Exploiting shot noise correlations in the photodetection of ultrashort optical pulse trains

Photocurrent shot noise represents the fundamental quantum limit for amplitude, phase and timing measurements of optical signals. It is generally assumed that non-classical states of light must be employed to alter the standard, timeinvariant shot noise detection limit. However, in the detection of periodic signals, correlations in the shot noise spectrum can impact the quantum limit of detection. Here, we show how these correlations can be exploited to improve shot noise-limited optical pulse timing measurements by several orders of magnitude. This has allowed us to realize a photodetected pulse train timing noise floor at an unprecedented 25 zs Hz (corresponding phase noise of 2179 dBc Hz on a 10 GHz carrier), ∼5 dB below the level predicted by the accepted time-invariant shot noise behaviour. This new understanding of the shot noise of time-varying signals can be used to greatly improve photonic systems, affecting a wide range of communication1, navigation2 and precision measurement3 applications. Shot noise results from the discrete nature of the detection of optical fields. There is a fundamental randomness of photon flux that, upon photodetection, is transformed into fluctuations in the photocurrent known as shot noise4. Whether the light is continuous wave or has a periodically varying intensity, the shot noise spectral density is a well-defined quantity, allowing a useful frequency-domain description. In either case, the shot noise current spectral density (units A Hz) is 2qIavg, where q is the fundamental charge and Iavg is the average photocurrent 5. In the absence of quantum-mechanically squeezed states of light6, this expression is considered to dictate the fundamental limit to the achievable signal-to-noise in photocurrent measurements. Even for classical light fields, however, 2qIavg is not a complete description of the shot noise, as it does not provide information on possible phase correlations in the noise spectrum. The impact of spectral correlations is often overlooked, because, for continuous wave signals, no correlations are present. On the other hand, signals with periodically varying intensity do produce spectral correlations in the shot noise, with consequences for the shot noise limit of measurements of the optical field. For example, such correlations have been shown to degrade the noise floor in some gravitational wave detectors by !2 dB (ref. 7). Until now, it has gone unrecognized that these correlations can result in orders of magnitude improvement in the quantum limit of the timing precision of a train of photodetected ultrashort pulses. Our measurements confirm our prediction that shot noise can be manipulated such that the pulse-to-pulse timing precision can be significantly improved simply by keeping the optical pulse width at the detector sufficiently short. Previous studies on the shot noise of time-varying signals have not addressed the detection of ultrashort pulses7–12. This is due in part to the fact that, until recently, the power-handling capability of high-speed photodetectors has been insufficient to operate well within the shot noise limit at microwave frequencies. Photodetection of a train of ultrashort pulses (for example, the output of a mode-locked laser) produces microwave tones at harmonics of the pulse repetition rate fr. The pulse train timing jitter is determined by measuring the phase noise sidebands of these harmonics13. Without sufficient microwave power, the signal-to-noise ratio is limited by thermal noise, and no optical pulse width dependence on phase noise is detectable. To understand the optical pulse width dependence of the photocurrent shot noise-limited timing precision, it is useful to start with the pulse-to-pulse timing jitter at the fundamental limit imposed by the discrete nature of photons. From pulse to pulse, there are random variations in both the number of photons per pulse and the photon distribution within a pulse. Randomness in the photon distribution produces small deviations in the time of arrival of the pulse, or timing jitter, which may be thought of as variations in the arrival of the pulse ‘centre of mass’. The shorter the optical pulse, the smaller the jitter, because there is a smaller pull on the pulse centre when the photons are more tightly packed14 (Supplementary Section S1). This is illustrated in the pulse ensemble of Fig. 1a by noting the difference in the thickness of the rising edge of the short and broad pulse ensembles. Photodetection produces a train of much broader current pulses with a minimum timing jitter due to shot noise. Conceptually, the pulse-to-pulse timing instability is revealed by comparing (multiplying) the photodetected signal with the zero-crossing of a timing reference15, as shown in Fig. 1b. The shot noise is not constant, but arrives in bursts along with the pulses. Multiplying the zero-crossings with the shot noise bursts results in lower noise power, and therefore improved timing precision, for shorter pulses. Although Fig. 1b suggests a pulse width dependence, it is not immediately clear whether the impact of photocurrent shot noise on timing measurements scales with the optical pulse width or the much broader electrical pulse. Because the phase noise of the photonically generated microwave harmonics represents the optical pulse timing jitter, it is the optical pulse width that must determine the fundamental timing precision in a photocurrent measurement. A time-varying photocurrent analysis at the shot noise limit shows that this is indeed the case, predicting an imbalance of the shot noise between amplitude and phase quadratures of the phototonically generated microwave harmonics (Supplementary Section S2). The unequal distribution between the amplitude and phase quadratures is key to understanding how shot noise limits the timing precision, so we have developed an intuitive interpretation of how this imbalance arises. Shot noise may be viewed as the result of heterodyne beat signals between the optical signal and vacuum fluctuations16,17 (Fig. 2). In the frequency domain, an optical pulse train is represented as a frequency comb with a fixed