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Europhysics Letters (EPL)

Dynamic two-level system in the presence of a delta-function pulse: Exactly solvable dynamic system

Er’el Granot

Published Jan 1, 2026• 1 citation
Publisher

Abstract

Abstract The Landau-Zener formula represents the only case where a dynamic quantum system was solved exactly analytically. There is no model of a system, which is perturbed by a temporal pulse, and can be solved exactly. It is suggested to use a delta-function perturbation, a model which has an exact analytical solution, to model the effect of a finite period pulse perturbation. Not only can this model be solved exactly and mimic the effect of a very narrow pulse, but it also presents a quantization rule for suppressed transmission ( i.e. , no change in initial state), which resembles qualitatively a similar quantization rule that appears due to the presence of a finite period pulse. Nevertheless, despite its simplicity, the model exhibits non-trivial dynamics that challenge approximation schemes. Specifically, it shows that the suppressed transmission criterion substantially changes due to an arbitrarily small change in potential value.

DOI
10.1209/0295-5075/ae3980

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Abstract The Landau-Zener formula represents the only case where a dynamic quantum system was solved exactly analytically. There is no model of a system, which is perturbed by a temporal pulse, and can be solved exactly. It is suggested to use a delta-function perturbation, a model which has an exact analytical solution, to model the effect of a finite period pulse perturbation. Not only can this model be solved exactly and mimic the effect of a very narrow pulse, but it also presents a quantization rule for suppressed transmission ( i.e. , no change in initial state), which resembles qualitatively a similar quantization rule that appears due to the presence of a finite period pulse. Nevertheless, despite its simplicity, the model exhibits non-trivial dynamics that challenge approximation schemes. Specifically, it shows that the suppressed transmission criterion substantially changes due to an arbitrarily small change in potential value.
Abstract The transmission probability in a generic two-quasi-state system is addressed. An approximate expression for the transmission probability between the two quasi-states is derived for the case where the energy gap between the two states varies. Unlike the Landau-Zener formula, the temporal change can have a generic form and can vanish several times. In particular, when the energy gap vanishes twice, an analytical quantization rule for the suppression of transmission between two quasi-states is derived.
Abstract Three analytic solutions to the Schrödinger equation for the time-dependent Landau–Zener (LZ) Hamiltonian are presented. They correspond to specific finite-time driving paths in a bounded parameter space of a two-level system. Two of these paths go through the avoided crossing of levels, either with a constant speed or with variable speed that decreases in the region of reduced energy gap, the third path bypasses the crossing such that the energy gap remains constant. The solutions yield exact time dependencies of the excitation probability for the system evolving from the ground state of the initial Hamiltonian. The LZ formula emerges as an approximation valid within a certain interval of driving times for the constant-speed driving through the avoided crossing. For long driving times, all solutions converge to the prediction of the adiabatic perturbation theory. The excitation probability vanishes at some discrete time instants.
A simple algorithm is presented based on a type of partial reverse engineering that generates an unlimited number of exact analytical solutions to the Schr\"odinger equation for a general time-dependent two-level Hamiltonian. I demonstrate this method by deriving exact solutions corresponding to fast control pulses that contain arbitrarily many tunable parameters. It is shown that the formalism is naturally suited to generating analytical control protocols that perform precise nonadiabatic rapid passage and Landau-Zener interferometry near the quantum speed limit. A general, exact formula for Landau-Zener interference patterns is derived.
The precise characterization of dynamics in open quantum systems often presents significant challenges, leading to the introduction of various approximations to simplify a model. One commonly used strategy involves Markovian approximations, assuming a memoryless environment. In this study, such approximations are not used and an analytical dynamical depiction of an open quantum system is provided. The system under consideration is an oscillator that is surrounded by a bath of oscillators. The resulting dynamics are characterized by an integrodifferential equation, which can be solved numerically. Remarkably, an exactly analytical solution exists when its reservoir spectral density is Lorentzian. Furthermore, this analytical solvability remains when rectangular pulse control is applied to the system, though with different parameters due to the pulses. Moreover, distinct dynamical regions emerge, depending on certain parameter values. Notably, the steady-state average excitation number (AEN) of the system shows rapid escalation with increasing non-Markovianity, reflecting the intricacies of real-world dynamics. In cases where there is detuning between the system frequency and the environmental central frequency within a non-Markovian regime, the AEN maintains its initial value for an extended period. Interestingly, the exact analytical solution with pulse control inherently protects the quantum system from decoherence effects without using approximations, while this protection fades as the system approaches the Markovian regime. The pulse control can not only prolong the relaxation time of the oscillator, but can also be used to speed up the relaxation process, depending on the specifications of the pulse.
The paper presents an analytic study of a two-state system which interacts with an external pulsed field. The excited state is supposed to decay out of the system with a decay rate \ensuremath{\Gamma}. We present two exact analytic solutions, viz., the finite Landau-Zener model and the Demkov-Kunike model. We also derive an approximate formula for arbitrary pulse shapes and detunings in the limit of a large decay rate. It gives the exact result for the original Landau-Zener model and the correct asymptotics for large \ensuremath{\Gamma} in the general case. We conclude that the independence of the final ground-state population on the decay rate, recently established for the original Landau-Zener model [V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992)], is only valid for the latter and not for any other model. The results show that the population loss ultimately decreases at large decay rates which can be viewed as a quantum analog of the overdamped oscillator.
We solve the problem of two discrete quantum levels which are coupled by a time-dependent radio-frequency pulse $W(t)=V(t) {e}^{i\ensuremath{\nu}t}$, where the envelope function is of a form suggested by Rosen and Zener: $V(t)={V}_{0}sech(\frac{\ensuremath{\pi}t}{T})$. When a level damping constant $\ensuremath{\gamma}$ is included, in the manner of Bethe-Lamb theory, the solutions show new features which are not expected on the basis of a sudden-approximation theory, where V(t)=constV(t)=\mathrm{const}V(t)=const over the pulse duration TTT. Various transient effects such as "ringing" are not present in the extended Rosen-Zener solution; these effects are related to the large impulsive forces at the step discontinuities in the sudden approximation. The final-state level amplitudes can be quite different depending on the size of the pulse rise time TTT as compared with the system Bohr period $\frac{1}{\ensuremath{\omega}}$. Our results allow a continuous and quantitatively exact comparison between the extremes of the sudden ($\ensuremath{\omega}T\ensuremath{\ll}1$) and adiabatic ($\ensuremath{\omega}T\ensuremath{\gg}1$) approximations. A model of a "quasisudden" step function is also constructed, and remarks are made on the validity of a certain conjecture by Rosen and Zener.