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arXiv:1208.5863v1 [physics.optics] 29 Aug 2012

Non-perturbative Interband Response of InSb

Driven Off-resonantly by Few-cycle Electromagnetic Transients

F. Junginger,1B. Mayer,1C. Schmidt,1O. Schubert,1,2S. M¨ ahrlein,1A. Leitenstorfer,1R. Huber,1,2and A. Pashkin1

1Department of Physics and Center for Applied Photonics,

University of Konstanz, Universit¨ atsstraße 10, 78464 Konstanz, Germany

2Department of Physics, University of Regensburg,

Universit¨ atsstraße 31, 93053 Regensburg, Germany

(Dated: August 30, 2012)

Intense multi-THz pulses are used to study the coherent nonlinear response of bulk InSb by means

of field-resolved four-wave mixing spectroscopy. At amplitudes above 5 MV/cm the signals show a

clear temporal substructure which is unexpected in perturbative nonlinear optics. Simulations based

on a two-level quantum system demonstrate that in spite of the strongly off-resonant character of

the excitation the high-field pulses drive the interband resonances into a non-perturbative regime

of Rabi flopping.

PACS numbers: 42.65.Re, 78.47.nj, 42.65.-k

Semiconductors form a uniquely well-defined labora-

tory to explore novel limits of nonlinear optics. A key

parameter for interaction of a coherent light field with

an electronic transition is given by the Rabi frequency

ΩR = µE/?.E is the electric field strength and µ

the transition dipole moment. When the dephasing rate

is negligible compared to ΩR, coherent Rabi flopping

governs the dynamics of electronic systems [1]. If the

detuning of the driving electromagnetic field is smaller

than ΩR the response of a system cannot be described

by the perturbative approach which usually depicts off-

resonant nonlinear optics. This non-perturbative exci-

tation regime provides access to many fascinating quan-

tum effects in semiconductors such as Rabi splitting, self-

induced transparency and generation of high harmonics

[2–8]. In particular, sufficiently intense and ultrashort

laser pulses have been exploited to implement ultimate

scenarios in which the duration of the light pulse, the

Rabi cycle and the oscillation period of the carrier wave

all become comparable in size [9]. Under such conditions,

a detailed insight into the nonlinear optical interaction

calls for complete phase and amplitude resolution of all

interacting light fields. However, in most of the cases, the

lack of phase-stable laser pulses and fast detectors does

not allow for capturing sub-cycle polarization dynamics

of a system.

The development of ultraintense THz laser systems

generating phase-stable transients with field amplitudes

above 1 MV/cm [10–12] paves the way towards a coher-

ent spectroscopy of extreme nonlinearities in condensed

matter systems with absolute sampling of amplitude and

phase. Recent experiments performed with high-field

multi-THz pulses have demonstrated a high potential of

this approach [13–16]. However, the regime of an off-

resonant excitation of Rabi flopping has remained almost

unexplored due to the lack of sufficiently intense and

phase-locked pulses. The latest breakthrough in genera-

tion and field-resolved detection of multi-THz pulses with

peak fields up to 100 MV/cm [17, 18] opens up the pos-

sibility to explore this highly non-perturbative regime.

In this Letter, we report the nonlinear response of the

bulk semiconductor indium antimonide (InSb) excited

far below interband resonance using our novel high-field

multi-THz laser source [18]:

mixing (FWM) signals are recorded with amplitude and

phase at different THz peak fields of the excitation pulses.

For the highest intensities, the Rabi frequency becomes

comparable to the detuning and the interband resonance

is driven into a non-perturbative regime of Rabi flop-

Time-resolved four-wave

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FIG. 1. (a) Schematic band structure of InSb with conduc-

tion (CB) and valence bands (VB). The blue arrow illustrates

a non-resonant excitation by a THz pulse.

sient generated by a 370-µm-thick GaSe crystal. (c) Ampli-

tude spectrum of the THz transient in b). (d) Scheme of the

FWM setup: The THz beam is split into two branches by

a Ge beam splitter (BS). Both branches are modulated by

mechanical choppers (CH) and delayed with respect to each

other by a translation stage (DS). The THz transients are

non-collinearly focused on the sample and the transmitted

signals are detected by electro-optical sampling (EOS).

(b) THz tran-

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ping. Our simulations of the FWM response based on

the Maxwell-Bloch equations provide a qualitative un-

derstanding of the phenomena observed experimentally.

The sample under study is a (100)-oriented undoped

single crystal of InSb (Fig. 1(a)) mechanically polished

to a thickness of 30 µm and kept at room temperature.

THz transients with a center frequency of f0= 23 THz,

a bandwidth of 8 THz (full width at half maximum,

FWHM) and variable peak fields between 2 MV/cm and

5.3 MV/cm are generated by difference frequency mix-

ing of near-infrared pulse trains with a repetition rate of

1 kHz in gallium selenide (GaSe) emitters (Figs. 1(b) and

1(c)) [18]. A specific feature of our experiment is a large

detuning of 18 THz between the THz transients and the

nearest interband resonance of InSb (Eg/h = 41.1 THz).

Thus, the entire THz spectrum is located well below the

band gap excluding the possibility of direct linear exci-

tation of electron-hole pairs.

We use a two-dimensional scheme of nonlinear spec-

troscopy (Fig. 1(d)). In order to perform a THz multi-

wave mixing experiment, a pair of mutually synchronized

pulses is obtained by splitting the THz beam after the

GaSe emitter crystal. In this way nonlinear mixing ef-

fects of two pulses within the emitter itself are precluded.

A germanium (Ge) beam splitter set at Brewster’s an-

gle and coated with a 6-nm-thick gold layer provides a

splitting ratio of 1:1. The transmitted electric field in

branch 2 (E2) is delayed by a retroreflector stage. The

transient in branch 1 (E1) propagates through a second

Ge wafer to match the dispersion of E2. The THz beams

in both branches are individually chopped with frequen-

cies of 500 Hz and 250 Hz, respectively, and the chopper

phases are locked to the 1 kHz pulse train. This con-

figuration enables a fast and efficient acquisition of the

transmitted signals from only branch 1, only branch 2 or

both branches at the same time (E12). The THz beams

are tightly focused onto the same spot on the sample

(FWHM: 85 µm). The emerging linear and nonlinear

fields are collected with large numerical aperture and

directed onto a 140-µm-thick GaSe electro-optic sensor

gated by near-infrared pulses with a duration of 8 fs.

Fig. 2(a) shows the total transmitted field E12 as a

function of the electro-optic sampling delay time t and

the relative temporal offset τ between the THz pulses.

The signal from branch 1 with a fixed temporal posi-

tion appears as a set of vertical lines of constant phase

centered around t = 0 ps, whereas the diagonal lines

correspond to the delayed signal from branch 2. The

external peak fields are 2 MV/cm per pulse. The field

strengths within InSb are attenuated by a Fresnel factor

˜t = 2/(n+1) = 0.4 defined by a refractive index of n = 4

in the frequency range of interest [19]. The nonlinear sig-

nal ENLshown in Fig. 2(b) is retrieved by subtracting the

contributions of individual transients E1and E2from the

total response E12: ENL= E12− E1− E2. The Fourier

transform of ENL(Fig. 2(c)) has a direct correspondence

with the wave vector space and, thus, allows to disentan-

gle different contributions to the total nonlinear field re-

sponse [14]. The inverse Fourier transform of selected re-

gions in frequency space depicted in Fig. 2(c) provides the

temporal fingerprints of multi-wave mixing signals of var-

ious orders. The pump-probe signals for each THz pulse,

located around wave vectors ±k1 (ft = ±f0, fτ = 0)

and ±k2 (ft = ±f0, fτ = ∓f0) (Fig. 2(c)), are shown

in Figs. 2(d) and 2(e), respectively. These signals corre-

spond to the transmission change of the sample excited

by the first and probed by the second THz transient.

They do not depend on the relative phase.

Most remarkably, a FWM signal at the wave vector

±k4 = ±(2k1− k2) (ft = ±f0, fτ = ±f0) becomes

Delay time τ (ps)

Total THz field amplitude

0

0.1

0.2

0.3

-0.1

-0.2

-0.3

0

0.2

-0.2

Delay time t (ps)

0

-1.2

1.2

E (MV/cm)

(a)

Nonlinear field response

0

0.2

-0.2

Delay time t (ps)

0

-0.8

0.8

E (MV/cm)

(b)

Fourier spectrum

k4

k1

k2

40 60

0

1

-40 -20

Frequency f (THz)

20

t

0

-40

-20

40

20

0

Frequency f (THz)

τ

(c)

Pump-probe signal 1

Delay time τ (ps)

0

0.1

0.2

0.3

-0.1

-0.2

-0.3

0

0.2

-0.2

Delay time t (ps)

0

-0.2

0.2

E (MV/cm)

(d)

Pump-probe signal 2

0

0.2

-0.2

Delay time t (ps)

0

-0.2

0.2

E (MV/cm)

(e)

Four-wave-mixing signal

Delay time τ (ps)

0

0.1

0.2

0.3

-0.1

-0.2

-0.3

0

0.2

-0.2

Delay time t (ps)

0

-0.1

0.1

E (MV/cm)

(f)

Delay time τ (ps)

0

0.1

0.2

0.3

-0.1

-0.2

-0.3

Delay time τ (ps)

0

0.1

0.2

0.3

-0.1

-0.2

-0.3

FIG. 2. (a) Electric field of two THz transients traversing

InSb plotted as a function of delay time τ and sampling time

t. (b) Nonlinear signal ENL. (c) 2D FT of b) revealing pump-

probe signals at wave vector positions k1 and k2 and the

FWM signature at k4. (d) Selective inverse FT of the pump-

probe signal emerging in the direction of k1. (e) Inverse FT

of the pump-probe signature at k2 only. (f) Inverse FT of the

FWM signal in the direction of k4 only.

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clearly discernible (Fig. 2(c)). This signal features lines

of constant phase that point along a downward diagonal

(Fig. 2(f)) indicating its dependence on the respective

phases of both THz transients. Thus, it bears informa-

tion about coherence induced in the sample. Therefore,

we will concentrate in the following on the FWM sig-

nal which is free from influence of incoherent excitation

processes dominating the pump-probe response such as

two-photon absorption [20] or impact ionization [21].

To study the field dependence of the FWM signal,

external peak fields of 2 MV/cm, 3.5 MV/cm and

5.3 MV/cm per pulse are selected.

ing pulse intensities are I0, 3I0 and 7I0 where I0 =

10.6 GW/cm2. For the lowest peak electric field we

observe an oval-shaped envelope of the FWM signal,

identical to the cross-correlation function of both pulses

(Fig. 3(a)). This result is as expected in the limit of

The correspond-

0

-0.1

0.1

E (MV/cm)

(d)

0

-0.2

0.2

E (MV/cm)

(e)

0

0.2

-0.2

Delay time t (ps)

0

-0.4

0.1

0.4

E (MV/cm)

-0.1

(f)

(a)

Delay time τ (ps)

0

0.1

0.2

-0.1

-0.2

0

-0.1

0.1

E (MV/cm)

Delay time τ (ps)

0

0.1

0.2

-0.1

-0.2

(b)

0

-0.3

0.3

E (MV/cm)

0

0.2

-0.2

Delay time t (ps)

0.1

-0.1

(c)

0

-0.4

0.4

E (MV/cm)

Delay time τ (ps)

0

0.1

0.2

-0.1

-0.2

Experiment

Simulation

FIG. 3. (a) Oval FWM signal by applying an external field of

2 MV/cm per pulse. (b) S-shaped FWM signature driven by

an external field of 3.5 MV/cm. (c) Splitted FWM signal at

an external exciting field strength of 5.3 MV/cm. (d-f) Cal-

culated FWM signatures reproducing the main features of the

corresponding measured signals on the left side of each image.

perturbative nonlinear optics far from resonance. In con-

trast, increasing the field strength up to 3.5 MV/cm leads

to a deviation from the symmetric profile resulting in an

S-shaped signal (Fig. 3(b)). Surprisingly, the maximum

field of 5.3 MV/cm leads to a splitting of the FWM signal

(Fig. 3(c)). A minimum appears in the temporal region

where the strongest total excitation field is present. This

signature is an unequivocal indication of an extremely

nonlinear interaction.

Our experimental results may be understood by using

a simplified model of a two-level system representing the

interband resonance in InSb. The simulation is based on

the Maxwell-Bloch equations [22] which are solved nu-

merically without applying the slowly-varying-envelope

and rotating-wave approximations. The solution is ob-

tained by an iterative predictor-corrector finite element

method. For the simulations we assume a dephasing time

of T2= 1 ps [23] and a depopulation time of T1= 10 ps.

Owing to the extremely short THz transients the choice

of the relaxation time barely affects the results of the

simulations as long as T1and T2are longer than the du-

ration of the THz pulse. The transition dipole moment

µ12 = 2.4 e˚ A and the density of the two-level systems

N = 2.9 × 1020cm−3were adjusted in order to provide

the best agreement with the shapes and intensities of the

FWM signals measured experimentally.

Figs. 3(d)-(f) show the time domain FWM signals sim-

ulated for the same peak fields as those used for the

experimental results depicted in the panels on the left.

For the simulation we assume Fourier-limited Gaussian

THz transients with a FWHM duration set to the pulse

width measured experimentally. A perturbative response

at moderate field strengths leads to an oval envelope of

the FWM signal (Fig. 3(d)). At the intermediate THz in-

tensity this envelope evolves into an S-shape (Fig. 3(e)).

Finally, external peak fields above 5 MV/cm induce a

minimum in the center of the signal, where the total

field is strongest. This results in two side-lobes similar

to those observed in the experiment (see Figs. 3(c) and

3(f)). As one can clearly see, the simulations with a sin-

gle energy electronic resonance allow us to reproduce the

essential features observed in the experiment. This result

is highly surprising since it is well known that interband

excitations in bulk semiconductors like InSb include a

continuum of electronic resonances with a broad distri-

bution of frequencies above the absorption edge. Never-

theless, simulations using a set of two-level systems with

a density distribution fitting the joint density of states in

InSb essentially lead to the same results as those shown

in Figs. 3(c) and 3(f). The reason for that is the steep de-

pendence of the non-perturbative response on the detun-

ing frequency. This finding is supported by the density of

the two-level systems N estimated from our simulations

which constitutes only about 1% of the total number of

available states in the conduction band of InSb. These

are the states near the band edges which constitute the

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interband transitions with the smallest detuning.

A qualitative physical understanding of the splitting

observed in the FWM signal can be obtained by consider-

ing the polarization response of a two-level system driven

by two THz transients with zero delay time (τ = 0 ps).

Fig. 4(a) depicts the simulated time-resolved polariza-

tion at the frequency of the driving pulses for different

peak electric fields Em. In case of moderate fields the

maximum population inversion wmaxshown in Fig. 4(b)

remains negative and the deflection of the Bloch vector

from the ground state is minor as illustrated in Fig. 4(d).

Therefore, the response of the two-level system is pertur-

bative and the shape of the polarization signal (Fig. 4(a))

follows the profile of the driving fields shown in Fig. 4(c).

The FWM signal, thus, can be described in terms of

an instantaneous third-order nonlinearity. However, this

picture breaks down as soon as field amplitudes exceed

3 MV/cm and a clear splitting of the temporal signature

of the FWM signal starts to develop. In this case the

maximum population inversion shown by the solid red

line in Fig. 4(b) becomes positive (wmax > 0), indicat-

ing the onset of a strongly non-perturbative regime. As

illustrated in Fig. 4(e), a THz peak field of 5.3 MV/cm

promotes the system almost to the limit of complete pop-

1

-1

0

0

0

Inversion w 1

-1-1

1

2

4

6

8

Peak field E (MV/cm)

m

2

0

-2

E/E m

2

4

6

8

Peak field E (MV/cm)

m

-0.1

Delay time t (ps)

-0.2

0.2 0.1

0

1

0

-1

-101

wmax

(a)

(b)

(c)

2 MV/cm

1

-1

0

0

0

Inversion w 1

-1-1

1

5.3 MV/cm

(d)

(e)

P (norm.)

NL

FIG. 4.

τ = 0 ps: (a) Normalized polarization of the two-level sys-

tem at the fundamental frequency of the driving field as a

function of the delay time t and the peak electric field Em.

(b) Maximum inversion wmax as a function of Em (red solid

line) and of the amplitude of a continuous wave excitation

(blue dashed line) (c) Driving field of the THz transients.

(d,e) Corresponding pathways of the Bloch vector for a mod-

erate (Em= 2 MV/cm) and a high (Em = 5.3 MV/cm) peak

electric field.

Simulated response of the two-level system for

ulation inversion. This surprising result radically differs

from the case of an off-resonant excitation by a continu-

ous wave where the complete population inversion can be

achieved only in the limit of infinitely high electric fields

(see the dashed blue line in Fig. 4(e)). This fact indi-

cates a clear violation of the slowly-varying-envelope ap-

proximation for our conditions. Finally, in the center of

the FWM signal, where the driving electric field reaches

its maximum, the polarization oscillates mainly at high

harmonic frequencies [24]. The response at the funda-

mental frequency becomes weak leading to the observed

minimum (Fig. 4(a)). This regime of a non-perturbative

excitation sets in when the maximum Rabi frequency

ΩR/2π = 2˜tµ12Em/h becomes comparable to the large

detuning of 18 THz at Em= 3 MV/cm.

In conclusion, we have studied the off-resonant FWM

response of bulk InSb which provides a direct access to

the coherent dynamics of the interband polarization re-

sponse at THz frequencies. The observed splitting of the

FWM signals for electric fields above 5 MV/cm manifest

the onset of a non-perturbative response of Rabi flop-

ping. This extremely nonlinear behavior underpins the

high potential of the novel high-field multi-THz technol-

ogy for high harmonics generation [25] and coherent con-

trol of quantum states in semiconductors [15, 26]. The

high fields and excellent signal-to-noise performance of

THz multi-wave mixing open a way for investigations of

the coherent response in a large variety of important res-

onances in complex systems such as intermolecular li-

brations in hydrogen-bonded liquids or energy gaps in

superconducting condensates.

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