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arXiv:hep-th/0308198

Time evolution of quantum systems in microcavities and in free space -- a non-perturbative approach

G. Flores-Hidalgo, Adolfo P. C. Malbouisson

Published Aug 28, 2003• 1 citation
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Abstract

We consider a system consisting of a particle in the harmonic approximation, having frequency $\barω$, coupled to a scalar field inside a spherical reflecting cavity of diameter $L$. By introducing {\it dressed} coordinates we define {\it dressed} states which allow a non-perturbative unified description of the radiation process, for weak and strong coupling regimes. We perform a study of the energy distribution in a small cavity, with the initial condition that the particle is in the first excited state. In the {\it weak} coupling regime, we conclude for the quasi-stability of the excited particle. For instance, for a frequency $\barω$ of the order $\barω\sim 4.00\times 10^{14}/s$ (in the visible red), starting from the initial condition that the particle is in the first excited level, we find that for a cavity with diameter $L\sim 1.0\times 10^{-6}m$, the probability that the particle be at any time still in the first excited level, will be of the order of 97%. For appropriate cavity dimensions, which are of the same order of those ensuring stability for weak coupling, we ensure for strong coupling the complete decay of the particle to the ground state in a small ellapsed time. Also we consider briefly the effects of a quartic interaction up to first order in the interaction parameter $λ$. We obtain for a large cavity an explicit $λ$-dependent expression for the particle radiation process. This formula is obtained in terms of the corresponding exact expression for the linear case and we conclude for the enhancement of the particle decay induced by the quartic interaction.

DOI
10.48550/arxiv.hep-th/0308198

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We consider a system consisting of a particle in the harmonic approximation, having frequency ωˉ\barωωˉ, coupled to a scalar field inside a spherical reflecting cavity of diameter LLL. By introducing {\it dressed} coordinates we define {\it dressed} states which allow a non-perturbative unified description of the radiation process, for weak and strong coupling regimes. We perform a study of the energy distribution in a small cavity, with the initial condition that the particle is in the first excited state. In the {\it weak} coupling regime, we conclude for the quasi-stability of the excited particle. For instance, for a frequency ωˉ\barωωˉ of the order ωˉ∼4.00×1014/s\barω\sim 4.00\times 10^{14}/sωˉ∼4.00×1014/s (in the visible red), starting from the initial condition that the particle is in the first excited level, we find that for a cavity with diameter L∼1.0×10−6mL\sim 1.0\times 10^{-6}mL∼1.0×10−6m, the probability that the particle be at any time still in the first excited level, will be of the order of 97%. For appropriate cavity dimensions, which are of the same order of those ensuring stability for weak coupling, we ensure for strong coupling the complete decay of the particle to the ground state in a small ellapsed time. Also we consider briefly the effects of a quartic interaction up to first order in the interaction parameter λλλ. We obtain for a large cavity an explicit λλλ-dependent expression for the particle radiation process. This formula is obtained in terms of the corresponding exact expression for the linear case and we conclude for the enhancement of the particle decay induced by the quartic interaction.
We consider a system consisting of an atom in the approximation of a harmonic oscillator of frequency $\overline{\ensuremath{\omega}},$ coupled to the scalar potential inside a spherical reflecting cavity of radius R. We use dressed states introduced in a previous publication [Andion, Malbouisson, and Mattos Neto, J. Phys. A 34, 3735 (2001)], which allow a nonperturbative unified description of the atom radiation process, in both cases, of a finite or an arbitrarily large cavity. We perform a study of the energy distribution in a small cavity, with the initial condition that the atom is in the first excited state and we conclude for the quasi-stability of the excited atom. For instance, for a frequency $\overline{\ensuremath{\omega}}$ of the order $\overline{\ensuremath{\omega}}\ensuremath{\sim}4.00\ifmmode\times\else\texttimes\fi{}{10}^{14}/\mathrm{s}$ (in the visible red), starting from the initial condition that the atom is in the first excited level, we find that for a cavity with diameter $2R\ensuremath{\sim}1.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6} \mathrm{m},$ the probability of the atom being at any time still in the first excited level, will be of the order of 97%. For a typical microwave frequency $\overline{\ensuremath{\omega}}\ensuremath{\sim}2.00\ifmmode\times\else\texttimes\fi{}{10}^{10}/\mathrm{s}$ we find stability in the first excited state also of the order of 97% for a cavity radius $R\ensuremath{\sim}1.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2} \mathrm{m}.$
We consider a system modeled by a harmonic oscillator of frequency , coupled to the scalar potential inside a reflecting sphere of radius R. We use dressed states introduced originally in [1] and recently employed in [2] to present a non-perturbative unified description of the decay process of the system, in free space and in the case of the system being confined in a finite cavity. In the situation that we start from the initial condition that the system is in the first excited state, we give exact formulas to describe its time evolution for a cavity of arbitrary size. In the particular case of a very large cavity (free space), we recover the behaviour expected from perturbation theory in the limit of the small coupling constant. In the case of a very small cavity, our results are in good agreement with experimental observations
An atom, coupled linearly to an environment, is considered in a harmonic approximation in thermal equilibrium inside a cavity. The environment is modeled by an infinite set of harmonic oscillators. We employ the notion of dressed states to investigate the time evolution of the atom initially in the first excited level. In a very large cavity (free space) for a long elapsed time, the atom decays and the value of its occupation number is the physically expected one at a given temperature. For a small cavity the excited atom never completely decays and the stability rate depends on temperature.
We consider a complete study of the influence of the cavity size on the spontaneous decay of an atom excited state, roughly approximated by a harmonic oscillator. We confine the oscillator-field system in a spherical cavity of radius RRR, perfectly reflective, and work in the formalism of dressed coordinates and states, which allows to perform non-perturbative calculations for the probability of the atom to decay spontaneously from the first excited state to the ground state. In free space, R→∞R\to\inftyR→∞, we obtain known exact results an for sufficiently small RRR we have developed a power expansion calculation in this parameter. Furthermore, for arbitrary cavity size radius, we developed numerical computations and showed complete agreement with the exact one for R→∞R\to\inftyR→∞ and the power expansion results for small cavities, in this way showing the robustness of our results. We have found that in general the spontaneous decay of an excited state of the atom increases with the cavity size radius and vice versa. For sufficiently small cavities the atom practically does not suffers spontaneous decay, whereas for large cavities the spontaneous decay approaches the free-space R→∞R\to\inftyR→∞ value. On the other hand, for some particular values of the cavity radius, in which the cavity is in resonance with the natural frequency of the atom, the spontaneous decay transition probability is increased compared to the free-space case. Finally, we showed how the probability spontaneous decay go from an oscillatory time behaviour, for finite cavity radius, to an almost exponential decay, for free space.
We study the time evolution of an atom suddenly coupled to a thermal radiation field. As a simplified model of the atom-electromagnetic field system we use a system composed by a harmonic oscillator linearly coupled to a scalar field in the framework of the recently introduced dressed coordinates and dressed states. We show that the time evolution of the thermal expectation values for the occupation number operators depend exclusively on the probabilities associated with the emission and absorption of field quanta. In particular, the time evolution of the number operator associated with the atom is given in terms of the probability of remaining in the first excited state and the decay probabilities from this state by emission of field quanta of frequencies ωkω_kωk​. Also, it is showed that independent of the initial state of the atom, it thermalizes with the thermal radiation field in a time scale of the order of the inverse coupling constant.
Using the nonperturbative method of dressed states introduced in a previous publication [N. P. Andion, A. P. C. Malbouisson, and A. Mattos Neto, J. Phys. A 34, 3735 (2001)], we study effects of the environment on a quantum-mechanical system, in the case in which the environment is modeled by an ensemble of noninteracting harmonic oscillators. This method makes it possible to separate the whole system into the dressed mechanical system and the dressed environment, in terms of which a nonperturbative approach is possible. When applied to the Brownian motion, we give explicit nonperturbative formulas for the classical path of the particle in the weak and strong coupling regimes. When applied to studying atomic behaviors in cavities, the method accounts for experimentally observed inhibition of atomic decay in small cavities.