Magnonic Analogue of Black/White Hole Horizon in Superfluid 3 He-B
aa r X i v : . [ c ond - m a t . o t h e r] A p r Magnonic Analogue of Black/White Hole Horizon in Superfluid He-B
M. ˇCloveˇcko, E. Gaˇzo, M. Kupka, and P. Skyba ∗ Institute of Experimental Physics, SAS and P. J. ˇSaf´arik University Koˇsice, Watsonova 47, 04001 Koˇsice, Slovakia. (Dated: April 19, 2019)We report on theoretical model and experimental results of the experiment made in a limit ofabsolute zero temperature ( ∼ µ K) studying the spin wave analogue of black/white hole horizonusing spin (magnonic) superfluidity in superfluid He-B. As an experimental tool simulating theproperties of the black/white horizon we used the spin-precession waves propagating on the back-ground of the spin super-currents between two Bose-Einstein condensates of magnons in form ofhomogeneously precessing domains. We provide experimental evidence of the white hole formationfor spin precession waves in this system, together with observation of an amplification effect. More-over, the estimated temperature of the spontaneous Hawking radiation in this system is about fourorders of magnitude lower than the system’s background temperature what makes it a promisingtool to study the effect of spontaneous Hawking radiation.
PACS numbers: 04.70.-s, 67.30.-n, 67.30.H-, 67.30.hj, 67.30.er
Recently, it has been shown that Hawking radiationshould be related not only to physics of the astronomicalblack holes [1], but should be viewed as a general phe-nomenon of a dynamic nature of various physical systemshaving capability, at certain conditions, to create andform a boundary - an event horizon [2]. According totheory, a fundamental dynamical property of any eventhorizon analogue is a spontaneous emission of thermalHawking radiation, the temperature of which dependson a velocity gradient at the horizon [3, 4] T = ¯ h πk B ∂v r ∂r ∼ − ∂v r ∂r . (1)Belong a set of physical systems used to modelblack/white hole horizon can be included: the soundwaves in trans-sonic fluid flow [3, 5], the surface waveson flowing fluid [6–9], hydraulic jumps in flowing liquids[10, 11], and light in optical fibre [12]. It turns out, how-ever, that temperature of the spontaneous Hawking radi-ation is typically a several orders of magnitude lower thanthe background temperature of the physical systems usedas an experimental tool to study this phenomenon. Per-haps, solution to this problem is to find another, suitablecondensed matter systems, which can model the eventhorizon, but with background temperature approachingabsolute zero temperature [13–16].In this Letter we present as a theoretical model, soexperimental results of the first experiment made in alimit of absolute zero temperature ( ∼ µ K) studyingthe black/white hole horizon analogue using a physicalsystem based on the spin (magnonic) superfluidity in su-perfluid He-B. The concept of the experiment is quitesimple [17]. The experimental cell with superfluid He-Bconsists of two cylinders mutually connected by a chan-nel (see Fig. 1). The cell is placed in steady magneticfield B and magnetic field gradient ∇ B, both orientedalong z -axis. Using cw-NMR technique, we created aBose-Einstein condensate of magnons in a form of the HPDSD HPDSD
SOURCE DOMAIN DETECTION DOMAINSPIN FLOWRESTRICTION SP I N W AVE B B NMR coils Thermometric volumeLongitudinal coilsPart of rf-shield
FIG. 1: (Color on line) Schematic 3-D cross section of theexperimental cell and concept of the experiment. The channeldimensions are: the width is 3 mm, the height is 0.4 mm andthe channel length is 2 mm. homogeneously precessing domain (HPD) in both cylin-ders [18–20]. The HPD is a dynamical spins structureformed in a part of the cell placed in lower magnetic fieldwhich, with an aid of the dipole-dipole interaction, co-herently precess around a steady magnetic field at theangular frequency ω rf . Within the rest of cell, the spinsare co-directional with the steady magnetic field and donot precess. These two spin domains are separated by aplanar domain wall, the position of which is determinedby the Larmor resonance condition ω rf = γ ( B + ∇ B.z ),where γ is the gyromagnetic ratio of the He nuclei.To model black/white hole horizon in superfluid He-B, we used two fundamental physical properties of theHPD: the spin superfluidity and the presence of theHPD’s collective oscillation modes in a form of the spinprecession waves [21–24]. The spin superfluidity allowsto create and manipulate the spin flow, i.e. the spinsuper-currents flowing between precessing domains as aconsequence of the phase difference ∆ α rf between thephases of spin precession in individual domains [25]. Thespin precession waves serve as a probe testing formationand presence of a black/white hole horizon inside thechannel: channel has a restriction allowing to reach thedifferent regimes of the velocity of the spin super-currentflow with respect to the group velocity of the travellingspin-precession waves. This is similar to the experimentsuggested by R. Sch¨utzhold and W. Unruh [6] and laterperformed by G. Rousseaux et al. [7–9] and S. Weinfurt-ner et al. [9]. In the presented experiment, the magneticfield gradient ∇ B plays the role of the gravitational accel-eration, the spin super-currents represent the water flow,and the spin-precession waves correspond to the gravitywaves on the water’s surface.In order to develop a mathematical model, we initiallyconsider a simplified problem - a volume of He-B placedinto a large steady magnetic field B with gradient field ∇ B applied in the z -direction and a small magnetic rf-field B rf , which rotates in the “horizontal” x − y planeat the angular frequency ω rf , with the phase of rotationvarying linearly with x . Cartesian components of the re-sultant magnetic field are B x = − B rf cos( ω rf t + ∇ α rf x ), B y = B rf sin( ω rf t + ∇ α rf x ) and B z = − B + ∇ B.z .This model problem is treated theoretically by adaptingthe method we presented elsewhere [24].The steady-state response is a layer of magnetizationprecessing with the frequency and local phase of the rf-field lying over a layer of stationary magnetization. Thedomain with precessing spins (magnetization) may oscil-late about its steady state. The principal variable de-scribing small oscillations is the perturbation a ( t, x, y, z )to the phase of spin precession. In the long wavelengthapproximation and for a thin domain of precessing spins,the perturbation propagating along the domain wall isfound to be governed by the equation (cid:18) ∂∂t − ∂∂x u (cid:19) (cid:18) ∂∂t − u ∂∂x (cid:19) a dw − ∂∂x (cid:18) c ∂a dw ∂x (cid:19) − ∂∂y (cid:18) c ∂a dw ∂y (cid:19) + r γ ∇ BB rf La dw = 0 . (2)Here, a dw ( t, x, y ) = a ( t, x, y, z dw ), where z dw denotes z -coordinate of the domain wall position and L is the thick-ness of precessing domain. Two terms u and c representthe spin flow and the group wave velocities, respectively, and they can be expressed as u = (5 c L − c T ) ∇ α rf ω rf , c = (5 c L + 3 c T ) γ ∇ BL ω rf . (3)where and c L and c T denote the longitudinal and trans-verse spin wave velocities with respect to the field orienta-tion, respectively. We shall assume that ∇ α rf is localizedon the length of the sharpest restriction in the channelof the order dl = 0.5 mm, therefore ∇ α rf ∼ ∆ α rf /dl .The long spin-precession waves travelling along thesurface of a thin layer of precessing and flowing spinsare governed by the same equation as a scalar field ina (2+1)-dimensional curved space-time. Thus, thesewaves experience the background as an effective space-time with the effective metric ds = c (cid:2) − c dt + ( dx + udt ) + dy (cid:3) . (4)As it is implied by this equation, an “event horizon” forthe long spin-precession waves is formed where and when u = c . For the sake of completeness, the exact depen-dence of the angular frequency ω of any spin-precessionwave on components k x and k y of its wave vector is( ω + u k x ) = γ ∇ B ω rf c κ tanh( κL ) , (5)where c = (5 c T − c L ) and κ is defined as κ = 32 c (cid:20) √ ω rf γ B rf + 13 (5 c L + 3 c T )( k x + k y ) (cid:21) . (6)It is important to note that the applied rf-field explicitlydetermines the “vacuum”, that is to say, it prescribes theangular frequency and the variation of the phase for theprecessing magnetization (spins) representing a steadystate of the system considered. But Eq. (2) for small os-cillations superposed on the steady state is determinedprimarily by the properties of that state, no matter howthe steady state was created (except for the “mass” termthat is determined by the external field explicitly). So,the Eq. (2) is capable of describing the perturbationsof the steady state in the absence of rf-field, if the pre-cession of background magnetization has maintained itsgiven angular frequency and phase. Although derivedfor a background represented by a uniform magnetiza-tion flow parallel to the flat top of the cell, the abovepresented relations can be used for qualitative analysisand quantitative estimation of the situation where ∇ α rf and L slightly vary on spatial scales larger than L .As mentioned above, we performed the experiment inthe cell shown in Fig. 1. The cell was attached toKoˇsice’s diffusion nuclear stage [26], filled with He at3 bars and using the adiabatic demagnetization cooleddown to temperature of ∼ c . The temperature ofthe He was measured using a powder Pt-NMR ther-mometer immersed in the liquid and calibrated against
Time, seconds H P D r e s pon s e , m V R M S P = 3 barsT = 0.5 T c FIG. 2: (Color on line) Voltage signals corresponding to spin-precession waves: the source domain (red), the detection do-main (blue). The rectangular signal shows the time windowwhen 8 sinusoidal excitation pulses were applied in order toexcite the spin-precession waves. Insets: the picture of theexperimental cell on the bench before rf-shield installation. the He superfluid transition temperature T c . The HPDswere simultaneously and independently excited in bothcells using cw-NMR method at angular frequency ω rf =2 π. × rad/sec. To achieve this, we used two rf-generators working in phase-locked mode and with zerophase difference ∆ α rf between excitation signals. Theinduced voltage signals from NMR coils were amplified bypreamplifiers and measured by two rf-lock-in amplifiers,each controlled by its own generator. In order to reducethe mutual crosstalk between the rf-coils, each rf-coil wascovered by a shield made of a copper foil. The longitu-dinal coils provided an additional alternating magneticfield used for the spin-precession waves generation [22].Once two HPDs were generated, the position of the do-main wall was adjusted into channel. This step is easy toaccomplish by means of the homogeneous field B (withaid of a small longitudinal oscillations), as the position ofthe domain wall follows the plane where the Larmor reso-nance condition is satisfied. Specifically, the precision ofthe domain wall adjustment is given by ∆ B / ∇ B , where∆ B = 0 . µ T is the field step controlled by the currentsource and for ∇ B =15 mT/m giving a spatial precisionof ∼ µ m [27]. For comparison, the domain wall thick-ness ( λ F = c / L / ( γ ∇ Bω rf ) / ) for the above parametersis ∼ L in the channelis L ∼ µ m ± µ m. The spin flow between HPDscan be established in both directions depending on thesign of the phase difference ∆ α rf , while the spin flow ve-locity u depended on the magnitude of ∆ α rf . The detailsof the experiment are provided in [29]. The spin-precession waves in the source domain weregenerated by 8 sinusoidal pulses at an appropriate lowfrequency using the separate generator. The low fre-quency response from the source and detection HPD fora particular value of ∆ α rf was extracted from rf-signalby a technique based on application of a rf-detector anda low-frequency filter. The low frequency signals werestored by a digital oscilloscope for the data analysis. Theexamples of the signals representing the excited spin-precession wave in the source domain and incoming wavein the detection domain are showed in Fig. 2. Whenthe pulse is finished, there are clear free decay signals ofthe spin-precession waves from both domains, and theseparts of the signals were analyzed by the methods of thespectral analysis as a function of the phase difference∆ α rf , i.e. as a function of the spin flow velocity u . -7 -8 F r equen cy , H z F r equen cy , H z P ha s e d i ff e r en c e ∆ α , deg r ee r f P ha s e d i ff e r en c e ∆ α , deg r ee r f Horizon is formed P o w e r s pe c t r a l den s i t y , V / H z P o w e r s pe c t r a l den s i t y , V / H z HPDSD HPDSD
SOURCE DOMAIN DETECTION DOMAINSPIN FLOW
HPDSD HPDSD
SOURCE DOMAIN DETECTION DOMAINSPIN FLOW
WHH
FIG. 3: (Color on line) The power spectral density (PSD)of the free decay signals as function of the phase differencemeasured from the source domain (upper) and the detectiondomain (bottom). Insets show a schematic illustration of thespin waves dynamics in channel.
Figure 3 shows the power spectral density (PSD) ofthe free decay signals for the source (upper) and detec-tion (lower) domains as a function of the phase difference∆ α rf . There are a few remarkable features presentedthere. Firstly, there are relatively strong PSD signals inthe source domain with an exception of a deep minimumat region of ∆ α rf corresponding to ∼ ◦ . Secondly, theweaker PSD signals are observed in the detection domainin the range of negative values of ∆ α rf up to 10 ◦ , abovewhich strong PSD signals were measured. Thirdly, noPSD signals within experimental resolution were mea-sured in the detection domain for values of ∆ α rf > ∼ ◦ .How can one interpret these data?For the negative values of ∆ α rf , the spin super-currents flow from the source domain towards the detec-tion one. Thus, the spin-precession waves excited in thesource domain are dragged by the spin super-currents andthey travel downstream to the detection domain, wherethey are detected. The amplitude of the detected waves isreduced by process of the energy dissipation inside chan-nel due to spin diffusion, and by the spin flow modifyingthe frequency of spin-precession waves which is slightlydifferent from the resonance frequency of the standingwaves.The deep signal minimum in the source domain andcorresponding maximum in the detection domain for∆ α rf ∼ ◦ - 15 ◦ is consequence of zero spin flow betweendomains that leads to a resonance match [30]. Therefore,when the spin-precession waves are excited in the sourcedomain at this condition, all energy is transferred to andabsorbed by the detection domain at once. For ∆ α rf > ◦ the direction of the spin flow is reversed, i.e. thespin super-currents flow towards the source domain andthe emitted spin-precession waves propagate against thisflow. The change in direction of the spin super-currentsis also seen on the phase of the decay signal from thesource domain as the gradual phase shift by 180 ◦ [ ? ].As one can see from Fig. 3, there are no PSD signalsdetected in the detection domain for ∆ α rf > ∼ ◦ . Weinterpret this as a formation of the white hole horizon inthe channel: spin-precession waves sent from the sourcedomain towards to the detection domain are blocked bythe spin flow and never reach the detection domain. Thisinterpretation is supported by calculation using abovepresented model: the white hole horizon is formed in aplace, where and when the condition c = u is satisfied.For given experimental parameters and assuming that ∇ α rf is localized on the length of dl = 0.5 mm, in orderto satisfy the condition c = u for the phase difference∆ α rf ∼ ◦ , estimated length of the precessing layer L is L ∼ µ m, what reasonably corresponds to theexperimental value [ ? ].Finally, Fig. 4 shows the cross power spectral densitybetween free decay signals measured in source and detec-tor domains as function of the phase difference ∆ α rf . Forvalues of ∆ α rf < ◦ , i.e. when the spin flow drags excitedspin-precession waves from the source domain towardsto the detection domain, the decay signals from both do-mains are correlated. For values of ∆ α rf > ◦ , reductionand and following reversion of the spin flow affects thedynamics of the propagation of the spin-precession wavesthat leads to the change in correlation between the decaysignals detected in both domains. When the spin flow ap-proaches zero value, due to resonance match between the -8 F r equen cy , H z P ha s e d i ff e r en c e ∆ α , deg r ee r f P o w e r s pe c t r a l den s i t y , V / H z HPDSD HPDSD
SOURCE DOMAIN DETECTION DOMAINSPIN FLOW
WHH
HPDSD HPDSD
SOURCE DOMAIN DETECTION DOMAINSPIN FLOW
Horizon is formed
FIG. 4: (Colour on line) The cross-correlation power spec-tral density of the source and detector free decay signals asfunction of the phase difference ∆ α rf . domains, the energy is transferred in both directions thatis manifested as correlation/anti-correlation peaks. How-ever, when the white horizon is formed (∆ α rf > ◦ ), thedecay signals are anti-correlated. We may interpret thisin way that the rise of the decay signal in the source do-main is paid by the spin flow flowing from the detectiondomain towards to the source domain - in agreement withtheoretically predicted amplification of the wave on thehorizon paid by the energy of the flow [31]. This inter-pretation is supported by dependence presented in Fig.3 (upper dependence), where a notable feature regard-ing to the absolute values is showed: when spin currentflows from the detection domain to the source domain,the waves in source domain have a tendency to have ahigher power spectral density amplitude than those forthe spin current flowing in opposite direction. However,to confirm the physical origin of the observed phenomenaadditional measurement have to be done.In conclusion, we performed the experiment in a limitof absolute zero temperature probing the black/whitehole horizon analogues in superfluid He-B using thespin-precession waves propagating on the background ofthe spin super-currents between two mutually connectedHPDs and provided the evidence of the white hole forma-tion for spin precession waves. Moreover, the presentedtheoretical model and experimental results demonstratethat the spin-precession waves propagating on the back-ground of the spin flow between two HPDs possess allphysical features needed to elucidate physics associatedwith the presence of the event horizons, e.g. to test thespontaneous Hawking process. In fact, assuming thatthe spin super-currents velocity of the order of u ∼ dl ∼ − m one can estimate thetemperature of the Hawking radiation in this system tobe of the order of 10 nK, what is a temperature only fourorders of magnitude lower that the background temper-ature, and this makes presented system a promising toolto investigate this radiation [ ? ].We acknowledge support from European MicrokelvinPlatform (H2020 project 824109), APVV-14-0605,VEGA-0128 and erdf-itms 26220120047 (Extrem-II). Wewish to thank ˇS. Bic´ak and G. Prist´aˇs for technical sup-port. 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