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arXiv:2309.06448 · Physical Review A

Noisy Demkov-Kunike model

Lin Chen, Zhaoxin Liang

Published Feb 16, 2024• 4 citations
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Abstract

The Demkov-Kunike (DK) model, characterized by a time-dependent Rabi coupling $J\phantom{\rule{0.16em}{0ex}}\text{sech}(t/T)$ and on-site detuning ${\mathrm{\ensuremath{\Delta}}}_{0}+{\mathrm{\ensuremath{\Delta}}}_{1}tanh(t/T)$, has one of the most general forms of an exactly solvable two-state quantum system and therefore it provides a paradigm for coherent manipulations of a qubit's quantum state. Despite its extensive application in noise-free cases, the exploration of the noisy DK model remains limited. Here we extend the coherent DK model to take into account a noisy coupling term $J\ensuremath{\rightarrow}{J}_{\text{noisy}}(t)$. We consider colored Markovian noise sources represented by telegraph noise and Gaussian noise. We present exact solutions for the survival probability ${Q}_{\text{DK}}^{\text{noisy}}$ of the noisy DK model, namely, the probability of the system to remain in its initial state. For slow telegraph noise, we identify parameter regimes where the survival probability ${Q}_{\text{DK}}^{\text{noisy}}$ is suppressed rather than enhanced by noise. In contrast, for slow Gaussian noise, the noise always enhances the survival probability ${Q}_{\text{DK}}^{\text{noisy}}$, due to the absorption of noise quanta across the energy gap. This study not only complements the existing research on the noisy Landau-Zener model but also provides valuable insights into the control of two-level quantum systems.

DOI
10.1103/physreva.109.022219

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The Demkov-Kunike (DK) model, characterized by a time-dependent Rabi coupling Jsech(t/T)J\phantom{\rule{0.16em}{0ex}}\text{sech}(t/T)Jsech(t/T) and on-site detuning \ensuremathΔ0+\ensuremathΔ1tanh(t/T){\mathrm{\ensuremath{\Delta}}}_{0}+{\mathrm{\ensuremath{\Delta}}}_{1}tanh(t/T)\ensuremathΔ0​+\ensuremathΔ1​tanh(t/T), has one of the most general forms of an exactly solvable two-state quantum system and therefore it provides a paradigm for coherent manipulations of a qubit's quantum state. Despite its extensive application in noise-free cases, the exploration of the noisy DK model remains limited. Here we extend the coherent DK model to take into account a noisy coupling term J\ensuremath→Jnoisy(t)J\ensuremath{\rightarrow}{J}_{\text{noisy}}(t)J\ensuremath→Jnoisy​(t). We consider colored Markovian noise sources represented by telegraph noise and Gaussian noise. We present exact solutions for the survival probability QDKnoisy{Q}_{\text{DK}}^{\text{noisy}}QDKnoisy​ of the noisy DK model, namely, the probability of the system to remain in its initial state. For slow telegraph noise, we identify parameter regimes where the survival probability QDKnoisy{Q}_{\text{DK}}^{\text{noisy}}QDKnoisy​ is suppressed rather than enhanced by noise. In contrast, for slow Gaussian noise, the noise always enhances the survival probability QDKnoisy{Q}_{\text{DK}}^{\text{noisy}}QDKnoisy​, due to the absorption of noise quanta across the energy gap. This study not only complements the existing research on the noisy Landau-Zener model but also provides valuable insights into the control of two-level quantum systems.
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We study Landau-Zener-like dynamics of a qubit influenced by transverse random telegraph noise. The telegraph noise is characterized by its coupling strength $\ensuremath{\upsilon}$ and switching rate $\ensuremath{\gamma}$. The qubit energy levels are driven nonlinearly in time, $\ensuremath{\propto}\mathrm{sgn}(t){\ensuremath{\mid}t\ensuremath{\mid}}^{\ensuremath{\nu}}$, and we derive the transition probability in the limit of sufficiently fast noise, for arbitrary exponent $\ensuremath{\nu}$. The level occupation after the transition depends strongly on $\ensuremath{\nu}$, and there exists a critical ${\ensuremath{\nu}}_{c}$ with qualitative difference between $\ensuremath{\nu}<{\ensuremath{\nu}}_{c}$ and $\ensuremath{\nu}>{\ensuremath{\nu}}_{c}$. When $\ensuremath{\nu}<{\ensuremath{\nu}}_{c}$, the final state is always fully incoherent with equal population of both quantum levels, even for arbitrarily weak noise. For $\ensuremath{\nu}>{\ensuremath{\nu}}_{c}$, the system keeps some coherence depending on the strength of the noise, and in the limit of weak noise, no transition takes place. For fast noise ${\ensuremath{\nu}}_{c}=1∕2$, while for slow noise ${\ensuremath{\nu}}_{c}<1∕2$ and it depends on $\ensuremath{\gamma}$. We also discuss phase coherence, which is relevant when the qubit has a nonzero minimum energy gap. The qualitative dependency on $\ensuremath{\nu}$ is the same for the phase coherence and level occupation. The state after the transition does, in general, depend on $\ensuremath{\gamma}$. For fixed $\ensuremath{\upsilon}$, increasing $\ensuremath{\gamma}$ decreases the final state coherence when $\ensuremath{\nu}<1$ and increases the final state coherence when $\ensuremath{\nu}>1$. Only the conventional linear driving is independent of $\ensuremath{\gamma}$.
The effect of a slow noise in nondiagonal matrix element J(t)J(t)J(t) that describes the diabatic level coupling on the probability of the Landau-Zener transition is studied. For slow noise, the correlation time ${\ensuremath{\tau}}_{c}$ of J(t)J(t)J(t) is much longer than the characteristic time of the transition. Existing theory for this case suggests that the average transition probability is the result of averaging of the conventional Landau-Zener probability, calculated for a given constant JJJ, over the distribution of JJJ. We calculate a finite-${\ensuremath{\tau}}_{c}$ correction for this classical result. Our main finding is that this correction is dominated by sparse realizations of noise for which J(t)J(t)J(t) passes through zero within a narrow time interval near the level crossing. Two models of noise, random telegraph noise and Gaussian noise, are considered. Naturally, in both models the average probability of transition decreases upon decreasing ${\ensuremath{\tau}}_{c}$. For Gaussian noise we identify two domains of this falloff with specific dependencies of average transition probability on ${\ensuremath{\tau}}_{c}$.
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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.
Abstract The Demkov-Kunike (DK) model, characterized by a hyperbolic tangent detuning and a hyperbolic secant coupling, serves as a universal framework for describing driven quantum systems. However, existing research has primarily focused on two-level systems, some gaps still persist in extending the model to three-level configurations to achieve coherent population transfer (CPT). This study constructs exact solutions for both two- and three-level DK models via a canonical transformation, providing a basis for the quantitative analysis of population transfer, and achieves effective regulation of CPT via a finite-time adiabatic driving mechanism under specific parameter constraints. Furthermore, we analyze the impact of pulse truncation—a common experimental constraint—as well as environmental dissipation on quantum state evolution. The results show that the proposed scheme achieves rapid, high-fidelity coherent population transfer not only under ideal conditions, but also in the presence of realistic pulse truncation and even when decoherence induced by system-environment coupling is included. This work achieves CPT in the three-level DK model, offering a promising approach for advancing quantum state control in practical applications.