magnon bec at room temperature and its spatio-temporal ......e-mail: [email protected]...

ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2020, Vol. 131, No. 1, pp. 83–94. © Pleiades Publishing, Inc., 2020.

Magnon BEC at Room Temperatureand Its Spatio-Temporal Dynamics

S. O. DemokritovInstitute for Applied Physics and Center for NanoTechnology, University of Muenster, Muenster, 48149 Germany

e-mail: [email protected] January 30, 2020; revised January 30, 2020; accepted March 13, 2020

Abstract—Recent advances in the studies of magnon gases have opened new horizons for the investigation ofroom-temperature macroscopic coherent states and discovery of Bose–Einstein condensation of magnons.Although the phenomenon has been discovered almost 15 years ago, a lot of important issues connected withmagnon Bose–Einstein condensation remain unclear. Here I review the recent experimental achievementsin investigations of this phenomenon. I show that the magnon condensate possesses high degrees of temporaland spatial coherency, the latter leading to observation of interference of two condensate. Discovery of secondsound in magnon condensate is also discussed. Finally, I demonstrate a practical way to realize magnon laser,which create a freely propagating cloud of magnons.

DOI: 10.1134/S1063776120070158

1. INTRODUCTIONBose–Einstein condensation (BEC) is one of the

most striking manifestations of quantum nature ofmatter on the macroscopic scale. It represents a for-mation of a collective quantum state of particles withinteger spin—bosons. In 1925 Einstein, using themethod proposed by Bose, has shown that in a gas ofnoninteracting bosons at the thermal equilibrium thedensity of particles is described by the occupationfunction:

(1)where ε is the energy of the particle, T is the tempera-ture, and kB is the Boltzmann constant [1]. The chem-ical potential μ is determined from the condition forthe total density of the particles, N:

(2)

where D(ε) is the density of states of the particles. Asthe density of the particles, N, increases at a given tem-perature T, the chemical potential μ increases as well.On the other hand, it is seen from Eq. (1) that μ cannotbe larger than the minimum energy of the particlesεmin. Thus, for a parabolic dispersion characterized bya mass m Eq. (2) with μ = εmin defines the critical den-sity Nc(T):

(3)

If the density of the particles in the system is largerthan Nc, BEC takes place: the gas is spontaneouslydivided into two fractions: particles with the density Ncdistributed according to Eq. (1) with μ = εmin and par-ticles accumulated in the ground state with ε = εmin.The latter fraction represents a Bose-Einstein conden-sate [1]. I took 70 years to realize the idea of Einsteinexperimentally: BEC was observed in diluted atomicalkali gases at ultra-low temperatures of 10–7 K [2, 3].

Gases of quasi-particles are very attractive objectsfor the observation of BEC, since such extreme exper-imental conditions are not needed for the transition:(i) because of the effective mass of quasi-particles canbe essentially smaller than that of atoms, the BECtransition should occur at smaller densities or highertemperatures; (ii) a large number of quasi-particlesexists at non-zero temperatures due to thermal f luctu-ations; (iii) if necessary, the density of quasi-particlescan be increased using different methods of externalexcitation such as microwave pumping [4] or illumina-tion with laser light [5]. At the same time a possibilityof BEC in quasi-particle gases is not evident from thepoint of view of thermodynamics, since quasi-parti-cles are characterized by a finite lifetime, which isoften comparable to the time needed to reach thermo-dynamic equilibrium [6]. Therefore, BEC in a gas ofquasi-particles can be only realized, if the mean life-time of the considered particles is much longer thantheir thermalization time defined by the scatteringprocesses between the particles [7].

−ε = ε − μ − 1B( ) (exp(( )/ ) 1) ,n k T

∞

ε

= ε ε εmin

( ) ( ) ,N D n d

= �

3/2

B 2( ) .3.31

cmN T k T

83

84 DEMOKRITOV

Fig. 1. (Color online) Energy spectrum of magnons in a ferromagnetic films magnetized by an in-plane static magnetic field. Thelow frequency part of the magnon spectrum is shown in the log-log-scale. The wavevector intervals of parametrically injectedmagnons are indicated by ellipses. Shown is also the wavevectror interval accessible for BLS measurements limited by ±K.

Frequency, GHz10

8

6

4

2

−106 −104 −102 102

Wavevector, cm−1104 106

Photon(2νp, 0)

Magnon(νp, kp)

Magnon(νp, −kp)

−K K

k || H k ⊥ H

2. EXPERIMENTAL OBSERVATION OF MAGNON BEC

The experiments on room-temperature BEC ofmagnons were performed on monocrystalline films ofyttrium iron garnet (YIG). YIG (Y3Fe2(FeO4)3) is acubic ferrimagnetic insulator [8]. The magnon lifetimein YIG is very long (>200 ns), it is much longer thanthe characteristic time of magnon-magnon interac-tion τss < 30 ns [9]. Thus, thermalization of injectedmagnons can be achieved in the magnon gas, which ispractically decoupled from the crystalline lattice.Under such circumstances a quasi-equilibrium mag-non gas can be characterized by its own temperatureand the chemical potential, which can deviate fromthose of the lattice.

Figure 1 shows the low-frequency part of the mag-non spectrum in a ferromagnetic film magnetized byan in-plane static magnetic field. The spectrum ofmagnons is determined by two interactions: theexchange interaction and the magnetic dipole-dipoleinteraction. The exchange interaction dominates formagnons with large wavevectors and leads to the qua-dratic dependence of frequency of magnons on theirwavevector (f ~ k2), which is reminiscent to the spec-trum of free massive particles (ε = (2m)–1p2). In con-trast, the long-range dipole-dipole interaction domi-nates in the region of relatively small wavevectors.Since this interaction is anisotropic, the spectrum ofmagnons in this wavevector interval is anisotropic aswell, i.e., it depends on the angle between the vector ofthe static magnetic field, H0, and the magnonwavevector, k. For k || H0 the dipole contribution

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decreases with increasing wavevector. Therefore, ifboth the exchange and the dipole-dipole interactionsare considered [10], the lowest-frequency dispersioncurve of magnons with k || H0 demonstrates a mini-mum at kmin ≠ 0. This minimum corresponding to theabsolute minimum of magnon frequency is clearlyseen in Fig. 1. For 5 μm thick YIG films used in theexperiment and a typical value of the applied magneticfield of H0 = 700 Oe the minimum magnon frequencyis fmin = 2.10 GHz. It corresponds to the wavevectorkmin ~ 3.5 × 104 cm–1.

Figure 1 also illustrates the process of parametricpumping of the magnon gas. To realize this pumpingone has to apply a dynamic microwave magnetic fieldoriented parallel to the direction of the applied staticmagnetic field H0. The spatially uniform pumpingmicrowave field with the frequency fmw = 2fp causesparametric excitation of two primary spin waves withthe frequency fp and equal and oppositely directedwavevectors ±kp. A typical setup for magnon exci-tation and observation of the room-temperature BECof magnons is schematically shown in Fig. 2. Single-crystalline YIG film with the thickness of 5 μm wasmounted onto a microstrip resonator with resonantfrequency of 8.10 GHz providing an intense electro-magnetic pumping field. The system was placed into aspatially uniform static magnetic field H0 oriented inthe film plane. To perform time-resolved measure-ments and to avoid an overheating of the YIG sampleby microwaves, the pumping was performed in thepulsed regime. The duration of pumping pulses wasτp = 1 μs, which was sufficient long to allow the inves-

D THEORETICAL PHYSICS Vol. 131 No. 1 2020

MAGNON BEC AT ROOM TEMPERATURE 85

Fig. 2. (Color online) The experimental setup for time-resolved BLS measurements of distribution of magnons over frequenciesin a parametrically driven magnon gas.

interferometerFabry-Perot

Pulsegenerator

MWrgenerat o

Single-frequency laser

Magnet

Objective

Pumping pulse

Sample

interferoFabry-P

Pulsgenera

ObjectiveOO

tigation of the evolution of the magnon gas during thepulse and to observe formation of the condensate. Therepetition period of tp = 20 μs was chosen to be largeenough in order the magnon gas to return to its initialthermal state between pulses.

In order to probe the distribution of magnons overfrequencies the Brillouin light scattering (BLS) spec-troscopy was used [11, 12]. A probing beam from a sin-gle-frequency laser with the wavelength of 514 or532 nm was illuminating the film. Using differentoptical tools light scattered in rather large interval ofscattering angles was collected and sent to a multi-passtandem Fabry-Perot interferometer for a frequencyanalysis. This geometry provided an efficient way todetect magnons in a wide interval of the in-planewavevectors up to the maximum wavevector of K = 2 ×105 cm–1 as indicated in Fig. 1. Thus, the measuredBLS intensity I(f) is proportional to the reduced spec-tral density of magnons, I(f) ~ (f)n(f), where (f) isthe density of magnon states taking into account onlythe magnons accessible for BLS (i.e., the magnonswith in-plane wavevectors |k| < K) and n(f) is the occu-pation function of magnon states. Since (f) can beobtained independently, the measured spectrum pro-vides a direct way to determine n(f) experimentally andto detect formation of BEC. As also indicated inFig. 2, the analysis of scattered light was performed ina stroboscopic regime allowing temporal resolution.

Figures 3a and 3b show the measured BLS spectraat different delay times reflecting the quasi-equilib-

�D �D

�D

JOURNAL OF EXPERIMENTAL AND THEORETICAL PH

rium distributions of magnons over the phase space attwo pumping powers P = 4 and 5.9 W, correspond-ingly. Tokens in the figures represent the experimentaldata, solid lines are the distribution calculated basedon the Bose–Einstein statistics, using μ as the fitparameter. As seen in Fig. 3a, at pumping power P =4 W the chemical potential grows with time, reachingsaturation at μ/h = 2.08 GHz. This value is close butstill below fmin. Apparently higher values of μ cannot bereached at this pumping power, since the pumpedmagnons leave the magnon gas due to spin-latticerelaxation. Figure 3b illustrates the process at P =5.9 W. For this pumping power the maximum valueμ/h = 2.10 GHz is reached already after 300 ns. Onecan conclude that the critical density of the magnongas Nc is achieved at 300 ns, and the correspondingdistribution can be considered as the critical distribu-tion nC(f). Further pumping leads to a phenomenon,which can be indeed interpreted as Bose–Einsteincondensation of magnons: additionally pumped mag-nons are collected at the bottom of the spectrum with-out changing the population of the states with higherfrequencies. The latter fact is demonstrated by Fig. 3bas well, showing the high-frequency parts of the mag-non distribution curves in an appropriate scale. Thesedata demonstrate that the BLS spectra for t > 300 nscannot be described just by increasing the temperaturein the Bose–Einstein population function, since ahigher temperature means higher magnon populationsat all frequencies. Thus, Fig. 3b indicates formation ofa Bose–Einstein condensate of magnons. One can

YSICS Vol. 131 No. 1 2020

86 DEMOKRITOV

Fig. 3. (Color online) (a) BLS spectra from pumped magnons at the pumping power 4 W at different delay times, as indicated.Solid lines show the results of the fit of the spectra based on the Bose–Einstein statistics with the chemical potential being a fitparameter. Note that the critical value of the chemical potential cannot be reached at the used power. (b) Same as (a) for thepumping power 5.9 W. The critical value of the chemical potential is reached at 300 ns.

0.4 τ = 1000 ns τ = 500 ns

×60400 ns

300 ns

200 ns

μ/h = 2.08 GHz

μ/h = 2.10 GHz

2.04 GHz

1.96 GHz

1.70 GHz2.05 GHz

600 ns

400 ns

200 ns

Inte

nsity

, cou

nts/

ms

0.2

01.5 2.0

Frequency, GHz2.5 3.0 3.5

2

Inte

nsity

, cou

nts/

ms

1

01.5 2.0

Frequency, GHz2.5 3.0 3.5

build a difference between a distribution at a giventime t > 300 ns and the critical one. One sees fromFig. 3b, that this difference is non-zero just in theregion close to fmin, the width of the region Δf ≈ 0.2–0.3 GHz being defined by the resolution of the spec-trometer. Optical measurements with the ultimate res-olution have shown that the intrinsic width of theregion is about 700 kHz [13], which corresponds to ahigh degree of coherence of magnons in the conden-sate, giving Δf < 10–6 kBT/h. Thus, the narrowing ofthe magnon distribution with respect to that deter-mined by the classical Boltzmann statistics is morethan six orders of the magnitude.

3. SPATIAL COHERENCE OF THE CONDENSATE AND QUANTIZED VORTICES

As seen in Fig. 1, the lowest-energy magnon state isdoubly degenerate. Therefore, the condensation spon-taneously occurs at two non-zero values of the wavevector k = ±kmin = ±kBEC. This leads to a degeneracyof the condensate ground state and coexistence of twospatially overlapping wave-functions ψ+ and ψ–, thatcorrespond to ±kBEC. The structure of these wave-functions, including the information about their rela-tive phases, can be probed by a visualization of the den-sity of the total wave-function |ψ|2 = |ψ+exp(ikBECz) +ψ–exp(–ikBECz)|2. If the two wave-functions arephase-locked to each other, their interference leads to


a standing wave of the condensate density in the realspace. From experimental point of view, however,observation of this standing wave is rather challenging,since one needs a sub-micrometer spatial resolutiondue to a large value of kBEC and small spatial period ofthe wave.

To probe the condensate density with spatial reso-lution, we use an experimental setup presented inFig. 4a, which is similar to that shown in Fig. 2. Themain difference between the setups is that here thelaser beam was tightly focused. The locally detectedBLS intensity is proportional to the squared amplitudeof the magnetization precession, i.e. to the total con-densate density |ψ|2 = |ψ+ + ψ–|2 at the point of obser-vation. Therefore, by scanning the probing laser spotin the two lateral directions and recording the BLSintensity, spatial distribution of the local condensatedensity can be visualized. The mapping was performedby repetitive scanning of the spatial area followed bythe averaging of the recorded data to improve the sig-nal-to-noise ratio. Besides, since the pumping micro-wave power was supplied continuously, the createdcondensate continuously existed over the entire mea-surement time, which last up to several days.

Figure 4b shows the results of a two-dimensionalmapping of the condensate density across an 8 ×5 μm2 area close to the resonator as a pseudo-colourplot with increasing density from blue via green to redlight. The map clearly demonstrates a periodic pattern



along the direction of the static magnetic field createdas a result of interference of the two components of themagnon condensate. The spatial period of the pattern0.9 ± 0.1 μm obtained from Fourier transform of therecorded map, agrees well with the period 0.90 μmcalculated based on the known value kBEC = 3.5 ×104 cm–1. The detected periodic modulation of thecondensate density clearly confirms the existence oftwo spatially coherent wave-functions in the magnoncondensate. Moreover, the observation of the patternin the long-term repetitive measurements clearlyshows that the two components of the condensate arephase locked. If this were not the case, the f luctuationsof the phase difference between the wave-functions ψ+

and ψ– would lead to changes in the spatial positionsof the maxima and minima in the pattern with time. Asa result, the pattern would be washed out due to theaveraging during the long-term measurements. Weassociate this phase locking with interaction of mag-nons belonging to the two condensate componentsresulting in their phase coherence.

Another interesting feature of the interference pat-tern in Fig. 4b is the presence of topological defectsmarked by dashed circles. These defects correspond tosingularities of the phase of the individual componentsψ+ and ψ–. To illustrate this, we draw contours aroundthe defects and calculate the phase shift over them. Itis obvious that the phase is constant as one movesalong a line of max/min density in the map and thatthe phase changes by 2π, as one moves from amax/min line to the neighboring one. When calculat-ing the phase shift over the shown contours accordingto these rules, one gets 2π in either ψ+ and ψ– for bothdefects. The experimentally observed patterns arerather stable. This was proven by repeating the map-ping at different pumping powers, as well as afterswitching the power on and off. These measurementshave shown that the vortex positions as well the posi-tions of the stripes in the patterns stay unchanged.

To understand the physical origin of the observedperiodic patterns and topological defects and the roleof the phase locking between the two condensate com-ponents for their formation, we model the condensateusing two coupled generalized Ginzburg–Landauequations following the approach previously appliedto non-equilibrium condensates in [14–16]. Interac-tions between condensate components are taken intoaccount by introducing additional terms describingfour-magnon interaction processes [17, 18] thatdestroy a pair of non-condensed magnons and createa pair of magnons with wave vectors kBEC and vice


versa. The resulting system of coupled equations forwave-functions ψ+ and ψ– becomes:

(4)

Here U0 is the strengths of the pseudo-potentialsresponsible for self-interaction (we neglect the cross-interaction between the components), m|| ≈ me andm⊥ ≈ me/15 are the longitudinal and transverse massesof magnons, respectively, with me being the free elec-tron mass. P(ψ±) describes the f low of magnons intothe condensate due to the parametric pumping andtheir annihilation due to magnon-magnon interac-tions and spin-lattice relaxation. Based on theapproach from [14, 15], P(ψ±) can be written phenom-enologically as P(ψ±) = γeff – i η∂t – Γ|ψ±|2, providingthe simplest form of a saturating gain. Here γeff is aneffective gain, that takes into account both the pump-ing and the linear annihilation of magnons, ηdescribes the energy relaxation via interactions of thecondensate with non-condensed magnons, andΓ describes the nonlinear reduction of the pumpingefficiency at large magnon densities. The last term inEq. (4) is responsible for the phase locking between ψ+and ψ–.

Epitaxial YIG films possess exceptionally highcrystallographic quality with a remarkably low numberof crystalline defects [19]. However, these defects playa decisive role in the pinning of topological structuresin the magnon condensate. In non-equilibrium sys-tems there always exist velocity f luxes connectingregions where the particles are created to the regionswhere they are annihilated. Therefore, topologicalstructures such as vortices and solitary waves are sta-tionary only, if they are pinned by forces counteractingthe f low drag. To model the effect of the crystallinedefects, we shall assume that in a certain region of thesample there exist some localized areas causing anadditional annihilation of magnons. We model thisincrease by a spatially dependent Γ = Γ(r) defined asΓ = 0.1 ⋅ U0 outside the defect area and Γ = 0.3 ⋅ U0inside the defect area. In the simulations we observedseveral possible stationary vortex configurations: onlyone component of the condensate had a vortex pinnedto one of the defects, each component had one vortex,and these two vortices were pinned either to the samedefect or to different defects. The latter case is appar-ently realized in the experiment, as corroborated bythe striking similarity between Figs. 4b and 4c showinga pseudo-colour plot of the density of the steady wave-function |ψ|2, obtained for this configuration bynumerical integration of Eq. (4) for γeff = μC, η = 0.1and J = 0.1μC. In addition to the interference patterndemonstrating phase locking between ψ+ and ψ–, one

±⊥

± ± ±

∂∂∂ ψ = − −

−μ + ψ + ψ ψ + ψ

∓

��

22

||

20

2 2

*| | ( ) .

yyzzt

C

im m

U iP J

�


88

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 131 No. 1 2020

DEMOKRITOV

Fig. 4. (Color online) Schematic of the experiment and results of two-dimensional imaging of the condensate density. (a) Experimentalsetup. Magnons are injected into the YIG film using a microwave resonator. After thermalization they create a Bose–Einstein conden-sate, which is imaged by scanning the probing laser light in two lateral directions. (b) Measured two-dimensional spatial map of the BLSintensity proportional to the condensate density, obtained at the maximum used pumping power. Dashed circles show the positions oftopological defects in the standing-wave pattern. (c) calculated spatial map of the condensate density. Dashed circles indicate the posi-tions of two defects causing an appearance of two vortices of positive circulation in different components of the condensate. The vorticesshow themselves as forks in the interference pattern.

H0

Microwave

(a)

(b)

(c)

4

3

2

1

00 2 4 6 8 10

1.03

1.01

0.99

0.97

0.95

y, μ

m

z, μm

0.5

1.0

1.5

2.5

2.0

3.5

3.0

4.5

4.0

5.0

0 21 43 6 7 8

1.00.90.80.70.60.50.40.30.20.10

5

y, μ

m

z, μm

resonator

Probinglight


Fig. 5. (Color online) (a) General view of the experimental system. Dielectric resonator creates a microwave-frequency mag-netic field, which parametrically excites primary magnons in the YIG film. Electric current in the control line placed betweenthe resonator and the YIG film, produces a non-uniform magnetic field, which adds to the uniform static field H0. The localdensity of condensed magnons is recorded by BLS with the probing laser light focused onto the surface of the YIG film.(b) Cross-section of the experimental system illustrating the field created by the control line. (c) Sketch of the spatial distri-bution of the horizontal component of the total magnetic field H0 + ΔH and the corresponding spatial profile of the conden-sate density caused by the inhomogeneity of the field.

x

y

zProbinglight

Dielectric resonator

Control

YIG

line

Dielectric resonator

YIGH0

H0

Probinglight

Condensatedensity

Magneticfield

ΔH

(a) (b)

(c)

ΔH

clearly sees in Fig. 4c two forks corresponding to the2π shift of the phase difference between ψ+ and ψ–.These forks mark two vortices, each of them existing inone condensate component only.

Thus, the above findings confirm a long-term spa-tial coherence of magnon. Caused by the double-degeneracy of the lowest-energy magnon state and thepresence of nonlinear magnon-magnon interactions,the ground state of the condensate appears as a real-space standing wave of the total condensate densityoriginating from the interference of the condensatecomponents. Apart from revealing the spatial coher-ence of BEC, this interference enables an easy obser-vation and clear identification of complex topologicalstructures, such as quantized vortices.

4. SECOND SOUND IN MAGNON GAS

An important step in investigation of spatio-tem-poral dynamics of magnon BEC is made by introduc-tion of a dielectric resonator as a source of microwavepumping field, as illustrated in Fig. 5a. In the investi-gation of second sound a dielectric resonator with theresonance frequency of ωmw/2π = fmw = 9.0 GHz isused to inject magnons of a frequency ωp/2π = fp =fmw/2 = 4.5 GHz in a relatively large area. The setupwas placed in the static magnetic field H0, which wasaligned in the film plane and was equal to 1.33 kOe for


the most of the experiments. As already mentionedabove, thermalization of the injected magnons resultsin the formation of a quasi-equilibrium magnon gaswith a non-zero chemical potential, and also in thelarge overpopulation of the low-energy magnon states,eventually leading to the formation of BEC. The cor-responding BLS spectrum obtained at a microwavepumping with the power of Pp = 0.25 W is shown inFig. 6a. In agreement with above consideration, twodistinct peaks are seen in the spectrum: a relativelyweak peak at fp = fmw/2 = 4.5 GHz, corresponding tothe primary magnons created in the YIG film by themicrowave pumping, and a substantially more inten-sive peak at the frequency fp = 3.8 GHz caused by thequasi-equilibrium, thermalized magnons accumu-lated at the ground sate of the magnon spectrum. Theintensity of each of these two peaks is proportional tothe density of magnons near the corresponding fre-quency. By varying the pumping power one can con-trol the relative density of the accumulated magnons atdifferent frequencies, as illustrated in Fig. 6b, wherethe ratio between the amplitudes of the two abovementioned BLS peaks is plotted versus the pumpingpower. One clearly sees that with the increase of thepumping power above 1 W the ratio between the num-ber of the ground state magnons and the number ofprimary magnons increases substantially. As demon-


90 DEMOKRITOV

Fig. 6. (Color online) (a) Experimentally measured BLS spectrum, comprising peaks corresponding to the pumped magnons(fp) and thermalized gas of magnons (fg = fmin) (cf. Fig. 1); (b) Ratio of intensities of the BLS peaks shown in Fig. 6a as a func-tion of the pumping power; (c) Spatio-temporal map (t—time from the start of the rf-signal, z—the distance from the controlline of the measured BLS intensity for the thermalized magnon gas with red/blue color corresponding to the highest/lowestintensity. Inclined lines indicate propagating second sound waves. (d) the same as (c) for primary magnons. No propagatingsecond sound waves are seen.

f, GHz P, W3.6

1.0n,

a.u

.z,

μm

z, μ

mn g

/np

0.8

0.6

0.4

0.2

70

60

50

40

30

20

10

70

60

50

40

30

20

10

0.5t, μs

1.0 0.5t, μs

1.0

0

0 0

0

50

100

150

200

250

0

4.0 4.4 4.8 1 10

(a)

fp

PthBEC

fg

(b)

(c) (d)

strated by a direct observation of the interference oftwo magnon condensates in the real space [20], thepoint of the drastic growth of the density of theground-state magnons marks the onset (or threshold)of magnon BEC.

The main advantage of a dielectric resonator as asource of microwave pumping field is the possibility tointroduce into the setup a control strip line (seeFigs. 5a, 5b), which allows to create an additional spa-tially inhomogeneous magnetic field ΔH induced byan electric current f lowing in the line. For one direc-tion of the current one obtains underneath of the con-trol line a potential well, whereas for another directionof the current a potential hill is formed. The createdgradient of the field is responsible for lateral f lows ofmagnons and corresponding variation of the conden-sate density: the density increases in the well, as illus-trated in Fig. 5c, whereas it decreases in the case of the


hill. If, however, a radio-frequency current f lowsthrough the control line, it causes a periodic oscilla-tion of the magnon density underneath of the controlline. The intriguing question here is whether this oscil-lation can excite waves of magnon density propagatingaway from the control line, which is nothing else as themagnetic second sound. Figure 6c gives a clear answerto this question. It shows a spatio-temporal map of thedensity of ground-state magnons, where the colorcode reflects the magnon density, with red/blue colorcorresponding to the highest/lowest density. obtainedby sweeping the laser beam away from the control line,and using the time-resolved BLS spectroscopy. Themap was recorded for the driving frequency frf =3 MHz and pumping power Pp = 10 W, correspondingto a well formed BEC. One observes in Fig. 6c propa-gating waves of the magnon density, with the slope ofthe lines of the constant density in Fig. 6c being pro-



Fig. 7. (Color online) Dispersion of the magnonic sound at two values of the pumping power (a) above (Pp = 2.5 W) and (b) below(Pp = 0.5 W) the BEC transition. Symbols are the experimental data for the real and imaginary parts of the magnonic sound wav-enumber. Solid lines are the best theoretical fits using the model developed in [21]. Dashed lines show the linear dispersion withconstant sound velocity and constant damping.

(b)(a)

Pp = 0.6 W

K '

K ', K '', 104 rad/m

K ''

Pp = 2.5 W

K '

K ''

3

6

9

12

0 1 2 3 4 5fK, MHz

K ', K '', 104 rad/m

3

6

9

12

15

18

0 1 2 3 4 5fK, MHz

portional to the phase velocity of the waves. In con-trast, in the ensemble of primary non-equilibriummagnons one observes just a weak periodic modula-tion of the magnon density, but no signature of propa-gating waves (see Fig. 6d). By analyzing the spatio-temporal maps similar to that shown in Fig. 6c weobtained the real and imaginary part of the complexwavenumber K = K ' + iK '' of the observed propagatingwaves at different frequencies, fK. The functions K '(fK)and K ''(fK) obtained from the fits of the maps corre-sponding to different fK are presented in Fig. 7, illus-trating the dispersion of the second sound for two val-ues of pumping powers. Figure 7a corresponds to awell-formed condensate, whereas Figure 7b illustratesthe magnonic gas with the density below the criticalone. In both cases K '(fK) and K ''(fK) are completelydifferent: whereas K '(fK) is an almost linear function,K ''(fK) is almost constant. Therefore, at high frequen-cies K '(fK) ≫ K ''(fK), the second sound is weekly dissi-pating. However, dispersion shown in Fig. 7a reveals atiny deviation of K '(fK) from the linear function andK ''(fK) from a constant. Contrarily, these deviation arerather large in Fig. 7b. This behavior can be explainedtheoretically [21], if one takes into account the factthat the momentum and energy relaxation rates in amagnon gas are different and their ratio depends onthe gas density. The same theory also explains that thephase velocity of the second sound for the pumpingpowers far above the critical one does not depend onthe condensate density at a given applied magneticfield, but essentially decreases with increasing field.


5. MAGNON LASER

Although the magnon BEC takes place at nonzerowavevectors, corresponding to a phase velocity aslarge, as several kilometers per second, the magnongroup velocity at the ground state is zero. Therefore,the condensate cloud is at rest, if a homogeneous andtime-independent magnetic field is applied. However,if one puts coherent condensate into constant-velocitymotion, a magnetic analogue of an optical [22] oratom laser [2, 23, 24] will be implemented. Here werely on the definitions of atom lasers, proposed byRobins et al. [25]: “a device that produces a guided orfreely propagating beam of bosons that have been out-coupled from one or more macroscopically-occupiedtrapped modes”. In fact, atomic and magnon conden-sates possess a lot of similarities: in both cases, a quasi-equilibrium bosonic gas is formed either by coolingand thermalization of atoms [2] or by injection ofadditional magnons and their thermalization [9, 17].An atom laser utilizes a transfer of trapped atoms intoan untrapped state by radio-frequency radiation [26,27]. In a magnon system, the condensate can bebrought into motion by shifting it away from the statewith zero group velocity by using a time-dependentlocally inhomogeneous magnetic field. It is importantto note that atom and magnon lasers differ dramati-cally from optical lasers, as the dispersion is con-cerned. The dispersion of optical laser pulses is veryweak, since, for light, the dependence of the frequencyon the wavevector is almost linear. On the contrary,the dispersion of atom and magnon laser pulses is sig-nificant, since the corresponding dependence is para-bolic.


92 DEMOKRITOV

Fig. 8. (Color online) Pseudo-color dependencies of the condensate density in time-frequency coordinates recorded underneathof the control line: (a) the field is locally decreased; (b) the field is locally increased. The temporal traces of the field pulse areshown by dashed lines.

(a) (b)2500

2000

1500

1700 1800f, MHz

1900f, MHz

1900 2000 2100

1000

500

0

2500

2000

1500

1000

500

0

t, ns

t, ns

The experimental setup used for this study is verysimilar to that shown in Fig. 5. However, contrary tosecond sound studies, where a radio-frequency alter-native current was applied to the control line, here weapplied current in shape of rectangular pulses.Figure 8 shows the BLS intensity (proportional to thedensity of the condensate) in a YIG films placed in thefield ΔH0 = 660 Oe measured underneath of the con-trol line. First, the field is lowered by ΔH = 63 Oe byapplying a current pulse at t ≈ 0 with a rise/fall time τ =500 ns and the duration at maximum of T = 500 ns(white dashed line Fig. 8a). As seen in Fig. 8a, the con-densate experiences a frequency shift Δf = 160 ±30 MHz. This value is in a good agreement with thatexpected from the variation of the total field: Δfth =γΔH = 175 MHz, where γ is the gyromagnetic ratio ofan electron. After the end of the current pulse, the fre-quency of the condensate returns to its initial value. Asimilar process was studied by Rezende et al. [28] fortravelling backward volume spin waves. As illustratedin Fig. 8b, the behavior changes drastically when thelocal field is increased. Although the frequency of thecondensate first follows the variation of the field, itsdensity strongly decreases with time. At the moment,when the field reaches its maximum value, the con-densate completely disappears. Only after the end ofthe field pulse, the condensate can be observed again.Additional measurements described below show that,in reality, the condensate does not disappear. Instead,it is pushed away from the conductor by the hill-likeinhomogeneous magnetic field profile.

We study the spatio-temporal dynamics of the con-densate by using time- and space-resolved BLS. Wemap the condensate density as a function of time andthe distance from the center of the conductor. Toemphasize the spatio-temporal dynamics of the con-densate, we use smaller duration of the field pulse and


the rise/fall times: T = 30 ns and τ = 10 ns. Figure 9ashows snapshots of the space-resolved one-dimen-sional normalized profiles of the condensate densityrecorded at different delay times with respect to thestart of the field pulse, as labelled. These data clearlyshow that the condensate cloud is pushed away fromthe area of the inhomogeneous magnetic field. Thecloud keeps moving even after the end of the fieldpulse. During the motion, the spatial width of thecloud increases due to the magnon dispersion.Figure 9b shows the temporal dependencies of theposition of the center of the cloud and of its spatialhalfwidth at half maximum determined from theGaussian fit of the measured profiles. The data clearlyshow that the cloud propagates with a constant veloc-ity, as emphasized by the straight dashed line. Fromthe slope of the line, we obtain the condensate velocityof 930 m/s. Additional experiments show that thisvelocity monotonically increases with increasing ΔH.It is important to note that the integral magnon den-sity shows almost no variation during propagation, asillustrated in Fig. 9c. In other words, the total numberof magnons in the cloud remains constant within theexperimental accuracy. The observed behaviors (con-stant velocity and constant number of magnons in thecloud) might be interpreted as an indication of mag-non superfluidity. However, one should take intoaccount that the moving condensate cloud is not iso-lated. In fact, it receives additional magnons due to thecontinuous pumping by the microwave field and thefollowing condensation of the pumped magnons,which compensates the magnon decay due to thedamping. Nevertheless, it is obvious from the experi-mental data that the moving cloud remains in a certainquasi-equilibrium state described by a constant num-ber of magnons.



Fig. 9. (Color online) (a) Spatial profiles of the condensate density at different delays with respect to the start of the field pulse,as labelled. (b) Position of the center of the magnon cloud and its spatial width as a function of the propagation time. The straightline shows the linear fit of the experimental data, emphasizing the constant-velocity movement of the cloud. (c) Temporal depen-dence of the total number of magnons in the cloud obtained from the integration of the condensate density. Dashed horizontalline is a guide for the eye.

7

6

(a) (b)

ExcitationHalfwidth

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(c)

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mal

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ate

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ity

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mIn

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al, a

.u.

0 50 100 200150

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t, ns

To gain further insight into physical processesunderlying the creation of a moving condensate cloudand determining the operation of the magnon laser, weconsider the magnon dispersion spectrum in thevicinity of the minimum-frequency state (Fig. 10).The middle curve in Fig. 10 shows the dispersioncurve for magnons at H0, which describes magnons inthe absence of the inhomogeneous field created by thecurrent in the conductor. It is also applicable at alltimes for magnons outside of the conductor area. Themiddle circle indicates the spectral state with thewavevector kBEC, at which the condensate is formedbefore the current pulse is applied. The lower curve isthe dispersion curve at H0 – ΔH. It describes magnonsclose to the center of the conductor, when the field islowered, whereas the lower circle marks the spectralstate of the condensate at this field. Note, that thisstate corresponds to the global minimum of the con-densate frequency in all spatial positions. Therefore, alocal decrease of the field creates a potential well trap-ping the condensate. Moreover, due to the gradient ofthe potential, one expects a f low of the condensatetoward the center of the well resulting in an increase ofthe condensate density underneath of the control line,in agreement with the experimental data shown inFig. 8a.


The upper curve in Fig. 10 is the dispersion curveat H0 + ΔH. It characterizes magnons close to the cen-ter of the control line, when the field is increased,whereas the upper circle marks the spectral state of thecondensate at this field. Apparently, the situation inthe case differs dramatically from that in the case ofthe local field reduction. The frequency of the con-densate close to the control line is higher than that inthe surrounding film. As a result, the condensate canevolve into states with the wavevectors which are largerand/or smaller than kBEC possessing a non-zero groupvelocity (arrows in Fig. 10). It is important to note thatthis evolution does not require changes in the energyof magnons building the condensate. In other words, alocal increase of the magnetic field results in an out-coupling of magnons from the condensate and cre-ation of moving cloud of condensed magnons, i.e.,realization of a magnon laser.

6. CONCLUSION

One can see from the above that discovery of room-temperature magnon BEC made 15 years ago, broughtabout a very unusual and interesting object for investi-gation of spatio-temporal dynamics of macroscopic


94 DEMOKRITOV

Fig. 10. (Color online) Calculated magnon dispersioncurves for three different values of the magnetic field. Mid-dle curve: dispersion at the field H0 without additionalinhomogeneous field. Lower curve: dispersion at the fieldH0 – ΔH. Upper curve: dispersion at H0 + ΔH. The circlesmark the lowest-frequency spectral states corresponding tothe equilibrium condensate at the corresponding field val-ues. Arrows illustrate transition of the condensate intostates with non-zero group velocity resulting in its motion.

Freq

uenc

y

kBEC Wavevector

H0

H0 − ΔH

H0 + ΔH

coherent state. It has also revealed very interestingconcepts of quasi-equilibrium bosonic systems.

ACKNOWLEDGMENTS

Finally, I would like to emphasize with many thanks a deci-sive role of A.S. Borovik-Romanov in my choice of magneticdynamics as the main direction of my scientific carrier.

FUNDING

This work is a short review of original works performed atMünster University during more than a decade with financialsupport of Deutsche Forschungsgemeinschaft and EuropeanUnion. This work supported in part by the Deutsche For-schungsgemeinschaft (project no. 416727653). I would likestress important contributions of V.E. Demidov,O. Dzyapko, G.A. Melkov, P. Nowik-Boltyk, N. Berloff,A.N. Slavin, and V. Tiberkevich to the above reported studies.

ADDITIONAL INFORMATION

This article was prepared for the special issue dedicatedto the centenary of A.S. Borovik-Romanov.


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