島野 亮Ryo Shimano

東京大学低温センター東京大学大学院理学系研究科物理学専攻

The University of Tokyo

非従来型超伝導体のヒッグスモードHiggs mode in unconventional superconductors

(1) Introduction to Higgs mode and Higgs mode in isotropic pairing(s-wave) superconductor (NbN)

(2) Higgs mode in anisotropic pairing (d-wave) High-Tc superconductor (Bi2Sr2CaCu2Ox )

(3) Higgs mode in multiband superconductor (FeSe0.5Te0.5)

Outline

National Institute of CommunicationTechnology

K. MakiseY. Uzawa

H. Terai

Z. Wang (SIMIT at present)

Dept. of Phys., Univ. of Tokyo

H. AokiN. Tsuji (Riken CEMS at present)

Coworkers and Collaborators

K. KatsumiK. Tomita

Y. MurotaniN. YoshikawaR. Matsunaga (ISSP at present)

NbN:

Brookhaven National Lab.R.D. Zhong,J.Schneeloch,G.D. Gu,

Univ. of Paris DiderotY. Gallais

FeSexTe1-x:

Dept. of Basic Sci., Univ. of Tokyo

A. MaedaF. NabeshimaT. IshikawaN. Shikama

Bi2212:

YBCO：

Osaka Univ.

S. TajimaS. Miyasaka

MgB2:

Kitami Inst. TechH. Shibata

BCS theory0)',()(Δ ''

'

kk ccVk

kkk

Particle Physics and BCS theory

Energy dispersion ofQuasiparticles in BCS

Energy dispersion of particleand antiparticle

1957 BCS theory(Bardeen, Cooper & Schrieffer)1960 Nambu’s theory of SSB1960-61 Nambu-Goldstone theorem1963-66 Anderson-Higgs theory(Anderson, Higgs)1967 Unified theory of electroweak interaction (Winberg& Salam)2012 Discovery of Higgs boson(CERN LHC)

SSB of U(1) gauge symmetry

青木秀夫： “南部理論と物性物理学” [日本物理学会誌、64, 80 (2009)]より

Spontaneous Symmetry breaking and collective modes

amplitude mode

When spontaneous symmetry breaking occurs, massless and massive collective modeswith respect to the order parameter appear.

ReΨImΨ

Free Energy

phase mode

（ Nambu-Goldstone mode）

E

k

Anderson-Higgs mechanism

“Anderson-Higgs mechanism” or “ Brout-Englert-Higgs mechanism”“ABEGHHK'tH mechanism “

[for Anderson, Brout, Englert, Guralnik, Hagen, Higgs, Kibble and 't Hooft]

amplitude mode (Higgs boson, Higgs mode)

E

k

phase mode

（ Nambu-Goldstone mode）

0

Z,W boson, p-mesons, plasmon

Massive gauge boson(photon)in superconductors

Meissner-Ochsenfeld effect 1933

T>Tc T

Free Energy

ReYImY

Y0

massive amplitude mode massive gauge boson

Local gauge transformation

Scalar field theory of the Anderson-Higgs mechanism

Note that the phase degree of freedom is gone. “Gauge boson has eaten the N-G boson.”

Quantum quench problem

Quenching the interaction U(t) much faster thanτΔ ~ ℏ/Δ （Δ:order parameter）

Volkov et al., Sov. Phys. JETP 38, 1018 (1974). Barankov et al., PRL 94, 160401 (2004).Yuzbashyan et al., PRL 96, 230404 (2006).Gurarie et al., PRL 103, 075301 (2009).Podolsky, PRB84, 174522 (2011).A. P. Schnyder et al., PRB84, 214513 (2011)N. Tsuji et al., PRB 88,165115 (2013).N. Tsuji et al., PRL 110, 136404 (2013).

●

●

●

Theoretical studies for dynamics of nonequilibrium BCS state after nonadiabatic excitation

Emergence of order parameter oscillation (Higgs mode)

0

Free energy

ReΨ

Higgs mode in superconductors

For a review, e.g., Lee, J. Phys. Chem. Sol. 59, 1682 (1998).G. E. Volovik, and M. A. Zubkov, J. Low Temp. Phys.175, 486 (2014)

BCS-CDW coexistent compound NbSe2

Cf.) p-wave superfluid 3He (not Higgs, but amplitude mode as there is no Higgs mechanism)

R. Sooryakumar and M. V. Klein, PRL 45, 660 (1980).P.B. Littlewood and C. M. Varma, PRL 47, 811 (1982).C. M. Varma, J. Low Temp. Phys.126, 901 (2002)

M.-A. Measson, et al., PRB 89, 060503 (2014).

For a recent review: David Pekker and C. M. Varma, Ann. Rev. Cond. Matt. Phys.6, 269(2015).

Quench by the injection of quasiparticles

Ene

rgy

DOS

2D(0) 2D(T)

Energy

DOS

EFCooper pair

quasiparticlephoton hn

The gap (order parameter) is determined self-consistently with the quasiparticle distribution f (e) through the gap equation

Quasiparticle injection by ultrafast optical pulse

Ene

rgy

DOS

What happens if one create quasiparticelinstantaneously, t

1.5

1.0

0.5

0.02.01.51.00.50.0

6x103

4

2

2.01.51.00.50.0

THz pump and THz probe experiment in NbN

Nb0.8Ti0.2N film (12nm)/Quartz

TC = 8.5 K, 2Δ(T=4 K) = 3.0 meV = 0.72 THz

response time : tD D-1 ~ 2.8 ps

Re

σ(Ω

-1cm

-1)

real-partoptical conductivity

Frequency (THz)

9K

4K

8K

6K7K

Inte

nsity

(ar

b. u

nits

)

pump pulse power spectrum

Frequency (THz)

5K

Sample

Center frequency 0.7THz～2D

pulse width: tpump ~ 1.5 ps

tpump/tD ～0.57 < 1

nonadiabatic excitation condition

THz pump pulse

1086420Delay Time (ps)P

ump

Ele

ctri

c F

ield

(arb

. un

its) temporal waveform

gate

LiNbO3

sample

wire grid×3

wire grid

ZnTeZnTe

balanced detection

pump THz pulse

probe THz pulse

electro-optic samplingtilted pulse-front scheme

Pump

Probe

wire-grid polarizer

wire-grid polarizer

sample

xy z

tpp

Pump ： Epump//xProbe： Eprobe//ytpp: pump-probe delay

Transmitted probe THz electric field:Free space EO samplingtgate: gate pulse delay

THz pump and THz probe experiment in NbN

Detection of order parameter dynamics

Temperature dependence of the probe E-field without pump

We fixed the gate delay at tgate=t0and measure the pump-probe delay dependence

Eprobe(tgate)

Pro

be E

lect

ric

Fie

ldE

pro

be(a

rb. u

nits

)

Gate Delay Time tgate (ps)

tgate=t0

At tgate=t0, the change in Eprobe is proportional to the change in the order parameter Δ.

Dynamics after the THz pump pulse

THz pump-induced change in the probe E-field δEprobe(tgate=t0)

τpump/τΔ=0.57

86420-2-4

1

0

nJ/cm2

9.6 8.5 7.9 7.2 6.4 5.6 4.8 4.0C

han

ge

of

Pro

be

Ele

ctri

c F

ield

δE

prob

e(t g

ate=t 0

) (

arb

. un

its)

Pump-probe Delay Time tpp (ps)

Order parameter dynamics

Pump Intensity (nJ/cm2)

Fre

quen

cy (

TH

z)86420-2-4

1

0

nJ/cm2

9.6 8.5 7.9 7.2 6.4 5.6 4.8 4.0

0.6

0.4

0.2

0.010864

Pump-probe Delay Time tpp (ps)

Cha

nge

of

Pro

be E

lect

ric

Fie

ldδE

pro

be(t

gate=t 0

) (

arb

. un

its)

U

ΔΔ∞ Δ0

0

f = 2Δ∞

f2Δ∞

R. Matsunaga et al., PRL111, 057002 (2013)

5

4

3

2

1

0

-1

-2

-32.01.51.00.5

5

4

3

2

1

0

-1

-2

-32.01.51.00.5

3x104

2

1

0

6x104

5

4

3

2

1

02.01.51.00.5

w/o pump 0.9 ps 1.3 ps 1.7 ps 2.1 ps 2.5 ps

2x104

1

02.01.51.00.5

w/o pump 0.7 ps 1.1 ps 1.5 ps 1.9 ps 2.3 ps

(a) σ1 (ω) (Ω-1 cm-1) (b) σ2 (ω) (Ω-1 cm-1)

σ1

(ω)

(Ω

-1cm

-1)

σ2

(ω)

(Ω

-1cm

-1)

Pu

mp

-Pro

be D

ela

y T

ime

t pp

(ps)

Pu

mp

-Pro

be D

ela

y T

ime

t pp

(ps)

Frequency ω (THz) Frequency ω (THz)

Frequency ω (THz) Frequency ω (THz)

(c) (d)

18

Time evolution of conductivity spectrum σ1(ω; tpp)

-0.1

0.0

1210864

Pump-Probe Delay Time (ps)

Data Power-law fit Exponential fit

δEpr

obe

(arb

. uni

ts)

exponential decay

t= 1.3 ps

power-law decay

b = 0.71

Tc=15K, 2Δ(T=4K)=1.3THz

c2= 3.6×10-4

c2= 2.8×10-4

f=1.35 THz

Power law decay

Volkov et al., Sov. Phys. JETP 38, 1018 (1974). Yuzbashyan et al., PRL 96, 097005 (2006).

Weak coupling case (BCS)

1.5

1.0

0.5

0.016141210864

Temperature (K)

Ele

ctric

Fie

ld (

arb

. u

nits

)

2520151050Delay Time (ps)

monocycle pulse monocolor pulse

Inte

nsi

ty (

arb

. u

nits

)

2.01.51.00.50.0

Frequency (THz)

monocycle pulse monocolor pulse

Coherent excitation regime with multicycle THz pulse

Quasi-monochromatic THz pulse（0.3THz, pulsewidth～13ps）

E-field waveform Power Spectrum

2Δ (T

Hz)

Photon energy vsBCS gap

How does the BCS ground state respond to the strong electromagnetic field with ℏω

1086420Delay Time (ps)

0

1086420Pump-Probe Delay Time (ps)

0

14 K 14.5 K 14.8 K 15.5 K

13 K 12 K 10 K 4 K

Epump

|Epump|2

δEp

rob

e(a

rb.

uni

ts)

ω=0.6THzE=3.5 kV/cm @ peak

Coherent Excitation Regime Experiments

R. Matsunaga et al., Science 345, 1145 (2014)

2Δ

(T)

ω=0.6THz

1

016141210864

Temperature (K)

δE

prob

e(a

rb. u

nits

) long-term component

oscillating component

16141210864

Temperature (K)

2ω=2D

Anderson’s pseudospin representation

where 𝒃𝒌 is the pseudo magnetic field

P.W. Anderson, PR 112, 1900 (1958)

normal state (T=0)

BCS state

Pseudospin up : (k, -k) both occupied

Pseudospin down: (k, -k) both empty

kFk

f

0

kFk

f

0

all ↓ (k>kF)all ↑ (k

Time evolution of BCS state is described by the motion of pseudospins under effective magnetic field

The time evolution of the pseudospin is given by the Heisenbergs’ equation of motion

𝒌 𝒌 𝒌 𝒌

𝒌 𝒌

𝒌

𝒌 𝒌

z

-Δ’

x

𝒌

𝒌

-Δ’(0)

𝒌

Let’s consider that D’ is suddenly quenched at t=0.

Each pseudospin 𝒌 starts the precession around the new 𝒌 .

In the presence of EM field (vector potential)

z-component of effective magnetic field oscillates at 2w⇒ precession of Anderson’s pseudospins

z

bk

σk

-Δ’x

εk

kkb e,",'eff DD

k

kkyx σiσUi "Δ'ΔΔ

.)(e2

)()()(22

1

,

422

22

,

422

)()(

ji

tiji

ji

jiji

jitete

AOEE

kk

e

AOtAtAkk

e

w

wee

eeee

kk

kkAkAk

Pseudospin dynamics under the presence of vector potential A(t)

kkkk σbσσ eff2,idtd

Pseudospin dynamics : simulation with BdG equation

Free Energy

ReYImY

Y0

Ginzburg-Landau picture

Local gauge transformation

H

A

A

A

A

THz THG by Higgs mode

Current density

dD(t)～ei2wt, A(t) ～ eiwt

)(Δ)(Δ

)(~

2

1)()(

2

linear

)(

tδtU

et

tσε

enet zte

Aj

kvj

kk

Ak

k kAk

London equation for nonlinear current jnl

j (t) ～ ei3wt

Does superconductor emit THz third harmonics?

Waveform of the transmitted pulse

Power spectrum of the transmitted pulse

121086420

Delay Time (ps)

10 K 15.5 K

Epu

mp

(arb

. u

nits

) Tc=15K

10-4

10-3

10-2

10-1

100

101

102

Inte

nsi

ty (

arb

. u

nits

)

210Frequency (THz)

10 K 15.5 K

3ω

ω

Efficient THG from superconductor

Nonlinear transmission experiment

5x10-2

4

3

2

1

0In

ten

sity

(a

rb.

un

its)

2.42.22.01.81.61.41.2

Frequency (THz)

4 K 8 K 10 K 11 K 12 K 13 K 14 K 15.5 K

R. Matsunaga, R.Shimano et al., Science 345, 1145 (2014)

1

01.21.00.80.60.4

1.5

1.0

0.5

0.0

0.8 THz1

01.21.00.80.60.4

1.5

1.0

0.5

0.0

0.3 THz

1

01.21.00.80.60.4

1.5

1.0

0.5

0.0

0.6 THz

TH

G I

nte

nsi

ty (

arb

. un

tis)

Experiment

0

1

1.21.00.80.60.4

0.8 THz 0.6 THz 0.3 THz

T/Tc

TH

G I

nte

nsi

ty (

arb

. un

tis)

Theory

1.2 THz

1.6 THz

0.6 THz2Δ

(T)

(TH

z)

T/Tc

Temperature dependence of THG

Experiments with different frequencies ω=0.3, 0.6, 0.8 THz

THG shows a peak at 2ω=2Δ(T),but not at ω=2Δ(T)!

Theory: N. Tsuji and H. Aoki, Phys. Rev. B 92, 064508(2015)

R. Matsunaga et al., Science 345, 1145 (2014)

H

A

A

A

A

Science 345, 1121 (2014)

(1) Introduction to Higgs mode and Higgs mode in isotropic pairing(s-wave) superconductor (NbN)

(2) Higgs mode in anisotropic pairing (d-wave) High-Tc superconductor (Bi2Sr2CaCu2Ox )

(3) Higgs mode in multiband superconductor (FeSe0.5Te0.5)

Outline

Higgs in High Tc cuprate

Phase diagram of Bi2Sr2CaCu2Ox

M. Hashimoto et al., Nat. Phys. 10, 483 (2014)

UD OP OD

Higgs modes in d-wave SC

Barlas and Varma, PRB 87, 054503 (2013)

A1g

A2g

B2g

B1g

THz pump and optical probe experiments in Bi2Sr2CaCu2Ox

400200

0-200

5.02.50.0Time (ps)

ba

Cu-O

Sample

Time Delay

THz Pump

Optical Probe(800 nm)

θ

1086420Delay Time (ps)

0

1086420Pump-Probe Delay Time (ps)

0

14 K 14.5 K 14.8 K 15.5 K

13 K 12 K 10 K 4 K

Epump

|Epump|2

δEp

rob

e(a

rb.

uni

ts)

ω=0.6THzE=3.5 kV/cm @ peak

Coherent Excitation Regime Experiments in NbN

R. Matsunaga et al., Science 345, 1145 (2014)

2Δ

(T)

ω=0.6THz

1

016141210864

Temperature (K)

δE

prob

e(a

rb. u

nits

) long-term component

oscillating component

16141210864

Temperature (K)

2ω

Transient reflectivity change

300

200

100

0

Te

mp

era

ture

(K

)

1050Time (ps)

840DR/R (10

-4)

TC

Cu-O

THz Pump

a

Probe °

Optical Probe

b

Symmetry of the signal

y

x

Cu

O

y

+

--

+

xEProbe

A1gB2g

B1g

2.0

1.6

1.2

0.8

0.4

0.0

DRM

ax/R

(10

-3)

1000

Probe (degree)

30 K 100 K 300 K

Pump = 0°

)2sin2sin2cos2cos(2

1

Re1

~),(

)3()3()3()3(

)3(0

1

211 probepumpBprobepumpBA

pumpl

pumpkijkl

probej

probei

ggg

EER

REE

R

R

cccc

cee

D

Bi2212: D4h point group

THz-induced Kerr effect

Pump polarization dependence

1.0

0.5

0.0

-0.5

-1.0DR

Max

/R (

no

rm.)

-50 0 50

Pump (degree)

A1g

B1g

)2sin2sin2cos2cos(2

1 )3()3()3()3(211 probepumpBprobepumpBA ggg

cccc

1050Time (ps)

1.50.0DR/R (10

-4) B1g

300

200

100

0Te

mp

era

ture

(K

)

1050Time (ps)

840DR/R (10

-4)

Tc

A1g

Temperature dependence of A1g and B1g

A1g: oscillatory(coherent) component + decay(incoherent) componentB1g: only oscillatory(coherent) component

Decomposition into coherent and incoherent part

5.0

2.5

0.0

DR

/R (

10-

4 )

100Time (ps)

10 K A1g

Temperature dependence of each component

Doping dependence

A1g signal is always dominant.

Polarization dependence of CDF mean field(BCS) theory with d-wave symmetry

W : 4meV w : 1.5 eV

The dominance of A1g signal cannot be explained by CDF.

CDF: B1g is dominantTHz pump-optical probe

Doping dependence of the oscillating component

A1g signal is attributed to Higgs.

B1g is most likely CDF.

200

150

100

50

0

Tem

pera

ture (K

)

1.0

0.8

0.6

0.4

0.2

0.0

B1g

/ A

1g

0.200.10Doping

TC

K. Katsumi et al, Phys. Rev. Lett. 120, 117001 (2018).

(1) Introduction to Higgs mode and Higgs mode in isotropic pairing(s-wave) superconductor (NbN)

(2) Higgs mode in anisotropic pairing (d-wave) High-Tc superconductor (Bi2Sr2CaCu2Ox )

(3) Higgs mode in multiband superconductor (FeSe0.5Te0.5)

Outline

(1) Higgs mode in s-wave SC (NbN)

(2) Higgs mode in d-wave High-Tc SC (Bi2Sr2CaCu2Ox )

(3) Higgs mode in multiband SC (FeSe0.5Te0.5)

86420-2-4

1

0

nJ/cm2

9.6 8.5 7.9 7.2 6.4 5.6 4.8 4.0

Ch

an

ge o

f Pro

be

Ele

ctric

Fie

ldδE

prob

e(t g

ate=t 0

) (

arb

. un

its)

Pump-probe Delay Time tpp (ps)

Summary

1050Time (ps)

1.50.0DR/R (10

-4)

B1g

300

200

100

0Te

mp

era

ture

(K

)

1050Time (ps)

840DR/R (10

-4)

Tc

A1g

Outlook: toward the Higgs spectroscopy in ”strongly” unconventional SCs, U(1) SU(2) T

H

A

A