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Theory of non-Markovian decay of a cascade atom in high-Q cavities and photonic band gap materials

Garraway, B M, Bryan J. Dalton

Published Feb 13, 2006• 17 citations
OpenAlex

Abstract

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.

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Theory of non-Markovian decay of a cascade atom in high-Q cavities and photonic band gap materials

B M Garraway, Bryan J. Dalton

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.

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Non-Markovian decay of a three-level cascade atom in a structured reservoir

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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-$Q$ cavities and photonic band-gap systems, is presented. An application to a high-$Q$-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-$Q$-cavity case of identical Lorentzian reservoir structure functions, and differs from that for the single reservoir situation.

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Non-Markovian behaviour in atomic systems coupled to a structured reservoir of quantum EM field modes, such as in high Q cavities, is treated using a quasimode description, and the pseudo mode theory for single quantum reservoir excitations is obtained via Fano diagonalisation. The atomic transitions are coupled to a discrete set of (cavity) quasimodes, which are also coupled to a continuum set of (external) quasimodes with slowly varying coupling constants. Each pseudomode corresponds to a cavity quasimode, and the original reservoir structure is obtained in expressions for the equivalent atom-true mode coupling constants. Cases of multiple excitation of the reservoir are now treatable via Markovian master equations for the atom-discrete quasimode system.

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Bryan J. Dalton, B. M. Garraway

This paper deals with atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in photonic band gap materials. The case of high Q cavities has been treated elsewhere using Fano diagonalization based on a quasimode approach, showing that the cavity quasimodes are responsible for pseudomodes introduced to treat non-Markovian behaviour. The paper considers a simple model of a photonic band gap case, where the spatially dependent permittivity consists of a constant term plus a small spatially periodic term that leads to a narrow band gap in the spectrum of mode frequencies. Most treatments of photonic band gap materials are based on the true modes, obtained numerically by solving the Helmholtz equation for the actual spatially periodic permittivity. Here the field modes are first treated in terms of a simpler quasimode approach, in which the quasimodes are plane waves associated with the constant permittivity term. Couplings between the quasimodes occur owing to the small periodic term in the permittivity, with selection rules for the coupled modes being related to the reciprocal lattice vectors. This produces a field Hamiltonian in quasimode form. A matrix diagonalization method may be applied to relate true mode annihilation operators to those for quasimodes. The atomic transitions are coupled to all the quasimodes, and the true mode atom-EM field coupling constants (one-photon Rabi frequencies) are related to those for the quasimodes and also expressions are obtained for the true mode density. The results for the one-photon Rabi frequencies differ from those assumed in other work. Expressions for atomic decay rates are obtained using the Fermi Golden rule, although these are valid only well away from the band gaps.

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Nonperturbative decay of an atomic system in a cavity

B. M. Garraway

Atoms can nowadays be placed in increasingly exotic environments such as microscopic cavities and materials with photonic band gaps. High-Q cavities can now easily result in a strong coupling between an atom and its environment where perturbation theory should no longer be appropriate. The purpose of this paper is to describe the dynamics of a multilevel V-type atomic system (including the case of a two-level system) which interacts with a reservoir modeled by a generalized density of states. A theoretical construct, the pseudomode, is utilized to develop general methods for solution. Without using perturbation theory the equivalent master equation is developed and the relationship between the master equation, the pseudomodes, and the generalized density of states function is explored with examples. Utilizing a straightforward definition of the pseudomode, it is found that many functions for the density of states lead to problematic non-Lindblad master equations. Several examples are given, and it is shown how to convert the non-Lindblad master equations into a Lindblad form in these cases. The examples include a non-Lorentzian resonance and a simple model of a photonic band gap.

Physical Review A•Mar 1, 1997•518 citations
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.
This paper deals with non-Markovian behavior in atomic systems coupled to a structured reservoir of quantum electromagnetic field modes, with particular relevance to atoms interacting with the field in high-Q cavities or photonic band-gap materials. In cases such as the former, we show that the pseudomode theory for single-quantum reservoir excitations can be obtained by applying the Fano diagonalization method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two, and many discrete quasimodes are made. For a simple photonic band-gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes.
Non-Markovian behaviour in atomic systems coupled to a structured reservoir of quantum EM field modes, such as in high Q cavities, is treated using a quasimode description, and the pseudo mode theory for single quantum reservoir excitations is obtained via Fano diagonalisation. The atomic transitions are coupled to a discrete set of (cavity) quasimodes, which are also coupled to a continuum set of (external) quasimodes with slowly varying coupling constants. Each pseudomode corresponds to a cavity quasimode, and the original reservoir structure is obtained in expressions for the equivalent atom-true mode coupling constants. Cases of multiple excitation of the reservoir are now treatable via Markovian master equations for the atom-discrete quasimode system.
This paper deals with atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in photonic band gap materials. The case of high Q cavities has been treated elsewhere using Fano diagonalization based on a quasimode approach, showing that the cavity quasimodes are responsible for pseudomodes introduced to treat non-Markovian behaviour. The paper considers a simple model of a photonic band gap case, where the spatially dependent permittivity consists of a constant term plus a small spatially periodic term that leads to a narrow band gap in the spectrum of mode frequencies. Most treatments of photonic band gap materials are based on the true modes, obtained numerically by solving the Helmholtz equation for the actual spatially periodic permittivity. Here the field modes are first treated in terms of a simpler quasimode approach, in which the quasimodes are plane waves associated with the constant permittivity term. Couplings between the quasimodes occur owing to the small periodic term in the permittivity, with selection rules for the coupled modes being related to the reciprocal lattice vectors. This produces a field Hamiltonian in quasimode form. A matrix diagonalization method may be applied to relate true mode annihilation operators to those for quasimodes. The atomic transitions are coupled to all the quasimodes, and the true mode atom-EM field coupling constants (one-photon Rabi frequencies) are related to those for the quasimodes and also expressions are obtained for the true mode density. The results for the one-photon Rabi frequencies differ from those assumed in other work. Expressions for atomic decay rates are obtained using the Fermi Golden rule, although these are valid only well away from the band gaps.
Atoms can nowadays be placed in increasingly exotic environments such as microscopic cavities and materials with photonic band gaps. High-Q cavities can now easily result in a strong coupling between an atom and its environment where perturbation theory should no longer be appropriate. The purpose of this paper is to describe the dynamics of a multilevel V-type atomic system (including the case of a two-level system) which interacts with a reservoir modeled by a generalized density of states. A theoretical construct, the pseudomode, is utilized to develop general methods for solution. Without using perturbation theory the equivalent master equation is developed and the relationship between the master equation, the pseudomodes, and the generalized density of states function is explored with examples. Utilizing a straightforward definition of the pseudomode, it is found that many functions for the density of states lead to problematic non-Lindblad master equations. Several examples are given, and it is shown how to convert the non-Lindblad master equations into a Lindblad form in these cases. The examples include a non-Lorentzian resonance and a simple model of a photonic band gap.