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arXiv:quant-ph/0510058 · Physical Review A

Bound-state eigenenergy outside and inside the continuum for unstable multilevel systems

Manabu Miyamoto

Published Dec 6, 2005• 49 citations
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Abstract

The eigenvalue problem for the dressed bound state of unstable multilevel systems is examined both outside and inside the continuum, based on the $N$-level Friedrichs model, which describes the couplings between the discrete levels and the continuous spectrum. It is shown that a bound-state eigenenergy always exists below each of the discrete levels that lie outside the continuum. Furthermore, by strengthening the couplings gradually, the eigenenergy corresponding to each of the discrete levels inside the continuum finally emerges. On the other hand, the absence of the eigenenergy inside the continuum is proved in weak but finite coupling regimes, provided that each of the form factors that determine the transition between some definite level and the continuum does not vanish at that energy level. An application to the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field is demonstrated.

DOI
10.1103/physreva.72.063405

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The eigenvalue problem for the dressed bound state of unstable multilevel systems is examined both outside and inside the continuum, based on the NNN-level Friedrichs model, which describes the couplings between the discrete levels and the continuous spectrum. It is shown that a bound-state eigenenergy always exists below each of the discrete levels that lie outside the continuum. Furthermore, by strengthening the couplings gradually, the eigenenergy corresponding to each of the discrete levels inside the continuum finally emerges. On the other hand, the absence of the eigenenergy inside the continuum is proved in weak but finite coupling regimes, provided that each of the form factors that determine the transition between some definite level and the continuum does not vanish at that energy level. An application to the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field is demonstrated.
In quantum field theory it is believed that the spontaneous decay of excited atomic or molecular level is due to the interaction with continuum of field modes. Besides, the atom makes a transition from upper level to lower one so that the probability to find the atom in the excited state tends to zero. In this paper it will be shown that the mathematical model in single-photon approximation may predict another behavior of this probability generally. Namely, the probability to find the atom in the excited state may tend to a nonzero constant so that the atom is not in the pure state finally. This effect is due to that the spectrum of the complete Hamiltonian is not purely absolutely continuous and has a discrete level outside the continuous part. Namely, we state that in the corresponding invariant subspace, determining the time evolution, the spectrum of the complete Hamiltonian when the field is considered in three dimensions may be not purely absolutely continuous and may have an eigenvalue. The appearance of eigenvalue has a threshold character. If the field is considered in two dimensions the spectrum always has an eigenvalue and the decay is absent.
Bound states and decay dynamics in NNN-level Friedrichs model with factorizable interactions
Considering an NNN-level system interacting factorizably with a continuous spectrum, we derive expressions for the bound states and the dynamical evolution within this single-excitation Friedrichs model by using the projection operator formalism. First, we establish explicit criteria to determine the number of bound states, whose existence suppresses the complete spontaneous decay of the system. Second, we derive the open system's decay dynamics, which is naturally described by an energy-independent non-Hermitian Hamiltonian in the Markovian limit. As an example, we apply our framework to a two-level atomic chain side-coupled to a photonic lattice, uncovering a rich variety of decay dynamics and realizing an anti-PT\mathcal{PT}PT-symmetric Hamiltonian in the system's evolution.
The long-time behavior of the survival probability for unstable multilevel systems that follows the power-decay law is studied based on the NNN-level Friedrichs model, and is shown to depend on the initial population in unstable states. A special initial state maximizing the asymptote of the survival probability at long times is found and examined by considering the spontaneous-emission process for the hydrogen atom interacting with the electromagnetic field.
The decaying behavior of both the survival S(t) and total P(t) probabilities for unstable multilevel systems at long times is investigated by using the N-level Friedrichs model. The long-time asymptotic-forms of both S(t) and P(t) are obtained for an arbitrary initial-state extending over the unstable levels. It is then clarified how the asymptotic forms depend on the initial population in unstable levels. In particular, a special initial state that maximizes the asymptotic form of both S(t) and P(t) is found. On the other hand, the initial states eliminating the first term of their asymptotic expnasions also exist, which implies that a faster decay rather than expected can be realized. This faster decay for S(t) is numerically confirmed by considering the spontaneous emission process for the hydrogen atom interacting with the electromagnetic field. It is demonstrated that the t-4-decay and a faster decay are realized depending on the initial states, where the latter is estimated as t-8.
Numerical and theoretical analysis of electron survival probability is presented for the system where a periodically driven discrete state is coupled with continuum states of the tight-binding model. The stability of the discrete state is examined by analyzing the survival probability in the weak-coupling case. Stable nondecaying states as well as usual unstable decaying states appear in the systems where the oscillating energy of the discrete state and the energy band of the continuum states overlap. Dynamical stable states caused by the energy oscillation are qualitatively explained by an approximate solution of a periodically driven two-level model. The decay is suppressed as a result of vanishing reduced coupling constant at the resonance.