than unity. The measured intracavity photon
numbers are consistently larger than the photon number made only by the cavity-enhanced
spontaneous emission (dashed line). When the
pump pulse area is 0.5p, corresponding to
jyatomi ≈ ½jei þ expðifÞjgi;=
, the observed
log-log slope is 1.66 ± 0.01. After subtracting
the contribution by the noncollective emission
corresponding to the dashed line, the recalculated
log-log slope becomes 1.94 ± 0.04 (inset of Fig.
3A), which indicates that the observed emission
is dominantly superradiance-proportional to the
square of the number of atoms. A near-quadratic
growth appears even in the negative inversion
case of Q = 0.3p, in which only 21% of atoms are
in the excited state with the rest in the ground
state. When Q > 0.5p, the atoms have positive
population inversion, and thus the photon number grows further by stimulated emission beyond
the level reached by the collective emission. In
this case, it is impossible to isolate the collective
emission effect clearly in the log-log plot.
It is also notable that the log-log slope is
almost invariant for a large range of hNci for
Q ≤ 0.5p. The theory predicts that the quadratic
dependence on hNci would be dominant in
the region of ð1 þ cosQÞ;1 < hNci < ðgtÞ;2 for
the perfectly phase-aligned atoms, although the
practical phase noise would make the domain
somewhat reduced. Such a broad-range quadratic
growth, occurring independently of hni values,
including hni ≪ 1 as in (23), is a distinctive feature
of the present superradiance compared with the
marked slope change occurring near the thresh-
old condition of hni ≈ 1 in the ordinary lasing
case. The absence of the usual lasing threshold
or thresholdless lasing in the present super-
radiance cannot be explained in terms of the
so-called b factor in ordinary lasers based on
noncollective emission (24). In our experiment,
b = (gt)2 ≈ 0.034 in the nanohole-array–aperture
case (Figs. 2A and 3A) and 0.011 in the rectangular-
aperture case (Figs. 2B and 3B) (17). The latter is
consistent with the large mean photon number
change occurring at the threshold in Fig. 3B
(Q > p/2). Additionally, the range of super-
radiance or the maximum number of atoms par-
ticipating in the collective emission can be easily
scaled up by choosing smaller gt values (fig. S5).
This feature may provide a new approach in
building thresholdless lasers.
The present single-atom superradiance can
be viewed as a consequence of one-sided inter-
action among a series of atoms separated by tens
of meters. The photon emitted by a preceding
atom interacts with the next atom after traveling
ct=hNi (about 30 m for hNi ¼ 1; c, speed of light
in a vacuum) when we unfold mirror reflections,
although their average distance in real space is
only hundreds of micrometers. Due to causality,
only the preceding atoms can then affect the
quantum states of the following atoms. This
interaction induces the emission rate of the atom
in the cavity to be twice the emission rate per
atom in the usual superradiance (17) (fig. S6).
The time-separated atoms linked by such one-
sided interaction can form atom-atom interac-
tion systems, which can serve as a testbed for
various quantum many-body physics (25).
The present study deepens our understanding
on matter-light collective interaction and provides
insight into field-mediated long-range (26, 27)
interactions. In addition, the phase-controlled
many-atom–field interaction based on the nanohole-
array technique can be used in nonclassical field
generation such as optical Schrödinger cat states
and highly-squeezed vacuum states (28), even in
a lossy cavity, contrary to previous studies in the
microwave region (29), as well as in realizing
superabsorption (30). The greatly enhanced single-
atom emission may be useful in constructing
efficient quantum interfaces (31).
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Fig. 3. Intracavity photon number dependence on the number of atoms. (A and B) Intracavity mean photon number hni versus excited-state mean
atom number hNiree for (A) the phase-aligned case and (B) the random-phase case. We use hNiree instead of just hNi for the horizontal axis to align the
noncollective emission contribution as a common baseline. The inset in (A) shows hni with the noncollective contribution subtracted for Q = 0.5p cases and a
linear fit to data with a log-log slope of 1.94 ± 0.04. Dashed lines correspond to the expected photon number made only by the cavity-enhanced
spontaneous emission of atoms. Solid lines denote theoretical predictions with the actual experimental parameters (17). Error bars indicate SD.