Principle of operation

The process at the basis of our findings is illustrated in Fig. 1a. A pair of comparatively strong continuous pump waves is launched into one end of a standard single-mode fibre [point (2) in the figure]. The two tones are of equal power P_p, and they are detuned by a frequency difference Δv ∼ 1 MHz, which can be precisely controlled. A pair of continuous optical probe tones is launched into the opposite end of the same fibre [point (1)]. The probe frequencies are separated by the same frequency difference Δv. The frequency of the upper (lower) pump tone is set to be higher than the one of the upper (lower) probe tone by the BFS v_B of the fibre [see Fig. 1b; in the figure, variations around v_B, relevant for the following discussion, are marked with Δv_B]. This choice leads to strong SBS interactions between pumps and probes, which introduce a controlled coupling between the two probe tones. The probes’ power levels are much weaker than the pumps, and the polarization states of all four waves are aligned throughout the fibre, since we consider a total length L shorter than the beat length of residual linear birefringence⁴⁴ to maintain polarization alignment. We also note that compared to other Brillouin-based anti-PT-symmetric approaches, our probe tones do not need to be spaced by the BFS^22,45, affording us great flexibility and, down the line, the freedom to add even more probes.

Fig. 1

Anti-PT-symmetric Brillouin-enhanced four-wave mixing process.

a Schematic view of the fibre platform at the basis of our experiments. All optical signals are synthesized from a single, narrow-band laser diode. Electro-optic modulation is used to prepare the input probe states, as well as the SBS pumps. Coherent detection fully recovers the output probe states. b Phase-matching diagram of the participating signals. Both resonant and off-resonant Brillouin interactions (solid and dotted black lines) take place in the fibre. Off-resonance interactions are significant since the detuning Δv is much smaller than the Brillouin linewidth. c Dependence of the probe eigenvalues σ_± on the system (ratio between frequency detuning Δv and SBS pump power P_p) and environmental (manifested through changes in the Brillouin resonance frequency Δv_B) conditions. The bold line refers to a system tuned to the SBS resonance (Δv_B = 0), highlighting the transition from anti-PT-symmetric to a broken-symmetry regime, through the EP (see also experimental results in Fig. 2). Measurements of σ_± as a function of Δv_B in the vicinity of the EP are shown in Fig. 3 below, and the potential use in Brillouin sensing applications is discussed further in the text.

Backward SBS interactions between pumps and probes generate longitudinal acoustic waves within the fibre at frequencies v_B, v_B − Δv and v_B + Δv (see Supplementary Information). The detuning Δv is much smaller than the SBS linewidth Γ_B/2π ~ 30 MHz, hence all three acoustic wave components are significant. The generated acoustic waves mediate the transfer of power from the pump to the probe tones. However, since the SBS gain is only few dB, we may safely assume that the pumps stay undepleted and their power is constant along the entire fibre.

The vector $$ \overrightarrow{A}\left(z\right)={\left[A_1\left(z\right),A_2\left(z\right)\right]}^{\mathrm{T}}$$ contains the complex amplitudes A_1,2 of the probe tones as a function of position z, and the wavenumber mismatch between the pair of probe tones is Δk = 2πnΔv/c, where c is the speed of light in vacuum and n is the effective refractive index of the optical mode. The evolution of $$ \overrightarrow{A}\left(z\right)$$ under steady-state conditions is described by $$ i\partial \overrightarrow{A}/\partial z={\mathcal{H}}_0\overrightarrow{A}$$, with the 2 × 2 anti-PT-symmetric Hamiltonian (see Supplementary Information):

Here, γ is the SBS gain coefficient in units of W⁻¹ × m⁻¹ and Δ ≡ 2(2 π · Δv)/Γ_B is the normalized detuning between probe tones. The Hamiltonian includes SBS amplification of each probe tone $$ ig\equiv i\gamma P_p$$, the coupling between probes $$ ig/2\equiv i\frac{\gamma }{2}{P}_p$$, and the total wavenumber mismatch $$ \widetilde{\mathrm{\Delta }k}\equiv \left(\mathrm{\Delta }k+\frac{\gamma }{2}{P}_p\mathrm{\Delta }\right)$$, which is the sum of Δk and the additional contribution stemming from complex-valued Brillouin gain induced by the two off-resonant acoustic waves. Equation (1) is valid under the slowly varying envelope approximation and neglects linear losses, which are much smaller than the Brillouin gain in our setup. The Hamiltonian is non-Hermitian and satisfies anti-PT symmetry^9,10,46,47, since its diagonal and off-diagonal terms obey the relations $$ \Im \mathrm{m}\left(h_{11}\right)=\Im \mathrm{m}\left(h_{22}\right)$$ and $$ h_{21}=-h_{12}^{*}$$, respectively. The imaginary coupling is due to the joint amplification or depletion of the probes through the acoustic modes, depending on their relative phases. The eigenvalues σ_± are the roots of the characteristic polynomial $$ \det \left({\mathcal{H}}_0-\sigma I\right)=0$$, where I is the identity matrix:

An EP singularity is reached when the wavenumber mismatch equals the coupling strength: $$ \widetilde{\mathrm{\Delta }k}=g/2$$. In our system, this requirement is met when pump power and frequency detuning are balanced:

In Fig. 1c, we show the real and imaginary parts of the probe eigenvalues σ_± as system parameters (ratio between frequency detuning Δv and SBS pump power P_p) and environmental conditions (manifested through changes in the Brillouin resonance frequency Δv_B) are varied, highlighting interesting topological features arising around the EP. In particular, we observe two interweaving Riemann surfaces centred around the EP and, when the system is tuned to the SBS resonance (Δv_B = 0, bold line in the figure), the eigenvalues traverse along the cuts across the Riemann sheets (black curves) as the ratio $$ \mathrm{\Delta }\nu /P_p$$ is varied. Adiabatically changing these parameters provides exotic topological features, which we explore and discuss in the following sections.

The BE-FWM process generally generates additional spectral sidebands of the input probe tones at frequency intervals of Δv^48,49, which grow as cascaded SBS processes accumulate^48,49. Hence, a complete description of the probe propagation would require considering additional tones in (1), which makes the system higher-dimensional. However, for our system parameters and the considered fibre length, the key predictions of our simplified 2D model hold very well, as seen by our experimental demonstration of EP singularity and phase transition presented in the next section.

Experimental demonstration of EP singularity and symmetry breaking

Figure 2 presents the real and imaginary parts of the two eigenvalues σ_± as a function of Δv, with P_p = 27 dBm, comparing their dispersion predicted by Eq. (2) (solid lines) with the retrieved data from measurements (dots). The measurement setup and protocols are described in detail in Methods, and the mean value $$ \overline{\sigma }=\left({\sigma }_{+}+{\sigma }_{-}\right)/2$$ has been subtracted for convenience. For large values of detuning $$ \mathrm{\Delta }\nu >\mathrm{\Delta }{\nu }_{EP}$$, the two eigenvalues have the same imaginary part, whereas their real parts differ. As the detuning is reduced, they coalesce at an EP singularity at Δv_EP = 1.145 MHz. Below the EP, the two eigenvalues acquire different imaginary parts, manifesting a phase transition and broken symmetry. The measured splitting of the eigenvalues agrees very well with our analytical predictions, with discrepancies in the order of 0.001 m⁻¹ that may be attributed to various issues: residual imbalance between power levels of the two pump tones, the onset of cascaded SBS processes, small-scale polarization variations affecting the Brillouin gain and coupling strength, and temperature instabilities modifying the BFS. The eigenvalues exhibit drastically enhanced sensitivity to variations in Δv in the EP proximity: the difference between imaginary (real) parts of σ_± scales with $$ \sqrt{∣\mathrm{\Delta }\nu -\mathrm{\Delta }{\nu }_{EP}∣}$$ when Δv approaches Δv_EP from below (above), in agreement with the predictions in other photonic platforms. Further away from the EP, the splitting of eigenvalues approaches the linear dependence $$ ∣\mathrm{\Delta }\nu -\mathrm{\Delta }{\nu }_{EP}∣$$, as expected. The onset of the EP, the phase transition and square-root-like dispersion around the EP are all remarkably retrieved in our experiments. Compared to previous experiments in other platforms, the simplicity of our layout and the robust control enabled by SBS processes allow us to resolve the dynamics around the EP with very sharp resolution.

Fig. 2

EP singularity and anti-PT-symmetry breaking.

Measured (dots) and theoretical (solid lines) ℜe (left) and ℑm (right) parts of the eigenvalues of the system. The means of the two eigenvalues have been subtracted. Each point is an average of four measurements and the error bars denote one standard deviation. The phase transition and square- root-like dispersion around the EP are evident. Insets show results over a larger, more coarse scanning range Δv, to highlight the transition towards linear dependence away from the EP.

Enhanced sensitivity to changes in Brillouin frequency shift

The enhanced response to perturbations of the system parameters in the vicinity of the EP, as observed in Fig. 2, can also help detect small changes in the BFS v_B. Consider a frequency offset between the pairs of pump and probe tones that is modified from precisely v_B to $$ {\nu }_B+\mathrm{\Delta }{\nu }_B$$ (see also Fig. 1b), where Δv_B is on the order of a few MHz or less. When the normalized detuning is sufficiently small, $$ {\mathrm{\Delta }}_B\equiv 2\left(2\pi \cdot \mathrm{\Delta }{\nu }_B\right)/{\mathrm{\Gamma }}_B\ll 1$$, the Hamiltonian of the modified system can be approximated as (see Supplementary Information):

For small $$ {\mathrm{\Delta }}_B\ne 0$$, the EP singularity cannot be reached using pump waves of equal powers, but when the pump power P_p and probe pair detuning Δv satisfy the EP condition of the unperturbed system (3), the difference between the two eigenvalues is proportional to $$ \sqrt{{\mathrm{\Delta }}_B}$$:

Therefore, the system in the vicinity of its EP is extremely sensitive not only to small changes in Δv, but also to changes in the BFS Δv_B, with similar square-root dependence. The splitting also scales with the Brillouin gain coefficient, in similarity to the SBS gain in uncoupled interactions. Changes to the BFS would be easier to observe in systems with larger Brillouin gain, such as highly nonlinear waveguides^26,27,30,48. Even greater sensitivity may be obtained with higher-order EPs, arising when considering a larger number of probe tones within the same experimental platform.

Figure 3 shows our observation of enhanced sensitivity to BFS variations in the vicinity of the EP, for a pump power P_p = 28 dBm. The system eigenvalues σ_± were retrieved for different values of Δv and of the frequency separation between pump and probe pairs, yielding a maximum sensitivity for v_B = 10.7583 GHz ± 50 kHz and probe detuning Δv_EP = 1.31±0.01 MHz. The EP shifts up relative to the value obtained in the experiment of Fig. 2 due to the higher pump power used in the current experiment. The residual uncertainty in the estimate of v_B corresponds to variations in temperature of ±0.05 degrees Kelvin³⁵. The uncontrolled laboratory temperature may well have changed over a comparable range during the data acquisition time of about 5 minutes, hence even smaller changes in BFS may be observed in a thermally stable environment. Likewise, a reduction of experimental uncertainty in Δv_EP below 10 kHz would require stabilization of the SBS gain within 0.1%. This condition is difficultly met in the open-loop operation of our platform. The observed precision of BFS measurements shown here therefore represents a lower bound on the attainable sensitivity in better controlled environments.

Fig. 3

Enhanced sensitivity of the eigenvalues to deviations from the BFS at the EP.

a Measured (left) and calculated (right) ℜe of the eigenvalues σ_± as a function of small offsets from the BFS. The frequency detuning between the two probe tones was adjusted above the EP (green, anti-PT-symmetric regime), at the EP (red), or below the EP (blue, broken-symmetry regime), see legends. The real parts of the eigenvalues repel each other at the anti-PT-symmetric regime and coalesce at the Brillouin resonance when the symmetry is broken. b Measured changes in the splitting of eigenvalues as a function of normalized detuning from the BFS: ℜe$$ \left[\mathrm{\Delta }\sigma \left({\mathrm{\Delta }}_B\right)-\mathrm{\Delta }\sigma \left(0\right)\right]$$ (logarithmic scale). The colours follow the same legend as in (a). The response is the largest when Δv approaches Δv_EP (red dots). The logarithmic slope at the EP is smaller than the slopes obtained for Δv below (blue dots) or above (green dots) the EP. The solid and dotted black lines mark the square root and linear dependence of the splitting on Δ_B/2, respectively.

Figure 3a shows measured (left) and calculated (right) real parts of σ_± as a function of a small offset Δv_B from the estimated BFS at three different settings of probe pair detuning Δv. In all traces, the minimum separation ℜe$$ \left(\mathrm{\Delta }\sigma \right)$$ is observed near Δv_B = 0. In the anti-PT-symmetric regime ($$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$ + 90 kHz, green trace), we observe an avoided crossing for the real parts of the two eigenvalues with a nonzero separation even with Δv_B = 0 (see also Fig. 1c). In the broken-symmetry regime ($$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$ – 80 kHz, blue trace), and at the EP ($$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$, red trace), the real parts of the two eigenvalues do coalesce at Δv_B = 0. The slopes of the eigenvalues at the crossing point are higher when the system is at the EP, highlighting the enhanced sensitivity to Δv_B. This property is further illustrated in Fig. 3b, which shows on a logarithmic scale the effect of Δv_B on the splitting ℜe$$ \left[\mathrm{\Delta }\sigma \left(\mathrm{\Delta }{\nu }_B\right)-\mathrm{\Delta }\sigma \left(0\right)\right]$$. The experimental uncertainty in the eigenvalues splitting is in the order of 0.0015 m⁻¹, hence smaller differences can be disregarded. The measured slope of the trace at $$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$ is smaller than the slopes obtained for Δv below or above the EP, as expected. Enhanced response to small Δv_B in the EP proximity was also found for the imaginary parts of the eigenvalues. However, a complete quantitative description of the imaginary parts of σ_± on such a fine scale requires that our model is extended to account for BE-FWM generation of additional probe sidebands, beyond the zeroth-order sidebands considered in our analysis.

Adiabatic encircling of the EP and its topological nature

Deviations from the BFS also allow exploring the topological features associated with the EP supported by our platform. Figure 4a shows the calculated Riemann sheets of our Hamiltonian as a function of the relative probe pair detuning $$ 2\widetilde{\mathrm{\Delta }k}/g$$ and offset Δ_B from the BFS. The surfaces representing the imaginary parts of the two eigenvalues intersect each other in the plane Δ_B = 0, whereas the real parts avoid crossing. This difference between real and imaginary parts is introduced by the offset Δ_B. As seen in Eq. (4), the perturbation due to Δ_B introduces real-valued coupling to the modified system Hamiltonian $$ \mathcal{H}$$, whereas the coupling term between the tones in $$ {\mathcal{H}}_0$$ is purely imaginary. As a result, the real part of the splitting between the roots of $$ \det \left(\mathcal{H}-\sigma I\right)=0$$ [Eq. (5)] is an even function of Δ_B, while the imaginary part is odd. In addition, the imaginary part has a discontinuity at $$ \left\{\widetilde{\mathrm{\Delta }k}<g/2,{\mathrm{\Delta }}_B=0\right\}$$, but the real parts are equal on this line. Hence, for the eigenvalues to continuously change across this line, they have to switch their values (red surface to blue surface). These considerations do not apply for $$ \left\{\widetilde{\mathrm{\Delta }k}>g/2,{\mathrm{\Delta }}_B=0\right\}$$, so for one turn around the EP in parameter space of Fig. 4a we expect one swap of the eigenvalues (see the rectangular-spring-shaped black curves in the figure).

Fig. 4

Topological encircling of EP by detuning from the BFS.

a ℜe (left) and ℑm (right) parts of the eigenvalues of the system σ₊ (red) and σ₋ (blue), as a function of the offset Δ_B from the BFS and the ratio between detuning and coupling $$ 2\widetilde{\mathrm{\Delta }k}/g$$. The means of the two eigenvalues have been subtracted for convenience. The solid black line denotes the clockwise trajectory of σ₊, showing the swap with σ_- after one round trip. b Traces of real and imaginary parts of the experimentally measured eigenvalues as the frequency difference between probes Δv and offset of the probes from the Brillouin resonance Δv_B are varied around the EP. The solid and dotted black lines denote the clockwise trajectories of σ₊ and σ₋, respectively. The eigenvalues swap after one round trip.

This observation is consistent with the topological nature of EPs, which manifests itself in the swapping of eigenmodes under an adiabatic encircling of the EP in parameter space^39–42. This encircling can be achieved in our system by varying Δv and Δv_B. From our experimentally recovered eigenvalues, we clearly observe this topological feature in Fig. 4b. As we track the eigenvalues around the EP, we are able to experimentally observe the swap. This observation confirms that robust topological modal transfer may be possible in our fibre platform. Note that in our setup we cannot continuously change the parameters under a single excitation, as in ref. ^41,42, but we retrieve the eigenvalues as we modify the driving conditions. We therefore do not directly observe handedness-dependent energy transfer, but, as evident in Fig. 4, our system nonetheless provides all the knobs to observe such a phenomenon. For instance, by introducing variations of backward SBS pumps as a function of z along the fibre, probe pulses may find different pump conditions as they propagate, enabling a direct observation of topological modal transfer.

Narrowing of eigenmode projections near the EP

Further post-processing of output probe states reveals an additional manifestation of degeneracies in non-Hermitian operators. Enhanced response to small-scale deviations from the BFS may be observed in the eigenmode projections around the EP. In the anti-PT-symmetric region ($$ \mathrm{\Delta }\nu \gg \mathrm{\Delta }{\nu }_{EP}$$), the output projection is Lorentzian with linewidth $$ {\Gamma }_B$$. As we approach and cross the EP, however, the spectral shape changes dramatically. Figure 5 shows measurements and calculations of the output probe vector $$ \overrightarrow{A}\left(L\right)$$, projected onto the basis spanned by the eigenmodes of the system (see Supplementary Information). We considered three distinct realizations, with $$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$ + 90 kHz, $$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$ and $$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$ - 80 kHz, as above. In each of them, the detuning Δ_B was scanned within the range ±0.23 (Δv_B between ±3.5 MHz), and the input probe vector was scanned over N = 18 states of the form $$ \overrightarrow{A}\left(0\right)={\left[\begin{array}{cc}\hfill 1\hfill & \hfill \exp \left(j{\phi }_m\right)\hfill \end{array}\right]}^{\mathrm{T}}$$, $$ {\phi }_m=2\pi m/N,$$ $$ m=1\dots N$$. All input states were chosen with equal magnitudes of the two tones to reduce the detrimental effects of residual nonlinearities in the optical modulators. This choice also addresses constructive and destructive interference between the two tones. The figure shows the squared moduli of the two output probe wave components $$ {∣A_{1,2}\left(L\right)∣}^2$$ in the eigenvector basis of $$ \mathcal{H}\left(\mathrm{\Delta }\nu ,{\mathrm{\Delta }}_B\right)$$, as a function of $$ {\phi }_m$$ and Δ_B.

Fig. 5

Enhanced response to small-scale variations from the BFS in eigenmode projections around the EP.

a Measured and b calculated squared moduli of the output probe wave projections as a function of the relative phase $$ {\phi }_m$$ between the input tones and Δ_B, in the eigenmode basis of $$ \mathcal{H}\left(\mathrm{\Delta }\nu ,{\mathrm{\Delta }}_B\right)$$. The top (bottom) row presents projection on the first (second) eigenmode. The frequency detuning between the two probe tones Δv was fixed above the EP (right, anti-PT-symmetric regime), at the EP (centre), or below the EP (left, broken-symmetry regime). The two output projections exhibit significant narrowing when the system approaches the EP from the anti-PT-symmetric regime. Past the EP, asymmetry appears between the two output projections with respect to Δ_B, with discontinuity at the BFS corresponding to a branch-cut in the Riemann sheets.

In the anti-PT-symmetric regime ($$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$ + 90 kHz), the projection amplitudes are enhanced with a linewidth reduced to 4.4 MHz due to the non-orthogonality of the eigenmodes. Unlike the uncoupled case, the projections become strongly dependent on the choice of $$ {\phi }_m$$ and may vary by two orders of magnitude, based on the orientation of the output state with respect to the eigenvectors. When the system operates at the EP (with an experimental uncertainty of ±10 kHz as discussed above), the output projections exhibit even more pronounced spectral squeezing, with the enhancement of projection amplitudes narrowing to a Brillouin detuning of 400 kHz. Furthermore, the projections vary by three orders of magnitude among different input phases $$ {\phi }_m$$. Below the EP ($$ \mathrm{\Delta }\nu =\mathrm{\Delta }{\nu }_{EP}$$ − 80 kHz), asymmetry appears between the two output projections with respect to Δ_B, with discontinuity at the BFS. These observations are well supported by the predictions of our model and confirm the EP singularity, providing evidence of a phase transition in the eigenmode spectra and not only in the spectra of eigenvalues. Anyhow, it should be stressed that the narrowing of eigenvector basis shown here may not necessarily lead to dramatic benefits in sensing operations, since the narrowing of the eigenvector basis makes also the measurement more sensitive to noise, partially offsetting the potential advantages highlighted in Fig. 5.

Experimental setup and procedures

The experimental setup is illustrated in Fig. 6. All optical signals were drawn from a single narrow-band laser diode of frequency v₀ in the 1550 nm wavelength range. In the input probe path, the laser light passed through a single-sideband, suppressed-carrier electro-optic modulator (SSB-SC EOM) that was driven by the output voltage of an arbitrary waveform generator (AWG). The modulation voltage consisted of two sine waves of radio frequencies $$ f_{IF}$$ ∓ $$ \mathrm{\Delta }\nu /2$$, where f_IF = 5.332 GHz is a fixed intermediate frequency. The SSB-SC EOM was adjusted to transmit the lower modulation sideband only, so that the optical frequencies of the two probe tones were $$ {\nu }_0-f_{IF}\pm \mathrm{\Delta }\nu /2$$. The magnitudes and phases of the two input probe tones could be controlled independently by the AWG. The optical power level of each probe wave component was 3 dBm. The probe tones were launched into one end of a standard single-mode fibre under test of length L = 10 m.

Fig. 6

Simplified schematic illustration of the experimental setup.

S/DSB-SC EOM: single/double-sideband suppressed-carrier electro-optic modulator, EDFA: erbium-doped fibre amplifier, BPR: balanced photoreceiver, OLO: optical local oscillator.

The two pump wave components were generated in two stages. Light in the pump branch passed first through a double-sideband suppressed-carrier electro-optic modulator (DSB-SC EOM), driven by a sine-wave voltage of frequency $$ \mathrm{\Delta }\nu /2$$. The modulated waveform consisted of two tones of equal amplitudes and phases, at optical frequencies $$ {\nu }_0\pm \mathrm{\Delta }\nu /2$$. The pump waves then passed through a second SSB-SC-EOM, driven by a sine-wave voltage of frequency $$ {\nu }_B-f_{IF}$$. The BFS v_B in the fibre under test was 10.762 GHz. That modulator was biased to transmit the upper modulation sideband only. The optical frequencies of the two pump waves were therefore upshifted to $$ {\nu }_0+{\nu }_B-f_{IF}\pm \mathrm{\Delta }\nu /2$$. The difference between the optical frequency of the upper (lower) pump wave and that of the upper (lower) probe wave equalled the BFS v_B. The pump wave was amplified by an erbium-doped fibre amplifier and launched into the opposite end of the fibre under test through a fibre-optic circulator. The power of each pump tone was varied between 27 and 28 dBm.

Following the BE-FWM process, the output probe wave was mixed with an optical local oscillator (OLO) in a coherent photoreceiver. The OLO was drawn from the same laser diode source, and down-shifted in frequency by an offset f_LO = 5.181 GHz in another DSB-SC EOM (see Fig. 6). The polarization of the OLO was aligned for maximum interference with the output probe. The mixed signal at the coherent receiver output was detected by a balanced photo-detector of 1.6 GHz bandwidth. The coherent detection recovered the complex envelope of the output probe wave: amplitude and phase⁷¹. The output voltage of the receiver was sampled by a real-time digitizing oscilloscope at 2 giga-samples per second rate, and the Fourier transform of the collected trace was calculated digitally. The Fourier components at frequencies $$ f_{IF}-f_{LO}\mp \mathrm{\Delta }\nu /2$$ were proportional to the complex envelopes $$ A_{1,2}\left(L\right)$$ of the two probe tones at the output of the fibre under test. All waveform generators and the sampling oscilloscope were synchronized to a common 10 MHz reference clock signal. The relative radio-frequency phases between the waveform sources and sampling oscilloscope were calibrated and adjusted.

Data analysis protocol

The transfer of the probe waves through the Brillouin-amplified fibre under test was characterized using the following protocol. First, the pump power P_p and probe tones detuning Δv were chosen and fixed. Next, the input probe vector $$ \overrightarrow{A}\left(0\right)$$ was scanned through N known states, between 8 and 18, depending on the specific experiment. The output probe vector $$ \overrightarrow{A}\left(L\right)$$ was recorded for each input state as explained above. The 2 × 2 transfer matrix $$ \overline{\overline{M}}\equiv \left[{\mu }_{11}{\mu }_{12};{\mu }_{21}{\mu }_{22}\right]$$ between z = 0 and z = L was estimated based on a least-squares solution to the following system of equations:

The eigenvalues and eigenvectors of the system are not observed directly on the oscilloscope. They were retrieved by first bringing the matrix $$ \overline{\overline{M}}$$ to a diagonal form $$ \overline{\overline{D}}$$. The two eigenvalues of $$ \overline{\overline{D}}$$ are related to those of $$ \mathcal{H}$$ by $$ \exp \left(-i{\sigma }_{\pm }L\right)$$, assuming the fibre is uniform. The procedure was repeated for different settings of $$ \left\{\mathrm{\Delta }\nu ,\mathrm{\Delta }{\nu }_B,P_p\right\}$$.