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arXiv:1307.0479 · Physical Review A

Uncertainty principle in a cavity at finite temperature

Adolfo P. C. Malbouisson

Published Jul 17, 2013• 1 citation
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Abstract

We employ a dressed-state approach to perform a study on the behavior of the uncertainty principle for a system in a heated cavity. We find, in a small cavity for a given temperature, an oscillatory behavior of the momentum-coordinate product $(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}p)\phantom{\rule{0.16em}{0ex}}(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}q)$, which attains periodically finite absolute minimum (maximum) values, no matter how large the elapsed time is. This behavior is in sharp contrast to what happens in free space, where the product $(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}p)\phantom{\rule{0.16em}{0ex}}(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}q)$ tends asymptotically, for each temperature, to a constant value, independent of time.

DOI
10.1103/physreva.88.014101

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We employ a dressed-state approach to perform a study on the behavior of the uncertainty principle for a system in a heated cavity. We find, in a small cavity for a given temperature, an oscillatory behavior of the momentum-coordinate product (\ensuremathΔp)(\ensuremathΔq)(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}p)\phantom{\rule{0.16em}{0ex}}(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}q)(\ensuremathΔp)(\ensuremathΔq), which attains periodically finite absolute minimum (maximum) values, no matter how large the elapsed time is. This behavior is in sharp contrast to what happens in free space, where the product (\ensuremathΔp)(\ensuremathΔq)(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}p)\phantom{\rule{0.16em}{0ex}}(\ensuremath{\Delta}\phantom{\rule{0.16em}{0ex}}q)(\ensuremathΔp)(\ensuremathΔq) tends asymptotically, for each temperature, to a constant value, independent of time.
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.
This note is an Addendum to our previous article [Phys. Rev. A 81, 053820 (2010)]. We show that under the assumption of a Bose-Einstein distribution for the thermal reservoir, zero-temperature properties of the entangled states considered there are not changed by heating, for temperatures up to the order of room temperatures. In this case, the system is dissipative in free space and presents stability for a small cavity, both for T=0T=0T=0 and for finite temperature.
We construct non-equilibrium states by coupling a large but finite system with quasifree evolution to two temperature baths of free fermions or bosons with different temperature. We are interested in the resulting behaviour of the temperature baths as well as in the consequences for the finite system, especially what happens in the limit when the finite system tends to infinity. As a special example we consider the Kronig–Penney model and the tight binding model. Here a heat current remains that changes the baths but does not allow a limiting behaviour when the finite system tends to infinity. For the random versions of these models, especially the Anderson model, the heat current disappears exponentially with the size of the system. We show that a limit state is attained that is independent of the time direction and does not show any kind of symmetry breaking.
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.