Abstract
Classically, optical solitons are robust nonlinear waves, capable of long-distance propagation with little or no distortion. Quantum mechanically, however, solitons experience phase diffusion and wave packet spreading during propagation. Progress in quantum optics has led to experimental confirmation of these effects, with demonstrations of entanglement, sub-shot noise and non-demolition measurements. These phenomena will be reviewed, along with recent results describing phase noise reduction in soliton collisions.

Introduction
Solitons are localized, shape-preserving waves characterized by ‘elastic’ collisions, in which no change in amplitude or velocity occurs. First observed as a water wave in 1834 by John Scott Russell in the Union Canal near Edinburgh, and subsequently recreated in the laboratory [1], solitons arise in a variety of physical systems, as both temporal pulses which counteract dispersion and spatial beams which counteract diffraction. For recent reviews of solitons, their interactions and applications, see Stegeman and Segev [2] and the book by Kivshar and Agrawal [3].
Temporal soliton pulses in optical fiber were first predicted in 1973 by Hasegawa and Tappert [4], originating from a balance between self-phase modulation and anomalous dispersion. The first experimental observation occurred in 1980 by a group in Bell Labs [5]. Since then, many theoretical and experimental advances in soliton science have occurred— among them, the investigation of the fundamental quantum character of these nonlinear waves. The inherent robustness of these pulses, low loss of optical fiber, and low noise of mode-locked laser systems have enabled experimental measurements of soliton pulses at the quantum level. In this article, we review the basic principles of quantum soliton theory and highlight relevant experimental results.

Figure 1: Schematic of quantum soliton noise evolution and squeezing process. Initial and propagation-induced noise distributions are shown on the left and right, respectively. Amplitude fluctuations DA remain constant, whereas phase noise Dq increases as a function of propagation distance z. A sub-shot noise, or squeezed, measurement in a phase-sensitive interferometric detection scheme can be made at angle j

Quantum Soliton Propagation, Squeezing, and Entanglement
A pulse in optical fiber undergoes dispersion, or temporal spreading, during propagation. This effect arises because the refractive index of the silica glass is not constant, but is rather a function of frequency. The pulse can be decomposed into a frequency range—the shorter the pulse, the broader its spectral width. The frequency dependence of the refractive index will cause the different frequencies of the pulse to propagate at different velocities, leading to an overall dispersion, or temporal spreading, of the pulse. As a result, the pulse develops a chirp, meaning that the individual frequency components are not evenly distributed throughout the pulse. There are two types of dispersion: anomalous and normal. If the longer wavelengths travel slower, the medium is said to have anomalous dispersion. If the opposite is true, the medium has normal dispersion.
In a Kerr nonlinear medium such as fiber, the refractive index depends on the intensity of the propagating field. Because the material responds almost instantaneously, on the order of femtoseconds, each component of an intense optical pulse sees a phase shift proportional to its intensity. The leading edge of the pulse is red-shifted, while the trailing edge is blue-shifted, an effect known as self-phase modulation (SPM). If the medium has anomalous dispersion, the pulse is compressed. Under the proper conditions, this pulse compression can exactly cancel the anomalous dispersion-induced chirp, resulting in distortionless soliton propagation. For more details, see the book by Agrawal [6].
Classically, dispersion and Kerr nonlinearity are described by Maxwell’s equations, completely deterministic phenomena. However, a full quantum mechanical description of these effects imposes fundamental uncertainties in the soliton parameters, including amplitude, phase, position, and frequency. Frequency fluctuations will cause an overall uncertainty in the amount of dispersion experienced by the pulse, and hence an increasing uncertainty in the pulse position as a function of propagation distance. Similarly, amplitude fluctuations will directly affect the amount of nonlinearly-induced phase shift, leading to an increasing uncertainty in the soliton phase as the pulse propagates. For reviews of quantum soliton theory and experiments, see [7, 8].
From the preceding discussion, it is clear the inherent quantum noise in solitons places a fundamental limit on the processes governing their propagation. Initially, the quantum noise distribution of the soliton can be viewed as a circle, as shown in the left of Fig. 1. Amplitude and phase fluctuations DA and Dq are depicted in the vertical and horizontal directions, respectively. In the absence of additional noise sources, amplitude fluctuations remain constant, as shown by the dashed lines. Balanced direct-detection will measure this noise level, which corresponds to the shot noise limit.
Due to the Kerr nonlinearity, the circle of Fig. 1 will evolve into an ellipse, as shown on the right-hand side of Fig. 1 after an arbitrary propagation distance z. Because of the amplitude-dependent phase shift—the higher the amplitude, the larger the phase shift—the higher amplitude fluctuations of the noise distribution will acquire a phase lead over smaller amplitude fluctuations, as shown by the tilted direction of the ellipse. Overall propagation-induced phase fluctuations, given by Dq (z), are larger than the initial phase noise level, as expected by the phase diffusion process. The major consequence of this evolution is the possibility of observing squeezing, in which the noise dips below the standard shot noise limit. A squeezed detection is possible using a phase-sensitive interferometer. In Fig. 1, maximum squeezing is achieved at a phase angle j to the amplitude quadrature. The first observation of soliton squeezing using this method measured a noise level 1.1 dB below the shot noise limit [9]. More recently, directly detectable amplitude squeezing 2.3 dB below the shot noise limit was observed using spectral filtering [10]. For a review of fiber-based squeezing experiments, see [8].
In addition to more accurate measurements, squeezed light can be used to generate entangled soliton pulses. In fact, the linear interference between two independently squeezed solitons yields entanglement. This effect was demonstrated by a group in Erlangen, Germany, in which an entanglement correlation of 4 dB below the shot noise limit was measured [11].

Solution Collisions and Quantum Non-Demolition Measurements
Solitons experience ‘elastic’ collisions, in which amplitude and velocity remain unchanged. However, collisions induce phase and position shifts. The phase shift experienced by each soliton in a collision is a function of two sets of parameters: the soliton amplitudes and velocities. Just as the optical intensity self-induces a phase change through SPM, each soliton ‘feels’ the intensity of the other in a collision to cause a phase shift. This type of cross-interaction is called, appropriately, cross-phase modulation (XPM). The magnitude of the phase shift depends also on the relative velocity; it increases as the relative velocity decreases, which is intuitively logical given the increased interaction length in this regime.
Because of quantum fluctuations in the soliton amplitudes and velocities, an uncertainty in the amount of collision-induced phase and position shift will result. In addition, a nonclassical correlation arises between solitons. For example, if two solitons collide—a signal and probe—a measurement of the probe phase will provide information on the signal amplitude, without the need for a direct measurement of the signal soliton. Moreover, the signal amplitude is undisturbed in the collision, with additional noise introduced only to the signal phase, the conjugate observable. Quantum theory thus allows a measurement of a quantum observable in which no noise, or back-action, is introduced to the measured quantity. No violation of Heisenberg’s uncertainty principle occurs, because the inevitable back-action is introduced to the conjugate observable only. Such a measurement is called a quantum non-demolition (QND) measurement (for reviews of QND measurements, see [12, 13]). Soliton-based QND measurements demonstrate several advantages over other kinds of QND schemes, including simplicity of implementation and compatibility with existing optical communications technologies.
An accurate reading of the probe soliton phase is therefore important, but two sources of noise tend to limit the effectiveness of the QND measurement. One noise source is guided acoustic-wave Brillouin scattering (GAWBS) [14], caused by thermal fluctuations of the refractive index. This effect can be efficiently suppressed using a closely spaced reference pulse in such a way that both the reference and probe solitons acquire the same GAWBS phase fluctuations [15, 16] or by working at liquid-nitrogen temperatures [9]. Experiments performed at NTT Laboratories in Japan found that the propagation-induced process of phase diffusion through SPM (Fig. 1) accounts for another major source of noise [15, 16]. In recent work [17], we have shown that phase noise can be reduced in a soliton collision because of a negative correlation that exists between phase fluctuations induced by propagation and collision. This effect can be exploited to improve QND measurement accuracy.

Figure 2: Contour plots of variance ratio (ratio of phase noise variances in collision with respect to no collision) as function of (a) propagation distance z with dp = 0.6 and (b) relative velocity dp with z = 10 as a function of signal amplitude As and fixed probe amplitude Ap = 1. These plots converge at z = 10 and dp = 0.6, and the dashed curves are contours of minimum variance ratio. All quantities are in dimensionless, normalized units.

In order to describe the effect of collision-induced phase noise reduction, we introduce a variance ratio, defined as the ratio between the post-collision phase noise variance and the variance in the absence of collision. In Fig. 2, the dependence of the variance ratio on propagation distance z and relative velocity dp is plotted as a function of signal amplitude As, keeping the probe amplitude constant at Ap = 1. From Fig. 2, we can see that by optimizing the relative amplitudes of the solitons, the propagation distance, and the relative velocities, a soliton collision can reduce the phase fluctuations in the probe by more than 20%. The increase in variance ratio for z > 10, shown in Fig. 2(a) with dp = 0.6, arises due to SPM-induced phase noise, which tends to drown out the noise reduction at long propagation distances. In addition, the variance reduction favors a larger probe for z < 16. For fixed z, phase noise can be minimized for small dp and, consequently, a small wavelength separation between probe and signal. This trend is shown in Fig. 2(b), in which z = 10. It should be noted that a collision-induced phase noise reduction exists also for the case of two-component vector solitons [18].
As discussed above, under the proper conditions, a soliton collision can reduce the fluctuations in the probe phase by more than 20%. This, in turn, can provide a more accurate reading of the signal amplitude in a QND measurement.
To quantify the effectiveness of a QND measurement, we calculate the normalized error variance S with which a measurement of the probe phase will infer the signal amplitude [19]. The best possible measurement corresponds to a value of S = 0, the minimum achievable inference error of the signal amplitude, and S = 1 indicates no correlation between the probe phase and signal amplitude. Contours of S are plotted in Fig. 3 with respect to z and dp as a function of As, again keeping Ap = 1. For short z, approximately equal probe and signal amplitudes are optimal, as seen in Fig. 3(a). At longer lengths, however, smaller probes favor a more accurate measurement. When z = 5, the dependence of S is plotted as a function of dp and As in Fig. 3(b), which demonstrates an optimal S for small relative velocity. Clearly, the performance of a soliton-based QND measurement can be optimized by choosing the experimental parameters properly. The reduction in phase fluctuations provided by the soliton collision can be leveraged to significantly reduce the error in the measurement of the signal amplitude.

Figure 3: Contour plots of normalized QND error variance S with respect to (a) propagation distance z with dp = 0.6 and (b) relative velocity dp with z = 5 as a function of signal amplitude As and fixed probe amplitude Ap = 1. These plots converge along the horizontal axis, and the dashed curves are contours of minimum S. All quantities are in dimensionless, normalized units.

Conclusions
Quantum information science initially developed from the interdisciplinary work of scientists in the fields of physics, computer science, mathematics, and information theory. The field has since experienced an overwhelming surge in interest, with much research focused on finding practical implementations to diverse applications in communications, cryptography, and computing. Remarkable features of fiber optics in quantum information include low noise and strong coherence. In particular, one benefit of using optical solitons is the fact that the same nonlinearity which allows soliton propagation is the driving force behind the quantum evolution of the soliton, enabling squeezing, entanglement, and non-demolition measurements, all of which have been demonstrated. In this paper, these phenomena were reviewed, along with recent work describing quantum phase noise reduction in soliton collisions, which arises due to a negative cross-correlation between propagation- and collision-induced phase fluctuations [17]. This effect may lead to improvements in the accuracy of quantum non-demolition measurements, taking the measurement further into the quantum regime and furthering the potential of soliton-based quantum information processing.

Acknowledgements
The authors acknowledge collaboration with Prof. Ken Steiglitz and many useful discussions with Prof. Jason Fleischer.

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