Aspects of coherence and decoherence are analyzed within the optical Bloch equations. By rewriting the analytic solution in an alternate form, we are able to emphasize a number of unusual features: (a) despite the Markovian nature of the bath, coherence at long times can be retained; (b) the long-time asymptotic degree of coherence in the system is intertwined with the asymptotic difference in level populations; (c) the traditional population-relaxation and decoherence times, T1 and T2, lose their meaning when the system is in the presence of an external field, and are replaced by more general overall time scales; (d) increasing the field strength, quantified by the Rabi frequency Omega, increases the rate of decoherence rather than reducing it, as one might expect; and (e) maximum asymptotic coherence is reached when the system parameters satisfy Omega2=1(T1T2).
Aspects of coherence and decoherence are analyzed within the optical Bloch equations. By rewriting the analytic solution in an alternate form, we are able to emphasize a number of unusual features: (a) despite the Markovian nature of the bath, coherence at long times can be retained; (b) the long-time asymptotic degree of coherence in the system is intertwined with the asymptotic difference in level populations; (c) the traditional population-relaxation and decoherence times, T1 and T2, lose their meaning when the system is in the presence of an external field, and are replaced by more general overall time scales; (d) increasing the field strength, quantified by the Rabi frequency Omega, increases the rate of decoherence rather than reducing it, as one might expect; and (e) maximum asymptotic coherence is reached when the system parameters satisfy Omega2=1(T1T2).
We present a comprehensive analytical study of the optical Bloch equations for a closed two-level atom. We obtain a map in phase space of the Rabi frequency and the laser frequency detuning, discriminating the characteristics of solutions to the optical Bloch equations separated by a boundary in phase space. Outside the boundary, oscillatory behavior is shown, while simple decaying behavior is shown inside the boundary. The behavior of the solutions is also found to be dependent on the relative magnitude of the decay rates of the optical coherence and the excited state population.
A systematic, rigorous, and complete investigation of the Bloch equations in time-harmonic driving classical field is performed. Our treatment is unique in that it takes full advantage of the partial fraction decomposition over real number field, which makes it possible to find and classify all analytic solutions. Torrey's analytic solution in the form of exponentially damped harmonic oscillations [Phys. Rev. {\bf 76}, 1059 (1949)] is found to dominate the parameter space, which justifies its use at numerous occasions in magnetic resonance and in quantum optics of atoms, molecules, and quantum dots. The unorthodox solutions of the Bloch equations, which do not have the form of exponentially damped harmonic oscillations, are confined to rather small detunings δ2≲(γ−γt)2/27 and small field strengths Ω2≲8(γ−γt)2/27, where γ and γt describe decay rates of the excited state (the total population relaxation rate) and of the coherence, respectively. The unorthodox solutions being readily accessible experimentally are characterized by rather featureless time dependence.
In this paper, we present an analytical solution to the Maxwell–Bloch equations of the two-level semiconductor quantum well, G a A s / A l G a A s . In addition, we discuss the effects of coherent Rabi oscillations Ω ( t ) and the frequency of the semiconductor system ν ( t ) on atomic occupation probabilities, ρ 11 ( t ) and ρ 22 ( t ) , population inversion, ρ z ( t ) , and information entropies, H ( σ x ) , H ( σ y ) , and H ( σ z ) . We observe clearly the emergence of long-lived quantum coherence and the decay in curves for some special cases of Ω ( t ) and ν ( t ) . Also, we show that the dynamic nonlinear properties of the system can be controlled by changing the values of the coherent Rabi oscillations Ω ( t ) and the frequency of the semiconductor system ν ( t ) . Due to the lack of mathematical treatment of such systems, our study promises significant advantages for a large number of nonlinear dynamic systems, opening up a wide range of applications for semiconductor quantum wells.
The loss of coherence in a general open quantum system interacting with a bosonic environment is analyzed. The reservoir is initially in a thermal state. The reduced dynamics is described by a non-Markovian time-local master equation. We consider spectral densities that are sub- or super-Ohmic at low frequencies and arbitrarily shaped at high frequencies. In the super-Ohmic regime, for noninteger frequency powers larger than 2, long time survival of coherence appears. In the latter regime, at vanishing temperature, the asymptotic amount of surviving coherence is stabilized to its initial value, up to a phase factor, by properly increasing the bandwidth and decreasing the low-frequency profile of the spectral density. For noninteger positive frequency powers less than 2, stretched exponential-like decoherence is found over long times. The relaxations to the asymptotic configurations become arbitrarily slow by approaching the frequency power 2 of the super-Ohmic regime. The same dependence on temperature, spectral density, and scale frequency appears for purity and concurrence of two qubits and coherence of a qubit.
Multi-qubit systems are essential for scalable quantum technologies, but their performance is often limited by decoherence from qubit--qubit interactions and environmental noise. Although environmental decoherence in single-qubit systems and gate fidelity in multi-qubit systems have been widely studied, a predictive framework connecting qubit interactions, concentration, spatial distribution, and bath occupation to relaxation and decoherence times remains lacking. Here, we develop a multi-qubit mean-field Lindblad master equation (MQMF-LME) framework for the population and coherence dynamics of a solid-state qubit in an interacting multi-qubit environment. The framework treats one qubit as the system of interest and the surrounding qubits as an effective bath, incorporating intrinsic relaxation and bidirectional excitation transfer between the system and the bath. Analytical solutions provide closed-form expressions for density-matrix dynamics, steady-state populations, relaxation time T1, and decoherence time T2, while numerical simulations extend the framework to concentration-dependent dynamics, 1/f-noise-induced dephasing, and material-specific excitation-transfer mechanisms. For a model system with Förster resonance energy transfer (FRET)-mediated excitation exchange, higher qubit concentrations reduce both T1 and T2, whereas 1/f noise reduces T2 without changing T1. Applied to Er3+-doped CeO2, the framework shows that long-range FRET-mediated excitation transfer reproduces the experimental decrease in relaxation time with dopant concentration, whereas short-range Dexter-type exchange does not, identifying FRET-mediated excitation transfer as the dominant mechanism. The MQMF-LME framework provides a modular route for linking microscopic interactions and environmental noise sources to measurable decoherence times in solid-state multi-qubit systems.
This paper investigates the coherence analysis of quantum systems utilizing the measurement of quantum coherence time. The core concept lies in directly quantifying the duration for which a quantum system maintains its superposition state, providing a fundamental measure of its coherence. We present a framework for analyzing coherence based on this time parameter, demonstrating its potential for characterizing and optimizing quantum systems. This approach offers a novel perspective on quantum system analysis, distinct from traditional methods relying solely on population or phase information. The methodology includes a detailed mathematical description of coherence and its relation to observable quantities, emphasizing the importance of accurate coherence time determination. Furthermore, we explore potential applications of this analysis, such as improving quantum control and enhancing the fidelity of quantum operations. The central equation defining coherence is expressed as: Ψ(t) = ∑i=1K ci * exp(-iωt/ħ) , where Ψ(t) is the time-dependent wave function, ci are the complex amplitudes, ω is the angular frequency, ħ is the reduced Planck constant, and K represents the number of coherent states. The coherence term, C(t), is then calculated as: C(t) = Re[<Ψ(t)|Ψ(t)>] = ∑i=1K |ci|2 cos(ωt/ħ). The analysis highlights the direct link between coherence time and the system's ability to perform quantum operations effectively. Finally, we discuss the challenges and future directions in utilizing quantum coherence time for advanced quantum system design and control.