The dynamics of a two-level system coupled to a particle bath with the Dirac dispersion relation is studied. We analytically show that closing the Dirac gap results in a transition of the survival probability of the two-level system from non-exponential to exponential decay in the long-time regime, while the short-time regime remains exponential. The exact time-evolving state is also calculated. With the Dirac gap closing smoothly, the time-evolving state converges to a time-evolving resonant state, which is normalizable due to causality. We numerically show that introducing a finite cutoff to the Dirac dispersion relation leads to a transition from exponential to non-exponential decay both in short- and long-time regimes, with the time-evolving resonant state resolved to a time-evolving state. Furthermore, we propose several experimental setups that act as a particle bath with the Dirac dispersion relation. We give a detailed calculation for one of them, namely an optical array in the Su-Schrieffer-Heeger configuration. In this case, we show that our theoretical results can be observed experimentally with realistic parameters in an existing experimental setup of an optical waveguide array.
The quantum dynamics of the dissipative two-level system with nonzero bias and sub-Ohmic bath is studied by means of the perturbation approach based on a unitary transformation. It has been shown that for the sub-Ohmic bath it is necessary to use the non-Markovian approach, especially for the short time behavior of the coupled system and environment. The nonequilibrium correlation
P(t) has been calculated to show that a finite bias may favor the short time coherence. The spectrum of the susceptibility ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega})$ of the sub-Ohmic case may have a double peak structure in the range of $\ensuremath{\omega}>0$ when the coupling $\ensuremath{\alpha}$ is relatively strong. Besides, the coherence-decoherence transition point ${\ensuremath{\alpha}}_{c}$ is determined for different $0<s\ensuremath{\le}1$ by the condition of ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega}=0)=\ensuremath{\infty}$ when $\ensuremath{\alpha}={\ensuremath{\alpha}}_{c}$. Finally, we show that Shiba's relation is exactly satisfied in our results.
An analytical solution for the time evolution of decay of two identical noninteracting quantum particles seated initially within a potential of finite range is derived using the formalism of resonant states. It is shown that the wave function, and hence also the survival and nonescape probabilities, for factorized symmetric and entangled symmetric or antisymmetric initial states evolve in a distinctive form along the exponentially decaying and nonexponential regimes. Our findings show the influence of the Pauli exclusion principle on decay. We exemplify our results by solving exactly the
s-wave $\ensuremath{\delta}$ shell potential model.
We study resonances and time decay in a quantum waveguide with two semitransparent barriers. The model is first analyzed as a one-dimensional Schrödinger operator with two delta interactions and then extended to a two-dimensional strip, where transverse Dirichlet modes reduce the problem, channel by channel, to the one-dimensional setting. We derive explicit scattering amplitudes and characterize resonance poles by analytic continuation. In the strong-barrier regime ρ=α−1→0+, the real parts of the resonances approach the eigenvalues of the decoupled cavity, while the imaginary parts are negative and of order α−2, so the lifetimes grow like α2. We also study the time evolution on short, intermediate, and long time scales: the survival probability is quadratic at short times, in accordance with the quantum Zeno effect; the intermediate regime is resonance-dominated and approximately exponential; and the long-time behavior is governed by the low-energy threshold and becomes algebraic. Numerical simulations complement the analysis by confirming these regimes and by showing the breakdown of exponential decay. They also provide an empirical fit for the breakdown time tbr in terms of the barrier strength α and the distance d1 between the barriers.
We analyze the dynamics of quantum correlations between two qubits coupled to a linear chain of oscillators. The chain mediates interactions between the qubits and acts as a non-Markovian reservoir. The model is amenable to an analytical solution when the initial state of the chain is Gaussian. We study the dynamics of the qubits' concurrence starting from a separable state and assuming that the chain spectrum is gapped and the chain is initially in a thermal state. We identify three relevant regimes that depend on the strength of the qubit-chain coupling in relation to the spectral gap. These are (i) the weak-coupling regime, where the qubits are entangled at the asymptotics, (ii) the strong-coupling regime, where the concurrence can exhibit collapses followed by revivals with exponentially attenuated amplitude, and (iii) the thermal damping regime, where the concurrence rapidly vanishes due to the chain's thermal excitations. In all cases, if entanglement is generated, this occurs after a finite time has elapsed. This time scale depends exponentially on the qubits' distance and is determined by the spectral properties of the chain. Entanglement irreversible decay, on the other hand, is due to the dissipative effect induced by the coupling with the chain and is controlled by the coupling strength between the chain and qubits. This study unravels the basic mechanisms leading to entanglement in a non-Markovian bath and allows one to identify the key resources for realizing quantum coherent dynamics of open systems.
The exponential decay of an unstable quantum state, as described by standard Markovian theories such as Fermi's Golden Rule, assumes a simple, structureless environment. However, in complex environments characterized by disorder, long-range interactions, or strong fluctuations, local decay rates fluctuate, leading to non-Markovian dynamics and power-law ``long-time tails.'' In this paper, we apply the recently proposed \textit{hyperstatistics} framework to the problem of quantum decay in such complex environments. By considering a
γ-distribution of local decay rates across mesoscopic domains, we derive a macroscopic survival probability governed by a
q-exponential function. We then use the
q-generalized Gamma function, defined via the Mellin transform of the
q-exponential, to calculate the moments of the decay-time distribution. We show that the mean quantum lifetime is
⟨t⟩=∫0∞P(t)dt=[⟨Γ⟩(2−q)]−1, and is therefore finite for
q<2. The convergence of the second moment, and hence of the lifetime variance, instead requires the stricter condition
q<3/2. These results provide a refined physical interpretation: for
1<q<3/2 both the mean lifetime and its variance are finite; for
3/2≤q<2 the mean lifetime remains finite but lifetime fluctuations become infinitely broad; and for
q≥2 the mean lifetime itself diverges, corresponding to a truly trapped, localized, or Griffiths-like regime.
The question, whether an open system dynamics is Markovian or non-Markovian can be answered by studying the direction of the information flow in the dynamics. In Markovian dynamics, information must always flow from the system to the environment. If the environment is interacting with only one of the subsystems of a bipartite system, the dynamics of the entanglement in the bipartite system can be used to identify the direction of information flow. Here we study the dynamics of a two-level system interacting with an environment, which is also a heat bath, and consists of a large number of two-level quantum systems. Our model can be seen as a close approximation to the 'spin bath' model at low temperatures. We analyze the Markovian nature of the dynamics, as we change the coupling between the system and the environment. We find the Kraus operators of the dynamics for certain classes of couplings. We show that any form of time-independent or time-polynomial coupling gives rise to non-Markovianity. Also, we witness non-Markovianity for certain parameter values of time-exponential coupling. Moreover, we study the transition from non-Markovian to Markovian dynamics as we change the value of coupling strength.