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

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. MAIN AIM
 Persistent spin current
 To CONTROL spin currents
“Directmeasurement”
(i.e. super spin current)

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. 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. 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. 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.  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. OUTLINE
 INTRODUCTION
 REVIEW
 SUMMARY
 RESULT
 Josephson effects
 Persistent magnon-BEC current (i.e. super spin current)
 Magnon-BEC Josephson junction (MJJ)
 SYSTEM

9.  REVIEW

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.  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. 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. 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. 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. [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. OUR WORK
SYSTEM

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. 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. 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.  RESULTS

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. [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.  dc Josephson Effects
𝜽 = 𝟎

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. 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. 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.  Persistent Magnon-BEC Currents

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. 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.  REMARKS

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. 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. 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. 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