BOSE-EINSTEIN CONDENSATION OF MAGNONS IN SUPERFLUID 3 He-B and its - - PowerPoint PPT Presentation

BOSE-EINSTEIN CONDENSATION OF MAGNONS IN SUPERFLUID 3He-B and its applications to vortex studies

V.B. Eltsov, S. Autti, Yu.M. Bunkov, P.J. Heikkinen, J.J. Hosio, M. Krusius, M. Silaev, G.E. Volovik, V.V. Zavjalov Low Temperature Laboratory Aalto University

OVERVIEW

• 1. Superfluid 3He-B and traps for magnon quasiparticles.
• 2. Filling the ground and excited levels in the trap with magnons and spec-

troscopy of the trap levels.

• 3. Coherent precession of the ground- and excited-level condensates.
• 4. Interaction of the magnon condensates with the trapping potential (self-

trapping).

• 5. Measurements of relaxation of magnon condensates in rotating 3He-B filled

with vortex lines: A tool to observe vortex-core bound fermions.

SUPERFLUID He

3

1.0 2.0 3.0 10 20 30 40

Solid Superfluids Normal fluid Temperature, mK Pressure, bar He-B

3 He-A

3

Fermi system, which goes superfluid through Cooper pairing with S = 1 and L = 1.

SUPERFLUID He

3

1.0 2.0 3.0 10 20 30 40

Solid Superfluids Normal fluid Temperature, mK Pressure, bar He-B

3 He-A

3

Fermi system, which goes superfluid through Cooper pairing with S = 1 and L = 1. Spin-orbit interaction: Orbital momentum ⇔ spin precession ∂S ∂t = γ S × H + RD

dipole torque

H S

SUPERFLUID He

3

1.0 2.0 3.0 10 20 30 40

Solid Superfluids Normal fluid Temperature, mK Pressure, bar He-B

3 He-A

3

Fermi system, which goes superfluid through Cooper pairing with S = 1 and L = 1. Spin-orbit interaction: Orbital momentum ⇔ spin precession ∂S ∂t = γ S × H + RD

dipole torque

H S In 3He-B in magnetic field net L and S = (χ/γ )H appear. Connection L ⇔ S is given by the order parameter ⇒ gradient energy ⇒ equilibrium texture and waves.

Texture of ˆ l = L/L in a cylindrical sample results from competition of L H and L ⊥ wall H

TRAPPED MAGNON CONDENSATES IN He-B

H S

3

Magnons with spin S⊥ = S sin βMeiωt+iα

S = χH/γ , ωL = γ H, mM ∼ 10−4mHe

−¯ h : ˆ Nm = S − ˆ Sz ¯ h , Nm ∝ 1 − cos βM = 2 sin2 βM 2 Spin waves:

βM

ω ≈ ωL + ¯ hk2/2mM

TRAPPED MAGNON CONDENSATES IN He-B

H S

3

Magnons with spin S⊥ = S sin βMeiωt+iα

S = χH/γ , ωL = γ H, mM ∼ 10−4mHe

−¯ h : ˆ Nm = S − ˆ Sz ¯ h , Nm ∝ 1 − cos βM = 2 sin2 βM 2 Spin waves:

βM

ω ≈ ωL + ¯ hk2/2mM

H M z H

Magnon condensate in 3He-B: coherently precessing magnetization (r) ∝ sin βM(r) 2 eiωt+iα(r)

N1/2

m

∝ M⊥ ω ≡ chemical potential α ≡ phase of wave function

Review: Bunkov and Volovik, arXiv:1003.4889

βM βl

TRAPPED MAGNON CONDENSATES IN He-B

H S

3

Magnons with spin S⊥ = S sin βMeiωt+iα

S = χH/γ , ωL = γ H, mM ∼ 10−4mHe

−¯ h : ˆ Nm = S − ˆ Sz ¯ h , Nm ∝ 1 − cos βM = 2 sin2 βM 2 Spin waves:

βM

ω ≈ ωL + ¯ hk2/2mM

H M z H

Magnon condensate in 3He-B: coherently precessing magnetization (r) ∝ sin βM(r) 2 eiωt+iα(r)

N1/2

m

∝ M⊥ ω ≡ chemical potential α ≡ phase of wave function

Review: Bunkov and Volovik, arXiv:1003.4889

βM βl

Axial trap FZ = (ω − ωL)||2

TRAPPED MAGNON CONDENSATES IN He-B

H S

3

Magnons with spin S⊥ = S sin βMeiωt+iα

S = χH/γ , ωL = γ H, mM ∼ 10−4mHe

−¯ h : ˆ Nm = S − ˆ Sz ¯ h , Nm ∝ 1 − cos βM = 2 sin2 βM 2 Spin waves:

βM

ω ≈ ωL + ¯ hk2/2mM

H M z H

Magnon condensate in 3He-B: coherently precessing magnetization (r) ∝ sin βM(r) 2 eiωt+iα(r)

N1/2

m

∝ M⊥ ω ≡ chemical potential α ≡ phase of wave function

Review: Bunkov and Volovik, arXiv:1003.4889

βM βl

Axial trap FZ = (ω − ωL)||2 ˆ l texture: radial trap Fso ∝ sin2 βl 2 ||2

"PERSISTENT" PRECESSION AT LOW TEMPERATURES

Discovered in Lancaster in pulsed NMR experiments at T < 0.2Tc

PRL 69, 3092 (1992)

tipping pulse spin precession at 1 MHz (creating magnons) NMR pick-up, arb. un. time, s

• Relaxation times up to ∼ 103 s.
• Precession frequency increases

during relaxation.

• Off-resonance excitation (even with

noise) at higher frequencies.

"PERSISTENT" PRECESSION AT LOW TEMPERATURES

Discovered in Lancaster in pulsed NMR experiments at T < 0.2Tc

PRL 69, 3092 (1992)

tipping pulse spin precession at 1 MHz (creating magnons) NMR pick-up, arb. un. time, s

• Relaxation times up to ∼ 103 s.
• Precession frequency increases

during relaxation.

• Off-resonance excitation (even with

noise) at higher frequencies. These features (and more) find explanations in the picture of the magnon BEC in the magneto-textural trap.

(Bunkov and Volovik, PRL 98, 265302 (2007))

10 20 30 40 1 2 3 20 40 60 80

SELF-TRAPPING OF THE MAGNON BEC

r, mm βl, degrees βM, degrees

H M

βM βl

P = 29 bar T = 0.24 Tc

Fso ∝ sin2(βM/2) sin2(βl/2)

10 20 30 40 1 2 3 20 40 60 80

SELF-TRAPPING OF THE MAGNON BEC

r, mm βl, degrees βM, degrees

H M

βM βl

Texture is flexible, when βM increases βl tends to decrease: Magnon con- densate forms a "bubble" with ˆ l H.

P = 29 bar T = 0.24 Tc

Fso ∝ sin2(βM/2) sin2(βl/2)

SELF-TRAPPING OF THE MAGNON BEC

r, mm βl, degrees βM, degrees

H M

βM βl

Texture is flexible, when βM increases βl tends to decrease: Magnon con- densate forms a "bubble" with ˆ l H.

10 20 30 40 1 2 3 20 40 60 80 fit to the wave function in a box

P = 29 bar T = 0.24 Tc

Fso ∝ sin2(βM/2) sin2(βl/2)

• Harmonic trap transforms to a

box with impenetrable walls. First example of BEC in a box.

• Texture-mediated interaction re-

sults in dµ/dNm < 0.

• Analog of the electron bubble in

helium and of the MIT bag model

• f hadrons.

PRL 108, 145303 (2012)

FILLING TRAP WITH MAGNONS AT THE GROUND LEVEL

CW NMR: downward frequency (upward field) sweep. Number of magnons Nm ∝ M 2

⊥

50 100 150 200 250

f − fL, Hz

1 2 3 4 5

M⊥ · 10

3/ Mmax

20 30 40 50 60 Excitation (µV)

⇒ ⇒ ⇒ ⇔ ⇔

P = 29 bar, T = 0.24 Tc = 0.8 rad/s (vortex-free) fL = 0.865 MHz H/H = 8.3 · 10−4

Chemical potential µ ∝ f − fL

dµ/dNm < 0

COHERENT PRECESSION OF THE MAGNON CONDENSATE

Condensation is demonstrated by long decay times of free precession.

100 150 200

f f − − f f

L L

, Hz , Hz

1 2 3

M⊥, arb. un. Time, s

cw NMR

P = 0.5 bar, T = 0.14 Tc

170 180 190 200 210 10 20 30 40 50 60 70

COHERENT PRECESSION OF THE MAGNON CONDENSATE

Condensation is demonstrated by long decay times of free precession.

100 150 200

f f − − f f

L L

, Hz , Hz

1 2 3

M⊥, arb. un. Time, s

cw NMR

P = 0.5 bar, T = 0.14 Tc

free precession

170 180 190 200 210 10 20 30 40 50 60 70

SCALING IN THE CYLINDRICAL BOX

E(Rb) = Nm ¯ h2λ2

m

2mMR2

b

+ 2πRbσ(Rb) → Similar to electron bubble: min Scaling: βl β0 βM r Rb ξH

kinetic energy of magnons

mM λm

• magnon mass
• root of the Bessel

function

surface energy ≡

• rbital gradient

energy: 2πRb ξH β0 ξH 2

σ ∝ β2

0 ∝ R2 b ⇒ Rb ∝ N1/5 m

10

12

10

13

10

14 Nm

0.5 1

Rb, mm

simulation ∝ N1/5

m

SCALING IN THE CYLINDRICAL BOX

E(Rb) = Nm ¯ h2λ2

m

2mMR2

b

+ 2πRbσ(Rb) → Similar to electron bubble: min Scaling: βl β0 βM r Rb ξH

kinetic energy of magnons

mM λm

• magnon mass
• root of the Bessel

function

surface energy ≡

• rbital gradient

energy: 2πRb ξH β0 ξH 2

σ ∝ β2

0 ∝ R2 b ⇒ Rb ∝ N1/5 m

50 100 200

f − fL, Hz

0.1 1 10

M⊥ · 10

3/ Mmax

experiment simulation ∝f

• 7/4

⇒ M⊥ ∝ N1/2

m Rb ∝ R7/2 b

∝ (f − fL)−7/4 For the magnetization: f − fL ∝ R−2

b

M⊥ ∝

• sin βMdV , Nm ∝
• (1 − cos βM)dV

PROBING EXCITED MAGNON LEVELS

f − fL, kHz M⊥ · 103/Mmax In the harmonic trap (small Nm): 2π(f − fL) = ωr(m + 1) + ωz(n + 1/2) Since dµ/dNm < 0, in cw NMR sweep excited levels in the trap are encoun- tered first and macroscopic population of an excited level can be built.

0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8

(0,0) ... (0,10) (m,n) ⇒ (2,0) ... (2,10) (4,0) ... (4,8) (6,0) ... (6,8)

= 0.6 rad/s H/H = 0.8·10−3 ωr/2π = 168 Hz ωz/2π = 22 Hz

P = 29 bar, T = 0.25 Tc fL = 0.865 MHz

PROBING EXCITED MAGNON LEVELS

f − fL, kHz M⊥ · 103/Mmax In the harmonic trap (small Nm): 2π(f − fL) = ωr(m + 1) + ωz(n + 1/2) Since dµ/dNm < 0, in cw NMR sweep excited levels in the trap are encoun- tered first and macroscopic population of an excited level can be built.

0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8

(0,0) ... (0,12) (m,n) ⇒ (2,0) ... (2,10) (4,0) ... (4,8)

= 0.6 → 0.8 rad/s H/H = 0.8·10−3 ωr/2π = 168 → 253 Hz ωz/2π = 22 Hz

PROBING EXCITED MAGNON LEVELS

f − fL, kHz M⊥ · 103/Mmax In the harmonic trap (small Nm): 2π(f − fL) = ωr(m + 1) + ωz(n + 1/2) Since dµ/dNm < 0, in cw NMR sweep excited levels in the trap are encoun- tered first and macroscopic population of an excited level can be built.

0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8

(0,0) ... (0,12) (m,n) ⇒ (2,0) ... (2,10) (4,0) ... (4,8)

= 0.8 rad/s H/H = 0.8·10−3 ωr/2π = 253 Hz ωz/2π = 22 Hz

PROBING EXCITED MAGNON LEVELS

f − fL, kHz M⊥ · 103/Mmax In the harmonic trap (small Nm): 2π(f − fL) = ωr(m + 1) + ωz(n + 1/2) Since dµ/dNm < 0, in cw NMR sweep excited levels in the trap are encoun- tered first and macroscopic population of an excited level can be built.

0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8

(0,0) ... (0,12) (m,n) ⇒ (2,0) ... (2,12) (4,0) ... (4,8)

= 0.8 rad/s H/H = (0.8→1.6)·10−3 ωr/2π = 253 Hz ωz/2π = 22 → 28 Hz

PROBING EXCITED MAGNON LEVELS

f − fL, kHz M⊥ · 103/Mmax In the harmonic trap (small Nm): 2π(f − fL) = ωr(m + 1) + ωz(n + 1/2) Since dµ/dNm < 0, in cw NMR sweep excited levels in the trap are encoun- tered first and macroscopic population of an excited level can be built.

0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8

(0,0) ... (0,12) (m,n) ⇒ (2,0) ... (2,12) (4,0) ... (4,8)

= 0.8 rad/s H/H = 1.6·10−3 ωr/2π = 253 Hz ωz/2π = 28 Hz

COHERENT PRECESSION ON THE EXCITED LEVEL

Spectral amplitude of the NMR pickup, arb.un. Time, s

• 5

0.2 0.4 0.6 0.8 1 1

f − fL, kHz

10 20

Time, s

0.5 1 1.5

P = 0.5 bar, T = 0.14 Tc, fL = 0.827 MHz = 0.9 rad/s (vortices)

ground level (2,0) level rf pumping

COHERENT PRECESSION ON THE EXCITED LEVEL

Spectral amplitude of the NMR pickup, arb.un. Time, s

• 5

0.2 0.4 0.6 0.8 1 1

f − fL, kHz

rf pumping off ground level (2,0) level 5 10 15 20

10 20

Time, s

0.5 1 1.5

ground level (2,0) level

P = 0.5 bar, T = 0.14 Tc, fL = 0.827 MHz = 0.9 rad/s (vortices)

rf pumping

COHERENT PRECESSION ON THE EXCITED LEVEL

Spectral amplitude of the NMR pickup, arb.un. Time, s

• 5

0.2 0.4 0.6 0.8 1 1

f − fL, kHz

rf pumping off ground level (2,0) level 5 10 15 20

10 20

Time, s

0.5 1 1.5

ground level, τ = 5.36 s (2,0) level, τ = 0.74 s Decay ∝ exp(-t/τ):

P = 0.5 bar, T = 0.14 Tc, fL = 0.827 MHz = 0.9 rad/s (vortices)

• Relaxation time in the excited state is longer than in linear NMR (∼ 10 ms).
• Ground state is filled simultaneously − two coexsisting condensates.

rf pumping

MOTIVATION FOR RELAXATION STUDIES H M z H in bulk H M at the free surface with vortex lines

Long life time of the magnon BEC in the T → 0 limit (exceeding the life time

• f atomic condenstates) makes them a sensitive probe for extra relaxation

sources.

• We hope to find the contribution from the Majorana fermion zero modes bound

to the surface of cores of quatized vortices by comparing relaxation of magnon condensates in different trap configurations.

BOUND FERMION STATES IN THE VORTEX CORE

Caroli, de Gennes, Matricon 1964

b

x y y x particle hole

a

a

Andreev reflection εn(pz = 0, µ)

• −

n = 0 nmax µ/pF 1 ¯ hω0

Radial quantum number n (nmax ∼ a/ξ).

µ/¯ h =

• m + 1/2, s-wave superconductors

m, superfluid 3He

Angular momentum µ = b p⊥, quantized. Minigap ω0 ∼ a pF ∼ 1 ¯ h 2 EF ≪ ¯ h . Anomalous (crossing zero) branch n = 0.

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6

Relaxation rate 1/τ, s

• 1

0.15 0.16 0.17 0.175

T/Tc

RELAXATION IN THE VORTEX STATE = 0 P = 0.5 bar

Thermometer fork width [∝ exp(−/T )], Hz Relaxation rate: 1/τ = 1/τ0 + C exp(−/T )

0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6

Relaxation rate 1/τ, s

• 1

0.15 0.16 0.17 0.175

T/Tc Relaxation rate: 1/τ = 1/τ0() + C() exp(−/T )

RELAXATION IN THE VORTEX STATE = 0 P = 0.5 bar

Thermometer fork width [∝ exp(−/T )], Hz

0.2 rad/s 0.6 rad/s 1.2 rad/s 1.6 rad/s

TEMPERATURE DEPENDENCE OF RELAXATION

0.1 0.2 0.3 0.4

ωr/2π, kHz

0.5 1 1.5 2

Slope 1/τ vs fork width [∝ exp(−/T )] Spin diffusion via normal component (bulk thermal quasiparticles): 1/τ ∝ ρn |∇|2 ∝ ρn R−2

b

∝ exp(−/T) ωr

DEPENDENCE OF RELAXATION ON VORTEX DENSITY

• 0.5

1 1.5

, rad/s

0.1 0.2

Relaxation rate at T = 0, s

• 1

P = 0.5 bar

Vortices definitely contribute to the relaxation of magnon condensates. Is the effect related to the fermions bound to vortex cores?

BROKEN SYMMETRY OF VORTEX CORES IN He-B

0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0

Broken symmetry core Axisymmetric core Solid 1 2 10 20 30 Normal A B

A A

P, bar T, mK ∼ few ξ

Ikkala, Hakonen, Bunkov, Krusius et al 1982- Salomaa, Volovik, Thuneberg et al

3

BROKEN SYMMETRY OF VORTEX CORES IN He-B

0.4 0.6 0.8 1.0 0.4 0.6 0.8 1.0

Broken symmetry core Axisymmetric core Solid 1 2 10 20 30 Normal A B

A A

P, bar T, mK ∼ few ξ

Ikkala, Hakonen, Bunkov, Krusius et al 1982- Salomaa, Volovik, Thuneberg et al

3

0.5 0.55 0.6 0.65 0.1 0.15 0.2 0.25

Ω↑↑H Ω↑↓H

transition at 29 bar ωr/2π, kHz T/Tc Effect on the textural potential well for magnons:

DAMPING OF SPIN PRECESSION VIA VORTEX CORES

Core of the non-axisymmetric vortex

M

Torque from precessing magnetic moment puts vortex core in twisting motion (oscillations / precession) ⇓ Transitions between the core-bound fermion states are triggered and the core gets overheated ⇓ Dissipation

(Kopnin and Volovik, 1998)

• For the coherently precessing magnon condensate in a magneto-textural

trap in 3He-B the trap transforms with increasing magnon number from a harmonic well to a cylindrical box: bosonic analogue of the electron bubble in helium and of the MIT bag model of hadrons.

• Unlike cold-atom case, in the magnon trap different excited levels can be

selectively populated with condensates.

• Relaxation rate of magnon condensates depends on temperature and the

trap size as expected for the spin diffusion relaxation mechanism.

• In the vortex state relaxation rate has an additional contribution, which

grows linearly with the density of vortices. Whether this contribution can be attributed to the Majorana fermions bound to vortex cores remains to be established.

CONCLUSIONS

PRL 108, 145303 (2012)