Extreme Sub-radiance: Can Quantum Effects Generate Dramatically Longer Atomic Lifetimes?

Abstract

The prolongation of lifetimes for an excited atom due to the presence of nearby atoms in the ground state is shown to follow simply from unitarity of the time evolution. We also discuss possible approaches to the detection and the overcoming of various technical obstacles.

Appendices

Appendix A: Avoiding the Paradox by Separating the Atoms

It is interesting to consider the following modification of the ideal two atom setup.⁴ Let us introduce a metal disc of radius $R > {\lambda }\hbox to0pt{\raise4.6pt\hbox{\kern-4.5pt\vrule width3pt height.5pt depth0pt}}> a$, that is perpendicular to, centered on, and bisects the vector ${\vec{\mathbf{r}}_{12}}$ connecting the two atoms. Photons of wavelength ${\lambda }\hbox to0pt{\raise4.6pt\hbox{\kern-4.5pt\vrule width3pt height.5pt depth0pt}}$ incident on either side of the disc will be reflected and diffraction around the disc is negligible for $R\gg {\lambda }\hbox to0pt{\raise4.6pt\hbox{\kern-4.5pt\vrule width3pt height.5pt depth0pt}}$. Exciton trapping by hopping between the two atoms, and the attendant Lifetime prolongation will then be avoided.

The same conclusion also follows from our original approach. In the presence of the metal disc each atom would not emit a spherical wave. Rather, each atom would emit “half a spherical wave,” namely, a collection of plane waves along rays pointing in the z>0 direction for atom 1 say, and in the z<0 direction for atom 2. The final states obtained by de-exciting atom 1 or 2 are therefore orthogonal and the paradox and ensuing conclusions are therefore avoided. The metal sheet should be thinner than a and hence much thinner than ${\lambda }\hbox to0pt{\raise4.6pt\hbox{\kern-4.5pt\vrule width3pt height.5pt depth0pt}}$. Yet such thin sheets may also perfectly reflect light of wavelength ${\lambda }\hbox to0pt{\raise4.6pt\hbox{\kern-4.5pt\vrule width3pt height.5pt depth0pt}}$. Hence the modified set-up with the metal disc is realistic and could be achieved experimentally. We could replace the uniform conducting sheet by a series of parallel wires. In this case light polarized along the direction of the wires would be reflected more strongly and the avoidance of the increased lifetime will occur mainly for such polarization.⁵

Appendix B: A Pedagogical Discussion of the Absence of Doppler Broadening in a Cavity

This argument presented above notwithstanding, one might wonder why we cannot view the atom as a wave packet moving in the cavity. In this case we might naively expect a strong Doppler broadening of the emitted photon line with alternating shifts to the red and blue due to an atom receding from and approaching the observer. This, however, is not the case. The amplitude for emitting a photon with energy hω is given by Fourier transforming the time dependent atomic current, which couples to the electromagnetic field. If the latter were purely harmonic (with frequency ω ₀), then if we assume an exponential decay of the ground state, we obtain the standard Lorentzian line shape, {1/[ω−ω ₀]²+(Γ/2)²}, with Γ=1/τ, the natural width.

The varying Doppler shift due to the slow center of mass motion modulates the frequency of the harmonic atomic and multiplies it by 1+βSW(t) where β=v _atom/c and SW(t) is a square wave time profile:

$$ \mathit{SW}(t)= \begin{cases} 1 & \mbox{for }2n T < t < (2n+1)T, \\ -1 & \mbox{for }(2n+1) T < t < (2n+2)T. \end{cases} $$

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For simplicity, we assume one-dimensional motion with a positive Doppler shift during the time interval T=2R/v, alternating with a negative Doppler shift with a time interval T when the atom’s velocity has reversed direction. The phase accumulated at time t is

$$ \varPhi(t)= \omega(0)t + \beta\omega(0)\!\int_ 0 ^t dt' \mathit{SW}(t'). $$

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The contribution of an even number of T intervals to the last integral over the square wave profile vanishes yielding:

$$ \int_0 ^t dt' \mathit{SW}(t')= \sin\bigl(t(\mathrm{mod}\ {T})\bigr). $$

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We then need to evaluate the Fourier transform:

$$ \int dt \exp\bigl[-i\omega-\omega(0)t\bigr] \exp\bigl[i\beta\omega(0)\bigl( t(\mathrm{mod}\ {T})\bigr)\bigr] \exp \biggl[\biggl(-\frac{\gamma}{2}\biggr)t\biggr], $$

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where the last factor indicates the exponential damping of the oscillations. The argument of the second exponent,

$$ \beta\omega(0)t (\mathrm{mod}\ {T}) < \beta\omega(0) T= (v/c)\omega(0) R/v= R/{\lambda }\hbox to0pt{\raise4.6pt\hbox{\kern-4.5pt\vrule width3pt height.5pt depth0pt}}, $$

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is then (by our basic assumption that $R<{\lambda }\hbox to0pt{\raise4.6pt\hbox{\kern-4.5pt\vrule width3pt height.5pt depth0pt}}$) smaller than one. This avoids the Doppler shift and broadening, and the ordinary Lorentzian shape is reproduced.