1. Introduction

Random bit strings are of great utility in modern information processing systems [1]. They are used in cryptography [2], statistical sampling [3], and Monte Carlo simulations [4]. However, the generation of truly random bit strings remains a non-trivial problem. Most methods in use today are pseudo-random. With sufficient processing power, or knowledge of the generation technique, the output of a pseudo-random number generator may become predictable, which can lead to undesirable computational failures or security breaches.

Quantum random number generation (QRNG) techniques offer a robust alternative to pseudo-random methods. These methods rely on the inherently probabilistic outcomes of measurements on suitable quantum systems: while the probabilities of a measurement result may be accurately predicted by quantum theory, the outcome of a given measurement is random for a suitably prepared state. Example QRNG sources include beam splitter partition noise for photons [5], the shot noise of vacuum states [6–8], laser phase noise [9–12], binary phase fluctuations in a parametric oscillator [13], fluorescence from entangled ions [14, 15], photon statistics [16–20], and radioactive decay [21]. We recently demonstrated that random numbers can be generated by measuring the phase of Stokes light produced using spontaneously-initiated stimulated Raman scattering (SISRS) [22]. In this letter, we use the pulse energy fluctuations of Stokes light from SISRS as a source of randomness for QRNG. We convert measured Stokes pulse energies to strings of unbiased random bits using a random Toeplitz matrix to implement a randomness extractor [23,24]. The use of SISRS energy fluctuations rather than SISRS phase noise for QRNG [22] enables a simpler detection setup that does not require an interferometer and reference pulse with which to measure the target quantum random variable. Instead, energy fluctuations are easily measured with high signal-to-noise ratio using fast, inexpensive detectors such as PIN photodiodes, making them a convenient source of entropy for QRNG; this simple detection setup can readily be scaled to higher repetition rates.

Raman scattering is the inelastic scattering of photons from vibrational, rotational, or electronic excitations in the Raman-active medium. Here we use Raman scattering in diamond, as shown in Fig. 1. The population is initially in the the ground state, and the excited state is the optical phonon branch. In a typical Raman scattering event, an incoming ‘pump’ photon is annhilated, and a red-shifted ‘Stokes’ photon is created; the remaining energy is transferred to the diamond as an optical phonon. We focus a strong pump pulse through a diamond sample with no input Stokes pulse and no pre-existing excitation in the medium. Stokes photons scattered spontaneously into the path of the pump pulse stimulate more scattering events so that the Stokes pulse grows by stimulated Raman scattering as it propagates through the diamond. Spontaneous Raman scattering is a quantum phenomenon caused by pump photons scattering from broadband vacuum fluctuations of the electromagnetic field [25, 26]. The quantum mechanical origin of the Stokes pulse is manifest in its statistics, with large fluctuations in both phase [27–29] and energy [30–34].

 

Fig. 1 Λ-level diagram for Stokes Raman scattering in diamond. Inelastic scattering annihilates the pump photon at 532 nm, creating an optical phonon in the diamond and a red-shifted Stokes photon at 573 nm.

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Our technique has the potential to generate very high bit-rates with rapid turn-on times because the non-resonant nature of the Raman interaction allows broad-bandwidth, ultrashort pulses to be used and because the rapid decay of the optical phonons promptly resets the vacuum state before each Stokes pulse is generated. Often in coherent optical experiments such rapid decoherence is problematic, but here it is an advantage. Few-ps Raman decay times are typical in bulk solids and liquids, indicating that the physical limit for uncorrelated, repeated energy measurements is hundreds of GHz across a wide range of possible Raman gain media. Furthermore, the Stokes pulse energy is a continuous variable so that a single measurement can generate multiple random bits: a higher precision measurement extracts more bits, up to a physical limit set by the number of photons in the pulse. A lower threshold practical limit is set by noise in the detection system and by the shape of the Stokes pulse energy distribution. Combining the potential repetition rate and the potential bit extraction depth, the estimated physical limit to data rates is in excess of 1 Tbps.

2. Experiment

Diamond is used as the scattering medium in our QRNG device, shown in Fig. 2. Diamond’s large Raman gain coefficient and broad transparency range make it an excellent optical material for use with short pulses. It has a face-centered cubic lattice, with two carbon atoms per unit cell. There is a triply degenerate Raman active optical phonon mode with vibrational symmetry $T_{2g}({\mathrm{\Gamma }}_5^{+})$, and with frequency Ω = 40THz. The phonon is an excitation of a collective vibrational mode in which the two sub-lattices of atoms comprising the diamond crystal move relative to one another. Our diamond is mounted on a steel heat sink at 295 ± 0.5K. At room temperature (T = 295K), thermal excitation of the optical phonon modes is negligible, with a Boltzmann ratio of exp(−hΩ/k_BT) = 1.5 × 10⁻³ between the optical phonon band and the ground state. The decoherence time of the optical phonons is Γ⁻¹ = 7ps, based on the Raman linewidth and transient coherent ultrafast phonon spectroscopy measurements [35, 36]. Phonon lifetime measurements have a decay rate of ≈ 2Γ, indicating that the decay mechanism in high grade diamonds is almost completely longitudinal, with negligible transverse dephasing [37].

 

Fig. 2 Experimental setup: the pump pulse is focussed into a diamond plate, generating a Stokes sideband by SISRS. The Stokes pulse is spectrally-filtered from the pump, and its energy is measured by photodiode PD2. Part of the pump pulse energy is separated using a beamsplitter in front of the diamond, and is measured by photodiode PD1.

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Our laser source is a Nd:YVO₄ slab amplifier operating at 1 kHz, seeded using a Time-Bandwidth GE-100 Nd:YVO₄ 80-MHz oscillator; amplified pulses are frequency-doubled to produce linearly polarized pump pulses with duration τ_p = 100ps, mean pulse energy W_p ≈ 1 μJ, and wavelength λ_p = 532nm. Pump pulses are focussed by a 4 cm singlet into a 3 mm synthetic diamond crystal, oriented along the 〈100〉 axis. The pump pulse drives SISRS, generating optical phonons with frequency Ω and correspondingly red-shifted Stokes photons with wavelength λ_S = 573nm. The product Γτ_p ≈ 14 shows that the pump pulse duration is longer than the phonon decay time. Crucially, however, the Raman gain satisfies gL > Γτ_p where g is the steady-state Raman gain coefficient and L is the effective propagation length; as a result, the SISRS is operating in the transient regime so that the dynamics are coherent and dominated by a single temporal mode [34]. The spatial coherence of the Stokes beam is high because the pump is tightly focussed to a near diffraction-limited spot, with confocal parameter b ≈ 0.2mm much shorter than the diamond sample; in this geometry the pump beam effectively acts as a spatial filter for the Stokes light [34]. After generation in the diamond crystal, the Stokes pulse is collimated, and spectrally filtered from the pump using a bandpass filter. The Stokes beam is focussed on to a fast reverse-biased photodiode (PD2; Thorlabs DET10A, Si detector, 1 ns rise time), whose response is electronically integrated using a gated boxcar integrator (Stanford Research Systems SR200); the integrated voltage is digitized using a Measurement Computing USB1608FS data aquisition card and recorded as a Stokes energy measurement W_S with 8 bit resolution.

Our acquisition rate is limited to the pump laser repetition frequency (1 kHz); use of high repetition rate lasers could significantly improve on this practical limitation. The physical limit on the repetition frequency for generation of statistically uncorrelated Stokes pulses is set by the time taken for the number of coherently excited phonons to decay below the quantum noise limit of one phonon per mode. Coherent phonons existing above this level after excitation by a pump pulse would seed scattering from a subsequent pulse, thereby inducing correlations.

The Raman gain is proportional to the pump pulse intensity so energy fluctuations due to laser noise will modify the gain from shot-to-shot. We therefore also partition part of the pump pulse energy using a beam splitter in front of the diamond and detect it with at a fast reverse-biased photodiode (PD1; Thorlabs DET210, Si detector, 1 ns rise time); the photodiode response is digitized using the same procedure as for PD2 and recorded as a reference measurement of the pump pulse energy W_p. The measurements are binned according to pump pulse energy, with each bin of width 0.7% of its mean. This allows us to track and account for the influence of noise from the pump laser on the measured Stokes energy distributions.

Figure 3 shows a typical measured probability distribution P(W_S|W_p) of Stokes energies W_S versus W_S/〈W_S〉 for a single pump pulse energy value W_p = 〈W_p〉, where 〈·〉 denotes the sample mean. The measured probability distribution plotted in Fig. 3 shows fluctuations in the Stokes pulse energy of up to five times the mean. The shape of the Stokes pulse energy distribution is similar to previous experimental measurements of transient SISRS energy statistics with negligible pump pulse depletion so that the Raman gain is unsaturated [32].

 

Fig. 3 Plot of the measured Stokes pulse energy distribution P(W_S|W_p) as a function of the normalized Stokes pulse energy W_S/〈W_S〉 for a single pump pulse energy value W_p = 〈W_p〉.

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3. Discussion

SISRS is a quantum mechanical process. The Stokes pulse energy is, therefore, a quantum mechanical observable, here denoted by the operator Ŵ_S. In general, the measured Stokes pulse energy fluctuates from shot-to-shot because of the inherent quantum mechanical uncertainty in SISRS. The shape of the measured statistical distribution of Stokes energies P(W_S|W_p) is determined by the Raman gain, the focussing geometry, and the influence of phonon decay on the scattering process [34]. No general analytic expression is known for the form of P(W_S|W_p). However, in the strongly transient regime with Γτ_p = 0 and unsaturated Raman gain, a one-dimensional model predicts a negative exponential probability distribution with P(W_S|W_p) ≈ 〈Ŵ_S〉⁻¹ exp(−W_S/〈Ŵ_S〉) [30].

In agreement with the one-dimensional model, the measured distribution shown in Fig. 3 is predominantly a negative exponential. The non-exponential behaviour in Fig. 3 at W_S ≈ 0 occurs because the Stokes light is not perfectly single mode. The scattered Stokes light contains weak contributions from other spatio-temporal modes due to phonon decay and coupling of the pump to more than one spatial mode [34, 38]. In SISRS sources where many spatio-temporal modes are excited, the coherence of the output light decreases and the Stokes pulse energy fluctuations diminish [34]. However, in the case of Fig. 3, large energy fluctations are easily resolved, allowing us to convert the raw data to random bits.

4. Data processing

Bias and classical noise are inevitably mixed in with the quantum randomnesss we use to generate random bits. There are numerous approaches to remove bias and extract unbiased random bits from raw data. We use Toeplitz-hashing as a randomness extractor [23, 24]. The extractor is applied by multiplying concatenated binary strings of raw data, each of length s bits, with a m × s random seed Toeplitz matrix to output m random bits; the seed matrix can be used repeatedly because the hash function is a strong extractor [24]. The value of m is set by the quantum random information content of the data as we discuss below.

The amount of randomness in the Stokes energy W_S can be quantified by its min-entropy,

We tested the statistical properties of our random binary strings using the Diehard test suite [39]. The Diehard tests run on 11 MB binary files, and each test returns a p-value on [0, 1). For a good source of random bits, the output p-values should be uniform on [0, 1); clear failure of a test is indicated by p-values close to 0 or 1, up to several significant figures. The generally-accepted significance level α for “passing” a test is 0.01 < α < 0.99 [16]. As is shown in Table 1, the data passes all of the Diehard tests within this range. This confirms that Stokes pulse energy fluctuations are an effective source of statistically uncorrelated, unbiased bit-strings.

Table 1. Results of the Diehard statistical tests applied to the Raman random number bit strings. The p-values are within the significance interval 0.01 < α < 0.99, indicating that the bit strings pass all the tests.

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5. Conclusion

We have introduced and demonstrated a technique for generating sequences of quantum random bits by measuring the randomly fluctuating pulse energies of Stokes pulses generated via SISRS. The randomness of the bits is guaranteed by the quantum mechanical origin of the Stokes pulse energy fluctuations. The energy fluctuations are visible to the naked eye, and are easily measured using high speed PIN photodiodes. Since the pulse energy is a continuous variable, multiple random bits can be extracted from each measurement by digitizing the output from a detector with sufficiently high resolution. After Stokes generation, optical phonons typically decay on picosecond timescales so the vacuum state is quickly reset; new, statistically independent Stokes pulses can therefore be generated in rapid succession.

Diamond was used as the scattering substrate due to its high Raman gain and broad transparency. Its large Stokes shift, and that of other similar Raman active media [40], permit the use of ultrafast picosecond and femtosecond pump pulses. Diamond’s rapid phonon decay means that GHz, or higher, pulse repetition rates are possible. However, the propagation length in bulk diamond is limited by diffraction of the pump beam, which in turn limits the SISRS gain available for a given pump pulse energy. The current implementation was therefore limited to 1 kHz by the need to boost the pump pulse energy using a laser amplifier. Increases to the SISRS gain above our operating value by further increasing the pump pulse energy were limited by the onset of optical damage to the substrate. As a result, the SISRS gain was unsaturated, restricting the width of the Stokes pulse energy distribution and, therefore, the number of bits extracted to 3.4 bits per measurement. A higher bit extraction per measurement could be obtained by partially saturating the SISRS gain to give a more favourable Stokes pulse energy distribution which is less biased toward W_S = 0. For example, a longer interaction region and therefore higher SISRS gain could be achieved by placing the substrate in a cavity, or by use of an integrated photonic structure such as a gas-filled photonic crystal fiber [41]. We expect that a longer interaction region will also enable the use of low power laser oscillators operating at GHz repetition rates, leading to commensurate high speed random bit production.