a piecewise time-independent procedure. We assume that some parameters of an external interaction potential can be rapidly switched at specified time instants, and then kept constant during small time intervals $\mathrm{\ensuremath{\Delta}}t$. By properly setting the parameter values, obtained from algebraic, instead of differential or integral, equations, we can drive the time evolution of arbitrary observables. We illustrate the approach with some examples and discuss important technical aspects, relevant in real concrete situations, such as the robustness of the method to errors in the control parameter values, the influence of the switching mechanism transient times, the appropriate choice for the $\mathrm{\ensuremath{\Delta}}t$'s, and so on.

DOI
10.1103/physreva.75.053410

Related papers

Tracking control of two qubit entanglement using piecewise time-independent method

Guilherme Jurkevicz Delben, Marcus W. Beims

Abstract The tracking control of quantum correlations between two coupled qubits is analyzed, where only one of the qubits is coupled to a Markovian environment. Such a system is a generalized model that can be used, for example, to study the problem of a nuclear spin coupled to the electron spin exposed to the environment. Using an external control field, we increase the time for which the system remains coherent and entangled. The control field is an external potential applied to the system, which contains two adjustable parameters, the intensity and the phase of the field. Furthermore, we present quantum control protocols with different targets. First, for population control of both qubits, second for the coherence control of both two-level systems, and finally for the control of entanglement. Due to analytical for X states the target function can directly be identified with the entanglement. Furthermore, we have shown that when smaller dissipation rates are considered, the control method generates larger stable entanglement between the qubits.

Physica Scripta•Dec 4, 2020•6 citations

Analytically Solvable Driven Time-Dependent Two-Level Quantum Systems

Edwin Barnes, S. Das Sarma

Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.

arXiv:1206.0297•Physical Review Letters•Aug 7, 2012•159 citations

Classical equations of motion and algebras of quantum observables

M. A. Sokolov

In this work simple and effective quantization procedure of classical dynamical systems is proposed and illustrated by a number of examples. The procedure is based entirely on differential equations which describe time evolution of systems.

arXiv:0911.4836•Nov 25, 2009•0 citations

Optimal Control of Quantum Systems and a Generalized Level Set Method

Fariel Shafee

We study the application of a generalized form of the level set method used in classical physical contexts to quantum optimal control situations. The set of OCT equations needed to keep the expectation value of an observable constant is first discussed and the dimensionality of the actual parameter space carefully considered. Then we see how concepts of level set methods emerge that may help solve the inverse problem associated with designing the control Hamiltonian with greater speed. The formal equations and the algorithm are presented.

arXiv:quant-ph/0411160•Nov 21, 2004•0 citations

Dynamical stability of the weakly nonharmonic propeller-shaped planar Brownian rotator

Igor Petrović, J. Jeknić-Dugić, M. Arsenijević +1 more

Dynamical stability is a prerequisite for control and functioning of desired nanomachines. We utilize the Caldeira-Leggett master equation to investigate dynamical stability of molecular cogwheels modeled as a rigid, propeller-shaped planar rotator. To match certain expected realistic physical situations, we consider a weakly nonharmonic external potential for the rotator. Two methods for investigating stability are used. First, we employ a quantum-mechanical counterpart of the so-called "first passage time" method. Second, we investigate time dependence of the standard deviation of the rotator for both the angle and angular momentum quantum observables. A perturbationlike procedure is introduced and implemented to provide the closed set of differential equations for the moments. Extensive analysis is performed for different combinations of the values of system parameters. The two methods are, in a sense, mutually complementary. Appropriate for the short time behavior, the first passage time exhibits a numerically relevant dependence only on the damping factor as well as on the rotator size. However, the standard deviations for both the angle and angular momentum observables exhibit strong dependence on the parameter values for both short and long time intervals. Contrary to our expectations, the time decrease of the standard deviations is found for certain parameter regimes. In addition, for certain parameter regimes nonmonotonic dependence on the rotator size is observed for the standard deviations and for the damping of the oscillation amplitude. Hence, nonfulfillment of the classical expectation that the size of the rotator can be reduced to the inertia of the rotator. In effect, the task of designing the desired protocols for the proper control of the molecular rotations becomes an optimization problem that requires further technical elaboration.

arXiv:1912.09756•Physical review. E•Jan 3, 2020•7 citations

Analytically solvable Hamiltonians for quantum two-level systems and their dynamics

ANTONINO MESSINA, Hiromichi Nakazato

A simple systematic way of obtaining analytically solvable Hamiltonians for quantum two-level systems is presented. In this method, a time-dependent Hamiltonian and the resulting unitary evolution operator are connected through an arbitrary function of time, furnishing us with new analytically solvable cases. The method is surprisingly simple, direct, and transparent and is applicable to a wide class of two-level Hamiltonians with no involved constraint on the input function. A few examples illustrate how the method leads to simple solvable Hamiltonians and dynamics.

Journal of Physics A Mathematical and Theoretical•Oct 20, 2014•57 citations
We present a general and mathematically simple method to control two-level systems. It is based on a piecewise time-independent procedure. We assume that some parameters of an external interaction potential can be rapidly switched at specified time instants, and then kept constant during small time intervals \ensuremathΔt\mathrm{\ensuremath{\Delta}}t\ensuremathΔt. By properly setting the parameter values, obtained from algebraic, instead of differential or integral, equations, we can drive the time evolution of arbitrary observables. We illustrate the approach with some examples and discuss important technical aspects, relevant in real concrete situations, such as the robustness of the method to errors in the control parameter values, the influence of the switching mechanism transient times, the appropriate choice for the \ensuremathΔt\mathrm{\ensuremath{\Delta}}t\ensuremathΔt's, and so on.
Abstract The tracking control of quantum correlations between two coupled qubits is analyzed, where only one of the qubits is coupled to a Markovian environment. Such a system is a generalized model that can be used, for example, to study the problem of a nuclear spin coupled to the electron spin exposed to the environment. Using an external control field, we increase the time for which the system remains coherent and entangled. The control field is an external potential applied to the system, which contains two adjustable parameters, the intensity and the phase of the field. Furthermore, we present quantum control protocols with different targets. First, for population control of both qubits, second for the coherence control of both two-level systems, and finally for the control of entanglement. Due to analytical for X states the target function can directly be identified with the entanglement. Furthermore, we have shown that when smaller dissipation rates are considered, the control method generates larger stable entanglement between the qubits.
Analytical solutions to the time-dependent Schrödinger equation describing a driven two-level system are invaluable to many areas of physics, but they are also extremely rare. Here, we present a simple algorithm that generates an unlimited number of exact analytical solutions. We show that a general single-axis driving term and its corresponding evolution operator are determined by a single real function which is constrained only by a certain inequality and initial conditions. Any function satisfying these constraints yields an exact analytical solution. We demonstrate this method by presenting several new exact solutions to the time-dependent Schrödinger equation. Our general method and many of the new solutions we present are particularly relevant to qubit control in quantum computing applications.
We study the application of a generalized form of the level set method used in classical physical contexts to quantum optimal control situations. The set of OCT equations needed to keep the expectation value of an observable constant is first discussed and the dimensionality of the actual parameter space carefully considered. Then we see how concepts of level set methods emerge that may help solve the inverse problem associated with designing the control Hamiltonian with greater speed. The formal equations and the algorithm are presented.
Dynamical stability is a prerequisite for control and functioning of desired nanomachines. We utilize the Caldeira-Leggett master equation to investigate dynamical stability of molecular cogwheels modeled as a rigid, propeller-shaped planar rotator. To match certain expected realistic physical situations, we consider a weakly nonharmonic external potential for the rotator. Two methods for investigating stability are used. First, we employ a quantum-mechanical counterpart of the so-called "first passage time" method. Second, we investigate time dependence of the standard deviation of the rotator for both the angle and angular momentum quantum observables. A perturbationlike procedure is introduced and implemented to provide the closed set of differential equations for the moments. Extensive analysis is performed for different combinations of the values of system parameters. The two methods are, in a sense, mutually complementary. Appropriate for the short time behavior, the first passage time exhibits a numerically relevant dependence only on the damping factor as well as on the rotator size. However, the standard deviations for both the angle and angular momentum observables exhibit strong dependence on the parameter values for both short and long time intervals. Contrary to our expectations, the time decrease of the standard deviations is found for certain parameter regimes. In addition, for certain parameter regimes nonmonotonic dependence on the rotator size is observed for the standard deviations and for the damping of the oscillation amplitude. Hence, nonfulfillment of the classical expectation that the size of the rotator can be reduced to the inertia of the rotator. In effect, the task of designing the desired protocols for the proper control of the molecular rotations becomes an optimization problem that requires further technical elaboration.
A simple systematic way of obtaining analytically solvable Hamiltonians for quantum two-level systems is presented. In this method, a time-dependent Hamiltonian and the resulting unitary evolution operator are connected through an arbitrary function of time, furnishing us with new analytically solvable cases. The method is surprisingly simple, direct, and transparent and is applicable to a wide class of two-level Hamiltonians with no involved constraint on the input function. A few examples illustrate how the method leads to simple solvable Hamiltonians and dynamics.