Macroscopic magnonic quantum states
Burkard Hillebrands
Fachbereich Physik and Landesforschungszentrum OPTIMAS Technische Universität Kaiserslautern, Germany
www.physik.uni-kl.de/hillebrands CollaboratorsTeam Kaiserslautern
Alexander Halyna Morteza Michael Philipp Vitaliy Oleksandr Kreil Musiienko-Shmarova Mohseni Schneider Pirro Vasyuchka Serha
Collaborators ▪ Victor L'vov and Anna Pomyalov (Weizmann Institute of Science, Israel) ▪ Gennadii Melkov (Taras Shevchenko National University of Kyiv, Ukraine) ▪ Andrei Slavin and Vasyl Tiberkevich Dmytro Bozhko Andrii Chumak (Oakland University, USA) (now Univ. of Colorado) (now Univ. of Vienna)
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 2 Macroscopic quantum states
Main idea: find macroscopic magnonic quantum states for information transfer and processing - analogous to superconductivity (Josephson currents) and to superfluidity in 3He and 4He - free of dissipation (apart from magnon-phonon and magnon-electron coupling)
This talk: ▪ From magnon Bose-Einstein Condensates (BEC) and supercurrents to Josephson oscillations in a room-temperature magnon BEC
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 3 Why do we name this a „Macroscopic Quantum State“ ?
Microscopic quantum phenomena Wave–particle duality → Microscopic scale of the system is comparable with the de Broglie wavelength of particles (electrons, Bose atoms, etc.)
Macroscopic quantum phenomena Wave–particle duality → Macroscopic scale of the system is comparable with the coherence length of the de Broglie wave
▪ Bose-Einstein condensate (BEC) of particles – spontaneous In quasi-classical limit (large number of population by a large number of Bose particles of a single quantum particles and occupation numbers of magnons) state with macroscopically-large coherence length described by the Gross-Pitaevskii equation for ▪ BECs of magnons (quanta of spin waves) – spontaneous formation de Broglie or spin waves of a coherent wave in a chaotic magnon system
Properties of both types of condensates are almost identical
„Macroscopic Quantum Phenomena“
make use of analogy with numerous phenomena in Bose-Einstein condensates of atoms: supercurrent, Bogoliubov waves, Josephson effects
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 4 Magnon gas
Magnon as a quanta of spin-wave ▪ Energy ==p2
▪ Momentum
▪ Mass
▪ Spin s =1
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 5 (Non-)linear Processes
▪ Linear process Two-magnon scattering e , p e , p 1 2 1 1
▪ Nonlinear processesMagnon gas of interacting quasiparticles Number of particles is conserved Three-magnon decay Three-magnon confluence e , p e2 , p2 1 1 e , p e , p 3 3 1 1 e , p e , p 3 3 2 2
Four-magnon scattering Cherenkov processes (푣gr magnon > 푣phonon) magnon e , p e , p 1 1 3 3 magnon e , p e , p 2 2 4 4 phonon Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 6 Magnon Bose-Einstein condensation
Bose-Einstein distribution D() () = − T,= min exp1 − kTB
µ: chemical potential T,0=
External injection of magnons beyond the thermal Magnons are bosons (s=1) and similar to other equilibrium level (about 3%) increases the chemical quasi-particles are described in thermal equilibrium potential to the bottom of magnon spectrum and by Bose-Einstein distribution with leads to Bose-Einstein condensation scenario zero chemical potential even at room temperature S.O. Demokritov et al., Nature 443, 430 (2006)
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 7 Medium: Yttrium Iron Garnet (YIG, Y3Fe5O12)
▪ Cubic crystal ▪ Room temperature ferrimagnet ▪ Lattice constant 12.376 Å (TС = 560 K) ▪ Unit cell – 80 atoms ▪ Low phonon damping
▪ The lowest magnon damping of V. Cherepanov, I. Kolokolov, and V. L’vov, any known material ! The saga of YIG: spectra, thermodynamics, interaction and relaxation of magnons in a complex magnet 3” YIG wafer Phys. Rep. 229, 81–144 (1993)
A.J. Princep et al., The full magnon spectrum of yttrium iron garnet, YIG monocrystal npj Quantum Mater. 2, 63 (2017) Minority octahedral iron atoms (spin 5/2 down) Majority tetrahedral iron atoms (spin 5/2 up) Scientific Research Company “Carat”, Lviv, Ukraine Magnetic moment of a unit cell is 20 Bohr magnetons µB at zero temperature
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 8 Magnon spectrum of in-plane magnetized YIG film
Landau-Lifshitz equation
dipolar interaction exchange interaction
q2
Thickness modes having a non-uniform harmonic distribution of dynamic magnetization along the film thickness
6 µm thick YIG film Calculations based on: Kalinikos and Slavin, J. Phys. C: Solid State Phys 19, 7013 (1986)
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 9 Control of magnon gas density by parametric pumping
fp Parametric pumping Energy and momentum by electromagnetic wave conservation laws for at microwave frequency parametric pumping
µ=0
fp/2 Bose-Einstein distribution Df() ()f = hf − µ: chemical exp− 1 kTB potential
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 10 Control of magnon gas density by parametric pumping
fp Parametric pumping Energy and momentum by electromagnetic wave conservation laws for at microwave frequency parametric pumping
Emin>µ>0
fp/2 Bose-Einstein Magnon thermalization distribution due to 4-particle scattering: Df() incoherent magnon gas ()f = hf − µ: chemical exp− 1 kTB potential
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 11 Bose-Einstein condensation of magnons
fp Parametric pumping Energy and momentum by electromagnetic wave conservation laws for at microwave frequency parametric pumping
S.O. Demokritov et al., Nature 443, 430 (2006)
µ=Emin
fp/2 Bose-Einstein Magnon thermalization distribution due to 4-particle scattering: Df() incoherent magnon gas ()f = hf − µ: chemical exp− 1 Bose-Einstein magnon kT potential B condensate
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 12 Time-, space- and wavevector-resolved pulsed BLS spectroscopy
▪ Pump-probe technique
Resolution (111) LPE YIG films 5 - 7 µm Time 1 ns Width of the pumping area 50 and 500 µm Frequency 50 MHz Length of the pumping area 1 mm Wavenumber 2×103 cm-1 Maximum microwave power 100 W Space 5 µm
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 13 Microwave-free creation of a magnon BEC
Are there other ways how to create a magnon BEC state in YIG?
▪ Yes, by rapid cooling of a magnon-carrying specimen
High-energy magnon states are populated and participate T ,= Bose-Einstein distribution minmin in the BEC formation process - Gilbert damping D() ()= T , = 0 − max Bose--4Einstein-magnon distribution scattering and exp− 1 function is crucial for the kT Cherenkov processes B quantitative description of the condensation
Tmin , = 0
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 14 BEC in rapidly cooled magnon gas
Current pulse Magnon Magnon population distribution thermal YIG heating up to spectrum 415 K Fast non-adiabatic YIG cooling and BEC formation ! T=300 K
TT= min BEC: = min 10
Magnon BEC ! 1
0.1
M. Schneider, et al., Nat. Nanotechnol. 15, 457 (2020) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 15 Magnon BEC: Experiment
fp/2
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 16 Magnon BEC: Experiment
pumping pulse
Freely evolving magnon BEC is formed after BEC
the pumping is off Pumping Pumping pulseend
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 17 Supercurrent in magnon BEC
Supercurrent: Flow of particles due to phase gradient of the condensate’s wavefunction
Complex BEC rt, Supercurrent wave function: ( ) 2 BEC density: Nc = || JrtN( , ) = m c BEC phase: = arg()
By changing probing laser power or laser pulse duration it is possible to control the phase of the magnon BEC
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 18 Dynamics of condensed magnons in thermal gradient Pump pulse 2 μs Magnon BEC Probing laser pulse
PLaser
tLaser
tLaser PLaser Magnon gas
No influence of a thermal gradient
D. A. Bozhko et al., Nat. Phys. 12, 1057 (2016)
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 19 Dynamics of condensed magnons in thermal gradient - comparison with theory
= c ()Tt Supercurrent velocity ( t) m v = D c 450 T x x s
Observed dynamics of magnon condensate can be understood taking into account Larger negative magnon supercurrents slope due to outflow of supercurrent Corresponding maximal temperature change 4.7 K
D. A. Bozhko et al., Nat. Phys. 12, 1057 (2016)
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 20 Supercurrent and anisotropy of spin-wave spectrum
2D supercurrent JN= D x c x x JN= D y c y y
Dispersion coefficients 1()2 q Dx = 2 2 qx 12 (q ) Dy = 2 2 qy
Dx 21Dy Js = J x J y 1D supercurrent
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 21 Non-local measurement: Supercurrent magnon transport
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 22 Non-local measurement: Supercurrent magnon transport
Heating laser switched off Heating laser power 116 mW
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 23 Second sound scenario
Framework: Gross-Pitaevskii equation
2 2 i +−=Dx 2 W ||0 t x 1 2()q Dispersion coefficient Dx = 2 2 qx
Amplitude of four-wave W 0 repulsive interaction Dzyapko et al., Phys. Rev. B 96, 064438 (2017)
Stationary solution: (x , tiWN )exp()=− tNc c
Magnon BEC density: Nc
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 24 Second sound scenario
Framework: Gross-Pitaevskii equation Solution: Bogoliubov waves () k (second sound)
2 2 2 i +−=Dx 2 W ||0 =+()/kc k s 12Dxk WNc t x 2 1 ()q 2 Dispersion coefficient Long-wavelength limit Dxk 2WN c Dx = 2 2 qx
Amplitude of four-wave W 0 repulsive interaction Linear dispersion =()k cs k Dzyapko et al., Phys. Rev. B 96, 064438 (2017)
Stationary solution: (x , tiWN )exp()=− tNc c Second sound velocity
cs = 2DxWNc Magnon BEC density: Nc
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 25 Second sound scenario
==(),kc kc s s 2DxWNc
▪ The sound velocity c s must be independent on the excitation conditions i.e. the heating laser power, which determine the BEC pulse width
D. A. Bozhko et al., Nat. Commun. 10:2460 (2019) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 26 Second sound scenario
==(),kc kc s s 2DxWNc
▪ The sound velocity c s must be independent on the excitation conditions i.e. the heating laser power, which Group velocity: determine the BEC pulse width
▪ During the pulse propagation, the amplitude of the background condensate decays and the sound velocity also has to decay
The sound velocity decrease is clear visible from parabolic fits !
Smaller decrease of the sound velocity than Low decay of the propagating BEC pulse expected from the decay of the condensate due to an efficient gathering of surrounding magnons D. A. Bozhko et al., Nat. Commun. 10:2460 (2019) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 27 Second sound in an incoherent magnon gas
Excitation of coherent waves Second sound waves in a magnon condensate and in a magnon gas in a dense magnon gas magnon gas excitations magnon gas
magnon condensate excitations magnon condensate
V. S. Tiberkevich et al., Sci. Rep. 9:9063 (2019)
General two-fluid model of a magnon gas combined with the Gross-Pitaevskii equation for the magnon BEC needs to be developed
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 28 Two-component magnon BEC
BLS intensities of the Stokes and anti-Stokes spectral peaks are related to the densities of
magnons with wavenumbers +qmin and –qmin (for fixed angle of wavevector-resolved BLS setup)
scattered LLmin=
qqqscattered LLmin=
Elastically scattered light
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 29 Magnon condensate density in a magnetic trench
BLS measurement point
*
(* Position relative to the antenna) (Magnetic trench formation for negative DC current)
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 30 Josephson oscillations in magnon condensate
(1) time, when magnons are excited by parametric pumping (2) time interval during which the BEC forms (3) time interval during which the first oscillation appears (4) the rest of the observation time
A.J.E. Kreil et al., arXiv:1911.07802v1 (2019) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 31 Time-dependence of spatial magnon distributions in magnetic trench
Time intervals
(1) Pumping time (2) BEC formation (3) First oscillation peak (4) Rest of time
Full lines: BEC at +q Dashed lines: BEC at -q I = - 500 mA A.J.E. Kreil et al., arXiv:1911.07802v1 (2019) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 32 Josephson oscillations in magnon condensate
(1) time, when magnons are excited by parametric pumping (2) time interval during which the BEC forms (3) time interval during which the first oscillation appears (4) the rest of the observation time
(1) (2) (3) (4)
A.J.E. Kreil et al., arXiv:1911.07802v1 (2019) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 33 Josephson oscillations in magnon condensate
Potential barrier formed by dc current Potential trench formed by dc current
ac Josephson effect in a magnon BEC carried by a room-temperature ferrimagnetic film
A.J.E. Kreil et al., arXiv:1911.07802v1 (2019) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 34 Josephson oscillations in magnon condensate
Potential trench formed by dc current
Frequency of the Josephson oscillations is in semi-quantitative agreement with the prediction of theory A.J.E. Kreil et al., arXiv:1911.07802v1 (2019) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 35 Magnon BEC in confined systems
YIG microconduit
MuMax 3.0 numerical calculations
▪ Parallel pumping and magnon scattering processes ▪ Magnon BEC during relaxation M. Mohseni, et al, New J. Phys., In press (2020) Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 36 Summary
▪ Non-local spin transport by magnon supercurrent in a room-temperature magnon BEC
▪ Bogoliubov second sound scenario for the distant supercurrent propagation
▪ Josephson oscillations in a room-temperature Bose- Einstein magnon condensate observed
▪ Advancing of the physics of room-temperature macroscopic quantum phenomena for their application in magnon spintronics devices
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 37 Outlook
▪ Spin transport by magnon supercurrent in 2D magnetic landscapes
▪ New ways to create the magnon BEC (rapid cooling, spin pumping)
▪ Non-viscose propagation of the magnon BEC
▪ Interference of Bogoliubov waves
▪ Computing with two-component magnon condensates
▪ Qubit representation using macroscopic magnonic quantum states
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 38 Appendix: phase of supercurrent in magnon BEC
Supercurrent: Flow of particles due to phase gradient of the condensate’s wave function
Complex BEC wave Supercurrent of particles function of particles: Ψp 풓, 푡 2 ℏ Particle BEC density: 푁p = Ψp 푱p 풓, 푡 = 푁p∇휑p 푀p Particle BEC phase: 휑p = arg Ψp
Complex BEC wave function of magnons: Ψm 풓, 푡 = 퐶m 풓, 푡 exp 푖(풒0 ∙ 풓 − 휔0푡)
Magnon BEC density 푁m and phase 휑m: 퐶m 풓, 푡 = 푁mexp 푖휑m
Phase 휑m of a spatially heterogeneous condensate at position 풓: we take the deviation of the angle of rotation of the spin at this position from the rotation for the case of a fully monochromatic spatially coherent spin wave. Thus, the phase of a spatially homogeneous condensate is zero.
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 39 Appendix: phase of supercurrent in magnon BEC
Supercurrent: Flow of particles due to phase gradient of the condensate’s wave function
Complex BEC wave Supercurrent of particles function of particles: Ψp 풓, 푡 2 ℏ Particle BEC density: 푁p = Ψp 푱p 풓, 푡 = 푁p∇휑p 푀p Particle BEC phase: 휑p = arg Ψp
Complex BEC wave function of magnons: Ψm 풓, 푡 = 퐶m 풓, 푡 exp 푖(풒0 ∙ 풓 − 휔0푡)
Magnon BEC density 푁m and phase 휑m: 퐶m 풓, 푡 = 푁mexp 푖휑m
휕푁 풓, 푡 휕 퐶 풓, 푡 2 휕퐶 풓, 푡 휕퐶∗ 풓, 푡 m ≈ m = m 퐶∗ 풓, 푡 + 퐶 풓, 푡 m 휕푡 휕푡 휕푡 m m 휕푡 휕 1 푖 + 푖푣 ∇ + 휔′′ ∇ ∇ − 푇 퐶 풓, 푡 2 퐶 풓, 푡 = 0 ← Gross-Pitaevskii equation 휕푡 gr 2 푖푗 푖 푗 m m 푑휔 풒 푑2휔 풒 Continuity 휕푁m 풓, 푡 ′′ 풗gr = 0 + ∇푱 = 0 푣gr = 휔푖푗 = equation: 휕푡 m 푑풒 풒 = 풒0 푑풒푖푑풒푗 풒 = 풒0 푖휔′′ 휔′′ Magnon 2 푖푗 ∗ ∗ 푖푗 휕휑m supercurrent: 퐽m푖 = 푣gr푖 퐶m + 퐶m∇푗퐶m − 퐶m∇푗퐶m 퐽m푖 = 푁m 2 2 휕푟푗
Burkard Hillebrands Online SPICE-SPIN+X Seminar July 29, 2020 40