Photon antibunching statistics

As pointed out in many papers photon antibunching is a purely quantum phenomenon (see e.g. Refs. (2, 15). The fluorescence of a single ion displays the additional nonclassical property that the variance of the photon number is smaller than its mean value (i.e. it is sub-Poissonian). This is because the single ion can emit only a single photon and has to be re-excited before it can emit the next one which leads to photon emissions at almost equal time intervals. The sub-Poissonian statistics of the fluorescence of a single ion has been measured in a previous experiment (14) (see also Ref. (16) for comparison). 

Of fundamental interest are measurements of the photon statistics in a three-level system, which can be performed by observing the statistics of quantum jumps. While the durations Af/ of the on-phases or the off-phases show an exponential distribution, the probability P m) of m quantum jumps per second exhibits a Poisson distribution (Fig. 9.48). In a two-level system the situation is different. Here, a second fluorescence photon can be emitted after a first emission only, when the upper state has been reexcited by absorption of a photon. The distribution P(AP) of the time intervals AT between successive emission of fluorescence photons shows a sub-Poisson distribution that tends to zero for AT 0 photon antibunching), because at least half of a Rabi period has to pass after the emission of a photon before a second photon can be emitted [1234]. 

Turning now to the nanosecond time regime (lower half of Fig. 11), the emitted photons from a single molecule can provide still more useful information. On the time scale of the excited state lifetime, the statistics of photon emission from a single quantum system are expected [84] to show photon antibunching, which means that the photons space themselves out in time , that is, the probability for two photons to arrive at the detector at the same time is small. This is a uniquely quantvun-mechanical effect, which was first observed for Na atoms in a low-density beam [85]. Antibunching is fundamentally measured by computing the second-order correlation of the electric field (r) (whieh is simply the normalized form of the intensity-... 

Fig. 10. Photon arrival time statistics of single emitters, (a) Schematic description of the temporal structure of single-emitter emission, (b) Simulated timetraces for different intersystem crossing rates as indicated, (c) Start-stop measurement yielding and anticorrelation, so called antibunching, at zero delay (the offset is due to different lengths of cables for both detectors), (d) Same measurement for pulsed excitation. Thick line Single emitter with missing peak at zero time delay. Thin line scattered laser light signal for comparison.

A considerable progress in this direction was achieved by investigating the fluorescence spectrum of ultra-cold atoms in an optical lattice in a heterodyne experiment (lessen et al. (3)). In these measurements a linewidth of 1 kHz was achieved, however, the quantum aspects of the resonance fluorescence such as antibunched photon statistics cannot be investigated under these conditions since they wash out when more than one atom is involved. 

In conclusion, we have presented the Hrst high-resolution heterodyne measurement of the elastic peak in resonance fluorescence of a single ion. At identical experimental parameters we have also measmed antibunching in the photon correlation of the scattered Held. Together, both measurements show that, in the limit of weak excitation, the fluorescence light differs from the excitation radiation in the second-order correlation but not in the first order correlation. However, the elastic component of resonance fluorescence combines an extremely narrow frequency spectrum with antibunched photon statistics, which means that the fluorescence radiation is not second-order coherent as expected from a classical point of view. This apparent contradiction can be explained easily by taking into accoimt the quantum nature of light, since first-order coherence does not imply second-order coherence for quantized fields (19). The heterodyne and the photon correlation measurement are complementary since they emphasize either the classical wave properties or the quantum properties of resonance fluorescence, respectively. 

Another application is the deflection of atoms by photon recoil. For sufficiently good beam collimation, the deflection from single photons can be detected. The distribution of the transverse-velocity components contains information about the statistics of photon absorption [1207]. Such experiments have successfully demonstrated the antibunching characteristics of photon absorption [1208]. The photon statistic is directly manifest in the momentum distribution of the deflected atoms [1209]. Optical collimation by radial recoil can considerably decrease the divergence of atomic beams and thus the beam intensity. This allows experiments in crossed beams that could not be performed before because of a lack of intensity. 

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