Terahertz ionic Kerr effect Two-phonon contribution to the nonlinear optical response in insulators M. Basini1 2M. Udina1 3M. Pancaldi4 5V. Unikandanunni2S. Bonetti2 4and L. Benfatto3

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Terahertz ionic Kerr eﬀect: Two-phonon contribution to the nonlinear optical

response in insulators

M. Basini,1, 2 M. Udina,1, 3 M. Pancaldi,4, 5 V. Unikandanunni,2S. Bonetti,2, 4 and L. Benfatto3

1These authors contributed equally to this work

2Department of Physics, Stockholm University, 10691 Stockholm, Sweden

3Department of Physics and ISC-CNR, “Sapienza” University of Rome, P.le A. Moro 5, 00185 Rome, Italy

4Department of Molecular Sciences and Nanosystems,

Ca’ Foscari University of Venice, 30172 Venice, Italy

5Elettra-Sincrotrone Trieste S.C.p.A., 34149 Basovizza, Trieste, Italy

(Dated: July 1, 2024)

The THz Kerr eﬀect measures the birefringence induced in an otherwise isotropic material by a

strong THz pulse driving the Raman-active excitations of the systems. Here we provide experimental

evidence of a sizable Kerr response in insulating SrTiO3due to infrared-active lattice vibrations.

Such a signal, named ionic Kerr eﬀect, is associated with the simultaneous excitation of multiple

phonons. Thanks to a theoretical modeling of the time, polarization and temperature dependence of

the birefringence we can disentangle the ionic Kerr eﬀect from the oﬀ-resonant electronic excitations,

providing an alternative tunable mechanism to modulate the refractive index on ultrashort time-

scales via infra-red active phonons.

The latest advances in the generation of intense ter-

ahertz (THz) ﬁeld pulses made it possible to investi-

gate the low-frequency counterpart of nonlinear optical

phenomena in condensed matter, conventionally studied

with visible light, as it is the case for the THz Kerr ef-

fect [1–3]. The DC Kerr eﬀect detects a birefringence

in an otherwise isotropic material proportional to the

square of the applied DC electric ﬁeld, and it is a stan-

dard measurement of the third-order χ(3) non-linear op-

tical response of the medium [4]. Basically, the four-

wave mixing between the AC probe EAC (ω) and the

DC pump EDC ﬁeld leads to a non-linear polarization

P(3) ∼χ(3)E2

DC EAC (space indexes are omitted). The

P(3) in turn modulates the refractive index at the same

frequency ωof the AC ﬁeld, with a space anisotropy

set by the direction of EDC . In its optical counterpart,

the spectral components around zero frequency of the

squared AC ﬁeld play the same role of the DC compo-

nent. More recently, THz and optical pulses have been

combined in a pump-probe setup to measure the so-called

THz Kerr eﬀect [2]. The main advantage over its all-

optical counterpart is that intense THz pump pulses can

strongly enhance the signal by matching Raman-like low-

lying excitations in the same frequency range, such as

lattice vibrations [5–8], or collective-modes in broken-

symmetry states, as for magnetic [9–13] or superconduct-

ing transitions [14–16]. Such a resonant response usually

adds up to the background response of electrons, and it

can be used to identify the microscopic mechanisms un-

derlying the coupling among diﬀerent degrees of freedom.

As a general rule, the THz Kerr response, scaling as

the THz electric ﬁeld squared, is not aﬀected by infrared-

active (IR-active) phonons, that correspond to lattice dis-

placements that are linear in the applied electric ﬁeld.

While this is strictly true to ﬁrst order [17], higher-order

IKE

Ω!

𝜔"#

Ω!$ + 2𝜔"#

2𝜔"#

Ω!$

EKE

Ω!$ + 2Ω!

Ω!$

a b

Figure 1. Terahertz electronic and ionic Kerr eﬀect in a wide-

band insulator. In the THz Kerr eﬀect the pump ﬁeld (red

arrows) drives an intermediate state by a sum-frequency two-

photons process. Such state subsequently scatters the visible

probe ﬁeld (blue arrows). In the EKE (a) the intermediate

electronic state (dashed line) is virtual, being a insulator, so it

relaxes back almost instantaneously giving a 2Ωpmodulation

of the emitted light. In the IKE (b) two IR-active phonons

(wavy lines) can be used to reach the virtual electronic state

(process labeled as (a) in the text), leading to a modulation

at 2øIR. This pathway only occurs when the THz pulse is

resonantly tuned to the phonon frequency, i.e. when øIR ≃

Øp.

processes are not excluded by symmetry. In this work,

we demonstrate that the non-linear excitation of the IR-

active soft phonon mode in the archetypal perovskite

SrTiO3(STO) leads to a sizeable contribution to the

Kerr signal, which we name ionic Kerr eﬀect (IKE). STO

is a quantum paraelectric [18] showing a large dielectric

constant at room temperature (ϵ0≈300), and with a

low-lying phonon mode that progressively soften as the

temperature is lowered [19–21]. Such excitation is an

IR-active transverse optical (TO1) phonon, and it is the

same phonon mode which is responsible for the paraelec-

tric to ferroelectric transition upon, e.g., Ca doping [22].

arXiv:2210.14053v3 [cond-mat.mtrl-sci] 28 Jun 2024

Furthermore, it has been recently pointed out its pos-

sible role in the superconducting transition in electron-

doped samples [23–25], and the possibility of realizing

dynamical multiferrocity upon driving it with circularly

polarized THz electric ﬁelds [26, 27]. Here we show that

besides the well-studied electronic Kerr eﬀect (EKE), due

to oﬀ-resonant electronic transitions in wide-band insu-

lating STO [28] (Fig. 1a), a ionic contribution, associated

with the second-order excitation of the TO1phonon, is

present (Fig. 1b). Such IKE manifests itself with a siz-

able temperature dependence of the Kerr response, which

is unexpected for the EKE in a wide-band insulator. In

contrast, the IKE rapidly disappears by decreasing tem-

perature, due to the phonon frequency softening far be-

low the central frequency of the pump ﬁeld. We are able

to clearly distinguish between EKE and IKE thanks to

a detailed theoretical description of the THz Kerr sig-

nal, which we retrieve experimentally as a function of

the light polarization and of the pump-probe time de-

lay tpp. So far, the regime of large lattice displacements

has been mainly investigated to exploit the ability of

infrared-active vibrational modes, driven by strong THz

pulses, to anharmonically couple to other phononic ex-

citations [17, 29–32]. Our work demonstrates that non-

linear phononic represents also a suitable knob to ma-

nipulate the refractive index of the material, providing

potentially an additional pathway to drive materials to-

wards metastable states which may not be accessible at

thermal equilibrium [33].

I. EXPERIMENTAL SETUP

Measurements are performed on a 500 µm-thick

SrTiO3crystal substrate (MTI Corporation), cut with

the [001] crystallographic direction out of plane. Broad-

band single-cycle THz radiation is generated in a DSTMS

crystal via optical rectiﬁcation of a 40 fs-long, 800 µJ

near-infrared laser pulse centered at a wavelength of 1300

nm. The near-infrared pulse is obtained by optical para-

metric ampliﬁcation from a 40 fs-long, 6.3 mJ pulse at

800 nm wavelength, produced by a 1 kHz regenerative

ampliﬁer. As schematically shown in Fig. 2, the broad-

band THz pulses are ﬁltered with a 3 THz band-pass

ﬁlter, resulting in a peak frequency of Øp/2π= 3 THz

(with peak amplitude of 330 kV/cm), and focused onto

the sample to a spot of approximately 500 µm in di-

ameter. The time-delayed probe beam, whose polariza-

tion is controlled by means of a nanoparticle linear ﬁlm

polarizer, is a 40 fs-long pulse at 800 nm wavelength

normally incident onto the sample surface. A 100 µm-

thick BBO crystal (β-BaB2O4, Newlight Photonics) and

a shortpass ﬁlter are used for converting the probe wave-

length to 400 nm and increasing the signal-to-noise ra-

tio in temperature-dependent measurements. The probe

size at the sample is approximately 100 µm, substan-

𝑥

𝑦

𝜃

HWP

THz filter

Wollaston

Photodetectors

𝐸!

"(𝑡)

-1-0.5 00.5 1

tpp [ps]

ΔΓ

Figure 2. Schematics of the experimental setup. The THz

pulse is in red while the optical probe is in blue. Inset a:

schematic of the polarization geometry. The sample is rotated

in the (S, P ) plane, such that the ycrystallographic direction

forms a variable angle θwith respect to S. Here the probe

(blue arrow) polarization is ﬁxed along S, while the pump

pulse (red arrow) is polarized along either Sor Pfor linearly

polarized light and in both directions for circularly polarized

light. Inset b: typical time-trace of ∆Γ at ﬁxed θ= 67.5◦

(blue line) compared with the intensity proﬁle of the linear

pump pulse (red line).

tially smaller than the THz pump. The half-wave plate

(HWP) located after the sample is used to detect the po-

larization rotation of the incoming ﬁeld induced by the

non-linear response, and a Wollaston prism is used to

implement a balanced detection scheme with two pho-

todiodes. The signals from the photodiodes are fed to

a lock-in ampliﬁer, whose reference frequency (500 Hz)

comes from a chopper mounted in the pump path before

the DSTMS crystal. Data are collected as a function of

the tpp time-delay as well as a function of the angle θ

that the probe polarization direction forms with respect

to the main crystallographic axes. For convenience, here

we assume that the probe beam is kept ﬁxed along the S

direction, such that Epr(t)=(Epr(t) sin θ, Epr(t) cos θ),

while the pump ﬁeld Epcan be either linearly polarized

(along Sor Pdirection) or circularly polarized [26], by

placing a quarter-wave plate after the band-pass ﬁlter

(see Appendix A), such that in general

Ep(t) = Ep,S (t) sin θ+Ep,P (t) cos θ

Ep,S (t) cos θ−Ep,P (t) sin θ.(1)

In the presence of Kerr rotation, the probe ﬁeld trans-

mitted through the sample ˜

E(t) acquires a ﬁnite Pcom-

ponent. The half-wave plate rotates ˜

E(t) by 45◦with

respect to Pdirection and the outgoing signal reaching

the two photodetectors (Γ1,Γ2) reads [34]

Γ1

Γ2∝1 1

1−1˜

EP.(2)

Exp (300 K)

Exp (150 K)

ΔΓe

-1-0.5 0 0.5 1

pp [ps]

ΔΓ(tpp) [a. u.]

|| ⟂circ

100 200 300

-5

ΔΓ

(θ) [10-5 rad]

|| ⟂circ

100 200 300

-5

[degrees]

ΔΓ

(θ) [10-5 rad]

Figure 3. Polarization and time dependence of the THz Kerr

eﬀect. aExperimental data as a function of θat 300 K

and tpp ≃0, where the signal is maximized, in the parallel

(∥, red curve), cross-polarized (⊥, blue curve) and circular

(circ, brown curve) conﬁguration. bCorresponding simula-

tions from Eq. (6), accounting for a ﬁnite polarization mis-

alignment ∆θ= 5◦between the pump and probe pulses. c

Experimental data as a function of tpp at ﬁxed angle θ= 67.5◦

in the ⊥conﬁguration at 300 K (orange curve) and 150 K

(blue curve), compared with the simulated oﬀ-resonant elec-

tronic contribution ∆Γe(gray curve).

In a pump-probe detection scheme, the use of a chopper

on the pump path allows one for measuring, with a lock-

in ampliﬁer, the diﬀerential intensity |Γ1|2− |Γ2|2with

(on) and without (oﬀ ) the pump, to obtain

∆Γ = ∆Γon −∆Γoﬀ ∝(˜

ES˜

EP)on −(˜

ES˜

EP)oﬀ .(3)

A typical time-trace of ∆Γ for a linearly polarized pump

pulse along Pand at ﬁxed angle θis shown in Fig. 2b,

together with E2

p(t). As we shall see below, the close

qualitative correspondence among the two signals is a di-

rect consequence of pumping a band insulator below the

band gap. To measure the angular dependence of the

response, we then choose tpp at the maximum ∆Γ ampli-

tude, and we record its change as a function of θ. The

results are shown in Fig. 3afor both linearly polarized, in

either the parallel and the cross-polarized conﬁguration,

or circularly polarized pump pulses. In all cases, the

signal displays a marked four-fold angular dependency,

along with a smaller two-fold periodicity.

II. ELECTRONIC KERR EFFECT

In order to assess such angular dependence, we ﬁrst

outline the theoretical description of the conventional

EKE. The transmitted probe ﬁeld ˜

E(t) will contain both

linear and non-linear components with respect to the to-

tal applied ﬁeld Ep+Epr. Since the detection is restricted

around the frequency range of the visible probe ﬁeld, the

linear response to the pump can be discarded, as well as

higher-harmonics of the probe generated inside the sam-

ple. We can then retain for ˜

ES∼P(1)

S, with P(1)

Sthe

linear response to the probe along S, while ˜

EPscales as

the third-order polarization for a centro-symmetric sys-

tem, i.e. ˜

EP∼P(3)

P. By decomposing P(3)

Pin its (x, y)

components one gets:

∆Γ ∝˜

ESP(3)

P=˜

EShcos θP (3)

x−sin θP (3)

yi.(4)

As mentioned above only contributions linear in Epr to

P(3)

P≃χ(3)EprE2

pmust be retained, where χ(3) is the

third-order susceptibility tensor. By making explicit the

space and time dependence one can ﬁnally write:

P(3)

i(t, tpp) = ˆdt′X

jkl

Epr,j (t)×

×χ(3)

ijkl(t+tpp −t′)¯

Ep,k(t′)¯

Ep,l(t′),

(5)

where we introduced the time shift tpp between the pump

and probe pulses, such that ¯

Ep(t+tpp)≡Ep(t), with

Ep(t) and Epr(t) centered around t= 0 (see Refs. [35, 36]

for a similar approach). Since the pump frequency is

much smaller than the threshold for electronic absorption

(Eg∼3 eV), the nonlinear electronic response should

be ascribed to oﬀ-resonant interband excitations, lead-

ing to a nearly instantaneous contribution to the third-

order susceptibility tensor χ(3), which can be well approx-

imated by a Dirac delta-function, i.e. χ(3)

ijkl(t)∼χ(3)

ijklδ(t),

with χ(3)

ijkl a constant value. Under this assumption,

Kleinman symmetry for the cubic m3mclass appro-

priate for STO above Ts≃105 K assures that the

oﬀ-diagonal components of the third-order susceptibil-

ity tensor have the same magnitude [4]. In the speciﬁc

case of insulating SrTiO3, at room temperature one ﬁnds

χ(3)

iijj =χ(3)

ijji =χ(3)

ijij ≃0.47χ(3)

iiii [37]. Finally, since the

measurement at the photodetectors corresponds to the

time-average ﬁeld intensity |Γi|2, one can replace Epr,j (t)

in Eq. (5) with its value at t= 0, so that the probe pulse

acts as an overall prefactor, relevant only for its polar-

ization dependence. With straightforward algebra, and

using the decomposition (1) and (4), one obtains the gen-

eral expression for the EKE:

∆Γe(tpp, θ)∝1

4h¯

p,P (tpp)−¯

p,S (tpp)i∆χsin (4θ) +

+2 ¯

Ep,P (tpp)¯

Ep,S (tpp)hχ(3)

xxyy +1

2∆χsin2(2θ)i,(6)

where ∆χ≡χ(3)

xxxx −3χ(3)

xxyy. Eq. (6) accounts very well

for the angular dependence of the signal reported in Fig.

3a. In particular, for linearly polarized pump pulses, one

recovers in Fig. 3bthe four-fold symmetric modulation,

and the overall sign change observed when going from the

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TerahertzionicKerreffect:Two-phononcontributiontothenonlinearopticalresponseininsulatorsM.Basini,1,2M.Udina,1,3M.Pancaldi,4,5V.Unikandanunni,2S.Bonetti,2,4andL.Benfatto31Theseauthorscontributedequallytothiswork2DepartmentofPhysics,StockholmUniversity,10691Stockholm,Sweden3DepartmentofPhysicsandISC-C...

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Terahertz ionic Kerr effect Two-phonon contribution to the nonlinear optical response in insulators M. Basini1 2M. Udina1 3M. Pancaldi4 5V. Unikandanunni2S. Bonetti2 4and L. Benfatto3.pdf

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Terahertz ionic Kerr effect Two-phonon contribution to the nonlinear optical response in insulators M. Basini1 2M. Udina1 3M. Pancaldi4 5V. Unikandanunni2S. Bonetti2 4and L. Benfatto3

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