Statistical parity-time-symmetric lasing in an optical fibre network

Parity-time (PT)-symmetry in optics is a condition whereby the real and imaginary parts of the refractive index across a photonic structure are deliberately balanced. This balance can lead to interesting optical phenomena, such as unidirectional invisibility, loss-induced lasing, single-mode lasing from multimode resonators, and non-reciprocal effects in conjunction with nonlinearities. Because PT-symmetry has been thought of as fragile, experimental realisations to date have been usually restricted to on-chip micro-devices. Here, we demonstrate that certain features of PT-symmetry are sufficiently robust to survive the statistical fluctuations associated with a macroscopic optical cavity. We examine the lasing dynamics in optical fibre-based coupled cavities more than a kilometre in length with balanced gain and loss. Although fluctuations can detune the cavity by more than the free spectral range, the behaviour of the lasing threshold and the laser power is that expected from a PT-stable system. Furthermore, we observe a statistical symmetry breaking upon varying the cavity loss.


Supplementary Figure 1 | Structured lasing cavity configuration.
We depict a schematic of the structured cavity as a two-port system. The mirror M1, M2 and M3 have the reflectivities , 2 and , respectively. The incoming amplitudes from the left and the right are L and R , respectively. The outgoing amplitudes to the left and the right are L and R , respectively.
Consider a structured optical cavity comprising two coupled subcavities that contain gain and loss sections, as shown schematically in Supplementary Figure 1. The mirrors used are assumed lossless but are not necessarily symmetric, so that they may be represented in general by a scattering matrix of the form [3] Here and are the (real) transmission and reflection coefficients, respectively, 2 + 2 = 1, and are the transmission and reflection phases for incidence from the left, respectively, and and 2 − are the transmission and reflection phases for incidence from the right, respectively. 1 The scattering matrices act on incoming fields � L R � to produce the outgoing fields � R L �. The reflection and transmission coefficients for the optical power are = 2 and = 2 , respectively. To satisfy the PT-symmetry condition, M 1 and M 3 must have mirror-symmetry with respect to the centre of the structure. If we take M 1 to be represented by the -matrix in Eq. S1.1, then this symmetry requirement dictates that the -matrix for M 3 has the form: Furthermore, the mirror M 2 must be symmetric, which adds the constraint to its -matrix that = − /2, such that its S-matrix has the simple form: Exploiting the above-described scattering matrices, we obtain the corresponding transfer matrices that act on the forward and backward fields � R R � on the right to produce the corresponding fields � L L � on the left. The transfer matrices 1 , 2 , and 3 representing the mirrors are given by The parameters in these transfer matrices are not subject to statistical fluctuations, except perhaps the phases if the mirrors are implemented by extended fibre Bragg gratings (FBGs).
The transfer matrices for the intervening gain and loss layers are given by where and ℒ are single-pass amplification and attenuation factors for the power traversing the gain and loss layers, respectively, and g and ℓ are the single-pass propagation phases in the gain or loss layers, respectively. In the exact PT-symmetric configuration, ℒ = 1 and g = ℓ . However, statistical fluctuations in a macroscopic cavity precludes the realization of the latter condition.
The total transfer matrix for the whole cavity is thus The cavity transmission, which shares the same poles as the field transmission coefficient, Therefore the cavity transmission is determined by the mirror reflectivities and 2 and the gain and the loss ℒ, and (3) two phases G and ℒ :

Lasing thresholds
Since the cavity length cavity is very large with respect to the optical wavelength , cavity ≫ , the single-pass phases may not be deterministic due to minute thermal or mechanical fluctuations in the optical fibres used. We thus assume that the cavity single-pass phases are random each with a uniform probability distribution in the interval from 0 to 2 . Consequently, lasing may occur at any wavelength within the gain and loss bandwidth. To obtain the lasing threshold, we hold fixed the physical parameters related to the optical loss and mirror reflectivities, while varying the gain. After obtaining the lasing gain threshold, we can find the special case for the PT-symmetric configuration by choosing the attenuation to satisfy ℒ = 1.
At any value of gain, lasing is initiated whenever the cavity phases produce a zero in 11 (a transmission pole). We therefore minimize 11 with respect to the phases, and then extract the zeros to find the gain corresponding to the lasing threshold. We thus first set 11 G = 11 ℒ = 0, from which we find a minimal value of 11 when G = 0 and ℒ = , (13) 3 From this minimal value of 11 , we obtain the zero at a threshold gain th given by This general expression for the gain threshold is the minimum required gain for lasing to occur, while all other parameters are fixed except for the phases that are assumed to vary randomly.
We can now obtain expressions for the gain threshold when special structures are of interest. First, for a pseudo-PT-symmetric structure with ℒ = 1, the following threshold holds for lasing: which was used in the main text (Eq. 1). Second, when we eliminate the attenuation in the loss subcavity ℒ = 1, the gain threshold 0 has the form

Supplementary Note 2. Non-Hermitian temporal coupled-mode equations
We now introduce the temporal coupled-mode equations used as a basis for the analysis presented in the main text. We consider an optical-fibre-based laser cavity, comprising two coupled subcavities (one including net gain and the other net loss) of length each and group velocity . We introduce time-dependent quantities to characterize the sub-cavities: a linear background loss per second 1 for the field in the gain sub-cavity (which only depends on the side mirror M 1 of reflectivity ) and a corresponding quantity 2 for the loss sub-cavity (which depends on the side mirror M 3 of reflectivity in addition to the imposed attenuation). Furthermore, we define a temporal small-signal gain for the gain sub-cavity, which is dictated by the SOA amplification. These temporal losses and the small-signal gain are given by where and Γ are the single-pass intensity amplification and attenuation factors that are set by the SOA and the VOA, respectively.
Based on these considerations, we define two temporal coupled-mode equations for the mean-field amplitudes � and � in the amplifying and attenuating fibre sub-cavities, respectively, which adequately capture all the essential features of the optical field dynamics in the coupled fibre-cavity system: We have introduced into this model the nonlinear gain-saturation, the loss mechanisms, and the coupling between the sub-cavities, where is the gain-saturation intensity and Δ is used to phenomenologically introduce detuning between the resonances of the sub-cavities. For simplicity, we normalize the field amplitudes with respect to the saturation value by introducing the scaled amplitudes � �, � � = � ( , ), whereupon we obtain the coupled-mode equations:

Supplementary Note 3. Extended Lamb model for coupled cavities
In traditional configurations of coupled cavities, such as evanescently coupled micro-ring resonators [4], one considers time-averaged energy amplitudes in each cavity. The temporal coupling between the cavities is dependent on the effective interaction region between them. In our setup for coupled fibre cavities, their interaction is mediated by the middle mirror M 2 , which is a lumped element ( Fig. 1 and Supplementary Fig. 2). The usual approach of extracting a distributed effective coupling is thus no longer valid. A more pertinent method was employed by W. E. Lamb [5,6] in treating coupled-laser configurations, in which the central mirror coupling two sub-cavities ( Supplementary Fig. 2a) was modelled as a bump of width in the permittivity distribution in space along the cavity axis (Fig. S2b). The permittivity of the so-called bump is defined as: where 0 is the refractive index of the background medium. Lamb's well-known model thus connects the sought-after coupling coefficient (Eqs. 2-3 in the main text and Supplementary Note 2 and Supplementary Notes 4-6 below) to the bump parameter : according to Eq. 18 of reference [4]; here is the length of a single sub-cavity, = 0 2 ⁄ , and g is the group velocity of light in the fibres. In our experiment, the bump represents the mirror M 2 of reflectivity 2 . By establishing a relationship between the bump parameter and the reflectivity 2 , we can then express in terms of 2 . This connection enables us to simulate the dynamics of our structure via coupled-mode theory after employing the proper coupling coefficient.
According to the Supplementary Fig. 2b, the discontinuity in the refractive index ( = � / ) results in Fresnel reflection at each interface, int : Here the subscript 'int' indicates interface and the approximation makes use of the fact that = � / grows in the limit → 0. The two interfaces of the bump define a Fabry-Pérot resonator with a reflection of tot ( Supplementary Fig. 2b): Taking the limit → 0 leads to the approximation sin( / ) = sin( � / ) ≅ � / , and substituting int from Supplementary Equation 21 into Supplementary Equation 24, so that the total reflection of the bump is: We associate the reflectivity of the mirror 2 with that of the bump-based model tot , tot → 2 . Therefore can now be expressed in terms of 2 , According to Lamb's model the coupling rate is related to (and thus to 2 ) via Coupled-mode theory simulations that make use of defined in this last expression do not quite agree with our measurements. The reason is that the Lamb model assumes that the outcoupling windows M 1 and M 3 are perfect, which is not the case in our configuration. In another seminal paper, W. E. Lamb considered the case of partially transmitting windows that radiate into free space [6]. The analysis revealed that for windows with a high enough reflectivity (as is the case in our experiment), only a small leakage and a change in the cavity free-spectral-range (FSR) are effectively introduced -whereas the modal profiles are left unchanged. The change in the FSR means that the effective length of the cavity is now dependent upon the out-coupling mirror reflectivity. We model this effect in the expression for the coupling between the two sub-cavities of the PT-laser as follows: This is the definition of used in our coupled-mode theory simulations throughout. The value of obtained as such, when used to find the lasing threshold th based on Equations (2-3) of the main text, agrees well with the threshold found based on a transfer matrix analysis as provided in section Supplementary Note 1. When one considers the scenario of Γ = 1, 2 is always greater than , in fact 2 = 1 + -see Supplementary Equations 19 and 20. Now the condition for lasing in an unbroken mode is given by, > ( 1 + 2 ) which can never be achieved (consider the imaginary part of Supplementary Equation 59). However, the lasing threshold in the broken mode might be surpassed. In this case when one enters the broken symmetry regime where the square root in Supplementary Equation 59 turns complex, the threshold can be obtained from the requirement: 1 − � � � 2 − 1 < 0. This then leads to, Translating this into a single-pass gain value using Eq. S2.3, one obtains, The value of this threshold is compared to (according to Supplementary Equation 11) for various values of 2 in the figure below. A good agreement between the two is apparent. Figure S3 | Comparison of the lasing thresholds for the two presented models. PT-symmetric lasing lasing thresholds are obtained from the transfer matrix method and the extended Lamb model, and are plotted as a function of 2 with both out-coupling mirror reflectivities assumed to be = 80%.

Supplementary Note 4. Temporal coupled-mode solutions in absence of detuning
We first obtain solutions to the coupled-mode equations (Supplementary Equations 23-24) in absence of detuning, Δ = 0 to obtain the coupled-mode equations: We seek a harmonic solution of the form � where, as before, s = 1+| 0 | 2 for the saturated gain, and we assume that | 0 |, | 0 | ≠ 0. Multiplying Supplementary Equations 37 and Supplementary Equations 37, we obtain a quadratic equation in , This quadratic equation has a real solution for only when 1 + 2 = s , whereupon and the solutions for take the simple form: which applies when 2 ≤ . By defining 2 = sin , we obtain a concise expression for the eigenvalues: 1,2 = ± cos .
To obtain the complex ratio between 0 and 0 , we define 0 = 0 and subtract Supplementary Equation 38 from Supplementary Equation 37, which leads to 2 sin = cos + sin − 1 cos + 1 sin .
We now have a pair of conditions = 1 → = and = −1 → = − . Hence, the steadystate solution is given by where is obtained from sin = 2 / . We therefore obtain a pure unbroken PT-symmetric phase when there is no detuning, that is valid when 2 ≤ and > 1 + 2 . The latter restriction is inferred from Supplementary Equation The solution within this regime is thus given by, which corresponds to a PT-symmetry-broken phase solution that is valid when 2 > and > (II) Broken: when > 1 + 2 / 2 and < 2 → � � = �

Supplementary Note 5. Proof for the absence of a pure unbroken-symmetry mode in presence of detuning
In the presence of detuning Δ ≠ 0, it can be shown that a formal unbroken PT-symmetric phase (that is, one with equal intensities of harmonic fields at the two output ports | 0 | = | 0 |) cannot be realized. This fact can be deduced from Supplementary Equation 23-24 where a harmonic solution � � = � 0 0 � with ∈ results in the following equations: where s = /(1 + | 0 | 2 ). From Supplementary Equation 52, we obtain the ratio between 0 and where = + Δ 2 . Substituting in Supplementary Equation 51, and assuming that 0 ≠ 0, we obtain The imaginary part of this equation implies that Hence for a non-zero detuning, the exact unbroken PT phase in the two coupled sub-cavities never appears.

Supplementary Note 6. Linear analysis in the presence of detuning
Before the onset of lasing in the coupled-cavity structure, a linear analysis of the field dynamics can be carried out because the intensities are small. Under these conditions, the dynamical equations can be cast in the form: Detuning leads to an avoided eigenvalue-coalescence, as we proceed to show. First, we consider the harmonic ansatz � ( ) ( ) � = � 0 0 � in absence of detuning, i.e. Δ = 0, to obtain: whereupon a second order bifurcation takes place in the imaginary parts of the eigenvalue at = 2 + 1 − 2 by gradually increasing the gain from a small value. The system is said to enter a PT-symmetry broken regime when the amount of gain increases beyond this point. That is, the unbroken PT-symmetry regime occurs when < 1 − 2 + 2 (the eigenvalues have not undergone a bifurcation) and the broken PT-symmetry regime when > 1 − 2 + 2 (the eigenvalues have bifurcated). This transition point between broken and unbroken PT-symmetry regimes is better known as an exceptional point (EP) [7][8][9]. At this point, the two eigenvalues and associated eigenvectors ( 0 , 0 ) T coalesce: In general, lasing occurs once the sign of the imaginary part of one of the eigenvalues 1,2 becomes negative, Im{ } < 0, per our harmonic ansatz. Because the imaginary parts of the eigenvalues drop monotonically with prior to the EP (Supplementary Equation 59), the expression for the eigenvalues at the EP (Supplementary Equation S60) dictates whether lasing is initiated in the unbroken or broken PT-symmetry regimes. If Im{ } < 0 at the EP because > 2 , this indicates that lasing has already started before the EP was reached. In other words, lasing initiates here in the unbroken PT-symmetry regime. On the other hand if Im{ } > 0 at the EP because < 2 , then lasing has not started when the EP is reached. Therefore, lasing initiates after the EP in the broken PT-symmetry regime. These two scenarios of lasing occurring in the unbroken and broken PT-symmetric phases are depicted in Supplementary Fig. 4.
This picture of pure unbroken-and broken-symmetry regimes is true only for a zerodetuned system (Δ = 0). Once detuning is introduced (Δ ≠ 0), the eigenvalues do not coalesce, as shown in Supplementary Fig. 5. In other words, an exact EP does not occur. Nevertheless, two regimes are still identifiable, one in which the eigenvalues approach each other (unbroken PTsymmetry) and the other where they diverge (broken PT-symmetry). In the presence of detuning Δ ≠ 0, the eigenvalues are (61) We can infer from the expression under the square root in Supplementary Equation 61 that 1 ≠ 2 when Δ ≠ 0 because the parameters , , and are all real. Using − 1 + 2 + Δ = 2 sin , where is now a complex angle, the eigenvectors associated with these eigenvalues are Note that the EP in the zero-detuning scenario corresponds to = /2. The real parts of the two eigenvalues are plotted in Supplementary Fig. 4 and Supplementary Fig. 5 against gain in the case where 2 is kept fixed at 2 = 3, the coupling is = 5, and the loss is 1 = 1. The two eigenvalues are initially well apart -but with increasing they gradually coalesce at the exceptional point (EP) where = 2 − 2 + 1 . In contrast, the imaginary components of the eigenvalues stay together before the EP and only bifurcate afterwards. If we choose a value of 2 higher than , 2 = 7, the location of the EP now shifts to the positive imaginary half. In Supplementary Fig. 4c, the initial separation between Re{ } is now smaller whereas in Supplementary Fig. 4d, the final distance between Im{ } is now much larger compared to the previous case. Indeed, Im{ } crossing the zero value and switching sign (the green circle) heralds the onset of lasing which takes place in the unbroken symmetry mode (U) in Supplementary Fig. 4a-b while in Supplementary Fig. 4c-d it instead occurs in the broken symmetry mode (B).
Interestingly, the transition point between these behaviours is still the same value of gain as the EP, i.e. = 2 + 1 − 2 , when detuning is introduced (Supplementary Fig. 5). We conclude that, even in the presence of detuning, lasing can be either initiated in an unbroken PTlike mode if the loss in the attenuating component of the coupled cavity configuration is less than the effective coupling, i.e. 2 < ; or in an a broken PT-like mode for the case 2 > . More importantly, the two PT-symmetric phases have a physical significance in the following sense: Unbroken symmetry implies that the fields in both components of the system behave in a similar fashion, i.e. with increasing gain (loss), | | 2 and | | 2 increase (decrease) in synchrony. However, in the broken symmetry domain, a disparity exists between the behaviours of the fields in the two components. Specifically, the ratio between the field intensity in the amplifying and lossy components | | 2 /| | 2 starts to increase with gain. The bifurcation in the imaginary parts of the eigenvalues after the EP indicates this, as shown in Supplementary Fig. 5.
The scope of the linear analysis presented in this section is not just limited to determining the lasing conditions, but also has important ramifications on the full nonlinear response of the system. The two distinct regimes of unbroken and broken PT-symmetry still manifest themselves in the presence of random frequency detuning and gain saturation. Remarkably, the transition between the two still arises at the point where the loss in the attenuating cavity equals the coupling strength 2 = . This behaviour is depicted in Fig. 5 of the main text where instead of gain, the loss 2 is gradually increased from a low to a very high value including a passage through the point

Supplementary Note 7. Probability distribution for detuning
The term that describes the resonance detuning between the coupled subcavities (that is, Δ in Supplementary Equations 23-24) occurs because of random perturbations in the fibres that have their origin in thermal or mechanical fluctuations. Since the fibres are relatively long (> 6 m), length expansions or contractions can easily lead to changes in the path length on the order of a micrometre, which in turn can cause a phase accumulation of 2 . This would result in a frequency detuning equivalent to half the free spectral range ( FSR ) of a single fibre subcavity, as shown in In principle, Δ can span the entire range − FSR /2 ≤ Δ ≤ FSR /2. Two possible probability distributions for Δ are the uniform (Fig. S7a) and Gaussian (Fig. S7b) distributions. Because the detuning-phase-accumulation spans the range (−∞, ∞) in the Gaussian distribution, we fold the probability distribution function after each period of equivalent to FSR /2.

Supplementary
In Supplementary Equation 63, a,b (ss) (Δ) is the steady-state (ss) intensity value obtained from a Runge-Kutta simulation for a given set of parameters ( 1 , 2 , , , Δ), i.e. while Δ is considered deterministic. It is only after we obtain a,b (ss) (Δ) for all values of Δ, that the averaging is carried out according to Supplementary Equation 63. For computational convenience, this integral can also be approximated by a summation after considering a finite number of sampling points for Δ in the interval [− FSR /2, FSR /2]. Upon analyzing the numerical results and the experimental data, we found the Gaussian distribution to be a significantly better match when compared with the uniform distribution. Moreover, the standard deviation was also found to be small, i.e. ~ 0.1 FSR . The main text shows results that correspond only to the Gaussian distribution, e.g. Fig. 5. A figure similar to that when one considers a uniform distribution for Δ, is shown in Supplementary Fig. 8. Qualitatively, the uniform distribution leads to a larger split in the gain and loss cavity intensities since the two cavities are in essence decoupled from each other most of the time. On the other hand, a phase transition around the exceptional point more clearly visible in the case of a narrow Gaussian distribution as shown in Fig. 5 of the main text.