We find that photons from light emitting diode injected with Poissonian electrons have super-Poissonian statistics when the differential efficiency is larger than the mean emission efficiency of the device. This result shows the limitation of a conventional theory which describes the generation of sub-Poissonian light in term of Poisson point process.

With the use of semiconductor devices such as laser diode (LD) and light emitting diode (LED), it is not difficult to generate sub-Poissonian light or nonclassical light by injecting quiet electrons. Compared to the scheme using nonlinear optical processes, the scheme using semiconductor devices has the advantage of simple experimental configuration, small energy consumption, and good controllability by changing the structure of the device and the statistics of injection current. However, there are some unsolved problems. Firstly, we do not have an established quantum mechanical theory which describes the process of generating sub-Poissonian light and we can not predict the quantum state or density matrix of sub-Poissonian light. Secondly, in the most of the experiment, the measured quantity is restricted to the magnitude of the photon number fluctuations. Other properties such as intensity correlation function and phase fluctuations remain unexplored. Thirdly, it is not easy to generate weak sub-Poissonian light over broad bandwidth using conventional commercially-available LED. To the best of our knowledge, the largest squeezing realized with LED was 3.1dB, the weakest intensity was 7W, and the broadest measured bandwidth was 10MHz [1]. The aim of our work is to experimentally investigate these issues.

The Fano factor gives us a convenient measure of the magnitude of number fluctuations. It is defined by the variance normalized by that of Poisson distribution. The variance of Poisson distribution is equal to its mean value, therefore the Fano factor W is defined as W=<n²>/<n>. In a simple point-process picture, the Fano factor of the output photons W_PH is given by using that of input electrons W_EL as follows [2,3],

W_PH=1+W_EL. (1)

Where is the emission efficiency. In this picture, the difference between the differential and mean efficiency is not considered. According to eq.(1), the magnitude of noise suppression is equal to when W_EL=0, wheres W_PH is always unity when W_EL=1 for all values of .

Fig.1 Experimental Arrangement. By switching from the resistor Rs tothe photodiode, sub-Poissonian and Poissonian noise power can be measured.

In our experiment, the noise power for WEL 0 (sub-Poissonian mode operation) is normalized by that for WEL1 (Poissonian mode operation). This normalization method is used in the most of the previous experiments because the change of experimental apparatus can be minimized. The experimental setup is shown in Fig.1. In sub-Poissonian mode operation, a high-impedance resister is used to suppress current fluctuations. In Poissonian mode operation, a photodiode illuminated by white light (light from low efficiency LED or electric bulb) is used as a stochastic regulator. By just switching from the resistor Rs to the photodiode, sub-Poissonian and Poissonian noise power can be measured.

Figure 2 shows the normalized noise power for various injection current levels. The normalization was performed after subtracting the amlifier noise. The type of LED used in this experiment was Hitachi model HLP40RD, which is a GaAlAs LED with a single heterojunction structure. Starting from the bottom curve, the photo-current detected by PD2 was 500A, 50A, and 5A respectively. By extrapolating the trace, the nomalized noise power at zero frequency is estimated and the results are shown in Table 1. If eq.(1) is correct, the normalized value should be equal to

IPD	Experiment	1
5 A	0.90	0.93	0.89
50 A	0.84	0.89	0.85
500 A	0.81	0.85	0.80

Table 1. The values of normalized noise power and theoretical predictions

(1)/1=1. However, in all cases, experimental values were smaller than 1. This means that the description in terms of photonic stochastic point process is insufficient in contrast to the argument by Edwards et al.[4].

In the point process description where photons and electrons are treated as classical particles, it is not easy to intuitively consider the effect of differential efficiency. Thus we here describe the situation in term of small signal transfer as is shown in Fig.3. The horizontal axis represents the input current and the vertical axis represents the output. In our experiment, the input-output characteristic of LED is not linear. In this case, the mean efficiency that is expressed by the slope of a straight line drawn from the origin is smaller than the differential efficiency _d that is expressed by the slope of a tangential line. Then, for an average input current of n, the average output is n. When the input current fluctuates by n, the output will fluctuate by _dn. Therefore, the Fano factor of the output W_OUT may be given by W_out =(_dn)²/(n) =_d²/(n)²/n =_d²/ W_IN. Taking into account the contribution of random deletion, we obtain

W_OUT=1+_d²/ W_IN. (2)

Fig.4 Measured noise power with and without pinhole

For W_IN =1, W_OUT =1+_d²/. When _d>, the statistics of the output become super-Poissonian. This behavior is in a sharp contrast to the previous predictions, as in the simple point process description, output distribution always moves to the Poisson distribution. Equation (2) predicts that the normalized noise power is equal to (1)/(1+_d²/) and this agrees well with the experimental results as shown in Table 1. Equation (2) was first derived by Shimizu and Fujisaki. In Ref. [5], they generalized the quantum Langevin equation including a nonradiative decay term which is dependent on injection current levels.

In order to more directly check the validity of eq.(2), we artificially decreased the efficiency by inserting a pinhole between LED and PD2 and measured noise power. Results are shown in Fig.4. The injection currents were in Poissonian mode, and their levels were adjusted so that the photo-current detected by PD2 became the same. Without a pinhole, the mean and differential efficiencies were 16.1% and 19.5%, respectively. With a pinhole they were 4.3% and 4.7%. Using eq.(2), the output Fano factors are calculated to be 1.058 for the former and 1.005 for the latter. According to eq.(1) they should be the same. As is clearly seen in Fig.4, the noise power for two cases are different. The upper trace shows the ratio of two noise powers and its magnitude coincides well with the theoretical prediction of 1.05 (shown by arrow).

We experimentally investigated the effect of differential efficiency on the generation of sub-Poissonian light by LED. We have found that the simple point process description is insufficient when the mean emission efficiency differs from the differential efficiency. Experimental results agreed well with a new formula. This suggests a new possible scheme for generation of sub-Poissonian light in which differential efficiency is made to be smaller than mean efficiency.

[1] T. Hirano and T. Kuga, IEEE J. QE., 31, 2236 (1995); G. Shinozaki, T. Hirano, T. Kuga and M. Yamanishi, CLEO/Pacific Rim '95, WG3 (July 1995).

[2] M.C. Teich and B.E.A. Saleh, Opt. Lett. 7, 365 (1982).

[3] Akira Shimizu, Ouyou Butsuri (in Japanese), 62, p.881-888, (1993).

[4] P.J. Edwards and G.H. Pollard, Phys. Rev. Lett., 69, 1757 (1992).

[5] Hiroshi Fujisaki and Akira Shimizu, CLEO/Pacific Rim'95, PD1.7.