Throwing Dice Really Fast

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Dec 2, 2009 - Throwing Dice. Really Fast. Igor Reidler ... For the RBG, we used a sampling rate of 2.5 GHz, as shown by the red dots. At each sampling point, ...
STATISTICAL OPTICS

Throwing Dice Really Fast Igor Reidler, Yaara Aviad, Michael Rosenbluh and Ido Kanter

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he rapid generation of random numbers or bits (random bit generation, or RBG) is of paramount importance in cryptography, computational physics and many other applications. To achieve true randomness, one must draw numbers from a source using a stochastic physical process, rather than a deterministic algorithm run on a computer. This generation is therefore usually based on random noise sources or statistically random events, such as radioactive decay or photon arrival times. The limitation of such sources has been the speed at which the random numbers can be generated. Moreover, the sources and amplifier bandwidths that are generally available have limited the generation speeds to the range of 100 Mbit/s. We have now demonstrated an RBG with a speed of 12.5 Gbit/s based on sampling the chaotic intensity fluctuations of a semiconductor laser with timedelayed optical feedback.1 Chaotic diode lasers have been shown to emit spikes of roughly 100-ps duration, obeying Poissonian statistics,2 for time durations shorter than the feedback delay time. For times longer than the feedback delay time, however, the pattern of chaotic fluctuations repeats itself with high fidelity. The demonstration of a useful RBG must eliminate this repetition at the delay time. We accomplish this in our experiments by using an incommensurate ratio of delay time and digitization rate as well as the derivative of the chaotic signal; we also truncate the digitized signal value to the five least significant bits out of the eight bits available. An additional requirement for any RBG is that, within statistical limits, there must be an equal number of ones and zeroes in the generated sequence, independent of sequence length. Thus, the distribution of random numbers corresponding to the sampled laser

34 | OPN Optics & Photonics News

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A schematic (a) of the random bit generator in which the chaotic signal is produced by a diode laser with delayed optical feedback as shown in (b). A 4-ns duration trace of the chaotic laser intensity is displayed in (c). The signal was digitized at 40 GHz (blue circles connected by a line to guide the eye). For the RBG, we used a sampling rate of 2.5 GHz, as shown by the red dots. At each sampling point, the five least significant bits obtained from the difference between the current and previous sampled point are attached to the random bit stream. The bits thus generated for this 4-ns time slot are shown in the strip at the bottom of the figure.

intensity must be divided evenly into two groups, and this generally requires a very accurate knowledge of the average intensity value, which also must be stable in time.3 In our experiments, this thresholding requirement is eliminated by taking the derivative of the digitized signal, which, by necessity, forms a highly symmetric, temporally stable and smooth distribution that can therefore be evenly divided into an unbiased sequence. Using this technique, we have generated sequences as long as a few Gbit at an overall bit rate

of 12.5 Gbit/s. For longer sequences, the nonlinearity of the analog-to-digital converter introduces a measurable bias that can be eliminated by periodically inverting the generated bit sequence. Using this technique, arbitrarily long sequences can be generated at the 12.5-Gbit/s rate.  Igor Reidler, Yaara Aviad, Michael Rosenbluh ([email protected]) and Ido Kanter are with the physics department, Bar-Ilan University, Ramat-Gan, Israel. References 1. I. Reidler et al. Phys. Rev. Lett. 103(2), 024102 (2009). 2. M. Rosenbluh et al. Phys. Rev. E 76, 046207 (2007). 3. A. Uchida et al. Nature Photon. 2(12), 728-32 (2008).

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STATISTICAL OPTICS ’09

The Weird Math of Photon Subtraction A. Zavatta, V. Parigi, M.S. Kim and M. Bellini

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ccurate single-photon-level manipulation of light is needed for the development of optical quantum technologies that overcome the limits of classical physics. Such technologies may usher in a revolution in the way we exchange and process information or perform accurate measurements. The basic processes of adding or subtracting single photons to or from a radiation field are described by photon creation and annihilation operators: When applied to states with a welldefined number of photons (Fock states), such operators increase or decrease the photon number by one unit. However, this intuitive behavior does not hold when one is dealing with general superpositions or mixtures of Fock states. A few years ago, we applied sequences of creation and annihilation operators to ordinary light pulses by making use of beam-splitters1 and nonlinear crystals,2 and verified fundamental quantum commutation rules by demonstrating that the order of the operations makes a big difference to the outcome.3 During those experiments, we also found that, under particular conditions, subtracting a single photon changed the quantum state of light to the extent that its mean number of photons increased instead of diminishing. Pushed by those surprising findings, we have recently decided to systematically analyze the action of photon annihilation upon some paradigmatic states of light.4 For Fock states, we have confirmed the intuitive decrease of the photon number by exactly one unit. However, we achieved interesting results when subtracting a single photon from a thermal state, the most common form of light (both the sun and ordinary light bulbs emit chaotic thermal light). In this case, taking one photon away exactly doubled the mean number of photons in

Simplified scheme for conditional single-photon subtraction from a light field. BS is a lowreflectivity beam-splitter. A click in the on/off photodetector heralds the success of the photon annihilation operation on the initial field state. Depending on the photon statistics of the input field, the output state may contain the same mean number of photons or a smaller or larger number.

the pulse. Finally, subtracting a photon from a coherent state (the most classical, wave-like, state of light) did not change it at all. This last result is an experimental demonstration of the fact that coherent states are invariant under photon annihilation. Since their introduction by Nobel laureate Roy Glauber in the 1960s, coherent states have been a cornerstone in the quantum description of light. However, their definition as eigenstates of the annihilation operator had not been verified so directly in an experiment. Although counterintuitive, the strange behavior of these quantum operations is not unphysical and does not put energy conservation at stake. Most of its weirdness simply derives from the misleading implicit assumption that a deterministic addition and subtraction of particles can be represented by the creation and annihilation operators, which work in a probabilistic way.5 Apart from providing some beautiful demonstrations of the inner working of quantum mechanics, the techniques used in these experiments may be used to arbi-

trarily engineer light at the most accurate levels by making the appropriate sequence of photon additions and subtractions. This capability will open the way to tailor-made quantum light for future technologies, such as the secure exchange of information or the development of novel protocols for quantum-enhanced measurements and communications.  M. Bellini ([email protected]) and A. Zavatta are with the Istituto Nazionale di Ottica Applicata in Firenze, Italy. V. Parigi is with the European Laboratory for Nonlinear Spectroscopy in Firenze, Italy. M.S. Kim is with the School of Mathematics and Physics, The Queen’s University, Belfast, Northern Ireland, United Kingdom. References 1. J. Wenger et al. “Non-Gaussian Statistics from Individual Pulses of Squeezed Light,” Phys. Rev. Lett. 92, 153601 (2004). 2. A. Zavatta et al. “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660-2 (2004). 3. V. Parigi et al. “Probing quantum commutation rules by addition and subtraction of single photons to/from a light field,” Science 317, 1890-3 (2007). 4. A. Zavatta et al. “Subtracting photons from arbitrary light fields: experimental test of coherent state invariance by single-photon annihilation,” New J. Phys. 10, 123006 (2008). 5. M.S. Kim. “Recent developments in photon-level operations on travelling light fields,” J. Phys. B 41, 133001 (2008).

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