NQS 2022

1

High-harmonic genera.on instrongly correlated systems
Yuta Murakami (RIKEN CEMS)
Ref

Y. M., M. Eckstein, and P. Werner, Phys. Rev. Le8. 121, 057405 (2018).
Y. M.,, S. Takayoshi, A. Koga, and P. Werner, Phys. Rev. B 103, 035110 (2021).
Y. M., K. Uchida, A. Koga, K. Tanaka, P. Werner, Phys. Rev. Le8. 129, 157401 (2022).

speaker: Y. Murakami

Acknowledgement

2

Uni. Fribourg

Uni. Erlangen

Konan University

P. Werner

M. Eckstein

S. Takayoshi

Kyoto University
K. Uchida
K. Tanaka

Tokyo Tech.
A. Kogaspeaker: Y. Murakami

Strong light-maBer coupling and emergent phenomena

3・ Floquet engineering・ Nonlinear phononics
・ High-harmonics genera6on (HHG)
etc…..

D. N. Basov, et. Al., Nat. Mat. 16 1077 (2017).

speaker: Y. Murakami

What is HHG?

4

n=1

⌦ ! n⌦
Input

n=2

Outputn=3maBerHighly non-perturba6ve eﬀect!

▷ Conversion of laser frequency▷ HHG spectroscopy

n=4

‥‥

Applica<ondecompose

speaker: Y. Murakami

ery imanalsignifogicalustrateI„, andEvidently, the atomic dipole moment is singular at thesaddle point. This behavior of the atomic matrix elements is the direct consequence of the asymptotic behavior of the ground-state wave function which is proportional to exp( —/2Ipr) for large r. Therefore, in order
to be more accurate one has to incorporate the effects ofthe finite size of the ground-state wave function in solving the saddle-point equations. This leads to corrections
which vanish as
to the cut-off law of the order of

HHG from atomic gas and semiconductors(t —7 ).

nal kiapproxthe inshould
ing theom theNATURE PHOTONICS DOI: 10.1038/NPHOTON.2013.349

aIntensity

10

b

100

10

I„=

ared toi) Since
at theLETTERS
GaSe
2

10−1

ls

Plateau region

10

cut oﬀ

4
6

10−2

7

10−3

0-22

10"-

Ea (MV cm−1)
2040

3
I13 (a.u.)

unctionn agreely as z
30pression
93 andpredicts
pression

10-10

S. Ghimire et al., Nat. Phys. 7, 138 (2011).
O. Schubert, et al., Nat. Photon, 8 119 (2014).

Semiconductors

c

104
103
102
8

10−4

9

10
20 30
Eint (MV cm−1)
10

11

d

60

12

1

0

Energy (eV)

(48)

5

Intensity (norm.)

U„,

Corkum, PRL 71, 1994 (1993).
Lewenstein, et al., PRA 49 2117 (1994).

Atomic gas

Intensity

numer-

Intensity (norm.)

44).

350

13

3

CB1,2

0

VB1

−3

360
370
Frequency (THz)

14

10−5

15

16

VB2,3
Г
Wavevector, k

K

K

17
18

10−6

10 28-

19

10−7

10

20

10−8

10"

0

25

50

75

2M+1
Harmonic order
Harmonic
order

100

125

U„:

10
a = I„. The curve with open squares is obtained for2!cutfor=U„=
Ip 20.
+ The↵Elocationsofthe one frequency:with black squares
and cutoﬀ
0
thethe phenomenological and exact cutoffs are indicated byarrows.
I : IonizaPon potenPal

E0: Electric ﬁeld
αE02 : PonderomoPve energy

10−9

EOS
0

FIG. 6. Harmonic spectra for the GBR model with I„=30,p

Eg/h
InGaAs
100

200

21

22

Si CCD
400

300

500

600

Frequency (THz)

Frequency (THz)

Figure 2 | CEP-stable terahertz HHG in bulk GaSe. a, HH intensity spectrum (solid line and shaded area) emitted from a GaSe single crystal (thickness,
220 mm; u ¼ 508) driven by the phase-locked terahertz pulse in Fig. 1a. A combination of electro-optic sampling (EOS), an InGaAs diode array and a silicon
CCD maps out the HH spectrum throughout the terahertz, far-infrared, mid-infrared, near-infrared and visible regimes. Eg /h marks the bandgap frequency(476 THz). The blue dashed curve shows the computed HH intensity spectrum, obtained from a five-band model (see Chapter 3 in the Supplementary
Information for details). b, Dependence of the intensity I13 of the 13th harmonic on the incident terahertz amplitude Ea (top scale). Bottom scale: internalterahertz amplitude Eint , obtained from Ea by accounting for reflection at the crystal surface. Symbols, experimental data; dashed line, scaling law I13 / Ea26;
dotted line, scaling law I13 / Ea. c, The spectral interference between the frequency-doubled sixth harmonic and the 12th harmonic confirms the CEP stabilityof the HH radiation. For experimental details of f 2 2f interferometry, see Supplementary Fig. 2. d, Model electronic band structure of GaSe between theG- and K-points, underlying the subsequent computations. Two conduction bands (CB1 and CB2) and three valence bands (VB1, VB2 and VB3) are taken
into account. Coherent excitation of electrons (symbols), for example, from the second valence to the lowest conduction band, can proceed via interfering
pathways with different scaling in powers of the terahertz field.

cutoﬀ frequency: !cut =Here, we use CEP-stable terahertz waveforms with amplitudes
comparable to atomic fields to explore a qualitatively novel regimeof ultrafast all-coherent charge transport. The field drives interband
polarization in a bulk semiconductor off-resonantly and accelerates
the excited electron–hole pairs to perform dynamical Bloch oscillations, generating CEP-stable radiation throughout the frequency
range between 0.1 and 675 THz. Quantum interference of multiple
excitation channels makes the dynamics highly sensitive to the CEPof the terahertz waveform.
Intense phase-locked terahertz transients are generated bydifference frequency mixing of two phase-correlated infrared
pulse trains15. Figure 1a shows the waveform detected electrooptically (centre frequency, 30 THz; peak field in air, Ea ¼
72 MV cm21 ¼ 0.7 V Å21). The field is focused into a bulk
crystal of the semiconductor gallium selenide (GaSe), whichforms a perfect host for coherent high-field transport. As thewidths of the valence and conduction bands are substantiallysmaller than the fundamental energy gap Eg , ultrafast carrier scattering by impact ionization11 is suppressed. The pulse emitted+ ↵E0

stems from second-harmonic generation (see Chapter 5 in the
Supplementary Information for phase-matching considerations).
The shape of the electro-optic waveform remains constant forconsecutive scans, underpinning the absolute phase stability of
all frequency components.
Beyond second-order nonlinearities, ultrabroadband electrooptic traces also exhibit peaks at the third (90 THz) and fourth(120 THz) harmonics (Fig. 2a). Using a grating spectrometerwith an InGaAs array detector, we record maxima at the sixth
to ninth harmonics, with decreasing intensity. A monochromatorand a cooled silicon charge-coupled device (CCD) monitor highorder harmonics (HH) with comparable intensities up to the 16thorder (480 THz), which nearly coincides with the fundamental
energy gap of GaSe at 476 THz. Interband photoluminescenceattests to the presence of terahertz-induced electron–hole pairs.
The existence of a plateau-like region is a hallmark of non-perturbative nonlinearities. HH generation (HHG) continues up to the22nd order, despite strong interband absorption. The total spectrumof 23 harmonic orders (including optical rectification) covers 12.7

speaker: Y. Murakami

Mechanism of HHG: Semiconductor
Semiconductors

0

Accelera<on to charge ( = j (t))à emission of light

IHHG (!) = |!j(!)|

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Atomic gas

Corkum, PRL 71, 1994 (1993).
Lewenstein, et al., PRA 49 2117 (1994).

Three step model

6

Intraband mechanism
e

e

h

h

①Carrier
creationInterband mechanism② intrabandmove eehh② intraband
dynamics &emission① e-hcreation③ recombination

Band info spectroscopyBand dispersion

Berry curvature

T. Luu, et al., Nat. 521, 498 (2015).

T. Luu, H. J. Worner,
Nat. Commun. 9, 916 (2018).

Dipole moment
K. Uchida, et.al., PRB, 103, L161406 (2021).

speaker: Y. Murakami

HHG in strongly correlated systems ?

7

Excita<ons cannot be described by normal fermionic par<cles …
MoL insulator @ Hubbard : doublon, holon
MoL insulator @ Mul<-Hubbard : string states
Dimer MoL : Kink & an<-kink (solitons)
Ques<ons▷ Basic HHG feature in MoL insulator ?
▷ Mechanism of HHG ?
▷ Diﬀerence from semiconductors ?
▷ HHG vs single par<cle spectrum or HHG vs many body excita<on ?
▷ Can we use HHG to detect something?
speaker: Y. Murakami

Milestone of single-band Hubbard model
H(t) =

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X

eirij ·A(t) c†i cj

hiji,

Lower
Hubbard band

doublon/holon hopping

+U

X

8

ni" ni#i

Upper
Hubbard band

doublon/holon recombina<onholon doublon

cf. intraband current

cf. interband current speaker: Y. Murakami

Message of this talkOrigin of HHG

9Three step model for many body states▷ Doublon-holon recombina<onkc▷ Kinema<cs of doublon-holon pairsi)

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Tc

doublon

HHG spectroscopy of many body states

ii)

or

Importance of strong correlates between spins and charges

k
iii)

c
i)
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Strong spin-charge coupling (string states)

Tc

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doublonAnomalous temperature dependence HHG

speaker: Y. Murakami

10Origin & basic featureof HHG in Mo4 insulators
Floquet Dynamical mean-ﬁeld theory (dim>1 system)
Y. M., M. Eckstein, and P. Werner, Phys. Rev. Le8. 121, 057405 (2018).
Y. M., and P. Werner, Phys. Rev. B 98, 075102 (2018).

Inﬁnite <me-evolving block decima<on (1d systems)
Y. M.,, S. Takayoshi, A. Koga, and P. Werner, Phys. Rev. B 103, 035110 (2021).

speaker: Y. Murakami

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Bb i?2+?2KB+H
2/ BM i?2 >>: Q7 i?2 >m##`/#2+QK2+H2`†7QHHQrBM; bBKmHiBQMb
iA(t)iA(t) †
h
X
AM
i?2
rBi?
Bh1".r2
TTHvzĵ(t) =h?2
iv Hbiei2`KĉBMe BM /2iBHĉi, ĉi+1,. UkV X
BM TT2M/Bt
iBH- 7`QK
r?B+?i?2
Bb b2iiQ U/2- BX2X- ?H7 }HHBM;X
i+1, ĉi,THBM2/
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KQ/2Hb/2`Bp2/iBK2@/2T2M/2Mi3
Q 2z2+iBp2
i?2 >>:
bT2+i`mKX
h?2+m`@bKHH bi;;2`2/
K;M2iB+ }2H/ hz = 10 v- r?B+? ?2HTey `2T`2b2Mib i?2 2z2+i(a)r2Q7  bi;;2`2/ K;M2iB+
(b)
1[X
URV
}2H/
aBKBH` bi`i2;B2b iQ K2bm`2 i?2 bBM
B2z2`@qQHzi`Mb7Q`KiBQMX
lbBM;
i?2b2
KQ/2Hb`b +M #2 +HbbB}2/ BMiQ M BMi2`@iQ +QMp2`;2 i?2 2[mBHB#`BmK bQHmiBQM Ui?2 BMBiBH bii2
Observable
hz X AMr?B+?
i?2 7QHHQrBM;
bBKmHiBQMb rBi?
Bh1".- i?2
r2 TTHv bT2+i`mKi`mKb#v ii+?BM; mtBHB`v #i? bBi2
r@?QH2i?iMMB?BHiBQMf+`2iBQMV
i?2 bTBM /vMKB+b?b MQ MHQ;m2
q2 2pHmi2
M/
M
bKHH2` +miQz /BK2MbBQMb
X q2 ?p2 +QM}`K2/
3 rBi? >>:bKHHbi;;2`2/
K;M2iB+}2H/ hz = 10 v- r?B+? ?2HTb
TQb2/ T`2pBQmbHv M/ BKTH2K2Mi2/ 7Q`
2KB+QM/m+iQ`
bvbi2Kb`2/m+2bi?2>>:BMi2MbBivX
M i?2 >m##`/ KQ/2H- r2 MHQ@@ Ĥ
i?i i?2 bBM;H2@T`iB+H2 M/>>: bT2+i` b?QrM #2HQr
2
iQCurrent：+QMp2`;2i?2
2[mBHB#`BmKi`Bt`2MQ`KHBxiBQM;`QmT
U.J_:Vej
HbQ `2p2H i?2 BKTQ`iM+2
Q7 i?2 HQM;@`M;2+QKTQ@ bQHmiBQM Ui?2 BMBiBH Ibii2V
ĵ
=
(!)
=
|!j(!)|,UjV
HHG
m``2Mi Q`B;BMiBM; 7`QK i?2 MMB@`2 ?`/Hv z2+i2/ #v i?Bb }2H/ i?`Qm;? i?2 +QKT`BbQM@A
Q7 i?2 ǳ/BTQH2 KQK2MiǴ
#2ir22M/Qm#HQM?QHQM
rBi?
bKHH2`+miQzM/
/BK2MbBQMb
X q2 ?p2 +QM}`K2/
Q/bX Hi?Qm;?4 i?2b2 bi`i2;B2b 2bb2MiBH/Qm#HQM
M/
?QHQM
j
M/
i?2
+m`@
rBi?
i?2
`2bmHib
7Q` hz = 10 vX h?2 Hb2` 2t+BiiBQM B
ac/b 7Q` i?2 >>: BMi2MbBivr2HHbBM;H2@T`iB+H2
b i?2 `QH2 Q7M/
i?2 >>:
i?ibi?2
bT2+i`i?ib?QrM
iQ2KBbbBQM
2tT2`BK2MibQm` K2i?Q/ Bb
bbmKBM;i?2 #2HQr
2KBii2/ `/BiBQM
Q`B;BMi2b
7`QK i?2
2 BM+Q`TQ`i2/ pB i?2 S2B2`Hb T?b2
Q7i?2?QTTBM;T`K2QTTBM;Q7/Qm#HQMb
Q`?QHQMbjj9hop(!)= i?`Qm;?
|!j(!)|
2Hi2/ ?QTTBM; Q7 /Qm#HQMb
M/intensity：?QHQMb7Q` Ii?2b?T2`2HHG?`/Hv
z2+i2/
#v
i?Bb
}2H/
i?2 Q7+QKT`BbQM
T`2pBQmbT`2pBQmb++2H2`iBQM+?`;2b #v i?2
2ti2`MHQM2bX}2H/Xh?2aBM+2
i?2 b+?2K2b 7Q
HHG
i2` M/ A(t)`2T`2b2Mib i?2
p2+iQ` TQi2MiBHX
LQi2 i?i r2
MmK#2`
i?2b2 +``B2`bX
QT@ 7Q` h = 10 4 vXMmK2`B+H?2 >>: Q7
bT2+i`mKX
iQ  }MBi2`M;22t+Bi2/ iQ i?
rBi? h?2i?2 `2bmHib
h?2 Hb2`bBKmHiBQMb
2t+BiiBQM `2
Bb `2bi`B+i2/
Q7 i?2 MmK#2`
Q7 iBK2T`iB+H2bz
speaker: Y. Murakami
iQ i?2b2+M b#27QHHQrbX
2tT`2bb2/?Bb
TT2` +m``2Mib
Bb Q`;MBx2/
AMpB
a2+X
r2 BM@T?b2[0,Bi Bb mb27mH
iQ BMi`Q/m+2
+QMM2+i2/:mbbBM iQ
rBM/Qr
BM+Q`TQ`i2/
i?2AAS2B2`Hb
Q7ti?2?QTTBM;T`K2@r?B+?
Bb
A(p, !)X PM i?2 Q
max ]E20 = 0.7(c) U = 10.0, ⌦ = 0.75,(tt)
0m+2 i?2 >m##`/ KQ/2HX
r2 `2T`2b2Mib#`B2~v 2tTHBMi2`h?2MM/ A(t)i?2 p2+iQ`
LQi2 i?i r2
`2+iHv K2bm`2  MQMHQ+H +Q``2HiBQM 7m
FTQi2MiBHX
0 2 )- r?B+? Bb rB/2 2MQm;? #mi
gauss (t) = exp(
2i
K2i?Q/hiQ† 2pHmi2i?2 bBM;H2@T`iB+H2
bT2+i`mK BM
b?Q`i2` i?M tmax X h?2M r2 +H+mHi2 i?2 6Qm`B2` i`Mb@†
iA(t)rij ˆˆe M/ /2`Bp2
di, ĥj,¯ĥi, dj,¯ KQ/2HbX,".
i?2+2z2+iBp2
h?2 MmK2`B+H
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sha1_base64="bs8QlLljyYde7WbNFuyzJJm+ZMk=">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</latexit>
sha1_base64="0OYh+yqc/dVTCseO/PbG17RKxYQ=">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</latexit>
sha1_base64="JgLQ3/d2Pdlw9dzxNgK0q3MVb+o=">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</latexit>
sha1_base64="thqG8bcfxXXUoZNEX/7Avz+atUI=">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</latexit>

HHG Spectrum(a)(a)⌦⌦==0.750.75

12<latexit sha1_base64="cgKn+qG0BHNscile4kv+nWUMJuM=">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</latexit><latexit sha1_base64="cgKn+qG0BHNscile4kv+nWUMJuM=">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</latexit>

IHHG(ω)(b)

Electric field (E0)(b)

cut oﬀ freq.

(b) ⌦ = 0.5<latexit sha1_base64="Xnqi0Nxg5WFQ3kfBdLza3M8Yp5w=">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</latexit>(b) ⌦ = 0.5<latexit sha1_base64="Xnqi0Nxg5WFQ3kfBdLza3M8Yp5w=">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</latexit>

full
doublon-holon hop
6A:X jX >>: bT2+i` BM i?2 THM2 Q7 7`2[m2M+v ! M/ }2H/
6A:X kX >>: bT2+i` Q7 i?2 JQii BMbmHiQ` 7Q` UV ⌦ = 0.75
Mottbi`2M;i? E0 7Q` UV ⌦ =0.75 M/ U#V ⌦ = 0.5X h?2 `2/ +B`+H2b
M/ U#V ⌦ = 0.5 i }2H/ bi`2M;i?b E0 = 0.5 M/ yXdX aQHB/
BM/B+i2
+miQz 7`2[m2M+B2b
M/ Q7i?27`2[m2M+v}HH2/ `2/ i`BM;H2
6A:X jX>>:i?2bT2+i`BM i?2 THM2! M/b?Qrb}2H/
BM/B+i2>>:bT2+i`- 7Q`
r?BH2>:HBM2bbT2+i`Q7 i?2i?2 7mHH
JQii
BMbmHiQ`
UVi?2⌦ /b?2/= 0.75HBM2b BM@
i?2
JQii
;TX
h?2
bQHB/
p2`iB+H
HBM2b+Q``2bTQM/
iQ
2p2M+QMi`B#miBQM
7`QK/Qm#HQM@?QHQMXbi`2M;i?
E0 7Q` UV ⌦ = 0.75 M/ U#V ⌦ = 0.5X h?2 `2/ +B`+H2b= /B+i20.5 ii?2}2H/
bi`2M;i?b
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Vampa, et al,
PRBi?291, 064302
emit (trec )V rBi? ✏g (p)SemiconductorA(t)<latexit sha1_base64="kQPUzDVYJskYJpawILxT8a+x2Gc=">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</latexit>

1[X UReV- r?BH2 i?2 `2/ K`F2`b +Q``2bTQM/ iQ i?Qb2 7Q` i?2/2b+`B#2/
rBi? i?2 2[miBQM
mM`2MQ`KHBx2//BbT2`bBQM-Q7UKQiBQM
4v sin(p)X h?2 TBMF K`F2`b`2 i?2 `2bmHib
mbBM;byi?2ﬁeld/BbT2`bBQM 2ti`+i2/ 7`QK
1) atQ#iBM2/
t0： Tunnelingi?2bBM;H2@T`iB+H2
bT2+i`mK- BX2X 6B;X R- M/ i?2v HKQbi Qp2`@
dxrel
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/
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=
@
✏
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UR3V
p g K`F2`bX
pair
HT dtrBi? i?2electron・hole#Hm2
h?2
HBM2b BM/B+i2
p=p(t)
bi?2 iBK2b r?2M E(t)= 0Xmin,h?2 xT`K2i2`b
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rel = 0
p(t)
=
p
A(t)
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A(t
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0
0
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i?2 TmKT }2H/ `2 i?2 bK2 b BM 6B;X kX UR3#V2) Dynamics under ﬁeld(t)
BM/B+i2bi?2`2HiBp2 /BbiM+2 #2ir22M i?2
v-l+M?QH2#2 /2b+`B#2/
rBi?
Q7 KQiBQM=+ ✏✏vh(p)( p)M/✏✏gg(p)(p)=✏✏di?2(p)2[miBQM
rBi?
✏c (p)=M/ i?2c(p)/2MQiBM;i?2 2M2`;v Q7 i?2 +QM/m+iBQM UpH2M+2V⌘
dxrel (t)2@+i`QM rBi? KQK2MimK
pX h?2Q7 M= @pp2HQ+Biv✏g (p) p=p(t), 2H2+@UR3V
dt
X Bb ;Bp2M #v i?2 /2`Bpi2 @p ✏c (p) U@p ✏v (p)V- M/
pH2Vp(t) = p0 A(t) + A(t0 ).
UR3#VpB i?2 b?B7i/i Q7 i?2 }2H/ Bb iF2M BMiQ ++QmMi
Q`
QK2MimKXBi
Bb bbmK2/
i?i BMBiBHHv
i t#2ir22M
0>2`2 x>2`2BM/B+i2bi?2`2HiBp2/BbiM+2
rel (t)3)
Recombina<on
at
t
where
x
(t
) = 0 i?2
rec
rel
rec
bi`QM@?QH2TB`
Bb+`2i2/
rBi?
p
M/
|x
(t
)%
4
rel 0✏v (p) rBi? ✏c (p)2H2+i`QM M/ i?2 ?QH2- M/ ✏g0(p) = ✏c (p)0-✏g (p)U✏#2+QK2bKBMBKmK2M2`;v
i p0 X Q7h?2
2KBb@ UpH2M+2V
i?2HB;?i+QM/m+iBQM
v (p)V /2MQiBM; i?2
✏
(t
)
is
emi\edgrec
B@ i?2#M/
K`2+QK#BMiBQM
Q++m`b
i trecpXU>
t0p2HQ+Biv
V r?2M Q7 M 2H2+@
2H2+i`QM rBi?
KQK2MimKh?2

vector poten]alZ. Lenarčič et al., PRL. 108, 196401 (2012).
T. Oka, PRB 86, 075148 (2012).

<latexit sha1_base64="m6frZZ/0CL2v4fhcvEkwfpxFlUk=">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</latexit><latexit sha1_base64="nFxmWbLLkfJAB4zf0YK1n80c6T4=">AAACfHicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQuIx6QWFGfmAJnpGiXx1TFFuQpFqcm1mvECygZ6BmCggMkwhDKUGaAgIF9gOUMMQwpDPkMyQylDLkMqQx5DCZCdw5DIUAyE0QyGDAYMBUCxWIZqoFgRkJUJlk9lqGXgAuotBapKBapIBIpmA8l0IC8aKpoH5IPMLAbrTgbakgPERUCdCgyqBlcNVhp8NjhhsNrgpcEfnGZVg80AuaUSSCdB9KYWxPN3SQR/J6grF0iXMGQgdOF1cwlDGoMF2K2ZQLcXgEVAvkiG6C+rmv452CpItVrNYJHBa6D7FxrcNDgM9EFe2ZfkpYGpQbMZuIARYIge3JiMMCM9QwM9w0ATZQczaFRwMEgzKDFoAMPbnMGBwYMhgCEUbO9ihjUMaxn/MakwaTPpQpQyMUL1CDOgACYzAMHMk04=</latexit>)| = 0Xh?2
7`2[m2M+v
Bb U@
b@p ✏v (p)V- M/i`QM
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Bb ;Bp2MQ7#vi?2i?2 2KBii2//2`Bpi2 HB;?i@p ✏c (p)Q2 #2 2[mHiQ i?2
Q7Bbi?22H2+i`QM@?QH2
TB`,
i?2 2z2+i
Q7 2M2`;v
i?2 }2H/iF2M
BMiQ ++QmMi
pB i?2 b?B7i

2

14

Q7 i?2 KQK2MimKX >2`2- Bi Bb bbmK2/ i?i BMBiBHHv i t0

speaker: Y. Murakami

Subcycle Analysisj(!; tpro ) =Zdt̄eit̄! j(t̄)W (t̄; tpro ) window

15

Ihh (!; tpro ) ⌘ |!j(!; tpro )|2E(t) = 0
U=7.0, Ω=0.3, E0=0.4

3 step model

U=10.0, Ω=0.5, E0=0.75

or

HHG@MoL Ins.

3 step model

Dispersion of many body elemental excita<onProbe of dispersion of elemental excita<ons
speaker: Y. Murakami

2t+Bi2/ bii2 mM/2` M /B#iB+ +?M;2 Q7 A M/ mb2 Bi
rBi?BM i?2 i?`22 bi2T KQ/2H iQ 2tTHBM i?2 >>: 72im`2bX
AoX *PL*GlaAPLa

Eﬀect of spin background to Single par1cle spectrum and HHG

16

3 BMbm@AM i?Bb TT2`- r2 bim/B2/ >>: BM i?2 R/ JQii
Spinstate=
AF typeSpin state= spin liquidHiQ`/2b+`B#2/#v i?2 bBM;H2@#M/ >m##`/ KQ/2H mb@
TTHB2biQTQBMi2/
Qi?2` ivT2b
BM; Bh1".
M/T2`BQ/
2t+ir?2M/B;QMHBxiBQMX
q2
Qmi?M+2//m`BM; i?2|E(t)| BM+`2b2bX
AMi2`2bi@
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BM;Hvi?2 g`2+QK#B@ `2+QK@
i?i i?2
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Q`B;BMi2bBM/B+i2b
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rBi? `2bT2+i iQ Bib `2HiBp2 KQK2MimK-AMr?B+?
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2t+BiiBQMbX
Pm`
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M/ 2t+iiBp2 KQ/2Hb 2tTHBM i?2 >>: BM i?2 R/ JQii
BMbmHiQ`(b)i?ii?2 >>:`2bmHib
BM/B+i2>>: BMT`Q+2bb2bX
JQii BMbmHiQ`b+M Q`B;BMi#2 T@(b)
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bBM;H2@T`iB+H2 bT2+i`mK
Q#iBM2/#v ii?2H2bi(c)
/2KQMbi`i2/
i?i i?2
bBQMofQ77Q`
i?2`2H2pMiKMv@#Q/v
2H2K2Mi`v
2t+BiiBQMbBh1".
i?2background
HQr2bi Q`/2`
KQ/2H Ĥe↵,1,1(t)X h?2 2[mB@▷ Diﬀerence
spin
r2HH +Tim`2/ #v i?2 b2?2`2 i?2
HB#`BmKbTBM/Qm#HQM@?QHQM+QM};m`iBQM Bb TB`bX
/2i2`KBM2/ 7`QK i?2 ;`QmM/
P/Qm#HQM@?QHQM TB`bX h
i
Eﬀect
on
singlepar6cle：JQ`2Qp2`r2
BMi`Q/m+2/
2z2+iBp2
KQ/2Hb
#b2/
bii2
Q7
Ĥspin ⌘Ĥspin,ex+ hBig()
ŝ
X
AM i?Bb
KQ/2Hzz,ii
TB`
Bb `mH2/#v QM
i?2i?2/Bb
E0=0.7i?2/BbT2`bBQM Q7 i?2
mTT2` M/ HQr2`>m##`/#M/ba+?`B2z2`@qQHzi`Mb7Q`KiBQMr?B+?
HHQrbmb
iQ
B/2M@
rBi? `2bT2+i iQ Bib `2H
U
Eﬀect
on
HHG：
Small
Ki+?2b
r2HH
rBi?
±
±
2v
sin(p)b22
6B;X
eUVX +Tim`2/
LQi2
iB7v T`Q+2bb2b bBKBH`iQ M/ /Bz2`2Mi2bb`BHv
7`QK
i?2
b2KB+QM@
2
#v i?
i?i
i?Bb
/BbT2`bBQM
U±2v
sin(p)VBb
2t+iHv
i?i
Q7
i?2
i?2Q7KMv@#Q/v
Mim`2/m+iQ` KQ/2Hb- M/ iQ /Bb+mbb i?2 `QH2
i?2b2 BM/BpB/mH/Qm#HQM UMiB?QHQMV M/ ?QHQM Q#iBM2/ 7`QK
i?2
"2i?2
`2bmHib BM/B+i2
i?i>T`Q+2bb2bX
q2
b?Qr2/i?i
i?2 bTBM/vMKB+br?B+?
?b
E0=0.5
HHG
is 1
directly
relatedto dispersionofedMbix 7Q`
U !
Q` 7`QK
i?2 /B`2+i+QMbi`m+iBQM
Q7
THB2/bm#biMiBHHv
7Q`  bT2+i`Qb+QT(c)
MQ MHQ;m2 BM b2KB+QM/m+iQ`
bvbi2Kb`2@de(c) U = 10.0, ⌦ = 0.5, E0 = 0.7
i?2 rp2 7mM+iBQMbX
AM 6B;XeU#V- r2 b?Qr
i?2
+Q``2@
many-bodyelemental
excita6on
bBQM
Q7
i?2
`2H2pMiK
/m+2b i?2>>:
BMi2MbBivX
BM/B+i2bi?i i?2
bTQM/BM;>>:bT2+i`mKX
h?2h?Bb>>:`2bmHi
BMi2MbBiv
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i?2?2`2Y.i?2/Qm#HQM@?QHQM
speaker:
Murakami>>:`2BMi2MbBiv
BM JQii+QKT`2/
bvbi2Kb+Mi?2#27mHH
b2MbBiBp2
iQ bTBM+miQz
mM/2`2biBKi2/
iQ
bBKmH@
JQ`2Qp2`- r2 BMi`Q/m/vMKB+bX
Ai rBHH#2 i?2BMi2`2biBM;
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i?2 2z2+ib
Q7
U/b?2/ HBM2bVr?BH2`2bTQMb2 i
7`2[m2M+B2b
6A:X 3X UV aBM;H2@T`iB+H2 bT2+i`mK A(p, !) Q7 i?2 2z2+@ iBQM
a+?`B2z2`@qQHzi`Mb7Q! =ivT2b⌦ Bb H`2/v/2b+`B#2/X
h?2
bm#+v+H2i?2/Bz2`2MiQ7 bTBM r2HH
+QmTHBM;b
M/iB7v
iQ T`Q+2bb2b/Bb+mbb+QM@i
bBKBH`
iBp2 KQ/2H ĤMott,1 BM 2[mBHB#`BmK 2pHmi2/ rBi? Bh1". 7Q` `QmM/
7Q` i?2Q72z2+iBp2
KQ/2H
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i?2 >>:
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2ti2`MH
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KQ/2HbM/ iQ
U = 10X >2`2- i?2 bTBM +QM};m`iBQM Q7 i?2 2[mBHB#`BmK bii2 MHvbBb
bB;MH
Bb+QMbBbi2Mi
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17

Eﬀect of strong spin-charge coupling on HHG

Y. M., et al., Phys. Rev. Le8. 129, 157401 (2022).

K. Uchida, et. al., PRL speaker:
128, 127401Y.
(2022).
Murakami

6Setup & Formula1on
Hamiltonian

H(t) =<latexit sha1_base64="Ad5wQB+ZQj+3maItCT9WyOheO1Q=">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</latexit>v

X

eirij ·A(t) c†i cj

hiji,Electric ﬁeld: E(t) =+U

X

Dynamical mean-ﬁeld theory(a)ni" ni#i@t A(t) =

18

Glat
ij

!
imp
Glat
[∆](t, t! )ii (t, t ) = G

格子模型
La^ce
model

E0 cos(⌦t)

Jij∆

有効不純物問題
Impuritymodel
自己無撞着ループloop
Self-consistency

t

t!
!

∆(t, t )U

Lower
Hubbard band

Upper
Hubbard band

HHG intensity： Ihh (!) = |! 2 j 2 (!)|

U =6, W=4 (Bethe lattice), Ω =0.5
Theory unit energy ≒ 0.5 eV

Σlat
ij

!
imp
Σlat
(t, t! )ij (t, t ) = δij ΣΣimp

Impurity solver(b) 松原 H. Aoki, et Kadanoﬀ-Baym
Keldyshal., Rev. !Mod. Phys.
86, 779 (2014).
Im
Im
Im
Re
Re
t

▷ MoL physics

Ret▷ Temperature eﬀects

ਤ 3 (a) ಈతฏ‫ۉ‬৔ཧ࿦ (DMFT) ʹ͓͚Δɺ֨ࢠ໛‫ͱܕ‬ෆ७෺໰୊Λ८Δࣗ‫ݾ‬ແ಍ணϧʔϓͷ֓೦ਤɻࠞ੒ؔ਺ ∆ ʹ͍ͭͯ͸ɺ͜͜Ͱ͸ɺҰൠʹඇฏߧͷ৔߹ʹରԠ͢ΔҾ਺ [(t, t! )] Ͱද͍ࣔͯ͠Δɻ(b) άϦʔϯؔ਺Λఆٛ͢Δࡾछྨͷ࣌ؒ‫ܦ‬࿏ɻ▷ NCA implementa<on2⇡෼͸ॳ‫ظ‬ঢ়ଶͰ͋Δฏߧঢ়ଶΛද͠ɺ࣮࣌ؒ෦෼Ͱ࣌ؒൃలΛଊ͑ΔɻҰํɺఆৗͳඇฏ
⇡ 16fs▷ Open Library: Nessi⌦ߧঢ়ଶͰ͸ॳ‫ظ‬ঢ়ଶ͸ॏཁͰͳ͘ͳΔͷͰɺ‫ؒ࣌ڏ‬෦෼Λল͍ͨ
Keldysh‫ܦ‬࿏͕࢖ΘΕΔ
M. Schüler, et al., Comp. Phys. Comm.
257,
107484(2020).
speaker:Y. Murakami
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Tc

Equilibrium Proper1es

19 k

k

Transi<on to an<ferromagne<c state (AF)doublonSingle par<cle spectrum and stringstates (spin polaron)
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Tc

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Mott

ii)

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▷ T

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MottTemperature dependence of HHG (DMFT results)
HHG temp. dependence

Photo carrier

20

HHG vs MoL gap

Occupa]on of upper Hubbard band▷▷▷<latexit sha1_base64="MQt4L2if/3+z4mD0WTxPMxYRlNI=">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</latexit><latexit sha1_base64="MQt4L2if/3+z4mD0WTxPMxYRlNI=">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</latexit>

6A:X RX MQKHQmb >>: 2M?M+2K2Mi BM JQii BM
bmHiQ`bX  1tT2`BK2MiH `2bmHib Q7 i?2 >>: BMi2MbBiv 
i?2 BM/B+i2/ >>: T2Fb b  7mM+iBQM Q7 i?2 QTiB+H ;T 7Q
*2 _mP4 UJQii BMbmHiQ`V M/ AMb Ub2KB+QM/m+iQ`V- `2T`Q(exponen<ally)
HHG
intensity/m+2/ 7`QK _27X jjX h?2 i2KT2`im`2 T Bb KQ/B}2/ BM i?2
Mott`M;2 Q7 T 2 [290 E- 50 E)- M/ i?2b2 bvbi2Kb b?Qr  KQMQ
iQMB+ BM+`2b2 Q7 i?2 ;T rBi? /2+`2bBM; i2KT2`im`2X h?2
2t+BiiBQM 7`2[m2M+v Bb ⌦ = 0.26 2oX # h?2Q`2iB+H `2bmHib 7Q
Larger increase for higher harmonics
i?2 BMi2MbBiv i i?2 BM/B+i2/ >>: T2Fb b  7mM+iBQM Q
i?2 JQii ;T U Mott V 7Q` i?2 bBM;H2@#M/ >m##`/ KQ/2H BM
Cf. K. Uchida, et. al.,
JQii
BMbmHiBM;
T?b2X HbQ BM i?2 +H+mHiBQMb- i?2 JQi
Qualita<vely same as experiment i?2
on Ca
2RuO
4
PRL 128,Y.
127401(2022).
Murakami;T BM+`2b2b rBi? /2+`2bBM; speaker:
i2KT2`im`2Xh?2 `2bmHib `2
6A
Q#iBM2/ rBi? MQM2[mBHB#`BmK /vMKB+H K2M@}2H/ i?2Q`v
2p
M/ r2 +QMbB/2` i?2 KQ/2H QM i?2 "2i?2 HiiB+2 rBi? #M/
T2hn̂
rB/i? W = 4 M/ >m##`/ BMi2`+iBQM U = 6X h?2 2t+BiiBQM
i2K
BQm
7`2[m2M+v Bb ⌦ = 0.5X
Mott

Photo carrieri?

j(!; tpro ) =Z

Origin of characteris1c HHG jbehavior?
dt̄eit̄! j(t̄)W (t̄; tpro )
10-7

10-9

E(t) = 0

10-7

21

10-9

one cycle

U=6▷ Subcycle analysis:

U=8

U=10

6A:X 9X am#+v+H2 72im`2b Q7 i?2 >>: bB;MH Q7 i?2i) Even for low T, coherence .me is very short (for U=6)
JQii BMbmHiQ`X -# am#+v+H2 bT2+i` IHHG (!, tp ) 7Q` U = 6ii) Trajectory of d-h pair is almost
the=same
PM andAF M/ i # T = 0.05 U6 T?b2VX 6Q`
i  T
0.2 inUSJ
T?b2Viii) Major diﬀerence is coherence<me of pairs
i?2 rBM/Qr
6Qm`B2`i`Mb7Q`KiBQM- r2 2KTHQv  :mbbBM
U=6
U=8
U=10
rBM/Qr r?Qb2 biM/`/ /2pBiBQM Bb 0 = 0.9X h?2 `2/ /b?2/
U#Hm2 /Qi@/b?2/V HBM2b BM/B+i2 i?2 KtBK Q7 IHHG (!, tp ) i
▷ Diﬀerence between PM and AFTis =liied
for large
0.05
UT =U(smaller
0.2V iJex) ;Bp2M ! b  7mM+iBQM
Q7 tp Y.`QmM/
speaker:
Murakami
tp = 80- M/ `2T`2b2Mi i?2 ivTB+H FBM2iB+b Q7 /Qm#HQM@?QHQM
TB`bXi?2 7mM+iBQM f! (tp )X h?2 p2`iB+H
6A:X jX UV-U#V am#+v+H2 bT2+i`
IHHGh?2b2(!, tp )+m`p2b
7Q` U /2}M2= 6 ii?2 iBK2b
UV T = 0.2 USJ T?b2V M/ i /b?2/
U#V T =HBM2b
0.05BM/B+i2
U6 T?b2VX
 r?2M i?2 2H2+i`B+ }2H/ E(t) =

H
r
i
/
+
H
?
T
i
/
i

_
i
Q
r
Q
K
i
Q
r
+
b
b

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Role of the spin-charge couplingi)zz
BstaggHubbardchain+ Bstaggab ﬁeld (Bzstagg)
1d Hubbard
modelwith staggered

e Thermal ensembles

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i)

z
Bstagg<latexit sha1_base64="i38OYhnFJ+kZSJHAy61mjinlaCU=">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</latexit>

ii)
ii)

10-3
10-3

10-5

10-5

10-5
10-5

ii)

10-7

ii)

10-7
tpair/Tp

tpair/Tp

iii)
iii)

iii)
iii)

z
▷ Emula<ngspin-charge
coupling
viaQM
staggered
ﬁeld
in 1d
6A:X
8X
1z2+ibQ7
bTBM@+?`;2+QmTHBM;>>:X
 magne<ca+?2KiB+
TB+im`2bi?2 2z2+ib Q7 TB+im`2b
i?2 bi;;2`2/
K;M2iB+}2H/Q7
Bstagg
6A:X
8X 1z2+ib
Q7 bTBM@+?`;2+QmTHBM;
QM >>:X
Q7a+?2KiB+
Q7 i?22z2+ibi?2 bi;;TTHB2/ iQ i?2 QM2@/BK2MbBQMH >m##`/ +?BM- r?B+? +M #2 /B`2+iHv +QKT`2/ iQ 6B;X k +X BV AM 2[mBHB#`BmK- 2+? bBi2 Bb
TTHB2/i?2 QM2@/BK2MbBQMH>m##`/+M#2BBV-/B`2+iHv
iQ 6B;X
k +X BV AM
Q++mTB2/#v QM2 iQ
2H2+i`QMM/ i?2
Bb HB;M2/
MiB@T`HH2H
iQ i?2 r?B+?
bi;;2`2/}2H/X
BBBV b i?2+QKT`2//Qm#HQM KQp2bBi T`Q/m+2b• bTBM
Suppressionof HHG +?BMz
Q++mTB2/#v i?2QM2bTBM
2H2+i`QMM/ M/i?2 i?2bTBM
Bb HB;M2/
MiB@T`HH2H
i?2bi;;2`2/}2H/X BBVBBBVBstagg
b i?2 /Q
 KBbKi+?
#2ir22M+QM};m`iBQM
bi;;2`2/
K;M2iB+}2H/ i i?2 iQ+Qbi
Q7 i?2
w22KM 2M2`;v#Qmi
z
B
▷
stagg
•
Sharper
peaks
in
HHG
Cf.
Slater-type(MF-type)
AF
UxB;x; HBM2bVX# h?2 #2ir22M>>: bT2+i`mK
i?2+QM};m`iBQM
QM2@/BK2MbBQMH M/>m##`/
2pHmi2/
rBi? Bh1".X
KBbKi+?
i?2 7Q`bTBM
i?2KQ/2Hbi;;2`2/
K;M2iB+}2H/+-/ih?2
i?2+Q``2bTQM/BM;+Qbi Q7 i?2 w22zz
bm#+v+H2 MHvbBb 7Q` Bstagg = 0.001 U+V M/ Bstagg = 0.2 U/VX 6Q` i?2 rBM/Qr 6Qm`B2` i`Mb7Q`KiBQMr2
2KTHQv
 :mbbBM
z KQ/2H 2pHmi2/• >>:Shortercoherent7Q`.meofQM2@/BK2MbBQMH
d-h pairs
UxB;x;
HBM2bVX#h?2
bT2+i`mK
i?2>m##`/
rBi?Q7Bh1".
B
does
not
aﬀect
coherence0
stagg
rBM/Qr r?Qb2 biM/`/ /2pBiBQM Bbz = 0.9X h?2 +QHQ`2/ K`F2`bi?2 `2+QK#BMiBQM
Y. Murakami
z BM/B+i2 i?2 2M2`;v 2KBii2/ i tp #v speaker:
bm#+v+H2 TB`MHvbBb
Bstagg =0.001
U+V M/ KQ/2H
BstaggmbBM;= 0.2
U/VX 6Q`
rBM/Qr
6Qm`B2`i`Mb7Q`KiBQ
 /Qm#HQM@?QHQM
r?B+? 7Q`
Bb T`2/B+i2/
7`QKi?2 i?`22@bi2T
i?2 /Qm#HQM
M/i?2?QHQM/BbT2`bBQMb
7`QK i?2"2i?20MbixrBM/Qr
7QHHQrBM; _27X
jyX h?2+QHQ` BM/B+i2bi?2 iBK2#2ir22M`2+QK#BMiBQM
M/BM/B+i2
i?2 +`2iBQM
Q7 i?2/Qm#HQM@?QHQM
r?Qb2
biM/`/
/2pBiBQM
Bb BMi2`pH= 0.9Xh?2i?2+QHQ`2/K`F2`bi?22M2`;v
2KBii2/ i tp
2⇡
TB` tpair- M/ Tp = ⌦ X 2 TB`a+?2KiB+
TB+im`2b
Q7 i?2 BMi2`THv
i?2`KH 2Mb2K#H2bbTBM@+?`;2+QmTHBM;XM/
SM2Hb
 /Qm#HQM@?QHQM
r?B+?
Bb T`2/B+i2/
7`QK#2ir22M
i?2 i?`22@bi2T
KQ/2HM/mbBM;
i?2 /Qm#HQM?QHQM /B
BV@BBBV b?Qr +b2b rBi? /Bz2`2Mi bTBM +QM};m`iBQMb +iBpi2/ #v i?2`KH ~m+imiBQMbX h?2 /Bz2`2M+2 #2ir22M +b2 BV M/ +b2

Mbix 7QHHQrBM; _27X jyX h?2 +QHQ` BM/B+i2b i?2 iBK2 BMi2`pH #2ir22M i?2 `2+QK#BMiBQM M/ i?2 +`2M/ i?2 2t+BiiBQM T`K2i2`b `2 ⌦ = 0.5, E0 = 0.8- t0 = 60 M/= 15X

biQQ/ b  +QQT2`iBp2 2z2+i Q7 i?2 bTBM@+?`;28 +QmTHBM;#v i2KT2`im`2 BM a*ab M
M/ i?2 i?2`KH 2Mb2K#H2X+QmTHBM;X LK2Hv- 23
7Q` bKH
EﬀectsiofMQMx2`Q
thermali2KT2`im`2bﬂuctua1ons
i?2 BMBiBH
2[mBHB#`BmK bii2
Bb /2b+`B#2/#v M 2Mb2K@
ir22M /Bz2`2Mi bTBM +QM};me Thermal
ensemblesfThermal#H2ensembles
Q7 2B;2Mbii2b- `2T`2b2Mi2/ #v i?2 /2MbBiv Ki`Bt
+QMbBbi2Mi rBi? i?2 `2bmHibi)Phase of`2/m+iBQM
J(nΩ)
Q7 i?2 2M?M+2K
⇢ˆ = e Ĥ /Z rBi? Z = Tr[e Ĥ ]X AM bm+?  bvbi2K- i?2
H`;2` U BM 6B;X j +X hQ 2
iQiH +m``2Mi BM/m+2/ #v i?2 }2H/ +M #2 +H+mHi2/ b+QmTHBM; +M BM/22/ T`QpB/
i?2 2Mb2K#H2 p2`;2 Qp2` i?2 BM/BpB/mH +m``2Mib 2pHm@z
7- r2 b?Qr i?2 Bstagg@/2T
7Q` overi?2b2 2B;2Mbii2bX qBi? BM+`2bBM; i2KT2`im`2J(t) i2/
= Averageii)
7Q` ! = n⌦ UrBi? n bQK2
10
10
10
i?2
r2B;?iQ7statesi?2 10?B;?@2M2`;vbii2b BM+`2b2bX AM Qm`
diﬀerent
ini.ali?i i?2 T?b2 Bb b2MbBiBp2
t /T+b2- i ?B;?2` i2KT2`im`2bbTBM +QM};m`iBQMb /Bz2`@
#Qp2 `;mK2MiX h?mb- r2
2Mi 7`QK i?2 6 ;`QmM/ bii2 `2 +iBpi2/- b22 6B;X 8 2X
States with diﬀerent
bvbi2K- i?2 KQ/B}+iBQM Q7
h?2 /vMKB+b Q7 i?2 /Qm#HQM Q` ?QHQM Bb /Bz2`2Mi 7Q`
spin conﬁgura<ons (for T<ΔMott)
iii)
bTBM@+?`;2 +QmTHBM; M/ i?
2+? +QM};m`iBQM- bBM+2 i?2
2M2`;v i`Mb72` iQ i?2 bTBM
Qp2` i?2 `2/m+iBQM Q7 i?2 im#+F;`QmM/ /m`BM; M 2t+m`bBQM /2T2M/b QM i?2 bTBM +QM@
BM;- H2/BM; iQ M 2M?M+2};m`iBQMX h?Bb b?QmH/ T`Q/m+2 2KBii2/ HB;?i rp2b rBi?
HQr2` i2KT2`im`2bX/Bz2`2Mi T?b2b 7Q` /Bz2`2Mi bTBM +QM};m`iBQMb`2bmHi@z>>:X  a+?2KiB+ TB+im`2b Q7 i?2 2z2+ib Q7 i?2 bi;;2`2/ K;M2iB+ }2H/ Bstagg
BM T?b2
+M+2HiBQMb
i?2 2Mb2K#H2
p2`;2M/
M- r?B+? +MBM;
#2 /B`2+iHv+QKT`2/
iQ 6B;X k +X 7i2`
BV AM 2[mBHB#`BmK2+? bBi2
Bb▷
Diﬀerent
spin
conﬁgura<onsà
Diﬀerent
<ming
of
pair
crea<onandrecombina<onà Dephasing.Bb+mbbBQM/ MiB@T`HH2H
iQ i?2`2/m+2bi;;2`2/i?2}2H/X +Q?2`2M+2
BBV- BBBV b i?2 /Qm#HQM
KQp2bBi T`Q/m+2bi?mb#2ir22M
i?2/Qm#HQM@?QHQM
z/ i?2 bi;;2`2/ K;M2iB+ }2H/ i i?2 +Qbi Q7 i?2 w22KM 2M2`;v- #Qmi Bstagg
h?2 bi`QM; i2KT2`im`2 /
TB`b
rBi?
i2KT2`im`2X
LQi2 i?i
i?Bb 27@▷ >m##`/Ac<vated
byBM+`2bBM;
spin-charge
coupling& thermal
à Should
be absent for large U case
M2@/BK2MbBQMH
KQ/2H
2pHmi2/ rBi?
Bh1".X+-/ h?2 ensembles+Q``2bTQM/BM;zi`mK Q#b2`p2/ BM JQii BMbm
i?2 rBM/Qr
6Qm`B2`QM
i`Mb7Q`KiBQMr2 K;M2iB+
2KTHQv  :mbbBM
72+i6Q`/Q2b
MQi `2Hv
HQM;@`M;2Q`/2`BM;M/ with
stagg = 0.2 U/VX
(ConsistentDMFT)
h?2 +QHQ`2/ K`F2`b BM/B+i2 i?2 2M2`;v 2KBii2/ i tp #v i?2 `2+QK#BMiBQM Q7
speaker: Y. Murakami
B+H
b2KB+QM/m+iQ`bX
AM i?
Bb
HbQ
`2H2pMiBM
i?2
SJ
T?b2X
h?Bb
2z2+i
Bb
bBKBH`
i?2 i?`22@bi2T KQ/2H mbBM; i?2 /Qm#HQM M/ ?QHQM /BbT2`bBQMb 7`QK i?2 "2i?2
BM b2KB+QM/m+iQ`b- QM2 Q7i
iQ#2ir22M
i?2 2z2+i
Q7 /BbQ`/2`
r?B+? `2/m+2b
2 iBK2 BMi2`pHi?2 `2+QK#BMiBQM
M/BM
i?2b2KB+QM/m+iQ`b+`2iBQM Q7 i?2 /Qm#HQM@?QHQM
iBK2 T2 Q7  72r 7b 7Q` M 2H2
Q7 i?2 BMi2`THv
i?2`KHM/ bTBM@+?`;2+QmTHBM;XSM2Hb  +`m+BH
i?2 #2ir22M+Q?2`2M+2
Q72Mb2K#H2b
2H2+i`QM@?QH2
TB`b9k
X >Qr2p2`QMb +iBpi2/ #v i?2`KH ~m+imiBQMbX h?2 /Bz2`2M+2 #2ir22M +b2 BV M/ +b2
Q`B;BM Q7 i?Bb b?Q`i /2T?bB/Bz2`2M+2 Bb i?i i?2 2z2+i BMp2biB;i2/ ?2`2 Bb +iBpi2/
-3

-5

-5

-7

pair

p

Discussion: Diﬀerence from semiconductors

24kVery small dephasing <me T2 (less than a few fs) is oien usedci)<latexit sha1_base64="q/gptXDofjc+Doo8s0eqAfREyyE=">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</latexit>

Tc

Non-intrinsic eﬀects• Propaga<on of light• Intensity distribu<on

Intrinsic eﬀects
ii)>

doublon• El-El scaLering• El-Ph scaLering

I. Floss, et al. PRA 97, 011401 (2018).
I. Lilen, et al. PRL 125, 083901 (2020).
K. Uchida, et. al., PRL 128, 127401 (2022).

:X RX MQKHQmb >>: 2M?M+2K2Mi BM JQii BM@

iii)

The characteris<c HHG feature is unexpected in typical semiconductors
6A:X kX 1[mBHB#`BmK T`QT2`iB2b Q7 i?2 >m##`/ KQ/2H
2pHmi2/ rBi?BM /vMKB+H K2M@}2H/ i?2Q`vX  h2K@
T2`im`2 /2T2M/2M+2 Q7 i?2 K;M2iBxiBQM- m = |hn̂i" ihn̂i# i|/2X h?2 p2`iB+H /b?2/ HBM2 BM/B+i2b i?2 i`MbBiBQM
speaker: Y. Murakami
i2KT2`im`2 Tc X # GQ+H bT2+i`H 7mM+iBQM- Aloc (!)- 7Q` p`@
BQmb i2KT2`im`2bX 6Q`  M/ #- r2 mb2 U = 6 M/ +QMbB/2`
i?2 "2i?2 HiiB+2X + a+?2KiB+ TB+im`2b Q7 i?2 bTBM@+?`;2+QmTHBM; ++QKTMvBM; i?2 FBM2KiB+b Q7  /Qm#HQM U+B`+H2VX
SM2H BV b?Qrb i?2 bTBM +QM};m`iBQM Q7 M MiB72``QK;M2iB+bii2- BBV b?Qrb  /Qm#HQM //2/ iQ i?Bb bii2- M/ BBBV b?Qrb

Discussion: Diﬀerence from semiconductors

25

Intrinsic very short coherence 6me + its strong temperature dependence
from strong spin-charge coupling

Anomalous temperature/gap-size dependence of HHG in MoL insulators
Proposal: Dimensional crossover of HHG
weaker•1 dim (spin charge separa<on)•2leg ladder••stronger•2dim

Spin-charge
coupling

Srn-1Cun+1O2n
speaker: Y. Murakami

Summary

26

Explora6on of interes6ng non-linear op6cal phenomena in strongly correlated systems
Three step model for many body states

Origin of HHG

k▷ Doublon-holon recombina<onc▷ Kinema<cs of doublon-holon pairsi)<latexit sha1_base64="q/gptXDofjc+Doo8s0eqAfREyyE=">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</latexit>Tc

3 step model for many body excita.ons (for 1d)
doublon

HHG spectroscopy of many body statesii)

Importance of strong correlates between spins and charges

k

or

iii)
c
i)
<latexit sha1_base64="q/gptXDofjc+Doo8s0eqAfREyyE=">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</latexit>

Strong spin-charge coupling (string states)

Tc

doublonii)

6A:X kX 1[mBHB#`BmK T`QT2`iB2b Q7 i?2 >m##`/ KQ/2H
2pHmi2/ rBi?BM /vMKB+H K2M@}2H/ i?2Q`vX  h2K@
T2`im`2 /2T2M/2M+2 Q7 i?2 K;M2iBxiBQM- m = |hn̂i" ihn̂i# i|/2X h?2 p2`iB+H /b?2/ HBM2 BM/B+i2b i?2 i`MbBiBQM
i2KT2`im`2 Tc X # GQ+H bT2+i`H 7mM+iBQM- Aloc (!)- 7Q` p`@
BQmb i2KT2`im`2bX 6Q`  M/ #- r2 mb2 U = 6 M/ +QMbB/2`
i?2 "2i?2 HiiB+2X + a+?2KiB+ TB+im`2b Q7 i?2 bTBM@+?`;2+QmTHBM; ++QKTMvBM; i?2 FBM2KiB+b Q7  /Qm#HQM U+B`+H2VX
SM2H BV b?Qrb i?2 bTBM +QM};m`iBQM Q7 M MiB72``QK;M2iB+bii2- BBV b?Qrb  /Qm#HQM //2/ iQ i?Bb bii2- M/ BBBV b?Qrb
i?2 /vMKB+b Q7 i?2 /Qm#HQM- r?B+? /Bbim`#b i?2 bTBM +QM};@m`iBQM UxB;x; HBM2bV i i?2 +Qbi Q7 KmHiBTH2b Q7 i?2 2t+?M;2

Anomalous temperature dependence HHG

speaker: Y. Murakami

iii)

6A:X RX MQKHQmb >>: 2M?M+2K2Mi BM JQii BM@
bmHiQ`bX  1tT2`BK2MiH `2bmHib Q7 i?2 >>: BMi2MbBiv i
i?2 BM/B+i2/ >>: T2Fb b  7mM+iBQM Q7 i?2 QTiB+H ;T 7Q`*2 _mP4 UJQii BMbmHiQ`V M/ AMb Ub2KB+QM/m+iQ`V- `2T`Q@

27

Supplementary informa0on

speaker: Y. Murakami

Related works on HHG in SCES
Many body eﬀects on HHG

HHG spectroscopy
for phases

Screening eﬀects
Photo-induced transi<ons

N. Tancogne-Dejean, et al.,PRL 121, 097402 (2018).

R. E. F. Silva, et al.,Nat. Photon. 12, 266 (2018).

O. Grånäs, et al., arXiv:2008.11115.

M. R. Bionta, et al. ,PRR 3, 023250 (2021)

HHG from many- body
excita<ons
MagnonS. Takayoshi, et al., PRB 99, 184303 (2019).

Solitons
S. Imai, et al., PRL 124, 157404 (2020).

ScaLering and edge

C. Orthodoxou, et al,
npj QM 6, 76 (2021).

Strings/Bosons

T. Hansen, et al.,
PRA 105, 053118 (2022).

Equilibrium phase transi<ons
C. Shao, et al.,PRL 128, 047401 (2022).

28

M. Lysne, et al., PRB 101, 195139 (2020).

Excitons
M. Udono, et al., PRB 105, L241108 (2022).

etc ... + some experiments (Kyoto Uni) are coming

Thank you very much for your a=en>on!

speaker: Y. Murakami

HHG spectrum in MoR insulators
Weaker ﬁeld regime

229Stronger ﬁeld regimeg. 1. (a,b) HHG spectra in the stronger-field regime (a) and in the weaker-field regime (b), and (c) HHasinathefunctionof regime
the harmonicenergy(n⌦)regime
fieldstrength
and
circ
Gctra
spectrastronger-field(a) and in theweaker-field(b),and (c)
HHGin the(E0 ). The arrows
Fig.
1. and(a,b)
HHG
spectrastronger-fieldregime (a)and thein thewhite
weaker-field
n of theshow
harmonic
and field White
strength (Earrows
the
white
asand
a functionof circlethen⌦harmonic
0 ). The0 ). Therkers
theenergy
cuto↵(n⌦)
energies.
lines
inspectrapanel(c)indicate= U +energy
mE0(n⌦)
andandthefield
redstrengthline is(Ethe
fit arrfouto↵ energies. White lines in panel (c) indicate n⌦ = U + mEmarkers
red lineis the fit
forshow
the cuto↵
energies.
White lines in panel (c) indicate n⌦ = U + mE0 an
0 and theweaker-fieldregime.
We
use
U
=
8.0,
=
2.0,
=
0.06,
⌦
=
0.5.
ime. We use U = 8.0, = 2.0, = 0.06, ⌦ = 0.5.
the weaker-field regime. We use U = 8.0, = 2.0, = 0.06, ⌦ = 0.5.
Single plateau

Mul6ple plateaus

trong-field regime, (b) HHG spectra in the weak-field FIG.
regime,andHHG(c) HHG
spectrain strong-fieldthe
1: (a)
spectrain theregime, (b) HHG s
E0 = 0.6
E0 = 4.0
d the harmonic energy (n⌦). The white circle markers
show
energies.
Theofthe harmonic energy (n⌦).
planeof cuto↵the field
strength(Einset0 ) andectric field during one period
at E0 = 4.0. We use U =panel
8.0, (a)= 2.0,
0.06,⌦ = 0.5.
shows=thecurrent
and electric field during one period
FullrecombinaIon
Hopping

Recombina6on process

複数のプラトー構造vsis dominant†2⇡
th a period of T ⌘ ⌦ ,j(t) = iq i,j, vij (t)(e= e0 with· j(t) a period of T ⌘ 2⇡
a time-periodic
steady0 · ri j )hci, (t)cj, (t)istate⌦ ,Rfield is balanced by thewheninjected byfield is balanced by theand j(n⌦) = T1 dt̄eit̄n⌦j(t̄).theAsenergy
an illustration,
we theshowin the inset of Fig. 1 dissipation
an exampletoofthe
thebath.
time evolution speaker: Y. Murakamiof the electric fieldtheinducedduringone extracted
tice withlatticeacurrent
hyper-cubic
latticewith latticeFig.and
2. We(a)(b)consider
Comparisonof the HHG
intensityfrom the spacing
current⇤ origin
mparisonof theHHG spacing
intensity⇤ extracted from the current originating from di↵erent provpv
period.
Becauseof theinversion
symmetry,
only
odddimensions
l dimensions(v the= stronger-field
a
the
limit
of
infinite
spatialcessesin intheweaker-fieldregime(a) andthe stronger-fieldregime (v(b). =“Full”pcorresr-field
regime (a) andregime(b).
“Full”
corresponds
to
the
full
FDMFT
2 d
2 d
calculation,jisthecurrent
fromtherecombination
processes,
andjis the checurrent
fromtherecombination
processes,
and
j
is
the
currentfromthehopping
frequency
components
show
up
in
the
HHG
spectrum.
rc
hop
hop
with d ! 1), which has a Gaussian density of statesussian density of statesPSummary of HHG spectrum in MoR insulator
Ihhg (n⌦)

230<latexit sha1_base64="SD3lmmGjfGlL2kF4dortaD7lpE8=">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</latexit>(Field strength)n⌦ = UE0n⌦ = U + E0n⌦ = U + 2E0(Frequency)▷ Stronger ﬁeld regime (E & 2 ) : ✏cut = U + mE00(b) HHG spectra in the weak-field regime, and(c) HHG spectra in the
nergy (n⌦). The white circle▷ Weakermarkersshow( Ecuto↵ﬁeld regime✏cut = +The
↵E0 inset of
0 . 1 ) :energies.
one period at E0 = 4.0. We use U = 8.0, = 2.0, = 0.06, ⌦ = 0.5.
speaker: Y. Murakami

⌘

2⇡

,

j(t) =iqPv (t)(e · r)hc† (t)c(t)i = e · j(t)2Mobility of Carrier

31

Aloc =ImGR
loc

i↵erent HHG plateau can then be underate from the a recombination of a doublonlized at the m-th nearest-neighbor sites.
mparable or larger than the width of thes, doublons and holons remain almost lo, the spectral functions in the NESS show
Stark peaks in the strong field regime, seelower Hubbard
upper Hubbard
recombinationa regime(
doublon-holon
pair on▷ Weakerof
ﬁeld
is not shrinked
E0 . 1 ) : Bandwidth
spectra in the weak-field regime, and (c) HHG spectra in the5
sites
themarkers
positive
direction). The alongwhite circle
show fieldcuto↵ energies.
The reinset of▷ Stronger
regime(: Bandwidth
0 & 2
d at E0 = 4.0.
We use Uﬁeld= 8.0,=E
2.0,=) 0.06,⌦ = 0.5.shrinking and Wanner-Stark side peaks focused site
gy
Unot
+ an
n|E(t)|
13) isexact at a certain time t. Thissidering
the linear the main
ently explainscharacteristiccf) Casewith staPc ﬁeld à feaU
P of†
Eq.j(t)(13)=in terms
2
iq ini,j,vijstrong-field(t)(e0 · ri j )hci,regime.
(t)cj, (t)iSince= e0 · j(t)GspectratheRpticalconductivity
Uand j(n⌦) = T1 dt̄eit̄n⌦ j(t̄). As an illustration, we show
+ E0
between
E
and
E
,
one
expects
that
the
0
0
2
in the inset of Fig. 1 an example of the time evolution
Ufromrecombinationof m-thnearestof thethe
electric
field and the induced
current
during one+ 2E0
speaker:Y.
Murakami
2c,eq (! + ⌦)period. Because
symmetry,
only odd FIG. 2: (a) Time averaged local spect
on-holonpairs ofisthe
in inversion
the triangle
regioncomponents
show up in the HHG spectrum.
feqfrequency(!
+U⌦)].
(18) , which
nonequilibrium steady states. (b) Field-st
⌦
When
+the
mE
naturally
explains
0field is strong, the HHG spectrum initiallyof the time-averaged doublon density in
HHG
in theonsymmetryleadsdominant m = 1 sector, andΩ=0.5, U=10, J=0

12
10

5

12

0

10

Ω=0.5, U=10, J=3

5
0

0

-30

0

10

20

30

40

Ĥ(t) = ĤkinnΩ (t) + Ĥloc

50

Hloc =U|nΩj(nΩ)|2E0E0|nΩj(nΩ)|2

imaginary parts of the frequency dependent
FIG. 2.-5 Spectral function for various boson distributions-5with
8
8), for !0 = 17, Uscr = 5 and = 20. The= 0, 1, 2, t = 1, Uscr
= 5 and = 20. For = 1, 31 bosons-10-10
d lines are the asymptotic
values for ReU in
are used and for =6
6 2, 61.
-15-15±1 and ! ! 0. 4Besides the single
boson
多軌道ハバード模型:軌道自由度の効果
4
-20
-20
also show the results for a distribution of22-25
dth defined by (seeEq. (8)).
M.
Lysne,-25Y.M, P. Werner in prep.
form
2軌道ハバード模型

X0 0

32

-30
X40
50
n↵" n↵# + (U nΩ 2J)
n1 n2¯
10

20

30

↵=1,2
model such a realistic situation we may
Xle boson model
to
a
model
with
a
distri2
FIG. 10. High-harmonic spectra |n⌦j(n⌦)| as a function of E0 for
J = 0 (left)+ di↵erent(U 3J)
n1andnJ2 = 3 (right). The gray dashed(9)
represent the
cuto↵s U +for
J ±eachnE0 .
ns, where thelines
coupling
constant

U+J +E2(!i !0 ))g 2 exp (2 21.
P(!i !0 )2!0)
0.01
i exp (
2 2

U+2J+2E with U the▷intra-orbital
and J the Hund couU + J +nE + repulsion(n-1)J にカットオフ
02|nΩ j(nΩ)|
高次高調波強度(log)inant bands in the U = 10, J = 3 spectrum (see Fig. 3).
pling.
2
基底Both figures confirm a shift of the cuto↵s to higher ener=(8)
Low-temperatureof thegies in thestatesmodel with
Hundhalf-filledcoupling. In system
particular,with
theJ > 0 have predominatly
doublyhighJ) in arecomcuto↵ line associatedwithEoccupied
next-nearest
neighbor
0 = 3(U sites
bination processes
is shifted
by J, while for for next-nextspin configuration.
Creating
a “singlon”-“triplon”
exci0.0001
oupling constants ensures that the renornearest
neighbor
recombinations,
we
find
both
evidence
外場励起
励起1tation in
thisforhalf-filledbackground, (", ")i (#,lines#)jwhich! (0,"shifts by J and 2J (see white dasheding (width of the main1e-06
Hubbard band) is)i ("#, #)j , costs
EE ==Urespectively).
+EJ.+ UHencethedicateandthe energy
shifts by J and
2J,
We
also
E
+
+
J
0
0
single-boson and multi-boson
case.
The
U +J
1e-08
noticeare
thatexpectedeven the nat= an
1 lineappears!to ⇡be ±shifted,Hubbard bands
energy.
ns for = 0, 1, 2, v = 1, Uscr = 5 andat least for large E0 . This indicates nearest-neighbor 2(inThe NCA spectral functionsof thehalf-filledprocesses
systemwith
are
1e-10
of the field)
recombination
0
10
20
wn in Fig 2 and the corresponding
U30(!)40 50 60 励起2 the direction外場励起
shown for U simultaneous= 10 andHunddi↵erent
J inonthe
left panel
de-excitationa neighboringsite.of
高次高調波次数(nΩ)
Fig. 1. [for the real part you canjustnΩadd
E
=
E
+
2E
+
U
+
2J
0
0 the10position
The above
complex argument
structures evident
in Fig.
suggestFig. 3. While theexplains
FIG. 11. High-harmonicfor E
0 = 8 and J = 3. The
ary part to broaden
the poles.]spectraHere,
weinterference e↵ects from other types of processes. At
Hubbardbands, we also recognize a strong
vertical lines show U + J + nE0 (gray dashed)of
andtheU + mainJ+
energies & 2U + 2J one can expect to see higher ormber of bosons
such
that
theblack).
energy sepnE0 +(n 1)J(solidnarrowing ofder
these
bands
and
the
appearanceof ")sideprocesses(",")i (#,#)j (",")k (#, #)l ! (0,ストリング状態を反映したHHG
àlike
UやJの推定i ("#n neighboring bosons is constant ▷[whatis, #)j (", 0)k (#, "#)If the narrowing
production of istheduetwosinglonspeaker:
Y. Murakami
bands with increasing
J.l . Theto thewhile the bosons cover the same portion of
triplon pairs occurs in the direction of the field, the maxblocking of the
motion of one spin species, while the satelimum cuto↵ value associated with the recombination isstribution. Note that in the multi-boson
lites arereassociatedspin(Hundpair
exwith Hund excitations. The corresponding2(U + J) with
+ 2E0 . local
However,theexcitations
second singlon-triplon
sidebands ofassociatedthemain
Hubbardband
getcombination processes yield cuto↵ values shiftedby mul- Looking
could alsoat
bethe
excitedin a directionspectral
perpendicular
to thecitations).
equilibrium
functionstiples of ±J, with the maximum cuto↵ energy shifted byfield in which case the cuto↵ becomes 2(U + J) + E0 . If it

背景と動機

33

メタマテリアルとプラズモニクスの発展▷ 強力な振動磁場 (1T, サブTHz)の生成
強力な振動磁場下でのスピン系のダイナミクス▷ 強力磁場によるスピン操作 à

S. Takayoshi, H. Aoki, T. Oka,
PRB, 90, 085150 (2014).
S. Takayoshi, M. Sato, T. Oka,
PRB, 90, 214413(2014).

Spin系のHHGの可能性?起源?
(a)

(c)

Laser magnetic
field B(t)Driving
force

Radiation frommagnetic dipole

Magnetic dipole M(t)Spin excitations

HHG
originSpin HHG intensity

rgy
ge

(b)

Electronic systems

Sp

Electric field : E(t)M

speaker:Y. Murakami
Intraband current,Interband polarization

M

Charge dynamics in bands

Spin ex

From a few to tens of eVFromsummation over all spins and S tot (t) ⌘ U 1 (t)S tot U (Rt 0(U (t) = T 0 dt exp[ iH(t0 )t0 ] is the time evolution oerator with T the time ordering). From this we o
To study quantum spin systems
in Fourier
the presenceof of theInmagnetization
the following 34
two
sect
tain
the
transform
M (!)
R 背景と動機
i!t we consider thea time-dependent magnetic fielddte
B(t)culations
M (t) and the radiation
power to study the HH
Hamiltonian
quantum spin systems.
Spin模型 + 振動磁場2
I / |! M (!)|2 .
H(t) = Hspin B(t) · S tot ,(1)
Thesymmetryof thesystem may impose constraincf.電子系
IV.
HHG
IN
XXZ
MODELS
III. HHG IN
IV. Hamiltonian
HHG
MODELS
where Hspin▷ isthe spin
and
the
last
iTEBD、厳密対角化、平均場で評価
I /HHG|! 2 P(!)|
onIN
theXXZ
structureof term
thesignal. For example, t(a)electron
represents the Zeeman couplinginversionof the spinsin the limits
ma- the HHG signal(a)symmetryinIn thissectionwewe
consideranothermodel
Letletususstartby investiga
terial withthemagnetic
field
producedby fundamental
thelaser. Now,In
thissectionconsideranotherfundamental
model
systems
to oddharmonics.
considerthe caof quantum
theferromagnetic
XXZ
model.
Theare among
which
simp
We calculate
the magnets,
time
evolutionof thethe time
magnetizationof quantum
magnets,theferromagnetic
XXZ model.
The
whendependent
Hamiltonian
has a thesymmetPof magnets.
In thisca
Hamiltonian
isSistot which
M (t) spin
⌘ spinhS具体例：XXZ(t)i, where=
S j represents
the aselsthe
模型@1次元
totHamiltonian
be represented
combinationof timj can
1z
in Eq.the(1) Sexplicitly
readssummation over all
spins and S(t)SU (t) aroundtranslation⇡
rotationaxis
XX
tot (t) ⌘ U and
tot
Rt 0 =yy yy
zz zz
x 0x S
xx
x
(
0
H
[J
(S
+
S
S
)
J
S
S
],
(15)
H
=
[J
(S
S
+
S
S
)
S
S
],
(15)
+B(t)S
XXZ
xy
z
XXZ
xy
z
X
j
j+1
j
j+1
j
j+1
j
j+1
j
j+1
(U (t) = T 0 dt exp[ iH(t )t ] is the time
evolution opj j+1
totj
HIsing = J (erator with T thej timeordering). From H(t)this !
weH(tob-+ Tper /2),
x
y
z
x
y
z
B<Bc の基底状態：z方向の強磁性
(S
,
S
,
S
)
!
(
S
,
S
,
S
),
J
=
2,
J
=
10
tain
the
Fouriertransformofthemagnetization
M
(!)
=
xy
z
R
where
Jz >
>0.0.The
Thedi↵erencedi↵erence
to theIsing
model
B>Bc の基底状態:
x方向の強磁性
xy
where
Jand
JxyJradiation
>
to
the
Ising
model
z >
dtei!t
M
(t)
the
power
y
y
+
+
1
x x
y
y
+
+ ),of Jthe
1 (SxSjxSj+1 +where
is
the
term
S
S
=
S
+
S
S
where
0laser.
is9.theferrom
FIG.
Time
evo
j
j+1
j
j+1
j
j+1
T
⌘
2⇡/⌦
is
the
periodThen,tis the term Sj Sj+1 + Sj Sj+1 per= 22(Sj Sj+1 + Sj Sj+1
), >
FIG.
9.
Time
Takayoshi,
YM, P. Werner,H
PRB
which
acts
akineticforthemagnons.
Note
2kinetic
2term S.
withexterna
Jxy =
>99,
0 ismodel
amodelstaticmagnetization
satisfies/
M (!)|
.
which
acts
asIas
a |!
term
for
the
magnons.
Note
with
Jxy
184303 (2019)zthat the spins are completely frozen
in the Ising model
S are spin-1/2
operators
that the spins are completely frozen inx the Ising modelx
speaker:Y.
MurakamiMEq.
(t +Tperis/2)=a ferromagnetic
M (t),
without oflaser.
The groundstate ofconstraints(15)
ais ferThe symmetry
the system
may impose
state (
without
laser.
The
ground
state
of
Eq.
(15)
is
a
fery
y
romagnetic
state
for signal.
Jz > Jxy
0MwhileitTperis /2)
a thisgapon the structureof the
HHG
For>example,
the
state
is perturbed
by
(t
+
=
M
(t),
the
shape changromagnetic
state
for
J
>
J
>
0
while
it
is
a
gapz
xy
the shapechaLuttinger
liquid|Jz |signal< |Jxy[?z ]. The lowpolarized
en- z tensity
inversion lesssymmetry
limitsthe for
HHG
in|M
electronic
pulse
laser
B(t
as
B
is
in
Tper /2)
M (t),
less Luttinger liquid for |J | < |J | [? (t].+ The
low=enzxywindow WB (t) = 0.42 0.5(0 < t < Tf ) and WB (t) =result clearly demonstrates
in both magnetization co
the system satisfies the symz (!) become 0 at ! = 2
an integer), respectively, fo
presented results are for t
tudes of |! 2 mx (!)|2 and |and ! = (2n + 1)⌦, respe
when B is small and ! arou
the excitation gap, as can band 4(f) where the spectrain mx (!) and at ! = 7, 9 inconsider the application ofa transient regime and doe
steady state. Hence the co
not necessarily satisfied. Tof the HHG spectra is supperturbation theory (Appefrom the perturbation theofor mx and at ! = H + J
pear at an arbitrary frequemultiple of ⌦) depending ovalidity of the time-depend
confirmed by the scaling o
the laser amplitude B. In|mz (!)| in the region of sm! = n⌦ scale as B n , while

FIG. 5. Scaling of the magni
for the magnetizations (a) |region of small B, in the Isin
The power of B agrees well wi
dependent perturbation theor

(a)

IV.
HHG
IN
(c)
IV.
HHG
INXXZ
XXZ MODELS
MODELS
methodwith
the Cash-Karpparameters,
and cal-(h)an (13)by Kutta
the
fifth
order
Kutta
methodwithRungethe Cash-Karpparameters,
and calxxzxxz
the dynamics(t) hS⌘ (t)ihS and(t)i andm (t).
Cash-Karpparameters,
and ofcal-mof(t)m ⌘culateculate
the dynamics
m (t).
z(a)f mxThe(t) ⌘discretized
hS xdiscretized(t)i and
m
(t).
The
time
step
is
t
=
0.05.
The
resulttime step is t = 0.05. The result is isIn
thissectionwe
consideranotherfundamental
model
2 x
2 |!
Incurves
thisweanother fundamental model
ep isalsot =0.05.
is4. The
also
showninresult
Fig.
ofm 2(!)|
and
shown
inThe
Fig.
4. Thecurvesof |!section
mx (!)|
and2consider
2
x
2
2
z
2
2 |!
The |!
curvesofm calculated(!)| by
andof
quantummagnets,model.The
The
m2|!
(!)|
by
singlespin
dynamicsagree
mz (!)|
calculated
thesinglespin dynamics
agreetheofthequantum
magnets,
theferromagnetic
ferromagnetic XXZ
XXZ model.
y thewellsinglespin
dynamics
agreeby iTEBD
(b)
(g)
well
with
those
calculated
by
iTEBD
theis
first HHG
with
those
calculated
up toupthetofirst
HHG
spinspin
Hamiltonian
Hamiltonianis35
FI
ed by threshold.
iTEBD
up
to
the
first
HHG
(d)
(i)
(b)
(g)
(d)
(i)
threshold.
no rescaling
the results
Note Notethat that
therethere
is no isrescalingof theofresults
log
(d)
(i)
here isandno rescalingoftheresults
Xdeviations
and agreement
the agreement
is quantitative.
The
theis quantitative.
TheX
deviations
be- beJz
y
y
y
y
x
x
z
z
x
x
z
z
x
uantitative.
The above
deviations
bexx
above
the
first
threshold.
indicates
that
HXXZ=(q,This[Jand(S
Sj+1++SSjjSSj+1) JzzS
SSj+1],],+B(t)S
(15)
come
largerthe first
threshold.
This
indicates
that
H
=
[J
(S
)
S
(15)
sol
xy
FIG. come7. larger
The Floquet
DSF
(a)
|Im
!)|
(b)S
XXZ
xy
j
j
j+1
j
j+1
j
j+1
F
tot
rst threshold.
This indicates
thatmagnons beyond mean-field theoryofcorrelations
betweencorrelations
betweenmagnons
beyondmean-field
theory|Im zz
(q,
!)|
for
the
Ising
model
with
J
=
2,
H
=
6,
and
F
j j
agnons beyond
mean-field
theoryma
are essential
forspontaneous
the spontaneousrecombinationof mulare
for therecombinationof mulB =essential
8.
7
ontaneous recombination of mul7
magnons.
7 = 10
tiple tiple
magnons.
7
B
where
J
>
J
>
0.
The
di↵erence
to
the
Ising
model
(a)
(f)
(c)
(h)
z
xy
B = 6(BtowhereJonz HHG>canJbe>0. They di↵erencec ) the Ising model
Another
perspectives
canobtained
be obtainedxy
Another
usefulusefulperspectives
on HHGy++1z can be obtainedzx equax
ectiveswhere
on
HHG
y
y
m
(t)
⌘
hS
(t)i.
We
solve
the
Schrödinger
+
1
xS
xSin
is
term
+
S
S
=
(S
S
+
S
S
from
the Floquet
picture[?
?the].termThe
spectrumthe
FloFIG.
9. 9.
Timeevolutionof (a)
fromtheFloquetpicture[?
?
].
The
spectrum
in
the
Floj
j+1
(e)
(j)+ ),),
j
j+1
j
j+1
j
j+1
(c)
(h)
2
(e)
(j)
is
the
S
S
+
S
S
=
(S
S
+
S
S
(a)
(f)
e [? ? tion]. Thespectruminthe
FloFIG.
Time
evolution
of m
(a
(e)
(j)
j
j+1
j
j+1
j
j+1
j
j+1
↵
2
with
the
Hamiltonian
(13)
by
the
fifth
order
Runge↵↵ derived
quet
theoryisfromtheFloquet
DSF
|
(q,
!)|,
which
acts
as
a
kinetic
term
for
the
magnons.
Note
quet
theoryisderivedfromtheFloquet
DSF
|
(q,
!)|,
model
with
J
=
2
and
J
=
10.
xy
z
F
F
om theFloquetDSF with| F (q,(a) for the magnons.
(f) Note
which
acts
as(q,
aand
kinetic
1マグノンバンドmodel with Jxy = 2 and Jz =Kutta
methodthe!)|,
Cash-Karpparameters,
cal- Let↵(a)term(f)modelwhichis↵calculated
in ax thatsimilarwayas| ↵!)|.
(q,z!)|.
(a)
(f)
which
isascalculated
in aLetsimilarwaytheasLetx | spins
arecompletely
frozenintheIsinga similar
way
|
(q,
!)|.
culatedynamicsof thatm (t) the⌘ hS(t)i and
(t).
spins
arem
frozen in the Ising model
us theconsiderthe ↵time-dependent
HamiltonianH(t;us consider
the time-dependent
HamiltonianH(t;↵) =↵0 ) =0completely
pendent
Hamiltonian
H(t;
)
=
0
laser.
The
groundstateof Eq. (15) is a ferThe H
discretized
time
step
t
=
0.05.
The
resultis
x is withoutx
B groundsin(⌦t+↵)Sofwithout
and represent
the groundstateof
H0 B0sin(⌦t+↵thegroundstate
ofgroundtot represent0 )Sand represent
the
state
tot0 and2 x The2
laser.
state
of
Eq.
(15)
isa agapferalso H(0;
shown
in
Fig.
4.
The
curves
of
|!
m
(!)|
and
romagnetic
state
for
J
>
J
>
0
while
it
is
↵2phase|↵0(0;
the phaseshift.
We↵z 0the) by| by(0;shift.
)i,↵
where
↵0 is↵theshift.
Wezxy
0)0 )i, where
0 isphasewhereH(0;↵20 isWetheespecially nq=⇡shape changes
(b)
(i)
q=0
|! mcalculate(!)| calculated
byromagnetic
thesingle correlation
spin
dynamics
agreeJz > J
state
for
> 0| [?
while
it (g)islow
a (d)
gapxy
the Floquet
retardedfunction
calculatethe function
Floquetretardedcorrelation
functiontheshapechanges
especiatardedcorrelation
lessLuttinger
liquid
for
|J
|
<
|J
].
The
enFIG.
11.
Colormapof thesubcyclezxy
tensityas|!2BHigh
well with those calculated by iTEBD up to the first HHG
x is increased.
2
log
m
(!;
t
)|
for
the
XXZ
mode
⇤
less
Luttinger
liquid
for
|J
|
<
|J
|
[?
].
The
low
ensub
10
z
xy
↵
↵
↵ ↵
↵
(i)
ergy
excitations
of
Eq.
(15)
are
magnonswith
dispersion
tensityasBisincreased.
(b)(g) (d)
Note
that
there
is
no
rescalingoftheresults
(r,
t;
↵
)
=
i#(t)h
(0;
↵
)|[S
(t;
↵
),
S
]|
(0;
↵
)i
(0; ↵threshold.
(t;
↵
),
S
]|
(0;
↵
)i
(r,
t;
↵
)
=
i#(t)h
(0;
↵
)|[S
(t;
↵
),
S
]|
(0;
↵
)i
J
=
10
under
the
laser
field
(a)
B
=
4a
aH
0
0
0
0
than
⌦
appearnear
thepeaks,z0 )|[S
0
0
r
0
0
0
0
0
0
r
0
F r F
FI
ergyE(q)excitationsof Eq.
(15)
aremagnonswith dispersion
and the agreement is (14)quantitative.
TheJxydeviations
besolidlines
show thenear
single-magnon
and=cos(q)+xx J(14)theHamiltonian
Eq. (15)
(14)
than
⌦!)|
appearthepeakz . Since(b)xx (g)zz (seezz
HHG
the
sub-cycle
analys
9
xx
zz
of
the
snapshot
Hamiltonian
at
time
tby
FIG.
8.
DSFs
(a)-(e)
|Im
(q,
!)|
and
(f)-(j)
|Im
(q,
!)|
FIG.
8.
DSFs
(a)-(e)
|Im
(q,
!)|
and
(f)-(j)
|Im
(q,
⇤
xx
FIG.
8.
DSFs
(a)-(e)
|Im
(q,
!)|
and
(f)-(j)|Im(q,!)|
largerabove the firstthreshold.
This
indicates+thatz
DSF come(a) |Im(b)E(q)=Jcos(q)
J
.
Since
the
Hamiltonian
Eq.
(15)
Th
z
xy
z
does
not
includethelongitudinal
static
field
HS
,
the
1F (q, !)| ↵ and ↵↵ for the XXZ
1↵Jxy = 2 and Jzfor
magnon
mode
in
(b).
1
↵
HHG
(see
the
sub-cycle
an
model
with
=
10.
the
XXZ
model
with
J
=
2
and
J
=
10.
tot
(t;
↵
)
=
U
(t;
↵
)S
U
(t;
↵
),
z
lution of m drastically change
2 andzJz = 10.
[cf.J (7)],
Eq.
(7)],
S=
(t;
↵
= (t;
U ↵0(t;
U01
(t;
0[cf.
0
Eq.
where
S
) beyond
=
)S0↵⌦
U
(t;
), ↵0 ),for the XXZ model with Jxy = xy
0 )U
0 )S
0= wherecorrelations
magnons
mean-field
theory
0↵
are
model
with
= 2,betweenandr 0(t; ↵r0
Bsin(⌦t)=R t6,B(t)R↵H
0
0 does00 ;↵ notxx
↵Zthe
t0 ;↵0 )t0zt (q,0 !;includelongitudinal
static
field
HS
,
the
↵
0
iH(t
)t
systemhasasymmetry,
and
thus
a
quantum
phase
0
iH(t
;↵
)t
0
.
The
DSF
↵
)
is
(b)
(g)
de
2
FIG.
Floquet
(a)
|Im
(q,
!)|
and
(b)
tot
=
T
dt
e
.
The
DSF
(q,
!;
↵
)
is
0DSF
are
the
spontaneousrecombinationofmul(b)(g)lutionofmdrasticallychFU(t;7.essential↵U0(t;
) The
=↵T
dt
e
.
The
DSF
(q,
!;
↵
)
is
on
whetherBissmallerorlarg
F0 ) for
0
0
F
0
F
0
(c)
(h)
zz
(e)
(j)
(a)
(f)
fro
ransformof
thiscorrelation
func|Imdefined
(q,
!)|
for
the
Ising
model
with
J
=
2,
H
=
6,
and
definedthe Fouriertransformof correlation
thiscorrelation
func- byandztiple
magnons.
system
has
acan
Z2be
symmetry,
thus a transverse
quantum field.
phase theFtransition
inducedapplyingas theasFouriertransformof thisfuncmagnitudeofmdecreasesonwhetherBissmallerorveragerelative
to
the
phase
shift
B =tion,
8.Anothertion,
and
we perspectives
takethe averagerelativethephase
shift case where J
usefulonHereHHG
canbeobtained
and wetakethe averagerelative
towe
thetoconsider
phase
shiftthe(a)(b)
zz
weak, Jxyfield.
⌧
transitioncanbeinducedby applyingxya istransverse
xx
We
solve
the
Schrödinger
equaisstrong,otherwise
m
'
1/2,
xx
s
the
magnitude
of
m
decre
↵
over
a
single
cycle
as
from
the
Floquetpicture[?
?
].
The
spectruminthe
FloDSF
(a)
|Im
(q,
!)|
and
(b)
↵
over
a
single
cycle
as
0
(c)
(h)
mnFloquet
(q,
!)|
and
(b)
(j)
0
ea
F by
(13)
the fifth order Runge-F HereJzFloquet, whichisfor (e)themodelingofquasi-one↵ relevantz sta
weconsider
the
case
where
J
is
weak,
J
⌧
xy
xy
the
state
follows
theground↵Isingbe1
quettheoryisderived
from
the
DSF
|
(q,
!)|,
is
strong,otherwise
m
'
the
model
with
J
=
2,
H
=
6,
and
z
z
↵ cal↵ energy
J whereH=
h=F 2,
(q,m
!;
↵
)i⌘
↵and
↵ with
0=
↵6,
(t)
(t)i.
We
solve
Schrödinger
equaCash-Karpparameters,
and
0 .hS
magnonmasswithin terms
of belowtran(q,!)=dimensional
h the
(q,
!;
↵0 0.↵below)i↵0magnetic.F the
(q,
!)
=
h
(q,
!;
↵
)i
insulators
such
as
CoNb
O
[?
].
energythemagnon
mass
in
terms
of
tran0
↵
F
F
ati
with
energybelow
themagnon
mass
in
terms
of
tranF
F
(c)
(h)
2
6
way
as | is
(q,
!)|.
Let
Jz , bywhich
relevantfor ofthemodelingof quasi-one- tonian,x which is xcalculated inz a similari.e., follows
the timethe
evolution
sitions
betweenFloquetsidebands
themagnonspecthestate
ground
m
(t)
⌘
hS
(t)i
and
m
(t).
tion
with
the
Hamiltonian
(13)
the
fifth
order
Rungethsitions
betweenFloquetsidebandsof magnon
the magnon/8 (nus= consider
0,Here1, . . . we, 15).
In ↵Fig.=7,n⇡/8The
sitions
betweenFloquetsidebands
of=thespec-specthe
Hamiltonian
H(t;
↵are
=set
z
parameters
to
J
=
2,
J
=
10,
and
⌦
1.
0 ) Fig.
take
(n
=
0,
1,
. .magnetic
.
,
15).
In
7,
xy
z
we
take
↵0time-dependent
=
(n
=
0,
1,
.
.
.
,
15).
In
Fig.
7,
trum.
0 Cash-Karpp Kutta
is Here
t =
0.05.
The
resultisdimensional
insulators
such
as
CoNb
O
[?
].
ever,
for
B
>
B
,
m
suddenlydmethodwiththen⇡/8
parameters,
andcalxx(zz)c
2 6ma! state of trum.trum. !
x = 8.
tonian,
i.e.,
the
time
evolutDSF |Im(q, !)|
B2tot
xx(zz)
H
B
sin(⌦t+↵)S
andrepresent
thegroundxx(zz)
2 xfor
0
0
(b)
(g)
x
x
z
F
(d)
(i)
We
take
both
J
and
J
to
be
larger
than
⌦
so
that
The
curves
|! m
(!)|
and
weofshowtheFloquetDSF|Im F(q,(q,!)|are
for
B
8. zJ
ad
xy
culatethedynamics
ofwith
mThe
(t)
(t)i
and
m
we
the
Floquet
DSF
|Im ⌘
!)|
for
B
=(t).
8. = to
shows
that
the
system
makes
tr
F hS
parameters
set
10, 11.
and
⌦ 10.
= HHG
1. thefrom
rance
of show
Floquetsubbandsxz , mz sudde
xy = 2, Jz = FIG.
H(0;
↵spin
by
| see
(0; the↵0 agree
)i,▷
where
↵q=0
the
phase subbands
shift.
We
Colormapofsubcycleradiation
spectrum
B<Bc:
マグノン
à
カットオフ
0 ) can
0 is
ever,
for
B
>
B
FIG.
(a)-(e)mand(f)-(j)
m
for
the
XXZ
z
マグノン消滅由来のFicthesingledynamics
We
appearanceof
Floquetwith2
Wediscretizedcantheappearance
Floquetsubbands
The
time
step
is of
t correlation
=
0.05.
The
resultisthe
HHG
plateaucontains
severalharmonics.
For
SSchrödinger
(t)i.
Wesee
solve
the
Schrödinger
equaequalog10 |! 2 mxsub(!; t⇤ )|theforJthe=XXZ
model
with
Jxy = 2lines
andcorrespond to⌦ratherthan⌦.
Thesubbands
In
thisbothsection,we withhave
focusedonthe model
with
of
the
snapshot
Hamiltonian.
calculatetheFloquetretardedfunction
model
with
2
and
J
=
10.
Dashed
xy
z
We
take
J
and
J
to
be
larger
than
⌦
so
that
tio
2
x
2
(c)
(h)
xy subbandsz In thisby
iTEBD
upenergy
to
the
first
an
splittingofrather2⌦
rather
than
The
In
this
section,have
focused
on
the
model
with
(c)
(h)
anat
energysplitting
ofHHG
2⌦
than
⌦.
The
subbandssection,we(Jhavefocused
the
model
Jwe
under
theon
laser
field
(a) B
=with
4 dashed-dottedand(b)
B = 10.
Thecorrespond
shows
that
the
system2tomake
also
shown
in
Fig.
4.
The
curvesoflongitudinal|! ofm⌦.the(!)|
and<z = 10
HHG
▷
B>Bc:
q=π
マグノン×２à
カットオフ
cated
(one
magnonband)±
strong
field
(H
J)
and
large
gap.
With
(timesinteger)
and
lines
x
analysismodel
withstrong
J
J
),
see
q=0
speaker:Y.
Murakami
miltonian
(13)
by
the
fifth
order
RungeED(省)
x
ere
fifth
order
Rungexxxyxyz(d)(i) several
12. (a) hspectra⇠(Hshown |Stot | n i and
z
↵|Im
solid
the
single-magnon
andFIG.
two-magnon
modes
isof2ofnorescalingoftheresultsof!)|arelocatedat ↵(oneband)
longitudinal
fieldJ) the
and
large
With
↵
The
HHG
|! (b)
mAn(![
zz 2xx
(q,
!)|
are
at
(one
magnon± ±stronglongitudinalfield(HlinesJ)
largegap. gap.
With
˜and
the
HHG
plateau
harmonics.
m(!)|
calculated
bylocatedthesinglespin
dynamics
2For
ose|!
|Im
are
locatedat
H,S0magnon
thegap
decreases
andstrongthe
HHG
behavior(r,t;F|Im↵!)|=(q,i#(t)h(0;↵0decreasing
)|[S
(t;
),
]| band)(0;agree↵contains
q=⇡ .
0) F
0
0 )i
of
the
snapshot
Hamiltonia
F (q,
r
by
ED
for
the
model
(16)
with
J
= 2,res
J
F
zz
xy
of
the
snapshot
Hamiltonian
at
time
t
in
(a)
and
the
twozz
Appendix
C
2.
In
Fig.
8,
we
show
the
DSF
in
the
ground⇤with
theCash-Karpparameters,
andcalantitative.
The
bearameters,
andcalxx while
integer)⌦
thoseof F|Im(q,!)|
arethelocateddecreasing
H, gapthe=decreases
gap decreases
and HHG
the HHG
behavior(odd
integer)⌦
while
those
of
|Im
(q,
!)|
are
locatedat atdecreasingH,
theand thebehaviorF<
well
with(odd
thosecalculated
by
iTEBD
up
to
the
first
HHG
even
integer)⌦.
Indeviations|Imchanges.
We
discuss
results
of
a
weak
H
model
(J
shown
in
Fig.
10.
Here
the
sa
(14)
The
arrowsshow
theenergyEE=n0ma
F (q, !)|,
magnonmode
in(b).
analysis
of
the
model
with
strong
J
(J
J
),
see
2
xy thexy
z weakxxxxxx± (evenz |Imx negative
zmagnonx m
tSthe
threshold.
This
indicatesthat(onemagnoninteger)⌦.
In
|Im
(q,
!)|,changes.
changes.
We
theresults
of(f)-(j)a weak
H zz
model
=HHG
(one
band)
±hS
(even
In
(q,xx
!)|,
We
discuss
resultsof(i)awhich
H |Im
model
(JThe
=(Jused
FIG.
8. aDSFs(a)-(e)|Im field
(q,⇠
!)|
and
(q,
!)|
state
the
XXZ
model
with
transverse
B,
spectra(d)arefor the statesin the h n|!
|Stot
|
F
amics
ofmagnonm(t)⌘band)(t)iandenergydis4,rescaling
H
=ofm
2)
in
Appendix
C 1.
The
emergenceof discuss
HHG
(t)i
and
m
(t).
threshold.
Note
that
there
is integer)⌦.
noof(t).
theF results
tem
used
as
in
the
Ising
case.
The
↵ the
1 Cenergy↵xxbeyondmean-field
theoryAppendix
2.
In
Fig.
8,
we
show
the
DSF
in
the
groundtheFloquetsubbandsofthenegativemagnon
dis4,
H
=
2)
in
AppendixC
1.
The
emergence
of
HHG
for
the
XXZ
model
with
J
=
2
and
J
=
10.
demonstratethecontribution
to
the
HHG
the
Floquetsubbandsofnegativeenergymagnon
dis4,
H
=
2)
in
AppendixC
1.
The
emergence
of
HHG
xy
z
[cf.
Eq.
(7)],
where
S
(t;
↵
)
=
U
(t;
↵
)S
U
(t;
↵
),
is
much
larger
than
that
of
as
seen
in
Fig.
8.
Thus
!nons
'
1.
These
FloquetDSFs
sugplateaus
with
a
close
relationtomagnonenergies
can
0
0
0
ca
r
0
and theagreementis quantitative.
The
deviations
be-snapshot(c)(h)
corresponds
theof theshown
in two-magnon
Fig.
10.
Hereitthtime
step
is appear
0.05.
The
result
isto
.05.
The
resultis
R tt =(e)Hamiltonian(j)toitsystem0fromcan↵ DSFstaneousrecombination
mulfor
the
weak
field
B
while
i
persion!0 also'1.
These
FloquetDSFs
sugplateaus
witha closerelationtomagnonenergies
0of around
iH(t
;↵
)t
iswhichmore
naturalto regard
itthisas asector.
process.
0 1.
persionappeararound!
'These
Floquetsugplateaus
with
a
close
relation
magnonenergiescanHH
terpretthehigh-harmonic
peaks
beobserved
there.
U
(t;
↵
)
=
T
dt
e
.
The
DSF
(q,
!;
↵
)
is
state
of
the
XXZ
model
with
a
transverse
field
B,
come2 larger
that
0
2underx This2 indicates
x0 above
2 0 the first threshold.
F irradiation,
laser

(a)

(f)

(g)

(h)

(b)

(c)

(d)

Fig. 4. The curves of |! m (!)| andotherwise the magnetization in the z direction recovers
to mz = 1/2. This demonstrates that the state of the system closely follows the ground state of the instantaneous
Hamiltonian at each time. In other words, the laser frequency is slow enough for an adiabatic time evolution of
the magnetization. In the present case of H
J, the gap
is large ( ⌦) even for B = 0, and it increases mono-

!

(e)

(a)

˜

HHG構造

基底(B=一定)
の励起構造

B=2

FIG. 4. The radiation power from (a)-(d) mx and (e)-(h) mz
in the Ising model with J = 2 and H = 6. The dashed lines
correspond to the mass of a magnon at q = 0 (times integer).

XXZ模型:マグノン励起とHHGの構造

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2H2K2Mi`v 2t@
2t@
i ?H7?H7 }HHBM;}HHBM;- i?2`2
i?2`2 2tBbi
2tBbi irQ
irQ ivT2b
B+iiBQMb,
UBV;TT2/bTBMH2bb
2t+BiiBQMb+HH2/?QHQMbB+iiBQMb, UBV ;TT2/ bTBMH2bb 2t+BiiBQMb +HH2/ ?QHQMb
M/
UBBV ;TH2bb;TH2bb +?`;2+?`;2 M2m@
M2m@
M/ MBiB?QHQMb
MBiB?QHQMb U/Qm#HQMbV
U/Qm#HQMbV M/
M/ UBBVi`H
2t+BiiBQMb+HH2/bTBMQMbX
S?vbB+H
2t+BiiBQMb`2i`H 2t+BiiBQMb +HH2/ bTBMQMbX S?vbB+H 2t+BiiBQMb `2+QMbi`m+i2/
7`QK
i?2b2
2H2K2Mi`v
2t+BiiBQMbX
AM
T`@+QMbi`m+i2/ 7`QK i?2b2 2H2K2Mi`v 2t+BiiBQMbX AM T`@iB+mH`#v  [mMiBiv[mMiBiv
iB+mH`-  ?QHQM?QHQM U/Qm#HQMVU/Qm#HQMV Bb
Bb T`K2i`Bx2/
T`K2i`Bx2/ #v+HH2/KQK2MimK+HH2/ `TB/Biv`TB/Biv kk- r?2`2
r?2`2 i?2
i?2 +Q``2bTQM/BM;+Q``2bTQM/BM; KQK2MimK
pphh(k)(k) U✏
U✏dd(k)V(k)V `2`2 ;Bp2M;Bp2M #v#v(k) UpUpdd(k)V(k)V M/M/ i?2i?2 2M2`;v
2M2`;v ✏✏hh (k)
Z
Z 1
1
⇡⇡
d! J
J00(!)(!)sin(!
sin(!sin
sink)
k)
d!
pphh(k)(k) == ppdd(k)(k) ++ ⇡⇡ == 2 kk 2
2
,,
U
!
U
!
! 11 ++ exp(
exp( 2 ))
2
!
0
0
2
UReV
UReV
✏✏hh(k)(k) == ✏✏dd(k)
(k)
Z 1
U!
UU
d! J 1(!) cos(!
cos(! sinsink)ek)e U4!4== ++ 22cos
cos kk + 2,,
22!
cosh(UU44!!))
cosh(
0
URe#V
URe#V

doublon・holon dispersion (Bethe Ansatz)

ii

4v 2
doublondispersion/2b+`B#2bi?2/2b+`B#2bi?2bTBMbTBM2t+?M;22t+?M;2- r?2`2
r?2`2 JJexex == U4vU - - ŝŝi i ==P
1 1 Pĉ†i,↵i,ĉi,- -M/ĉ†i,↵ ↵,M/ /2MQi2b/2MQi2bi?2
i?2SmHB
SmHB Ki`B+2bX
Ki`B+2bX
Peakof↵,ĉA(p,ω)
2 2 ↵,↵,Ai AiBb BbMQi2rQ`i?v
i?i
i?2
bTBM
2t+?M;2i2`K
MQi2rQ`i?v i?i i?2 bTBM 2t+?M;2 i2`K /Q2b
/Q2b MQi
MQi
?p2
Mv
/B`2+i+Q``2bTQM/2M+2
BMi?2
b2KB+QM/m+iQ`?p2 Mv /B`2+i +Q``2bTQM/2M+2 BM i?2 b2KB+QM/m+iQ`bvb@
bvb@
i2KbX
(t)
i2KbXHbQ
HbQMQi2
MQi2i?i
i?iĤĤ(t)#2+QK2b#2+QK2bi?2i?2r2HH@FMQrM
r2HH@FMQrM
Mott,2
Mott,2
t@Jt@JKQ/2H
KQ/2H7i2`
7i2`bm#i`+iBM;
bm#i`+iBM;b2p2`H
b2p2`Hi2`Kb
i2`KbB``2H2pMi
B``2H2pMi 7Q`
7Q` bKHH
bKHHMmK#2`
MmK#2`Q7Q7?QH2bX?QH2bXAMAM;2M2`H;2M2`H-rBi?
rBi?BM+`2bBM;
BM+`2bBM;NN11- ĤĤ(t) bi`ib iQ /2p2HQT N1 @i?
M2B;?#Q`U+Q``2Hi2/V
U+Q``2Hi2/V
Mott,N
Mott,N
1 @i?M2B;?#Q`
1 1 (t) bi`ib iQ /2p2HQT N?QTTBM;bQ7/Qm#HQMb
M/?QHQMbX
b7Q`
Ĥ(t)- i?2?QTTBM;b Q7 /Qm#HQMb M/ ?QHQMbX b 7Q` Ĥex,Nex,N
2 2 (t)- i?2?B;?2`?B;?2`Q`/2`
Q`/2`+Q``2+iBQMb+Q``2+iBQMbiQiQD̂(t)D̂(t)BM+Hm/2
BM+Hm/2i?2
i?2+`2iBQM+`2iBQM Q7Q7
/Qm#HQM@?QHQM/Qm#HQM@?QHQMTB`b
TB`bQMQMNN
M2`2bibBi2b
bBi2bHi?Qm;?
Hi?Qm;?i?2
i?2
2 @i?
2 @i?M2`2bi
+Q2{+B2Mi#2+QK2b#2+QK2bbKHH2`X
bKHH2`XAMAMi?2
i?27QHHQrBM;
7QHHQrBM;/Bb+mbbBQM/Bb+mbbBQM+Q2{+B2Mi}M/i?ii?ii?Bb
i?BbHQM;@`M;2/
HQM;@`M;2//BTQH2/BTQH2KQK2Mi
KQK2MiBbBbBKTQ`iMi
BKTQ`iMir2r2}M/i?2>>:>>:T`Q+2bbX
T`Q+2bbX
7Q`7Q`i?2
2▷ Doublon dispersion appears in single par6cle`2bT2+iBp2HvX>2`2
7mM+iBQMX q2
q2 b?HH
b?HH b22b22
`2bT2+iBp2HvX spectrum>2`2 JJ Bb i?2 "2bb2H 7mM+iBQMX
i?i
FBM2iB+b Q7
Q7  /Qm#HQM/Qm#HQM
i?i i?Bb
i?Bb /BbT2`bBQM/BbT2`bBQM /2b+`B#2b i?2 FBM2iB+bmM/2`bi`QM;}2H/b
i?mb `2Hi2/`2Hi2/iQM/  ?QHQM?QHQM
mM/2` to
M/ Bb later)
i?mb
iQ
This is not general M/
(come
back
this
point
speaker:Y. Murakami>>:- b22 a2+X AAA *Xnn>>:- b22 a2+X AAA *X
q2 b?Qrb?Qr ?Qr?Qr i?2b2
i?2b2 KMv@#Q/v 2H2K2Mi`v
q2
2H2K2Mi`v 2t+BiiBQMb
2t+BiiBQMb`2 `2~2+i2/`2~2+i2/ BM
BM i?2i?2 bBM;H2@T`iB+H2 bT2+i`mK`2
bT2+i`mK A(p,A(p,!)!)- b22b22
6B;X RXRX h?2h?2 bBM;H2@T`iB+H2
bBM;H2@T`iB+H2 2t+BiiBQM +QMbBbib
6B;X+QMbBbib Q7
Q7 ?QHQMb?QHQMbbTBMQMb- M/M/ /Qm#HQMb/Qm#HQMb M/ b  `2bmHi Bi 2t?B#Bib
bTBMQMb2t?B#Bib KmHiBTH2
KmHiBTH2dy#M/bdyrBi?  /B`2+i/B`2+i ;T M2` p = ⇡/2X
#M/b
rBi?
⇡/2X AMAM T`iB+mH`T`iB+mH`-Subcycle Analysisj(!; tpro ) =Zdt̄eit̄! j(t̄)W (t̄; tpro ) window

37

Another parameter: U=7

iQ i?2 `2HiBp2 K
bm;;2bib i?i i?2
i?2 /Qm#HQM M/
KBM2/ #v i?Bb /Q
U = 10.0, ⌦ = 0.5, E0 = 0.7
q2 i2bi i?Bb B/Qm#HQM@?QHQM T
iBM2/ 7`QK i?2
biBimiBM; Bi BMiQ
7Q` ;Bp2M t0 Bb i?
h?2 `2bmHiBM; !
6B;bX 9U+V M/ 9(b) U = 7.0, ⌦ = 0.3, E0 = 0.4
bB+H MHvbBb 2t
HB;?i Bb 2KBii2/
2`b TT`QtBKi2;BQM BM IHHG (!,vbBb i2M/b iQ mM
KtBK Q7 IHH
ir22M IHHG (!, tor7Q` ?B;?2` 7`2[m2T`BbQM #2ir22M
A(p,w) peak
Bethe ansatz
or
K2Mi Bb 2tT2+i2/
rQ`b2 7Q` ?B;? 7`
Dispersion of U à ∞, -2v0sin(p)6A:X 8X UV aBM;H2@T`iB+H2 bT2+i`mK A(p, !) BM 2[mBHB#`BmK
K2Mi #2ir22M I+H+mHi2/ #v Bh1". 7Q` U = 7 M/ hz = 0.001X h?2 `2/#2+QK2b #2ii2`/b?2/ HBM2b BM/B+i2 i?2 T2F TQbBiBQMb Q7 i?2 bT2+i`mK i
i?i i?2 ;`22K
2+? pX h?2 #H+F /Qi@/b?2/ HBM2 BM/B+i2b ✏d (pd ) U1[X UReVVX
HHG
p Ins.
HHG@MoL
Dispersion of many body elemental
excita<on
3step
model
R
;QQ/ b i?i #2ir
hQ Q#iBM A(p, !) 7`QK G (p, t)- r2 TTHv  :mbbBM rBM/QrM/ i?2 +Q``2bTQ0
t2
exp( 2 0 2 ) rBi?
= 5.0 7Q` i?2 6Qm`B2` i`Mb7Q`KiBQMX U#V
TT2M/Bt *X >2
am#+v+H2 MHvbBb Q7 i?2 >>:- IHHG (!, tp ) 7Q` U = 7 M/
rec
g
cf.
Imai-san’stalk+HbbB+H MHvbBb
Probe of dispersion of elemental
excita<ons
hz = 0.001X h?2 T`K2i2`b Q7 i?2 TmKT `2speaker:⌦ = 0.3, E0 =Y. Murakami
TQ`iMi bT2+ib Q
0.4, t0 = 100 M/ = 30X h?2 #Hm2 K`F2`b b?Qr i?2 `2bmHib`2+QK#BMiBQMX
Q7 i?2 b2KB+HbbB+H MHvbBb U!emit (trec )V rBi? ✏g (p) Q#iBM2/
7`QK i?2 "2i?2 Mbix `2bmHib 1[X UReV- r?BH2 i?2 `2/ K`F2`b
q2 HbQ b?Qr
+Q``2bTQM/ iQ i?Qb2 7Q` i?2 mM`2MQ`KHBx2/ /BbT2`bBQM- U
mbBM; i?2 /BbT2`4v sin(p)X h?2 TBMF K`F2`b `2 i?2 `2bmHib Q#iBM2/ mbBM;
i`mK Ui?2 `2/ /
i?2 /BbT2`bBQM 2ti`+i2/ 7`QK i?2 bBM;H2@T`iB+H2 bT2+i`mK BM
TBMF K`F2`b BM

(d)

or

(a)

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6A:X 9X U-#V TTHB2/ 2H2+i`B+ }2H/ M/ BM/m+2/ +m``2Mi 7Q`
UV ⌦ = 0.75 M/ U#V ⌦ = 0.5X U+-/V am#+v+H2 MHvbBb Q7 i?2>>:- I(!, t )- 7Q` U+V ⌦ = 0.75 M/ U/V ⌦ = 0.5X h?2#Hm2 K`F2`b b?Qr i?2 `2bmHib Q7 i?2 b2KB+HbbB+H MHvbBb
U!emit (t )V rBi? ✏ (p) Q#iBM2/ 7`QK i?2 "2i?2 Mbix `2bmHib
1[X UReV- r?BH2 i?2 `2/ K`F2`b +Q``2bTQM/ iQ i?Qb2 7Q` i?2
mM`2MQ`KHBx2/ /BbT2`bBQM- U 4v sin(p)X h?2 TBMF K`F2`b`2 i?2 `2bmHib Q#iBM2/ mbBM; i?2 /BbT2`bBQM 2ti`+i2/ 7`QK

38

(a)

(b)

speaker: Y. Murakami

HHG in strongly correlated systems

39• doublon/holon hopping
Upper
Hubbard bandcf. intraband current• doublon/holon recombina<on
Lower
Hubbard bandcf. interband current▷ Eﬀects of correla6on and sca\erings?
▷ Origin? : doublon/holon hopping VS recombina6on
YM, M. Eckstein, P. Werner, arXiv:1712.06460 [to be published from PRL].
YM, P. Werner, arXiv:1804.08257 [to be published from PRB].
cf. Silva. et al.,Nat. Photo.12, 266 (2018).
Tancogne-Dejean, Sentef, Rubio, arXiv:1712.01067.

speaker: Y. Murakami

強相関電子系におけるHHG
HHG の 起源40

HHG spectroscopy for 光誘起相転移

Hubbard模型

絶縁体à 金属 転移

ダブロン・ホロン再結合

R. E. F. Silva, et al, Nat. Photon. 12, 266 (2018).

Y. M. et al, PRL. 121, 057405 (2018).
Y. M. and P. Werner, PRB 98, 075102 (2018).
M. Lysne, et al , PRB 101, 195139 (2020).

ダイマーMo\キンク・アンチキンク再結合

実験
日本物理学会2020秋:11pH1-10
Ca2RuO4からのHHG

S. Imai, et al, PRL. 124, 157404 (2020).

2

method is adopted [38]composition is utilized
doublon-holon
wavefunction |ψ(t + δta b(b)Pair annihilation
Pair creationinkkdifference δt and the ticreaPon-ant
Kinkknmost of the numerical
Antikink
Ki
moving aroundthe matrix dimension (QCP
difference are chosen t
E
Adiabatic dynamics
tively, which are enougtMott insulator Charge ordershown later [see inset(d) recombinaPon(e)speaker:diagonalization
Y. Murakami(ED) mfor finite size clusters,
HHG spectra in the strong-field regime, (b) HHG spectra in the weak-field regime, and (c) HHG spectra in theis N = 16 and 18 withfield strength (E0 ) and the harmonic energy (n⌦). The white circle markers show cuto↵ energies. The inset of
optical absorption spec
ows the current and electric field during one period at E0 = 4.0. We use U = 8.0, = 2.0, = 0.06, ⌦ = 0.5.
Material (SM)] is sche(a)EnergyFloquet DMFT

(c)

Charge-ordered ground state