.O. Demokritov niversität Münster, Germany

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1 Quantum Thermodynamics of Magnons.O. Demokritov niversität Münster, Germany Magnon Frequency Population BEC-condensates k z k y Group of NonLinea Magnetic Dynamic

2 Spin dynamics B M M () t

3 B M M () t Spin dynamics precession frequency = d /dt p dl=ld L=I m D r F=mg Angular momentum L = Iω Acting torque: D dl D= r F = r mg dt dl D Since: D L L = const; dl = Ld dl dϕ D = = L = Lω p dt dt D mg ω p = = L Iω Frequency of precession

4 Spin dynamics B M M () t M () t B precession frequency = d /dt D p r dl=ld m L=I F=mg

5 B M Spin dynamics i t mrt (, ) = m( r) e ω M () t 0 dynamic magnetization M () t B 1 dm ( t) γ dt = M () t B precession frequency = d /dt D p r dl=ld m L=I F=mg Spin dynamics is precessional dynamics!!!

6 Spin waves i( t kr) mrt (, ) = m( r) e ω 0 q 1 dm ( t) γ dt q = M () t B eff B = B + B + B + B eff a dip ex an

7 Magnons S = Ns, Ns 1, Ns 2,... z z z z Spin waves are quantized Spin-wave quanta: MAGNONS Sz = Nsz Sz = Nsz 1 Creation of one magnon Magnons carry out spin 1 => Magnons are Bose-particles Magnons Phonons Magnon spectrum is anisotropic Frequency Phonons Wavevector Magnons

8 Quantum statistics Quantum statistics is important, if: Temperature is low or Particle density is high

9 Widely spread confusion about spin waves Frequency (Hz) exchange quantum dipole classic Wavevector (1/cm)

10 Widely spread confusion about spin waves Frequency (Hz) exchange quantum dipole classic quantum Wavevector (1/cm) Spin waves are quantum objects even at small wave vectors

11 Brillouin light scattering (classic consideration) ( ω ) n exp i t + cc.. ω ω LS-satellites result from phase-(or amplitude, polarization plane) modulation of the cident light wave hey are symmetric, their amplitude being proportional to n

12 Brillouin light scattering (quantum consideration = inelastic scattering of photons by magnons I S N+1 I AS N BLS-satellites result from absorption (radiation) of a magnon by photon They are symmetric in frequency, but different in amplitude N + 1 ω = exp N kt

13 Experimental observation N + 1 ω = exp N kt T = 300 K, ω = 2 GHz ω 1 kt 3000 N + 1 = N T = 4 K, ω = 40GHz ω 0.5 N + 1 = 1.5 kt N

Experimental observation N + 1 ω = exp N kt T = 300 K, ω = 2 GHz ω 1 kt 3000 N + 1 = 1.0003 N 500 400 300 200 100 EuTe T = 4 K B = 2.

14 Experimental observation N + 1 ω = exp N kt T = 300 K, ω = 2 GHz ω 1 kt 3000 N + 1 = N EuTe T = 4 K B = 2.5 T q = 10 5 cm -1 T = 4 K, ω = 40GHz 0 ω 0.5 N = 1.5 kt N Frequency shift (GHz) Demokritov et al. JETP Lett. 8

15 Bose-Einstein-Condensation of atoms classical gas quantum gas De Broglie wavelength: 2π λ = kt p p 2 = 2mkT p 2 2m 2π 2π λ = = 2 p 2mkT

16 Bose-Einstein-Condensation of atoms classical gas quantum gas De Broglie wavelength: 2π λ = kt p p 2 2m Condition of BEC transition: p 2 = 2mkT 2π 2π λ = = 2 p 2mkT λ λ c c r = 2 1/3 2π = 2mkT c N N 1/3 ktc 2π m N

17 Bose-Einstein-Condensation of atoms classical gas quantum gas BEC Condition of BEC transition: A. Einstein: λ λ c c r = 2 1/3 2π = 2mkT c N N 1/3 ktc = m N 2 3

18 Bose-Einstein-Condensation of atoms classical gas quantum gas BEC Condition of BEC transition: 2 λ λ c c r = 2 1/3 2π = 2mkT c N N 1/3 kt N c c = 3.31 N m m = kt

19 Magnons in ferromagnetic films 5 YIG (yttrium-iron-garnet) H= 700 Oe k H Transparent ferromagnet Films 5 µm thick No domains Frequency (GHz) f k H min E = h GHz = min 2 = k 100mK = 10µ ev B Wavevector (1/cm) 5 1 km cm

20 Magnons in ferromagnetic films 5 YIG (yttrium-iron-garnet) H= 700 Oe k H Transparent ferromagnet Films 5 µm thick No domains Frequency (GHz) f k H min Wavevector (1/cm) 5 1 km cm m eff m e Emin = h 2GHz = = k 100mK = 10µ ev ktc B 2 = 3.31 m N 2 3

21 In equilibrium: 20 3 RT : N 10 cm ktc 2 = 3.31 m N 2 3 (Thermo)dynamic of magnons Magnons are quasi-particles with variable N.

22 n In equilibrium: = 1 E µ exp 1 kt (Thermo)dynamic of magnons Magnons are quasi-particles with variable N. In equilibrium (F=F min ). Therefore: E min > 0. µ F = = N 0 No BEC possible.

23 In equilibrium: µ = 0 < Emin at any temperature (Thermo)dynamic of magnons Magnons are quasi-particles with variable N. In equilibrium E min > 0. µ = 0 No BEC possible. In quasi-equilibrium: e τ ss s τ sp ph We can change N Two important time scales: τ ss In YIG: τ ss 10 50ns τ µ s sp τ sp Under external influence magnon gas first thermalizes itself to a quasi-equilibrium with µ 0 and then relax as a whole.

24 Strategy to reach BEC of magnons 1. Inject ultracold magnons (parametric pumping) 2. Wait and see what happens with magnon distribution 2f P MW Photon Frequency Magnons Magnon f P Population BEC-condensates -5x10 4-1x10 5 Magnons created by microwaves and detected by light scattering with time (and space) resolution k z 0 5x10 k y, cm -1

25 xperimental setup for time-resolved measurement

26 Pumping ultracold magnons 5 Pumping k H Pumping frequency 2f p = 8 GH Frequency (GHz) 4 3 k H f p Pumping power up to 6 W. Absorption <0.1% Pumped density: cm 2 f min Wavevector (1/cm) E = h 4GHz = k 200mK magnon B

27 BLS spectroscopy 5 4 Frequency (GHz) A well defined wavevecto Photon Photon Magnon 3 2 0,0 0,5 1,0 1,5 2,0 Wavevector (10 5 cm -1 )

28 BLS spectroscopy 5 4 Frequency (GHz) A well defined wavevecto Photon Photon Magnon 3 2 0,0 0,5 1,0 1,5 2,0 Wavevector (10 5 cm -1 )

29 Integrated BLS spectrum Integration over wavevectors f Frequency (GHz) Frequency BLS-intensity ~ n DOS k z k y 3 f min 2 0,0 0,5 1,0 1,5 2,0 Wavevector (10 5 cm -1 )

30 Thermal magnons Intensity (counts/s) 0,020 0,015 0,010 0,005 µ =0 Thermally excited magnons are measured to determine DOS and to calibrate the setup Equilibrium: µ = 0 0, Frequency, GHz n = 1 hν exp 1 kt BLS-intensity ~ n DOS

31 Magnons in equilibrium Intensity (counts/s) 0,020 0,015 0,010 0,005 0,000 µ = Frequency, GHz Population function, n BLS-intensity ~ n DOS Thermally excited magnons are measured to check DOS Equilibrium: µ = 0 n = 1 hν exp 1 kt

32 Thermalization: BEC of atoms Livetime 30 s, thermalization time 2 s

33 Mechanisms of magnon thermalization Two-magnon scattering ω1 = ω2 k k 1 2 Impurity-scattering, linear effect (independent of the magnon dens Elastic, k-thermalization Four-magnon scattering: Nonlinear effect ω1+ ω2 = ω3+ ω4 k + k = k + k (increase with increasing density) Inelastic, ω,k-thermalization Thermalization keeps the number of particles constant. Thermalization time should decrease with increasing pumping power/magnon density.

34 Magnon thermalization (step-like pumping) P=0.7 W 0 τ t Magnon population τ=50 ns τ=40 ns τ=0 ns (thermal) τ=30 ns 60 ns GHz 2,0 2,5 3,0 3,5 4,0 Frequency (GHz) 10 2 Time (ns)

35 Magnon thermalization (step-like pumping) P=0.7 W 0 τ t Relaxation happens wave-like via multiscattering processes Magnon population τ=50 ns τ=40 ns τ=0 ns (thermal) τ=30 ns 60 ns ,0 2,5 3,0 3,5 4,0 Frequency (GHz) 10 2 Time (ns)

36 Magnon thermalization (step-like pumping) 0 τ t Magnon population (a.u.) 8 6 P=0.7 W Thermalization happens wave-like 4 via multiscattering processes 2 0 2,0 f min 2,5 3,0 3,5 Frequency (GHz) f 0 4,0 f p 4, Time (ns)

37 Thermalization time Thermalization time depends on the pumping power/magnon density At high magnon densities it is below 50 ns. Magnon population f 100 f min Phys. Rev. Lett ,0 0,2 0,4 0,6 0,8 1,0 1,2 Pumping power (W) 0

38 Pumped magnons (step-like pumping) P = 4 W 0 τ t Time development of magnon distribution ( ) n ω Known DOS: fit with µ Bose-statistics with non-zero µ < µ max Intensity, counts/ms 0,6 0,4 0,2 0,0 τ= 200 ns 400 ns 600 ns 800 ns 1000 ns µ/h= Theory: µ/h=2.10 GHz max. value 1.70 GHz 1.96 GHz 2.04 GHz 2.07 GHz 2.08 GHz 1,0 1,5 2,0 2,5 3,0 3,5 4 Frequency, GHz

39 Time dependence of the chemical potential Chemical Potential E min P = 5.9 W P = 4.5 W P = 4 W P = 2.5 W Stationary state due to spin-lattice relaxation For high pumping power one can reach the critical density of magnons t, ns J. Appl. Phys.

40 Pumped magnons (step-like pumping) P = 5.9 W 0 τ t Time development of magnon distribution Known DOS: fit with µ ( ) n ω Bose-statistics. Intensity, counts/ms At 300 ns critical density: µ µ max τ= 100 ns 200 ns 300 ns 400 ns 500 ns nc µ/h= ( f ) n/d 2.05 GHz 2.10 GHz 1,0 1,5 2,0 2,5 3,0 3,5 4 Frequency, GHz max. value

41 Pumped magnons (step-like pumping) P = 5.9 W 0 τ t Time development of magnon distribution Known DOS: fit with µ ( ) n ω Intensity, counts/ms 2 1 τ= 100 ns 200 ns 300 ns 400 ns 500 ns µ/h= n/d 2.05 GHz 2.10 GHz max. value 30 Bose-statistics. At 300 ns critical density: µ µ max 0 1,0 1,5 2,0 2,5 3,0 3,5 4 Frequency, GHz

42 Difference between reality and BE-statistics The difference is only at f min. Width of the peak? 2,1 1,4 τ=500 ns ( ) ( ) n f n f C 0,7 0,

43 Experiments with ultimate resolution The addition to the critical density is of δ type (width is 1.5 mk, i.e kt). A condensate is created! 2,1 1,4 0,7 τ=500 ns ( ) ( ) n f n f C normal measurements measurements with ultimative frequency resolution HWHM = 30 MHz 0,0 Nature

44 Spatial coherence Frequency Two condensation points in the k-space: standing wave of the condensate density. Magnon Population Check the spatial coherence space resolved measurements BEC-condensates k z k y Magnon mapping Resolution 250 nm

45 Space coherence of the condensate mw k=0.6*10 5 cm -1 BLS intensity (a.u.) mw 250 mw Fourier intensity Space coordinate (µm) 5.0x x x10 5 Wavevector The created condensate is COHERENT in space

46 Spectrum of magnetic precession t 1,0 ump magnons ait 50 ns nalyze the ringing of the sample uring next 1 µs using MW pectrum-analyzer. Spectral power, a.u. 0,8 0,6 0,4 0,2 BLS HWHM =5 MHz Appl. Phys. Lett. 08 0, Frequency shift, MHz

47 Monochroamtic radiation due to condensate 1,0 0,8 3,09 Pumping frequency, GHz 8,18 8,19 8,20 8,21 8,22 8,23 8,24 8,25 0,6 0,4 0,2 HWHM =5 MHz 0, Frequency shift, MHz Frequency, GHz 3,08 3,07 3,06 3,05 3, Magnetic field, Oe Frequency of the radiation depends on the applied field, it is independent of the pumping freqeuncy

Lifetime of magnons BLS Magnons are pumped just for 30 ns. 0 τ t 1 @f P P = 2.

48 Lifetime of magnons BLS Magnons are pumped just for 30 ns. 0 τ t P P = 2.0 W Delay (ns) f min f p Normalized occupation 0,1 f min lifetime: 250 ns Delay time (ns) Lifetime is determined by interaction with the lattice: τ sp

49 Thermalization at different densities P = 2.5 W P = 3 W P = 5 W Lifetime depends on the magnon density

50 Scattering from (non)coherent states BLS-intensity ~ n DOS(?) I=<(E 1 +E 2 ) 2 >= <(E 1 ) 2 > + <(E 2 ) 2 >+2<E 1 E 2 > Incoherent scatterers: <E 1 E 2 > = 0 I = 2I 1 Coherent scatterers: <E 1 E 2 > 0 I = 4I 1 For coherent scatterers the scattering intensity n 2! Independent measurements of n are needed

51 Light scattering from magnons For incoherent magnons we are sensitive to For coherent magnons we are sensitive to n 2 n Number of magnons decays Scattering intensity n n 2 Coherent Incoherent The decay rate is determined by the livetime α = 1/ τ sp BLS-intensity from incoherent magnons n exp( αt) 10 0 BLS-intensity from coherent 2 magnons n exp 2α t ( ) Time

52 Scattering from the bottom of the spectrum 100 EXPERIMENT: The measured decay rate increases with pumping power (total magnon density) Phys. Rev. Lett. 08. Normalized BLS intensity W 4 W 5 W 6 W Time (ns)

53 Doubling of the decay rate 8 7 2α Decay rate µs α Pumping power (W)

54 Scattering intensity 8 Decay rate µs α Maximum scattering intensity 4 α Pumping power (W) Experimental confirmation of the temporal coherency of the condensate 10 0

55 What is BEC useful for Monochromatic microwave radiation is widely used for technical applications (Sat-TV, Mobiles, chip-to-chip communication, spin quantum computing etc.) Common ways to produce microwave radiation: 1. Vacuum microwave sources (clystrons, magnetrons etc.) 2. Semiconducting microwave sources (Han-diode etc.) 3. Spin-torque-oscillators (on board nanodevices) Conversion of the direct current power into microwave power Can we use nonmonochromatic radiation to produce microwaves?

56 Our vision Approach to create monochromatic microwave radiation from free-of-charge nonmonochromatic radiation: 1. Focus nonmonocromatic radiation from the enviroment to your ferromagnetic sample 2. Fullfil conditions of BEC 3. Use the powerfull oscillations of the condensate to radiate monocromatic radiation 4. Enjoy your life

57 Summary Magnons obey quantum statistics Quasi-equilibrium gas of magnons with chemical potential µ 0 can be created BEC of magnons is found at room temperature Coherence of the observed state is experimentally confirmed

58 Summary Magnons obey quantum statistics Quasi-equilibrium gas of magnons with chemical potential µ 0 can be created BEC of magnons is found at room temperature Coherence of the observed state is experimentally confirmed New young collegues are wellcome!!!

Bottleneck accumulation of hybrid bosons in a ferrimagnet

Bottleneck accumulation of hybrid bosons in a ferrimagnet Alexander A. Serga Fachbereich Physik and Landesforschungszentrum OPTIMAS Technische Universität Kaiserslautern, Germany www.physik.uni-kl.de/hillebrands