Results in Physics

Magnus expansion applied to a dissipative driven two-level system

Tuguldur Begzjav, Hichem Eleuch

Published Apr 8, 2020• 10 citations

Abstract

One of the most powerful and potentially useful methods of time-dependent matrix differential equations is the Magnus expansion. In this work, we applied the Magnus expansion for Schrödinger and master equations of a dissipative two-level system interacting with a time-dependent π pulse. Two different models of dissipative two-level systems are considered, namely (i) the system decays from two considered levels into some other levels with given decay rates; (ii) the system decays from its excited level to its ground level. The obtained approximate solutions of a time dynamics by the first-order Magnus expansion are well consistent with the results of numerical calculations obtained by the fourth-order Runge-Kutta method.

DOI: 10.1016/j.rinp.2020.103098

Related papers

Magnus expansion method for two-level atom interacting with few-cycle pulse

Tuguldur Begzjav, J. Ben-Benjamin, Hichem Eleuch +3 more

Using the Magnus expansion to the fourth order, we obtain analytic expressions for the atomic state of a two-level system driven by a laser pulse of arbitrary shape with small pulse area. We also determine the limitation of our obtained formulas due to limited range of convergence of the Magnus series. We compare our method to the recently developed method of Rostovtsev et al. (PRA 2009, 79, 063833) for several detunings. Our analysis shows that our technique based on the Magnus expansion can be used as a complementary method to the one in PRA 2009.

arXiv:1809.02642Journal of Modern OpticsJul 14, 20173 citations

Higher order Magnus expansions for driven two-level quantum dynamics

Chen Wei, Frank Großmann

We investigate the Magnus expansion for a generic time-dependent two-level system under single-axis driving.By virtue of the su(2) Lie algebra, the expansion is decomposed into a commutator-free form. To illustrate the usefulness of the gained expression, we then revisit the Landau-Zener-Stückelberg-Majorana model, with a focus on non-adiabatic transitions as well as the Stokes phase. In addition, the semiclassical Rabi model is systematically treated by determining the Floquet quasienergy up to different orders. We demonstrate how to employ suitable picture transformations as well as on how to enforce the symmetry of the underlying model in order to guarantee convergence of the expansion as well as to achieve satisfactory agreement with the exact results. For both models that we studied it turns out that a third order approximation yields results that are in next to perfect agreement with exact analytical ones. Surprisingly, in the case of the semiclassical Rabi model, even the second order Magnus approximation in the adiabatic picture produces almost exact results for a large parameter range.

quant-pharXiv:2603.13821The Journal of Chemical PhysicsMar 17, 20260 citations

Demkov-Kunike models with decay

M. B. Kenmoe, A. B. Tchapda, Lukong Cornelius Fai

Exact analytical solutions to the dissipative time-dependent Schrödinger equation are obtained for a decaying two-state system with decay rates Γ1 and Γ2 for the levels with extremal spin projections. The system is coherently driven with a pulse whose detuning is made up of two parts: a time-dependent part (chirp) of hyperbolic-tangent shape and a static part with real and imaginary terms. This gives us a wide range of possibilities to arbitrarily select the interaction terms. We considered two versions which led to decaying Demkov-Kunike (DK) models; the version in which the Rabi frequency (interaction) is a time-dependent hyperbolic-secant function (called decaying DK1 model) and the case when it is constant in time and never turns off (decaying DK2 model). Our analytical solutions account for all possible initial moments instead of only t0 = 0 or t0 = − ∞ as for non-decaying models and may be useful for the experiments on level crossings. Two complementary limits of the pulse detuning are considered and explored: the limit of fast (i) and slow rise (ii). In the case (i), the coupling between level positions in the first DK model collapses, while the second DK model reduces to a Rabi model (constant Hamiltonian), in the case (ii), both DK models reduce to the LZ model. In both cases (i) and (ii), analytical approximated solutions which conveniently approach the exact solutions are derived.

arXiv:1307.3878Journal of Mathematical PhysicsDec 1, 20167 citations

Magnus expansion for a chirped quantum two-level system

P. Nalbach, Vicente Leyton‐Ortega

We derive a Magnus expansion for a frequency chirped quantum two-level system. We obtain a time-independent effective Hamiltonian which generates a stroboscopic time evolution. At lowest order the according dynamics is identical to results from using a rotating wave approximation. We determine, furthermore, also the next higher-order corrections within our expansion scheme in correspondence to the Bloch-Siegert shifts for harmonically driven systems. Importantly, our scheme can be extended to more complicated systems, i.e., even many-body systems.

arXiv:1806.02738Physical Review AAug 28, 201810 citations

Simple and General Unitarity Conserving Numerical Real-Time Propagators of the Time-Dependent Schrödinger Equation Based on Magnus Expansion

Taner M. Ture, Seogjoo Jang

Magnus expansion (ME) provides a general way to expand the real-time propagator of a time-dependent Hamiltonian within the exponential such that the unitarity is satisfied at any order. We use this property and explicit integration of Lagrange interpolation formulas for the time-dependent Hamiltonian within each time interval and derive approximations that preserve unitarity for the differential time evolution operators of general time-dependent Hamiltonians. The resulting second-order approximation is the same as using the average of Hamiltonians for two end points of time. We identify three fourth-order approximations involving commutators of Hamiltonians at different times and also derive a sixth-order expression. A test of these approximations along with other available expressions for a two-state time-dependent Hamiltonian with sinusoidal time dependences provides information on the relative performance of these approximations and suggests that the derived expressions can serve as useful numerical tools for time evolution in time-resolved spectroscopy, quantum control, quantum sensing, real-time ab initio quantum dynamics, and open system quantum dynamics.

arXiv:2312.01115The Journal of Physical Chemistry AApr 2, 20243 citations

Numerical Case Study of an Atom-Photon Interaction in a Cavity Exploring Quantum Control

Javier Jalandoni

We study Magnus expansion (ME) approximation scheme for the interaction between an atom and a single quantized cavity mode (Jaynes-Cumming model) in a closed quantum system in resonance or near resonance for a time-dependent coupling coefficient g(t) in both the interaction and rotating picture by implementing a novel numerical method called MG4 and compare out results to the Runge-Kutta 4th (RK4) order solution to demonstrate the conservation of unitary evolution of the ME. A cursory study of open quantum system is given to encourage the study of ME for dissipative systems. Furthermore, we assume that our time-dependent coupling coefficient g(t) can take on two forms, Gaussian and sinusoidal, which are introduced as pulses to study the behavior and response of the cavity. Our results show that ME is a sufficient approximation scheme in our study of closed quantum systems which may have applications in quantum control.

Jan 1, 20161 citation