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Josephson and Persistent Spin Currents in Bose-Einstein Condensates of Magnons

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Josephson and Persistent Spin Currents in Bose-Einstein Condensates of Magnons

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Phys. Rev. B 90, 144419 (2014)
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.144419
arXiv:1406.7004
http://arxiv.org/abs/1406.7004

Phys. Rev. B 90, 144419 (2014)
http://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.144419
arXiv:1406.7004
http://arxiv.org/abs/1406.7004

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Josephson and Persistent Spin Currents in Bose-Einstein Condensates of Magnons

  1. 1. Kouki Nakata Josephson Effects & Persistent Spin Currents in Magnon-BEC due to Berry Phase University of Basel, Switzerland 仲田光樹 Based on [arXiv:1406.7004] [Note] All the responsibility of this slide rests with “Kouki Nakata”; Sep. 2014.
  2. 2. MAIN AIM  Persistent spin current  To CONTROL spin currents “Directmeasurement” (i.e. super spin current)
  3. 3.  Rapid PROGRESS of experiments BACKGROUND  Spin-wave spin current  Quasi-equilibrium magnon-BEC  Achieved even at “room temperature” by using microwavepumping (Low temperature is not required.) Ferromagnetic insulator (YIG) [Y. Kajiwara et al., Nature 464, 262 (2010)] [S. O. Demokritov et al., Nature 443, 430 (2006)]
  4. 4. BEC BEC “Bose–Einstein condensation of quasi-equilibrium magnons at room temperature under pumping” [S. O. Demokritov et al., Nature 443, 430 (2006)] Based on  Can be semi-classically treated  Canonically conjugate variables; [𝓝, 𝝑] 𝒂 = 𝓝 𝒆 𝒊𝝑 𝒂 ~ 𝐒+ = 𝐒 𝐱 + 𝐢𝐒 𝐲 𝝑~ Direction of spin 𝓝~ Length of macroscopicspin Semantic issue; Y. M. Bunkov and G. E. Volovik,arXiv:1003.4889. Textbook by Leggett D. Snoke, Nature 443, 403 (2006). C. D. Batista et al., RMP. 86, 563 (2014). Macroscopiccoherent stateQuantum effectsIndividual spins/quasi-particles Condensed time: over a few hundred ns. 𝓝;Magnon-BEC 𝝑; PhaseBEC Quasi-equilibrium Magnon-BEC Magnon picture Spin picture
  5. 5. BEC BEC “Bose–Einstein condensation of quasi-equilibrium magnons at room temperature under pumping” [S. O. Demokritov et al., Nature 443, 430 (2006)] Based on 𝒂 = 𝓝 𝒆 𝒊𝝑 𝒂 ~ 𝐒+ = 𝐒 𝐱 + 𝐢𝐒 𝐲 Semantic issue; Y. M. Bunkov and G. E. Volovik,arXiv:1003.4889. Textbook by Leggett D. Snoke, Nature 443, 403 (2006). C. D. Batista et al., RMP. 86, 563 (2014). Macro scopic Spin BEC Quasi-equilibrium Magnon-BEC Magnon picture Macroscopiccoherent stateQuantum effectsIndividual spins/quasi-particles Condensed time: over a few hundred ns. 𝓝;Magnon-BEC 𝝑; Phase 𝝑~ Direction of spin 𝓝~ Length of macroscopicspin  Can be semi-classically treated  Canonically conjugate variables; [𝓝, 𝝑] Spin picture
  6. 6. HOW TO ACHIEVE  Berry phase(Geometric phase)  Quasi-equilibrium magnon-BEC Persistent magnon-BEC current To electro-magnetically control spin currents “Macroscopic quantum effect (coherence)”  Spin currents; drastically ENHANCED !!
  7. 7.  Spin Current ”Persistent magnon-BEC current” Under our control Directmeasurement  Electromagnet Towardthe direct measurementof spin (magnon) current  Berry Phase Aharonov-Casher(A-C)位相 Magnon-BEC (Ferromagnetic insulator) 「MacroscopicEffect」 CONCEPT
  8. 8. OUTLINE  INTRODUCTION  REVIEW  SUMMARY  RESULT  Josephson effects  Persistent magnon-BEC current (i.e. super spin current)  Magnon-BEC Josephson junction (MJJ)  SYSTEM
  9. 9.  REVIEW
  10. 10.  Superconductors (SC) [Cooper pair] = [Boson] B. D. Josephson, [Phys.Lett.1,251 (1962)] 1962~ 𝑑 𝑑𝑡 ∆𝑁 𝑡 = 2J ℏ sin ∆𝜙(𝑡) 𝑑 𝑑𝑡 ∆𝜙 𝑡 = − 2𝑒𝑉(𝑡) ℏ Josephsonequations in SC  dc Josephson effect; 𝑑 𝑑𝑡 ∆𝜙 𝑡 ∝ 𝑉 𝑡 = 0  Relative phase is time-independent; 𝑑 𝑑𝑡 ∆𝜙 𝑡 = 0 ; Josephson current ”charge current” Josephson current J (tunneling)(1973) Textbook by Leggett w.f. w.f. Josephson Effects Universal Phenomenon of bosonic particles 𝑉(𝑡); the external voltage applied across thejunction J (> 0); the tunneling amplitude, ∆𝑁 ≔ the relative population, ∆𝜙; the relative phase• • • Figby [Fa Wang and Dung-Hai Lee, Science, 332 (2011) 200] Fig by[J. Q. You andF. Nori, Nature, 474, 589 (2011)] Picture by Googlesearch (HPfornovel prize).
  11. 11.  Universal Phenomenon of bosonic particles Anderson et al., Science (‘95) Atomic BEC  Atomic BEC  Magnon BEC [Magnon] = [Bosonic quasi-particle] Josephson Effects B. D. Josephson, [Phys.Lett.1,251 (1962)] Berry Phase (Aharonov-Casher phase) 1962~ 1997~ Now We (Our present work)  Superconductors (SC) [Cooper pair] = [Boson] A. Smerzi etal., [PRL. 79, 4950 (1997)] [PRA, 59, 620 (1999)] [PRL.84, 4521 (2000)] Leggett[Rev.Mod.Phys. 73, 307 (2001)] M. Albiezetal.[PRL.95, 010402 (2005)] S. Levyetal. [Nature 449, 579 (2007)] (2001)(1973) Figby [Fa Wang and Dung-Hai Lee, Science, 332 (2011) 200] Picture by Googlesearch (HPfornovel prize).
  12. 12. Berry Phases Aharonov- Bohm phase [Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959)] Aharonov- Casher phase [Y. Aharonov and A. Casher, PRL, 53, 319 (1984)] Charged particle; 𝑒 Magnetic dipole; 𝜇 𝜇 = 𝑔𝜇 𝐵 𝒆 𝑧 ;(Magnon) Magnetic vector potential 𝐴 [Electric field]×[Magnetic dipole]; 𝐸 × 𝜇 𝜑A−B = 𝑒 ℏ𝑐 𝐴 ∙ 𝑑𝑠 =: 𝑒 ℏ𝑐 𝛷A−B 𝜑A−C = 𝑔𝜇 𝐵 ℏ𝑐2 (𝐸 × 𝜇) ∙ 𝑑𝑠 𝐸 𝜇 𝐴 𝑒 𝛷A−B  Special casesof Berryphase [R.Mignani,J.Phys.A:Math. Gen. 24, L421 (1991)] [X.-G.Hea and B. McKellarb,Phys.Lett.B264, 129(1991)] A special case of Berry phase
  13. 13. Microwave Pumping Magnon Magnon-BEC (macroscopicstate) Magnon pumping Room temperature [S.O. Demokritov etal.,Nature 443, 430 (2006)]  Excite additionalmagnons.  Create a gas of quasi-equilibriummagnons with a non-zerochemical potential.  A Bose condensate of magnons is formed. Microwave pumping  We can directlyinject magnons so that it becomes a macroscopicnumber(BEC). [K.Nakataand G. Tatara, J.Phys.Soc. Jpn. 80, 054602 (2011).] [K.Nakata,Doctoral Thesis,KyotoUniversity(2014).]
  14. 14. Magnon Magnon-BEC (macroscopicstate) Magnon pumping Room temperature 𝒂 = 𝑵 𝐁𝐄𝐂 𝒆𝒊𝝁𝒕+𝒊𝜶  BEC order parameter Quasi-equilibrium Magnon-BEC [S.O. Demokritov etal.,Nature 443, 430 (2006)] 𝑛BEC~1018 − 1919cm−3 [Y. M. Bunkov andG. E. Volovik,arXiv:1003.4889.]
  15. 15. [C.D. Batistaet al., Rev.Mod. Phys., 86, 563 (2014).] [Y. M. Bunkov andG. E. Volovik,arXiv:1003.4889.] Quasi-equilibrium Magnon-BEC  [Metastable state]≠[Groundstate] [J. Hick et al., Phys. Rev. B 86, 184417 (2012)] [T. Kloss et al., Phys. Rev. B 81, 104308 (2010)] [S. M. Rezende, Phys. Rev. B 79, 174411(2009)] [F. S. Vannucchi et al., Phys. Rev. B 82, 140404(R) (2010)] [F. S. Vannucchi et al., EPJB 86 (2013) 463] [S. M. Rezende, Phys. Rev. B 79, 174411 (2009)] Thermalizationprocess 𝒂 = 𝑵 𝐁𝐄𝐂 𝒆𝒊𝝁𝒕+𝒊𝜶  BEC order parameter [K.Nakataand G. Tatara, J.Phys.Soc. Jpn. 80, 054602 (2011).] [K.Nakata,Doctoral Thesis,KyotoUniversity(2014).]
  16. 16. OUR WORK SYSTEM
  17. 17. E = E𝐞 𝐲 𝒏 𝐋, 𝝑 𝐋 𝒏 𝐑, 𝝑 𝐑 𝐉 𝐞𝐱 J J 𝒏 𝐋, 𝝑 𝐋 𝒏 𝐑, 𝝑 𝐑 ΓL ΓR A-C phase: Magnon BEC Josephson Junction  Tunneling Hamiltonian (boundary spins)  Hamiltonian of each single FIs (Magnon BECs) with Diag 𝐉 = J{1, 1, 𝜂}, J < 0 E = E𝐞 𝐲 ℋH =(Gross-Pitaevskii Hamiltonian;ℋGP) Microscopic spin model Electric field (E = E𝐞 𝐲) Magnon picture −𝜽 𝐀−𝐂 𝜽 𝐀−𝐂 ( Jex ≪ J ) Magnon BEC (Holstein-Primakoff tr.); ~ 𝐒+ = 𝐒 𝐱 + 𝐢𝐒 𝐲
  18. 18. Magnon picture BEC 𝒂 = 𝓝 𝒆𝒊𝝑 ~ 𝐒+ = 𝐒 𝐱 + 𝐢𝐒 𝐲 𝐉 𝐞𝐱 𝐉 𝐞𝐱 𝓝 𝐋, 𝝑 𝐋 𝓝 𝐑, 𝝑 𝐑 CALCULATION PROCEDURE Spin picture Canonically conjugate variables [𝓝, 𝝑] BEC BEC Macro scopic Spin Macro scopic Spin 𝓝;Magnon-BEC 𝝑; Phase 𝒩T: = 𝒩L + 𝒩R• Population imbalance; 𝐳 ≔ (𝒩L − 𝒩R)/𝒩T, • Relative phase; 𝛉 ≔ ϑR − ϑL ~Two macroscopicspins interact with each other through 𝐉 𝐞𝐱
  19. 19. EACH VALUE E = E𝐞 𝐲 Each Value Our estimation The exchange interaction between the two FIs Jex = 1μeV The exchange interaction between the neighboring spins in a single FI J ≈ 0.1eV The density of magnpn-BECs [S. O. Demokritov et al., Nature (2006).] nBEC = 1019 cm−3 The applied magnetic field 𝐵 ≈ 1mT The applied electric field to the interface 𝐸 ≈ 5GV/m The width of the interface Δ𝑥 ≈ 10Å The lattice constant of a FI 𝛼 ≈ 1Å
  20. 20.  RESULTS
  21. 21. Josephson Equations in MJJ ;Renormalized time 𝜏 = 1 ↔ 𝑡 = 1ns (ex. K0/𝑆 ≡ Jex = 1μeV)  Josephson spin current ∝ nL, ϑL E = E𝐞 𝐲 nR, ϑR −𝜽 𝐀−𝐂 𝜽 𝐀−𝐂 Δ𝐸 & Λ; renormalized magnetic field difference & mag-mag interactionin terms of K0 (K0 ; tunnelingmagnitude)  nT: = nL + nR • Population imbalance; z ≔ (nL − nR)/nT • Relativephase; θ ≔ ϑR − ϑL • A-C phase; • ∆x; the width of the interface (~Å)
  22. 22. [Period]~𝟔ns ac Josephson Effect ;Renormalized time 𝜏 = 1 ↔ 𝑡 = 1ns (ex. K0/𝑆 ≡ Jex = 1μeV)  No Aharonov-Casherphase; ~(Chemical potential difference) Condensed time: over a few hundreds ns. S. O.Demokritovetal., Nature 443, 430 (2006).
  23. 23.  dc Josephson Effects 𝜽 = 𝟎
  24. 24. Time-dependent Magnetic Field i) Increasing rate; 𝑩 𝟎 ii) Josephson equation (weak coupling) ∝ 𝑬; electric field ∝ 𝑩 𝟎; magnetic field (increasing rate) dc effect 𝜽 = 𝟎 (steady-state solution) θ(τ = 0) = 0 E = E𝐞 𝐲 θ ≔ ϑR − ϑL z ≔ (nL − nR)/nT nL, ϑL nR, ϑR −𝜽 𝐀−𝐂 𝜽 𝐀−𝐂Λ; renormalized mag-mag interaction in terms of K0 (K0 ; tunneling magnitude) UL UR Jex ≪ J
  25. 25. dc Josephson Effect 0 𝜏 = 1 ⟷ 𝑡~1ns 𝜽 ≠ 𝟎 𝜽 = 𝟎(+ small oscillationin “z”)  AtomicBEC;A. Smerzi etal., [PRL. 79, 4950 (1997)] [PRA, 59, 620 (1999)] [PRL.84, 4521 (2000)]
  26. 26. dc-ac Transition; 𝐳 𝟎 = 𝐁 𝟎/𝚲 0  (c) dc-ac transition; 𝒛 𝟎 ≈ 𝟎. 𝟕𝟐𝟓 𝑧0 = 0.10 𝑧0 = 0.724 𝑧0 = 0.726 𝑧0 = 1.1 dc-ac Transition  (d) dc-ac transition; 𝒛 𝟎 ≈ 𝟏 𝑧0 = 0.726 𝑧0 = 0.726 Recovery due to A-C phase 𝜏 = 1 ⟷ 𝑡~1ns𝜏 = 1 ⟷ 𝑡~1ns dc effect ac effect dc effect ac effect  Atomic BEC; A. Smerzi et al., [PRL. 79, 4950 (1997)] [PRA, 59, 620 (1999)] [PRL. 84, 4521 (2000)]
  27. 27.  Persistent Magnon-BEC Currents
  28. 28. Magnon-BEC Ring ・Electric-gradient flux  Single-valuednessof the BEC wave function In analogy to superconductingrings 𝑝 ∈ ℤ; phase winding number ・Electric flux quantum  Persistent magnon-BEC current The A-C phase in the ring Quantized electric-gradientflux 𝐄(𝜌, 𝜑) =
  29. 29. Direct Measurement ≫ 10−13 V nBEC = 1019cm−3 J ≈ 0.1eV [F. Meier and D. L., PRL 90, 167204 (2003).]  [Persistentmagnon-BEC current 𝐈 𝐁𝐄𝐂] = [Steady flow of the magnetic dipoles] (i.e. magnons or magnetic moment 𝑔𝜇 𝐵 𝒆 𝑧)  Moving magnetic dipoles  “Electric dipole fields 𝐄 𝐦”  Voltage drop 𝐕 𝐦. S. O. Demokritov et al., Nature (2006). ; Spin chains Largely enhanced  due to “Macroscopic coherence” [D.Loss and P.M. Goldbart,PLA215, 197 (1996)] 𝐕 𝐦 𝜌0 = 1mm 𝑟0 = 1mm Vm~1nV 𝑔 = 2, 𝑆 = 1/2 × 𝟏𝟎, 𝟎𝟎𝟎 times!! 𝑔𝜇 𝐵 𝒆 𝑧 𝑅 ≈ 10mm 𝑝 ≈ 50 (Phase winding number;𝜙 = 𝑝𝜙0 )
  30. 30.  REMARKS
  31. 31. Analogous Phenomenon Magnon Josephson effect Magnon Hall effect Dzyaloshinskii-Moriya interaction Temperature gradient ≈ Onose et al. [Science 329, 297 (2010)] [Josephson spin current] ⊥ [Electric field] [Thermal spin current] ⊥ [temperature gradient] ≈ Key point; Transverse spin currents Picture from [Science 329, 297 (2010)]
  32. 32. SIGNIFICANCE The Bose Josephson junction (BJJ) of atomic BEC 𝜽 𝐀−𝐂 = 𝟎 The magnon Josephson junction (MJJ) M. Albiez et al. [PRL. 95, 010402 (2005)] S. Levy et al. [Nature 449, 579 (2007)] Leggett [Rev. Mod. Phys. 73, 307 (2001)] A. Smerzi et al., [PRL. 79, 4950 (1997)] [PRA, 59, 620 (1999)] [PRL. 84, 4521 (2000)] [Theory] [Experiment]  Exact dc Josephson effect ・Time-dependent magnetic field ・Aharonov-Casher phase ・ac-dc transition ・Persistent magnon-BEC current  Magnon-interferenceAharonov-Casher phase Cold atom [Our work on MJJ] = [The generalization ofthe preceding studies on BJJ] Picture byGoogle search.
  33. 33. LAST MESSAGE Phys. Lett. A, 96 (1983), p. 365  Our work [arXiv:1406.7004] Persistent (charge) current due to the Aharonov-Bohm phase Persistent “magnon-BEC” current due to the Aharonov-Casher phase K. N., K. A. van Hoogdalem, P. Simon, and D. Loss
  34. 34. SUMMARY “JosephsonEffects& PersistentSpinCurrentsin Magnon-BECduetoBerryPhase” I). How to electromagnetically control Josephson spin currents  [Period of ac Josephson effect]~10ns III). How to directly measurethe Josephson magnon-BEC currents  The resulting voltage drop from the flow of the magnons (i.e. magnetic dipoles).  It is largely enhanced due the macroscopic coherence of magnon-BECs; Vm~1nV ≫ 10−13 V  This method is applicable to Josephson junction; 0 ≤ Vm ≤ 1𝜇V due to ac or dc effects. II). Persistentmagnon-BEC current (i.e. super spin current) due to the Berry phase  It is quantized in the magnon-BEC ring. Regarding macroscopic quantumself-trapping, please see the preprint [arXiv:1406.7004]. Each Value Our estimation The exchange interactionbetweenthe twoFIs Jex = 1μeV The exchange interactionbetweenthe neighboringspinsinasingle FI J ≈ 0.1eV The densityof magnpn-BECs[S.O.Demokritov etal.,Nature (2006).] nBEC = 1019cm−3 The appliedmagneticfield 𝐵 ≈ 1mT The appliedelectricfieldtothe interface 𝐸 ≈ 5GV/m The widthof the interface (The lattice constant 𝛼 ≈ 1Å) Δ𝑥 ≈ 10Å Based on [arXiv:1406.7004] K. N., K. A. van Hoogdalem, P. Simon, and D. Loss

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