Higgs amplitude mode and charge density fluctuationsin the nonlinear optical response of superconductors

PHYSICAL RE

PRL 111, 057002 (2013)
(a)

(b)

sample

WGP

tpp

Pump

4

15 K
14 K
13 K
12 K
10 K
8K
4K

(c)

Naoto Tsuji (RIKEN CEMS)
1 (

1

0
0.0

0.5

1.0

1.5

y

Probe

2.0

2.5

z

1.5

2 (THz)-1cm-1)

2x10

x

Im

Re

WGP

A
B
C

1.0
0.5
0.0

(d)

0

5

10

15

Collaborators[ Theory ]Y. Murakami (Fribourg Univ.)
H. Aoki (Univ. Tokyo, AIST)[ Experiment ]R. Shimano (Univ. Tokyo)
R. Matsunaga (Univ. Tokyo)
K. Makise (NICT)
H. Terai (NICT)[1] Tsuji, Murakami, Aoki, Phys. Rev. B 94, 224519 (2016).
[2] Matsunaga, Tsuji, Makise, Terai, Aoki, Shimano, arXiv:1703.02815.
(To be published in Phys. Rev. B, Rapid Communications.)

Outline of the talk• Introduction: Higgs amplitude mode in superconductors• Higgs mode vs charge density fluctuation (CDF)• Experimental discrimination by polarization-resolved THG• A mechanism of enhancing the light-Higgs coupling— Phonon retardation effectOutline of the talk• Introduction: Higgs amplitude mode in superconductors• Higgs mode vs charge density fluctuation (CDF)• Experimental discrimination by polarization-resolved THG• A mechanism of enhancing the light-Higgs coupling— Phonon retardation effectCollective modes in superconductors
Anderson (1958), Volkov, Kogan (1973), Littlewood,Varma (1981,1982), …

PHYSICAL RE

PRL 111, 057002 (2013)

E

(a)

(b)

WGP

samplepquasiparticle
excitations

Re

1

(r) =
(c)

x
2
Probe

n s (r)ei (r)

1.5

y

0
15 K
14 K
superfluid density.
• Higgs mode = collective amplitude oscillation
13 K of the1.0
12 K• One of the lowest energy excitation modes.
K
broken phases• Universal phenomenon in spontaneous10symmetry
8K
0.5

2 (THz)1 (-1cm-1)2x10

4Nambu-Goldstonemode

Higgs mode

Pump
Im

tpp

WGP

z
k

A
B
C

—wave (CDW) distortion at 33 K which induces
These are joined in the superconducting state at
se in energy to the BCS gap 24. Magnetic fields
s and enhance that of the CDW-induced modes, thus
the superconducting-gap

Observation of Higgs mode

excitations and the CD%.

74. 70. Lp

defects and impurities inhibit the formation of
CDW's.
2H-NbSe
2 (SC+CDW), Raman
From Figs. 1(c) and 1(d), where all curves
have therelative
intensities,
proper
we find 25that 1980
8
PHYSICALREVIEW
LETTERS
4$,
the CDW modes lose intensity when the new "gap"Raman Scattering by Superconducting-Gap
and Their Coupling
modes appear.
This
direct Excitations
coupling bebveen
toCharge-Density%VavesM. V. Klein
modes C and 6R.isSooryakumar
shown andmore
dramatically in
Department of 2%ysics and Materials Research Laboratory, University of Illinois at Urbana C-hampaign,•

ud- VOLUME

-

T„

-

d

in
e
K

PRL 111, 057002 (2013)

PACS numbers: 78.30.Er, 74.30.6n, 74. 70. Lp
Edefects and impurities inhibit the formation of
StructuralOP phase transitions involving chargeCDW's.
density waves(CDW) in layered transition-metalCA
From Figs. 1(c) and 1(d), where all curves
dichalcogenides have been studied extensively inhave the proper relative intensities, we find thatthe last several years. ' Neutron diffraction studO
ies' on 2tI-NbSe,
the CDW modes lose intensity when the new "gap"show a transition from a norO
inmodes appear. This direct coupling bebveen
mal lattice to one with a three-wave-vector&-50—
6 is shown more dramatically in
CDW at the onset temperature T„
commensurate
SAMPLEb modes C and M
of 33 K. The CDW is only a few percent out of
4Gthe neutron data show thatandcommensurability
IICcO
it remains LLJincommensurate down to 5 K. From
SAMPLE M 80 —LE
I
SAMPLE B
the modulus ' measurements of Barmatz, Testardi,
AAZ.
and DiSalvo' it is concluded that incommensuraE
o70bility persists at least to 1.3 K. 2H-NbSe, is ahighly anisotropic type-II superconductor below
OO
7.2 K.' The upper critical fields at 2 K may be&-50—
b SAMPLE M 50—
estimated from published data' and are found to4G
be 1Q5 and 42 ko for fields parallel and perpen80—
I-'dicular to the layers, respectively. MagnetoreA
Z.
Ii
I
o
on
2H-NbSe,
have
been
carriedsistance studies
30—
30—
IO
40—
out by Morris, Coleman, and Bhandari. 'Figure 1 shows four pairs of Raman spectra
RAMAN SHIFT (cm )[(a)-(d)J from two different samples of 2H-NbSe„
Ii
o
IO
60
20
M and B, at two different temperatures, 9 K (low60
2060
RAMAN SHIFT (cm )2 K (upper curves)er curves in each pair) andRaman
samples M andThe
for A and E Raman symmetries. The characterFIG. l. Raman spectrum of samples M and B. The
lower amplitude modeseach
9K
and] is the] of each pair
istic CDW-induced
lower curve[{a)-{d)(C) are nearat 9 K and the'upper at 2 K. Raman symmetries polarizations] are33
first
ap40 cm-'. On cooling below
they
K,
2
Raman
upper
E l{xy)l and AI, t(xx) —{&y)]. C labels CDW modes;
pear, then harden, and get stronger. ' The mainG, C
gap excitations; and I, the interlayer mode charto report
A that when theandpurpose of this paper isacteristic of the 2HCDW
polytype. Incident laser beam at
sample is immersed in superfluid helium at 2 K
5145
A
and
30
mW
was
power
spread into a line 40-50
mode
gap modes are seen at and18 cm ' thetwo new Raman-active
p, m wide.
Light was incident at the pseudo Brewster'
in energythe(A) and at 15 cm(E), close thethe scattered light collected
angle;Incident
along the c axis.
2H topolytype.
beam
BCS gap at 24. These are labeled G in Fig. 1. It
Resolution was 3 cm
Curves (a) and (b) were drawn

low-

ear
p-

Department of Physics, The University of Tokyo, Tokyo 113-0033, Japan
National Institute of Information and Communications Technology, 588-2 Iwaoka, Nishi-ku, Kobe 651-2492, Japan
3
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan(Received 2 May 2013; published 29 July 2013)

A

50—

80—

30—

30—

OP

40—

CA

LLJ

Se„

r-

Ryusuke Matsunaga,1 Yuki I. Hamada,1 Kazumasa Makise,2 Yoshinori Uzawa,3Hirotaka Terai,2 Zhen Wang,2 and Ryo Shimano1
2

80 c

I-

60

I

20

I

60

60

20
I

I

I

FIG. l.
spectrum ofB.
curve ofpair [{a)-{d) is at
See alsoat Measson
etsymmetriesal., PRB 89,
060503 (2014).
K.
polarizations]are—
E l{xy)lt(xx) {&y)]. labels
modes;
excitations;charinterlayerG,I,
acteristic of
laser
at
'.
I,

week ending
2 AUGUST 2013

Higgs Amplitude Mode in the BCS Superconductors Nb1-x Tix N Induced
by Terahertz Pulse Excitation

PHYSICAL REVIEW L

Ultrafast responses of BCS superconductor Nb1-x Tix N films in a nonadiabatic excitation regime were
investigated by using terahertz (THz) pump-THz probe spectroscopy.
0.8After an instantaneous excitation
with the monocycle THz pump pulse, a transient oscillation emerges in the electromagnetic responsein the BCS(a)gap energy
The oscillation frequency coincides with the asymptotic value of the BCS/τ =0.57
τ pumpregion.
gap energy, indicating the appearance of the theoretically anticipated collective amplitude mode of theorder parameter, namely the Higgs amplitude mode. Our result opens
0.6 a new pathway to the ultrafast
manipulation of the superconducting order parameter by optical means.

The secodecrease
thermalizdescribes(b)
DOI: 10.1103/PhysRevLett.111.057002
PACS numbers: 74.40.Gh, 74.25.Gz, 78.47.J!
law decay
0.4
shows the2of continuous symmetry,state, while it still remains experimentally challenging.
With spontaneous breaking
nJ/cm
1
An alternative way to induce the transient oscillationof p
two types of collective
excitations associated with thevarious
9.6is the gapless phase mode the order parameter has been proposed in conventional
order parameter emerge. One
2!is nonat tpp
0.2SCs [19]. When a BCS ground state
8.5 mode, and the other is the metallic BCS
called as the Nambu-Goldstone
adiabatically excited byfa short laser pulse, the indicatescoherence
gapped amplitude mode also
referred to as the Higgs mode
7.9
betweendifferent
quasiparticle(QP)
statesleads
to thefrom the analogy to the Higgs
boson
in
particle
physics
7.2
oscillation of the order2parameter. Such a nonadiabatic[1,2], as schematically shown in Fig. 1(a). Recently, thethepum
6.4been observed in strongly excitation 0.0
for BCS superconductivity requires a short
Higgs amplitude mode has
paramete
5.6of bosonic ultracold atoms pump pulse with the duration !pump small enoughcominteracting superfluid phases
3
in optical lattices by means4.8
of Bragg spectroscopy [3] andpared to the response time of the BCS state characterized
from
the
(c)
!1 . Here a near-visible
lattice modulation [4]. The studies of the Higgs mode
by
the
BCS
gap
!
as
!
¼
"=!
!
4.0
2
spondenc
realized on tabletop experiments would provide substantial
femtosecond optical pulse is not applicable, becausetheplatforms for exploring the nature of symmetry-broken
huge excess energies of photoexcited hot electrons in theof the pum
1states in quantum many-body physics. In condensed matterorder of electronvolts
are transferred to the generation
systems, the amplitude mode has been widely observed inof large amounts of high-frequency phonons (@!
> 2!), tbecause0 (CDW) systems by Raman or pump- which in turn0induce the Cooper pair breaking. This procharged density wave
5
10
gapevenreduc
2
4 by6 cess8 destroys the nonadiabatic
probe spectroscopy-4[5–8] -2and in 0
an antiferromagnet
excitation
condition2
neutron spectroscopy [9]. However, theof theif one uses
the laser
pulse (nJ/cm
much shorter
tppobservation
(ps)
Pump
Energy) than !dependen! [25,26].
amplitude mode in fermionic condensates has been limited
Therefore, to ensure the nonadiabatic excitation, it is neca charact
to the specific cases of superconducting CDW compound
essary to use a short pump pulse with its photon
energy3
NbSe2 [10,11]
and p-wave
superfluid
He [12,13].
resonant
to theBCS thegap whichis typically located in
FIG.
2 (color
online).
(a)Then,
Theweopen
circlesshow
temporalpredicted
can pose a question as to whether the Higgs mode in a pureterahertz (THz) frequency range [19]. With the recent
,
at
t
¼
t
evolutionofthechange
of
the
probe
E
field,!E
gate such
0 an intense
metallic BCS superconductor (SC), which does not couple
development of probe
THz technology,
and str
resultto the radiation
field,
canbeobserved
experimentally.
monocyclelike
THz
pulse
has
becomeavailable[27], makas a function of tpp in sample A at 4 K. The solid curves show
The amplitude mode in the BCS order parameter has
ing it possible to investigate the THz nonlinearfrom
responsethe
Eprobe(tgate=t0) (arb. units)o

PHYSICAL REVIEW LETTERS

1a charge-density
II —wave (CDW) distortion at 33 K which induces
A and E Raman-active phonon modes. These are joined in the superconducting state at
C in energy to the BCS gap 24. Magnetic fields—
2 K by O
new ASAMPLE
and E Raman M
modes closeLE
suppress theI intensity of the new modes and enhance that of the CDW-induced modes, thus
SAMPLE B
providing evidence of coupling between the superconducting-gap
excitations and the CD%.
2H-Nbse~ undergoes

o70-

x

PRL 111, 057002 (2013)

glinois 61801(Received 24 March 1980)

m
rdi,

w
e

AUGUsv

1-x

Uybana,hat

a

NUMBER• Nb Ti N (SC only), THz pump-probef (THz)inb-Light-Higgs coupling
PRL 111, 057002 (2013)• Ginzburg-Landau theoryb 4
1
F = a| | + | | +
|( ir
⇤
2
2m

e⇤ A) |2

Pump

• Amplitude oscillation

ND HIDEO AOKI∗

entials, and e and m are the effective
and effective mass, respectively. The La(1) is invariant under the gauge transformaie∗ χ(r,t)
e"(r,t), φ(r,t) → φ(r,t) − ∂t χ (r,t),) + ∇χ (r,t). At temperatures T < Tc , a =mes negative, and the global U (1) symmetry"(r,t) with a constant χ ] is spontaneously
er coefficients are taken to be positive. To4(c)e⇤2 20 2 e⇤2 0 2
A +A H + ···
⇤
⇤
2m
m
1

mass of EM fieldd d are coefficients, φ and A are the scalar (Meissner effect)∗

2x10-1cm-1)F=

0 + H(r, t)

1
2
2
2aH +
(rH)
2m⇤

Im

Re

PHYSICAL REVIEW B 92, 064508 (2015

Coupling between Higgs and EM fields
1 ((r, t) =

(b)

A

𝜔
𝜔
A

0
0.0

0.5

4

H

2

(e)

probe (arb. units)

2

(a)

2𝜔
0

-2

1.0 1.5 2.0
Frequency (THz)Third-harmonic generation from Higgs

PHYSICAL REV

PRL 111, 057002 (2013)
(a)

(b)

sampleWGP

3𝜔tpp

2𝜔 Pump
Probe

4

15 K
14 K
13 K
12 K
10 K
8K
4K

y

z

1.5

2 (THz)

(c)

1.0

A
B
C

(e)

8K

z)

nits)

2

0.6

-2.0

Eprobee⇤2 n s (t)j(t) =
A(t) (London equation)
0.5
⇤
m
(d)
⇤2
2e
0
0
0.0
jNL (t) 0.0= 0.5 ⇤1.0 H(t)A(t)
1.5 2.0 2.5
05 10 15mTemperature (K)Frequency(THz)
2𝜔 + 𝜔 = 3𝜔 oscillation
0.8
-1.5
4
1

1 (

• Current:-1cm-1)

𝜔

2x10

x

Im

Re

WGP

a monocycle THz pulse (16); previous observaPump
tions were in a special case where the supertively, coexists
with their power
spectradisplayedin Fig. wave
n ordered
phasestime delay of another optical gate pulse andconductivity
withchargedensity
1B. These photon energies are all below the sumetry breaking areusing the electrooptic (EO) sampling method. Inthat makes
the Higgsgapmode(17, 18).
perconductingof ourRaman-active
NbN sample in thee modes [Nambuthis experiment, we fixed the timing of the
low-temperature
limit,
which
is
1.3
THz
(Fig.
1C);
gapped amplitude
optical gate pulse such that, in the absence ofHence, many questions regarding the Higgs mode
this implies that the pump pulse does not gencharged-particle
the pump, Eprobe at this timing monotonicallyin longsuperconductors
remainerate quasi-particles(QPs) inunresolved:
one-photon pro- Howctors withchanges with temperature, reflecting the change
low temperatures.
Theelectromagnetic
sample is an, the gaplessof the order parameter. Temporal evolution of the
doesNGthe cesses
modeatcouple
to strongs-wave superconductor NbN thin film with 24-nm
s, its energy is eleorder parameter induced by the THz pump isfields in athickness
nonlinearregime? Is it possible to dygrown on an MgO substrate (24) withas a result of thesensitively monitored through the change of
namically
control thecritical
Higgs
mode and
therefore
superconducting
temperature(Tc) = 15
hoton field),which
Eprobe relative to its value in the absence of theK.
Higgs mechanism
Theultrafast
dynamicsofthesuperconducting
pump as a function of the pump-probe delay
the superconducting order parameter?
ode in superconorder parameter driven by the multicycle pumptime, tpp (16, 22); we denote this change as
RecentadvancesintheintenseTHz
generaWGP
ied theoreticallypulse is then probed through the transmittanceIn the present case,dEprobe. For details, see (25).BPF
accompaniedof a monocycle
THzopenpulse athat
entersthewetion bytechnique(19, 20)newavenueforinvestigated the order parameter dynamics
couple directly tosample collinearly with the pump pulse within the presence of coherently oscillating multistudying
phasesnonequilibriumnear response
reamattervariable timedelay. in
In general,
we can detect concycle pump fields. The temporal waveform of themode inditions.
conven- Amplitudethe temporal andwaveformof the transmitted
pump THz electric field Epump is displayed inphase-resolved
spectrosrs was observed
probe THz electric field, Eprobe, by varying the
Fig. 1D for the center frequency of w = 0.6 THz
copy using multi-THz pulses has been realized
tic excitation with
0.3 THzprevious (21),observaenabling the study of coherent transientsω=0.6 THz
Pump
where the super0.6 THz
in many-body systems in low-energy ranges. The
arge density wave
0.8 THzpurposeof the present work is to explore coman-active(17, 18).
ng the Higgs
mode nonlinear interplay between collective
herentunresolved: Howmode in a superconductor and THz light fieldg electromagnetic
s it possible
dy02
bytoinvestigating
the real-time evolution of the
WGP
NbN/MgO
ode and therefore
ameter? order parameter under the driving field of a mul0
1
2ense THzticycle
genera- (as opposed
WGPTHz pulse.
to monocycle)
BPF
Probe
Frequency (THz)a new avenue for
In
order
to
study
evolutions
onapicosecond
nequilibrium conresolved time
spectrosscale, we performed THz pump–THz probe
has been realized
Epump
ω>2∆(T)
spectroscopy (16, 22) (Fig.
1A). To generate an
0.3 THz
herent transientsω=0.6 THzintense
0.6 THzas a coherent drivnergy ranges.
The multicycle THz pulse
0
0.8
THz
is to explore
coing source, we first created an intense monocycle
etween collective|Epump|2
THz
pulse by the tilted-pulse front method withd THz light
field
18
0
2
4
6
10
evolution
the
a of
LiNbO
3 crystal (19, 23). The monocycle pulseω<2∆(T)
Delay Time (ps) 0.8 THz
ing field of a mul0
1
2
then
goes
throughaband-passfilterto
produceacle) THz pulse.
(THz)s on a picosecond
14 K
narrow-spectrumFrequency
multicycle
pulse. Three bandpump–THz probe
14.5 K
0.6 THzpass filters are used to generate theω>2∆(T)
different). To generate an
14.8 K
centerrespecs a coherent
driv- frequencies at 0.3, 0.6, or 0.8 THz,
015.5 K
ntense monocycle
0.3 THzront method with
10
monocycle pulseω<2∆(T)
0.8 THz1481216
filter to produceaDepartmentof Physics, University of Tokyo, Hongo, Tokyo,
0

THz laser experiment
WGP
NbN/MgO
Matsunaga, Tsuji, Aoki, Shimano, et al., Science 345, 1145 (2014)
Probeulse. Threeband- Japan. 2Kobe Advanced Research Center, National
113-0033,
0.6 THzate the different
Institute of Information
and Communications Technology,
r 0.8 THz, respec3
National Astronomical Observatory
Hyogo 651-2492, Japan.
0.3 THzof Japan, Osawa,0 Mitaka, Tokyo 181-8588, Japan. 4High
4 Research81216EnergyOrganization(KEK),
okyo, Hongo,
Tokyo, Accelerator
0 Tsukuba,

Epump|Epump|2
4

6

8

Delay Time (ps)

10

14 K
14.5 K
14.8 K
15.5 KδE probe (arb. units)2∆( T ) (THz)δE probe (arb. units)2∆( T ) (THz)Intensity (arb. units)

Intensity (arb. units)• THz pump-THz probe spectroscopy

13 K

Temperature
12 K(K)
10 K
4K

0

2

13 K
12 K
10 K
4K

4

6

8

10

Pump -Probe Delay Time tpp (ps)2𝜔 oscillation of the superfluid density

Fig. 1. THz pump–THz probe spectroscopy. (A) Schematic experimental setup for the THz pump–THz
probe spectroscopy, where BPF is a metal-mesh band-pass filter and WGP a wire-grid polarizer. (B) Power
spectra of the pump THz pulse with the center frequencies of w = 0.3, 0.6, and 0.8 THz. (C) Temperature5
earch Center, National
Cryogenic
Ibaraki 305-0801, Japan.
Temperature(K) Research Center,
0
2
4
6
8
10
ations Technology,
University of Tokyo, Yayoi, Tokyo 113-0032, Japan. Pump -Probe Delay Time tpp (ps)onomical Observatory
88, Japan. 4High

2

4
6
8
10
Delay Time (ps)Intensity (arb. units)

THG resonance experiment
012

Intensity (Epump (

5x10

4

-2

10

-4

10

0

1
Frequency (Matsunaga, Tsuji, Aoki, Shimano, et al., Science 345, 1145
3 (2014)

12

-2

10

-4

10
Figure 3012Frequency (THz)2∆(T) (THz)

m

p

|6

0
1.2

0.1

0.01

pu

0

1

|E

10

15.5 K

1

2

I∝

5K

Intensity (arb. units)

10 K

K

Intensity (arb. units)

10

2

THG Intensity (arb. units)Pump

THG Intensity (arb. units)

NbN/MgO

could increase2ω=1.6 THz
0.2 mm, the pe
0.6 THz (30),1.0
1.2 THzconversion eff
0.5
with temperat
0.6 THzis attributed t
0.0[2D(T)] towar
0.8 THz
Experiment1
before the sof
0.6 THz
0.3 THz
onant enhanc
D(T ) moves ou
Tc=15 K
incident pump
The 2w osci
the THG were a
pumping. Fig
0ature depende
0.4
0.6
0.8
1.0
1.2
0.3, 0.6, and 0
T/Tc
the THG sign
Fig.
3.
THG
inwhe
tr
0.8 THzspectively,
Theory1
0.6 THzmonotonicallyA0.3schematicofTHzperature. Comof the(Borderpaexperiment.
anthe pump freq
tra of the
onetransm
can dedu
fall on 2w = 2
(10
K)
and
above
0
depends
on th
0.4
0.6
0.8
1.0
1.2
amplitude,wh(D) Power
spectr
T/Tc
2w approache
pumpmode 2D(T). t
Fig. 4. Temperature dependence ofpulse
the THGat various
dence of the T
intensity. (A) Temperature dependence of the
1.5

E

2

3
4
(kV/cm)Resonance at 2𝜔=2Δ

1.4FOutline of the talk• Introduction: Higgs amplitude mode in superconductors• Higgs mode vs charge density fluctuation (CDF)• Experimental discrimination by polarization-resolved THG• A mechanism of enhancing the light-Higgs coupling— Phonon retardation effectHiggs mode vs CDF• According to BCS theory, Higgs-mode energy is degenerate with thelower bound of the quasiparticle continuum (2Δ).
E
quasiparticle
continuum

Higgs modepair breaking

2𝜔

2

→ CDF

0

k

• Laser may excite not only Higgs but also charge density fluctuation (CDF).

Polarization dependence of THG• CDF and Higgs have different polarization-angle dependence for THG
resonance.

Cea, Castellani, Benfatto, Phys. Rev. B 93, 180507 (2016)3𝜔

eIsingle crystal thin film𝜔Polarization dependence of CDF• In general, CDF has polarization dependence relative to the crystal
axis since it depends on the band structure of the material.

• CDF also has a THG component with polarization perpendicular tothat of the incoming wave.

Matsunaga, Tsuji et al., arXiv:1703.02815.

eO

eI

3𝜔

eI
𝜔

Polarization dependence of Higgs• Higgs is a scalar boson, and does not have polarization dependence.
F=

1
2
2
2aH +
(rH)
⇤
2m

e⇤2 20 2 e⇤2 0 2
A +A H + ···⇤⇤
2m
m

• Higgs always induces THG with polarization parallel to that of theincoming wave.

jNL (t) =2e⇤2 0H(t) A(t)
⇤
m

3𝜔

eI
𝜔

eI = eO

General Polarization dependence
Matsunaga, Tsuji et al., arXiv:1703.02815.

CDF

eOeθ
eI e=I ekO =A(ω)+2B(ω)sin22θ

O
eI = eeθ,I e?
=eeOθ+90°

Higgs

C(ω)

0

B(ω)sin4θ

eO

Table 13𝜔eI𝜔Outline of the talk• Introduction: Higgs amplitude mode in superconductors• Higgs mode vs charge density fluctuation (CDF)• Experimental discrimination by polarization-resolved THG• A mechanism of enhancing the light-Higgs coupling— Phonon retardation effectPolarization-resolved spectroscopy
Matsunaga, Tsuji et al., arXiv:1703.02815.

WGP4

(a)
WGP3

x

z
y

NbN/MgO
[100]

WGP2
θ4

WGP1

θ3

BPF

θ2

x

[100]

θ2

3.0

10

5

ω
3

θ2=22.5°
20 K, //
11.5 K, //11.5 K, ⊥2.5

ed |χ(3)|2(arb. units)

10

2.0

1.5

(c)

[110]

[100]
y

θ1
(b)

ETHz

θ3

BPF

Polarization
dependence
of
THG
[100]
[100]
θ
2

Matsunaga, Tsuji et al., arXiv:1703.02815.
y• Experiment:θ

1

3.0
10
10
10
10
10

5

θ2=22.5°
20 K, //
11.5 K, //11.5 K, ⊥

ω
3

1

3ω

-1

-3

0.0

0.5

1.0

1.5

2.0

Frequency (THz)

(c)

2.5

Normalized |χ(3)|2Intensity (arb. units)

(b)

2.5

2.0

1.5
1.0
0.5
0.0

0

[100]

15

30

θ2 (deg.)

45

[110]

THG intensity is quite isotropic, and does not depend on the polarization.
•
Figure 3← It implies that the Higgs is a dominant contribution to THG.

Band structure of NbN
Matsunaga, Tsuji et al., arXiv:1703.02815.

t = -0.72 eV
t’ = -0.15 eVt’’ = 0.12 eV
μ = -0.6 eVθ3Polarization dependence of THG[100]

CDF(θ=45o)
CDF(θ=0o)

1.0
2.0
0.0
1.5

0.0

0.5

1.0
0.5

(b) DMFT

1.0
ω/Δ

1.5

0

1.0

[100]

0.0
0.0

CDF(θ=45o)
CDF(θ=0o)

15

30

45

θ2 (deg.)
0.5

2.0

1.0
ω/Δ

3.0

(c) BCS
CDF
Higgs

2.0
1.0
0.0

0

15

30

45

30

45

θ (deg.)2.0

0.0

Normalized |χ|2

y

3.0
3.0
(a) BCS
(c)
Exp.
2.0
2.5

3.0

Matsunaga, Tsuji et al., arXiv:1703.02815.

[110]
1.5

2.0

Normalized |χ|2

2.5

|χ|2 (arb. units)(3)|2Normalized |χ
2

°
/
, //
,⊥

|χ| (arb. units)• Theory:

[100]

3.0

(d) DMFT
CDF
Higgs

2.0
1.0
0.0

0

15

θ (deg.)Outline of the talk• Introduction: Higgs amplitude mode in superconductors• Higgs mode vs charge density fluctuation (CDF)• Experimental discrimination by polarization-resolved THG• A mechanism of enhancing the light-Higgs coupling— Phonon retardation effectHiggs mode vs CDF• In BCS theory, for a general lattice with a general laser polarization
CDF has much stronger coupling to light than Higgs.

Cea, Castellani, Benfatto, Phys. Rev. B 93, 180507 (2016)• Why is the Higgs mode contribution to THG so dominant for NbN?
• NbN is known to have a strong electron-phonon coupling (λ~1).
Kihlstrom, Simon, Wolf, PRB 32, 1843 (1985); Bronson et al., PRL 64, 2172 (1990);
Chockalingam et al., PRB 77, 214503 (2008).

• Is there any mechanism to enhance the light-Higgs coupling?

Retardation effect• Un-retarded pairing• Retarded pairing

A(t)2

non-resonant
light-matter couplingA(t)A(t’ )

resonant
light-matter couplingHiggs diagrams

BCS (clean limit)(a) Non-resonant coupling✔
(b) Mixed coupling🚫
(c) Resonant coupling

⌦

3⌦

🚫
⌦

⌦

Beyond BCS

and the mixed [Fig. 1(b)] and resonant (c)THG exactly vanish. This is confirmed by e
ing the convolution of the three electron pro
Zd!
Tr[⌧1Ĝ k (! + 2⌦)Ĝ k (! + ⌦)Ĝ k (!)]
2⇡✔

However, this does not guarantee that thwould remain small if one goes beyond thetion. For example, the real part of the op(⌦) vanishes for ⌦ , 0 within the BCS theZd!
Tr[Ĝ k (! + ⌦)Ĝ k (!)]< = 0
2⇡✔
✔

in much the same way as in Eq. (5). In rea
hand, the real part of the optical conductivi
not even small.9,60 They become nonzero w
count of dynamical correlations such as theOptical conductivity

BCS (clean limit)

Beyond BCS

✔

✔

🚫

✔

=0

,0

(a) Diamagnetic term(b) Paramagnetic term

Re (! , 0)Model of el-ph system• HolsteinXmodelti j (c†i c j

H=+ h.c.)

i j,

µ

X
i

8

X †
X
ni + !0bi bi + g(bi + b†i )(ni

Higgs mode

i

1)

i

R
-Im χpair(ω) : dynamical pair susceptibilityA(ω/2) (×4) : single particle spectrum

6

4

2

2Δ
0
0.0

0.1

0.2

0.3

0.4

ω
g = 0.8, !0 = 0.6, T = 0.02, = 0.2, = 0.005 ( = 0.77)

Tsuji, Murakami, Aoki, Phys. Rev. B 94, 224519 (2016)

FIG. 5: The single-particle spectrum A(!/2) and the dynamical pairR

FIG. 6: The TH(i)For the notation of < for products of nonequilibrium Green’s functions, see Appendix A. Note that 0 (i = 1, . . . , 5) do not¨ so that they can be computed independently of the dotted DMFT. The vertex corrections are also explicitly derived as
contain ⌃,X Z d!
i
(1)
<
¨
(⌦)
=
[¨
✏
G
(!
+
2⌦)
⌃(!;
⌦)G
(!)]
,
(49)
k
k
k
vc
2
2⇡
k
X Z d!
i(2)¨ + ⌦; ⌦)G k (! + ⌦)˙✏ kG k (!) + ✏˙ kG k (! + 3⌦)˙✏ kG k (! + 2⌦)⌃(!;¨[˙✏ kG k (! + 3⌦)⌃(!
⌦)G k (!)]< . (50)vc (⌦) =2
2⇡

THG susceptibility (DMFT calc.)k

Higgs modeCDFχ0(1) =

χ0(2) =

is1.4replaced by

χ0(3) =

1.2

(1)
χvc
=

χ0(5) =

¨
Σ

(2)
χvc
=

|χ 2

χ0(4) =

90.03

¨ <,> (t; ⌦)⌧ .
⌃¨ˆ <,> (t; ⌦) = ig2 D<,>
30 (t)⌧3 Ĝ
0.02

(1)

(51) χ(2)0Let us finally comment on the generality of the present for- χ0(3)1.0
mulation. Although we describe
0.01 the dotted DMFT formula- χ(4)
0
tion
for
the
THG
susceptibility,
it
is
not
restricted
to
THG
but
0.8
(5)
χ
0
can be generalized to arbitrary dynamical
responsefunctions.
00.05 field 0.1(1)
One(not neces- χvc
0.6 can introduce an infinitesimal external
sarily an electric field), and take the derivative with respect to (2)
χvc
0.4

CDF
it for observables.
It may contain the derivative of the selfenergy, which can be evaluated by the corresponding dotted
0.2
DMFT,
where the DMFT self-consistency equations are differentiated with respect to the external field. For example, the
0.0
40,48
dynamical
pair susceptibility,
0.00
0.05
0.10
0.15
Z 1
Ω
R
i⌦t
i
dte
h[B0 (t), B0 (0)]i,(52)pair (⌦) =¨ΣHiggs mode

χ0

0.20

FIG. 4: Feynman diagrams for the susceptibility for the thirdharmonic generation. There are five (two) topologically di↵erent di0
agramsgfor
thebaresusceptibility(vertexcorrection
).
Solid
0
vc
= 0.8, !A(!/2)
T = the0.02,= 0.2, pair
= 0.005 FIG. 6:
0 = 0.6, and
The THG
decomposed
into
the baresusceparticle
spectrum
dynamical
Tsuji,
Murakami,
Phys.amplitude
Rev. B 94,(2016)
is definedas thesusceptibility
responseof Aoki,the pairing
hB224519
0 i against
and
wavy
lines
representtheelectronandexternal
photonpropaga(i)
(i)
R
i⌦t
tibilities
(i
=
1,
.
.
.
,
5)
and
vertexcorrections(!)
calculatedbythe(dotted)
DMFT
for
the
an
externalpairpotential
H
(t)
=
"B
e
,
where
Polarization:= shaded
0.5 box represents the vertex correcvc (i = 1, 2) for
tors, respectively,
while↵the
ex
0
air
0
¨ the
Among
the fourmodel
photon lines,
the superconducting phase Xof themodel calculated with thesetionof⌃.
Holstein
withoneg is=outgoing
0.8, !with
0 =an energy† Holstein
†
B
=
(c
c
+
c
c
)
(53)
0
i"
other three are incoming with an energy ⌦. The photoni"i# thei#same
dotted DMFT. The parameters
are
as
in
Fig.
5. The po2,3⌦,
andand the= 0.005.
ilines attached directly to the vertex are incoming.

Higgs mode can contribute to THG
resonance
an
order
of
magnitude
larization
direction ofwith
the laserfield is taken
to be ↵ = 0.5. The
is the bosonic pairing operator withmo⇤ the⇤ center-of-mass
2
susceptibilities
arenormalized
by
=
t
/d
.
The
inset
is a blowup
comparable or even larger than the(1)contribution
ofquantity
CDF.detects collective amplitude
mentum q = 0. Thisof.
vc oscillationsof the superconducting order parameter. The dot-Comparison to BCS theory
Tsuji, Murakami, Aoki, Phys. Rev. B 94, 224519 (2016)

10

8

40
DMFT
DMFT

6

Higgs contribution

4

|χ 2

|χ 2

BCS
BCS

χ0 +χvc + Higgs
CDF
χ0
30
CDF

20

2

10

0
0.0

0
0.90

0.5

1.0

1.5

2.0

2.5

χ0 +χvc+ Higgs
CDF
χ0
CDF

0.95

1.00

1.05

1.10

Ω/Δ

Ω/Δ

FIG.intensity
8: The intensityof the third-harmonic
generation
for the su
FIG.
8:
The
of
the
third-harmonic
generationforthesuperFIG.
7:
The
intensityofthethird-harmonic
generationforthesuThe intensityof0.8,
the!third-harmonic
generation
for thesuconducting
phaseof the Holsteinmodel calculated
withinPolarization:
g
=
=
0.6,
T
=
0.02,
=
0.2,
=
0.005
↵ =of0.5
0conductingphase
the Holsteinmodel calculated
within the
BCStheperconducting
phaseof the model
Holsteinmodel calculatedby the dottedctingphase of theHolsteincalculatedby the dotted
approximation.
Thecontribution
bubble contribution( 0 ) and the total susc
approximation.
The
bubble(
DMFT.
The
bubblecontribution
(
)
and
the
total
susceptibilityin0 ) and the total suscepti0
The bubble contribution ( 0 ) and the total susceptibility inbility including
thecorrections
vertex corrections( 0 + vc ) are plotted res
bilityincludingthevertex(cluding
thevertexcorrections
(
+
)
are
plottedrespectively.
The
0 + vc ) are plotted respec0vc
he vertex corrections ( 0 + vc ) are plotted respectively. The
tively.
The parameters
are
taken
tosamebe thesame
as 5.
in Fig.
5.
tively.
Theparameters
are
taken
to
be
the
as
in
Fig.
The
parameters
are
taken
to
be
the
same
as
those
of
Fig.
5.
The
polarizars are taken to be the same as those of Fig. 5. The polarizapolarization
directionof thelaser
field istaken
tobe ↵ Note= 0.5.
tion
direction
of
the
laser
field
is
taken
to
be
↵
=
0.5.
polarization
direction
of
the
laser
field
is
taken
to
be
↵
=
0.5.
tion of the laser field is taken to be ↵ = 0.5.
the di↵erencein theof theaxes
the di↵erencein the scales
ofscalesthe axes
fromFig.from
7. Fig. 7.

In DMFT the Higgs-mode contribution is enhanced due to the retardation
effect mediated by phonons, which is in sharp contrast to BCS.

by reducing
We thatcan seeboth exandd removedby reducing. We can. seeboththat and
0

0 ex-

from the resonant diagram [Fig. 1(c)], in further studies ofthe THG resona
comparable to ccharge density fl
BCS result. The
can be mediatedthe retarded inte
3.0This is confirme
(3)
2
|CDF
(
=
)
THG resonance0
2.5
the phonon frequ(2)2|Higgs(=)
vc
phonon couplin
2.0
enhance the Hig
renormalization
1.5
effects from the
1.0
beyond DMFT,
With relevan
0.5
to investigate thDMFT
ceptibility. How
0.0
on the details o
0.3
0.4
0.5
0.6
0.7
0.8
0.9
only includes a0
situations acous(2) 2
low temperatur
FIG. 9. Phonon-frequency dependence of |χ0(3) |2 and |χvc
| at theTsuji, Murakami, Aoki, Phys. Rev. B 94,(2016)by
T = 0.02,= 0.2,
0.5
be224519
affectedresonance($ ==%)0.005,
for the =superconducting
phase of the Holstein
model
with
T = 00.02,= 0.2,and δ = 0.005. gap
Theispolarization
g for
We tuneeachsuch γthatthe superconducting
fixed to be 2 paper,
0.12 . we haveby the broadeni
direction of the laser field is taken to be α = 0.5. We tune g forcomplicated in r
each ω0 such that the superconducting gap is fixed to a constantthe electron-phonon coupling g such that the superconducting
gap is fixed to a constant (2% ≈ 0.12). In Fig. 9, we plot the(2)height of the resonance peak for |χ0(3) ($)|2 and |χvc($)|2 as|2

Phonon frequency dependence

8

2.0
Enhancement
of
Higgs
mode
due to the•
CDF(θ=0 )1.0phonon retardation effect.
CDF(θ=45o)
o

0.0
0.0

(a)
0.5

• Strong light-Higgs interaction through theresonant coupling.

|χ|2 (arb. units)x3.0(b) DMFT

2.0

1.0
ω/Δ
z

y

1.5

2.0

WGP3

NbN/MgOo)
CDF(θ=45
WGP2

CDF(θ=0 )Polarization-angle
independence
of
THG
•
BPF
1.0
resonance in NbN superconductor.
0.0
0.0

WGP1

o

DMFT
3.0 DMFT(c) BCS
6

4

0

2

0.0

0
WGP4
0.00

0.5

1.5 30151.0
Ω/Δθ (deg.)

2.0

2.5
45

FIG. 7: The intensity of the third-harmonic generation for the superconducting
phase of the Holstein model calculated by the dotted
3.0 (d)DMFT
DMFT. The bubble
contribution [100]( 0 ) and the total susceptibility including the vertex corrections ( 0 + vc ) are plotted respectively. The
THz
parameters areθtakenCDF
to be the same as those ofE
Fig.
5. The polariza2.0
4tion direction of the laser
xfield is taken to beθ↵2 = 0.5. [110]
Higgs

DMFT

θ3

Normalized |χ(3)|2Intensity (arb. units)Exp.

20 K, //

FIG
con
app
bili
tive
pol
the

1.0

removed by reducing . We can see that both and 0 exhibit a conspicuous resonance[100]peak at ⌦ = . Although[100] theθ20.0
position and shape of the peak does not change so much be0.5
1.0
1.5
2.0
15height does. 30
45tween 0Withythe parameters
0 and , the peakθ1taken in the present case, theof is enhanced as large
ω/Δθ height(deg.)as about four times that of 0 due to the resonance with the
3.0
Higgs mode. The
main contribution comes from (2)vc , as can(b)5
be seen in Fig. 6. The resonanceasθ2=22.5°(c) width of 0 is broadened
10
(3)
(5)
compared to that2.5
of 0 due to the spectral weight of 0 dis20 K,// around the peak.
ω
tributed
The amplitude ratio between 0 and
11.5
K,
//
3
WGP4
(a)
various model parameters, but at least there
vc can depend on
2.0
10
WGP3
is such11.5
K, ⊥a possibility in a certain realistic parameter regime thatxthevertex correction has a non-negligible e↵ect.
z
NbN/MgO
1
We are now in1.5
position to make a comparison between the3ω
10y WGP2
results [100]of DMFT and BCS theory by calculating the THG
E
THz the BCS approximation for the same
1.0
θ4 susceptibility θwithin
[110]
2
WGP1
x
parameter
set
as
those
of DMFT. The electron-phonon cou-1
θ3pling is translatedintoa
BPF 100.5 static attractive interaction with U =222
2g !0 /(!0 + ) [see Eqs. (9) and (10)]. In the gap equation,
we perform-3[100]the momentum[100] integral in the range of |✏ k |  !0 .
θ2
0.0
10
The results are depicted in Fig. 8, indicating that the e↵ect
y 0 is rather15
0.0θ1 0.5 1.0 1.5 2.0
2.5 correction
30 to the 45of the vertex
small comparedbaresusceptibility 0 in the BCS theory if one considers a general[100]θ2 (deg.)[110]
Frequency (THz)3.0polarization direction(b)(↵ = 0.5 here).
The resonance width5θ2=22.5°(c)10
2.5is much sharper and the peak height is higher in BCS thans)[1] Tsuji, Murakami, Aoki, Phys. Rev.
B 94, 224519 (2016).
Figure 2[2] Matsunaga, Tsuji et al., arXiv:
1703.02815. (To be published inPhys. Rev. B, Rapid Comm.)

vc

1.0

• Dominant contribution to THG from Higgs
mode rather than from CDF.

χ +χCDF+Higgs
χ
CDF
0

CDF
Higgs

2.0

|χ 2

Normalized |χ|2(a) BCS

Normalized |χ|2

|χ|2 (arb. units)

3.0

Summary

fro
TH
late

T
spo
pho
DM
sul
ap
the
com
cha
BC
can
by
plin
the
ma
fro
DM

Big challenge• Can we directly observe the Higgs mode by tr-ARPES?
Noneq. DMFT calculation: Kemper et al., Phys. Rev.PHYSICAL
B 92, 224517(2015).
REVIEW B 92, 224517 (2015)

KEMPER, SENTEF, MORITZ, FREERICKS, AND DEVEREAUX

(a)

(b)

(c)

(f )
(g)
THz
pump(?).
Energyresolution~ 1meV.
•• We are waiting for the news!

(d)

(e)

(h)