Backreaction of stimulated Hawking radiation in an optical analogue

Backreaction of stimulated Hawking radiation in an optical analogue

Lorenzo M. Procopio, Raul Aguero-Santacruz, David Bermudez and Ulf Leonhardt.

Department of Physics, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany, and Institute for Photonic Quantum Systems (PhoQS), Paderborn University · Department of Physics of Complex Systems, Weizmann Institute of Science, 7610001 Rehovot, Israel · Department of Physics, Cinvestav, A.P. 14-740, 07000 Ciudad de Mexico, Mexico.

arXiv:2607.01118v1 [gr-qc], 1 July 2026.

Abstract

Hawking radiation — the emission of quantum particles at the event horizon of a black hole — connects gravity with quantum mechanics and thermodynamics. But Hawking radiation has never been observed in astronomy, only in laboratory analogues and the chances of ever observing it in space are astronomically small. The energy of Hawking radiation must come from the gravitational field around the black hole, but how field quanta generate Hawking quanta has been unknown. Here we report on experimental and theoretical evidence for the process that generates Hawking radiation in a fibre–optical analogue of the event horizon. There, as in gravity, it has been believed that Hawking radiation comes from a complicated, cascaded process; here we have identified theoretically a simple, direct process and observed experimentally how this process reacts back onto the field. Our findings suggest an equally direct process for other laboratory analogues and perhaps also for gravitational fields, shedding light on how black holes might radiate.

Article

Laboratory analogues are based on a simple yet powerful idea: imagine waves — sound waves or light waves — in a medium moving with variable speed or having variable wave velocity. Suppose the medium moves slower than the waves. In this case they can travel freely. Suppose now the flow velocity begins to exceed the wave velocity: waves are swept along and trapped. The area where the flow speed exactly equals the wave velocity establishes the analogue of the event horizon. Note that this is not just an analogy, but an exact mathematical equivalence, as the propagation equation for waves in moving media is equivalent to the scalar wave equation in general relativity. In fibre optics, light pulses modify the speed of light in the fibre due to the Kerr effect: the refractive index n acquires a small additional contribution δn being proportional to the intensity and moving with the group velocity u of the pulse. In a frame co–moving with the pulse, the glass of the fibre appears as a medium moving with velocity −u while the speed of light c is reduced by n + δn. For the analogy with gravity we thus need to consider the co–moving frame. Let us give the mathematically simplest argument for the generation of Hawking radiation there (as illustrated in Fig. 1).

Suppose a weak probe pulse interacts with the strong pulse establishing the moving medium, called pump pulse. In fibre optics, the co–moving frame is defined by the retarded time τ = t − z/u and the propagation time ζ = z/u where z denotes the propagation distance in the fibre and t the laboratory time. The frequency ω′ in the co–moving frame is related to the frequency ω in the laboratory frame by the Doppler effect. For probe light with phase φ we obtain, from the definitions of the co–moving frequency, the laboratory frequency and the wavenumber, together with k = nω/c, the Doppler formula

ω′ = (1 − n u/c) ω. (1)

As the refractive index n depends on ω the Doppler shift varies with frequency (Fig. 2). Note that the Doppler–shifted frequency is negative when the speed of the moving medium exceeds the phase velocity of light, that is when u exceeds c/n. For gravitational black holes, Hawking radiation consists of particle pairs, one corresponding to a wave with positive frequency escaping into space and the other to a wave of negative frequency falling into the black hole (Fig. 1). In fibre optics, the Hawking quanta are pairs with frequencies ω′ and −ω′. In our case (Fig. 2) the positive–frequency Hawking waves appear in the laboratory frame in the infrared (IR, with wavelengths 1100 nm to 1860 nm) while their negative–frequency Hawking partners are in the ultraviolet (UV, around 233 nm).

Figure 1: Analogue of the event horizon. a: Schematic diagram illustrating the Hawking radiation of an astrophysical black hole. Pairs of quanta are emitted from the event horizon: quanta of positive–frequency waves (indicated in red) escape into space, while their negative–frequency partners (indicated in blue) fall into the singularity of the black hole. The energy for creating the radiation must come from the energy of the gravitational field around the black hole in a process as yet unknown. b: Fibre–optical analogue of the event horizon. A light pulse (indicated in yellow) in a fibre changes the local refractive index, acting as a moving medium in a frame co–moving with the pulse. Its front establishes the analogue of a black–hole horizon for probe light (waves) at the point (black dot) where the group velocity of the incident probe (red wave) matches the speed of the pulse. The probe incident with positive co–moving frequency stimulates a wave (red) with the same co–moving frequency and another wave (blue) with negative co–moving frequency, the analogue of the Hawking partner. Due to the Doppler effect (Fig. 2) the Hawking partner appears in the UV (233 nm wavelength, Fig. 3) for probes in the IR (1100 nm to 1600 nm). In nonlinear fibre optics all elementary processes are known in principle. We identify theoretically which ones are responsible for creating Hawking radiation and measure experimentally both the stimulated Hawking radiation and its backreaction (Fig. 4).

Let us derive the dynamics of the light in the co–moving frame. We describe the optical field by the dimensionless amplitudes A1 for the pump and A2 for the probe. Pump and probe are distinct light fields of different frequency range and possibly different polarization. In our case, they are separated in frequency ω by about an octave but share the same polarization (for maximizing their interaction). Classical fields consist of the analytic signals, with the pump and probe amplitudes each written as one half of the analytic signal plus its complex conjugate, where the analytic signals oscillate with positive laboratory frequencies. For quantum fields the analytic signals correspond to the local annihilation operator and their conjugates to its Hermitian conjugate. Writing the analytic signal in terms of quadratures we obtain from Hamilton’s equations the propagation equation

i times the derivative of the analytic signal with respect to the propagation time equals the derivative of the Hamiltonian with respect to the conjugate amplitude, (2)

where the dot denotes differentiation with respect to propagation time ζ. The Hamiltonian H has units of frequency here and it consists of three parts, the independent–propagation Hamiltonians of each field and the interaction (Methods Sec. A). In nonlinear fibre optics, the interaction is due to the Kerr effect: the interaction Hamiltonian is 16 κ times the square of the pump amplitude times the square of the probe amplitude, with coupling constant κ. Expanding the fields in terms of the analytic signals and their conjugates gives all the elementary interactions, and among them the processes described by the Hermitian Hamiltonians

HR = 2κ times the pump intensity times the sum of the squared probe amplitude and its conjugate square, and HS = 4κ times the pump intensity times the probe intensity. (3)

In the propagation equation (2) for the probe amplitude the process HS adds a contribution linear in the probe amplitude and proportional to the pump intensity. This corresponds to the shift δn in the refractive index known in fibre optics as the cross phase modulation. This shift in refractive index is essential for establishing horizons.

But the refractive–index shifting HS is not the process generating Hawking radiation: that is HR. To see this, consider a probe wave incident with co–moving frequency ω2′ and interacting with a pump wave of frequency ω1′. In Eq. (2) for the probe amplitude the Hamiltonian HR generates a term oscillating with minus ω2′. A probe wave of positive co–moving frequency ω2′ thus stimulates a wave with negative minus ω2′, the Hawking partner (Fig. 1). In the limit of zero average probe intensity, we need to replace the amplitudes by quantum amplitudes. The quantum Hamiltonian should be normally ordered, because otherwise it would create radiation even if the pump were in the vacuum state. For an intense pump reaching the classical limit, HR of Eq. (3) is a standard pair–creation Hamiltonian. It spontaneously generates an entangled two–mode squeezed state of Hawking quanta from the quantum vacuum, one with positive and the other with negative co–moving frequency. This shows that HR is responsible for creating Hawking radiation. In our experiment, we do not yet observe the fully quantum spontaneous Hawking radiation, but classical stimulated Hawking radiation. While this classical radiation shares many of the features of quantum Hawking radiation, in particular its spectrum, we would not be able to measure quantum entanglement.

Note that in the experiment HR requires a pump pulse with near–single–cycle features and hence ultrabroad spectrum, as pump and probe are not frequency matched (Methods Sec. B). For this reason, HR is normally neglected in nonlinear fibre optics. One notable exception is Negative–frequency Resonant Radiation (NRR, Fig. 2c) that was predicted and observed as the result of a similar process. The Hamiltonian HR was also neglected in the previous discussions of fibre–optical event horizons. But there it is essential: instead of a complicated, cascaded process, Hawking radiation is made directly by the simple Hamiltonian HR. Note also that the generation of Hawking radiation only becomes effective when the probe wave travels with the pump wave, i.e. when it has the same group index as the pump, including the change δn due to HS. This is the horizon condition: the optical analogue of the event horizon is established when the group velocity of the probe reaches the speed of the moving medium made by the pump pulse. Both aspects of the interaction must act in unison, but the actual pair creation is done by HR. Note also that in our case the difference between group index and phase index is an order of magnitude larger than the perturbation δn caused by the Kerr effect of the pump, which is of order 10^-3. This implies that we are in the regime of strong dispersion. For weak dispersion, Hawking radiation has a thermal spectrum with Bekenstein–Hawking temperature given by kB T = h-bar divided by 2 pi τ0, where the inverse of τ0 is the derivative of the logarithm of δn with respect to the retarded time, evaluated at the horizon. We found (Methods Sec. B) that for strong dispersion and short pump pulses τ0 is the asymptotic exponent of the Fourier transform of δn for large frequencies.

Figure 2: Doppler shift. Co–moving frequency versus laboratory–frame frequency according to the Doppler formula (1) with n(ω) measured and modeled for the fibre used and u/c fitted to match the measured Hawking frequency (Fig. 3). In the UV the Doppler–shifted co–moving frequency is negative, because there the velocity u of the effective moving medium (Fig. 1b) exceeds the phase velocity c/n(ω) of the probe. Subfigure a shows the entire Doppler curve while b and c focus on the IR and the UV regions relevant in our experiment. b: The light of the pump pulse lies in a local minimum of the Doppler curve, as the derivative of the co–moving frequency with respect to the laboratory frequency equals 1 − u/v with group velocity v and u = v for the pump, while the horizon of the probe lies at a local maximum. The probe gets red–shifted along the Doppler curve over the horizon while keeping its co–moving frequency. c: In the UV the pump pulse generates Negative–frequency Resonant Radiation with co–moving frequency equal to minus that of the pump. The probe stimulates a Hawking partner with the negative co–moving frequency equal to minus the co–moving frequency of the probe. We show that, as backreaction to the Hawking process, the pump acquires a wave whose co–moving frequency is minus the probe’s co–moving frequency minus the difference between the probe and pump co–moving frequencies, and we observe it further in the UV (Figs. 3 and 4).

We have identified the elementary process generating Hawking radiation in our optical analogue, the Hamiltonian HR that describes the transfer of energy between pump and probe, which corresponds to the energy transfer between the gravitational field and Hawking radiation. But as actio equals reactio this Hamiltonian also describes the backreaction of the Hawking radiation onto the pump. As HR is proportional to the pump intensity it does not change the total photon number of the pump field, but redistributes the original pump photons to a different frequency. Given probe and pump with co–moving frequencies ω1′ and ω2′ we obtain from Eq. (2) for the pump amplitude a term oscillating with minus ω2′ minus δ′, where δ′ = ω2′ − ω1′. As ω2′ is greater than ω1′ (Fig. 2) this co–moving frequency is more negative than minus ω2′, which implies (Fig. 2) that the corresponding laboratory frequency is further shifted to the UV (by 0.5 nm to 1 nm in wavelength). As the backreaction of Hawking radiation is a secondary process it goes with the square of the probe power.

Note that our backreaction resembles the backreaction in the Unruh effect. There, an accelerated observer receives the equivalent of Hawking radiation from the quantum vacuum in empty Minkowski space. For an observer at rest, the detection of an Unruh quantum appears as the emission of a Minkowski quantum. Multiple such events slow the acceleration down. In our case, the backreaction plays the role of the emission in Minkowski space while the eventual evaporation of the pump pulse (or the black hole) corresponds to the eventual deceleration in the Unruh effect.

In order to observe the backreaction experimentally, we used the same setup as in our previous observation of stimulated Hawking radiation, but improved the stability and the data taking such that we could increase the signal–to–noise ratio to the required level. Both pump and probe pulses are split off from the same train of ultrashort light pulses (of 8 fs duration, 800 nm carrier wavelength, 80 MHz repetition rate, from a Thorlabs Octavius laser). The probe undergoes frequency shifting in a 1 m photonic–crystal fibre (NKT Photonics NL-1.9-765) by the Raman effect, depending on its intensity that we control. For the pump we use chirped mirrors and a wedge to compensate and control its chirp. Pump and probe are recombined using broad–band optics (parabolic mirrors) and interact in a 7 mm piece of photonic–crystal fibre (NKT Photonics NL-1.5-590). From the outgoing light we remove the pump contribution by dichroics and either detect its IR or UV spectrum. In the IR we can observe the frequency shifting due to the analogue of the event horizon, in the UV we count the photons of Hawking partners with negative co–moving frequency and the photons of the presumed backreaction. We use a combination of filters to reduce the background noise. For the UV detection, we send the beam via a rotable prism and through a pinhole to a photomultiplier tube (Hamamatsu H8259) for counting the photons of the wavelength set by the angle of the prism and the pinhole. The apparatus is calibrated with a Mercury lamp. While taking data we continuously monitor the interaction between probe and pump by measuring with a photodiode the frequency shifting of the probe in the IR. A large part of the UV photons comes from Negative–frequency Resonant Radiation of the pump itself (Fig. 2c). We remove this feature by chopping the train of probe pulses and subtracting from the spectrum with pump–probe interaction the spectrum without interaction (Fig. 3). The pump is easily perturbed by probes with intensities as low as 10^-2 of the pump as the pump is primarily a soliton that reacts rigidly to perturbations. In particular, the carrier frequency gets shifted (by 10^-3) which results in a shift of the NRR we fit and correct for (Methods Sec. D). Without the correction, the difference signal may become negative; some remnants are still visible (Fig. 4).

Figure 3: Experimental data. We measure the UV spectra of the pump–probe interaction and the pump beam alone for several probe wavelengths: 1100 nm, 1200 nm, 1300 nm, 1400 nm, 1450 nm, and 1600 nm, displayed in a-f, respectively. The subfigures show the counts of light quanta per second for bins of wavelength (dots). Blue: both pump and probe are present and interact with each other. Black: the probe is chopped out such that the pump interacts only with itself, producing Negative–frequency Resonant Radiation (Fig. 2c). Red: corrected difference between the counts shown in detail and compared with theory in Fig. 4. We correct the difference by shifting the spectra for the pump by a small frequency we determine by fitting to the pump–probe spectra around and to the right of the main peaks (Methods Sec. D). Each experiment (except f) was independently repeated on another day with consistent results. Statistical errors are smaller than the size of the data points.

In the absence of a systematic study of the signal as a function of probe power that would allow one to unambiguously separate genuine Hawking backreaction from direct probe-induced effects, it remains difficult to exclude the possibility that effects beyond a simple frequency shift contribute to the observed backreaction signal. These may include probe-induced modifications of phase-matching conditions, conversion efficiency, pump dynamics, or additional nonlinear mixing. Such measurements are experimentally very demanding, requiring long acquisition times (up to 12 hours) and stability against slow parameter drifts during the whole measurement run, which are not currently achievable in our setup. Nevertheless, the removal of excess signal near the NRR after applying the spectral shift (Fig. 4), together with the emergence of a sideband structure consistent with our model, suggests that the frequency shift constitutes a significant contribution to the probe–induced backreaction effects.

Our probe wavelengths (1100 nm, 1200 nm, 1300 nm, 1400 nm, 1450 nm, 1600 nm) span the spectrum from the edge (1100 nm) of the capture range of the horizon through the spectral region where the probe is redshifted to the beginning (1600 nm) of the blueshifting. In all cases we clearly see (Methods Sec. D) the peaks of the stimulated Hawking radiation and the backreaction at the correct positions (Fig. 2). But we also see sidebands (Fig. 4, in particular 4a and 4d) with co–moving frequencies integer multiples m of δ′ away. They are caused by modulation with the beat frequency of the pump and the probe, similar to the vibrato on a musical string instrument. Normally, frequency modulation of strength χ generates a spectrum of Bessel functions behaving as χ to the power m divided by m factorial for small χ and non-negative m. They go with χ to the power m because the generation of the m–th sideband requires m excitations. As those are indistinguishable, we need to divide by the number of all permutations. In our case of extremely short pulses, we have a pure power law in the modulus of m, with a base that is a function of χ (Methods Sec. C), as the modulation couples many, distinguishable modes. Note that the sidebands of the Hawking radiation and the backreaction lie exactly on top of each other, apart from a shift by δ′, which causes an asymmetry in the spectrum. This asymmetry is the characteristic feature of the backreaction.

Figure 4: Theory versus experiment. We fit the signals (Fig. 3) with the theoretical curve for each probe wavelength: 1100 nm, 1200 nm, 1300 nm, 1400 nm, 1450 nm, and 1600 nm, shown in a-f, respectively. The circles correspond to the corrected differences in the UV spectra between the pump–probe and pump data (Fig. 3) while the black curves are fits with theory (Methods Sec. C-D). The fits agree well with the data, except in the spectral region where the signal interferes with the Negative–frequency Resonant Radiation. In each subfigure we see a central peak of Hawking radiation and several sidebands (in particular in a and d) due to modulation at the beat frequency of pump and probe. The parameters of the theory curves are fitted to the left and at the Hawking peaks, but in a and d also reproduce the right sidebands, which indicates the consistency of our correction procedure. Without the backreaction, the sidebands were symmetric; their visible asymmetry reveals the backreaction.

We clearly see the asymmetry (Fig. 4). The Hawking and backreaction intensities for all data sets we retrieve by fitting (Methods Sec. D). The agreement between the observed asymmetry and the analytic predictions is significant and supports the proposed interpretation, although alternative nonlinear processes producing similar effects cannot be explicitly ruled out. Stimulated Hawking radiation should be linear in the probe intensity and the backreaction quadratic. Hence we normalize the Hawking intensity with respect to the measured probe intensities (expressed in photon counts) and the backreaction intensity with their squares. Given the normalized intensities, we can test whether they establish a thermal spectrum. If so, we expect that the Hawking temperature is determined by the shortest duration τ0 of the pump during propagation, but τ0 varies from day to day due to imperceptible changes. To take this into account, we use the NRR as a thermometer. The NRR is essentially Hawking radiation stimulated by the pump itself and so it should have the same temperature. If this hypothesis and our normalizations are correct, the logarithms of the Hawking intensity and the backreaction intensity, each divided by the NRR intensity raised to the frequency ratio, plotted over that frequency ratio, should lie on straight lines with equal slopes. Given the simplicity of our model, it is remarkable how well they do this (Fig. 5). The relative root–mean–squares of the straight–line fits of the Hawking radiation and the backreaction are 0.015 and 0.010, respectively. For the ratios of the slopes we get 1.02: the Hawking and backreaction temperatures are equal within the accuracy of the linear fits.

Figure 5: Thermal spectra of Hawking radiation and backreaction. Logarithm of the normalised Hawking ratio (green dots) and of the normalised backreaction ratio (red dots) for the normalized Hawking radiation and backreaction counts and the NRR counts, plotted over the frequency ratio of the redshifted probe to the pump for all six probes in our experiment. For a spectrum with Bekenstein–Hawking temperature they would approximate straight lines with equal slopes. They do (lines) and the slopes of those lines agree with 2 percent accuracy.

We have stimulated Hawking radiation by classical light, but one might use nonclassical light as well, in particular heralded single photons as probes, for testing the entanglement of the Hawking partners. Our findings can be generalized to other analogues of gravity, too. In particular, in Bose–Einstein condensates (BECs) the interaction Hamiltonian is also bi–quadratic (Methods Sec. E) like in our case of the Kerr interaction in nonlinear fibre optics. In BECs the backreaction from acoustic horizons was studied theoretically using a similar phenomenology as in the description of black–hole evaporation; the backreaction from suddenly switching on the interaction and from the analogue of cosmic inflation was studied theoretically, too. The backreaction of water waves on a draining bathtub vortex mimicking a rotating black hole was measured and so was the one on fluids of light, but so far all these studies were either theoretical or did not consider the backreaction due to Hawking radiation.

Our experiment and the underlying theory show that Hawking radiation is the result of a direct process, if the interaction between the radiation and the equivalent of the gravitational field is bi–quadratic. In general relativity, the gravitational field is described by the metric tensor, which can be written in a quadratic form in terms of tetrads such that the interaction between field and radiation becomes bi–quadratic, as in our case (Methods Sec. E). Maybe astrophysical black holes radiate by a process as simple and direct as ours. The resulting backreaction would describe in microscopic detail how black holes evaporate, which was the subject of Hawking’s 1974 paper. Our experiment also shows that Hawking radiation has a thermal spectrum even in the regime of strong dispersion where the notion of the event horizon loses sense and where the temperature is no longer given by the surface gravity. All this could shed light on the information paradox, a problem Hawking struggled with until his very last, 2018 paper.

Methods

A. Unidirectional Hamiltonian

We briefly sketch the Hamiltonian method for nonlinear fibre optics. Our starting point is the one–dimensional wave equation for light with fixed transversal mode structure, where E denotes the electric field strength, n0 the effective refractive index in the fibre and PNL the nonlinear polarization. We write the left–hand side as a product of two first–order operators and approximate one of them by twice i n0 ω E, and the second time derivative of the nonlinear polarization by minus ω squared times the polarization. Expressing the result in the co–moving coordinates τ = t − z/u and ζ = z/u we obtain the unidirectional wave equation (A.1), with ω = i times the derivative with respect to τ and the co–moving frequency given by Eq. (1).

The nonlinear polarization on the right–hand side may act as a source for new frequencies or it may modulate existing frequencies — we are going to encounter both cases. The electric field is real and so its Fourier transform comprises both positive and negative frequencies. We express the field as the analytic signal plus its conjugate, the analytic signal describing the positive laboratory frequencies while its conjugate accounts for the negative ones. Approximating the differential operator ω on the right–hand side by the carrier frequency we obtain for the analytic signal Eq. (A.2), where the plus sign denotes a projection to positive frequencies and the interaction Hamiltonian appears through its derivative with respect to the conjugate amplitude.

For the Kerr effect this Hamiltonian is proportional to the fourth power of the electric field. Expanding that fourth power in terms of the analytic signal and its conjugate gives all processes derived from the Kerr nonlinearity. These include the familiar self–phase and cross–phase modulation, third–harmonic generation, the generation of dispersive waves and supercontinua, but also Negative–frequency Resonant Radiation (NRR) and stimulated Hawking radiation. Each individual Hermitian sub–Hamiltonian defines, for the corresponding sub–process, a conservation law of the energy in the co–moving frame, Eq. (A.3). Indeed, we obtain a vanishing derivative of that energy with respect to the propagation time, as the co–moving frequency is a Hermitian operator with respect to τ. In our paper, we decompose the light field into two parts, the first containing the pump pulse (and new frequencies generated) and the second the probe (and the results of its interaction with the pump). The Hamiltonian describes then how energy from the first is converted to energy in the second and vice versa, actio et reactio.

B. Hawking radiation and backreaction

Let us derive the creation of stimulated Hawking radiation and its backreaction from the optical propagation equation (A.2) and show in what respect fibre-optical Hawking radiation is thermal. Consider first the sub–Hamiltonian HS, proportional to the pump intensity times the probe intensity. This Hamiltonian generates for the probe a nonlinear polarization of the form twice n0 δn times the probe amplitude, where δn is proportional to the pump intensity. Comparing this term in Eq. (A.2) with Eq. (1) for the co–moving frequency we see that δn is the nonlinear contribution to the refractive index. This process describes the red– or blue–shifting of the probe at the horizon. For redshifting the probe shifts to a lower laboratory frequency (and for the opposite case it blueshifts to a higher frequency) while keeping its co–moving frequency. The resulting wave is the positive–frequency Hawking radiation (Fig. 1).

Consider now the process with sub–Hamiltonian HR, proportional to the pump intensity times the sum of the squared probe amplitude and its conjugate square, generating a nonlinear polarization proportional to twice n0 δn times the conjugate probe amplitude. The conjugate of the initial probe oscillates with negative co–moving frequency and thus generates the Hawking partner with negative co–moving frequency (Fig. 1). In terms of the laboratory frequency, it corresponds to the frequency whose co–moving value is minus that of the probe. We will argue that above the horizon frequency we should take the redshifted probe as initial wave. We thus have Eq. (B.1), a wave equation in space and time, as the left–hand side depends on the frequency operator via Eq. (1) while the right–hand side contains the moving index perturbation. We Fourier–transform Eq. (B.1) with respect to τ. The product of the index perturbation and the conjugate probe turns into a convolution which reduces to the Fourier transform of the index perturbation times the conjugate field at the position of the pump, as the spectrum of the redshifted probe is peaked and much narrower than the spectrum of the pump.

Integrating the Fourier–transformed propagation equation (B.1) we obtain Eq. (B.2). Consider, for example, a soliton–like pump of duration τ0 where the index perturbation has a squared hyperbolic-secant profile. In our experiment, the pump undergoes soliton fission but a sech pulse is still a good model for the most intense fraction. As the Fourier transform of that profile falls off exponentially with an exponent of minus pi τ0 ω over 2 for large frequencies, the intensity of the Hawking radiation goes with the exponential of minus pi τ0 times the redshifted probe frequency. The Hawking rate for the initial probe frequency would be exponentially lower. Hence it is the redshifted pulse that stimulates the lion’s share of the Hawking radiation. For astrophysical black holes, the Hawking radiation is determined by the frequencies of the outgoing light, being redshifted from Planck–scale initial fluctuations, whereas for white–holes it depends on the initial frequencies. This is also the case in our fibre–optical analogue where, below the horizon frequency, the blue–shifted probe contributes exponentially less.

Moreover, in either case we can relate the exponent of the optical Hawking rate directly to the astrophysical Hawking temperature T. There T is given by the surface gravity. Translated into optics, kB T equals h-bar over 2 pi times the modulus of the derivative of the logarithm of δn with respect to the retarded time, evaluated at the horizon. For a sech pump this derivative is 2 over τ0 times the modulus of the hyperbolic tangent, which gives kB T equal to h-bar over pi τ0 for large δn where the horizon lies in the wings of the sech profile. This temperature gives a rate proportional to the exponential of minus h-bar times the frequency over kB T for large frequencies, which is exactly our rate. Note that for general pulse profiles the optical Hawking temperature is given by the asymptotic exponent of the Fourier–transformed index perturbation, where pi τ0 is the lowest pole on the upper complex half plane (unless there is no pole, as for a Gaussian pulse). This pole and hence the Hawking temperature depends on the entire index profile and not solely on the logarithmic derivative at the horizon, which distinguishes our case of weak index perturbation and strong dispersion from the ideal astrophysical case.

We have calculated the negative–frequency Hawking radiation generated by the pump and stimulated by the probe. Turn now to its backreaction on the pump with Hamiltonian HR. Equation (A.2) with that interaction implies that the total photon number of the pump field is conserved. The pump photons are not lost but some are redistributed within the pump field. To find out at which frequency they appear, differentiate HR with respect to the conjugate pump amplitude. One term oscillates with the pump co–moving frequency plus twice the probe co–moving frequency, which lies outside the dispersion curve (Fig. 2a) and so cannot be excited. The other term oscillates with the pump co–moving frequency minus twice the probe co–moving frequency, which equals minus the probe co–moving frequency minus δ′, and corresponds to a UV wave of slightly higher frequency than the Hawking radiation (Fig. 2c). For calculating the amplitude we may use the same procedure as for the Hawking radiation and obtain Eq. (B.3).

The backreaction is quadratic in the probe intensity, which would make it significantly weaker than the Hawking radiation, but it goes with the same exponential rate for a pump pulse with a given lowest pole. This rate is exponentially larger than the Hawking rate. Consequently, one can observe the backreaction, but there is another complication: overlapping sidebands.

C. Sidebands

The moving refractive–index perturbation is proportional to the total intensity, which for a weak probe is the pump intensity plus the nonlinear interference term made of the pump amplitude times the conjugate probe amplitude and its conjugate. That interference term oscillates with the co–moving frequency difference δ′ and the oscillating index perturbation generates sidebands at regular intervals of δ′. Here we work out their amplitudes.

The sidebands lie at the co–moving frequencies given by the source frequency plus integer multiples of δ′ around the source. We follow the same procedure as in Sec. B to derive for their Fourier–transformed amplitudes Eq. (C.1), where the source term describes either the source of the Hawking radiation or the source of the backreaction. The parameter Ω is proportional to the Fourier transform of the pump–probe product at the difference frequency. As that product is short in time it has a wide, nearly uniform spectrum that gives the same Ω for each coupling. Solving Eq. (C.1) by expanding in a Fourier series gives Eq. (C.2), which is an exact solution with vanishing initial condition, and summing the amplitudes gives a closed form. It remains to determine the Fourier coefficients, Eq. (C.3), in which the sideband amplitudes fall off as a pure power law in the modulus of the sideband index, with a base determined by the ratio of Ω to δ′.

The sidebands would obey an exact power law symmetric around the Hawking peak — were it not for the backreaction. The backreaction is generated exactly one sideband to the left of the main Hawking peak. Therefore, the bands of the backreaction align perfectly with the Hawking bands, but are shifted by minus δ′. Both bands are coherent; we need to add their amplitudes, and obtain Eq. (C.4), in which the effective source for negative sideband index is the Hawking source plus i times the backreaction source divided by the power–law base, and for non-negative index it is the Hawking source minus i times the backreaction source times that base.

We see that the intensities of the total radiation bands around the main Hawking peak are given by simple power laws with asymmetric prefactors. For zero modulation, the spectrum reduces to the Hawking peak and the backreaction one interval away. For non–zero modulation, the backreaction still appears in the asymmetry of the sidebands. Given a measured spectrum, we may retrieve the Hawking–radiation and backreaction intensities from the data, as we show next.

D. Data analysis

First we discuss the dispersion data (Fig. 2). They are determined by measurements of the group velocity in the IR combined with modeling based on electron microscopy of the fibre cross section, as described previously. We noticed a systematic shift in the measured wavelength of Hawking radiation due to months of light exposure of the fibre and corrected for it by adding to the propagation constant a small, constant offset that does not change the group velocity. We determine that offset by requiring that it reproduces both the measured Hawking partner frequency and the measured horizon (with a horizon wavelength of 1551 nm for a 1450 nm probe).

We take the UV data (Fig. 3) with both pump and probe present and the UV data without the probe, each 50 percent of the time, in order to subtract the background of Negative–frequency Resonant Radiation (NRR). Taking the difference between the two spectra should give us our signal. However, the probe perturbs the pump and hence the NRR, despite it being a hundred times less intense. The pump pulse fractures into solitons that, like black holes in General Relativity, are characterized by only a few parameters. These are the carrier frequency and the duration. The intensity is locked to the duration as the inverse square and the group velocity is set by the carrier frequency. Both the carrier frequency and the velocity define the NRR frequency in the dispersion diagram (Fig. 2). So if the carrier frequency changes the NRR frequency changes (in the order of one part in a thousand). The spectral shape of the NRR is less related to the pump spectrum and remains intact. If we do not correct for the spectral shift of the NRR we may get negative values of the signal intensity. For performing the correction, we select in each data set a relevant range around the NRR peak, minimize the root mean square of the corrected difference for the frequency offset, and take that corrected difference for all wavelengths as our signal.

The spectral shapes of the Hawking and backreaction bands and their sidebands are identical, as the Fourier–transformed field is given by the Fourier–transformed amplitude of the Hawking radiation (emitted with unity strength) with the coefficients of Eq. (C.4). The peak frequencies are determined by the co–moving sideband condition, which translates into laboratory frequencies spaced by the difference between the backreaction and the Hawking partner frequency, as the dispersion curve (Fig. 2c) is nearly linear there. Due to variations in the pump, the propagation time varies more than one modulation period: we need to average. As a result, the intensity is an incoherent sum of the Hawking spectra, despite the bands being coherent, Eq. (D.1).

The Hawking spectrum is emitted while the probe is being redshifted. We found empirically that it interpolates between the squared hyperbolic secant of a soliton and a parabolic profile, through an exponent µ that is 1 for a soliton and tends to infinity for a parabola. For our six data sets (Fig. 4) we found the exponents 10.0, 1.2, 1.1, 10.0, 1.0 and 1.05. We determine the Hawking partner frequency, the backreaction frequency, the spectral width, the power–law base and the two intensities by fitting. For each of the six data sets we determine u/c in the Doppler formula (1) by requiring that it reproduces the determined Hawking partner frequency. We need to do this due to the Raman shifting of the pump at the point of emission of Hawking radiation that is difficult to control. Then the backreaction frequencies agree well with their predicted values from the dispersion data (Fig. 2).

The theoretical curve, Eq. (D.1), fits the UV data well (Fig. 4) except in the region of the first sideband. There the co–moving frequency coincides with the negative co–moving frequency of the pump, i.e. with the frequency of the NRR (Fig. 2). As the pump and the probe are coherent, the NRR and this sideband are also coherent and interfere. As they oscillate with the same frequency this interference is not averaged out, but remains visible in the data (Fig. 4). To avoid the interference, the fitting is done only with data to the left of and including the Hawking peak, but it still gives a remarkably good fit for the second sideband (Fig. 4a and 4d). Without our modified subtraction procedure this sideband would be drowned in negative values of the raw difference. The fact that it appears and agrees with a fit done on the other end of the spectrum shows the consistency of the data analysis.

The NRR intensity varies greatly from day to day (up to a factor of 3) due to imperceptible variations in the incoupling efficiency and so does the intensity of the Hawking radiation. Yet they are related: NRR is Hawking radiation made by the pump pulse and stimulated by the pump itself. This implies that its intensity falls exponentially in frequency with the same exponent as Hawking radiation, Eq. (D.2), where the effective frequency characterizes the Boltzmannian tail of the Planck spectrum of Hawking radiation in the limit of large frequencies. For each data set we apply our fitting procedure to extract the photon flux of the stimulated Hawking radiation and divide it by the stimulant, the measured average coupled probe power (0.66 mW, 0.99 mW, 1.35 mW, 1.2 mW, 1.2 mW, 0.81 mW) divided by the original probe frequency, for converting energies to photon numbers. The backreaction should also fall with the same exponential and be quadratic in the probe intensity. We obtain the relations of Eq. (D.3), with unknown constants and the known frequency ratios for the six data sets (Fig. 4).

The logarithms of the two normalised ratios should lie on straight lines in the frequency ratio with the same slope. In the linear fits we exclude the 1100 nm data. There the probe lies at the edge of the capture range where the red–shifted efficiency drops (we see this by comparing the signal with the NRR in Fig. 3). As we normalize with respect to the incident probe intensity (not the red–shifted one) we would otherwise make a systematic error. The other data points lie on straight lines (Fig. 5) with relative root–mean squares of the deviations (RMS divided by the signal) of 0.015 for the Hawking radiation and 0.010 for the backreaction. The slopes agree within 2 percent, i.e. within the combined error of the slopes. This shows the thermality of the Hawking–radiation and backreaction spectra and also the fact that the backreaction goes quadratically with probe intensity. Figure 3c also shows that the experiment is run in a regime where both effects are of roughly equal magnitude.

E. Bose Einstein condensates and tetrads

Here we show how the concept of a Hawking Hamiltonian applies to other analogues of gravity and possibly to gravity itself. The best–understood analogue of gravity is based on Bose–Einstein condensates. There the theory starts from the grand–canonical Hamiltonian for a Bose gas of atoms with mass m confined by the trapping potential U and interacting by point collisions with strength g, Eq. (E.1), where µ is the chemical potential (the energy per atom). The atoms shall be Bose–condensed with a macroscopic wave function. We split the quantum field into the condensate and the field of elementary excitations, and expand the Hamiltonian (E.1) up to quadratic order in the excitation field. We obtain three Hamiltonians: the zeroth–order classical Hamiltonian that generates the Gross–Pitaevskii equation of the condensate, the first–order Hamiltonian that vanishes under the Gross–Pitaevskii equation, and the quadratic Hamiltonian that generates the Bogoliubov–de Gennes equation of the elementary excitations, Eq. (E.2), with c the speed of sound (not light).

Both the speed of sound and the condensate flow may vary. The simplest way of seeing how the Bogoliubov–de Gennes dynamics represents an analogue of gravity is considering an excitation that is a modulated plane wave with wavevector k and frequency ω. Setting the trapping potential to zero and choosing the chemical potential appropriately, we obtain from Eq. (E.2) a dispersion relation in which the combination of frequency and wavevector represents the Doppler shift due to the motion of the condensate — the dispersion relation is simply Bogoliubov’s dispersion relation in a locally co–moving frame. In the regime of small wavevectors we can give it a completely relativistic form, Eq. (E.3), adopting Einstein’s summation convention. To the inverse metric tensor corresponds Unruh’s famous acoustic metric, Eq. (E.4). The metric shows that, when the condensate reaches the speed of sound, time appears to stand still for the elementary excitations and so a horizon may be established.

The Hamiltonian for the resulting Hawking radiation we read off from the grand–canonical Hamiltonian (E.1) by splitting the atomic field into the condensate and the excitation, assuming now that the condensate is as quantum as the excitations (since it needs to provide the energy for Hawking radiation). Expressing the excitations in terms of the Bogoliubov-de Gennes modes, we see that the grand–canonical Hamiltonian (E.1) contains processes completely analogous to our optical case. Energy conservation requires that one of the two modes must have the exact negative frequency of the other, which is the hallmark of Hawking radiation. The backaction of this Hamiltonian should be noticeable in Bose–Einstein condensates as well, but has not been observed yet.

Other analogues of gravity share similar bi–quadratic interactions, and possibly gravity itself. This is because the central quantity of gravity is the metric tensor of the space–time geometry and one can always express it in a quadratic form. This is done in terms of tetrads, the metric being the scalar product of two tetrads with respect to the Minkowski metric. The inverse metric tensor is expressed in terms of the inverse tetrads. To give a simple example of tetrads, for the Unruh metric (E.4) the line element becomes the flat form in a shifted radial coordinate, and we thus obtain the tetrad and its inverse given in Eq. (E.5).

In the tetrad formalism, the interaction of gravity with other fields is typically bi–quadratic. To see this, consider a scalar field. The interaction Lagrangian with gravity is proportional to the inverse metric contracted with two derivatives of the field, which written in tetrads is bi–quadratic. Einsteinian gravity is not a renormalizable quantum field theory (as the Einstein–Hilbert action depends on second derivatives of the metric tensor and hence goes with the inverse second power of a length, which blows up in a perturbative expansion). Yet it could establish a low–energy effective field theory. There are several options for such a theory. Our analysis of Hawking radiation suggests that it might be wise to take the tetrad formalism as the starting point of an effective theory of quantum gravity.

Data availability

Data is available upon request.

Acknowledgements

We are grateful to S. Amiranashvili, U. Bandelow, H. Cohen, J. Drori, M. Gelvan, F. König, Y. Rosenberg, S. Rotter, and most of all to the late Y. Silberberg for help, advice and scientific discussions.

Funding

L.M.P. and U.L. were supported by the Israel Science Foundation, the Murray B. Koffler Professorial Chair and (U.L) by a Global Fellowship of the Vienna Institute of Technology. R.A-S. and D.B. acknowledge the support of Conahcyt (Mexico) Ciencia de Frontera 51458-2019 and the Marcos Moshinsky Chair (2023).

Author contributions

L.M.P. was conducting, analysing and leading the experiment, R.A.-S. and D.B. were performing numerical simulations, U.L. developed the analytical theory, D.B. and U.L. did the data analysis, supervised the project, and conceived the idea of this paper.

Competing interests

There are no competing interests.

(The displayed equations of the Methods and the reference list of 53 items are omitted here; the complete text with its equations, figures and references is at the source.)