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arXiv:2111.09730 · Physical Review A

Non-Markovian dynamics of a single excitation within many-body dissipative systems

Adam Burgess, Marian Florescu A

Published Jun 7, 2022• 13 citations
arXivHTMLPDF

Abstract

We explore the dynamics of $N$ coupled atomic two-level systems embedded within a generic bosonic reservoir under specific system symmetries. In the regime of many atoms identically coupled to a single reservoir, we identify remarkable effects, notably that the initial configuration of the atomic excited-state amplitudes strongly impacts the dynamics of the system and can even fully sever the system from its environment. Additionally, we find that steady-state amplitudes of the excited states become independent of the specific structure of the bosonic reservoirs considered. The framework introduced is deployed to model a structured photonic reservoir associated with a photonic crystal, where it recaptures previous theoretical and experimental results for the superradiance rates even within the single-excitation regime. For the photonic band-gap system, our formalism predicts the generation of pairwise entanglement between initially uncorrelated atomic systems. Furthermore, it suggests that---with respect to a non-Markovian metric---the non-Markovianity of the aggregated many-atom system is modulated by the total number of atoms. This is due to a stark interplay between the Lamb shifting of the atomic transition energies due to their varying number and the increased number of atomic systems with resonant transition energies.

DOI
10.1103/physreva.105.062207

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We explore the dynamics of NNN coupled atomic two-level systems embedded within a generic bosonic reservoir under specific system symmetries. In the regime of many atoms identically coupled to a single reservoir, we identify remarkable effects, notably that the initial configuration of the atomic excited-state amplitudes strongly impacts the dynamics of the system and can even fully sever the system from its environment. Additionally, we find that steady-state amplitudes of the excited states become independent of the specific structure of the bosonic reservoirs considered. The framework introduced is deployed to model a structured photonic reservoir associated with a photonic crystal, where it recaptures previous theoretical and experimental results for the superradiance rates even within the single-excitation regime. For the photonic band-gap system, our formalism predicts the generation of pairwise entanglement between initially uncorrelated atomic systems. Furthermore, it suggests that---with respect to a non-Markovian metric---the non-Markovianity of the aggregated many-atom system is modulated by the total number of atoms. This is due to a stark interplay between the Lamb shifting of the atomic transition energies due to their varying number and the increased number of atomic systems with resonant transition energies.
This article explores the dynamics of many-body atomic systems symmetrically coupled to Lorentzian photonic cavity systems. Our study reveals interesting dynamical characteristics, including non-zero steady states, super-radiant decay, enhanced energy transfer, and the ability to modulate oscillations in the atomic system by tuning environmental degrees of freedom. We also analyze a configuration consisting of a three-atom chain embedded in a photonic cavity. Similarly, we find a strong enhancement of the energy transfer rate between the two ends of the chain and identified specific initial conditions that lead to significantly reduced dissipation between the two atoms at the end of the chain. Another configuration of interest consists of two symmetrical detuned reservoirs with respect to the atomic system. In the single atom case, we show that it is possible to enhance the decay rate of the system by modulating its reservoir detuning. In contrast, in the many-atom case, this results in dynamics akin to the on-resonant cavity. Finally, we examine the validity of the rotating wave approximation through a direct comparison against the numerically exact hierarchical equations of motion. We find good agreement in the weak coupling regime, while in the intermediate coupling regime, we identify qualitative similarities, but the rotating wave approximation becomes less reliable. In the moderate coupling regime, we find deviations of the steady states due to the formation of mixed photon-atom states.
In this work, taking NNN noninteracting two-level atoms coupled to an anisotropic three-dimensional photonic crystal into account, we study exact non-Markovian dynamics of the atoms with the photonic crystal as an environment. We find that there exists a threshold number NCR{N}_{\mathrm{CR}}NCR​ for the identical atoms, beyond which the system environment has a bound state regardless of how weak the system-environment coupling is. This means all identical atoms do not dissipate to the ground state. Further, when the whole system has no bound state, we find that the amplitude of collective excitation approaches a limit value 2/32/32/3 as the number of atoms approaches to infinity. For nonidentical atoms with different transition frequencies, there exist NNN bound states at most that lead to NNN-wave mixing in the system dynamics. The present prediction is possible to observe in the recent circuit QED experiment, and the results suggest that coherence in a many-body system can be maintained in bound states by manipulating the environment.
Entanglement evolution for a classically pumped three-level atom interacting with an anisotropic structured reservoir is studied. We demonstrate that entanglement decay is much slower in the reservoir with the memory effects. This point is beneficial for the implementation of quantum computation in the non-Markovian environments. Unlike the free-space case, the unusual reservoir stimulates stationary entanglement with amplitude depends strongly on the relative phase between the control laser coupling the two upper levels and the pump laser pulse used to create an excited state of the atom in a coherent superposition of the two upper levels. An optimal stationary degree of entanglement is achieved for negative relative phase, where both detunings - from the upper band gap and driving laser - are the same. Moreover, a significant stationary degre e of entanglement is reached, for high detuned upper levels with the pump laser pulse, on resonant frequency near the edge of the PBG. For entanglement in stationary regime, the Rabi frequency of the control laser field exceeds the rate of entanglement degradation or e nhancement for tuned or detuned frequency from the band gap, depending on the relative phase value.
The dynamics of a three-level atom in a cascade configuration with both transitions coupled to a single structured reservoir of quantized field modes is treated using Laplace transform methods applied to the coupled amplitude equations. Results are also obtained from master equations by two different approaches, that is, involving either pseudomodes or quasimodes. Two different types of reservoir are considered, namely a high-Q cavity and a photonic band-gap system, in which the respective reservoir structure functions involve Lorentzians. Non-resonant transitions are included in the model. In all cases non-Markovian behaviour for the atomic system can be found, such as oscillatory decay for the high-Q cavity case and population trapping for the photonic band-gap case. In the master equation approaches, the atomic system is augmented by a small number of pseudomodes or quasimodes, which in the quasimode approach themselves undergo Markovian relaxation into a flat reservoir of continuum quasimodes. Results from these methods are found to be identical to those from the Laplace transform method including two-photon excitation of the reservoir with both emitting sequences. This shows that complicated non-Markovian decays of an atomic system into structured EM field reservoirs can be described by Markovian models for the atomic system coupled to a small number of pseudomodes or quasimodes.
The dynamics of a three-level atom in a cascade configuration with both transitions coupled to a single structured reservoir of quantized field modes is treated using Laplace transform methods applied to the coupled amplitude equations. Results are also obtained from master equations by two different approaches, that is, involving either pseudomodes or quasimodes. Two different types of reservoir are considered, namely a high-Q cavity and a photonic band-gap system, in which the respective reservoir structure functions involve Lorentzians. Non-resonant transitions are included in the model. In all cases non-Markovian behaviour for the atomic system can be found, such as oscillatory decay for the high-Q cavity case and population trapping for the photonic band-gap case. In the master equation approaches, the atomic system is augmented by a small number of pseudomodes or quasimodes, which in the quasimode approach themselves undergo Markovian relaxation into a flat reservoir of continuum quasimodes. Results from these methods are found to be identical to those from the Laplace transform method including two-photon excitation of the reservoir with both emitting sequences. This shows that complicated non-Markovian decays of an atomic system into structured EM field reservoirs can be described by Markovian models for the atomic system coupled to a small number of pseudomodes or quasimodes.
The dynamics of a three-level atom in a cascade (or ladder) configuration with both transitions coupled to a single structured reservoir of quantized electromagnetic field modes is treated using Laplace transform methods applied to the coupled amplitude equations. In this system two-photon excitation of the reservoir occurs, and both sequences for emitting the two photons are allowed and included in the theory. An integral equation is found to govern the complex amplitudes of interest. It is shown that the dynamics of the atomic system is completely determined in terms of reservoir structure functions, which are products of the mode density with the coupling constant squared. This dependence on reservoir structure functions rather than on the mode density or coupling constants alone, shows that it may be possible to extend pseudomode theory to treat multiphoton excitation of a structured reservoir---pseudomodes being introduced in one-one correspondence with the poles of reservoir structure functions in the complex frequency plane. A general numerical method for solving the integral equations based on discretizing frequency space, and applicable to different structured reservoirs such as high-QQQ cavities and photonic band-gap systems, is presented. An application to a high-QQQ-cavity case with identical Lorentzian reservoir structure functions is made, and the non-Markovian decay of the excited state shown. A formal solution to the integral equations in terms of right and left eigenfunctions of a non-Hermitian kernel is also given. The dynamics of the cascade atom, with the two transitions coupled to two separate structured reservoirs of quantized electromagnetic field modes, is treated similarly to the single structured reservoir situation. Again the dynamics only depends on reservoir structure functions. As only one sequence of emitting the two photons now occurs, the integral equation for the amplitudes can be solved analytically. The non-Markovian decay of the excited state is shown for the same high-QQQ-cavity case of identical Lorentzian reservoir structure functions, and differs from that for the single reservoir situation.