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Physical Review Letters

Photon Statistics of a Laser with Slow Inversion

Yngve Lien, S. M. de Vries, N. J. van Druten, Martin P. van Exter, J. P. Woerdman

Published Mar 26, 2001• 21 citations
Publisher

Abstract

We have measured the photon number probability distribution of a laser in which the inversion is not slaved to the field. For the experiments, we have used a Nd(3+):YVO(4) laser which has a sufficiently slow inversion to allow measurement of the photon fluctuations at a time scale much shorter than that of the relaxation oscillations. The photon distribution function becomes highly nonstandard (i.e., non-Poissonian) in such a laser; this is consistent with available theoretical work. We point out the relevance of our results for the case of the semiconductor microlaser.

DOI
10.1103/physrevlett.86.2786

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We have measured the photon number probability distribution of a laser in which the inversion is not slaved to the field. For the experiments, we have used a Nd(3+):YVO(4) laser which has a sufficiently slow inversion to allow measurement of the photon fluctuations at a time scale much shorter than that of the relaxation oscillations. The photon distribution function becomes highly nonstandard (i.e., non-Poissonian) in such a laser; this is consistent with available theoretical work. We point out the relevance of our results for the case of the semiconductor microlaser.
A laser with a Poissonian random pump, operated far above threshold, has Poissonian statistics both for the intracavity photon number and for extracavity photocounts. We analyze a pump-noise-suppressed laser using the positive P representation. When the pump noise is suppressed below a critical level, the intracavity photon-number distribution becomes sub-Poissonian.1,2 When the spontaneous decay of the upper lasing level is also eliminated, intracavity photon-number fluctuations can be diminished by a factor up to half. For counting times much shorter than the cavity decay time, the extracavity photocount statistics are essentially Poissonian, since cavity decay dominates. For counting times comparable with or larger than the cavity lifetime, the extracavity photocount statistics become sub-Poissonian. In the limit that intracavity photon-number fluctuations are diminished by one half, the extracavity photocount noise can be completely suppressed for counting times much longer than the cavity lifetime. The cavity mode occupies an incoherent superposition of two states 180° out of phase, both of which diffuse in phase. In the microlaser regime, quantum diffusion between these two states can result in 180° phase jumps, which are in addition to ordinary phase diffusion.
We study the field correlation functions and the natural laser spectrum for a slow-inversion laser with the thresholdless intensity fluctuations. In such a laser, the spontaneous-emission-driven relaxation oscillations are responsible for some unique features: The field correlation function oscillates with the frequency of relaxation oscillations ${\ensuremath{\omega}}_{r}$ and the laser line changes its profile. We show that appreciably above threshold the laser spectrum consists of three lines. In addition to the main line at the frequency of the lasing mode ${\ensuremath{\omega}}_{c}$ appear two lines at frequencies ${\ensuremath{\omega}}_{c}+{\ensuremath{\omega}}_{r}$ and ${\ensuremath{\omega}}_{c}\ensuremath{-}{\ensuremath{\omega}}_{r}$. The spontaneous emission noise in such lasers gives rise to nonlinear stochastic effects: Quantum noise affects the oscillation frequency ${\ensuremath{\omega}}_{r}$ and the decay rates of the field correlation function.
If a beam of light is coherent, then the photons constituting the beam arrive at random moments in time. Consequently, the photon counting statistics in a given time interval below the coherence time, should in theory follow a Poissonian distribution. When the measurement interval exceeds the coherence time, the distribution should broaden and become a super-Poissonian distribution. The aim of this study is to experimentally demonstrate the Poissonian nature of light, as well as quantifying the transition to a super-Poissonian distribution. In the experiment, a low-intensity laser beam was measured with single-photon detectors, able to capture individual photons. The arrival time of the photons was detected with an oscilloscope, and the data further analysed with our own Python code. The probability distribution of the arrival time of photons was constructed and fitted with a Poissonian distribution for different time intervals. The results showed a clear Poissonian distribution for all intervals from 0.2 μs up to 2000 μs. We did however, observe tendencies of randomness for the longest measured intervals. A Poissonian distribution is defined by the variance being equal to the mean, whilst for a super-Poissonian distribution the variance is higher. The biggest measured ratio between the variance and the mean in our experiment was 1.06 for interval time 2000 μs, which is too small to infer super-Poissonian. Due to limited data memory, our dataset was restricted in length and resolution. Therefore, longer intervals could not be studied without statistical uncertainty. To cross-check the result we also calculated the probability distribution for the arrival time itself. The result perfectly fitted an exponential distribution which was in accordance with the theory of coherent light. Hence, within the given measurement accuracy and statistical uncertainty, it is safe to conclude that the laser beam is coherent for all tested time intervals. However, this does not allow us to conclude that it is a single-mode laser.
A recent analysis of the influence of the pump statistics on the laser output fluctuations is applied to the experimentally feasible case of a laser being pumped by light coming from a separate pump field cavity that is driven by a broad-band squeezed vacuum plus a coherent driving field. In contrast to the 'squeezed-reservoir laser', where a reservoir of squeezed modes coming from all directions coupled to the dipole moment of the lasing transition and leads to atomic phase decay inhibition and phase locking of the laser field, here the squeezed light merely pumps the gain atoms, that is gives rise to inversion of the laser transition. In analogy to photodetection theory the photon statistics of the laser output is calculated as a function of the photon statistics of the absorbed squeezed pump light. For the case of a squeezed pump field chosen to display sub-Poissonian statistics, it is demonstrated that the Mandel Q-parameter of the laser cavity output can theoretically reach the optimum value of Q=- 1 / 2 , similar to the case of a deterministic pumping mechanism.
We have measured the second-order correlation function of the cavity-QED microlaser output and observed a transition from photon bunching to antibunching with increasing average number of intracavity atoms. The observed correlation times and the transition from super- to sub-Poisson photon statistics can be well described by gain-loss feedback or enhanced-reduced restoring action against fluctuations in photon number in the context of a quantum microlaser theory and a photon rate equation picture. However, the theory predicts a degree of antibunching several times larger than that observed, which may indicate the inadequacy of its treatment of atomic velocity distributions.