Observation of Stimulated Hawking Radiation in Optics
Jonathan Drori, Yuval Rosenberg, David Bermudez, Yaron Silberberg, Ulf Leonhardt
OObservation of Stimulated Hawking Radiation in an Optical Analogue
Jonathan Drori , Yuval Rosenberg , David Bermudez , Yaron Silberberg , and Ulf Leonhardt Weizmann Institute of Science, Rehovot 7610001, Israel Departamento de F´ısica, Cinvestav, A.P. 14-740, 07000 Ciudad de M´exico, Mexico (Dated: January 15, 2019)The theory of Hawking radiation can be tested in laboratory analogues of black holes. We uselight pulses in nonlinear fiber optics to establish artificial event horizons. Each pulse generates amoving perturbation of the refractive index via the Kerr effect. Probe light perceives this as anevent horizon when its group velocity, slowed down by the perturbation, matches the speed of thepulse. We have observed in our experiment that the probe stimulates Hawking radiation, whichoccurs in a regime of extreme nonlinear fiber optics where positive and negative frequencies mix.
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In 1974 Stephen Hawking published his best–knownpaper [1] where he theorized that black holes are notentirely black, but radiate due to the quantum natureof fields. Hawking’s paper confirmed Jacob Bekenstein’sidea [2] of black-hole thermodynamics that subsequentlybecame a decisive test for theories of quantum gravity.Yet Hawking radiation has been a theoretical idea itself;its chances of observation in astrophysics are astronomi-cally small indeed [3].In 1981 William Unruh suggested [4] an analogue ofHawking’s effect [1] that, in principle, is observable inthe laboratory. Unruh argued that a moving quantumfluid with nonuniform velocity — liquid Helium [5] wasthe only choice at the time — establishes the analogueof the event horizon when the fluid exceeds the speed ofsound. This is because sound waves propagating againstthe current can escape subsonic flow, but are draggedalong in spatial regions of supersonic flow. Mathemat-ically, the moving fluid establishes a space–time metricthat is equivalent to the geometry of event horizons [4–6].So the analogue to quantum fields in space–time geome-tries should exhibit the equivalent of Hawking radiationas well, Hawking sound in Unruh’s case [4].With this [4] and other [6] analogues one can inves-tigate the influence of the extreme frequency shift athorizons, shifts beyond the Planck scale [7]. The par-ticles of Hawking radiation appear to originate from ex-treme frequency regions where the physics is unknown.In analogues of the event horizon, instead of the un-known physics beyond the Planck scale, the known fre-quency response of the materials involved regularize theextreme frequency shift [7, 8]. Analogues are thus a test-ing ground for the potential influence of trans-Planckianphysics on the Hawking effect.In 2000 horizon analogues began to become the subjectof serious experimental effort and to diversify into variousareas of modern physics. While none of the first proposals[9, 10] were directly feasible, they inspired experimentson horizons in optics [11–16], ultra-cold quantum gases[17–20], polaritons [21] and water waves [22–27]. Yetdespite admirable experimental progress, there is still no clear–cut demonstration of quantum Hawking radi-ation. The optical demonstration [12] turned out to bethe horizon–less emission from a superluminal refractive–index perturbation [28]. The demonstration [18] of blackhole lasing in Bose-Einstein condensates (BEC) was dis-puted [29, 30] with overwhelming arguments [29], andthe demonstration of Hawking radiation in BECs [19]appears to suffer from similar problems [31, 32], with thepossible exception of Ref. [20].Here we report on clear measurements of the stim-ulated Hawking effect in optics. This is not a fulldemonstration of quantum Hawking radiation yet, butit already gives quantitative experimental results on thespontaneous Hawking effect. It represents the next mile-stone following the demonstration of frequency shifting athorizons [11] and negative resonant radiation [13]. Froman optics perspective, it establishes the regime of extremenonlinear fiber optics where controlled conversions be-tween positive and negative frequencies occur.In optical analogues [11], an intense ultrashort lightpulse in a transparent medium creates a perturbation δn of the refractive index due to the Kerr effect [33] thattravels with the pulse. In a co–moving frame the pulsestands still and increases the local refractive index n ,reducing the velocity of itself and other light, while thematerial appears to be moving against the pulse. Forprobe light present, the pulse establishes horizons whereits group velocity u matches the group velocity c/ ( n + ω dn/dω ) of the probe. A black–hole horizon is formedin the leading end of the pulse, and a white–hole horizonin the trailing end [34]. In the spontaneous Hawkingeffect, the probe consists of vacuum fluctuations of theelectromagnetic field, while in the stimulated effect, theprobe has a coherent amplitude.Consider the probe in the co–moving frame. There thepulse is stationary and hence the co–moving frequency ω (cid:48) of the probe is a conserved quantity. A pair of Hawkingquanta thus consists of a photon with positive ω (cid:48) and apartner with the exact opposite, − ω (cid:48) , such that the sumis zero. The time–dependent annihilation operators ˆ b ± ofthe outgoing radiation are given [34] by the Bogoliubov a r X i v : . [ g r- q c ] J a n transformations of the time–dependent ingoing ˆ a ± asˆ b ± = α ˆ a ± + β ˆ a †∓ (1)with constant α, β and | α | − | β | = 1, where ± refers tothe sign of ω (cid:48) . The ingoing field is incident in the materialat rest, i.e. in the laboratory frame. This implies [34] thatthe ˆ a ± oscillate with positive laboratory frequencies ω .To see how and for which positive ω negative co–movingfrequencies ω (cid:48) appear, consider the Doppler effect: ω (cid:48) = γ (cid:16) − n uc (cid:17) ω (2)where u denotes the group velocity of the pulse and γ − = 1 − u /c . For positive laboratory frequencies, ω (cid:48) is positive when the phase velocity c/n is faster thanthe pulse, which in our system is the case in the in-frared (IR) (Fig. 1). The co–moving frequency ω (cid:48) is neg-ative when u exceeds c/n , which occurs in the ultraviolet(UV) (Fig. 1). Making measurements in these spectralregions gives us data on the Hawking effect. In particu-lar, an IR probe with (cid:104) ˆ a − (cid:105) = 0 stimulates the UV signal (cid:104) ˆ b − (cid:105) = β (cid:104) ˆ a + (cid:105) ∗ that proves the existence of the effect andgives the spontaneous photon number | β | if the ampli-tude (cid:104) ˆ a + (cid:105) interacting with the pulse is known. [PHz]-0.2-0.100.10.2 C o m o v i ng F r equen cy ' [ P H z ] FIG. 1: Doppler curve. Plot of ω (cid:48) given by Eq. (2) for n ( ω )and u of our fiber (solid curve: n determined from measure-ments [35], dashed–dotted: n extrapolated). The pump pulsesits at a local minimum and the horizon at a local maximum[13]; ω (cid:48) is conserved during pump–probe interaction (hori-zontal lines). The probe light (black and white diamonds)is incident with frequencies lower or higher than the horizon,experiencing the analogue of a black or a white hole. Bothincident and outgoing Hawking partner has − ω (cid:48) of the probe(lower line) intersecting the Doppler curve where we expectnegative Hawking radiation (NHR, Fig. 3c). The pump itselfcreates negative resonant radiation (NRR) at the intersectionof its − ω (cid:48) (lower dotted line) with the Doppler curve [13]. Furthermore, while the ingoing modes oscillate withpositive laboratory frequencies, the outgoing modes mustcontain negative–frequency contributions due to the Her-mitian conjugation in the Bogoliubov transformation,Eq. (1). This combination of positive and negative fre-quencies in the laboratory frame differs from ordinaryoptical parametric amplification [36] and is only possiblein a regime of extreme nonlinear optics with few–cyclepulses beyond the slowly–varying envelope approxima-tion [33]. Only in this extreme regime β is sufficientlylarge to be detectable.Figure 2 shows our experimental setup. We perform apump–probe experiment: the pulse creating the movingrefractive–index perturbation is called pump, and its ef-fect is probed by a probe pulse we derive from the samesource as the pump. The pump pulses are of 8 fs dura-tion at 800 nm free-space carrier wavelength (producedby a Thorlabs Octavius oscillator). They are coupledinto a 7 mm photonic-crystal fiber (PCF) (NKT NL-1.5-590). In this fiber, probe pulses of ≈
50 fs durationand tuneable carrier wavelength may interact with thepump. The probe pulses have been generated by Ra-man shifting [33] in a 1 m PCF (NKT NL-1.7-765) af- Δ τ DC …
Probe PCF P… Int. PCF
BS DM
IR detection UV detection DM
12 3 … Wavelength [nm] P o w e r [ a . u ]
600 800 100000.51
Wavelength [nm] P o w e r [ a . u ] FIG. 2: Experimental setup. The starting point (left) arelight pulses of 8 fs duration at 800 nm carrier wavelength. Theinset shows the pulse spectrum. At the 50:50 beam splitter(BS) each pulse is distributed to two channels. In 1 (cid:13) the pulseis dispersion–compensated and delayed before being combinedwith the probe pulse that is prepared in the other arm. Theintensity of the other split pulse is tuned 2 (cid:13) by a half–waveplate and a polarizer. It is coupled into the Probe PhotonicCrystal Fiber (PCF) by a parabolic mirror to be Raman–shifted in its wavelength depending on the initial intensity.The right inset shows a typical spectrum after Raman shift-ing. In 3 (cid:13) the train of probe pulses is modulated by a chop-per wheel before being combined with the pump pulse at adichroic mirror (DM). Pump and probe enter the InteractionPCF via a parabolic mirror. The resulting light is collimatedand distributed (DM) to the IR and UV detection stations. ter reflection off the original master pulses by a 50:50beamsplitter. We take advantage of the intensity depen-dence of the Raman effect to tune them over the wave-length range from 800 nm to 1620 nm by small intensitychanges. The output of the pump–probe interaction isdistributed via a dichroic mirror and spectral and spatialfilters to two detection stations, a commercial spectrom-eter (Avantes AvaSpec-NIR256-1.7) for the IR and, forthe UV, a prism–based tuneable monochromator and aphotomultiplier tube (Hamamatsu H8259) as detector.Some representative detection results are shown inFig. 3. To understand them we note the following. Forthe probe the group–velocity dispersion is normal — thegroup velocity increases with increasing wavelength. Atthe horizon wavelength the group velocity of the probematches the pulse velocity, so the probe is faster than thepump for longer wavelengths and slower for shorter wave-lengths. Therefore, when the probe is tuned to the redside of the horizon wavelength (Fig. 3a) it runs into thewhite–hole horizon and is blue–shifted [11, 37] (Fig. 3a).The probe on the blue side (Fig. 3b) is slower then thepulse, experiences a black–hole horizon and is red-shifted[11, 37] (Fig. 3b). The red–shifting produces a clearersignal than the blue–shifting, although of lower magni-tude, because, due to the Raman effect [33], the pumppulse de–accelerates [33] such that the blue–shifted probelight interacts longer with the white–hole horizon, andwith more complicated dynamics (producing the spec-tral modulations of Fig. 3a). In extreme cases, the probemay even get trapped by the pulse [38].Figure 3c shows results of the UV detection with andwithout the probe. With the probe off, one sees a clearpeak at 231 . n ( ω ) that depends on both thematerial and the microstructure of the fiber [33]. Weobtain n ( ω ) from measurements of the group index inthe IR and Visible interpolated to the material refractiveindex in the UV [35] checked and fine–tuned with ourmeasurement of the previously known negative resonantradiation [13]. The group velocity u of the pump was fit-ted and corresponds to a carrier wavelength of 818 . P o w e r [ A . U .] ProbePump & ProbeHorizonWavelength a P o w e r [ A . U .] ProbePump& ProbeHorizonWavelength b
226 228 230 232 234
Wavelength [nm] C oun t s [ k H z ] Pump & ProbePumpSignal c FIG. 3: Experimental results. a : Spectrum of the IR probe(solid curve) after the interaction with the pump for an initialprobe (dashed curve) tuned to the red side of the horizonwavelength (dotted line). The probe has been blue–shifted(arrow) and also spectrally modulated. b : Spectrum (solidcurve) after interaction for an initial probe (dashed line) tunedto the blue side of the horizon (dotted line). The figure showsa distinct red shift (arrow). c : UV spectrum for the 1450 nmprobe shown in b interacting with the pump (solid curve)and for the pump alone (dashed and dotted). The differenceproduces a clear signal (curve with 2 σ error bars) we interpretas stimulated negative Hawking radiation (Fig. 1, NHR). by numerical calculations of the pump–probe interactionin the negative ω (cid:48) range [35].Additionally, we have also varied the probe power whilekeeping everything else constant. Figure 4b shows thatthe power of the UV peak due to the probe is linearin the probe intensity for low probe power until it sat-urates for a probe power of ≈ .
5% of the pump peak
Probe Wavelength [nm]
NHR W a v e l eng t h [ n m ] a Probe Power [%] P o w e r [ A . U .] b FIG. 4: Experimental checks. a : Comparison with theory.Measured smoothed peaks (circles) of the stimulated negativeHawking radiation (NHR, see e.g. Fig. 3c) for various probewavelengths versus theory (curve, from Fig. 1). The error barsindicate our spectral resolution. The outlier is due to a com-plicated peak structure. b: Linearity of stimulated Hawkingradiation. Power of the UV signal (Fig. 3c) for 1600 nm probewavelength as a function of probe power (circles and crosses)versus a linear fit (for the circles). The maximal probe powerreaches ≈
2% of the peak intensity of the pump. The figureshows that the power of the generated radiation is linear inthe probe power for low intensities, whereas for higher powerit saturates. power. The linearity is another important feature of astimulated effect, while the saturation indicates a regimeknown from numerical simulations [39] where the probeis able to influence the pump with relatively low power— where probe and pump switch sides.We have thus strong reasons for the correct interpre-tation of the observed UV peak (Fig. 3c) as stimulatedHawking radiation: the agreement with the Doppler for-mula (2) for negative frequencies in the co–moving framewith the measured and calculated refractive–index data[35], supplemented with numerical simulations [35], andthe linearity of the stimulated signal (Fig. 4b) for lowprobe power. The measurements show empirically wherethe spectrum of the stimulated Hawking radiation lies,and hence also where the spontaneous Hawking radia-tion is expected. However, our pump–probe techniquedoes not allow us to make precise measurements of the Hawking spectrum, as the probe spectrum is too wide.We do not expect [40] a Planck spectrum there, as we arein a Hawking regime of strong dispersion [41]. We alsofound [35] that the UV part of the stimulated Hawkingradiation consists of multiple modes.We can estimate how many Hawking quanta are spon-taneously produced in the mode we detect with our cur-rent apparatus. For a probe intensity of 1 kW we have2 × photons [42] stimulating 41 ,
000 additional countsper second between 229 . . × − spontaneous UV Hawkingpartners per second (detected with our current efficiencyof about 10 − ). In the IR the fiber is single–mode, con-centrating all light into a guided wave, and there numer-ics [35] indicate a 10 times higher Hawking rate.Our measurements prove that the optical analogue ofthe event horizon [11] does indeed describe our observa-tions, despite other effects present in nonlinear fiber op-tics [33] such as third–harmonic generation and the Ra-man effect. Third harmonics [33] are produced, becausethe nonlinear polarization is proportional to the cube ofthe instant electric field. This gives two contributions:while one appears as the refractive–index perturbation δn we use for generating Hawking radiation, the otheroscillates at trice the carrier frequency of the pulse andgenerates third harmonics. We have seen the wide, un-structured range of non–resonant third harmonics overthe third of the wavelength range of the pulse, but boththe negative–frequency peak of the pump and the stim-ulated Hawking radiation of the probe lie at the tail ofthis range and are clearly distinguishable (Fig. 3c).The Raman effect [33] de–accellerates the pump pulse,which makes the pulse velocity intensity–dependent, andhence also the horizon. However, most of the stimulatedHawking radiation is generated during the first 1 mm ofpropagation in the fiber. There the pump pulse, of solitonnumber N = 2 . Acknowledgements.—
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