## Abstract

We identify and discuss nonlinear phase noise arising in Kerr self-phase modulation of a coherent light pulse propagating through an attenuating medium with third-order nonlinearity in a dispersion-free setting. This phenomenon, accompanying the standard unitary Kerr transformation of the optical field, is described with high accuracy as Gaussian phase diffusion with parameters given by closed expressions in terms of system properties. We show that the irreversibility of the nonlinear phase noise ultimately limits the ability to transmit classical information in the phase variable over a lossy single-mode bosonic channel with Kerr-type nonlinearity. Our model can be also used to estimate the amount of squeezing attainable through self-phase modulation in a Kerr medium with distributed attenuation.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Third-order nonlinearity in optical media makes the effective refractive index dependent on light intensity. One of the consequences of this phenomenon, known as the Kerr effect, is self-phase modulation, which consists of an intensity-dependent contribution to the phase shift acquired by a light beam propagating trough the medium. In the single-mode description, such propagation can be described by a nonlinear oscillator with 1 degree of freedom [1–3]. The quantum version of this model exhibits a wealth of nonclassical effects, such as squeezing [4] and generation of Schrödinger cat states [5]. However, their experimental realization requires suppression of the accompanying decoherence mechanisms with a prominent role played by photon loss. While it is feasible to experimentally produce squeezed states using third-order nonlinearity in optical fibers [6–8], Schrödinger cat states turn out to be much more elusive owing to higher nonlinearities necessary for generation and greater susceptibility to losses [9].

In parallel, optical nonlinearities have been recognized as a key factor limiting the information capacity of fiber optical communication links [10–13]. To date, most studies in this area have used the classical description of electromagnetic fields in terms of complex amplitudes governed by deterministic equations of motion. In this approach, noise appears as a result of signal amplification, averaging over signals transmitted through other channels in a multiplexed system or the functioning of the detection stage. The performance of conventional optical detection is limited by the shot noise, which is a manifestation of quantum fluctuations in measured electromagnetic fields [14]. However, if the propagation of electromagnetic fields is both nonlinear and lossy, quantum fluctuations in the optical medium undergo a nonunitary transformation, which may result in additional noise that would not emerge in a classical description. This eventuality motivates a full quantum mechanical study of nonlinear effects in optical signal propagation.

The purpose of the present paper is to analyze a single-mode model for self-phase modulation of a coherent light pulse propagating through a lossy nonlinear dispersion-free medium. Our central objective will be to identify and characterize quantitatively excess noise, which occurs as a result of nonlinear transformation of quantum fluctuations in the course of lossy propagation and cannot be compensated at the output with the help of a reversible unitary transformation. Specifically, we show that lossy Kerr propagation can be mathematically decomposed into a sequence of three distinct processes that are formally applied to the input coherent state one after another. This decomposition is represented pictorially in Fig. 1. The first process is standard attenuation, which lowers the complex amplitude of the input coherent state. The last one is unitary Kerr evolution, which would have occurred in the absence of loss. The intermediary process we will focus our attention on arises solely as a result of a nontrivial interplay between the Kerr nonlinearity and the optical loss. We demonstrate that it has the form of nonlinear phase noise, which can be approximated with high accuracy by a Gaussian distribution for the additive random phase. We provide closed analytical expressions for parameters that characterize the properties of this decoherence mechanism.

The derived effective description of the nonlinear phase noise is employed to discuss its impact on the transmission of classical information in the phase variable. We choose continuous phase modulation with constant intensity as the encoding format [15] and characterize accessible information using the Holevo quantity [16–18], which takes into account the most general quantum mechanical measurement strategies at the channel output, including compensation of reversible nonlinear effects. It is shown that, for the propagation over a lossy Kerr medium, the Holevo quantity exhibits a maximum in the signal intensity, and that the nonlinear phase noise renders the phase variable useless for information encoding at high signal powers. Consequently, this effect can be expected to substantially alter the ultimate communication capacity of an optical channel compared with the linear case, which has been analyzed rigorously in recent years [19]. As another application of the presented description, we study generation of squeezing using the Kerr effect and derive a simple estimate for the impact of distributed loss in the squeezing medium on the output quadrature variance.

This paper is organized as follows. In Section 2, we introduce the theoretical model for the nonlinear propagation. In Section 3, we derive the effective description for the excess noise generated in the model. Based on this description, we demonstrate in Section 4 how the excess noise has an impact on the accessible information for the phase variable. In Section 5, we discuss the generation of squeezed states using a lossy Kerr medium and derive a simple estimate for the attainable reduction of quadrature noise in the case of distributed attenuation. Finally, Section 6 closes the paper with some concluding remarks.

## 2. LOSSY NONLINEAR PROPAGATION

In a single-mode model describing dispersion-free propagation of an optical pulse in a lossy Kerr medium along a direction $z$, the evolution of the system is given by the master equation

In order to separate reversible phenomena, we will carry out the analysis in the interaction picture defined by the nonlinear Hamiltonian $\mu {\widehat{n}}^{2}$ of the Kerr interaction and represent the field density operator $\widehat{\varrho}(z)$ as

It will be convenient to characterize the medium with two dimensionless parameters. The first one is the transmission $\tau =\mathrm{exp}(-\alpha z)$, which specifies the fraction of optical energy $\tau \overline{n}$ retained in the field at the channel output. The second one is the ratio $\varkappa =\mu /\alpha $ describing the strength of the nonlinear interaction with respect to attenuation. Note that $\varkappa $ is an intensive parameter characterizing the bulk medium itself, while $\tau $ depends on the channel length. Using this parameterization, the solution of the master equation can be written compactly as

## 3. NONLINEAR PHASE NOISE

Let us now derive effective parameters that describe features seen in Fig. 2. The starting point will be the inspection of the real and imaginary parts of the function ${f}_{\tau}(\varkappa )$ defined in Eq. (7), shown in Fig. 3 for transmissions $\tau ={10}^{-8}$ and $\tau =0.8$ as a function of the dimensionless nonlinearity $\varkappa $. As seen in Fig. 3(a), for $\tau \ll 1$ the real part $\mathrm{Re}\text{\hspace{0.17em}}{f}_{\tau}(\varkappa )$ has the form of a dip centered around $\varkappa =0$. Using Eq. (7), this dip can be modelled by an inverted Lorentzian proportional to $1/(1+4{\varkappa}^{2})$, whose width is of the order of 1. The relevant range of the argument of ${f}_{\tau}$ in Eq. (6) is defined by the range of the Fock basis indices $m$ and $n$ for which the product of the scalar products $\u27e8m|\sqrt{\tau}{\zeta}_{0}\u27e9\u27e8\sqrt{\tau}{\zeta}_{0}|n\u27e9$ substantially differs from zero. Because $|\u27e8m|\sqrt{\tau}{\zeta}_{0}\u27e9{|}^{2}$ is a Poissonian distribution in the integer variable $m$ with an average $\tau \overline{n}$ and hence standard deviation $\sqrt{\tau \overline{n}}$, this condition will be met for $|m-n|\lesssim \sqrt{\tau \overline{n}}$. Consequently, the argument of the function ${f}_{\tau}$ has the order of magnitude $\varkappa |m-n|\lesssim \varkappa \sqrt{\tau \overline{n}}$. If the product $\varkappa \sqrt{\tau \overline{n}}\ll 1$, which is the case for the numerical example shown in the preceding section, it is justified to expand the function ${f}_{\tau}(\varkappa )$ up to the second order in $\varkappa $, which gives

After the above approximation, the exponent multiplying Fock basis elements of the coherent state density matrix in Eq. (6) contains terms linear and quadratic in $m-n$ with, respectively, imaginary and real multiplicative constants. It is easily verified by a direct calculation that such a density operator can be equivalently written as a statistical mixture of coherent states with a fixed mean photon number and the phase averaged using a Gaussian distribution with a mean ${\varphi}_{0}$ and a standard deviation $\sigma $:

As a side remark, let us note that the validity of the expansion in Eq. (8) can be extended beyond the regime discussed above. This is because the function ${f}_{\tau}(\varkappa )$ appears in the exponent in Eq. (6) multiplied by the input intensity $\overline{n}$. If this product is large compared with 1, the exponential factor effectively suppresses the respective off-diagonal elements of the density matrix regardless of its specific form. In Fig. 3(b) we plot the real and imaginary parts of ${f}_{\tau}(\varkappa )$ for $\tau =0.8$ compared with the corresponding quadratic expansion. Because the asymptotic value for $|\varkappa |\gg 1$ is ${f}_{\tau}(\varkappa )\approx 1-\tau $, the rule of thumb for the quadratic expansion to hold using the above reasoning is $(1-\tau )\overline{n}\gg 1$. The accuracy of this approximation can be assessed by comparing the infidelity $1-F=1-\mathrm{Tr}\sqrt{\sqrt{{\widehat{\varrho}}^{\prime}}{\widehat{\varrho}}_{G}^{\prime}\sqrt{{\widehat{\varrho}}^{\prime}}}$ between the actual density matrix ${\widehat{\varrho}}^{\prime}$ and its approximation ${\widehat{\varrho}}_{G}^{\prime}$ introduced in Eq. (9) with parameters of the Gaussian phase noise given in Eqs. (10) and (11). The infidelity is shown in Fig. 4 for $\tau =0.8$ as a function of the mean output photon number $\tau \overline{n}$ and the strength of the nonlinear interaction $\varkappa $. It is seen that the departure from the Gaussian approximation in the parameter space is localized in the region of high $\varkappa $ and moderate output mean photon numbers $\tau \overline{n}$ and works very well for higher pulse intensities. Note that the upper limit for the range of $\varkappa $ in Fig. 4 has been taken well above realistic values for optical fibers. Notice also the small values of the infidelity $1-F$, ensuring that the physical properties of the phase diffused state are close to those of the state subject to lossy nonlinear propagation [23–25].

## 4. PHASE SHIFT KEYING

Nonlinear phase noise analyzed in the preceding section is a nondeterministic phenomenon that cannot be compensated at the channel output in contrast with unitary Kerr transformation. This affects the ability to transmit classical information in the phase variable, which is routinely used in keying constellations for fiber optic communication, such as quadriphase shift keying [26]. Without phase noise, the amount of information that can be transmitted by modulating the phase of a light pulse grows with its intensity. Fixing intensity defines a ring in the complex amplitude plane, which, with increasing radius, can accommodate more coherent states whose distinguishability is limited by the shot noise. This picture changes dramatically when channel nonlinearities induce phase noise. As seen in Eq. (11), the variance of the phase noise grows linearly with the input field intensity, which may actually obliterate information keyed in the phase.

We will analyze quantitatively the above phenomenon by considering a keying constellation, which consists of coherent states $|{e}^{i\varphi}{\zeta}_{0}\u27e9$ continuously distributed on a circle with a uniform distribution ${p}_{\varphi}=1/2\pi $ [15], as shown in Fig. 5(a). As a measure of accessible information, we will take the Holevo quantity $\chi $, which takes into account the most general measurement strategies permitted in quantum mechanics [16–18]. The Holevo quantity is given as a difference between the von Neumann entropy $S(\widehat{\varrho})=-\mathrm{Tr}(\widehat{\varrho}\text{\hspace{0.17em}}\mathrm{log}\widehat{\varrho})$ of the average state emerging from the channel and the average entropy of individual states, and it is given in our case by

In Fig. 5(b), we depict the Holevo quantity as a function of the mean photon number in the output pulse for several values of the dimensionless nonlinearity parameter $\varkappa $. In the case of a linear channel, when $\varkappa =0$, the Holevo quantity grows monotonically with the signal intensity. However, in the nonlinear case, the Holevo quantity exhibits a maximum, which can be understood as an interplay between two effects. For low intensities, the nonlinear phase noise has minute impact, and increasing the mean photon number allows us to encode more information in the phase variable thanks to the lower extent of the shot noise relative to the circumference of the ring. For high intensities, the nonlinear phase noise makes coherent states diffused on the ring, as seen in Figs. 2(c) and 2(d), which effectively scrambles any information encoded in the phase variable. In general, one can consider an input ensemble of coherent states, which modulated both the intensity and the phase parameters. The quantification of the attainable transmission rate needs to take into account the nontrivial dependence of the information capacity of the phase variable on the signal intensity.

Let us stress that the nonlinear phase noise is irreversible, and it cannot be compensated at the channel output. Its origin can be understood intuitively as follows. The amplitude of the coherent pulse entering the nonlinear medium exhibits vacuum-level fluctuations. The nonlinearity deforms these fluctuations in a nontrivial manner, which for short distances can be visualized as squeezing one field quadrature at the cost of expanding the conjugate one [27]. In the lossless case, variances of these quadratures saturate the Heisenberg uncertainty relation, which indicates that the state remains pure, and the transformation is reversible. However, in the case of distributed attenuation, the propagating field exchanges fluctuations with the environment. The contribution to field fluctuations acquired that way is given at the vacuum level, which introduces excess noise [28–30] that makes the state no longer Heisenberg-limited. Because the effect of losses is to lower the mean photon number of the Poissonian statistics of the pulse, and the Kerr nonlinearity affects only the off-diagonal elements of the density matrix in the Fock basis that characterize phase coherence, this mixedness occurs for the phase variable.

## 5. KERR SQUEEZING IN A LOSSY MEDIUM

The Kerr effect in optical fibers can be used to generate squeezed states of light, which exhibit quadrature fluctuations reduced below the shot noise level [4,6–8]. We will now use the effective description of self-phase modulation in a lossy nonlinear medium developed in preceding sections to analyze the attainable degree of squeezing in the presence of distributed attenuation.

In order to describe the output field emerging from the medium, we will take the phase diffused state evaluated using the Gaussian phase noise profile derived in Eq. (9):

The effective limit on attainable squeezing can be estimated using the following simple calculation. When exponents ${e}^{-{\sigma}^{2}}$ and ${e}^{-2{\sigma}^{2}}$ in Eqs. (25)–(27) are expanded up to the first order in ${\sigma}^{2}$, the dominant effect of phase noise is the addition of an excess noise term ${(\mathrm{\Delta}p)}_{\mathrm{exc}}^{2}=4{\sigma}^{2}{\zeta}^{2}$ to the variance of $\widehat{p}$. All other terms including ${\sigma}^{2}$ can be neglected, as they involve factors $\mathrm{sinh}\text{\hspace{0.17em}}r$ or ${\mathrm{sinh}}^{2}\text{\hspace{0.17em}}r$ that are small compared to ${\zeta}^{2}$. This excess noise can be written as ${(\mathrm{\Delta}p)}_{\mathrm{exc}}^{2}=4\text{\hspace{0.17em}}\mathrm{g}(\tau ){\mathrm{sinh}}^{2}\text{\hspace{0.17em}}r$, where

## 6. CONCLUSIONS

We have presented a theoretical model for the propagation of coherent states in a lossy medium with third-order nonlinearity in a dispersion-free setting. Within this model, we have identified nonlinear phase noise emerging from the interplay between optical loss and the Kerr nonlinearity. This excess noise is nonunitary and cannot be compensated at the output. It has been shown that, for a broad range of system parameters, an effective description of this excess noise is given by a Gaussian distribution introducing also a nonlinear phase shift for the input state. While the phase shift can be taken into account at the preparation stage, the Gaussian phase diffusion process is nonunitary and cannot be compensated. Identification of the nonlinear phase noise allows us to formally decompose the description of propagation in a lossy nonlinear medium into a sequence of three distinct processes, namely, standard attenuation followed by Gaussian phase diffusion and then unitary Kerr evolution. The presented decomposition is valid for coherent states. It can be generalized to arbitrary states of the electromagnetic field using the Glauber–Sudarshan $P$ quasiprobability distribution for the density operator [32,33], but it should be kept in mind that the effective transformation of individual coherent states in this representation depends in a nontrivial manner on their amplitudes through the parameters of the Gaussian phase diffusion process. As a side note, the quadratic approximation used in the derivation of the nonlinear phase noise parameters is reminiscent of certain simplified models of decoherence in quantum mechanics [34], which can be tested also in spatial propagation of optical fields through dense, multiple-scattering, random media [35].

We also demonstrated that the nonlinear phase noise severely impairs the ability to encode classical information in the phase variable. Specifically, we considered phase shift keying using a continuous constellation of coherent states distributed on a ring with a fixed intensity. In the case of nonzero nonlinearity, the Holevo quantity, used as a measure of accessible information, exhibits a maximum for a finite input mean photon number. This is because, for input coherent states with a sufficiently high mean photon number, the phase becomes completely scrambled due to increasing diffusion. Consequently, when designing an optimal input ensemble of coherent states in both phase and intensity variables for transmission of classical information, the contribution from the high intensity region should approach that of a Poisson channel [36]. It would be also interesting to investigate the ultimate quantum limit on the classical capacity of an optical channel with nonlinear phase noise, generalizing recently results obtained for linear Gaussian models [19]. Finally, we investigated the interplay between nonlinearity and losses in the generation of squeezed states utilizing a Kerr medium. We showed that the nonlinear phase noise can be used to estimate the attainable squeezing in the presence of distributed attenuation.

One should note that, in the context of optical communication, nonlinear transformation of spontaneous emission contributed by signal amplification is known to introduce excess phase noise [37,38]. This effect arises in the classical propagation model of electromagnetic fields. Our analysis incorporates fully quantum fluctuations contributed by both the input field and the loss mechanism, which are not taken into account in classical propagation equations. Hence, nonlinear phase noise discussed here appears even without signal amplification built into the optical link. Furthermore, when considering accessible information, we assumed that the reversible effects of the nonlinear propagation have been compensated by applying an appropriate unitary transformation at the channel output. Our quantum mechanical model can be extended to include amplifier noise using methods developed in [2].

## APPENDIX A: SOLUTION OF THE MASTER EQUATION

In this appendix, we present a detailed calculation of the factor ${c}_{j}(z)$ used in the ansatz in Eq. (5). With the transformed operator ${\widehat{a}}^{\prime}(z)=\mathrm{exp}(-i\mu z{\widehat{n}}^{2})\widehat{a}\text{\hspace{0.17em}}\mathrm{exp}(i\mu z{\widehat{n}}^{2})$ and the number operator $\widehat{n}$ the master equation,

Inserting the ansatz given in Eq. (5) into Eq. (A2) yields a differential equation for the factor ${c}_{j}(z)$ with $j=m-n$, which can be written as

This first-order differential equation is solved by standard separation of variables, which yields## Funding

Fundacja na rzecz Nauki Polskiej (FNP); European Regional Development Fund (ERDF).

## Acknowledgment

We acknowledge insightful discussions with Ivan H. Deutsch, R. García-Patrón, and Ch. Marquardt. This work is part of the project “Quantum Optical Communication Systems” carried out within the TEAM program of the Foundation for Polish Science co-financed by the European Union under the ERDF.

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